The set of parameters can change depending on the vacuum values of quark condensate $\langle \bar{\psi}\psi \rangle$, the pion mass $m_\pi$ and pion decay constant $f_\pi$ [198]. However, the general qualitative behavior is preserved.

The quark AMM for each flavor is determined to match the experimental values of the magnetic moments of the proton and neutron [730]. Then, one can obtain two different sets of parameters, which depends on $k_f$, defined in Eq. (29.2) by $a_f = q_f \alpha_f \mu_B \rightarrow a_f = q_f e k_f$. For the set 1, we can consider the higher values of AMM as

$$ \begin{aligned} k_u^{[1]} &= 0.29016 \text{ GeV}^{-1}, & k_d^{[1]} &= 0.35986 \text{ GeV}^{-1}, \\ \alpha_u^{[1]} &= 0.242, & \alpha_d^{[1]} &= 0.304, \end{aligned} \qquad (29.19) $$

while the second set 2 is given by

$$ \begin{aligned} k_u^{[2]} &= 0.00995 \text{ GeV}^{-1}, & k_d^{[2]} &= 0.07975 \text{ GeV}^{-1}, \\ \alpha_u^{[2]} &= 0.006, & \alpha_d^{[2]} &= 0.056. \end{aligned} \qquad (29.20) $$

The set 2 of quark AMM values induces almost no difference when compared to the zero AMM case for the effective quark masses as a function of the magnetic field in which the magnetic catalysis effect is observed [753]. The main ideas behind the application of FF regularization with quark AMM are represented in Figure 29.1. The first aspect we want to evidence is the nonphysical oscillations observed in the effective quark masses, which are present due the non-MFIR regularizations, already observed in the zero AMM case [131]. The second aspect concerns to phase transitions. For the set 1, the FF₁ regularization induces a possible first-order phase transitions close to $eB \sim 0.6 \text{ GeV}^2$, while the FF₂ the transition occurs when $eB \sim 0.4 \text{ GeV}^2$ [730, 740]. Therefore, the basic aspects with quark AMM in the two-flavor NJL model are in qualitative agreement between the two form-factor regularizations explored in this work. Nevertheless, first-order phase transitions at low temperatures are predicted to happen at very strong magnetic fields, e.g., $eB_c \sim 8 \text{ GeV}^2$ [203], which is far beyond the validity range of the model, since $eB_c \gg \Lambda^2$. In this way, we believe that the evaluations with FF regularization could not be allowing to reproduce well-known aspects of the magnetized QCD phase diagram and further developments are necessary in order to improve the applications with FF and quark AMM.

<figure><img src="image_91.png" /><figcaption>Figure 29.1: Effective quark masses as a function of the magnetic field for the two sets of quark AMM with the form-factor regularizations FF₁ and FF₂.</figcaption></figure>
### 29.4.2. MFIR/VMR procedures

In the MFIR/VMR prescription, some seminal results have been obtained in Refs. [200, 747], in which the quark AMM effect has less prominent results when compared to the FF case. We adopt two sets of parameters for the VMR scheme with 3-D sharp cutoff in Ref. [200] and Proper-Time Ref. [747], given in Table 2.

<table><caption>Table 2: In VMR₁ we follow Ref. ([200]). In VMR₂ we follow Ref. ([747]).</caption><thead><tr><th>Regularization</th><th>Λ[MeV]</th><th>GΛ<sup>2</sup></th><th>m<sub>0</sub>[MeV]</th></tr></thead><tbody><tr><td>VMR<sub>1</sub></td><td>591.60</td><td>2.404</td><td>5.723</td></tr><tr><td>VMR<sub>2</sub></td><td>886.62</td><td>4.001</td><td>7.383</td></tr></tbody></table>

The first interesting result is that we can observe no oscillations in the quark condensate as a function of the magnetic field for zero temperatures when considering both sets of AMM parameters as observed in the left plot of Figure 29.2. This is a signature of MFIR procedures as explored in detail in Ref. [131]. One can verify that there is no first-order phase transitions in the region of magnetic fields explored [200]. In what concerns to inverse magnetic catalysis, we can look to the right plot of Figure 29.2. For the set 2, there is almost no difference with the zero AMM case, in which one obtains the usual magnetic catalysis effect. For the set 1, with sizable quark AMM, a small window in which the pseudocritical temperature decreases with the magnetic field is observed [200].

