# Strongly interacting matter in extreme magnetic fields

Prabal Adhikari, Martin Ammon, Sidney S Avancini, Alejandro Ayala, Aritra

Bandyopadhyay, David Blaschke, Fabio L Braghin, Pavel Buividovich, Rafael

P Cardoso, Casey Cartwright, et al.

<figure><img src="image_1.png" /></figure>

## To cite this version:

Prabal Adhikari, Martin Ammon, Sidney S Avancini, Alejandro Ayala, Aritra Bandyopadhyay, et al. "Strongly interacting matter in extreme magnetic fields. Progress in Particle and Nuclear Physics, 2026, 146 (Part 1), pp.104199. 10.1016/j.ppnp.2025.104199. hal-04883151

HAL Id: hal-04883151

https://hal.science/hal-04883151v1

Submitted on 6 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>
Strongly interacting matter in extreme magnetic fields

Prabal Adhikaria<sup>a</sup>, Martin Ammon<sup>b</sup>, Sidney S. Avancini<sup>c</sup>, Alejandro Ayala<sup>d, *,</sup> Aritra Bandyopadhyay<sup>e</sup>,
David Blaschke<sup>f, g, h</sup>, Fabio L. Braghin<sup>i</sup>, Pavel Buividovich<sup>j</sup>, Rafael P. Cardoso<sup>k</sup>, Casey Cartwright<sup>l</sup>,
Jorge David Castaño-Yepes<sup>m</sup>, Maxim Chernodub<sup>n, o</sup>, M. Coppola<sup>p</sup>, Mayusree Das<sup>q</sup>, Mariana Dutra<sup>r, s</sup>,
Gergely Endrődi<sup>t, u</sup>, Jianjun Fang<sup>v</sup>, Ricardo L. S. Farias<sup>w</sup>, Eduardo S. Fraga<sup>x</sup>, Arthur Frazon<sup>y</sup>, Kenji Fukushima<sup>z</sup>,
Juan D. García-Muñoz<sup>aa</sup>, Eduardo Garnacho-Velasco<sup>u</sup>, D. Gomez Dumm<sup>ab</sup>, Sebastian Grieninger<sup>ac</sup>,
Francesca Gulminelli<sup>ad</sup>, Juan Hernandez<sup>ae</sup>, Chowdhury Aminul Islam<sup>af</sup>, Matthias Kaminski<sup>ag</sup>, Andrey Kotov<sup>ah</sup>, Gastão
Krein<sup>y</sup>, Jing Li<sup>v</sup>, Pok Man Lo<sup>ai</sup>, Marcelo Loewe<sup>aj, ak</sup>, Odilon Lourenço<sup>r, s</sup>, Gergely Markó<sup>u</sup>, Kau D. Marquez<sup>am</sup>,
Ana Mizher<sup>y, an, ao</sup>, Banibrata Mukhopadhyay<sup>q</sup>, Enrique Muñoz<sup>m, 1</sup>, S. Noguera<sup>ap</sup>, Rodrigo M. Nunes<sup>w</sup>, Helena Pais<sup>aq</sup>,
Letícia F. Palhares<sup>ar</sup>, Constança Providência<sup>aq</sup>, Alfredo Raya<sup>as, ao</sup>, Tulio Restrepo<sup>x</sup>, Juan Cristóbal Rojas<sup>at</sup>, N.N.
Scoccola<sup>p, au</sup>, Luigi Scurto<sup>aq</sup>, Armen Sedrakian<sup>f, av</sup>, Dominik Smith<sup>aw</sup>, William Rafael Tavares<sup>ar</sup>, Maria E.
Tejeda-Yeomans<sup>ax</sup>, Varese S. Timóteo<sup>ay</sup>, Laura Tolos<sup>az, ba, av</sup>, Cristian Villavicencio<sup>ao</sup>, Fridolin Weber<sup>bb, bc</sup>,
Shigehiro Yasui<sup>bd, be, bf</sup>, Renato Zamora<sup>bg, bh</sup>, Zenia Zuraiq<sup>q</sup>

<sup>a</sup>Physics Department, Faculty of Natural Sciences and Mathematics, St. Olaf College, Northfield, MN 55057, USA.

<sup>b</sup>Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany.

<sup>c</sup>Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Santa Catarina, Brazil.

<sup>d</sup>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Circuito Ext. S.N., Ciudad Universitaria, CdMx 04510, Mexico.

<sup>e</sup>Institut fur Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany.

<sup>f</sup>Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-204 Wrocław, Poland.

<sup>g</sup>Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Bautzener Landstrasse 400, 01328 Dresden, Germany.

<sup>h</sup>Center for Advanced Systems Understanding (CASUS), Untermarkt 20, 02826 Görlitz, Germany.

<sup>i</sup>Instituto de Física - Federal University of Goias, Av. Esperança, s/n, 74690-900, Goiânia, GO, Brazil.

<sup>j</sup>Department of Mathematical Sciences, University of Liverpool, UK.

<sup>k</sup>Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil.

<sup>l</sup>Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands.

<sup>m</sup>Facultad de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile.

<sup>n</sup>Institut Denis Poisson, Université de Tours, Tours 37200, France.

<sup>o</sup>Department of Physics, West University of Timişoara, Bd. Vasile Pârvan 4, Timişoara 300223, Romania.

<sup>p</sup>Departamento de Física Teórica, Comisión Nacional de Energía Atómica, Av. Libertador 8250, 1429 Buenos Aires, Argentina.

<sup>q</sup>Department of Physics, Indian Institute of Science, Bengaluru 560012, India.

<sup>r</sup>Departamento de Física e Laboratório de Computação Científica Avançada e Modelamento (Lab-CCAM), Instituto Tecnológico de
Aeronáutica, DCTA, 12228-900, São José dos Campos, SP, Brazil.

<sup>s</sup>Institut de Physique des 2 infinis de Lyon, CNRS/IN2P3, Université de Lyon, Université Claude Bernard Lyon 1,
F-69622 Villeurbanne Cedex, France.

<sup>t</sup>Institute of Physics and Astronomy, ELTE Eötvös Loránd University, Pázmány P. sétány 1/A, H-1117 Budapest, Hungary.

<sup>u</sup>Faculty of Physics, Universität Bielefeld, Bielefeld, Germany.

<sup>v</sup>School of Physics and Physical Engineering, Qufu Normal University, 273165 Qufu, China.

<sup>w</sup>Departamento de Física, Universidade Federal de Santa Maria, zero5-900, Santa Maria, RS, Brazil.

<sup>x</sup>Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972, Rio de Janeiro, RJ, Brazil.

<sup>y</sup>Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II
01140-070 São Paulo, SP, Brazil.

<sup>z</sup>Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan.

<sup>aa</sup>Physics Department, CINVESTAV, P.O.B. 14-740, 0700, Mexico City, Mexico.

<sup>ab</sup>IFLP, CONICET – Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 1900 La Plata, Argentina.

<sup>ac</sup>Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA.

<sup>ad</sup>Normandie Univ., ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, F-14000 Caen, France.

<sup>ae</sup>Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB) and The International Solvay Institutes,
Pleinlaan 2, B-1050 Brussels, Belgium.

<sup>af</sup>Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany.

<sup>ag</sup>Department of Physics and Astronomy, University of Alabama, 514 University Boulevard, Tuscaloosa, AL 35487, USA.

<sup>ah</sup>Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52428 Jülich, Germany.

<sup>ai</sup>Institute of Theoretical Physics, University of Wrocław, plac Maksa Borna 9, PL-50204 Wrocław, Poland.

<sup>aj</sup>Centre for Theoretical and Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa.

<sup>ak</sup>Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián, Santiago, Chile.

<sup>al</sup>Center for Nanotechnology and Advanced Materials CIEN-UC, Avenida Vicuña Mackenna 4860, Santiago, Chile.

<sup>am</sup>Departamento de Física e Laboratório de Computação Científica Avançada e Modelamento (Lab-CCAM), Instituto Tecnológico de
Aeronáutica, DCTA, 12228-900 São José dos Campos/SP, Brazil.

<sup>an</sup>Laboratório de Física Teórica e Computacional, Universidade Cidade de São Paulo, R. Galvão Bueno,
868, Liberdade, 01506-000, São Paulo, Brazil.

<sup>ao</sup>Centro de Ciencias Exactas, Universidad del Bío-Bío, Casilla 447, Chillán, Chile.

<sup>ap</sup>Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain.

<sup>aq</sup>CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal.

<footnote>*Corresponding author
Email address: ayala@nucleares.unam.mx (Alejandro Ayala)</footnote>
ar Universidade do Estado do Rio de Janeiro, Instituto de Física, Departamento de Física Teórica, Rua São Francisco Xavier 524,
20550-013 Maracanã, Rio de Janeiro, Brasil.

as Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Ciudad Universitaria. Francisco J.
Mújica s/n. Col. Felícitas del Río. 58040 Morelia, Michoacán, México.

at Departamento de Física, Universidad Católica del Norte, Angamos 610, Antofagasta, Chile.

au CONICET, Godoy Cruz 2290 (C1425FQB), Buenos Aires, Argentina

av Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany.

aw Facility for Antiproton and Ion Research in Europe GmbH (FAIR GmbH), 64291 Darmstadt, Germany.

ax Facultad de Ciencias-CUICBAS, Universidad de Colima, Bernal Díaz del Castillo Número 340,
Colonia Villas San Sebastián, Colima, 28045, Mexico.

ay Grupo de Óptica e Modelagem Numérica - GOMNI, Faculdade de Tecnologia - FT,
Universidade Estadual de Campinas - UNICAMP, 13484-332 Limeira, SP, Brazil.

az Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, 08193 Barcelona, Spain.

ba Institut d'Estudis Espacials de Catalunya (IEEC), 08860 Castelldefels (Barcelona), Spain.

bb Department of Physics, San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA.

bc Department of Physics. University of California at San Diego, La Jolla, California 92093, USA.

bd Nishogakusha University, 6-16, Sanbancho, Chiyoda, Tokyo 102-8356, Japan.

be International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM²), Hiroshima University,
Higashi-Hiroshima, Hiroshima 739-8526, Japan.

bf Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan.

bg Centro de Investigación y Desarrollo en Ciencias Aeroespaciales (CIDCA), Fuerza Aérea de Chile, Casilla 8020744, Santiago, Chile.

bh Instituto de Ciencias Básicas, Universidad Diego Portales, Casilla 298-V, Santiago, Chile.

Abstract

Magnetic fields are ubiquitous across different physical systems of current interest; from the early Universe, compact astrophysical objects and heavy-ion collisions to condensed matter systems. A proper treatment of the effects produced by magnetic fields during the dynamical evolution of these systems, can help to understand observables that otherwise show a puzzling behavior. Furthermore, when these fields are comparable to or stronger than Λ<sub>QCD</sub>, they serve as excellent probes to help elucidate the physics of strongly interacting matter under extreme conditions of temperature and density. In this work we provide a comprehensive review of recent developments on the description of QED and QCD systems where magnetic field driven effects are important. These include the modification of meson static properties such as masses and form factors, the chiral magnetic effect, the description of anomalous transport coefficients, superconductivity in extreme magnetic fields, the properties of neutron stars, the evolution of heavy-ion collisions, as well as effects on the QCD phase diagram. We describe recent theory and phenomenological developments using effective models as well as LQCD methods. The work represents a state-of-the-art review of the field, motivated by presentations and discussions during the “Workshop on Strongly Interacting Matter in Strong Electromagnetic Fields” that took place in the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) in the city of Trento, Italy, September 25-29, 2023.

Keywords: Magnetic Fields, Quantum Chromodynamics, Quantum Electrodynamics, LatSuggestion: tice Quantum Chromodynamics, Neutron Stars
# Contents

<table><tr><td>I</td><td>Introduction</td><td>9</td></tr><tr><td>II</td><td>Theory Developments</td><td>11</td></tr><tr><td>1</td><td>Introduction</td><td>11</td></tr><tr><td>2</td><td>Magnetized mesons in the Nambu-Jona-Lasinio model</td><td>12</td></tr><tr><td>2.1</td><td>Effective Lagrangian and mean field quantities</td><td>13</td></tr><tr><td>2.2</td><td>Meson masses in the NJL model at finite magnetic field</td><td>14</td></tr><tr><td>2.3</td><td>Summary</td><td>19</td></tr><tr><td>3</td><td>Magnetic field effects in form factors and in the meson exchange potential</td><td>20</td></tr><tr><td>3.1</td><td>Meson and constituent quark form factors in constant magnetic fields</td><td>21</td></tr><tr><td>3.2</td><td>One-pion exchange potential</td><td>23</td></tr><tr><td>3.3</td><td>Discussion</td><td>24</td></tr><tr><td>4</td><td>Magnetic fields in heavy-ion collisions: a catalyst of photo-production via gluon fusion</td><td>25</td></tr><tr><td>4.1</td><td>Introduction</td><td>25</td></tr><tr><td>4.2</td><td>New photo-production channel available in pre-equilibrium stage</td><td>26</td></tr><tr><td>4.3</td><td>Concluding remarks</td><td>31</td></tr><tr><td>5</td><td>Hot and dense perturbative QCD in a very strong magnetic background</td><td>32</td></tr><tr><td>5.1</td><td>Introduction</td><td>32</td></tr><tr><td>5.2</td><td>Thermal pQCD in a strong magnetic background</td><td>32</td></tr><tr><td>5.3</td><td>Cold and dense pQCD in a strong magnetic background</td><td>35</td></tr><tr><td>6</td><td>Strongly Interacting Matter in Magnetic Fields</td><td>39</td></tr><tr><td>6.1</td><td>Chiral perturbation theory</td><td>39</td></tr><tr><td>6.2</td><td>Topological cumulants in a magnetic field</td><td>40</td></tr><tr><td>6.3</td><td>Finite volume thermodynamics</td><td>43</td></tr><tr><td>7</td><td>QED Fermions in a noisy magnetic field background: Perturbation theory and renormalized quasi-particles</td><td>47</td></tr><tr><td>7.1</td><td>The model</td><td>47</td></tr><tr><td>7.2</td><td>Perturbation theory: Self-energy and dressed propagator</td><td>50</td></tr><tr><td>7.3</td><td>Results</td><td>56</td></tr><tr><td>7.4</td><td>Conclusions</td><td>58</td></tr><tr><td>8</td><td>QED Fermions in a noisy magnetic field background from the perspective of the effective action approach</td><td>60</td></tr><tr><td>8.1</td><td>Introduction</td><td>60</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>8.2</td>
      <td>Auxiliary boson fields</td>
      <td>64</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Saddle-point approximation (Mean-field)</td>
      <td>65</td>
    </tr>
    <tr>
      <td>8.4</td>
      <td>The Schwinger propagator</td>
      <td>67</td>
    </tr>
    <tr>
      <td>8.5</td>
      <td>Results and Discussion</td>
      <td>68</td>
    </tr>
    <tr>
      <td>8.6</td>
      <td>Case 1: q<sub>3</sub><sup>2</sup> ≡ q<sub>⊥</sub><sup>2</sup>, with q<sub>1</sub> = q<sub>2</sub> = 0</td>
      <td>69</td>
    </tr>
    <tr>
      <td>8.7</td>
      <td>Case 2: q<sub>1</sub><sup>2</sup> = q<sub>2</sub><sup>2</sup> ≡ q<sub>⊥</sub><sup>2</sup>, with q<sub>3</sub> = 0</td>
      <td>71</td>
    </tr>
    <tr>
      <td>8.8</td>
      <td>Physical interpretation of the order parameter Q<sub>j</sub></td>
      <td>73</td>
    </tr>
    <tr>
      <td>8.9</td>
      <td>Summary</td>
      <td>74</td>
    </tr>
    <tr>
      <td><b>9</b></td>
      <td><b>Photon mass generation from magnetic fluctuations</b></td>
      <td><b>75</b></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Introduction</td>
      <td>75</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>The one-loop polarization tensor</td>
      <td>76</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>The one-loop polarization tensor in the limit p<sup>μ</sup> → 0</td>
      <td>77</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>Mass of gauge fields by magnetic fluctuations</td>
      <td>78</td>
    </tr>
    <tr>
      <td>9.5</td>
      <td>Summary</td>
      <td>79</td>
    </tr>
    <tr>
      <td>9.6</td>
      <td>Conclusions and outlook</td>
      <td>79</td>
    </tr>
    <tr>
      <td><b>III</b></td>
      <td><b>Anomalous transport and non-perturbative phenomena on and off the lattice</b></td>
      <td><b>81</b></td>
    </tr>
    <tr>
      <td><b>10</b></td>
      <td><b>Introduction</b></td>
      <td><b>81</b></td>
    </tr>
    <tr>
      <td><b>11</b></td>
      <td><b>Anomalous transport coefficients from lattice QCD</b></td>
      <td><b>82</b></td>
    </tr>
    <tr>
      <td>11.1</td>
      <td>Introduction</td>
      <td>82</td>
    </tr>
    <tr>
      <td>11.2</td>
      <td>Conductivities</td>
      <td>83</td>
    </tr>
    <tr>
      <td>11.3</td>
      <td>Non-interacting fermions</td>
      <td>83</td>
    </tr>
    <tr>
      <td>11.4</td>
      <td>Results</td>
      <td>87</td>
    </tr>
    <tr>
      <td>11.5</td>
      <td>Summary and outlook</td>
      <td>89</td>
    </tr>
    <tr>
      <td><b>12</b></td>
      <td><b>Enhancement of Chiral Separation Effect due to heavy quark flavours: a signature of QCD Kondo effect?</b></td>
      <td><b>90</b></td>
    </tr>
    <tr>
      <td>12.1</td>
      <td>Introduction</td>
      <td>90</td>
    </tr>
    <tr>
      <td>12.2</td>
      <td>CSE and the QCD Kondo effect</td>
      <td>91</td>
    </tr>
    <tr>
      <td>12.3</td>
      <td>Lattice setup: SU (2) lattice gauge theory with N<sub>f</sub> = 2 + 1 light and heavy quark flavours</td>
      <td>91</td>
    </tr>
    <tr>
      <td>12.4</td>
      <td>CSE enhancement in N<sub>f</sub> = 2 + 1 SU (2) gauge theory</td>
      <td>93</td>
    </tr>
    <tr>
      <td>12.5</td>
      <td>Discussion and conclusions</td>
      <td>94</td>
    </tr>
    <tr>
      <td><b>13</b></td>
      <td><b>Novel hydrodynamic transport coefficients in extreme magnetic fields &amp; the CME far from equilibrium</b></td>
      <td><b>95</b></td>
    </tr>
    <tr>
      <td>13.1</td>
      <td>Introduction</td>
      <td>95</td>
    </tr>
    <tr>
      <td>13.2</td>
      <td>Novel transport coefficients in extreme magnetic fields from chiral hydrodynamics [1]</td>
      <td>96</td>
    </tr>
    <tr>
      <td>13.3</td>
      <td>The chiral magnetic effect far from equilibrium [2, 3]</td>
      <td>99</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>13.4 Discussion and Conclusions</td>
      <td>102</td>
    </tr>
    <tr>
      <td><b>14 Anomalous transport in low dimensions</b></td>
      <td><b>103</b></td>
    </tr>
    <tr>
      <td>14.1 Introduction</td>
      <td>103</td>
    </tr>
    <tr>
      <td>14.2 Honeycomb lattices</td>
      <td>104</td>
    </tr>
    <tr>
      <td>14.3 Reduced-QED</td>
      <td>107</td>
    </tr>
    <tr>
      <td>14.4 Anomalous Hall effect</td>
      <td>107</td>
    </tr>
    <tr>
      <td>14.5 External fields generating anomalous transport</td>
      <td>108</td>
    </tr>
    <tr>
      <td>14.6 Summary</td>
      <td>110</td>
    </tr>
    <tr>
      <td><b>15 Magnetic fields, nonzero densities and QCD on the lattice</b></td>
      <td><b>111</b></td>
    </tr>
    <tr>
      <td>15.1 Introduction</td>
      <td>111</td>
    </tr>
    <tr>
      <td>15.2 Phase diagram</td>
      <td>112</td>
    </tr>
    <tr>
      <td>15.3 Equation of State</td>
      <td>113</td>
    </tr>
    <tr>
      <td>15.4 Fluctuations of the conserved charges</td>
      <td>115</td>
    </tr>
    <tr>
      <td>15.5 Summary</td>
      <td>115</td>
    </tr>
    <tr>
      <td><b>16 Superconductivity of vacuum in an extreme magnetic field: evidence from the Electroweak sector of the Standard model and implications for QCD</b></td>
      <td><b>116</b></td>
    </tr>
    <tr>
      <td>16.1 Introduction and Motivation</td>
      <td>116</td>
    </tr>
    <tr>
      <td>16.2 Magnetic-field induced vortex phase in the electroweak model</td>
      <td>119</td>
    </tr>
    <tr>
      <td>16.3 Magnetic-field induced vortex phase in QCD</td>
      <td>121</td>
    </tr>
    <tr>
      <td>16.4 Conclusions</td>
      <td>123</td>
    </tr>
    <tr>
      <td><b>17 Outlook</b></td>
      <td><b>124</b></td>
    </tr>
    <tr>
      <td><b>IV Effective Models</b></td>
      <td><b>126</b></td>
    </tr>
    <tr>
      <td><b>18 Introduction</b></td>
      <td><b>126</b></td>
    </tr>
    <tr>
      <td><b>19 Modified NJL models</b></td>
      <td><b>126</b></td>
    </tr>
    <tr>
      <td><b>20 Background magnetic field helps fixing model parameters</b></td>
      <td><b>131</b></td>
    </tr>
    <tr>
      <td>20.1 Motivation</td>
      <td>131</td>
    </tr>
    <tr>
      <td>20.2 Determining the strength of the 't Hooft determinant term</td>
      <td>131</td>
    </tr>
    <tr>
      <td>20.3 Topological susceptibility</td>
      <td>134</td>
    </tr>
    <tr>
      <td>20.4 QCD corrections to the pion mass difference</td>
      <td>135</td>
    </tr>
    <tr>
      <td><b>21 van der Waals-type models</b></td>
      <td><b>137</b></td>
    </tr>
    <tr>
      <td>21.1 Overview</td>
      <td>137</td>
    </tr>
    <tr>
      <td>21.2 Causality</td>
      <td>138</td>
    </tr>
    <tr>
      <td>21.3 Short-range correlations</td>
      <td>138</td>
    </tr>
    <tr>
      <td>21.4 More improvements and new challenges</td>
      <td>140</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>22 Supersymmetric Quantum Mechanics and the Dynamics of Dirac Matter Under External Electro-magnetic Fields</td>
      <td>141</td>
    </tr>
    <tr>
      <td>22.1 SUSY and the Dirac equation</td>
      <td>143</td>
    </tr>
    <tr>
      <td>22.2 Bound states in external electromagnetic fields</td>
      <td>144</td>
    </tr>
    <tr>
      <td>22.3 Green Functions</td>
      <td>144</td>
    </tr>
    <tr>
      <td>23 Quark condensate nonequilibrium dynamics under a strong magnetic field</td>
      <td>147</td>
    </tr>
    <tr>
      <td>23.1 Langevin equation from the LSMq</td>
      <td>147</td>
    </tr>
    <tr>
      <td>23.2 Explicit results</td>
      <td>150</td>
    </tr>
    <tr>
      <td>23.3 Conclusions and perspectives</td>
      <td>152</td>
    </tr>
    <tr>
      <td>V Dense Magnetized Matter and Phase Diagram</td>
      <td>154</td>
    </tr>
    <tr>
      <td>24 Introduction</td>
      <td>154</td>
    </tr>
    <tr>
      <td>25 Axial-vector current and implications to the phase diagram and the pulsar kick</td>
      <td>156</td>
    </tr>
    <tr>
      <td>25.1 Induced vector and axial-vector currents</td>
      <td>156</td>
    </tr>
    <tr>
      <td>25.2 Inhomogeneous condensation and the axial-vector interaction</td>
      <td>158</td>
    </tr>
    <tr>
      <td>25.3 Axial-vector background scattering for the pulsar kick</td>
      <td>161</td>
    </tr>
    <tr>
      <td>26 Driving chiral phase transition with ring diagram</td>
      <td>163</td>
    </tr>
    <tr>
      <td>26.1 Introduction</td>
      <td>163</td>
    </tr>
    <tr>
      <td>26.2 Chiral Quark Model</td>
      <td>163</td>
    </tr>
    <tr>
      <td>26.3 Towards Dense Matter</td>
      <td>165</td>
    </tr>
    <tr>
      <td>27 Nucleon axial coupling constant under magnetars conditions</td>
      <td>167</td>
    </tr>
    <tr>
      <td>27.1 Introduction</td>
      <td>167</td>
    </tr>
    <tr>
      <td>27.2 In-medium nucleon-nucleon correlator</td>
      <td>168</td>
    </tr>
    <tr>
      <td>27.3 Axial-vector coupling</td>
      <td>169</td>
    </tr>
    <tr>
      <td>27.4 Results</td>
      <td>170</td>
    </tr>
    <tr>
      <td>28 Dynamics of Heavy quarks in a magnetised medium</td>
      <td>172</td>
    </tr>
    <tr>
      <td>28.1 Introduction</td>
      <td>172</td>
    </tr>
    <tr>
      <td>28.2 Formalism</td>
      <td>172</td>
    </tr>
    <tr>
      <td>28.3 Results</td>
      <td>173</td>
    </tr>
    <tr>
      <td>28.4 Summary</td>
      <td>176</td>
    </tr>
    <tr>
      <td>29 The role of quark anomalous magnetic moment in magnetized quark matter</td>
      <td>177</td>
    </tr>
    <tr>
      <td>29.1 Introduction</td>
      <td>177</td>
    </tr>
    <tr>
      <td>29.2 Nambu-Jona-Lasinio SU(2) with quark AMM in a constant magnetic field</td>
      <td>178</td>
    </tr>
    <tr>
      <td>29.3 Regularization</td>
      <td>179</td>
    </tr>
    <tr>
      <td>29.4 Recent Achievements</td>
      <td>181</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>29.5 Conclusions</td>
      <td>184</td>
    </tr>
    <tr>
      <td><b>30 Summary and outlook</b></td>
      <td><b>185</b></td>
    </tr>
    <tr>
      <td><b>VI Neutron Stars</b></td>
      <td><b>186</b></td>
    </tr>
    <tr>
      <td><b>31 Introduction</b></td>
      <td><b>186</b></td>
    </tr>
    <tr>
      <td><b>32 Effective hadronic models applied to compact stars</b></td>
      <td><b>187</b></td>
    </tr>
    <tr>
      <td>32.1 Overview</td>
      <td>187</td>
    </tr>
    <tr>
      <td>32.2 Symmetry energy</td>
      <td>188</td>
    </tr>
    <tr>
      <td>32.3 Neutron stars macroscopic properties</td>
      <td>189</td>
    </tr>
    <tr>
      <td><b>33 Neutron-star matter under strong magnetic fields and magnetars</b></td>
      <td><b>191</b></td>
    </tr>
    <tr>
      <td>33.1 Introduction</td>
      <td>191</td>
    </tr>
    <tr>
      <td>33.2 Microphysics</td>
      <td>192</td>
    </tr>
    <tr>
      <td>33.3 Macrophysics</td>
      <td>195</td>
    </tr>
    <tr>
      <td>33.4 Conclusion</td>
      <td>197</td>
    </tr>
    <tr>
      <td><b>34 Hyperons in Magnetized Neutron Stars</b></td>
      <td><b>198</b></td>
    </tr>
    <tr>
      <td><b>35 Massive, magnetized neutron stars</b></td>
      <td><b>201</b></td>
    </tr>
    <tr>
      <td>35.1 Introduction</td>
      <td>201</td>
    </tr>
    <tr>
      <td>35.2 Massive neutron star structure</td>
      <td>203</td>
    </tr>
    <tr>
      <td>35.3 Results</td>
      <td>204</td>
    </tr>
    <tr>
      <td>35.4 Plausible detection through continuous gravitational waves</td>
      <td>208</td>
    </tr>
    <tr>
      <td>35.5 Summary and outlook</td>
      <td>210</td>
    </tr>
    <tr>
      <td><b>36 Strong magnetic fields and the inner crust of neutron stars</b></td>
      <td><b>211</b></td>
    </tr>
    <tr>
      <td>36.1 Introduction</td>
      <td>211</td>
    </tr>
    <tr>
      <td>36.2 Theoretical Framework</td>
      <td>212</td>
    </tr>
    <tr>
      <td>36.3 Results and discussion</td>
      <td>215</td>
    </tr>
    <tr>
      <td>36.4 Conclusions</td>
      <td>220</td>
    </tr>
    <tr>
      <td><b>37 Superconductivity and superfluidity in strong magnetic fields</b></td>
      <td><b>220</b></td>
    </tr>
    <tr>
      <td>37.1 Introductory remarks</td>
      <td>220</td>
    </tr>
    <tr>
      <td>37.2 Destruction of S-wave neutron superfluidity by strong magnetic fields</td>
      <td>221</td>
    </tr>
    <tr>
      <td>37.3 Destruction of S-wave superconductivity of protons in strong magnetic field</td>
      <td>222</td>
    </tr>
    <tr>
      <td>37.4 Astrophysical implications: Precessing magnetars and fast radio bursts (FRBs)</td>
      <td>224</td>
    </tr>
    <tr>
      <td>37.5 Neutrino radiation from magnetars</td>
      <td>224</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td><b>38</b> Neutron P-wave superfluids in neutron stars: topological surface defects</td>
      <td><b>225</b></td>
    </tr>
    <tr>
      <td>38.1 Introduction</td>
      <td>225</td>
    </tr>
    <tr>
      <td>38.2 Phase diagram of neutron <sup>3</sup>P<sub>2</sub> superfluids</td>
      <td>226</td>
    </tr>
    <tr>
      <td>38.3 Topological surface defects on neutron star and magnetar</td>
      <td>231</td>
    </tr>
    <tr>
      <td>38.4 Summary</td>
      <td>234</td>
    </tr>
    <tr>
      <td><b>39</b> Neutron star kicks: rocket effects in strong B-fields</td>
      <td><b>234</b></td>
    </tr>
    <tr>
      <td>39.1 Introduction</td>
      <td>234</td>
    </tr>
    <tr>
      <td>39.2 Neutrino beaming by magnetic vortices and cooling delay</td>
      <td>235</td>
    </tr>
    <tr>
      <td>39.3 Pulsar kicks from color superconductivity parity violation</td>
      <td>237</td>
    </tr>
    <tr>
      <td>39.4 Electromagnetic rocket effect</td>
      <td>238</td>
    </tr>
    <tr>
      <td>39.5 Eccentric and isolated MSPs from accretion-induced twin star transition</td>
      <td>239</td>
    </tr>
    <tr>
      <td>39.6 Summary and outlook</td>
      <td>241</td>
    </tr>
    <tr>
      <td><b>VII</b> Conclusions</td>
      <td><b>243</b></td>
    </tr>
  </tbody>
</table>
## Introduction