<figure><img src="image_92.png" /><figcaption>**Figure 29.2:** Left: Effective quark masses as a function of the magnetic field for different values of quark AMM. Right: Pseudocritical temperature as a function of the magnetic field for different values of quark AMM. We have defined the zero AMM case as $κ^{[0]}$.</figcaption></figure>

As it is discussed in Ref. [747], in FF regularization there are due mass-dependent (MD) terms in the ultraviolet limit of the model, as described in Eq. (29.13). These terms could change the gap equation, $\partial\Omega/\partial M = 0$, and, therefore, induce artificial first-order phase transitions. One way to circumvent this problem is to apply the VMR prescription. However, it is not easy to treat the regularization of these mass-dependent terms in a general way. In fact, in the context of effective QED, the Heisenberg-Weisskopf lagrangian has the Schwinger ansatz, which is valid in the region where $eB/m_e^2 \ll 1$, with $m_e$ being the electron mass. In our case, therefore, the limit is given by $eB/M_0^2 \ll 1$, which is the first term, $n = 0$, in the series expansion Eq. (29.12). When considering this situation, we obtain a mass-independent (MI) regularization, given by Eq. (29.15) by using the $F_f^0(\tau)$ function given in Eq. (29.17), and the resulting effective potential is analogous with that obtained in Ref. [752], but applied in the context of the two-flavor NJL model [200]. The full analysis is given in Ref. [747] and in the Figure 29.3 we have the comparison between the thermodynamic potential, Eq. (29.5), with MD
(left) regularization and MI regularization (right) scenarios at $eB = 0.3\text{GeV}^2$.

<figure><img src="image_93.png" /><figcaption>**Figure 29.3:** Effective potential as a function of the effective quark mass at $eB = 0.3\text{GeV}^2$. Left: Mass-dependent regularization. Right: Mass-independent regularization.</figcaption></figure>

There are some attempts similar to the VMR/MFIR procedures that are based in the subtraction of divergences, as explored in Ref. [733, 734]. Recently, the VMR/MFIR prescription has been applied to some aspects of the chiral phase transition [737], at zero temperature [743] and to investigate the anisotropy structure of magnetization in the context of Polyakov-Nambu-Jona-Lasinio model with quark AMM [749].

## 29.5. Conclusions

In this section we have analyzed the basic aspects chiral phase transitions in a magnetized two-flavor Nambu-Jona-Lasinio model with the inclusion of quark AMM. The Schwinger *ansatz* is used in order to include the quark AMM as a set of new parameters in the model, which is studied in the mean field approximation at zero temperature. Then, we explore two central groups of regularization, the MFIR/VMR procedures and the non-MFIR procedures with FF regularizations. We conclude that FF regularizations can induce non-physical results as, oscillations in the effective quark masses, first-order phase transitions and IMC at low temperatures. These effects can be avoided by the application of VMR procedures in the mass-independent regularization case.

It is possible that further studies will be necessary to understand how to apply properly the FF regulators in order to avoid possible nonphysical results. Also, some non-MFIR procedures also need more specific and detailed analysis to test the consistency of the results. Future studies can be made in order to understand the dependence of the quark AMM as a function of magnetic field, temperature and density, as well as the thermodynamics, meson masses and equations of state in the context of MFIR/VMR procedures. The quark AMM connection to the IMC phenomena is an important topic to forthcoming work.
## 30. Summary and outlook

In conclusion, this Part elucidates significant theoretical advancements in the exploration of magnetic field effects on the QCD phase diagram and dense magnetized matter. We have introduced a novel mechanism for the acceleration of proto-neutron stars through the chiral separation effect, underscoring the critical role of anisotropy in generating observable recoil effects. The examination of ring diagrams has demonstrated their essential function in modulating the chiral transition temperature and has indicated potential instances of inverse magnetic catalysis.

Furthermore, our analysis employing finite energy sum rules in dense nuclear matter has facilitated the extraction of pivotal parameters related to nucleon interactions, thereby enhancing our understanding of dense matter behavior in strong magnetic fields. The investigation into momentum diffusion coefficients for heavy quarks in a hot, magnetized medium has broadened our comprehension of quark dynamics under varying magnetic field strengths.

The implications of these findings are profound, as they bridge theoretical predictions with the physical conditions characteristic of magnetars, the primordial universe, and ultrarelativistic heavy-ion collisions. Additionally, the exploration of quark anomalous magnetic moments provides further insight into emergent phenomena within the magnetized QCD phase diagram, emphasizing the necessity for effective models to navigate the complexities of non-perturbative QCD.

Collectively, these results contribute significantly to the ongoing discourse regarding the interplay between magnetic fields and QCD, advancing our understanding of fundamental properties of matter in extreme environments.
# Neutron Stars

## 31. Introduction

In this Part, we report on selected topics, organized in eight sections, on neutron star (NS) physics for which the consideration of strong magnetic fields is essential. For magnetic fields to have a direct effect on QCD properties, i.e. the hadron structure and the equation of state (EOS) of dense nuclear matter, the scale of $\Lambda_{QCD} \sim 200$ MeV is relevant which corresponds to field strengths of $\sim 10^{18}$ G. These field strengths influence integral properties of NSs such as masses, radii and tidal deformabilities. They may explicitly occur in the general relativistic equations of hydrodynamic stability of neutron star configurations and thus modify the standard Tolman-Oppenheimer-Volkoff (TOV) equations. More subtle effects are occurring on the scale of nucleonic pairing gaps $\Delta \lesssim 1$ MeV for field strengths of the order $\sim 10^{15}$ G and influence transport phenomena like neutron star cooling. For the occurrence of these extreme field strengths, observational evidence is reported for a special class of neutron stars, the magnetars. But also for the more typical populations of young pulsars like Crab or Vela and old, recycled millisecond pulsars in the period-derivative vs. period diagram, with surface magnetic fields of $\sim 10^{12}$ G and $\sim 10^8$ G, respectively, there may prevail superstrong magnetic fields in their interiors due to a density-dependent field profile and/or collimation of field lines in superfluid or superconducting vortices.

We begin this part with Sec. 32 where the stage is set for discussing the present status of observational constraints for observational properties of NSs and their relation to properties of the equation of state of dense QCD matter described by relativistic (Walecka-type) or nonrelativistic (Skyrme-type) density functionals. These models parametrize the phenomenology of nuclear saturation properties and allow for an extrapolation to supranuclear densities and isospin asymmetry as they are relevant for the description of NS interiors in β-equilibrium with electrons and muons. These constraints concern also the symmetry energy and its slope at the nuclear saturation density. Using the one-to-one relationship between the EOS of NS matter and the mass-radius relation obtained by integrating the TOV equations one may discuss constraints on models for the EOS from a comparison to the actual status of NS mass and radius measurements. Starting from this baseline, in the following section 33 the NS matter model is extended to include heavier baryon states such as hyperons and resonances as well as the effect of a strong B-field on the EOS and thus the structure of the NS and its global properties as mass and radius when field strengths exceed $10^{17}$ G. Spherical harmonics of the B-field are considered in sect. 33, while in Sects. 34 and 35 a density-dependent profile of the B-field inside the NS are considered. In Sec. 35 the modifications of the TOV equations due to extremely strong radially and transversely oriented B-fields are presented and the possibility to see effects on gravitational wave signals in present and future laser interferometer instruments are discussed. Turning to more subtle effects, in Sec. 36 the influence of string B-fields on nuclei in the inner crust and pasta structures at the crust-core interface are considered. While modifications of integral NS properties (mass, radius, tidal deformability etc.) require extremely strong magnetic fields, phenomena like cooling and glitches as well as neutrino and electromagnetic neutron star kicks are obtained in less extreme fields. Section 37 discusses that strong B-fields destroy the superfluidity and superconductivity in the neutron and proton pairing channels which results in astrophysical implications not only for neutron star cooling, but also for pulsar precession and fast radio bursts. In Sec. 38 the topological surface effects are explained, which occur in the neutron p-wave superfluids structures of NSs. The final section, 39 considers the complex phenomenon of NS kicks that may be inherently related to the presence of
strong magnetic fields as a precondition. Two types of kicks are distinguished: the natal kicks which NS obtain at birth in a supernova due to a neutrino rocket effect and the less strong kicks of cold recycled pulsars which are subject to an electromagnetic rocket effect due to radiation bursts in off-center magnetic fields. The latter may play a role in explaining why only a fraction of millisecond pulsars (MSPs) in binaries have eccentric orbits and how isolated MSPs originate.

Future observational campaigns may provide more precise data, but may also discover more interesting compact stars that challenge the scenarios which have been developed in order to explain the rich NS phenomenology largely based on the interplay of strong and electroweak matter with extreme magnetic fields.