The properties of physical systems influenced by the presence of strong magnetic fields have become a subject of intense research over the recent past. It has been known already for a long time, that the magnetic field leads to dimensional reduction in the description of particle dynamics, which is at the core of a myriad of important effects involving the physical properties of systems subject to the presence of magnetic fields. Among these, an important effect is magnetic catalysis, namely, the dynamical generation of mass and the enhancement of the chiral condensate in the presence of a magnetic field. Nevertheless, it is by now known that in the simultaneous presence of a heat bath and a magnetic field, strongly interacting matter exhibits an interesting phenomenon, whereby the chiral condensate melts for temperatures close to the chiral transition, and the transition temperature itself also decreases. This effect is dubbed inverse magnetic catalysis and was discovered about a decade ago by means of lattice QCD simulations. It is fair to say that this unexpected phenomenon sparked a great deal of activity, mainly among practitioners performing research on the properties of strongly interacting matter under extreme conditions, since it was realized that a magnetic field of a strength similar or larger than $\Lambda_{QCD}$ offered an excellent probe to explore QCD matter in this regime. The exploration includes studying the pattern of modifications of mass and other static properties, such as form factors, in the meson and baryon sectors. On the other hand, the intense magnetic fields present in semi-central heavy-ion collisions at large energies, opened up the possibility to produce and detect chirally imbalanced matter in this kind of systems, a phenomenon dubbed the Chiral Magnetic Effect (CME). This has further motivated exploring the effects of these fields not only during the hot and dense phases of the reaction but also at the beginning, when matter is not yet equilibrated but the field is the strongest. Although the CME has proven difficult to be experimentally confirmed, an analogous effect has been successfully detected in condensed matter systems. This finding has also motivated another intense line of research aiming to explore two-dimensional systems of electromagnetically interacting particles, subject to the presence of magnetic fields. In another front of research, the advent of gravitational wave detection has strengthened the era of precision cosmology and astrophysics, motivating more detailed studies to elucidate the role played by the intense magnetic fields, present in compact astrophysical objects, on the QCD equation of state. The goal in this context is to find possible explanations for the anomalous mass-radius relations found for some of these objects, or the role that these fields may play as catalyzers of a rocket effect to help understand the neutron star kick velocities.

In this work we present a comprehensive review of the state of the art of research aimed to understand the properties of a range of physical systems subject to the influence of strong magnetic fields. The work was motivated by presentations and discussions during the “Workshop on Strongly Interacting Matter in Strong Electromagnetic Fields” that took place in the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) in the city of Trento, Italy, September 25-29, 2023. The work is organized as follows: In Part II we discuss recent developments in the theory front, covering a wide range of topics, from the description of mixing of charged states and the proper treatment of Schwinger phases to account for the magnetic field driven mass and form factor modifications for mesons and baryons, photon production during pre-equilibrium in heavy-ion collisions, the thermomagnetic properties of QCD, including finite volume effects, and finally the description of magnetic field effects in QED matter beyond the constant field approximation, considering the field as a background of white noise around an average value. Part III is devoted to describing recent developments in the description of non-perturbative phenomena in QCD and the electroweak theory with magnetic fields.
In particular, we study anomalous transport phenomena in QCD as well as in low-dimensional QED, a subject that is relevant for certain condensed matter systems. Given the non-perturbative nature of these phenomena, they require either first-principles lattice field theory simulations or effective approaches, for instance hydrodynamics, which are also discussed in this part. In Part IV we present several effective models that have been developed for the description of few-nucleon systems, nuclear matter, nuclei and QCD matter, extended to describe strongly interacting magnetized matter. Although these models were conceived to address specific features, they can still be used to gain insight into the properties of strong interactions in the presence of magnetic fields or nonzero densities, which otherwise cannot be obtained from first-principles calculations. In Part V we present recent theoretical advancements in the exploration of magnetic field effects on the QCD phase diagram and dense magnetized matter. We discuss a new mechanism for accelerating proto-neutron stars involving the chiral separation effect. We examine the impact of incorporating ring diagrams into the four-quark interaction on the quark gap equation within a chiral model. Moreover, we use finite energy sum rules in dense nuclear matter and in the presence of a constant and uniform external magnetic field to obtain information on different hadronic and QCD parameters. We also study the momentum diffusion coefficients of heavy quarks in a hot, magnetized medium, examining a wide spectrum of external magnetic field strengths. In addition, we address the effects of considering a quark anomalous magnetic moment to probe new phenomena in the context of the magnetized QCD phase diagram. Part VI continues the discussion with neutron stars, which are among the densest objects in astrophysics, formed from massive stars after a supernova explosion. They possess a large mass in a relatively small volume as well as strong magnetic fields in their interior. These objects provide laboratories for nuclear physics under extreme conditions and therefore in this Part we discuss the their physics and the way magnetic fields can influence their properties. Finally in Part VII we summarize and conclude.
## Theory Developments

### 1. Introduction

Electromagnetic fields are a unique tool to explore the properties of the QCD vacuum, thermodynamics and phase structure. When the energy associated to the field strength is larger than $\Lambda_{QCD}$, the field is able to probe length scales smaller than the hadron size, potentially revealing aspects of the dynamics associated to confinement and chiral symmetry breaking. For example, it is by now well known that at zero temperature magnetic fields catalyze the breaking of chiral symmetry, producing a stronger light quark-antiquark condensate [4, 5, 6, 7, 8]. However, contrary to the first predictions from chiral models [9, 10, 11] and preliminary results from LQCD [12], at finite temperature magnetic fields inhibit the condensate formation, reducing the critical temperature for chiral symmetry restoration and thus giving rise to inverse magnetic catalysis (IMC), as found in LQCD analysis and described using effective models with magnetic field-dependent couplings [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. This behavior has motivated an intense activity aimed to search for the influence of magnetic fields on hadron dynamics [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94]. The possibility to generate large intensity, albeit short-lived, magnetic fields in peripheral heavy-ion collisions at high energies [95, 96, 97, 98, 99, 100] has further stimulated the experimental search for signals associated to the presence of these fields in that environment. In fact, experimentally, a peak intensity of order $B \sim 10^{19}$ G has been inferred at RHIC energies [101, 102]. The strength of these fields is usually taken as uniform, although it can in principle be also treated as white noise around a peak strength [103, 104, 105].

In this Part, we describe recent theoretical developments that contribute to the ongoing exploration of the properties of the QCD vacuum using magnetic fields as a probe. We approach the study both from the point of view of hadron, as well as from fundamental QCD degrees of freedom. Hadron properties are addressed from the perspective of effective models, in particular the Nambu-Jona-Lasinio (NJL) model and the constituent quark model (CQM). In Sec. 2, the mass spectrum of light mesons in the presence of a constant and homogeneous magnetic field is described. The importance of including the mixing between the pseudoscalar, vector and axial-vector channels is emphasized. The description of this mixing is made using a two-flavor NJL-like model which includes isoscalar and isovector couplings together with a flavor mixing 't Hooft-like term. Attention is also payed to the proper account of the effects of the Schwinger phase charged particles. In Sec. 3, the magnetic field modification of constituent quark-meson form factors and of the one-meson exchange amplitude in the weak field limit, are explored. In Sec. 4, photon production from gluon fusion and splitting induced by magnetic fields in relativistic heavy-ion collisions at pre-equilibrium is studied as a possible venue to solve the photon puzzle. Next, in Sec. 5, thermo-magnetic properties of QCD are explored, first from a perturbative computation of the pressure, the chiral condensate and the strange quark number susceptibility up to two-loops for very large values of the field strength and for physical quark masses. The convergence of the perturbative series for the pressure for different choices of the renormalization scale in the running coupling is also studied and results are compared with recent LQCD data, away from the chiral transition. Effects of a finite baryon density and of the renormalization scale in the running coupling and strange quark mass are also studied and the results are used to build a simple analytical model for the
equation of state of pure quark magnetars. In Sec. 6, the impact of a uniform magnetic field on the lowest two QCD topological cumulants is explored using chiral perturbation theory. The nature of the corrections is discussed and low energy theorems that connect the topological susceptibility with the chiral condensate and the fourth cumulant with the chiral condensate and susceptibilities, are deduced. The nature of finite volume thermodynamics with a particular focus on the chiral condensate, renormalized magnetization and the anisotropic pressure, is investigated. It is found that the local chiral condensate is doubly periodic on the plane transverse to the magnetic field. Both the renormalized magnetization and (spatially averaged) chiral condensate exhibit significant corrections, with the former being particularly sensitive to finite volume corrections due to its small size.

The effects of magnetic fields on QCD dynamics are usually explored assuming that the field intensity is constant and uniform. Nevertheless this assumption is an idealization of physical systems where the magnetic fields may at least fluctuate in intensity and even in direction in an extended space-time region. In order to include space and time fluctuations into a field theoretical framework, Secs. 7, 8 and 9 are devoted to the description of the properties of fermions and gauge fields immersed in a noisy magnetic field background. In Sec. 7, the effective quasi-particle fermion propagator, dressed with magnetic noise effects, is obtained as a solution of the corresponding Dyson equation. The perturbative results at first-order in the magnetic noise indicate a renormalization of charge as well as the emergence of an effective index of refraction. The picture that emerges corresponds to quasiparticles propagating in a dispersive medium induced by the fluctuations of the background field. In Sec. 8, a vector order parameter, representing an ensemble average over magnetic noise of the fermion currents, is introduced together with the replica formalism to study the modification of the fermion propagator induced by white noise at mean field level. The region where this order parameter becomes finite, thus breaking the $U(1)$-symmetry due to the magnetic noise, is identified. In Sec. 9, the one-loop polarization tensor for photons and gluons in a magnetized medium, where the magnetic field experiences fluctuations, is investigated. These fluctuations are described using the replica method. The resulting polarization tensor is not transverse and this property is shown to be connected with the dynamical generation of a magnetic mass for otherwise massless gauge bosons (photons or gluons).

## 2. Magnetized mesons in the Nambu–Jona-Lasinio model

As already mentioned, the effects caused by magnetic fields, $|eB|$ larger than $\sim \Lambda_{QCD}^2$, on the properties of strong-interacting matter have attracted a lot of attention along the past decades. In this context, the understanding of the way in which the properties of light hadrons are modified by the presence of an intense magnetic field becomes a very relevant task. Clearly, this is a nontrivial problem, since first-principle theoretical calculations require to deal in general with QCD in a low energy non-perturbative regime. As a consequence, the corresponding theoretical analyses have been carried out using a variety of approaches. In particular, in the framework of Nambu-Jona-Lasinio (NJL)-like models the effect of intense external magnetic fields on $\pi$ meson properties has been extensively studied [106, 107, 63, 46, 61, 108, 109, 53, 47, 52, 49, 58, 56, 110, 111, 112, 113, 114, 115, 116]. In addition, several results for the $\pi$ meson spectrum in the presence of background magnetic fields have been obtained from lattice QCD (LQCD) calculations [13, 117, 118, 119, 120, 121]. Regarding the $\rho$ meson sector, studies of magnetized $\rho$ meson masses in the framework of NJL-like models and LQCD can be found in Refs. [61, 52, 111, 115, 122, 77, 123] and Refs. [117, 118, 119, 124, 125, 126], respectively. In most of the existing calculations the mixing between states of different spin/isospin has been neglected. Although such mixing contributions are usually forbidden by isospin and/or angular momentum conservation, they can be nonzero in the presence
of the external magnetic field. In the framework of NJL-like models effects of this kind have been studied recently in Refs. [127, 128, 129]. The aim of the present contribution is to review the main results of those works.

## 2.1. Effective Lagrangian and mean field quantities

We consider the Lagrangian for an extended SU(2) NJL model in the presence of an electromagnetic field. We have

$$ \mathcal{L} = \bar{\psi}(x) (i\mathcal{D} - m_c) \psi(x) + g_S \sum_{a=0}^{3} \left[ \left( \bar{\psi}(x) \tau_b \psi(x) \right)^2 + \left( \bar{\psi}(x) i\gamma_5 \tau_b \psi(x) \right)^2 \right] - g_V \left[ \left( \bar{\psi}(x) \gamma_\mu \tau \psi(x) \right)^2 + \left( \bar{\psi}(x) \gamma_\mu \gamma_5 \tau \psi(x) \right)^2 \right] \\ - g_{V_0} \left( \bar{\psi}(x) \gamma_\mu \psi(x) \right)^2 - g_{A_0} \left( \bar{\psi}(x) \gamma_\mu \gamma_5 \psi(x) \right)^2 + 2g_D \left( \det[\bar{\psi}(x)(1+\gamma_5)\psi(x)] + \det[\bar{\psi}(x)(1-\gamma_5)\psi(x)] \right), \quad (2.1) $$

where $\psi = (u \ d)^T$, $\tau_b = (1, \tau)$, $\tau$ being the usual Pauli-matrix vector, and $m_c$ is the current quark mass, which is assumed to be equal for u and d quarks. The interaction between the fermions and the electromagnetic field $A_\mu$ is driven by the covariant derivative $D_\mu = \partial_\mu + i\hat{Q}A_\mu$, where $\hat{Q} = \text{diag}(Q_u, Q_d)$, with $Q_u = 2e/3$ and $Q_d = -e/3$, $e$ being the proton electric charge. We consider the case of a homogeneous stationary magnetic field $B$ orientated along the axis 3, or $z$. To write down the explicit form of $A^\mu$ one has to choose a specific gauge. Some commonly used (“standard”) gauges are the symmetric gauge, in which $A^\mu(x) = (0, -Bx^2/2, Bx^1/2, 0)$, the Landau gauge 1, in which $A^\mu(x) = (0, -Bx^2, 0, 0)$ and the Landau gauge 2, in which $A^\mu(x) = (0, 0, Bx^1, 0)$. To test the gauge independence of our results, all these gauges will be considered in our analysis.

Since we are interested in studying meson properties, it is convenient to bosonize the fermionic theory, introducing scalar, pseudoscalar, vector and axial vector fields $\sigma_b(x)$, $\pi_b(x)$, $\rho_b^\mu(x)$, $a_b^\mu(x)$, with $b = 0, 1, 2, 3$, and integrating out the fermion fields. The bosonized action can be written as

$$ S_{\text{bos}} = -i \ln \det(i\mathcal{D}) - \frac{1}{4g} \int d^4 x \left[ \sigma_0(x) \sigma_0(x) + \boldsymbol{\pi}(x) \cdot \boldsymbol{\pi}(x) \right] - \frac{1}{4g(1-2\alpha)} \int d^4 x \left[ \boldsymbol{\sigma}(x) \cdot \boldsymbol{\sigma}(x) + \pi_0(x) \pi_0(x) \right] \\ + \frac{1}{4g_V} \int d^4 x \left[ \boldsymbol{\rho}_\mu(x) \cdot \boldsymbol{\rho}^\mu(x) + \boldsymbol{a}_\mu(x) \cdot \boldsymbol{a}^\mu(x) \right] + \frac{1}{4g_{V_0}} \int d^4 x \rho_{0\mu}(x) \rho_0^\mu(x) + \frac{1}{4g_{A_0}} \int d^4 x a_{0\mu}(x) a_0^\mu(x), \quad (2.2) $$

with

$$ i\mathcal{D}_{x,x'} = \delta^{(4)}(x-x') \left\{ i\mathcal{D} - m_0 - \tau_b [\sigma_b(x) + i\gamma_5 \pi_b(x) + \gamma_\mu \rho_b^\mu(x) + \gamma_\mu \gamma_5 a_b^\mu(x)] \right\}. \quad (2.3) $$

In Eq. (2.2) we have introduced $g \equiv g_S + g_D$ and $\alpha \equiv g_D/(g_S + g_D)$, in such a way that the flavor mixing turns out to be regulated by the parameter $\alpha$.

We proceed by expanding $S_{\text{bos}}$ in powers of the fluctuations of the boson fields around the corresponding mean field (MF) values. We assume that the fields $\sigma_b(x)$ have nontrivial translational invariant MF values given by $\tau_b \bar{\sigma}_b = \text{diag}(\bar{\sigma}_u, \bar{\sigma}_d)$, while vacuum expectation values of other boson fields are zero; thus, we write

$$ \mathcal{D}_{x,x'} = \text{diag}(\mathcal{D}_{x,x'}^{\text{MF},u}, \mathcal{D}_{x,x'}^{\text{MF},d}) + \delta \mathcal{D}_{x,x'}, \quad (2.4) $$

where $\mathcal{D}_{x,x'}^{\text{MF},f} = -i \delta^{(4)}(x-x') (i\partial\phi + Q_f A(x) - M_f)$, with $f = u, d$. Here $M_f = m_c + \bar{\sigma}_f$ is the quark effective mass for each flavor $f$.

The MF action per unit volume is given by

$$ \frac{S_{\text{bos}}^{\text{MF}}}{V^{(4)}} = - \frac{(1-\alpha)(\bar{\sigma}_u^2 + \bar{\sigma}_d^2) - 2\alpha \bar{\sigma}_u \bar{\sigma}_d}{8g(1-2\alpha)} - \frac{iN_c}{V^{(4)}} \sum_{f=u,d} \int d^4x \, d^4x' \, \text{tr}_D \ln (S_{x,x'}^{\text{MF},f})^{-1}, \quad (2.5) $$

where $\text{tr}_D$ stands for the trace over Dirac space and $S_{x,x'}^{\text{MF},f} = (i\mathcal{D}_{x,x'}^{\text{MF},f})^{-1}$ is the MF quark propagator in the presence of the magnetic field. Its explicit expression can be written as

$$ S_{x,y}^{\text{MF},f} = e^{i\Phi_{Q_f}(x,y)} \int \frac{d^4 p}{(2\pi)^4} e^{-ip \cdot (x-y)} \bar{S}^f(p_\parallel, p_\perp), \quad (2.6) $$
where the function $\Phi_Q(x, y)$ is the so-called Schwinger phase, which is shown to be a gauge dependent quantity. In addition, introducing $B_f = |BQ_f|$, $s_f = \text{sign}(BQ_f)$ and $t_f = \tan(\sigma B_f)$, one has

$$ \bar{S}^f(p_\parallel, p_\perp) = -i \int_0^\infty d\sigma \exp \left[ -i\sigma \left( M_f^2 - p_\parallel^2 + \frac{t_f \mathbf{p}_\perp^2}{\sigma B_f} - i\epsilon \right) \right] \left[ (p_\parallel \cdot \gamma_\parallel + M_f) (1 - s_f t_f \gamma^1 \gamma^2) - (1 - t_f^2) \mathbf{p}_\perp \cdot \gamma_\perp \right] . \quad (2.7) $$

Here, we have defined $p_\parallel^\mu = (p^0, 0, 0, p^3)$, $p_\perp^\mu = (0, p^1, p^2, 0)$, and equivalent definitions have been used for $\gamma_\parallel$, $\gamma_\perp$.

The gap equations needed to determine the effective masses $M_f$ are obtained from $\partial S_{\text{bos}}^{\text{MF}}/\partial \bar{\sigma}_f = 0$. It is seen that they can be written in terms of the quark condensates $\phi_f \equiv \langle \bar{\psi}_f \psi_f \rangle$. As usual in NJL-like models, the expressions for the condensates involve divergent loop integrals that have to be properly regularized. We use here the magnetic field independent regularization (MFIR) scheme [130, 131]: for a given unregularized quantity, the corresponding (divergent) $B \to 0$ limit is subtracted and then it is added in a regularized form. Thus, the quantities can be separated into a (finite) “$B=0$” part and a “magnetic” piece. To regularize the “$B=0$” terms (which still keep an implicit dependence on the magnetic field, e.g. through the masses $M_f$) we use here a proper time (PT) scheme. This requires the introduction of a dimensionful parameter $\Lambda$, which plays the role of an ultraviolet cutoff.

## 2.2. Meson masses in the NJL model at finite magnetic field

The quadratic piece of the bosonized action corresponding to the neutral mesons can be written as

$$ S_{\text{bos}}^{\text{quad, neutral}} = -\frac{1}{2} \int d^4 x \, d^4 x' \sum_{M, M'} \delta M(x)^{\dagger} \mathcal{G}_{MM'}(x, x') \delta M(x') . \quad (2.8) $$

Here $M.M' = \sigma_0, \pi_0, \rho_0^\mu, a_0^\mu, \sigma_3, \pi_3, \rho_3^\mu, a_3^\mu$. Note that $\sigma_0, \pi_0, \rho_0$ and $a_0$ correspond to the isoscalar states $\sigma, \eta, \omega$ and $f_1$, while $\sigma_3, \pi_3, \rho_3$ and $a_3$ stand for the neutral components of the isovector triplets $a_0, \pi, \rho$ and $a_1$, respectively. The meson indices $M, M'$, as well as the functions $\mathcal{G}_{MM'}$, include Lorentz indices in the case of vector mesons. This also holds for the functions $\delta_{MM'}$, $\mathcal{J}_{MM'}$, $c_{MM'}^{ff'}$, etc., introduced below. In the corresponding expressions, a contraction of Lorentz indices is understood when appropriate. The functions $\mathcal{G}_{MM'}(x, x')$ can be separated in two terms, namely

$$ \mathcal{G}_{MM'}(x, x') = \frac{1}{2g_M} \delta_{MM'} \delta^{(4)}(x - x') - \mathcal{J}_{MM'}(x, x'), \quad (2.9) $$

where

$$ \frac{1}{g_M} \delta_{MM'} = \begin{cases} 1/g & \text{for } M = M' = \sigma_0, \pi_3 \\ 1/[g(1-2\alpha)] & \text{for } M = M' = \sigma_3, \pi_0 \\ -\eta^{\mu\nu}/g_V & \text{for } MM' = \rho_3^\mu \rho_3^\nu, a_3^\mu a_3^\nu \\ -\eta^{\mu\nu}/g_{V_0} & \text{for } MM' = \rho_0^\mu \rho_0^\nu \\ -\eta^{\mu\nu}/g_{A_0} & \text{for } MM' = a_0^\mu a_0^\nu \end{cases} \quad (2.10) $$

and $\delta_{MM'} = 0$ otherwise. Here $\eta^{\mu\nu}$ is the Minkowski metric tensor, $\eta^{\mu\nu} = \text{diag}(1, -1, -1, -1)$. In turn, the polarization functions $\mathcal{J}_{MM'}(x, x')$ can be separated into u and d quark pieces,

$$ \mathcal{J}_{MM'}(x, x') = c_{MM'}^{uu}(x, x') + \varepsilon_M \varepsilon_{M'} c_{MM'}^{dd}(x, x') . \quad (2.11) $$

Here, $\varepsilon_M = 1$ for $M = \sigma_0, \pi_0, \rho_0^\mu, a_0^\mu$ and $\varepsilon_M = -1$ for $M = \sigma_3, \pi_3, \rho_3^\mu, a_3^\mu$, while the functions $c_{MM'}^{ff'}(x, x')$ are given by

$$ c_{MM'}^{ff'}(x, x') = -i N_c \text{tr}_D \left[ i S_{x,x'}^{\text{MF},f} \Gamma^{M'} i S_{x',x}^{\text{MF},f'} \Gamma^M \right], \quad (2.12) $$

with $\Gamma^\sigma = 1$, $\Gamma^\pi = i\gamma^5$, $\Gamma^{\rho\mu} = \gamma^\mu$ and $\Gamma^{a\mu} = \gamma^\mu \gamma^5$.
We now address the charged mesons case, i.e. the states $s^{\pm} = (s_1 \mp is_2)/\sqrt{2}$ and $v^{\pm\mu} = (v_1^{\mu} \mp iv_2^{\mu})/\sqrt{2}$, with $s = \sigma, \pi$ and $v = \rho, a$. We concentrate on the positive charge sector, noticing that the analysis of negatively charged mesons is completely equivalent. The corresponding quadratic piece of the bosonized action can be written as

$$S_{\text{bos}}^{\text{quad,+}} = -\frac{1}{2} \int d^4 x \, d^4 x' \sum_{M,M'} \delta M(x)^{\dagger} \mathcal{G}_{MM'}(x, x') \delta M'(x') , \qquad (2.13)$$

where, for notational convenience, we simply denote the positively charged states by $M, M' = \sigma, \pi, \rho^{\mu}, a^{\mu}$ (a proper contraction of Lorentz indices of vector mesons is understood). Once again, the inverse propagator $\mathcal{G}_{MM'}(x, x')$ can be separated in a form as that given by Eq. (2.9). In this case one has

$$\mathcal{J}_{MM'}(x, x') = 2 c_{MM'}^{ud}(x, x') \qquad (2.14)$$

and

$$\frac{1}{g_M} \delta_{MM'} = \begin{cases} 1/g & \text{for } M = M' = \pi \\ 1/[g(1 - 2\alpha)] & \text{for } M = M' = \sigma \\ -\eta^{\mu\nu}/g_V & \text{for } MM' = \rho^\mu \rho^\nu, a^\mu a^\nu \end{cases}, \qquad (2.15)$$

with $\delta_{MM'} = 0$ otherwise.

To proceed we have to evaluate the functions $c_{MM'}^{ff'}(x, x')$ in Eqs. (2.11) and (2.14). From Eqs. (2.12) and (2.6) we have

$$c_{MM'}^{ff'}(x, x') = e^{i\Phi_M(x, x')} \int \frac{d^4 t}{(2\pi)^4} e^{-it(x-x')} \bar{c}_{MM'}^{ff'}(t_\parallel, t_\perp), \qquad (2.16)$$

where

$$\bar{c}_{MM'}^{ff'}(t_\parallel, t_\perp) = -iN_c \int \frac{d^4 p}{(2\pi)^4} \text{tr}_D \left[ i\bar{S}f(p_\parallel^+, p_\perp^+) \Gamma^{M'} i\bar{S}f'(p_\parallel^-, p_\perp^-) \Gamma^M \right]. \qquad (2.17)$$

Here we have defined $p_a^{\pm} = p_a \pm t_a/2$, where $a = \parallel, \perp$. In addition, we have used $\Phi_M(x, x') = \Phi_{Q_f}(x, x') + \Phi_{Q_{f'}}(x', x)$. It can be seen that $\Phi_M$ vanishes in the case of the neutral mesons, while it does not for charged mesons.

In what follows we consider the particular case $M = M' = \pi$. The procedure is similar for the all the other cases, although, in general, the Lorentz structure is more complicated. Details can be found in Ref. [129]. We introduce

$$c_{\pi\pi}^{ff'}(\bar{q}, \bar{q}') = \int d^4 x \, d^4 x' \mathcal{F}_Q(x, \bar{q})^* c_{\pi\pi}^{ff'}(x, x') \mathcal{F}_Q(x', \bar{q}'), \qquad (2.18)$$

where the functions $\mathcal{F}_Q(x, \bar{q})$ are solutions of the meson field equations in the presence of an external constant and homogenous magnetic field. For the neutral pion $\bar{q}$ stands for the usual four-momentum, $\bar{q} = q = (q^0, \mathbf{q})$, while $\mathcal{F}_Q(x, \bar{q}) = \exp[-iq \cdot x]$. On the other hand, for charged pions $\mathcal{F}_Q(x, \bar{q})$ is a gauge dependent Ritus-like function; in this case one has $\bar{q} = (q^0, \ell, \chi, q^3)$, where $\ell$ is an integer related with the so-called Landau level, and the fourth quantum number $\chi$ can be chosen according to the gauge fixing. Details on this issue can be found in Ref. [132], where the explicit form of $\mathcal{F}_Q(x, \bar{q})$ for the standard gauges is given. Replacing Eq. (2.16) into Eq. (2.18) we find

$$c_{\pi\pi}^{ff'}(\bar{q}, \bar{q}') = \int \frac{d^4 t}{(2\pi)^4} \bar{c}_{\pi\pi}^{ff'}(t_\parallel, t_\perp) h_Q(\bar{q}, \bar{q}', t), \qquad (2.19)$$

where $h_Q(\bar{q}, \bar{q}', t)$ is a gauge invariant quantity given by

$$h_Q(\bar{q}, \bar{q}', t) = \int d^4 x \, d^4 x' \mathcal{F}_Q(x, \bar{q})^* \mathcal{F}_Q(x', \bar{q}') e^{i\Phi_Q(x, x')} e^{-it(x-x')} . \qquad (2.20)$$

For neutral pions it is easy to show that $h_{Q=0}(q, q', t) = (2\pi)^4 \delta^{(4)}(q-q') (2\pi)^4 \delta^{(4)}(q-t)$. Thus, one has $c_{\pi\pi}^{ff,0}(q, q') = (2\pi)^4 \delta^{(4)}(q-q') C_{\pi\pi}^{ff,0}(q_\parallel, q_\perp)$, where $C_{\pi\pi}^{ff,0}(q_\parallel, q_\perp) = \bar{c}_{\pi\pi}^{ff}(q_\parallel, q_\perp)$. Therefore, from Eqs. (2.9), (2.11) and (2.18) we see
that for neutral mesons $G_{MM'}(x, x')$ becomes diagonal when transformed into “Fourier space”. In turn, for charged pions it is possible to show [132] that for all the standard gauges one has

$$h_{Q=e}(\bar{q}, \bar{q}', t) = \delta_{\chi\chi'} (2\pi)^4 \delta^{(2)}(q_{\parallel} - q'_{\parallel}) (2\pi)^2 \delta^{(2)}(q_{\parallel} - t_{\parallel}) f_{\ell\ell'}(t_{\perp}), \quad (2.21)$$

where

$$f_{\ell\ell'}(t_{\perp}) = \frac{4\pi(-i)^{\ell+\ell'}}{B_e} \sqrt{\frac{\ell!}{\ell'!}} \left(\frac{2t_{\perp}^2}{B_e}\right)^{\frac{\ell'-\ell}{2}} L_{\ell}^{\ell'-\ell} \left(\frac{2t_{\perp}^2}{B_e}\right) e^{-t_{\perp}^2/B_e} e^{is(\ell-\ell')\varphi_{\perp}}. \quad (2.22)$$

Here $B_e = |eB|$, $s = \text{sign}(eB)$, $L_k^m(x)$ are generalized Laguerre polynomials, and $t_{\perp}^\mu = |t_{\perp}|(0, \cos\varphi_{\perp}, \sin\varphi_{\perp}, 0)$. We can take these expressions and integrate over $\varphi_{\perp}$, using the fact that, due to invariance under rotations around the 3-axis and boosts in the 0-3 plane, $c_{\pi\pi}^{ff'}(t_{\parallel}, t_{\perp})$ can only depend on $t_{\parallel}^2$ and $t_{\perp}^2$. One gets in this way $c_{\pi\pi}^{ud,+}(\bar{q}, \bar{q}') = (2\pi)^4 \delta_{\chi\chi'} \delta_{\ell\ell'} \delta^{(2)}(q_{\parallel} - q'_{\parallel}) C_{\pi\pi}^{ud,+}(\ell, q_{\parallel})$, where

$$C_{\pi\pi}^{ud,+}(\ell, q_{\parallel}) = \int_{0}^{\infty} dt_{\perp}^2 \bar{c}_{\pi\pi}^{ud}(q_{\parallel}, t_{\perp}) \rho_{\ell}(t_{\perp}^2), \quad (2.23)$$

with $\rho_{\ell}(t_{\perp}^2) = (-1)^{\ell} e^{-t_{\perp}^2/B_e} L_{\ell}(2t_{\perp}^2/B_e)/B_e$. Thus, from Eqs. (2.9), (2.14) and (2.18) we see that for charged mesons $G_{MM'}(x, x')$ becomes diagonal when transformed into “Ritus space”. In this case, even for $\ell = 0$, we cannot set $t_{\perp} = 0$, i.e. we cannot take a charged meson to be rest.

As already mentioned, when vector or axial vector mesons are involved, the Lorentz structure of the polarization function is more complicated. For example, in the case of the pion-axial vector mixing polarization functions one gets

$$C_{\pi a \mu}^{ff,0}(q_{\parallel}, q_{\perp}) = d_{\pi a, 1}^{ff,0}(q_{\parallel}, q_{\perp}) q_{\parallel}^{\mu} + d_{\pi a, 2}^{ff,0}(q_{\parallel}, q_{\perp}) q_{\perp}^{\mu} \quad (2.24)$$

for the neutral $\pi^0 - a_1^0$ meson mixing, and

$$C_{\pi a \mu}^{ud,+}(\ell, q_{\parallel}) = d_{\pi a, 1}^{ud,+}(\ell, q_{\parallel}) \Pi_{\parallel}^{\mu*} + d_{\pi a, 2}^{ud,+}(\ell, q_{\parallel}) \Pi_{\perp}^{\mu*} - d_{\pi a, 3}^{ud,+}(\ell, q_{\parallel}) \text{ is } F^{\mu\nu} \Pi_{\perp,\nu}^{*}/B \quad (2.25)$$

for the $\pi^+ - a_1^+$ mixing, where $\Pi_{\parallel}^{\mu} = q_{\parallel}^{\mu}$ and $\Pi_{\perp}^{\mu} = \left(0, i\sqrt{\frac{B_M}{2}}(\sqrt{\ell+1}-\sqrt{\ell}), -s\sqrt{\frac{B_M}{2}}(\sqrt{\ell+1}+\sqrt{\ell}), 0\right)$. The tensor structure of all possible polarization functions, as well as the explicit form of the corresponding coefficients, can be found in Ref. [129]. It should be noted that some of these coefficients involve divergent integrals. To regularize them we use the MFIR method discussed in the previous section.

To proceed we have to contract the tensors $G_{MM'}$ that involve vector or axial vector mesons with the appropriate polarization vectors. In the case of neutral mesons there are 4 independent polarization vectors: one of them, that we indicate by $L$, is timelike, while the other three are spacelike and will be labelled with $c = 1, 2, 3$. For charged mesons the number of independent polarization vectors depends on the Landau level $\ell$. For $\ell = -1$ there is no timelike vector, whereas only one spacelike vector exists ($c = 1$); for $\ell = 0$ we have one timelike vector and two spacelike vectors ($c = 1, 2$), while for $\ell > 0$ all four polarization vectors exist. The explicit expressions of these vectors can be found in Ref. [129]. After performing the contractions, one is faced with a neutral meson inverse propagator given by a $20 \times 20$ matrix $G_{N,N'}(q_{\parallel}, q_{\perp})$, where $N, N' = \sigma_b, \pi_b, \rho_{b,L}, \rho_{b,c}, a_{b,L}, a_{b,c}, b = 0, 3$. For the positively charged meson sector, the size of the corresponding matrix depends on $\ell$, since, as stated, not all polarizations are available for $\ell = -1, 0$. In any case, one finds at most a $10 \times 10$ matrix with elements $G_{N,N'}(\ell, q_{\parallel})$, where $N, N' = \sigma, \pi, \rho_L, \rho_c, a_L, a_c$ with $c = 1, 2, 3$. Once the above described matrices are obtained, the meson masses can be determined by solving the equations

$$\det G(q_{\parallel}, q_{\perp})|_{q^0=m, q=0} = 0 \quad \text{for neutral mesons,} \quad (2.26)$$

$$\det G(\ell, q_{\perp})|_{q^0=E, q^3=0} = 0 \quad \text{for charged mesons,} \quad (2.27)$$
where $E = \sqrt{m^2 + (2\ell + 1)B_e}$.

At this point we note that the matrices become simplified due to an extra symmetry of the system, which corresponds to a parity operation followed by a $\pi$-rotation around the direction of the $B$ [129]. For charged mesons with $\ell = -1$, this symmetry implies that $\rho_1$ and $a_1$ (the only states in this sector) do not mix, while for $\ell = 0$ it is seen that the lowest energy charged pion state only mixes with $\rho_2$, $a_L$ and $a_1$. In the case of neutral mesons, the $20 \times 20$ matrix is given by a direct sum of four submatrices. One of them corresponds to the states $\pi_b, \rho_{b,2}, a_{b,L}$, one to $\sigma_b, \rho_{b,L}, a_{b,2}$, one to the (degenerated) $S_z = \pm 1$ $\rho$ meson states and, finally, one to the (degenerated) $S_z = \pm 1$ $a$ meson states.

To obtain numerical results it is necessary to fix the model parameters. As stated, in our framework divergent quantities are regularized using the MFIR scheme, with a proper time cutoff $\Lambda$. Within this scenario, we take $m_c = 7.01$ MeV, $\Lambda = 842$ MeV, $g = 5.94/\Lambda^2$, $g_V = 3.947/\Lambda^2$ and $\alpha = 0.114$. At $B = 0$, this leads to $M_{u,d} = 400$ MeV, $\phi_{u,d} = (-227 \text{ MeV})^3$, $m_\pi = 140$ MeV, $f_\pi = 92.2$ MeV, $m_\eta = 548$ MeV, $m_\rho = 775$ MeV and to a phenomenologically acceptable value of about 1020 MeV for the $a_1$ mass (in fact, as usual in this type of model, the $a_1$ mass is found to lie above the $q\bar{q}$ production threshold and can be determined only after some extrapolation). For simplicity, the remaining coupling constants are taken to be $g_{V_0} = g_{A_0} = g_V$, which leads to $m_\omega = m_\rho$ and $m_{f_1} = m_{a_1}$.

Now, while most NJL-like models are able to reproduce the effect of magnetic catalysis at vanishing temperature, they fail to describe the inverse magnetic catalysis effect observed in lattice QCD at finite temperature. One of the simplest approaches to partially cure this behavior consists of allowing the model couplings to depend on the magnetic field [21, 17, 18]. Thus, we consider here both the situation in which the couplings are constant and the one in which they vary with $B$. For definiteness, we adopt for $g(B)$ the form proposed in Ref. [108], namely

$$g(B) = g \mathcal{F}(B), \quad \mathcal{F}(B) = \kappa_1 + (1 - \kappa_1) e^{-\kappa_2(eB)^2}, \qquad (2.28)$$

with $\kappa_1 = 0.321$ and $\kappa_2 = 1.31 \text{ GeV}^{-2}$. Concerning the vector couplings, given the common gluonic origin of $g$ and $g_V$, we assume that they get affected in a similar way by the magnetic field; hence, we take $g_V(B) = g_V \mathcal{F}(B)$.

In what follows we present and analyze some selected results for the effect of the magnetic field on meson masses. Let us start with the neutral sector. As well known, for $B = 0$ pseudoscalar mesons mix with “longitudinal” axial vector mesons. For nonzero $B$ the mixing also involves neutral vector mesons with spin projection $S_z = 0$ (corresponding to the polarization state $c = 2$). The four lowest mass states of this sector are to be identified with the physical states $\tilde{\pi}^0$, $\tilde{\eta}$, $\tilde{\rho}^0$ and $\tilde{\omega}$, where the particle names are chosen according to the spin-isospin composition of the states in the limit of vanishing external field. In addition, the neutral sector includes states with spin projections $S_z = \pm 1$, i.e., spin parallel to the direction of the magnetic field. Here we concentrate only on the mass of the lowest state, $\tilde{\pi}^0$. Results for other states can be found in Ref. [129]. In Fig. 2.1 we compare our predictions with LQCD results that have been obtained using different methods and various values of the $B = 0$ pion mass. For comparison, we include the results obtained for NJL-like models in which different meson sectors have been taken into account. In the case where one considers just the pseudoscalar sector, the result is quite different depending on whether $g$ is kept constant or not. In the first case, the behavior of $m_{\tilde{\pi}^0}$ with the magnetic field is found to be non-monotonic, deviating just slightly from its value at $B = 0$ (green dotted line). In contrast, for a $B$-dependent $g$ one can get a monotonic decrease, in good agreement with LQCD results [108]. When the mixing with the vector sector is considered, the results for both constant and $B$-dependent couplings are similar to each other (red dash-dotted line) and monotonically decreasing, lying however quite below LQCD predictions [127]. Finally, if the mixing with $a_1$ mesons is also included, we obtain, for both constant (blue solid line)
and B-dependent couplings (blue dotted line), a monotonic decrease which is in good qualitative agreement with LQCD calculations. One may infer that the incorporation of axial vector mesons, being the chiral partners of vector mesons, leads to cancellations that help to alleviate the magnitude of the neutral pion mass suppression. Their inclusion into the full picture leads to relatively more robust results and improves the agreement with LQCD predictions.

<figure><img src="image_3.png" /><figcaption>**Figure 2.1:** Normalized mass of the $\pi^0$ meson as a function of $eB$.</figcaption></figure>

We turn now to the charged meson sector. To study the lowest lying states one has to consider the Landau modes $\ell = -1$ and $\ell = 0$. For $\ell = -1$, the lowest mass state is the one that we have denoted as $\rho_1$ and its energy is given by $E_{\rho^+} = \sqrt{m_{\rho^+}^2 - B_e}$, where $m_{\rho^+}$ is obtained from Eq. (2.27). The left panel of Fig. 2.2 displays our numerical results for $E_{\rho^+}$ as a function of $eB$, normalized by $m_{\rho^+}|_{B=0}$. It is seen that the results for constant $g$ (blue solid line) and for $g(B)$ (blue dashed line) are quite similar. In turn, they differ considerably from those obtained in a similar model [128] that does not take into account the presence of axial vector mesons (red dash-dotted line). In fact, here the differences between models that include or not axial vector mesons do not arise from direct mixing effects but from the fact that axial vector states mix with pions already for $B = 0$; this leads to some change in the model parameters so as to get consistency with the phenomenological inputs. In any case, it is found that —as in the case of neutral mesons— the results from the full model (blue solid and dashed lines) appear to be rather robust: they show a similar behavior either for constant or B-dependent couplings, and this behavior is shown to be in good agreement with LQCD calculations of Refs. [119, 125, 55]. We point out that our results, as those from LQCD, are not consistent with $\rho^+$ condensation for the considered range of values of $eB$. The curve corresponding to a pointlike $\rho^+$ meson is shown for comparison. It is worth mentioning that our results are qualitatively different from those obtained in other works [63, 111], which do find $\rho^+$ meson condensation for $eB \sim 0.2$ to $0.6 \text{ GeV}^2$ in the framework of two-flavor NJL-like models. Contrary to what is done here, in those works Schwinger phases are neglected and it is assumed that charged $\pi$ and $\rho$ mesons lie in zero three-momentum states.

In the case of the mode $\ell = 0$, the lowest mass state $\pi^+$ is given in general by a mixing between the states that we have denoted as $\pi, \rho_2, a_L$ and $a_1$. Its energy is given by $E_{\pi^+} = \sqrt{m_{\pi^+}^2 + B_e}$ where $m_{\pi^+}$ is obtained from the lowest $\ell = 0$ solution of Eq. (2.27). Our numerical results are shown in the right panel of Fig. 2.2 where we plot the values of the difference $E_{\pi^+}(B)^2 - E_{\pi^+}(0)^2$ as a function of $eB$. Blue solid and dashed lines correspond to the cases of constant and B-dependent couplings, respectively. We also include for comparison the results obtained from similar NJL-like models that just include the pseudoscalar meson sector (green dotted line), or just include the mixing between the pseudoscalar
and vector meson sectors (red full and dashed line), neglecting the effect of the presence of axial vector mesons. It can be seen that the inclusion of the axial vector meson sector leads to an improvement of the agreement with LQCD data. We remark, however, that our numerical results indicate a monotonic enhancement of the charged pion energy with the magnetic field, in contrast with the nonmonotonic behavior found by recent LQCD simulations, see Ref. [120]. It would be interesting to get more insight on this open issue from other effective models and further LQCD calculations.

<figure><img src="image_4.png" /><figcaption>**Figure 2.2:** Left panel: Normalized energy of the lowest $\rho^+$ meson state as a function of $eB$. Right panel: Difference $E_{\pi^+}(B)^2 - E_{\pi^+}(0)^2$ for the lowest $\pi^+$ meson state as a function of $eB$.</figcaption></figure>

## 2.3. Summary

We have considered the effect of a constant magnetic field $B$ on light meson masses, taking into account the various possible mixing effects. This has been done in the framework of a two-flavor NJL-like model that includes isoscalar and isovector couplings in the scalar-pseudoscalar and vector-axial vector sector, as well as a flavor mixing term in the scalar-pseudoscalar sector. The divergences associated with the non-renormalizability of the model have been regularized using the “magnetic field independent regularization” method. We have also explored the possibility of using $B$-dependent couplings to account for the effect of the magnetic field on sea quarks. As is well known, for $B=0$ pseudoscalar mesons mix with “longitudinal” axial vector mesons. The presence of an external uniform $B$ breaks isospin and full rotational symmetry, allowing for a more complex meson mixing pattern. The latter is constrained by the remaining unbroken symmetries, in such a way that the mass matrices can be separated into several “boxes”.

In the case of neutral mesons, Schwinger phases cancel and the polarization functions become diagonal in the usual momentum basis. We have found that when both vector and axial-vector interactions are taken into account, the mass of the lightest state in this sector, $\tilde{\pi}^0$, displays a monotonic decreasing behavior with $B$ in the studied range $eB < 1 \text{ GeV}^2$, which is in good qualitative agreement with LQCD calculations for both $g = \text{constant}$ and $g(B)$. This represents an improvement over the results that take into account just the mixing with the vector meson sector, or no mixing at all.

In the case of charged mesons, the polarization functions are diagonalized by expanding meson fields in appropriate Ritus-like bases, so as to account for the effect of non-vanishing Schwinger phases. Once again, the symmetries of the system constrain the mixing matrices, which also depend on the value of the meson Landau level $\ell$. For $\ell = -1$ one has only one vector and one axial vector polarization states that do not mix with each other. Thus, in this case the effect of
the inclusion of axial vector mesons on the $\rho^+$ mass comes solely from the model parametrization, which is affected by the presence of $\pi$-a₁ mixing at $B = 0$. Our results show that when the axial-vector sector is included, the energy $E_{\rho^+}$ of this state undergoes a considerable reduction, leading to a decreasing behavior that is in qualitative agreement with LQCD predictions, independently of the assumption of a B-dependent coupling constant. We notice that, in accordance with LQCD calculations, $E_{\rho^+}$ does not vanish for any considered value of $B$. For $\ell = 0$, the pion mixing subspace is given by $\pi^+ - \rho^+ - a_1^+$ for only certain polarizations states. Here, the lowest mass state can be identified with the $\pi^+$. We find that, even though vector mixing already induces a softening in the enhancement of the $E_{\pi^+}$ with $B$, the inclusion of the axial vector meson sector reinforces this softening, leading to an improved agreement with LQCD predictions.