# 32. Effective hadronic models applied to compact stars

## 32.1. Overview

Nuclear matter can exist in various temperature and density regimes, and the equation of state (EoS) is critical for describing macroscopic nuclear and astrophysical phenomena. As a result, it is essential to improve our understanding of this concept. To study nuclear matter beyond saturation density ($ρ_0 ≈ 0.16 \text{ fm}^{-3}$) in laboratory settings, nuclear reactions are necessary. Consequently, the improvement of the EoS has been one of the studies of the heavy-ion collisions experiments at intermediate energy [754, 755, 756, 757, 758]. In recent years, the determination of bulk properties of nuclear matter such as the symmetry energy (isovector component of the EoS) received a lot of attention from researchers, propelled by advancements in multimessenger astronomy with the NICER (Neutron star Interior Composition ExploreR) [759, 760, 761, 762] and the Ligo-Vigo collaboration [763, 764, 765]. These advances have significantly improved our ability to measure the properties of neutron stars, offering new approaches to studying the matter inside these compact objects. However, the manifestation of exotic degrees of freedom in the interior of massive neutron stars, where the density exceeds several times the saturation density, appears theoretically feasible but remains an unresolved question. For example, the existence of hyperons [766, 767] and the possibility of the presence of a quark matter core [768, 769].

In finite systems, properties are very well established for stable nuclei, in which the number of protons and neutrons is not that different. With the current experiments, measurements of nuclear masses and radii of unstable heavy nuclei with large excess neutrons will help in a better understanding of the theory for finite nuclei and neutron stars crust [770, 771]. In this sense, neutron stars represent one of the densest and most intriguing objects in astrophysics, with formed from the remnants of massive stars undergoing gravitational collapse after a supernova explosion. These stellar remnants pack an extraordinary amount of mass around $2M_⊙$ [772, 773, 774, 775, 776]. They also are known to be associated with strong magnetic field, which has a maximum value, estimated by the virial theorem ≈ $10^{18}$ G [777]. Neutron stars serve as critical laboratories for nuclear physics under extreme conditions, providing insights into the behavior of matter under pressures and densities unattainable in laboratories.

Theoretically, such systems can be studied using two distinct phenomenological approaches: (i) The microscopic approach, where the description of nucleon-nucleon (NN) interaction is carried out through a potential whose parameters are adjusted based on few-nucleon nuclear physics observables, such as deuteron properties and scattering data [778, 779]. To account for nuclear medium effects, many-body formalisms such as the (nonrelativistic) (Dirac-)Brueckner-Hartree-Fock approximation [780, 781, 782, 783, 784] or variational methods [785] are applied. (ii) The macroscopic approach, where the description of nuclear matter is conducted using relativistic and non-relativistic effective models.

The relativistic mean-field models (RMF) have their simplest version given by the Walecka model [786], and other
nonlinear versions (NLW)[787]. Another class of relativistic models is the point-coupling models[788], assuming zero-range nucleon-nucleon interactions (fermions do not interact via meson exchange). Its linear version is directly derived from the Walecka model [789, 790], and the nonlinear version from NLW [791, 788, 792]. Although these models have different EoS structures compared to the Walecka model, they yield the same relevant physics for the description of nuclear matter [793, 794, 795] and neutron stars [796].

For the nonrelativistic case, a representative class of models is the Skyrme model, initially developed by T. H. R. Skyrme and further studied by D. Vautherin and D. M. Brink [797, 798]. Later, this type of model was developed with different extensions [799, 800, 801, 802]. The Skyrme interaction, initially formulated for finite nuclei and nuclear matter at saturation density, represents a low-momentum expansion of the effective two-body nucleon-nucleon (NN) interaction in momentum space. The precise boundaries of its applicability—both lower and upper limits—remain to be definitively determined. A significant aspect of the Skyrme interaction is that it incorporates certain correlation effects via its parameters. Consequently, while it is theoretically represented as a zero-range interaction in coordinate space [798], it manifests certain finite-range characteristics [803]. For more details on the behavior of these models, see [804, 805, 806].

## 32.2. Symmetry energy

The EoS used to describe matter in neutron stars is strongly influenced by symmetry energy $S$, a crucial term in the nuclear equation of state that represents the energy difference between symmetric nuclear matter (equal number of protons and neutrons) and matter with excess neutrons [807, 808].

$$S(\rho) = E(\rho, y = 0.5) - E(\rho, y = 0), \qquad (32.1)$$

where $E(\rho, y)$ is the energy per particle in asymmetric matter, $y = Z/(Z + N)$ the proton fraction, and $n$ the barionic density.

The nuclear symmetry energy, and particularly its dependence on nuclear density, has been the focus of extensive research over the past decade. This quantity provides critical insights into the isospin dependence of nuclear forces, which are significant both in the context of nuclear matter and in finite nuclei. Its quadratic form can be right as

$$S_2(\rho) = \frac{1}{8} \left. \frac{\partial^2 E(\rho, y)}{\partial y^2} \right|_{y=0.5}, \qquad (32.2)$$

The symmetry energy can be also expanded in terms of the density parameter $x = (\rho - \rho_0)/(3\rho_0)$ as

$$S_2 = E_{\text{sym},2} + L_{\text{sym},2} x + \frac{1}{2} K_{\text{sym},2} x^2 + \dots \qquad (32.3)$$

where the $E_{\text{sym},2}$, $L_{\text{sym},2}$, $K_{\text{sym},2}$ are the quadratic nuclear empirical parameters.