### 3. Magnetic field effects in form factors and in the meson exchange potential

Hadron interactions have been shown to present different types of modifications in background magnetic fields, mostly due to the non-trivial hadron structure with quarks and gluons [48, 50, 133, 134, 135, 136, 137, 138, 139]. Experimental measurements and theoretical descriptions of form factors provide a powerful tool to understand hadron structure and interactions as described for example in Ref. [140]. Currently, there are several quite refined calculations for different types of form factors, mostly for the nucleon, based in ab initio or continuous in-vacuum methods. However, it may be useful, and conceptually enriching, to decompose such type of calculations in a CQM framework, such that the role of each degree of freedom may be tested or further understood [141, 142]. Besides its historical importance, the CQM is usually very good to describe global hadron properties, such as masses, overall interactions and decay constants. In the CQM, in different versions, hadrons are built out from colorless states of valence quarks. A dynamical derivation and investigation of constituent quark and quark-antiquark meson form factors in constant magnetic fields was initiated in the last years, starting with a one-gluon exchange quark-antiquark interaction as the leading QCD quark effective action [48, 50, 135]. By applying the background field method and the auxiliary field method, it is possible to describe the formation of quark-antiquark meson multiplets and the chiral condensate, and the development of constituent quark currents that emerge as a dressing of quark currents by a gluon cloud due to components of a non-perturbative gluon propagator. The resulting one-loop form factors, that may be considered for constant or running quark masses, have a structure akin to the Schwinger-Dyson equations (SDE) in the rainbow ladder approach [143, 144]. This becomes more appealing when embedding hadrons in an external magnetic field, in which case, the description of hadron structure and dynamics become more intricate. Expectations for the strength of magnetic fields in relativistic heavy ion collisions and in magnetars may be of the order of $eB \sim 10^{12} - 10^{13}$ T that may be weak with respect to quark or hadron masses, $eB \sim 0.1m_{\pi}^2 - 0.1M^2$, where $m_{\pi}$ and $M \sim 0.35$ GeV are the pion and a constituent quark masses [137, 145, 146, 147]. The weak magnetic field offers the possibility to allow (semi)analytical calculations since, in this limit, quark and meson propagators usually have well established analytical expressions. In the CQM, mesons couple to baryons via the baryon constituent quark that in turn couples to the magnetic field by means of insertions. In a slightly larger distance scale, the knowledge of hadron interactions and, consequently nuclear interactions, has been the subject of a wide and extensive investigation, under the assumptions of meson exchange dominated processes from which nucleon and nuclear potentials can be parameterized [148, 149]. In meson exchange potentials, the meson propagator also couples to the magnetic field, which can easily lead to non-trivial contributions for nucleon and nuclear interactions. In this section, some results obtained in Refs. [50, 150] are reviewed and highlighted: the leading weak magnetic field corrections for the one-loop axial and pseudoscalar pion-constituent form factors are discussed for space-like momenta. With the magnetic field contribution
<figure><img src="image_5.png" /><figcaption>**Figure 3.1:** (a) - (e): One loop meson-constituent quark form factor with leading magnetic field insertions in the quark propagator. (f) - (h): Meson exchange process, with the leading one or two weak magnetic field insertions in the meson propagator and the leading insertion in the pseudoscalar form factor (h). Dashed lines stand for meson, solid (wiggly with a big dot) lines for quarks (gluons) and small wavy lines, with a small dot, for magnetic field insertions.</figcaption></figure>

for the pseudoscalar form factor and the corresponding leading contribution for a (pseudoscalar) meson field propagator, the Fourier transform of the meson exchange process is performed for off-shell pions and on-shell constituent quarks.

## 3.1. Meson and constituent quark form factors in constant magnetic fields

The one-gluon exchange quark-antiquark interaction is a leading term of the QCD quark effective action. It is convenient to describe the structure of an effective gluon propagator in terms of longitudinal and transverse components: $\tilde{R}_{ab}^{\mu\nu}(k) \equiv \delta_{ab} \left[ \left(g^{\mu\nu} - \frac{k^\mu k^\nu}{k^2}\right) R_T(k) + \frac{k^\mu k^\nu}{k^2} R_L(k) \right]$. The one loop background field method (BFM) can be applied such that the quantum quark currents can be arranged into multiplets with the same quantum numbers as the quark-antiquark mesons. The auxiliary field method (AFM) makes it possible to introduce multiplets of quark-antiquark local meson fields, although non-local meson fields can also be defined. By integrating out the quantum quark field, a quark determinant is obtained with background quark currents and local meson fields arranged in flavor multiplets. Given that the dressed gluon propagator and corresponding running quark-gluon coupling constant has enough strength, Dynamical Chiral Symmetry Breaking (DChSB) takes place with the formation of a scalar quark-antiquark chiral condensate in the ground state. This leads to a large correction of the Lagrangian quark mass leading to an effective quark mass that can be considered at the basis of the CQM.

A large quark mass expansion leads to one-loop equations for meson-constituent quark current couplings that are well established in phenomenological investigations. In the presence of the magnetic field, leading effects in the quark propagator can be computed. Hereby, effects in the gluon propagator and in the quark-gluon running coupling constant will not be discussed. Magnetic field effects in the propagator require numerical calculations, which make it more difficult to be analyzed. However, by resorting to the weak magnetic field limit, weak with respect to the constituent quark mass, $M \sim 0.30 - 40$ GeV, analytical or semi-analytical results can be obtained.

The computation of weak magnetic field effects in the one-loop quark determinant, including in the resulting form factors, can be done by considering at least two methods: The first one, more convenient for weak constant background magnetic fields, is the expansion based in the Fock-Schwinger gauge [151]. A more commonly used approach is to consider the weak magnetic field limit of the quark (or meson) propagator, usually calculated in the proper time Schwinger representation [152]. These contributions can be translated into the corresponding diagrams, with the leading ones containing one or two magnetic field insertions, as shown in diagrams (a)-(e) of Fig. (3.1). Notice however, that the meson coupling to quark in these diagrams happens through the flavor-Dirac current with the same quantum numbers as the meson. Accordingly, the wiggly line with a big dot is not a full (dressed) gluon propagator but instead only a component of it that can be written in terms of $R_L(k)$ and $R_T(k)$. Each magnetic field insertion is of order $(eB/M^2)$,
with M being the constituent quark mass.

The dynamically generated terms for the leading magnetic field corrections for the dimensionless axial and pseudoscalar form factors, with canonical normalization of the isotriplet pion local field $P_i$, can be written in momentum space as

$$ \mathcal{L}_{\pi}^{B} = c_i \left(\frac{eB_0}{M^2}\right)^2 [F_{ps}^{B}(Q, K) + F_{ps}^{B,a}(Q, K)] P_i(Q) (j_{ps}^{i}(K, Q))^{\dagger} + d \epsilon^{0\rho\mu 3} \left(\frac{eB_0}{M^2}\right) \frac{F_A^{B}(Q, K)}{F} \partial_{\rho} P_i(Q) (j_i^{A,\mu}(K, Q))^{\dagger}, (3.1) $$
where $F_{ps}^{B}(Q, K)$ ($F_{ps}^{B,a}(Q, K)$) is the isotropic (anisotropic) contribution for the pseudoscalar form factor and $F_A^{B}(Q, K)$ the one for the axial form factor. $Q$ and $K$ are the pion and incoming constituent quark momenta, respectively. $F = 92$ MeV is the pion normalization constant, namely, the pion decay constant, and the constituent quark currents are $j_{ps}^{i} = \bar{\psi}i\gamma_5\lambda_i\psi$ and $j_{A,\mu}^{i} = \bar{\psi}\gamma_\mu\gamma_5\lambda_i\psi$ with $i = 1,2$ for charged and $i = 3$ neutral pions. The following numerical coefficients were defined: $c_1 = c_2 = -4/9$ and $c_3 = 5/9$, respectively, for the charged and neutral pion pseudoscalar couplings, and unique $d = 1/3$ for the axial coupling. $\epsilon^{0\rho\mu 3}$ stands for some of the components of the Levi-Civita tensor that picks up the transverse pion and axial current components. The leading contributions for the pseudoscalar form factor is that of one magnetic field insertion for each of the internal quark lines, which is of order $(eB)^2$, in diagram (e). The two magnetic field insertions in a single quark propagator, diagrams (c) and (d), and one single magnetic field insertion in only one quark propagator, diagrams (a) and (b), are either vanishing or sub-leading and will not be taken into account. The axial form factor goes with $(eB)$ corresponding to diagrams (a) and (b), leading to a linear behavior of the axial coupling with magnetic fields [50] that is seemingly present in the pion-nucleon axial coupling [136]. However, it turns out that diagram (a) provides a vanishing contribution and diagram (b) is responsible for the above result.

The leading magnetic field corrections to the corresponding form factors can be written as

$$ \begin{aligned} [F_{ps}^{B}(Q, K); F_{ps}^{B,a}(Q, K)] &= iC_{PS}^{B} M^{*4} \int \frac{d^4 k}{(2\pi)^4} \frac{[\bar{-}k \cdot (k+Q) + M^2]; [\bar{-}k_{\perp} \cdot (k_{\perp} + Q_{\perp})]}{[k^2 - M^2]^2 [(k+Q)^2 - M^2]^2} \frac{R_L(-k-K)}{K_g} \\ F_A^{B}(Q, K) &= iC_{PS}^{B} F M^3 \frac{g_{\mu\nu}}{10} \int \frac{d^4 k}{(2\pi)^4} \frac{1}{[k^2 - M^2][(k+Q)^2 - M^2]^2} \frac{\bar{R}^{\mu\nu}(-k-K)}{K_g}, \end{aligned} \quad (3.2) $$

where for the pseudoscalar coupling $R(-k) = 3R_T(-k) + R_L(-k)$, and for the axial coupling $\bar{R}^{\mu\nu}(-k) = g^{\mu\nu}(R_T(-k) + R_L(-k)) + 2\frac{k^{\mu}k^{\nu}}{k^2}(R_T(-k) - R_L)$, where $R(-k)$ is the effective gluon propagator (EGP), with a normalization $K_g$ discussed below, and the constant $C_{PS}^{B} = 8N_c\alpha K_g$ was defined. Hereby, an effective confining gluon propagator inspired in Ref. [153] is considered. This leads to Dynamical Chiral Symmetry Breaking, given by $R_L(k) = R(k) = \frac{K_g}{(k^2 - M_G^2+i\epsilon)^2}$, where $K_g, M_G$ are a dimensionful normalization constant, that incorporates the running quark gluon coupling constant, and an effective (constant) mass, respectively. The normalization of the effective gluon propagator was fixed by the pseudoscalar pion coupling constant in the vacuum, such that it reproduces the phenomenological value $G_{ps} \simeq 13$ [154] for a reasonably large gluon effective mass $M_G = 0.5$ GeV [150]. With this EGP, it is possible to carry out overall analytical calculations, yielding form factors free of UV divergences, in spite of the need of a renormalization, that settles the scale of $K_g$. Besides that, it provides numerical results for mesons-constituent quark form factors that are similar to results obtained using other effective gluon propagator extracted from SDE. These diagrams are in the class of triangle diagrams analyzed in [155] for which the Schwinger phase does not contribute. In Fig. (3.2) the ratios of the magnetic field correction to the pseudoscalar and axial form factors, with respect to their values in the vacuum, calculated by the same method [156], are presented in the low momentum space-like region for different values of $Q_{perp}^2$ that manifests in the anisotropic part of $G_{ps}^B$ where in the figure caption $\beta = (eB)/M^2$. The magnetic field correction $G_{ps}^B(Q^2) \equiv \beta^2 F_{ps}^B(Q^2)/F_{ps}(Q^2)$ (multiplied by a factor 10) is of order $(eB)^2$ and smaller than $G_A^B(Q^2) \equiv \beta F_A^B(Q^2)/F_A(Q^2)$. The anisotropy of $G_{ps}^B$ for different values of $Q_{perp}^2$ was not exploited before, and it is small.
<figure><img src="image_6.png" /><figcaption>Figure 3.2: Ratio of leading magnetic field corrections to the pseudoscalar and axial form factors $10 \times \beta^2 F_{ps}^B(Q^2)/F_{ps}(Q^2)$ and $\beta F_A^B(Q^2)/F_A(Q^2)$ in the pion space-like region with $K^2 = M_q^2$.</figcaption></figure>

## 3.2. One-pion exchange potential

The leading magnetic field effects will be considered for the Feynman diagrams of Figs. (3.1) (f), (g) and (h). The magnetic field contributions to the pion-constituent quark (nucleon) coupling is accounted for by means of its pseudoscalar form factor, shown above, with the pion propagator obtained in Ref. [157]. In the weak field limit, the leading terms of the meson exchange amplitude in momentum space can be written as

$$ \begin{aligned} \tilde{V}(Q, Q_z) & \approx [g_{ps} + \beta^2 F_{ps}^B(Q, Q_\perp)] [D_0(Q) + \beta^2 D_1^B(Q, Q_\perp)] [g_{ps} + \beta^2 F_{ps}^B(Q, Q_\perp)] \\ &= V_0(Q^2) + V_\pi^B(Q, Q_z) + V_{FF}^B(Q^2, Q_z) + \dots \end{aligned} \quad (3.3) $$

where it has been assumed $K^2 = M^2$, $D_0(Q^2)$ is the pion propagator in vacuum and $D_1^B(Q, Q_\perp)$ the leading magnetic field correction that contains an anisotropy expressed as a dependence on $Q_\perp$. The first term, $V_0(Q^2)$, yields the (isotropic) Yukawa potential, and the two types of corrections considered here are due to the pion propagator ($V_\pi^B(Q, Q_z)$) and to the form factor ($V_{FF}^B(Q, Q_z)$), respectively, which contain anisotropic terms expressed as a dependence on $Q_z$. Due to the linearization above, these two terms add to each other. The Fourier transform of the four-point Feynman diagrams amplitude, with off-shell meson propagation, $Q^2 = -Q^2$, can be performed semi-analytically and is given by

$$ V(R, R_z) = \int \frac{d^3 k}{(2\pi)^3} e^{-iQ \cdot R} [V_0(Q^2) + V_\pi^B(Q, Q_z) + V_{FF}^B(Q, Q_z) + \dots], \quad (3.4) $$

where, due to the Fourier transform, the anisotropic can be written in terms of the distance between two sources and the longitudinal distance between sources along the magnetic field direction $R_z$.

The (semi)analytical result of this Fourier transformation was exhibited in [150] and it will not be written here. There are mostly terms with two residual integrations in two Feynman parameters to be performed numerically, although some anisotropic terms need a numerical integration in the pion transversal momentum $Q_\perp$. Overall, the anisotropies in all contributions are very small. In Fig. (3.3) the resulting ratio of the magnetic field corrections to the Yukawa potential with respect to the Yukawa potential is presented for $R_z = 1$ fm for the anisotropic terms of the potential. The anisotropy noted for the pseudoscalar form factor in Fig. (3.2) is washed out in the calculation of the potential such that, by fixing
<figure><img src="image_7.png" /><figcaption>**Figure 3.3:** Ratio of magnetic field corrections to the Yukawa potential as a function of distance $R$, for the neutral and charged pions with corresponding magnetic field corrections to their masses and three different constituent quark masses as described in the text. (eB) = 0.1 $M^2 \simeq 0.01$ GeV$^2$ and $R_z = 1$ fm.</figcaption></figure>

$R_z \neq 1$ fm, the difference with respect to the results with another value is hardly noted in this figure. It can be seen that results for neutral (pne) and charged (pch) pions are non-degenerate, and there are two different reasons for that: First, the magnetic field contributions for the neutral (charged) pions in the range analyzed (eB) ~ 0.01 GeV$^2$, are of the order of -20 MeV (+20 MeV) [158, 159]. Second, Eq. (3.1) showed that the neutral and charged pion form factors have different coefficients $c_i$. Three different cases for the constituent quark masses were considered. First, a typical constituent quark mass with a correction due to the magnetic field $M = (M_q + 0.02)GeV = (0.35 + 0.02) GeV$ [160]. Second, a larger mass, as if the constituent quark carried all the nucleon mass, of order $M \sim 3M_q$. The overall contributions are amplified for large quark masses due to the structure of the magnetic field contributions to the form factor. Finally, a mixed case in which the normalization of the gluon propagator was considered to reproduce $G_{ps} = 13$ with $M = M_q$, as described above. However, in the magnetic field correction to the Yukawa potential, the quark mass was considered to be three times larger so that the role of the form factors (the part that depends on the constituent quark mass) was reduced, in agreement with expectations that an infinite quark mass leads to point-like interaction. The leading magnetic field contributions for the case of neutral pion exchange leads to an increase of the strength of the potential with distance and consequently a larger range of the interaction. For the case of the charged pion, results are more ambiguous since they strongly depend on the (considered) value of the constituent quark mass. This is mostly due to a possible cancellation of different contributions from different effects, since the charged pion form factor receives a contribution with opposite sign from the one of the neutral pion.

### 3.3. Discussion

The limit of constant magnetic fields, weak with respect to a hadronic mass scale such as the pion mass ($m_π$), or the constituent quark mass ($M$), was shown to make possible to develop analytical or semi-analytical calculations for intricate hadron processes from the point of view of the fundamental constituents, quarks and gluons. The strength of such magnetic fields is nearly in the range expected to appear in relativistic heavy-ion collisions and magnetars. For that, a leading term
of the QCD (quark) effective action, that is the quark-antiquark interaction mediated by one (non-perturbative) gluon exchange, is shown to produce, dynamically, meson-constituent quark interactions described by one loop form factors. The form factor was calculated and used in the one-loop level, being akin to the rainbow ladder approximation of the SDE. The pion pseudoscalar and axial form factors receive corrections with (leading) strengths that are of order $(eB/M^2)^2$ and $(eB/M^2)$, respectively. Whereas the pseudoscalar form factor presents some sizeable anisotropy, manifested in the dependence on the perpendicular component of the pion momentum, $Q_\perp$, the leading anisotropy of the axial form factor is mostly present in the shape of the interaction. Each of these pion couplings occur in color-singlet quark currents that, nevertheless, were shown to be extracted from a color-quark current interaction. The pseudoscalar form factor was employed in the calculation of magnetic field corrections to the one-pion exchange potential which requires an off-shell pion. The limit of weak magnetic field allowed to linearize the potential in two different contributions: one from the magnetic field dependent meson propagator and one from the form factor, since both are of order of magnitude $(eB/M^2)^2$. If a deuteron is formed in a region with such weak magnetic field, when leaving this region it would go into an excited state or even disintegrate. It should be possible to detect both processes by means of either the excitation emission spectrum or the outgoing neutron and proton. The CQM offers a framework for the prediction of observable hadron dynamics that must be tested and whose precise role must be understood in the more complete picture of baryon interactions. The use of the meson form factor in the meson exchange process also, hopefully, will make possible to probe nucleon and meson quark and gluon structures as participants of the nucleon and nuclear interactions. A more complete analysis of all the ingredients of these calculations is to be done with more complete account of the dependence on the magnetic field, for example for gluon propagator and running coupling constant. It is therefore possible to use magnetic fields (even relatively weak ones) as probes of the quark-gluon structure of hadrons and of the nucleon (and consequently nuclear) interaction.

# 4. Magnetic fields in heavy-ion collisions: a catalyst of photo-production via gluon fusion

## 4.1. Introduction

Most of the photons produced in heavy-ion collisions are *decay* photons, a by-product of the fireball expansion, cooling and eventual hadronization. The rest of the photons produced in heavy-ion collisions are the so called *direct* photons: *prompt* photons, just as the ones produced in proton-proton collisions, so we expect they scale as the number of binary collisions and *non-prompt* photons which can come from pre-equilibrium, jet-medium interactions or they can be purely thermal from the QGP phase or the hadron gas phase. We have learned from recent experiments that direct photons produced in heavy-ion collisions, have a large elliptic flow coefficient $v_2$, in some cases as large as that of hadrons [161, 162, 163].

The evidence from data could be interpreted as direct photons being produced in the later stages of the collision, since it is then when flow is already leaving an imprint in hadrons. But, looking closely at the photon yields, we can see that there is a large thermal component that dominates over the prompt one, for low values of $p_T$, the transverse momentum. Now the $p_T$ dependence of $v_2$ seems to confirm that most of the direct photons are really *early* photons, since $v_2$ for large $p_T$ is consistent with zero [164]. These observations initiated a discussion in the community a few years ago, that was named as the *direct photon puzzle*.

On top of this puzzle, the excess of low $p_T$ photons with respect to known sources, as reported by the PHENIX and the
<figure><img src="image_8.png" /><figcaption>**Figure 4.1:** Distinct epochs in a heavy-ion collision and its evolution [170].</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 4.2:** New photo-production channel available in pre-equilibrium stage: gluon fusion into a photon, mediated by a quark loop in the presence of an external magnetic field *B*.</figcaption></figure>

STAR Collaborations, has eluded explanations and models that work well for other electromagnetic probes. In summary, these recent data analysis show that the yield of these low $p_T$ photons scales with a given power of the number of binary collisions, both in large and small colliding systems, which suggests that the source of these photons is similar for different colliding species and beam energies. The data also shows tension between the yields measured by PHENIX and STAR, so further measurements of photon-production in the low energy regime will complement these and will push the discussion towards elucidating the source of these soft photons or making the effect disappear altogether [165, 166, 167, 168, 169].

## 4.2. New photo-production channel available in pre-equilibrium stage

Figure 4.1 shows a diagram of the system created in heavy-ion collisions and its evolution in time. In the pre-equilibrium stage, there are large number of gluons (glasma) available, unlike quarks which are suppressed due to the parametric proportionality in $\alpha_s$.

In Refs. [171, 172, 173], we have reported how fusion and splitting processes involving quarks and gluons at this stage, can give rise to an excess (over conventional sources) of photons. The posited source of this excess of photons is a new production channel via gluon fusion which is possible by the presence of an intense external magnetic field *B*. Figure 4.2 shows the Feynman diagram for this process where all quark propagators include the effect of the external magnetic field.

The magnetic field acts as a catalyst of the newly open channel that would otherwise not be available, since in the $B \to 0$ limit we recover Furry's theorem and QED and QCD are back to having charge conjugation symmetry. Altogether the idea we put forward is to focus on processes that are catalysed by these external magnetic fields in a heavy-ion collision
<figure><img src="image_10.png" /><figcaption>**Figure 4.3:** Evolution in time of the average dominant component of the magnetic field pulse, that would be created in heavy-ion collisions [174].</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 4.4:** Evolution in time of the average dominant component of the magnetic field pulse, that would be created under diverse heavy-ion collision conditions (colliding energy, impact parameter or centrality), as reported in Refs. [175, 176].</figcaption></figure>

epoch where there is a large abundance of soft pre-equilibrium gluons and test weather this allows an enhancement of the photon yield and a large $v_2$.

An immediate consequence of the presence of an external magnetic field in this pre-equilibrium stage is the breaking of Lorentz invariance, so calculation of observables in space-time have to be done with tensorial structures that are now differentiated: parallel and perpendicular with respect the external magnetic field.

Now, there is an initial limitation to this approach: the presence of an intense external magnetic field means that $qB$ is the dominant scale, in particular it is larger than the loop momenta, so the analysis of results should be constrained to the low photon momenta regime.
### 4.2.1. Estimation and simulation of magnetic fields created in heavy-ion collisions

The classical approach to estimate the magnetic field $B(\mathbf{r}_i, t)$ produced by non-accelerated charges moving along the beam direction in a heavy-ion collision event, can be done using the Liénard-Wiechert potential as [177]

$$e\mathbf{B}(\mathbf{r}_i, t) = \alpha_{em} \sum_j \frac{(1 - v_j^2) \mathbf{v}_j \times \mathbf{R}_j}{R_j^3 \left[ 1 - \frac{(\mathbf{v}_i \times \mathbf{R}_j)^2}{R_j^2} \right]^{3/2}},$$

where $\mathbf{R}_j = \mathbf{r} - \mathbf{r}_j(t)$, $\mathbf{x}_j(t)$ is the position of the j-th charge moving with velocity $\mathbf{v}_j$ and the sum runs over charged particles in each event. Figure 4.3 shows the magnetic field pulse using the Liénard-Wiechert approach for Bi+Bi collisions at both $\sqrt{s_{NN}} = 9$ GeV and $\sqrt{s_{NN}} = 200$ GeV with an impact parameter $b = 7$ fm. We can appreciate how the intensity of the pulse is of the order of $3m_{\pi}^2$ at RHIC energies and about $m_{\pi}^2/10$ for smaller collision energies. This is contrast with the duration of the pulse which is approximately 3 fm in the low energy collision and approximately 1/10 fm for the RHIC energy collision. Figure 4.4 shows similar results reported in some of the recent reviews devoted to the topic of understanding the emergence and effects on observables of electromagnetic fields produced in heavy-ion collisions [175, 176, 178].

Clearly the interplay of the height and width of the magnetic field is key to have in mind when implementing phenomenological test to look for such effects. Using simulations, we have been able to include the magnetic field effects into relevant observables for photo-production through gluon fusion so that effects of participant or spectator nucleons from the centrality of the collision, can also be taken into account [172, 171, 179].

### 4.2.2. Magnetic-field induced gluon-fusion to produce photons in the early stages of the collision

The lowest order gluon fusion process comes from an amplitude made out of a quark triangle diagram with two gluons and one photon attached each at one of the vertices of the triangle, as shown in Figure 4.2. The over occupied gluon state can be modeled as a quasi-particles system and perturbative methods are applicable [180]. In this context, the coordinate representation of the quark Schwinger propagator is

$$S(x, x') = \Phi(x, x') \int \frac{d^4 p}{(2\pi)^4} e^{-p \cdot (x-x')} S(p), \quad (4.1)$$

where $S(p)$ is the Fourier transform to momentum space of the translational invariant part of the propagator, given by

$$iS(p) = \int_0^\infty \frac{d\tau}{\cos(|eq_f B|\tau)} e^{i\tau \left[ p_\parallel^2 - p_\perp^2 \frac{\tan\left(\frac{|eq_f B|\tau}{|eq_f B|\tau}\right)}{|eq_f B|\tau} - m_f^2 + i\epsilon \right]} \quad (4.2) \\ \times \left\{ \left[ \cos(|eq_f B|\tau) + \gamma_1\gamma_2 \sin(|eq_f B|\tau) \right] (m_f + \not p_\parallel) - \frac{\not p_\perp}{\cos(|eq_f B|\tau)} \right\},$$

and $\Phi(x, x')$ is the phase factor with exclusive coordinate dependence

$$\Phi(x, x') = \exp \left\{ i \epsilon q_f \int_{x'}^x dv^\mu \left[ A_\mu + \frac{1}{2} F_{\mu\nu} (v - x')^\nu \right] \right\}, \quad (4.3)$$

with $eq_f$ the quark charge in units of $e$, the absolute value of the electron charge. We assume a constant magnetic field that points in the z-direction obtained from a vector potential $A^\mu$ in the symmetric gauge $A^\mu = \frac{B}{2}(0, -y, x, 0)$. Note that as we announced previously, Lorentz invariance breaks and so from a four-momentum $p^\mu$, we define $p_\perp^\mu = (0, p_1, p_2, 0)$, $p_\parallel = (p_0, 0, 0, p_3)$, $p_\perp^2 = p_1^2 + p_2^2$, $p_\parallel^2 = p_0^2 - p_3^2$, and therefore $p^2 = p_\parallel^2 - p_\perp^2$. Then we can write the expression for the
<figure><img src="image_12.png" /><figcaption>**Figure 4.5:** Mean magnetic field strength produced by (a) spectators and (b) spectators and participants at the middle of the interaction region as a function of time for three centrality classes 0-20%, 20-40% and 40-60% in Au+Au collisions and one centrality class 0-40% in Cu+Cu collisions at $\sqrt{s_{NN}}$ = 200 GeV, as reported in Refs. [171, 172].</figcaption></figure>

amplitude for the gluon fusion channel $\tilde{\mathcal{M}}_{gg\to\gamma}$ as

$$ \begin{aligned} \tilde{\mathcal{M}}_{gg\to\gamma} = & -\int d^4x d^4y d^4z \int \frac{d^4r}{(2\pi)^4} \frac{d^4s}{(2\pi)^4} \frac{d^4t}{(2\pi)^4} e^{-it\cdot(y-x)} e^{-is\cdot(x-z)} e^{-ir\cdot(z-y)} e^{-ip\cdot z} e^{-ik\cdot y} e^{iq\cdot x} \\ & \times \left\{ \mathrm{Tr} [ieq_f\gamma_\alpha iS_{ac}(s)ig\gamma_\mu t^c iS_{cd}(r)ig\gamma_\nu t^d iS_{da}(t)] + \mathrm{Tr} [ieq_f\gamma_\alpha iS_{ad}(t)ig\gamma_\nu t^d iS_{dc}(r)ig\gamma_\mu t^c iS_{ca}(s)] \right\} \\ & \times \Phi(x,y)\Phi(y,z)\Phi(z,x)\epsilon^\mu(\lambda_p)\epsilon^\nu(\lambda_k)\epsilon^\alpha(\lambda_q), \end{aligned} \quad (4.4) $$

where $p$ and $k$ are the gluon and $q$ the photon four-momenta, $t^c, t^d$ are Gell-Mann matrices, and the polarization vectors for the gluons and the photon are $\epsilon^\mu(\lambda_p)$, $\epsilon^\nu(\lambda_k)$, $\epsilon^\alpha(\lambda_q)$, respectively. The Lorentz indices $\mu, \nu, \alpha$ and the space-time coordinates $x, y, z$ associated to each vertex, are also depicted in Fig. 4.2. The product of phase factors is $\Phi(x, y)\Phi(y, z)\Phi(z, x) = \exp\left\{\frac{i|eq_fB|}{2}\epsilon_{mj}(z-x)_m(x-y)_j\right\}$, where the indices $m, j = 1, 2$ and $\epsilon_{mj}$ is the Levi-Civita tensor.

If we pursue the idea that at the earliest times, when thermalization has not been achieved, the magnetic field is the dominant internal energy scale in the process. This means that in the absence of thermal effects, and with $|eq_fB| \gg m_f^2$, we can work using the quark propagators in the lowest Landau level. Notice that the first non-vanishing term is where one of the quarks in the loop is in the first excited Landau level [171, 172, 179]. Once we have the matrix elements for both gluon fusion and splitting we can build the invariant photon momentum distribution as follows

$$ \omega_q \frac{dN^{\text{mag}}}{d^3q} = \frac{V\Delta\tau}{2(2\pi)^3} \int \frac{d^3p}{(2\pi)^3 2\omega_p} \int \frac{d^3k}{(2\pi)^3 2\omega_k} (2\pi)^4 \left\{ \delta^{(4)}(q-k-p) n(\omega_p) n(\omega_k) \overline{\sum_{c,p,f}} |\mathcal{M}_{gg\to\gamma}|^2 \right. \\ \left. + \delta^{(4)}(q+k-p) n(\omega_p) [1+n(\omega_k)] \overline{\sum_{c,p,f}} |\mathcal{M}_{g\to g\gamma}|^2 \right\} \quad (4.5) $$

where $V\Delta\tau$ is the space-time volume where the reaction takes place, spatial volume of the nuclear overlap region $V(t)$ at time $t$ and the time interval $\Delta\tau$ is where the magnetic field is taken as having a constant intensity $B(t)$ as shown in Fig. 4.5. As we commented back in Sec. 4.2.1, the overall lifetime $\Delta T$ can be estimated calculating the duration of the magnetic pulse using simulations. Once we calculate the invariant momentum distribution we can extract the second harmonic coefficient $v_2$ taking into account the contribution from direct photons reported in the literature [181] as a
<figure><img src="image_13.png" /><figcaption>Figure 4.6: Second harmonic coefficient and invariant photon spectra for Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV with 20-40% centrality events compared with PHENIX data, as reported in Ref. [172].</figcaption></figure>

weighted average

$$v_2(\omega_q) = \frac{\sum_{i=1}^{m} \left[ \frac{dN}{d\omega_q} \right]_i [v_2^{\text{mag}}(\omega_q)]_i + \frac{dN^{\text{direct}}}{d\omega_q}(\omega_q) v_2^{\text{direct}}(\omega_q)}{\sum_{i=1}^{m} \left[ \frac{dN}{d\omega_q} \right]_i + \frac{dN^{\text{direct}}}{d\omega_q}(\omega_q)} \quad (4.6)$$

We can finally compare with data and analyze the effectiveness of our model in explaining both the excess of low $p_T$ photon yield and large $v_2$. Figure 4.6 shows the second harmonic coefficient and invariant photon spectra for Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV with 20-40% centrality events compared with PHENIX data, as reported in Ref. [172]. Our work indicates that the magnetic field contribution improves the agreement with experimental data for the low part of the spectrum helping to describe the rise of $v_2$ as the photon energy decreases, but requires improvements as a systematic effect for the behavior of flow at larger $p_T$. At this point, the limitation is the kinematic hierarchy imposed by us, when we claim that the magnetic field drives the dynamics and dominates the loop momenta in gluon fusion. So we embarked on an approach where we can improve our calculation and extend its range of applicability.

### 4.2.3. Recent improvements

In order to go beyond the access only to low $p_T$ photons and to push back on implementing the hierarchy between the magnetic field and the loop momenta as $2|eq_f B| \gg t_\parallel^2, s_\parallel^2, r_\parallel^2$, we pursue a calculation where now we do a fully strong B-field approximation, as reported in Ref. [179]. The physical picture that emerges is the following: under a strong magnetic field two of the vector particles occupy parallel polarization states [182, 183]. So, when the vertex involves a third vector particle, invariance under charge conjugation and conservation of angular momentum require that its polarization is transverse. In a practical sense, using the Schwinger propagators this time, one of the quarks that make up the loop needs to be placed not in the lowest Landau level but instead in the first Landau level. There is plenty of work in the literature where the tensor and analytic structure of magnetized vertices is well understood, so that properties and identities need to be satisfied under controlled conditions of the external strong magnetic field [184, 182, 185, 186]. In the approximation where the magnetic field is the largest of the kinematical energy (squared) variables, the tensor structure
can be expressed as $\Gamma_{ab}^{\mu\nu\alpha} = \delta_{ab}\Gamma^{\mu\nu\alpha}$, where

$$ \begin{aligned} \Gamma^{\mu\nu\alpha} &= \sum_{n=1}^{3} \Gamma_n(\omega_q, \omega_k, q^2) \Gamma_n^{\mu\nu\alpha} \\ &= \Gamma_1(\omega_q, \omega_k, q^2) \frac{\epsilon_{ij} q_{\perp}^i g_{\perp}^{j\mu}}{\sqrt{q_{\perp}^2}} \left( g_{\parallel}^{\nu\alpha} - \frac{q_{\parallel}^{\nu} q_{\parallel}^{\alpha}}{q_{\parallel}^2} \right) + \Gamma_2(\omega_q, \omega_k, q^2) \frac{\epsilon_{ij} q_{\perp}^i g_{\perp}^{j\nu}}{\sqrt{q_{\perp}^2}} \left( g_{\parallel}^{\mu\alpha} - \frac{q_{\parallel}^{\mu} q_{\parallel}^{\alpha}}{q_{\parallel}^2} \right) \\ &\quad + \Gamma_3(\omega_q, \omega_k, q^2) \frac{\epsilon_{ij} q_{\perp}^i g_{\perp}^{j\nu}}{\sqrt{q_{\perp}^2}} \left( g_{\parallel}^{\mu\nu} - \frac{q_{\parallel}^{\mu} q_{\parallel}^{\nu}}{q_{\parallel}^2} \right) \end{aligned} \quad (4.7) $$

where $\Gamma_n(\omega_q, \omega_k, q^2)$, $n = 1, 2, 3$, are scalar coefficients and $\epsilon_{ij}$ is the Levy-Civita symbol in the transverse components: $\epsilon_{12} = -\epsilon_{21} = 1$. The contributing Feynman diagrams, obtained when placing two fermion propagators in the LLL and the other in the 1LL, after integration of the configuration space variables, the vertex is given by

$$ \Gamma_{ab}^{\mu\nu\alpha} = -\delta^{(4)}(q-k-p)\mathrm{Tr}[t_a t_b] \frac{8\pi^4 q_f g^2}{|eq_f B|} q_{\parallel}^2 e^{f(p_{\perp}, k_{\perp})} \sum_{i=1}^{3} D_i^{\mu\nu\alpha}. \quad (4.8) $$

We calculate $\Gamma_n(\omega_q, \omega_k, q^2)$ for $n = 1, 2, 3$ by projecting onto the basis

$$ \{v_{\perp}^{\mu} \Pi_{\parallel}^{\nu\alpha}, v_{\perp}^{\nu} \Pi_{\parallel}^{\mu\alpha}, v_{\perp}^{\alpha} \Pi_{\parallel}^{\mu\nu}\}. \quad (4.9) $$

Using these results, we can go back to the vertex and collect the contributions that correspond to each coefficient $\Gamma_n(\omega_q, \omega_k, q^2)$. The projection with $v_{\perp}^{\mu} \Pi_{\parallel}^{\nu\alpha}$ contributes to

$$ \Gamma_n \equiv \frac{8\pi^4 q_f g^2}{|eq_f B|} e^{f(p_{\perp}, k_{\perp})} |q_{\perp}| \tilde{\Gamma}_n(\omega_p, \omega_q, \theta). \quad (4.10) $$

As we reported in Ref. [179], we perform the calculation of $\sum |\tilde{\Gamma}_n|^2$ and with a thorough numerical analysis we find that it is larger for a photon propagation within the reaction plane ($\theta = \pi/2$) and that for $\theta = \pi/2$ it is also larger for smaller field intensities. We also note that $|q_{\perp}|e^{f(p_{\perp}, k_{\perp})}$ also depends on $\theta$ and peaks at small angles for large gluon energies, but here the square amplitude is highly suppressed close to the reaction plane. So for small gluon energies and/or a large field strength, the pre-factor is dominated by emission angles close to the reaction plane and vanishes for $\theta = 0$, which prevents photons from being emitted along the direction of the magnetic field. Since at pre-equilibrium the largest gluon abundance happens for small energies, a positive $v_2$ coefficient may be expected. The final steps to verify this result are work in progress.

### 4.3. Concluding remarks

Here we have reviewed the photon puzzle in the context of a new approach to understanding photon production at the earliest stages of a heavy-ion collision, where intense magnetic fields could be present. We summarize the results in the literature where we have computed the contribution to the photon yield and $v_2$ from gluon fusion by a magnetic field during the early stages of a relativistic heavy-ion collision, where there is a large gluon occupation number below the saturation scale. We note an agreement of the calculation with data in the lowest part of the spectrum: observed experimental fall between 0.5 and 1 GeV. Above 1 GeV the calculation overshoots the data spectrum and $v_2$ peaks for energy values $\sim \sqrt{eB}$. We further analyze how we have improved this approach by constructing a one-loop two-gluon and one-photon vertex where we fully include the effects of the magnetic field in the tensor structure: we construct the vertex as $\Pi_{\parallel}^{\mu\nu} \times \tilde{\epsilon}_{\perp}^{\alpha\beta}$ which is required to have components in the transverse plane with respect to the B-field and we
place two of the loop quarks in the lowest Landau level and the other one in the first Landau level. We are in the process of obtaining and updated result for both the photon yield and the second harmonic coefficient, to further test the range of our model in describing pre-equilibrium photon production in heavy-ion collisions.

# 5. Hot and dense perturbative QCD in a very strong magnetic background

## 5.1. Introduction

One of the systematic approaches to describe QCD matter at high temperatures and densities and extremely large magnetic fields is perturbative QCD (pQCD) in a non-perturbative magnetic background. Thermodynamic quantities of magnetic QCD like the pressure, chiral condensate and strange quark number susceptibility can be computed from first principles within perturbative QCD at finite temperature and very high magnetic fields. This indeed has been accomplished up to two-loop order and for physical quark masses [187, 188, 189], including a full implementation of the effects of the renormalization scale in the running coupling and running strange quark mass. Within this framework, the one-loop QCD correction to the photon-quark-antiquark vertex has also been computed, yielding a significantly altered contribution to the quark anomalous magnetic moment [190].

In this section, we summarize the technical approach and discuss the main findings for observables. In particular, these perturbative magnetic QCD results for the chiral condensate and strange quark number susceptibility can be directly compared to recent lattice QCD data away from the chiral transition. Furthermore, the calculation of the pressure at finite baryon density is discussed, as well as the simplifications that come about in the chiral limit. In fact, the effectively negligible contribution of the exchange diagram allows for building a simple analytic model for the equation of state for pure quark magnetars. The latter, however valid only for very large baryon densities, provides constraints on the behavior of the maximum mass and radius directly from in-medium perturbative QCD.

## 5.2. Thermal pQCD in a strong magnetic background

To compute the thermal pressure and chiral condensate in the lowest Landau level approximation up to two loops in perturbative QCD, we assume that the system is embedded in a uniform, very large magnetic field $B = B\hat{z}$, where the field strength $B$ is much larger than the temperature and all masses.

The one-loop (free) contribution to the pressure from the quark sector is given by the following renormalized expression, where the pure vacuum term has been subtracted [191, 192, 193, 194]:

$$ \frac{P_{\text{free}}^q}{N_c} = \sum_f \frac{(q_f B)^2}{2\pi^2} \left[ \zeta'(-1, x_f) - \zeta'(-1, 0) + \frac{1}{2}(x_f - x_f^2) \ln x_f + \frac{x_f^2}{2} \right] + T \sum_{n,f} \frac{q_f B}{\pi} (1 - \delta_{n,0}/2) \int \frac{dp_z}{2\pi} \left\{ \ln \left(1 + e^{-\beta[E(n,p_z) - \mu_f]}\right) + \ln \left(1 + e^{-\beta[E(n,p_z) + \mu_f]}\right) \right\}, \quad (5.1) $$

where $E^2(n, p_z) = p_z^2 + m_f^2 + 2q_f B n$, $x_f \equiv m_f^2/2q_f B$, $T = 1/\beta$ is the temperature, $\mu$ is the quark chemical potential, $N_c$ is the number of colors, $f$ labels quark flavors, $q_f$ is the quark electric charge, and $n = 0, 1, 2, \dots$ stands for the Landau levels. In this expression, Matsubara sums have already been performed in the medium contribution. One should notice that there is an inherent arbitrariness in the renormalization procedure (see Refs. [9, 10, 195, 196, 197, 198, 199, 200] for
further discussion). Taking the limit of very high magnetic fields ($m_s \ll T \ll \sqrt{eB}$), one ends up with the lowest Landau level (LLL) expression

$$ \frac{P_{\text{free}}^{\text{LLL}}}{N_c} = -\sum_f \frac{(q_f B)^2}{2\pi^2} [x_f \ln \sqrt{x_f}] + T \sum_f \frac{q_f B}{2\pi} \int \frac{dp_z}{2\pi} \left\{ \ln \left(1 + e^{-\beta[E(0,p_z)-\mu_f]}\right) + \ln \left(1 + e^{-\beta[E(0,p_z)+\mu_f]}\right) \right\}. \quad (5.2) $$

Since gluons do not carry electric charge, their one-loop contribution has the usual Stefan-Boltzmann form [201]

$$ P_{\text{free}}^{G} = 2(N_c^2 - 1) \frac{\pi^2 T^4}{90}. \qquad (5.3) $$

Strong interactions may then be introduced perturbatively, with the first corrections appearing for the pressure at two loops. The purely gluonic contribution at this order is given by the well-known formula [201]:

$$ P_2^G = -N_c(N_c^2 - 1) \frac{g^2 T^4}{144}. \qquad (5.4) $$

On the other hand, the exchange diagram for the pressure in the quark sector was first computed in Ref. [187] in the lowest Landau level approximation, displaying a dimensionally reduced structure for the matter part. For numerical purposes, however, it is convenient to recast the result found in that reference in a different fashion, where one first evaluates the momentum integrals and then carries out the Matsubara sums numerically at $\mu = 0$, as detailed in Ref. [188]. The final expression for the exchange pressure has the form

$$ \frac{P_{\text{exch}}^{\text{LLL}}}{N_c} = \frac{1}{2} g^2 \left( \frac{N_c^2 - 1}{2N_c} \right) T^2 \sum_f m_f^2 \left( \frac{q_f B}{2\pi} \right) \sum_{\ell, n_2} \int \frac{dm_k}{2\pi} m_k e^{-\frac{m_k^2}{2q_f B}} \frac{\mathcal{E}_\ell - \mathcal{E}_{n_2}}{\mathcal{E}_\ell \mathcal{E}_{n_1} \mathcal{E}_{n_2} |\mathcal{E}_\ell - \mathcal{E}_{n_2}| (|\mathcal{E}_\ell - \mathcal{E}_{n_2}| + \mathcal{E}_{n_1})}, \quad (5.5) $$

where $\mathcal{E}_\ell = \sqrt{\omega_\ell^2 + m_k^2}$, $\mathcal{E}_{n_1} = \sqrt{(\omega_{n_2} + \omega_\ell)^2 + m_f^2}$, and $\mathcal{E}_{n_2} = \sqrt{\omega_{n_2}^2 + m_f^2}$, with the Matsubara frequencies given by $\omega_\ell = 2\pi\ell T$ and $\omega_{n_2} = (2n_2 + 1)\pi T$. This expression has the advantage of being numerically simple. Its downside, however, is that it only holds for $\mu = 0$ and can not be used for cold and dense QCD, as will be discussed in the sequel.

Besides the pressure, one may also extract other observables from these calculations. The chiral condensate represents a very relevant physical quantity in the investigation of the phase diagram for strong interactions, since it can be considered a pseudo order parameter for the chiral transition in this case. Of course, a perturbative analysis is reliable only for very large temperatures and even larger magnetic fields, so that it can not bring information on the region near the phase transition or crossover. Nevertheless, since there are lattice results for high temperatures and magnetic fields, the comparison of these two first-principle calculations in this regime is certainly relevant. The condensate is obtained from the pressure as a derivative with respect to the quark mass. So, the f-flavor condensate is given by

$$ \langle \bar{\psi}_f \psi_f \rangle = - \frac{\partial P_f}{\partial m_f} = - \frac{\partial P_{\text{free}}^{\text{LLL}}}{\partial m_f} - \frac{\partial P_{\text{exch}}^{\text{LLL}}}{\partial m_f}. \qquad (5.6) $$

On the lattice, one computes the f-flavor renormalized condensate

$$ \Sigma_f^r(B, T) = \frac{m_f}{m_\pi^2 f_\pi^2} \left[ \langle \bar{\psi}_f \psi_f \rangle_{B,T} - \langle \bar{\psi}_f \psi_f \rangle_{0,0} \right], \qquad (5.7) $$

which eliminates additive and multiplicative divergences. Here, $m_\pi = 135$ MeV, $f_\pi = 86$ MeV, and $m_f = 5$ MeV for the light quarks. Within perturbative QCD in a nonperturbative magnetic background, however, one can not simply take the zero-field limit and obtain the vacuum condensate, since very large fields are assumed from the outset [187].

A different observable that can also be computed and directly compared to available lattice data is the strange quark number susceptibility

$$ \chi^s = \frac{1}{T^2} \frac{\partial^2 P}{\partial \mu_s^2}. \qquad (5.8) $$
Given the presence of a derivative with respect to the chemical potential, pure vacuum terms are excluded. This presents an advantage when comparing lattice results to pQCD, even if the temperature range in the simulations is still far from optimal for this purpose [202, 203].

The pressure and chiral condensate to this order depend also on the renormalization scale $\bar{\Lambda}$ via the scale dependence of the strong coupling $\alpha_s(\bar{\Lambda})$ and the strange quark mass $m_s(\bar{\Lambda})$. Computing the pressure and chiral condensate consistently up to order $\alpha_s$, the running coupling to be adopted reads [204]

$$ \alpha_s(\bar{\Lambda}) = \frac{4\pi}{\beta_0 L} \left( 1 - \frac{2\beta_1 \ln L}{\beta_0^2 L} \right), \qquad (5.9) $$

where $\beta_0 = 11 - 2N_f/3$, $\beta_1 = 51 - 19N_f/3$, $L = 2 \ln (\bar{\Lambda}/\Lambda_{\overline{MS}})$ and $N_f$ is the number of quark flavors. Since $\alpha_s$ depends on $N_f$, fixing the mass of the quark at some energy scale also depends on the number of flavors. For the strange quark mass, one has

$$ m_s(\bar{\Lambda}) = \hat{m}_s \left(\frac{\alpha_s}{\pi}\right)^{4/9} \left[1 + 0.895062 \left(\frac{\alpha_s}{\pi}\right)\right], \qquad (5.10) $$

with $\hat{m}_s$ being the renormalization group invariant strange quark mass, i.e. $\bar{\Lambda}$ independent. Since Eq. (5.9) for $\alpha_s$ implies that different values of $N_f$ yield different values of $\Lambda_{\overline{MS}}$, by choosing $\alpha_s(\bar{\Lambda} = 1.5 \text{ GeV}, N_f = 3) = 0.336^{+0.012}_{-0.008}$ [205], one obtains $\Lambda_{\overline{MS}}^{2+1} = 343^{+18}_{-12} \text{ MeV}$. Fixing the strange quark mass at $m_s(2 \text{ GeV}, N_f = 3) = 92.4(1.5) \text{ MeV}$ [206] gives $\hat{m}_s^{2+1} \approx 248.7 \text{ MeV}$ when using $\alpha_s^{2+1}$ in Eq. (5.10).

As usual, there is arbitrariness in the way one should connect the renormalization scale $\bar{\Lambda}$ to a physical mass scale of the system under consideration [201]. In thermal QCD where, besides quark masses, the only scale is given by the temperature, and $T \gg m_f$, the usual choice is the Matsubara frequency $2\pi T$ with a band around it, i.e. $\pi T < \bar{\Lambda} < 4\pi T$. In the present case, where the magnetic field also provides a relevant mass scale given by $\sqrt{eB}$, the choice becomes more ambiguous. Therefore, one can find a few different assumptions for the form of the running coupling in this case, ranging from different choices of the renormalization scale to entirely modified running coupling expressions [27, 207, 208]. The case with the standard pQCD running and $\bar{\Lambda} = \sqrt{(2\pi T)^2 + eB}$ seems to be the most physical, but results for a few relevant choices will also be presented below.

Fig. 5.1 displays the full pressure as a function of the magnetic field for two different values of the temperature, and as a function of the temperature for two different values of the magnetic field. The bands, as usual, correspond to increasing/decreasing the central renormalization running scale by a factor of 2. Notice that the green band in the last panel is divergent for small temperatures. Deviations between the computations within different choices of the renormalization scale are large in the behavior of $\alpha_s$, $m_s$ and the exchange term [188]. Here they seem to be small because the exchange correction becomes negligible for very high magnetic fields, except in the case of the nonstandard $\alpha_s(|eB|)$ that increases with $B$ [27].

In Fig. 5.2 the renormalized light quark chiral condensate is shown as a function of the temperature for $eB = 4 \text{ GeV}^2$ and $eB = 9 \text{ GeV}^2$ computed using pQCD [188]. We also show points obtained via lattice simulations for comparison [203]. In Fig. 5.3 the same is shown for the strange quark number susceptibility. Even though the temperature range for lattice results is still well below the ideal for a fair comparison to pQCD, one can see that perturbative results are in the right ballpark for the upper end of temperatures. In order to verify whether the perturbative calculations capture the qualitative trend at high temperatures, Lattice results for higher temperatures, and even higher magnetic fields, are necessary. Since the strange quark number susceptibility represents a better observable for our comparison, one can see
<figure><img src="image_14.png" /><figcaption>**Figure 5.1:** Full pressure as function of the magnetic field at $T = 0.3$ GeV (top-left) and $T = 0.6$ GeV (top-right), and as function of temperature at $eB = 2$ GeV$^2$ (bottom-left) and $eB = 9$ GeV$^2$ (bottom-right). The bands correspond to changes in the central scale by a factor of two. The running $\alpha_s(|eB|)$ is not the standard pQCD one and has been proposed in Ref. [27]. Figure from Ref. [188].</figcaption></figure>

from the figures that the results display a more promising trend for temperatures above the ones currently simulated. One should recall that this framework is valid only if the hierarchy of scales $m_s \ll T \ll \sqrt{eB}$ is satisfied.

<figure><img src="image_15.png" /><figcaption>**Figure 5.2:** Renormalized light quark chiral condensate as a function of the temperature for $eB = 4$ GeV$^2$ (left) and $eB = 9$ GeV$^2$ (right). Figure from Ref. [188].</figcaption></figure>

## 5.3. Cold and dense pQCD in a strong magnetic background

One can follow an analogous procedure to study the complementary sector of the phase diagram, namely the one of vanishing temperature and high baryon density [189]. Cold and dense magnetic QCD, however, has a major difference
<figure><img src="image_16.png" /><figcaption>Figure 5.3: Strange quark number susceptibility as a function of the temperature for $eB = 4 \text{ GeV}^2$ (left) and $eB = 9 \text{ GeV}^2$ (right). Figure from Ref. [188].</figcaption></figure>

with respect to its counterpart at finite temperature and zero net baryon density: the Sign Problem [209]. Monte Carlo simulations in the parameter regime that is relevant for compact star physics – corresponding to large values of the baryon chemical potential – are hindered. Besides magnetic pQCD, one can approach the equation of state in the limit of a large number of colors $N_c$ [210], via chiral perturbation theory [65], using holographic models [211, 212] and, of course, within effective models. For a detailed discussion and list of references, see Refs. [191, 192, 193, 194]. Effects from color superconductivity [213] in the presence of a strong magnetic field [214, 215, 216, 217, 218], which might be relevant to transport phenomena in the core of neutron stars and magnetars, are not discussed here.