Several experimental constraints on the symmetry energy in finite nuclei have been established [809, 810]. The Fig. 32.1 illustrates a selection of these constraints.

In Ref. [820] the study, using 415 relativistic mean field and nonrelativistic Skyrme-type interactions, finds that the better reproduction of low-energy nuclear physics data by nuclear models only weakly impacts the global properties of canonical mass neutron stars. The experimental constraint on the symmetry energy is identified as the most effective means for reducing uncertainties in neutron star matter. The authors isolate a group of interactions ($D_{4\text{sym}}$) that exhibit a strong correlation between the symmetry energy ($E_{\text{sym},2}$) and its slope ($L_{\text{sym},2}$). The estimate found was $E_{\text{sym},2} = 31.8 \pm 0.7$ MeV and $L_{\text{sym},2} = 58.1 \pm 9.0$ MeV, indicating a significant relationship between these parameters.
<figure><img src="image_94.png" /><figcaption>**Figure 32.1:** Correlation between symmetry energy and its slope at saturation density from different constraints: “HIC” [811], “Polarizability” [812], “$\Delta r_{np}(Sn)$” (neutron skin thickness) [813], “FRDM” (constraint from the finite-range droplet mass model calculations [814]; “IAS” (isobaric analog state) [815], “IAS + $\Delta r_{np}$” [815], “Neutron Stars” (constraint obtained from a Bayesian analysis of mass and radius observations of neutron star by considering the 95% confidence values for the slope of symmetry energy) [816], “Unitary Gas” [817], and PVES (derived from the PREX-II [818] and CREX [819]). Figure extracted from Ref. [820].</figcaption></figure>

## 32.3. Neutron stars macroscopic properties

The macroscopic properties of neutron stars, including mass, radius, angular momentum, and tidal deformability, are intricately linked to nuclear data. While low-energy nuclear physics provides essential constraints, the complexities of high-density matter and the density dependence of the EoS introduce significant challenges in accurately predicting these properties [820]. The density dependence of the EoS plays a crucial role in determining the mass and radius. To evaluate these quantities it is necessary to solve the Tolman-Oppenheimer-Volkoff (TOV) equations [821, 822]. The TOV equations describe the structure of a static, spherically symmetric star in hydrostatic equilibrium. They are essential for understanding how matter behaves under extreme gravitational forces, such as those found in neutron stars.

In the description of neutron stars, the RMF model and the Skyrme-type model are widely used in the literature. The EoS for pressure and energy density for both are given, respectively, by

$$ \epsilon_{\text{RMF}} = \frac{1}{2}m_{\sigma}^{2}\sigma^{2} + \frac{A}{3}\sigma^{3} + \frac{B}{4}\sigma^{4} - \frac{1}{2}m_{\omega}^{2}\omega_{0}^{2} - \frac{C}{4}(g_{\omega}^{2}\omega_{0}^{2})^{2} + g_{\omega}\omega_{0}\rho - \frac{1}{2}m_{\rho}^{2}\bar{\rho}_{0(3)}^{2} + \frac{g_{\rho}}{2}\bar{\rho}_{0(3)}\rho_{3} - \frac{1}{2}\alpha'_{3}g_{\omega}^{2}g_{\rho}^{2}\omega_{0}^{2}\bar{\rho}_{0(3)}^{2} \\ - g_{\sigma}g_{\omega}^{2}\sigma\omega_{0}^{2}\left(\alpha_{1} + \frac{\alpha'_{1}g_{\sigma}\sigma}{2}\right) - g_{\sigma}g_{\rho}^{2}\sigma\bar{\rho}_{0(3)}^{2}\left(\alpha_{2} + \frac{\alpha'_{2}g_{\sigma}\sigma}{2}\right) + \epsilon_{\text{kin}}^{p} + \epsilon_{\text{kin}}^{n}, \quad (32.4) $$