The equation of state for a system composed by up, down and strange quarks at zero temperature and nonzero baryon chemical potential, i.e. cold and dense quark matter, was first obtained within perturbative QCD more than four decades ago by Freedman and McLerran [219, 220], and also by Baluni [221] and Toimela [222]. Since then, it has been systematically improved [223, 224, 204, 225, 226, 227, 228, 229, 230, 231, 232, 233]. One relevant feature of perturbative QCD for cold and dense quark matter is that it seems to be much better behaved, compared to its thermal analog, as far as the convergence of the series is concerned [234]. Nevertheless, if one considers the exchange (two-loop) contribution to the pressure, including the renormalization group running of $\alpha_s$ and $m_s$, it brings corrections $\sim 30\%$ for $\mu_q \sim 600$ MeV [204]. Therefore, the enormous reduction in the size of the exchange term observed in thermal magnetic QCD [187, 188] has remarkable effects on the perturbative series. In fact, for cold magnetic QCD this feature is even more pronounced, making the exchange diagram effectively negligible [189]. Therefore, one can build a simple, and yet nontrivial, (analytic) description for the high-density sector of the equation of state, provided that the magnetic background is strong enough to justify the lowest-Landau level description.

In the case with zero temperature, there is only the contribution coming from the quark sector. The free part contribution is given by the following renormalized expression (subtracting the pure vacuum term) in the lowest Landau level (LLL) [235, 191, 195, 187, 192, 193, 194, 188]:

$$ \frac{P_{\text{free}}^{\text{LLL}}}{N_c} = -\sum_f \frac{(q_f B)^2}{2\pi^2} [x_f \ln \sqrt{x_f}] + \sum_f \frac{(q_f B)}{4\pi^2} \left[ \mu_f P_F - m_f^2 \log \left( \frac{\mu_f + P_F}{m_f} \right) \right], \quad (5.11) $$
where $P_F = \sqrt{\mu_f^2 - m_f^2}$ is the Fermi momentum. The exchange contribution reads [189]

$$ \frac{P_{\text{exch}}^{\text{LLL}}}{N_c} = -\frac{1}{2}g^2 \left( \frac{N_c^2 - 1}{2N_c} \right) m_f^2 \left( \frac{q_f B}{2\pi} \right) \int \frac{dm_k}{2\pi} m_k e^{-\frac{m_k^2}{2q_f B}} \int \frac{dp_z dq_z dk_z}{(2\pi)^3} (2\pi) \delta(k_z - p_z + q_z) \\ \times \frac{1}{\omega E_p E_q} \left\{ \frac{\omega}{E_-^2 - \omega^2} \Theta(\mu_f - E_p) \Theta(\mu_f - E_q) - \left[ \frac{2(E_q + \omega)}{(E_- - \omega)(E_+ + \omega)} \right] \Theta(\mu_f - E_p) - \frac{1}{E_+ + \omega} \right\}. \quad (5.12) $$

where

$$ E_{\pm} \equiv E_{\mathbf{p}} \pm E_{\mathbf{q}} = \sqrt{\mathbf{p}^2 + m_f^2} \pm \sqrt{\mathbf{q}^2 + m_f^2}, \quad (5.13) $$

and $\omega = \sqrt{k_z^2 + m_k^2}$, where $m_k^2$ accounts for the gluon momentum components transverse to the magnetic field [187, 235].

As before, there is arbitrariness in the way one should connect the renormalization scale $\bar{\Lambda}$ to a physical mass scale of the system under consideration [201]. In cold and dense QCD, besides quark masses, the only available scale is given by the quark chemical potential $\mu_f \gg m_f$. The usual choice, then, is $2\mu_f$ with a band around it, i.e. $\mu_f < \bar{\Lambda} < 4\mu_f$. Following the discussion in Refs. [188, 189], the most physical case is given by $\bar{\Lambda} = \sqrt{(2\mu_f)^2 + eB}$.

Figure 5.4 exhibits the free and full pressures, including their renormalization group running bands, for $eB = 1$ GeV$^2$. For comparison, the pressure in the chiral limit is also displayed. One can see that, for large magnetic fields, all cases are essentially indistinguishable unless one goes to values of the chemical potential below $\mu_s = 300$ MeV. This fact suggests that one could build a pQCD-based simple analytic model for the equation of state of cold and dense quark matter under very large magnetic fields.

<figure><img src="image_17.png" /><figcaption>**Figure 5.4:** $P_{\text{free}}$ (left) and total pressure $P$ (right) as functions of the chemical potential for $eB = 1$ GeV$^2$. The bands correspond to changes in the central scale by a factor of 2. For comparison, we also display the pressure in the chiral limit. Figure from Ref. [189].</figcaption></figure>

As discussed previously, the contribution of the two-loop exchange diagram is negligible when compared to the one-loop free term in the case of very high magnetic fields. For magnetic fields expected to be found in the core of magnetars and higher, one can show that the (two-loop) exchange contribution remains small, but renormalization group running effects are significant. This point is illustrated in Figure 5.5, that shows the free pressure $P_{\text{free}}$ and how it compares to the total pressure $P$ as functions of the chemical potential for $B = 10^{19}$ Gauss. One should notice the thicker running bands that appear for such values of magnetic field, indicating the increasing error in the pQCD prediction at low densities.

Hence, as discussed in Ref. [189], one can build a simple analytic pQCD-based model for the equation of state by using

$$ \frac{P_{\text{eff}}}{N_c} = -\sum_f \frac{(q_f B)^2}{2\pi^2} [x_f \ln \sqrt{x_f}] + \sum_f \frac{(q_f B)}{4\pi^2} \left[ \mu_f P_F - m_f^2 \log \left( \frac{\mu_f + P_F}{m_f} \right) \right], \quad (5.14) $$
<figure><img src="image_18.png" /><figcaption>**Figure 5.5:** $P_{\text{free}}$ (left) and $P/P_{\text{free}}$ (right) as functions of the quark chemical potential for $B = 10^{19}$ Gauss ($eB = 0.059 \text{ GeV}^2$). The bands correspond to changes in the central scale by a factor of 2. For comparison, we also display the pressure in the chiral limit. Figure from Ref. [189].</figcaption></figure>

including the renormalization group running. This simple expression represents an excellent approximation to the two-loop pQCD result at very large magnetic fields and intermediate to high densities. Once again, one will notice the larger bands for the pressure as pQCD is applied for smaller values of the magnetic field, essentially leaving the strict region of validity for the full calculation. Nevertheless, these bands provide a measure of the uncertainty in the perturbative calculation, a useful feature that is usually lacking in model calculations. In fact, one can use the perturbative band at a given chemical potential to restrict possible equations of state, in the same fashion as performed in Ref. [227] for neutron star matter<fnref n="1" />.

The utility of this formulation can be illustrated by the computation of the maximum mass and total radius of pure quark magnetars (strange magnetars) as functions of the magnetic field $B$. These results are shown in Figure 5.6. The different curves correspond to changes in the central scale by a factor of 2. The figure also shows results from the magnetic bag model for comparison. These results place constraints on the behavior of the maximum mass and associated radius from perturbative QCD coming down from very high values of the magnetic field. Any given model description should, ideally, approach these constraints for high enough values of $B$.

It is clear that the equation of state extracted from perturbative QCD can not provide a good description of the low- and intermediate-density sectors all by itself. However, this equation of state is obtained from first principles and, thus, should guide the description of the high-density regime. For instance, model calculations of the equation of state for quark matter in the presence of strong magnetic fields should approach this behavior for high $\mu_B$ and very high $B$. A more realistic description of the equation of state in magnetar matter, even for very large magnetic fields, can only be achieved by matching a low-density equation of state onto the equation of state from perturbative QCD.

Even though the window of applicability of hot and dense pQCD in a very strong magnetic field is still narrow, this framework yields results obtained from a clean first-principle calculation, one that can be systematically improved and used to provide solid bounds for other approaches. Furthermore, medium loop corrections seem to become essentially negligible as compared to the free term for very high magnetic fields and physical choices of the renormalization running

<footnote n="1">¹Even though the exchange contribution is small for $B = 10^{19}$ Gauss, such field is not high enough to justify the LLL approximation. Thus, the equation of state above should not be naively applied to magnetars. Reliable results in the current description are only obtained for higher values of the magnetic field.</footnote>
<figure><img src="image_19.png" /><figcaption>**Figure 5.6:** Maximum mass (left) and its respective radius (right) as functions of the magnetic field *B* obtained from perturbative QCD. The different curves correspond to changes in the central scale by a factor of 2. We also show results from the bag model for comparison. Figure from Ref. [189].</figcaption></figure>

scale. This fact can, in principle, improve the convergence of the theory in this limit and suggest simple effective models.

# 6. Strongly Interacting Matter in Magnetic Fields

In this section, we review recent developments characterizing the impact of a uniform magnetic field, assumed to point in the longitudinal direction, on the lowest two topological cumulants in a model independent approach using chiral perturbation theory. We identify low energy theorems that connect the topological susceptibility to the chiral condensate and the fourth cumulant to chiral condensates and susceptibilities. We also investigate the nature of finite volume thermodynamics with a particular focus on the chiral condensate, renormalized magnetization and the pressure, which becomes anisotropic. The local chiral condensate is doubly periodic on the plane transverse to the magnetic field, while the renormalized magnetization and (spatially averaged) chiral condensate exhibit significant finite volume corrections. The former is particularly sensitive to finite volume effects due to its small size.

## 6.1. Chiral perturbation theory

Due to its versatility, chiral perturbation theory allows for model-independent investigations of a wide range of physical phenomena. In the presence of magnetic fields, these include the study of pion polarizabilities [236, 237, 238], magnetic catalysis [239, 240, 241, 242, 243, 244, 245, 246], vacuum magnetization [247, 248], pion superconductivity [249, 250, 251], magnetic vortices [252, 253, 254] and chiral soliton lattices [255, 256, 257, 258, 259]. Since the primary objective of this section is to illustrate theory developments and the focus of this section is on recent developments of the impact of magnetic fields on topological cumulants and finite volume thermodynamics in a magnetic background, we refer readers interested in a thorough review of chiral perturbation theory to seminal papers [260, 261] and review articles [262, 263].

Chiral perturbation theory is an effective theory of pseudo-Goldstone modes associated with chiral symmetry breaking by the QCD vacuum. The resulting Goldstone manifold characterizes field fluctuations of these modes, which in two-flavor QCD are pions. Observables and condensates constructed from chiral perturbation theory are model-independent and valid in the low energy regime of QCD characterized by $p \ll \Lambda_\chi$. The mass dimension one quantity, $p$, in the context of a magnetic background is either the pion mass, or the renormalization invariant quantity, $\sqrt{eB}$. Throughout the discussion, we assume $e > 0$ and use *B* to denote the external magnetic field. $\Lambda_\chi \equiv 4\pi F_\pi$ is a characteristic hadronic scale
<figure><img src="image_20.png" /><figcaption>Figure 6.1: (Left) Plot of the tree-level ground state value of α as a function of the vacuum angle θ. (Right) Plot of tree-level pion mass as a function of θ (normalized by the corresponding θ = 0 value).</figcaption></figure>

that is a function of the pion decay constant, $F_π$, and emerges in one-loop calculations. In the context of finite volume thermodynamics [264], a further scale, the box size, $L$, is required. Since modern lattice calculations are conducted in the $p$-regime, for which box sizes fit a large number of pions, $m_π L \gg 1$, standard power counting rules of chiral perturbation theory apply.

## 6.2. Topological cumulants in a magnetic field

The QCD vacuum is characterized by CP-even topological cumulants constructed out of the gluon field tensor, $G_{\mu\nu}^a$ and its dual, $\tilde{G}_{\mu\nu}^a = \frac{1}{2}\epsilon_{\mu\nu\alpha\beta}G^{\alpha\beta}$. The lowest topological cumulant, the topological susceptibility, is a time-ordered correlation function of $\tilde{GG}$

$$ \chi_t(0) = -i \int d^4x \left\langle \mathcal{T} \frac{g^2 \tilde{G}(x)}{32\pi^2} \frac{g^2 \tilde{G}(0)}{32\pi^2} \right\rangle \qquad (6.1) $$

that is a measure of the susceptibility of the QCD-vacuum to second order changes in $\theta$ near the QCD ground state. Assuming a spacetime invariant angle $\theta$, the topological susceptibility can be expressed as a partial derivative, $\chi_t = \frac{\partial^2 F}{\partial \theta^2}|_{\theta=0}$. The fourth cumulant, on the other hand, is a four-point correlation function of $\tilde{GG}$ and a measure of fourth order changes to the free energy as a function of the vacuum angle. These topological cumulants are non-vanishing as a consequence of the axial $U(1)$ anomaly [265, 266, 267, 268, 269] that prevents the vacuum angle from being rotated away. The experimental observation of the neutron dipole moment constrains the angle to $\theta \lesssim 10^{-11}$ [270, 271]. Values of $\theta$ many orders of magnitude larger than the upper bound do not affect observables significantly [272, 273] – in the absence of an adequate explanation for the zero value of the angle, $\theta$ is for all practical purposes zero. This is the strong CP problem.

The $U(1)$ axial anomaly, however, is better understood. Besides the color gauge symmetry, the $n$-flavor QCD Lagrangian in the chiral limit has an $U(n)_L \times U(n)_R$ flavor symmetry. The QCD vacuum possesses a non-zero chiral condensate, which breaks the vector subgroup $SU(n)_A$ leaving $SU(n)_V \times U(1)_A \times U(1)_V$ intact. However, the low-energy spectrum of QCD only possesses $n^2 - 1$ Goldstone bosons (pseudoscalars). Naively, if $U(1)_A$ were intact, one expects to observe parity partners (scalsars) associated with the pions (3), kaons (4) and eta (1). However, no parity partners or even approximate parity partners are present in the spectrum. Additionally, if $U(1)_A$ were spontaneously broken, one expects to observe an isosinglet with an independent pion decay constant with a mass saturated at $\sqrt{3}m_\pi$.
The breaking of $U(1)_A$ is rather strong (and expected to occur for instance through instantons [274, 275, 276]) – the integration measure of the QCD partition function unlike the the QCD Lagrangian (in the massless limit) is not invariant under $U(1)_A$. Consequently the $\theta$-term of QCD

$$ \mathcal{L}_{\theta} = -\frac{g^2 \theta}{32\pi^2} \tilde{G}G \qquad \mathcal{D}\bar{q}\mathcal{D}q \rightarrow \exp\left[-i\int d^4x \frac{g^2\Theta n}{16\pi^2}\right] \mathcal{D}\bar{q}\mathcal{D}q, \qquad (6.2) $$

which is permitted by color gauge symmetry, cannot be rotated away through an axial $U(1)$ rotation of the quark field $q \rightarrow e^{-i\Theta\gamma_5}q$. Under the transformation, the measure rotates [277, 278] (as shown above) with a choice of $\Theta = -\frac{\theta}{2n}$ eliminating the $\theta$-term at the expense of introducing an imaginary quark mass term in the QCD Lagrangian proportional to the quark mass, $m_f$ for each flavor $f$

$$ \mathcal{L}_{\text{mass}} = -\sum_{f=1}^{n} m_f \bar{q}_f q_f \rightarrow -\sum_{f=1}^{n} m_f \bar{q}_f e^{-\theta \gamma_5} q_f. \qquad (6.3) $$

For a chiral perturbation theory analysis of the cumulants [279, 280, 281], one requires, in addition to the external magnetic field, an introduction of the $\theta$-term through a scalar source.

The leading chiral perturbation theory Lagrangian

$$ \mathcal{L}_2 = \frac{1}{4} F^2 \text{Tr}[D_\mu \Sigma^\dagger D^\mu \Sigma] + \frac{1}{4} F^2 \text{Tr}[\chi \Sigma^\dagger + \chi^\dagger \Sigma]. \qquad (6.4) $$

is parameterized in terms of $SU(n)$ fields, $\Sigma$, the scalar source $\chi = 2B_0Me^{-i\theta/n}$ and the external magnetic field that enters through the covariant derivative, $D_\mu\Sigma = \partial\Sigma + i[Q, ieA_\mu^{\text{ext}}]$. The presence of a $\theta$ vacuum rotates the vacuum that points in the (isospin) direction of the identity operator. For $n=2$, the rotation of the vacuum, characterized by $\alpha$, is understood from the isospin structure of the quark mass matrix, $M = \frac{1}{2}(m_u + m_d)1 + \frac{1}{2}(m_u - m_d)\tau_3$, which favors the third direction in the presence of an isospin splitting

$$ \Sigma_{\alpha} = \cos \alpha + i \sin \alpha \hat{\phi}_{a} \tau_{a} \qquad \hat{\phi}_{a} \hat{\phi}_{a} = 1. \qquad (6.5) $$

An obvious corollary is that in the strong isospin limit, the vacuum remains unrotated. Since the effect of the magnetic field on the free energy first enters at next-to-leading order, one requires the construction of the relevant Lagrangian and counterterms in terms of the correct field fluctuations. Since the rotation of the vacuum is axial, an appropriate rotation of the pion field fluctuations is required, $\Sigma = e^{i\frac{\alpha}{2}\tau_3}e^{i\frac{\phi_a\tau_a}{F}}e^{i\frac{\alpha}{2}\tau_3}$. A non-trivial check to the validity of this construction is the cancellation of divergences that arise in one-loop diagrams by the counterterms of chiral perturbation theory – there are both magnetic field independent and dependent divergences. The latter leads to charge renormalization or equivalently a magnetic field renormalization that keeps $eB$ invariant. Additionally, the construction of the leading field-dependent free energy only requires the tree-level ground state even though the magnetic field alters the ground state orientation, a result that is evident upon noting the perturbative nature of the free energy, which allows for a perturbative evaluation of the ground state orientation. The resulting magnetic-field-dependent free energy has an analogous structure to that of the $\theta=0$ result

$$ \mathcal{F}_B(\theta) = \frac{1}{2} B_R^2 - \frac{1}{(4\pi)^2} \int_0^\infty \frac{ds}{s^3} e^{-\dot{m}_\pi^2(\theta)s} \left[ \frac{eBs}{\sinh eBs} - 1 + \frac{(eBs)^2}{6} \right] \qquad (6.6) $$

$$ B_R = Z_B B \qquad Z_B = 1 + 4e^2h_2^r + \frac{e^2}{6(4\pi)^2} \left( \log \frac{\Lambda^2}{\dot{m}_\pi^2(\theta)} - 1 \right) \qquad (6.7) $$
<figure><img src="image_21.png" /><figcaption>Figure 6.2: Plot of the relative shifts of the topological susceptibility (left) and fourth cumulant (right) compared to tree-level values. The plots also include the small field approximations (up to quadratic order) to the relative shifts.</figcaption></figure>

with the tree-level pion mass, $\dot{m}_{\pi}(0)$ replaced by its $\theta$-vacuum value $\dot{m}_{\pi}(\theta)$

$$ \dot{m}_{\pi}^{2}(\theta) = \dot{m}_{\pi}^{2}(0) \sqrt{1 - \frac{4m_{u}m_{d}}{(m_{u}+m_{d})^{2}} \sin^{2} \frac{\theta}{2}} \qquad (6.8) $$

that is shown in Fig. 6.1. The mass decreases as a function of the vacuum angle reaching a minimum at $\theta = \pi$ and is symmetric about the angle. Also shown in Fig. 6.1 is the tree-level orientation of $\Sigma$ that is odd about $\theta = \pi$. The high-energy constant, $h_2^r(\Lambda)$, depends on the $\overline{\text{MS}}$-bar scale $\Lambda$ and arises through field renormalization.

Since the pion mass squared appears in the negative exponential of the free energy integrand, the free energy has its largest value at $\theta = 0$. The calculation of the impact of the magnetic field is now straightforward – they are derivatives with respect to $\theta$ though one does need to take care to incorporate the $\theta$-dependent contribution of the renormalized magnetic field. The expressions for the shifts for the topological susceptibility and topological cumulants and their weak field behaviors are presented below in terms of the reduced mass $\overline{m} = (\frac{1}{m_u} + \frac{1}{m_d})^{-1}$

$$ \chi_{t,B} = -B_0 \overline{m} \int_0^\infty \frac{ds}{(4\pi s)^2} e^{-m_\pi^2 s} \left[ \frac{eBs}{\sinh eBs} - 1 \right] = B_0 \overline{m} \left[ \frac{1}{6(4\pi m_\pi)^2} \right] (eB)^2 + \mathcal{O}\{(eB)^4\}, \qquad (6.9) $$

$$ c_{4,B} = B_0 \overline{m}^4 \left( \frac{1}{m_u^3} + \frac{1}{m_d^3} \right) \int_0^\infty \frac{ds}{(4\pi s)^2} e^{-m_\pi^2 s} \left[ \frac{eBs}{\sinh eBs} - 1 \right] - 3B_0^2 \overline{m}^2 \int_0^\infty \frac{ds}{(4\pi)^2 s} e^{-m_\pi^2 s} \left[ \frac{eBs}{\sinh eBs} - 1 \right] \qquad (6.10) $$

$$ = \left[ -B_0 \overline{m}^4 \left( \frac{1}{m_u^3} + \frac{1}{m_d^3} \right) + \frac{3B_0^2 \overline{m}^2}{m_\pi^2} \right] \frac{1}{6(4\pi m_\pi)^2} (eB)^2 + \mathcal{O}\{(eB)^4\}. \qquad (6.11) $$

A plot of the cumulants and their weak field behavior is presented in Fig. 6.2 with the two-flavor approximations exhibiting a larger range of validity compared to three-flavor. We conclude this subsection by presenting low-energy theorems that relate the shift of the cumulants to quark condensates and susceptibilities,

$$ (n=2) \qquad \chi_{t,B} = -\overline{m} \langle \bar{q}_f q_f \rangle_B \qquad c_{4,B} = \overline{m}^4 \sum_{q_f=u,d} \frac{\langle \bar{q}_f q_f \rangle_B}{m_{q_f}^3} + 3\overline{m}^2 \chi_{q_f,B}. \qquad (6.12) $$

The shift in the topological susceptibility is proportional to the shift of each quark condensate, $q_f = u, d$. The fourth cumulant is also proportional to the shift of the quark condensates and to the quark susceptibility, which measures the change in the free energy to second order changes in the quark mass.

Fascinatingly, these are manifestations of Ward-Takahashi identities that hold more generally in QCD and are constructed through functional derivatives and subsequent space-time integration of the axial rotated and unrotated versions
of the QCD partition function. In particular, one can use functional derivatives to relate the topological susceptibility to the quark condensate and a space-time integral of a time-ordered two-point correlation of the CP-odd operator $\bar{q}M\gamma_5q$

$$ \chi_{t,B} = -\frac{1}{n^2} \langle \bar{q} M q \rangle_B + \frac{i}{n^2} \int d^4 x \langle T \bar{q}(x) M \gamma_5 q(x) \bar{q}(0) M \gamma_5 q(0) \rangle_B , $$

allowing for the determination of the latter, which vanishes in the isospin limit

$$ (n=2) \qquad i \int d^4 x \langle T \bar{q}(x) M \gamma_5 q(x) \bar{q}(0) M \gamma_5 q(0) \rangle_B = \frac{m_d - m_u}{m_u + m_d} [-m_u \langle \bar{u}u \rangle_B + m_d \langle \bar{d}d \rangle_B] . $$

While we forgo the discussion of the three-flavor low energy theorem, we refer the interested reader to Ref. [279]. The difference from the two-flavor case arises due to the interaction of charged kaon loops with the external field. The resulting low energy theorem for the topological susceptibility is simultaneously proportional to all three quark condensates though the expression for the fourth cumulant is rather more involved. Nevertheless, in the limit of infinite strange quark masses, which is equivalent to integrating out heavy mesons (excluding the pions), the low energy theorems reduce to that of two-flavor chiral perturbation theory.

## 6.3. Finite volume thermodynamics

Lattice QCD calculations offer a more comprehensive avenue to investigate the impact of magnetic fields on topological cumulants [282]. Surprisingly, until very recently, systematic characterization of finite volume effects on lattice thermodynamic observables and condensates in magnetic backgrounds did not exist [283]. In this section, we discuss the most important aspects of this recent development beginning with a discussion of magnetic periodic boundary conditions and the resulting finite volume Green's function that encodes finite volume QCD thermodynamics.

Naturally, we work in Euclidean space with $x_\mu = (x_0, x_1, x_2, x_3)$ and assume zero temperature or equivalently infinite extent in the Euclidean time direction. Utilizing the fully asymmetric gauge $A_\mu = (0, -Bx_2, 0, 0)$, it is clear that the background gauge field is only periodic up to a gauge transformation [284]

$$ A_{\mu}(x + L_{2}\hat{x}_{2}) = A_{\mu}(x) + \partial_{\mu}(\theta_{2} + \Lambda_{2}) \qquad \phi(x + L_{2}\hat{x}_{2}) = e^{-iQ(\theta_{2}+\Lambda_{2})}\phi(x) \qquad (6.13) $$

that is characterized by $\Lambda_2 = -BL_2x_1$ and a twist angle $\theta_2$. Protecting gauge invariance under periodic transformations, therefore, requires an imposition of a boundary gauge transformation of the (positively charged) pion field, which we have denoted $\phi$ for notational simplicity. There are further such periodic gauge transformations induced by twist angles in the remaining Euclidean directions. A consequence of magnetic periodicity is the existence of gauge invariant objects, beyond electric and magnetic fields, namely Wilson lines,

$$ W_2(x_1) = e^{-iQ(\theta_2 - BL_2x_1)} \qquad W_1(x_2) = e^{-iQ(\theta_1 + BL_1x_2)}. \qquad (6.14) $$

that wrap around the first and second spatial directions, respectively.