and

$$ p_{\text{RMF}} = -\frac{1}{2}m_{\sigma}^{2}\sigma^{2} - \frac{A}{3}\sigma^{3} - \frac{B}{4}\sigma^{4} + \frac{1}{2}m_{\omega}^{2}\omega_{0}^{2} + \frac{C}{4}(g_{\omega}^{2}\omega_{0}^{2})^{2} + g_{\sigma}g_{\omega}^{2}\sigma\omega_{0}^{2}\left(\alpha_{1} + \frac{\alpha'_{1}g_{\sigma}\sigma}{2}\right) + g_{\sigma}g_{\rho}^{2}\sigma\bar{\rho}_{0(3)}^{2}\left(\alpha_{2} + \frac{\alpha'_{2}g_{\sigma}\sigma}{2}\right) \\ + \frac{1}{2}m_{\rho}^{2}\bar{\rho}_{0(3)}^{2} + \frac{1}{2}\alpha'_{3}g_{\omega}^{2}g_{\rho}^{2}\omega_{0}^{2}\bar{\rho}_{0(3)}^{2} + p_{\text{kin}}^{p} + p_{\text{kin}}^{n}, \quad (32.5) $$

with

$$ \epsilon_{\text{kin}}^{p,n} = \frac{\gamma}{2\pi^2} \int_{0}^{k_{F,p,n}} k^2 \quad \text{and} \quad p_{\text{kin}}^{p,n} = \frac{\gamma}{6\pi^2} \int_{0}^{k_{F,p,n}} \frac{k^4 dk}{(k^2 + M^{*2})^{1/2}}, (k^2 + M^{*2})^{1/2} dk \quad (32.6) $$
where the Fermi momentum for proton/neutron is given by $k_{Fp,n}$. The zero component ($\sigma, \omega_0$) and isospin space third component ($\bar{\rho}_{0(3)}$) are the expectation values of the mesons fields in the expressions above. $M^* = M_{nuc} - g_\sigma\sigma$ and the is the effective nucleon mass and $\gamma = 2$ is the degeneracy factor for asymmetric matter. The self-consistency of the model imposes to $M^*$ the condition of

$$M^* - M_{nuc} + \frac{g_\sigma^2}{m_\sigma^2}(\rho_{sp} + \rho_{sn}) - \frac{A}{m_\sigma^2}\sigma^2 - \frac{B}{m_\sigma^2}\sigma^3 = 0, \quad \text{with} \quad \rho_{sp,n} = \frac{\gamma M^*}{2\pi^2} \int_0^{k_{Fp,n}} \frac{k^2 dk}{(k^2 + M^{*2})^{1/2}} \quad (32.7)$$

and

$$\epsilon_{\text{Sky}} = \frac{3}{10M_{\text{nuc}}} \left(\frac{3\pi^2}{2}\right)^{2/3} \rho^{5/3} H_{5/3}(y) + \frac{t_0}{8} \rho^2 [2(x_0 + 2) - (2x_0 + 1)H_2(y)] \\ + \frac{1}{48} \sum_{i=1}^{3} t_{3i} \rho^{\sigma_i+2} [2(x_{3i} + 2) - (2x_{3i} + 1)H_2(y)] + \frac{3}{40} \left(\frac{3\pi^2}{2}\right)^{2/3} \rho^{8/3} [aH_{5/3}(y) + bH_{8/3}(y)], \quad (32.8)$$

with

$$a = t_1(x_1 + 2) + t_2(x_2 + 2), \quad b = \frac{1}{2}[t_2(2x_2 + 1) - t_1(2x_1 + 1)], \quad \text{and} \quad H_l(y) = 2^{l-1}[y^l + (1-y)^l]. \quad (32.9)$$

The pressure of the model as

$$p_{\text{Sky}} = \frac{1}{5M_{\text{nuc}}} \left(\frac{3\pi^2}{2}\right)^{2/3} \rho^{5/3} H_{5/3}(y) + \frac{t_0}{8} \rho^2 [2(x_0 + 2) - (2x_0 + 1)H_2(y)] \\ + \frac{1}{48} \sum_{i=1}^{3} t_{3i}(\sigma_i + 1) \rho^{\sigma_i+2} [2(x_{3i} + 2) - (2x_{3i} + 1)H_2(y)] + \frac{1}{8} \left(\frac{3\pi^2}{2}\right)^{2/3} \rho^{8/3} [aH_{5/3}(y) + bH_{8/3}(y)], \quad (32.10)$$

The nucleon rest mass $M_{nuc} = 939$ MeV is the same for both models used. A particular parametrization is defined by a specific set of the following free parameters.

In order to determine the macroscopic properties of neutron stars, a series of essential steps must be undertaken [823, 824]: (i) the combination of the EoS for hadronic matter with the EoS for free leptons; (ii) the imposition of conditions regarding charge neutrality and chemical equilibrium; (iii) the incorporation of the crust description, for example, the Baym-Pethick-Sutherland (BPS) EoS [825] for low densities into the EoS for hadrons and leptons; (iv) the utilization of the resulting EoS as an input for the TOV, which represent the differential equations governing the structure of a static, spherically symmetric star in hydrostatic equilibrium. Subsequently, the leptonic EoS only takes into account electrons and muons, as we specifically focus on the zero-temperature deleptonized phase of stellar evolution.

An example of the results for the mentioned EoSs is shown in Fig. 32.2. In this figure, were used the parametrizations SLy4 [826] and BRS8 [827] ($D_{4\text{sym}}$), are utilized for the hadronic component of the system. These specific parameterizations have undergone recent scrutiny [820], revealing their compatibility with neutron stars and the properties of finite nuclei. In the latter context [820], delved into an analysis of data on various aspects of spherical nuclei, such as ground state binding energies, charge radii, and giant monopole resonances. The nuclei studied included 16O, 34Si, 40Ca, 48Ca, 52Ca, 54Ca, 48Ni, 56Ni, 78Ni, 90Zr, 100Sn, 132Sn, and 208Pb.

It is important to mention that different studies present in the literature provide information about the contribution of the slope of the symmetry energy in the mass-radius diagram [830, 820, 809], with some additional contribution from the second-order derivative of the symmetry energy. The value of $K_{\text{sym},2}$ determines the stiffness of the EoS, and the softer the EoS, the smaller the radius and the greater the central density for a given mass of the star. To some extent, the stiffness of the EoS is determined by the value of $L_{\text{sym},2}$. The values of these quantities for these two parameterizations are indicated in Table 32.1.
<figure><img src="image_95.png" /><figcaption>**Figure 32.2:** Mass-radius relations for SLy4 and BSR2 parametrizations. The contours are related to data from the NICER mission, namely, PSR J0030+0451 [759, 760] and PSR J0740+6620 [761, 762], the GW170817 event [763, 828], and the GW190425 event [765], all of them at 90% credible level. The red (brown) horizontal lines are related to PSR J0740+6620 [776] (PSR J0952+0607 [829]). The recent observational constraint on neutron star mass, GW190814 [764], is shown as the violet horizontal lines.</figcaption></figure>

<table><caption>**Table 32.1:** Some properties for the parametrizations showed. $E_0$ (binding energy), $E_{sym,2}$, $L_{sym,2}$, and $K_{sym,2}$ are given in MeV, except for $n_0$ given in fm$^{-3}$, the maximum NS mass, $M_{max}$ is given in units of solar mass ($M_\odot$). Both radius are given in km.</caption><thead><tr><th>Parametrization</th><th>n<sub>0</sub></th><th>E<sub>0</sub></th><th>E<sub>sym,2</sub></th><th>L<sub>sym,2</sub></th><th>K<sub>sym,2</sub></th><th>M<sub>max</sub></th><th>R<sub>max</sub></th><th>R<sub>1.4</sub></th></tr></thead><tbody><tr><td>SLy4 [826]</td><td>0.160</td><td>-15.97</td><td>32.00</td><td>45.94</td><td>-119.73</td><td>2.05</td><td>10.01</td><td>11.83</td></tr><tr><td>BSR8 [827]</td><td>0.147</td><td>-16.04</td><td>31.08</td><td>60.25</td><td>-0.74</td><td>1.97</td><td>11.56</td><td>13.15</td></tr></tbody></table>

As a remark, future research should focus on refining nuclear models, incorporating more experimental data, exploring the density dependence of the energy density functional, investigating phase transitions, and utilizing advanced statistical methods to enhance the understanding of neutron star properties. These efforts could lead to more accurate predictions and a deeper understanding of the fundamental physics governing neutron stars [820]. Also, the authors from Ref. [796] call for significant efforts to address the current theoretical uncertainties and to explore the implications of their findings for various astrophysical phenomena, including the dynamics of neutron star binary mergers and the nucleosynthesis of heavy elements.

# 33. Neutron-star matter under strong magnetic fields and magnetars

## 33.1. Introduction

The description of the macroscopic structure of compact stars – including neutron stars and their various subclasses – begins at the microscopic level, with the choice of the composition of dense stellar matter to be considered and of the microscopic model to accurately characterize the behavior of these components. These two ingredients are the starting
point for obtaining the equation of state (EoS) of dense matter. The extraordinarily high densities within neutron star
lead to the creation of heavier particle species, beyond the conventional proton-neutron-electron composition assumed in
earlier models of pulsars, that are governed by some model of the strong force interactions. The calculation of the stellar
matter EoS subsequently defines the distribution of baryons and leptons, constrained by equilibrium conditions such as
β-stability and charge neutrality, while establishing relationships between energy density, pressure, and baryon density.
Going from micro to macrophysics requires applying the EoS that describes dense matter to conditions of mechanical (or
hydrostatic) equilibrium. Compact stars are characterized by their exceptionally strong gravitational fields, necessitating
the establishment of equilibrium within the framework of general relativity [831, 832].