Uniqueness of the pion field upon periodic translations in the transverse directions leads to quantization of magnetic flux. More transparently, we consider the equalities

$$ \phi(x + L_1 \hat{x}_1 + L_2 \hat{x}_2) = W_2(x_1 + L_1) \phi(x + L_1 \hat{x}_1) = e^{-iQ\theta_1} W_2(x_1 + L_1) \phi(x) \qquad (6.15) $$

$$ \phi(x + L_2 \hat{x}_2 + L_1 \hat{x}_1) = e^{-iQ\theta_1} \phi(x_2 + L_2 \hat{x}_2) = e^{-iQ\theta_1} W_2(x_1) \phi(x) . \qquad (6.16) $$

derived from imposing magnetic translations in different orders. Requiring the final results be identical leads to the condition that magnetic flux, $\Phi = eBL_1L_2$ is quantized in units of $2\pi N_\Phi$.
The study of thermodynamics is conducted through the construction of the relevant Green's function. The infinite volume, charged Green's function, $G_+^\infty(x', x) \equiv \langle \pi^+(x')\pi^-(x) \rangle$ in a magnetic background was constructed in the middle of the previous century [285] and has been utilized extensively within chiral perturbation theory literature to construct the free energy, chiral condensate and the renormalized magnetization. For a finite volume analysis, we require its finite volume counterpart, $G_+(x'x)$ that satisfies the Green's function relation

$$(-D_{\mu}D_{\mu} + m^2)G_+(x', x) = \prod_{\mu=0}^{3} \delta_{L_{\mu}}(x'_{\mu} - x_{\mu}) \qquad (6.17)$$

in addition to magnetic periodic boundary conditions. The $\delta$-functions are finite volume counterparts and possess compact support, i.e. they are non-vanishing for coincident points modulo periodicity. The resulting Green's function is related to its infinite volume counterpart through a sum over images [286]

$$G_+(x', x) = \sum_{\nu_{\mu} \in \mathbb{Z}} e^{iQ\theta_1\nu_1} [W_2^{\dagger}(x_1)]^{\nu_2} G_+^{\infty}(x' + \nu_{\mu}L_{\mu}, x), \qquad (6.18)$$

that unlike the zero field counterpart is dressed by translation symmetry breaking inverse Wilson lines. The infinite volume Green's function, $G_\infty(x', x) \equiv e^{ieB\Delta x_1\bar{x}_2} g_\infty(\Delta x)$, contains a translational symmetry breaking phase and the following translationally symmetric contribution

$$g_\infty(\Delta x) = \frac{1}{(4\pi)^2} \int_0^\infty \frac{ds}{s^2} \frac{eBs}{\sinh eBs} e^{-m_\pi^2 s} \exp \left( -\frac{1}{4s} \left[ \frac{eBs}{\tanh eBs} \Delta x_\perp^2 + \Delta x_\parallel^2 \right] \right) \qquad (6.19)$$

with the full expression being coordinate independent in the coincident limit. We have used for notational compactness $\Delta x_\mu = x'_\mu - x_\mu$, $\bar{x}_\mu = \frac{1}{2}(x'_\mu + x_\mu)$.

The periodic Green's function, however, does not share this property with translational symmetry broken down to $Z_{N_\Phi}$ in each of the transverse directions, a property encoded in the translational properties of the Wilson lines, which for integer $n$ satisfies

$$W_1(x_2 + \frac{n}{N_\Phi} L_2) = W_1(x_2) \qquad W_2(x_1 + \frac{n}{N_\Phi} L_2) = W_2(x_1). \qquad (6.20)$$

Discrete translation symmetry in the transverse directions is also evident in the (local) chiral condensate, defined in terms of the QCD partition function $Z$ and a scalar source $S(x)$, $\langle \bar{\psi}(x)\psi(x) \rangle = -\frac{\delta \log Z}{\delta S(x)}|_{S(x) \to m_q}$. We construct the dimensionless ratio

$$R(x_\perp) = \frac{\langle \bar{\psi}(x)\psi(x) \rangle - \langle \bar{\psi}\psi \rangle^\infty}{\langle \bar{\psi}\psi \rangle_0^\infty} \qquad (6.21)$$

that characterizes deviations of the chiral condensate from its infinite volume counterpart, normalized by the tree level condensate. Chiral perturbation theory predicts a doubly periodic structure in the transverse directions

$$R(x_\perp) = - \int_0^\infty \frac{ds}{(4\pi s F_\pi)^2} e^{-m_\pi^2 s} \left[ 3e^{-\frac{L^2}{4s}} + \frac{eBs}{\sinh eBs} \left\{ 2e^{-\frac{L^2}{4s}} + 2\cos\left(\frac{2\pi N_\Phi x_1}{L}\right) + 2\cos\left(\frac{2\pi N_\Phi x_2}{L}\right) \right\} \right], \qquad (6.22)$$

with leading finite volume corrections arising through transverse windings with magnitude one, $|\nu| = 1$. While twist angles have been set to zero, their presence add independent phases in the periodic functions that appear in the transverse directions. The first contribution arises from the neutral sector while the second contribution is due to charged pions – the latter contains a spatially homogeneous piece that arises through windings in the longitudinal direction. The large volume limit is defined through $m_\pi L \to \infty$ and taken assuming fixed flux, $\frac{eBs}{\sinh eBs} \to 1$, with the leading contribution characterized by

$$R(x_\perp) = - \left[ \frac{5}{2} + \cos\left(\frac{2\pi N_\Phi x_1}{L}\right) + \cos\left(\frac{2\pi N_\Phi x_2}{L}\right) \right] \times \frac{m_\pi^2}{F_\pi^2} \frac{e^{-m_\pi L}}{(2\pi m_\pi L)^{3/2}} \qquad (6.23)$$
<figure><img src="image_22.png" /><figcaption>Figure 6.3: Density plots of the local chiral condensate, $N_\Phi = 1, 2, 3$ respectively, on the transverse plane (normalized by the size $L$).</figcaption></figure>

In Fig. 6.3 we plot the local chiral condensate for the lowest three flux quanta, $N_\Phi = 1, 2, 3$, which is periodic on the transverse plane. The periodic structure vanishes upon averaging due to projection to contributions arising from zero windings. The resulting average shift (normalized by the tree level condensate)

$$ \langle R \rangle - \langle R \rangle_{B=0} = - \int_0^\infty \frac{ds}{(4\pi s F_\pi)^2} e^{-m_\pi^2 s} \left[ \left( \frac{eBs}{\sinh eBs} - 1 \right) \left( \vartheta_3(0, e^{-\frac{L_3^2}{4s}}) - 1 \right) \right]. \quad (6.24) $$

characterizes the size of the finite volume effect on the shift of the chiral condensate induced by the magnetic field. Comparison with its infinite volume counterpart

$$ R^\infty - R_{B=0}^\infty = - \int_0^\infty \frac{ds}{(4\pi s F_\pi)^2} e^{-m_\pi^2 s} \left[ \left( \frac{eBs}{\sinh eBs} - 1 \right) \right], \quad (6.25) $$

is performed through their ratio that is plotted in Fig. 6.4,

$$ \frac{\Delta \langle R \rangle}{\Delta R^\infty} = \frac{\langle R \rangle - \langle R \rangle_{B=0}}{R^\infty - R_{B=0}^\infty} = e^{-m_\pi L} \sqrt{2\pi m_\pi L} \left[ 1 + \mathcal{O}\left(\frac{1}{m_\pi L}\right) + \mathcal{O}\left(\frac{N_\Phi^2}{(m_\pi L)^2}\right) + \mathcal{O}(e^{-m_\pi L}) \right]. \quad (6.26) $$

The asymptotic structure is notably independent of flux quanta as is the leading power law correction with flux quanta dependence first appearing in the subleading correction. The asymptotic behavior most closely agrees with the lowest flux quanta result with agreement already excellent for box sizes with $m_\pi L = 2.5$ which is in significant contrast to higher fluxes. For $m_\pi L = 4$, the size of finite volume corrections is approximately $\gtrsim 10\%$ for the lowest flux while for higher fluxes, finite size effects are significantly smaller as the Landau wavefunctions fit more comfortably in the finite box.

In addition to finite volume corrections on the chiral condensate, it is also possible to consider such corrections on the matter contribution to the free energy, $\mathcal{F}_B$ and renormalized magnetization, which measures the response of the free energy to infinitesimal variations of the magnetic field, $M_r = -\frac{\partial \mathcal{F}_B}{\partial B}$. The latter is studied in lattice QCD through its relation to pressure anisotropy [287]. Since magnetic flux is quantized on the lattice, it is more natural and computationally cheaper to fix flux rather than the magnetic field when varying the size of the box

$$ p_i = - \frac{L_i}{V} \frac{\partial F_B}{\partial L_i} \Big|_{B, L_{j \neq i}} , \quad \tilde{p}_i = - \frac{L_i}{V} \frac{\partial F_B}{\partial L_i} \Big|_{\Phi, L_{j \neq i}} . \quad (6.27) $$

Pressure characterizes the response of the free energy, $F_B = V \mathcal{F}_B$ to changes in the size of the system. The finite volume correction, $\mathcal{F}_B^{FV}$ to the infinite volume free energy, $\mathcal{F}_B^\infty$, that excludes the pure gauge, renormalized contribution,

$$ \mathcal{F}_B^\infty = - \frac{1}{(4\pi)^2} \int_0^\infty \frac{ds}{s^3} e^{-m^2 s} \left[ \frac{eBs}{\sinh eBs} - 1 - \frac{1}{6}(eBs)^2 \right] \quad (6.28) $$

$$ \mathcal{F}_B^{FV} = - \frac{1}{(4\pi)^2} \int_0^\infty \frac{ds}{s^3} e^{-m^2 s} \left[ \theta_3(\frac{L_3^2}{4s}) - 1 \right] \left[ \frac{eBs}{\sinh eBs} - 1 \right], \quad (6.29) $$
<figure><img src="image_23.png" /><figcaption>**Figure 6.4:** (Left) Plot of the magnetic field dependence of the finite volume effect on the chiral condensate, Eq (6.26), as function of $m_\pi L$. (Right) Plot of finite volume effect on the magnetic pressure anisotropy, Eq. 6.32, as function of $m_\pi L$. The exact results for presented in solid lines while the leading asymptotic behavior are shown in dashed lines. The asymptotic behavior that incorporates subleading corrections are shown in dotted lines.</figcaption></figure>

requires no further renormalization. Rather fascinatingly, finite volume effects from the transverse plane only appears through flux quantization – transverse lengths do not appear explicitly in the finite volume correction. The longitudinal extent, however, appears in the finite volume free energy through a Jacobi theta function. The two pressures in the longitudinal direction depend on the longitudinal extent and are identical,

$$ \tilde{p}_3 = p_3 = -\mathcal{F}_B - L_3 \frac{\partial \mathcal{F}_B}{\partial L_3} \qquad \tilde{p}_{\perp} - p_{\perp} = -B\mathcal{M}_r. \qquad (6.30) $$

However, the transverse pressures, $\tilde{p}_{\perp}$, and $p_{\perp}$, in the two definitions differ due to an additional contribution proportional to renormalized magnetization that appears when the flux is fixed. The pressure anisotropy, $\Delta\tilde{p}$, characterizes the pressure difference at fixed flux between the transverse and longitudinal directions

$$ \Delta \tilde{p} = \tilde{p}_{\perp} - \tilde{p}_3 \qquad \Delta \tilde{p}^{\infty} = \lim_{L_3 \to \infty} (\tilde{p}_{\perp} - \tilde{p}_3) = -B\mathcal{M}_r^{\infty}. \qquad (6.31) $$

We compare this pressure anisotropy with its infinite volume counterpart, $\Delta\tilde{p}^{\infty}$, constructed by taking the infinite longitudinal extent limit while keeping $eB$ fixed. This is equivalent to comparing the finite volume magnetization with its infinite volume counterpart. For this comparison, we plot the quantity

$$ R(\Delta \tilde{p}) = \frac{\Delta \tilde{p} - \Delta \tilde{p}^{\infty}}{\Delta \tilde{p}^{\infty}} = \frac{30}{7(2\pi)^{3/2}N_{\Phi}^2} (m_{\pi} L)^{9/2} e^{-m_{\pi} L} \left[ 1 + \mathcal{O}\left(\frac{1}{m_{\pi} L}\right) + \mathcal{O}\left(e^{-m_{\pi} L}\right) \right]. \qquad (6.32) $$

and its large volume, asymptotic structure in Fig. 6.4. The size of finite volume corrections are substantial, significantly more so than the chiral condensate even for the largest magnetic fields. Unlike the chiral condensate, the renormalized magnetization is small and scales cubically with the magnetic field for small fields. Additionally, the asymptotic value of $R(\Delta\tilde{p})$ is reached slowly. The leading asymptotic behavior arises exclusively through the term proportional to the longitudinal derivative of the finite volume free energy, Eq. (6.31). The leading relative correction is a power law (with coefficient $-13/8$) that arises through the finite volume contribution to $-B\mathcal{M}_r$. Its inclusion significantly improves agreement with the exact result (shown in dotted lines).

Finite volume corrections to thermodynamic observables and condensates in a magnetic field are substantial since the effect of a magnetic field enters chiral perturbation theory at the same order as finite size effects. As such significantly
large volumes are required to minimize the effect – for the lowest quanta, the chiral condensate exhibits corrections of size ~ 10% for $m_πL = 4$ and an analysis of the neutral pion magnetic polarization, omitted in this review, presents corrections of a similar size, ~ 10% for $m_πL = 4$ [283]. The corrections to the renormalized magnetization are very significant even for $m_πL = 8$ due to their smallness. These results, while obviously relevant to lattice QCD calculations, may also be of theoretical and experimental interest in heavy ion collisions where finite size effects may be important for certain observables.

# 7. QED Fermions in a noisy magnetic field background: Perturbation theory and renormalized quasiparticles

Inhomogeneous and rapidly fluctuating magnetic fields are present in important physical scenarios, particularly in early stages of heavy-ion collisions [288, 179, 289, 172, 171] and the corresponding genesis of the quark-gluon plasma [290, 291, 292, 293]. Their effects can influence the physical properties of both charged as well as neutral particles, the latter due to the quantum mechanical fluctuations of the vacuum, leading to the creation of virtual charged fermion-antifermion pairs. Classical, static and uniform background magnetic fields have been extensively studied in the context of the gluon polarization tensor [294, 291, 295, 292] in relation to the emergence of the vacuum birefringence phenomena [294, 295], and the propagation of fermions in such magnetized background [296], as expressed by the self-energy, that leads to the definition of a magnetic mass and to an spectral broadening [297] due to Landau level mixing. Nevertheless, since spatio-temporal fluctuations with respect to a finite background magnetic field may indeed exist in the different aforementioned physical scenarios [289], in the present work we shall study their effect on the renormalization of the fermion propagator itself in QED.

As discussed in a recent work [298], a perturbative treatment of such fluctuations in the framework of the replica method [299, 300] allows us to show that their effect leads to a renormalization of the charge $e \to z_3e$ and an effective refraction index $v'/c = z^{-1}$. We further show [298] that $z_3$ and $z$ depend not only on the energy scale, as usual, but also on the magnitude of the average magnetic field $|B|$, as well as on the strength of its spatial fluctuations, that we define as $\Delta_B$.

## 7.1. The model

Let us start by assuming a system of QED fermions $\psi(x)$ immersed in a classical static magnetic field background, that nevertheless possesses random spatial fluctuations with respect to an otherwise constant mean value. For the gauge fields $A^\mu(x)$, we shall distinguish three physically different contributions [298]

$$A^{\mu}(x) \to A^{\mu}(x) + A_{\text{BG}}^{\mu}(x) + \delta A_{\text{BG}}^{\mu}(x). \qquad (7.1)$$

Here, $A^\mu(x)$ represents the dynamical photonic quantum field, while BG stands for “background”, representing the presence of a classical external field imposed by the experimental conditions. Moreover, for this BG contribution, we consider the effect of static (quenched) white noise spatial fluctuations $\delta A_{\text{BG}}^\mu(x)$ with respect to the mean value $A_{\text{BG}}^\mu(x)$, satisfying the statistical properties

$$ \begin{aligned} \langle \delta A_{\text{BG}}^{j}(\mathbf{x}) \delta A_{\text{BG}}^{k}(\mathbf{x}') \rangle &= \Delta_B \delta_{j,k} \delta^3(\mathbf{x} - \mathbf{x}'), \\ \langle \delta A_{\text{BG}}^{\mu}(\mathbf{x}) \rangle &= 0. \end{aligned} \qquad (7.2) $$
These statistical properties are represented by a Gaussian functional distribution of the form

$$dP[\delta A_{\text{BG}}^{\mu}] = \mathcal{N} e^{-\int d^3x \frac{[\delta A_{\text{BG}}^{\mu}(x)]^2}{2\Delta_B}} \mathcal{D}[\delta A_{\text{BG}}^{\mu}(\mathbf{x})]. \quad (7.3)$$

In the context of heavy-ion collisions (HIC), we expect strong spatial fluctuations in the strong magnetic fields $\mathbf{B} = \nabla \times \mathbf{A}_{BG}$ generated within small spatial regions in the early stages of individual collisions, where the characteristic length scale is $L \sim \sqrt{\sigma}$, with $\sigma$ being the effective cross-section. In these collisions, the axial (z-direction) component of the magnetic field dominates, such that on average we have $\langle \mathbf{B} \rangle = \hat{e}_3 B$. However, there are also smaller transverse components $\delta B_x$ and $\delta B_y$, such that the fluctuation of the field within the small collision region can be estimated to be on the order $(\delta B)^2 \sim (\delta B_x)^2 + (\delta B_y)^2$. Since many such collisions occur at different points in space, an approximate model for this physical scenario is provided by the spatial random noise Eq. (7.2). By dimensional analysis, the magnitude of $\Delta_B$ is of the order:

$$\Delta_B \sim (\delta B)^2 L^5 \sim (\delta B)^2 \sigma^{5/2}. \quad (7.4)$$

As discussed in our recent work [298], this analysis leads to the following estimation

$$\Delta_B \sim \pi^{5/2} (\delta B)^2 r_0^5 N^{5/3} \left( \frac{N_{\text{part}}}{2N} \right)^{5/3}. \quad (7.5)$$

In peripheral heavy-ion collisions, the magnetic field fluctuations along the transverse plane are approximately $|e \delta B| \sim m_\pi^2/4$, where $m_\pi$ is the pion mass [301, 302]. For instance, in a Au+Au collision with $N = 197$, and if $N_{\text{part}}/N = 1/2$, we obtain $\Delta \equiv e^2 \Delta_B \sim 2.6 \text{ MeV}^{-1}$, while for less central collisions with $N_{\text{part}}/N = 1/8$ we have $\Delta \sim 0.26 \text{ MeV}^{-1}$.

The Lagrangian for this model is a superposition of two contributions

$$\mathcal{L} = \mathcal{L}_{\text{FBG}} + \mathcal{L}_{\text{NBG}}, \quad (7.6)$$

where the first one represents the system of Fermions (and photons) immersed in the deterministic background field (FBG)

$$\mathcal{L}_{\text{FBG}} = \bar{\psi} (i \not\partial - e A_{\text{BG}} - e A - m) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}, \quad (7.7)$$

with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ the electromagnetic tensor for the quantum photon gauge fields. In contrast, the second term represents the interaction between the Fermions and the static classical noise (NBG), represented by the spatial fluctuations $\delta A_{\text{BG}}^\mu(x)$

$$\mathcal{L}_{\text{NBG}} = \bar{\psi} (-e \delta A_{\text{BG}}) \psi. \quad (7.8)$$

The generating functional (in the absence of sources) for a given realization of the noisy fields is given by

$$Z[A] = \int \mathcal{D}[\bar{\psi}, \psi] e^{i \int d^4 x [\mathcal{L}_{\text{FBG}} + \mathcal{L}_{\text{NBG}}]}. \quad (7.9)$$

To study the physics of this system, we need to calculate the statistical average over the magnetic background noise $\delta A_{\text{BG}}^\mu$ of the $\overline{\ln Z}$. For this purpose, we apply the replica method, which is based on the following identity [299]

$$\overline{\ln Z[A]} = \lim_{n \to 0} \frac{\overline{Z^n[A]} - 1}{n}. \quad (7.10)$$

Here, we defined the statistical average according to the Gaussian functional measure in Eq. (7.3), and $Z^n$ is obtained by incorporating an additional “replica” component for each of the Fermion fields, i.e. $\psi(x) \to \psi^a(x)$, for $1 \le a \le n$.
The “replicated" Lagrangian has the same form as Eqs. (7.7) and (7.8), but with an additional sum over the replica components of the Fermion fields. Therefore, the averaging procedure leads to

$$
\begin{align}
\overline{Z^n[A]} &= \int \prod_{a=1}^{n} \mathcal{D}[\bar{\psi}^a, \psi^a] \int \mathcal{D}[\delta A_{\text{BG}}^{\mu}] e^{-\int d^3x \frac{[\delta A_{\text{BG}}^{\mu}(x)]^2}{2\Delta_B}} \nonumber \\
&\quad \times e^{i \int d^4x \sum_{a=1}^{n} (\mathcal{L}_{\text{FBG}}[\bar{\psi}^a, \psi^a] + \mathcal{L}_{\text{NBG}}[\bar{\psi}^a, \psi^a])} \nonumber \\
&= \int \prod_{a=1}^{n} \mathcal{D}[\bar{\psi}^a, \psi^a] e^{i\bar{S}[\bar{\psi}^a, \psi^a; A]}, \tag{7.11}
\end{align}
$$

where in the last step the Gaussian integral over the background noise was explicitly calculated, leading to the definition
of the effective averaged action for the replica system [298]

$$
\begin{equation}
\begin{split}
\bar{S}[\bar{\psi}^a, \psi^a; A] = \int d^4 x \left( & \sum_a \bar{\psi}^a (i\not\partial - eA_{\text{BG}} - eA - m) \psi^a - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right) \\
& + i \frac{e^2 \Delta_B}{2} \int d^4 x \int d^4 y \sum_{a,b} \sum_{j=1}^3 \bar{\psi}^a(x) \gamma^j \psi^a(x) \bar{\psi}^b(y) \gamma_j \psi^b(y) \delta^3(\mathbf{x}-\mathbf{y}).
\end{split}
\tag{7.12}
\end{equation}
$$

The result of the averaging procedure is an effective interacting theory, with an instantaneous local interaction pro-
portional to the fluctuation amplitude Δ<sub>B</sub> that characterizes the magnetic noise, as defined in Eq. (7.2). The “free” part
of the action corresponds to Fermions in the average background classical field A<sub>BG</sub><sup>μ</sup>(x). We choose this background to
represent a uniform, static magnetic field along the z-direction **B** = *ê*<sub>3</sub><b>B</b>, using the gauge [303]

$$
A_{\text{BG}}^{\mu}(x) = \frac{1}{2}(0, -Bx^2, Bx^1, 0). \tag{7.13}
$$

Therefore, this allows us to use directly the Schwinger proper-time representation of the free-Fermion propagator dressed
by the background field [304, 303],

$$
S_F(x, x') = \Phi(x, x') \int \frac{d^4 p}{(2\pi)^4} e^{-ip \cdot (x-x')} S_F(p), \quad (7.14)
$$

Since the effective interaction is strictly local (i.e. proportional to $\delta^{(3)}(\mathbf{x} - \mathbf{y})$), the Schwinger phase is a global common factor in the corresponding Dyson equation for the dressed propagator. Therefore, we only need to consider the translational-invariant part of the Schwinger propagator

$$
[S_F(k)]_{a,b} = -i\delta_{a,b} \int_0^\infty \frac{d\tau}{\cos(eB\tau)} e^{i\tau(k_\parallel^2 - \mathbf{k}_\perp^2 \frac{\tan(eB\tau)}{eB\tau} - m^2 + i\epsilon)} \left\{ \left[\cos(eB\tau) + i\gamma^1\gamma^2 \sin(eB\tau)\right] (m + \mathbf{k}_\parallel) + \frac{\mathbf{k}_\perp}{\cos(eB\tau)} \right\}, (7.15)
$$

which is diagonal in the replica space. Moreover, since it is an explicit function of the average magnetic field **B** = ∇ × **A**<sub>BG</sub>
(rather than a function of the background gauge field A<sub>BG</sub><sup>μ</sup>), it is also gauge-invariant. Here, as usual, we separated the
parallel from the perpendicular directions with respect to the background external magnetic field by splitting the metric
tensor as g<sup>μν</sup> = g<sub>||</sub><sup>μν</sup> + g<sub>⊥</sub><sup>μν</sup>, with

$$
\begin{align}
g_{\parallel}^{\mu\nu} &= \operatorname{diag}(1, 0, 0, -1), \nonumber \\
g_{\perp}^{\mu\nu} &= \operatorname{diag}(0, -1, -1, 0), \tag{7.16}
\end{align}
$$

thus implying that for any 4-vector, such as the momentum $k^\mu$, we write $\not{k} = \not{k}_\perp + \not{k}_\parallel$, and $k^2 = k_\parallel^2 - k_\perp^2$. In particular,
$\not{k}_\parallel^2 = k_0^2 - k_3^2$, while $\mathbf{k}_\perp = (k^1, k^2)$ is the Euclidean 2-vector lying in the plane perpendicular to the field, such that its
square-norm is $\mathbf{k}_\perp^2 = k_1^2 + k_2^2$.
As we showed in our previous work [298], the Schwinger propagator can be expressed exactly as follows,

$$[S_F(k)]_{a,b} = -i\delta_{a,b} \left[ (m + \not k_\parallel) \mathcal{A}_1 + i\gamma^1\gamma^2 (m + \not k_\parallel) \mathcal{A}_2 + \mathcal{A}_3 \not k_\perp \right]$$

where we defined the scalar function

$$\mathcal{A}_1(k, B) = \int_0^\infty d\tau e^{i\tau(k_\parallel^2 - m^2 + i\epsilon)} - i \frac{\not k_\parallel^2}{eB} \tan(eB\tau), \qquad (7.17)$$

that clearly reproduces the inverse scalar propagator (with Feynman prescription) in the zero-field limit

$$\lim_{B \to 0} \mathcal{A}_1(k, B) = \frac{i}{k^2 - m^2 + i\epsilon}. \qquad (7.18)$$

The other coefficients are nothing but momentum-derivatives of this basic scalar function, defined as follows

$$\mathcal{A}_2(k, B) \equiv \int_0^\infty d\tau \tan(eB\tau) e^{i\tau(k_\parallel^2 - t_B(\tau)\mathbf{k}_\perp^2 - m^2 + i\epsilon)} = ieB \frac{\partial \mathcal{A}_1}{\partial(\mathbf{k}_\perp^2)}, \qquad (7.19a)$$

$$\mathcal{A}_3(k, B) \equiv \int_0^\infty \frac{d\tau}{\cos^2(eB\tau)} e^{i\tau(k_\parallel^2 - t_B(\tau)\mathbf{k}_\perp^2 - m^2 + i\epsilon)} = \mathcal{A}_1 + (ieB)^2 \frac{\partial^2 \mathcal{A}_1}{\partial(\mathbf{k}_\perp^2)^2}. \qquad (7.19b)$$

Moreover, with these definitions it is straightforward to calculate the inverse of the Schwinger propagator Eq. (7.17) as [298]:

$$\hat{S}_F^{-1}(k) = \frac{i}{\mathcal{D}(k)} \left[ (m - \not k_\parallel) \mathcal{A}_1 - i\gamma^1\gamma^2 (m - \not k_\parallel) \mathcal{A}_2 - \mathcal{A}_3 \not k_\perp \right], \qquad (7.20)$$

where

$$\mathcal{D}(k) = \mathcal{A}_3^2 \mathbf{k}_\perp^2 - (\mathcal{A}_1^2 - \mathcal{A}_2^2) (\mathbf{k}_\parallel^2 - m^2). \qquad (7.21)$$

Then, all the relevant expressions will be given in terms of $\mathcal{A}_1$.

## 7.2. Perturbation theory: Self-energy and dressed propagator

The effective action in Eq. (7.12) reveals a resulting fermion-fermion interaction arising from the average over the background magnetic noise. Starting from a free Fermion propagator, as defined by Eq. (7.15), we include the magnetic noise-induced interaction effects by “dressing” the propagator with a self-energy, as shown diagrammatically in the Dyson equation depicted in Fig. 7.1. We remark that for this theory, the skeleton diagram for the self-energy is represented in Fig. 7.2.

<figure><img src="image_24.png" /><figcaption>Figure 7.1: Dyson equation for the "dressed" propagator (double-line), in terms of the free propagator (single-line) and the self-energy $\Sigma$.</figcaption></figure>

At first-order in $\Delta \equiv e^2\Delta_B$, the self-energy depicted in the Feynman diagram in Fig. (7.3) is given by the integral expression [298]

$$\hat{\Sigma}_{\Delta}(q) = (i\Delta) \int \frac{d^3p}{(2\pi)^3} \gamma^j \hat{S}_F(p+q; p_0=0)\gamma_j \\ = \frac{i(i\Delta)}{(2\pi)^3} \int d^3p \left\{ 3(\gamma^0 q_0 - m) \mathcal{A}_1(q_0, p_3; \mathbf{p}_\perp) + i\gamma^1\gamma^2(m - q_0\gamma^0) (ieB) \frac{\partial}{\partial \mathbf{p}_\perp^2} \mathcal{A}_1(q_0, p_3; \mathbf{p}_\perp) \right\}.$$
<figure><img src="image_25.png" /><figcaption>**Figure 7.2:** Skeleton diagram representing the self-energy for the effective interacting theory. The dashed line is the disorder-induced interaction $\Delta_B$, while the box $\hat{\Gamma}$ represents the 4-point vertex function.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 7.3:** Self-energy diagram at first order in $\Delta = e^2\Delta_B$.</figcaption></figure>

After some elementary transformations [298], the self-energy (valid at all orders in the background average magnetic field B) is

$$ \hat{\Sigma}_{\Delta}(q) = \frac{i(i\Delta)}{(2\pi)^3} \left[ 3(\gamma^0 q_0 - m) \tilde{\mathcal{A}}_1(q_0) - i\gamma^1\gamma^2(i\pi eB)(m - q_0\gamma^0) \tilde{\mathcal{A}}_2(q_0) \right], \quad (7.22) $$

where we have defined:

$$ \begin{aligned} \tilde{\mathcal{A}}_1(q_0) &\equiv \int d^3p \mathcal{A}_1(q_0, p_3; \mathbf{p}_{\perp}), \\ \tilde{\mathcal{A}}_2(q_0) &\equiv \int_{-\infty}^{+\infty} dp_3 \mathcal{A}_1(q_0, p_3; \mathbf{p}_{\perp} = 0). \end{aligned} \quad (7.23) $$

Inserting this self-energy into the Dyson equation, as depicted diagrammatically in Fig. 7.1, we obtain the dressed inverse propagator at first-order in $\Delta$

$$ \hat{S}_{\Delta}^{-1}(k) = \hat{S}_{F}^{-1}(k) - \hat{\Sigma}_{\Delta}, \quad (7.24) $$

so that by using Eqs. (7.20) and (7.22) we explicitly obtain [298]

$$ \begin{aligned} S_{\Delta}^{-1}(q) &= \frac{i z}{\mathcal{D}(q)} \left[ (m - q_0 \gamma^0 - z^{-1} q_3 \gamma^3) \mathcal{A}_1(q) - z_3 (i \gamma^1 \gamma^2) (m - q_0 \gamma^0 - z^{-1} q_3 \gamma^3) \mathcal{A}_2(q) - i \mathcal{A}_3(q) z^{-1} \notag \\ & \phantom{=} - \mathcal{A}_3(q) z^{-1} \notag \\ & \phantom{=} \left[ (m - \tilde{q}_{\parallel}) \mathcal{A}_1(q) - z_3 (i \gamma^1 \gamma^2) (m - \tilde{q}_{\parallel}) \mathcal{A}_2(q) - i \mathcal{A}_3(q) \tilde{q}_{\perp} \right] \right], \end{aligned} \quad (7.25) $$

where in the last line we defined the four-vector $\tilde{q}^{\mu} = (q^0, z^{-1}q)$ that incorporates the definition of the effective refraction index $v'/c = z^{-1}$ due to the random magnetic fluctuations. Here, $z$ and $z_3$ are the renormalization factors for the wave fermion field and the charge, respectively. While $z$ will emerge as a global factor in the dressed propagator, the factor $z_3$ will only be associated to the tensor structures involving the spin-magnetic field interaction $e\sigma_{\mu\nu}F_{BG}^{\mu\nu} = i\gamma_1\gamma_2eB$. The explicit algebraic expressions, at arbitrary order in the background magnetic field, are [298]

$$ z = 1 + \frac{3i\Delta}{(2\pi)^3} \frac{\tilde{\mathcal{A}}_1(q_0)}{\mathcal{A}_1(q)} \mathcal{D}(q), \quad (7.26a) $$

$$ z_3 = \frac{1 - \frac{i\pi(i\Delta)(eB)}{(2\pi)^3} \frac{\tilde{\mathcal{A}}_2(q_0)}{\mathcal{A}_2(q)} \mathcal{D}(q)}{1 + \frac{3i\Delta}{(2\pi)^3} \frac{\tilde{\mathcal{A}}_1(q_0)}{\mathcal{A}_1(q)} \mathcal{D}(q)}, \quad (7.26b) $$
$$m' = m, \qquad (7.26c)$$

and

$$\frac{v'}{c} = z^{-1} = \left( 1 + \frac{3i\Delta}{(2\pi)^3} \frac{\tilde{\mathcal{A}}_1(q_0)}{\mathcal{A}_1(q)} \mathcal{D}(q) \right)^{-1} \qquad (7.26d)$$

By comparing Eq. (7.25) with Eq. (7.20), it is clear that they possess the same tensor structure. Therefore, by means of the elementary properties of the Dirac matrices, this expression can be readily inverted to obtain the “magnetic noise-dressed” fermion propagator [298]

$$S_{\Delta}(q) = -iz^{-1} \frac{\mathcal{D}(q)}{\tilde{\mathcal{D}}(q)} \left[ (m + \tilde{q}_{\parallel}) \mathcal{A}_1(q) + iz_3 \gamma^1 \gamma^2 (m + \tilde{q}_{\parallel}) \mathcal{A}_2(q) + \mathcal{A}_3(q) \tilde{q}_{\perp} \right], \qquad (7.27)$$

where $\mathcal{D}(q)$ was defined in Eq. (7.21), and

$$\tilde{\mathcal{D}}(q) = \mathcal{A}_3^2 z^{-2} \mathbf{q}_{\perp}^2 - (\mathcal{A}_1^2 - \mathcal{A}_2^2) (z^{-2} q_{\parallel}^2 - m^2) \qquad (7.28)$$

Let us now discuss the explicit magnetic field and magnetic noise dependence of the renormalized parameters defined in Eqs. (7.26a) and (7.26b), i.e $z$ and $z_3$. For this purpose, we shall distinguish three different regimes, corresponding to the very weak, the intermediate and the ultra-intense magnetic field, respectively.

### 7.2.1. Very weak field $eB/m^2 \ll 1$

As shown in detail in [298], for very weak fields $eB/m^2 \ll 1$ the function $\mathcal{A}_1(k, B)$ can be expanded in terms of the power series

$$\mathcal{A}_1(k, B) = \frac{i}{\mathcal{D}_{\parallel}} \left( 1 + \sum_{j=1}^{\infty} \left( \frac{ieB}{\mathcal{D}_{\parallel}} \right)^j \mathcal{E}_j(x) \right), \qquad (7.29)$$

where for notational simplicity, we defined the “parallel” inverse scalar propagator

$$\mathcal{D}_{\parallel} = k_{\parallel}^2 - m^2 + i\epsilon, \qquad (7.30)$$

and the dimensionless variable $x = k_{\perp}^2/eB$. We also defined the polynomials $\mathcal{E}_j(x)$, as those generated by the function $e^{-ix \tan v}$, i.e.

$$\mathcal{E}_j(x) = \lim_{v \to 0} \frac{\partial^j}{\partial v^j} (e^{-ix \tan v}). \qquad (7.31)$$

At the lowest order, and after subtracting the divergent vacuum contribution from Eq. (7.29), we have [298]

$$\mathcal{A}_1(k, B) - \mathcal{A}_1(k, 0) = \frac{-2i (eB)^2 \mathbf{k}_{\perp}^2}{[k^2 - m^2 + i\epsilon]^4} + O((eB)^4). \qquad (7.32)$$

Therefore, using this weak field expansion of the propagator, we obtain [298]

$$\tilde{\mathcal{A}}_1(q_0) = -\frac{\pi^2}{6} \frac{(eB)^2}{(q_0^2 - m^2)^{3/2}} + \mathcal{O}((eB)^4). \qquad (7.33)$$

In addition, at order $\mathcal{O}((eB)^2)$ we also obtain [298]

$$\tilde{\mathcal{A}}_2(q_0) = \frac{\pi}{\sqrt{q_0^2 - m^2}} + \mathcal{O}((eB)^4). \qquad (7.34)$$
Therefore, from Eqs. (7.26a), (7.32), (7.33), and (7.34), we can directly evaluate the renormalization parameters to obtain

$$ 
\begin{aligned}
z &= 1 + \frac{3i\Delta}{(2\pi)^3} \frac{\tilde{\mathcal{A}}_1(q_0)}{\mathcal{A}_1(q)} \mathcal{D}(q) \\
  &= 1 + O((eB)^4),
\end{aligned} 
\qquad (7.35) $$

and similarly from Eq. (7.26b)

$$ z_3 = 1 + O((eB)^4). \qquad (7.36) $$

### 7.2.2. Intermediate field

For intermediate magnetic field intensities, we calculate the integral $\mathcal{A}_1$ by means of an expansion in terms of Landau levels [298]. For this purpose, let us consider the generating function of the Laguerre polynomials[305]

$$ e^{-\frac{x}{2} \frac{1-t}{1+t}} = (1+t)e^{-x/2} \sum_{n=0}^{\infty} (-t)^n L_n^0(x), \qquad (7.37) $$

since

$$ 
\begin{aligned}
e^{-ix \tan v} &= \exp \left[ -x \frac{(1 - e^{-2iv})}{(1 + e^{-2iv})} \right] \\
&= (1 + e^{-2iv}) e^{-x} \sum_{n=0}^{\infty} (-1)^n e^{-2inv} L_n^0(2x)
\end{aligned} 
\qquad (7.38) $$

Therefore, we have (for $x = k_{\perp}^2/eB$) after evaluating the corresponding integrals [298]

$$ \mathcal{A}_1(k) = i \frac{e^{-x}}{\mathcal{D}_{\parallel}} \left[ 1 + \sum_{n=1}^{\infty} \frac{(-1)^n \left[ L_n^0(2x) - L_{n-1}^0(2x) \right]}{1 - 2n \frac{eB}{\mathcal{D}_{\parallel}}} \right]. $$

Inserting this expression into the definition of $\tilde{\mathcal{A}}_1$ of Eq. (7.23), we obtain

$$ 
\begin{aligned}
\tilde{\mathcal{A}}_1(q_0) &= \int d^3 p \mathcal{A}_1(q_0, p_3; \mathbf{p}_{\perp}) \\
&= \mathcal{I}_1 + \sum_{n=1}^{\infty} (-1)^n \mathcal{I}_{2,n}
\end{aligned} 
\qquad (7.39) $$

Here, we defined [298]

$$ 
\begin{aligned}
\mathcal{I}_1 &= i \int d^3 p \frac{e^{-\mathbf{p}_\perp^2 / eB}}{q_0^2 - p_3^2 - m^2 + i\epsilon} \\
&= \frac{\pi^2 eB}{\sqrt{q_0^2 - m^2 + i\epsilon}},
\end{aligned} 
\qquad (7.40) $$

and

$$ 
\begin{aligned}
\mathcal{I}_{2,n} &= i \int d^3 p e^{-\mathbf{p}_\perp^2 / eB} \frac{L_n\left(\frac{2\mathbf{p}_\perp^2}{eB}\right) - L_{n-1}\left(\frac{2\mathbf{p}_\perp^2}{eB}\right)}{q_0^2 - p_3^2 - m^2 - 2neB + i\epsilon} \\
&= 2\pi eB (-1)^n \int_{-\infty}^{\infty} \frac{dp_3}{q_0^2 - m^2 - p_3^2 - 2nqB + i\epsilon}
\end{aligned} 
\qquad (7.41) $$

Inserting Eq. (7.40) and Eq. (7.41) into Eq. (7.39), we obtain (after shifting the index $n \to n + 1$)

$$ 
\begin{aligned}
\tilde{\mathcal{A}}_1(q_0) &= \frac{\pi^2(eB)}{\sqrt{q_0^2 - m^2 + i\epsilon}} \\
&\quad + 2\pi eB \int_{-\infty}^{+\infty} dp_3 \sum_{n=0}^{\infty} \frac{1}{q_0^2 - m^2 - p_3^2 - 2(n+1)eB + i\epsilon}
\end{aligned} 
\qquad (7.42) $$
<figure><img src="image_27.png" /><figcaption>Figure 7.4: Spectral density for the Landau level spectrum $\rho(E)$, as a function of the dimensionless energy scale $E/m$, for different values of the average background magnetic field $eB/m^2$. The inset is shown in order to appreciate in detail the staircase pattern produced by the discrete Landau levels.</figcaption></figure>

Let us introduce the density of states for Landau levels

$$ \rho(E) = \int_{-\infty}^{\infty} \frac{dp_3}{2\pi} \sum_{n=0}^{\infty} \delta(E - E_n(p_3)), \qquad (7.43) $$

with the dispersion relation for the spectrum

$$ E_n(p_3) = \sqrt{p_3^2 + m^2 + 2(n+1)eB}. \qquad (7.44) $$

As shown in detail in [298]

$$ \rho(E) = \Theta(E - \sqrt{m^2 + 2eB}) \frac{E}{\pi\sqrt{eB}} \\ \times \left[ \zeta\left(\frac{1}{2}, \frac{E^2 - m^2 - 2eB}{2eB} - N_{max}(E)\right) \right. \\ \left. - \zeta\left(\frac{1}{2}, \frac{E^2 - m^2}{2eB}\right) \right], \qquad (7.45) $$

where we defined

$$ N_{max}(E) = \lfloor \frac{E^2 - m^2}{2eB} - 1 \rfloor, \qquad (7.46) $$

with $\lfloor x \rfloor$ the integer part of $x$, and $\zeta(n, z)$ is the Riemann zeta function. The spectral density Eq. (7.45) is represented in Fig. 7.4, as a function of the dimensionless energy scale $E/m$. For large magnetic fields $eB/m^2 \gg 1$, the spectral density displays a clear staircase pattern, where each step represents the contribution arising from a single Landau level $n = 0, 1, \dots$. On the other hand, for weak magnetic fields $eB/m^2 \ll 1$, the spectral density exhibits a denser, quasi-continuum behavior.

With these definitions, we obtain from Eq. (7.43) the exact expression [298]

$$ \tilde{A}_1(q_0) = \frac{\pi^2 eB}{\sqrt{q_0^2 - m^2 + i\epsilon}} + 4\pi^2 eB \int_{-\infty}^{+\infty} dE \frac{\rho(E)}{q_0^2 - E^2 + i\epsilon}, \qquad (7.47) $$
On the other hand, from Eq. (7.23) we have [298]

$$ 
\begin{aligned}
\tilde{\mathcal{A}}_2(q_0) &= \int_{-\infty}^{\infty} dp_3 p \mathcal{A}_1(q_0, p_3; \mathbf{p}_{\perp} = 0) && (7.48) \\
&= \frac{\pi}{\sqrt{q_0^2 - m^2 + i\epsilon}}.
\end{aligned}
$$

In order to evaluate the formulas for $z$ in Eq. (7.26a) and $z_3$ in Eq. (7.26b), we obtain explicit expressions for the following coefficients (with $x = \mathbf{k}_{\perp}^2/(eB)$) [298]

$$ \mathcal{A}_2 = i e B \frac{\partial \mathcal{A}_1}{\partial \mathbf{k}_{\perp}^2} = \frac{e^{-x}}{D_{\parallel}} \left[ 1 + \sum_{n=1}^{\infty} \frac{(-1)^n \left( L_n(2x) + 2L_{n-1}^{(1)}(2x) - L_{n-1}(2x) - 2\theta_{n-2} \cdot L_{n-2}^{(1)}(2x) \right)}{1 - 2 \frac{neB}{D_{\parallel}}} \right] \quad (7.49) $$

$$ 
\begin{aligned}
\mathcal{A}_3 &= \mathcal{A}_1 + (ieB)^2 \frac{\partial^2 \mathcal{A}_1}{\partial(\mathbf{k}_{\perp}^2)^2} \\
&= i \frac{e^{-x}}{D_{\parallel}} \sum_{n=1}^{\infty} \frac{(-1)^n \left( L_n(2x) - L_{n-1}(2x) + 2L_{n-1}^{(1)}(2x) + 4\theta_{n-2} \cdot L_{n-2}^{(2)}(2x) - 2L_{n-1}^{(1)}(2x) - 4\theta_{n-3} \cdot L_{n-3}^{(2)}(2x) \right)}{1 - 2 \frac{neB}{D_{\parallel}}}
\end{aligned}
$$

where $\theta_{n-k}$ is the Heaviside step function

$$ \theta_{n-k} = \begin{cases} 1, & n \ge k \\ 0, & \text{otherwise} \end{cases} \qquad (7.50) $$

Using these expressions, we evaluate

$$ \mathcal{D}(k) = \mathcal{A}_3^2 \mathbf{k}_{\perp}^2 - (\mathcal{A}_1^2 - \mathcal{A}_2^2) (k_{\parallel}^2 - m^2), \quad (7.51) $$

and finally calculate $z$ and $z_3$ with Eq. (7.26a) and Eq. (7.26b), respectively. These results can be appreciated in Figs. 7.5–7.10, as a function of the energy scale $q_0/m$, as well as the magnitude of the average background magnetic field $eB/m^2$, respectively.

### 7.2.3. Ultra-intense (LLL) field $eB/m^2 \gg 1$

In the ultra-intense magnetic field regime $eB/m^2 \gg 1$, we obtain the corresponding asymptotic expression for $\mathcal{A}_1(q)$ by considering only the lowest-Landau level (LLL) $n=0$ in Eq. (7.39). Therefore, we have [298]

$$ \mathcal{A}_1(q) \sim i \frac{e^{-\mathbf{q}_{\perp}^2 / eB}}{\mathbf{q}_{\parallel}^2 - m^2}, \quad (7.52) $$

and the corresponding expressions for its derivatives are [298]

$$ \mathcal{A}_2(q) = \frac{e^{-\mathbf{q}_{\perp}^2 / eB}}{\mathbf{q}_{\parallel}^2 - m^2}, \quad (7.53) $$

and [298]

$$ \mathcal{A}_3(q) = 0. \quad (7.54) $$

Similarly, we also obtained [298]

$$ \mathcal{D}(q) = 2 \frac{e^{-2\mathbf{q}_{\perp}^2 / eB}}{\mathbf{q}_{\parallel}^2 - m^2}. \quad (7.55) $$

Finally, the integrals of $\mathcal{A}_1(q)$ are given, in this approximation, by the expressions [298]

$$ \tilde{\mathcal{A}}_1(q) = \frac{\pi^2 eB}{\sqrt{q_0^2 - m^2}}, \quad (7.56a) $$
$$ \tilde{A}_2(q_0) = \frac{\pi}{\sqrt{q_0^2 - m^2}}. \qquad (7.56b) $$

Applying these asymptotic results for the ultra-strong field regime, and substituting into the general definitions Eq. (7.26a) and Eq. (7.26b), we obtain explicit analytical expressions for the renormalization factors z and $z_3$, respectively, as follows [298]

$$ z = 1 + \frac{3 \Delta(eB) e^{-q_{\perp}^2 / eB}}{4 \pi \sqrt{q_0^2 - m^2}}, \qquad (7.57) $$

and

$$ \begin{aligned} z_3 &= \left( 1 + \frac{\Delta(eB)e^{-q_{\perp}^2/eB}}{4\pi\sqrt{q_0^2 - m^2}} \right) z^{-1} \\ &= \frac{1 + \frac{\Delta(eB)e^{-q_{\perp}^2/(eB)}}{4\pi\sqrt{q_0^2 - m^2}}}{1 + \frac{3}{4} \frac{\Delta(eB)e^{-q_{\perp}^2/(eB)}}{\pi\sqrt{q_0^2 - m^2}}} \end{aligned} \qquad (7.58) $$

Remarkably, while for very large magnetic fields $z \sim eB/m^2$ grows linearly, $z_3$ instead converges asymptotically to the constant, field-independent limit

$$ \lim_{\frac{eB}{m^2} \to \infty} z_3 = \frac{1}{3}. \qquad (7.59) $$

<figure><img src="image_28.png" /><figcaption>Figure 7.5: Wavefunction renormalization factor z as a function of the dimensionless energy scale $q_0/m$. Here $q_3 = 0$.</figcaption></figure>

## 7.3. Results

In Figs. 7.5–7.8, we display the behavior of the renormalization factors z and $z_3$ as a function of the energy scale $q_0/m$, and the average background magnetic field $eB/m^2$, respectively, in the whole range of both parameters. As shown in Fig. 7.5, z presents a monotonically decreasing behaviour as a function of the energy $q_0/m$, that asymptotically reaches the limit $z \to 1$ as $q_0/m \gg 1$, for all values of the average background magnetic field $eB/m^2$. The physical interpretation is that the quasi-particle renormalization due to the random magnetic field fluctuations tends to be negligible as the
<figure><img src="image_29.png" /><figcaption>**Figure 7.6:** Wavefunction renormalization factor $z$ as a function of the dimensionless magnetic field scale $eB/m^2$. Here $q_3 = 0$, and $eB/m^2 \in [10, 500]$.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>**Figure 7.7:** Charge renormalization factor $z_3$ as a function of the dimensionless energy scale $q_0/m$. Here $q_3 = 0$.</figcaption></figure>

energy of the propagating fermions becomes very large, but in contrast it can be quite strong at low energy scales. This trend is also consistent with the effective refraction index $v'/c = z^{-1}$, as shown in Figs. 7.9,7.10 for low energy scales, $v'/c < 1$, indicating a strong renormalization of the effective group velocity of the propagating quasi-particles due to the presence of the magnetic background fluctuations. In contrast, for larger energies the renormalization is weaker, thus recovering the asymptotic limit $v'/c \to 1$ as $q_0/m \gg 1$. Since low energy and momentum components in the Fourier representation of the propagator correspond to long-wavelength components in configuration space, our results indicate that such long-wavelength components are more sensitive to the spatial distribution of the magnetic fluctuations, and hence experience a higher degree of decoherence, thus reducing the corresponding group velocity. In contrast, the high-
<figure><img src="image_31.png" /><figcaption>**Figure 7.8:** Charge renormalization factor $z_3$ as a function of the dimensionless magnetic field scale $eB/m^2$. Here $q_3 = 0$, and $eB/m^2 \in [10, 500]$</figcaption></figure>

energy Fourier modes of the propagator, corresponding to short-wavelength components in the configuration space, are less sensitive to the presence of spatial fluctuations of the background magnetic field.

Concerning the charge renormalization factor $z_3$ displayed in Fig.7.7, it experiences a strong effect $z_3 < 1$ at low quasi-particle energies $q_0/m$, but in contrast this effect becomes negligible a large energy scales $q_0/m \gg 1$, since $z_3 \to 1$ as an asymptotic limit. In physical terms, we interpret this as a charge screening due to the spatial magnetic fluctuations in the background, consistent with the aforementioned interpretation for the effective index of refraction as a function of energy. On the other hand, as displayed in Fig. 7.8, $z_3$ decreases as a function of the average background magnetic field intensity, thus achieving an asymptotic limit $z_3 \to 1/3$ as shown in Eq. (7.59).

## 7.4. Conclusions

In this section, we have studied the effects of quenched, white noise spatial fluctuations in an otherwise uniform background magnetic field, over the properties of the QED fermion propagator. This configuration is important in different physical scenarios, including heavy-ion collisions and the quark-gluon plasma, where spatial anisotropies of the background magnetic field may be present. Upon averaging over magnetic fluctuations using the replica formalism, we obtained an effective action in the replica fields, leading to an effective particle-particle interaction proportional to the strength $\Delta$ of the spatial auto-correlation function of the background noise. Our perturbative results show that, up to first order in $\Delta$, the propagator retains its form, thus representing renormalized quasi-particles with the same mass $m' = m$, but propagating in the medium with a magnetic field and noise-dependent index of refraction $v'/c = z^{-1}$, and effective charge $e' = z_3e$, where $z$ and $z_3$ are renormalization factors. We showed that $z$ presents a monotonically decreasing behaviour as a function of the energy $q_0/m$, that reaches the asymptotic limit $z \to 1$ as $q_0/m \gg 1$, for all values of the average background magnetic field $eB/m^2$. In physical terms, this shows that the quasi-particle renormalization due to the random magnetic field fluctuations, while being significant at low energy scales, tends to be negligible as the energy of the propagating fermions becomes very large. This trend is also observed in the effective refraction index $v'/c = z^{-1}$,
<figure><img src="image_32.png" /><figcaption>**Figure 7.9:** Effective refraction index $v'/c$ as a function of the dimensionless energy scale $q_0/m$. Here $q_3 = 0$.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>**Figure 7.10:** Effective refraction index $v'/c$ as a function of the dimensionless magnetic field scale $eB/m^2$. Here $q_3 = 0$, and $eB/m^2 \in [10, 500]$.</figcaption></figure>