Magnetars are a unique class of neutron stars distinguished by having the most intense stable magnetic fields known
in the nature. Surface field strengths, primarily of the poloidal component, are estimated to range from 10¹¹ to 10¹⁵ G,
with interior values surpassing these by more than an order of magnitude [833]. The magnetic field intensity in the central
regions remains uncertain but could potentially reach up to 10¹⁸ G according to the scalar virial theorem [834, 835].

The extreme magnetic field strength *B* found in magnetars impacts the formalism describing compact stars in two
significant ways. At the microscopic level, we must consider how the particles interact with the external electromagnetic
field in addition on how they interact with each other. So, the energy spectra of the particles will be altered by
their interaction with *B*. At the macroscopic level, magnetic fields exceeding ~ 10¹⁶ G, lead to deformation of the
stellar geometry away from spherical symmetry [836]. Consequently, the standard relativistic hydrostatic equations
used for describing non-magnetized stars, such as the Tolman-Oppenheimer-Volkoff (TOV) equations [837, 838], which
assume spherical symmetry in their derivation from general relativity equations, become inadequate. Addressing these
non-spherical configurations and the anisotropies introduced by magnetic fields also leads to the determination of the
magnetic field profile in the star interior, linking the surface value of *B* with the higher values expected deeper within the
star.

In the following, we summarize the formalism developed for the description of magnetars in our previous studies –
for more detailed expositions and discussions, we refer to [839], [840] and, mainly, [841]. The formalism employed in the
microscopic description of magnetized neutron-star matter is presented in Section 33.2, and the procedure of going from
the EoS to the macroscopic description of a compact star through General Relativity is discussed in Section 33.3. The
main conclusions are drawn in Section 33.4.

**33.2. Microphysics**

The high energies found in neutron star cores can yield heavier particle species beyond the conventional proton-neutron-electron mix. The spin-1/2 baryon octet has been extensively studied in literature to the point of being almost standard procedure to include them in the stellar matter composition, and recently attention has also been directed towards the spin-3/2 decuplet, also called baryon resonances – see, e.g., Refs. [842, 843, 844, 845]. The lightest members of this decuplet, the Δ baryons, are only about 30% heavier than nucleons and lighter than the heaviest spin-1/2 baryons, as shown in Tab. 33.1. The relative strengths of the coupling constants are determined by SU(3) symmetry arguments proposed in [846], which sets the complete hyperon-meson coupling scheme from a single free parameter α_v. Despite the value of α_v, hyperons are always present in the neutron-star matter and the sequence of hyperon thresholds are always the same, with an inversely proportional relationship between α_v and the stiffness of the EoS. In this work, we choose to use α_v = 0.5, which results in hyperon potential depths of U_Λ = -28 MeV, U_Σ = +21.8 MeV and U_Ξ = +35.3 MeV. On
<table><caption>**Table 33.1:** Vacuum mass, electric charge, isospin 3rd component, spin, and normalized anomalous magnetic moment of baryons considered in this work. Electric charge is shown in units of the electron charge and $\mu_N$ is the nuclear magneton.</caption><thead><tr><th></th><th>M<sub>b</sub> (MeV)</th><th>q<sub>b</sub>(e)</th><th>I<sub>3b</sub></th><th>S<sub>b</sub></th><th>&kappa;<sub>b</sub>/&mu;<sub>N</sub></th></tr></thead><tbody><tr><td>p</td><td>939</td><td>+1</td><td>+1/2</td><td>1/2</td><td>1.79</td></tr><tr><td>n</td><td>939</td><td>0</td><td>-1/2</td><td>1/2</td><td>-1.91</td></tr><tr><td>&Lambda;</td><td>1116</td><td>0</td><td>0</td><td>1/2</td><td>-0.61</td></tr><tr><td>&Sigma;<sup>+</sup></td><td>1193</td><td>+1</td><td>+1</td><td>1/2</td><td>1.67</td></tr><tr><td>&Sigma;<sup>0</sup></td><td>1193</td><td>0</td><td>0</td><td>1/2</td><td>1.61</td></tr><tr><td>&Sigma;<sup>-</sup></td><td>1193</td><td>-1</td><td>-1</td><td>1/2</td><td>-0.37</td></tr><tr><td>&Xi;<sup>0</sup></td><td>1315</td><td>0</td><td>+1/2</td><td>1/2</td><td>-1.25</td></tr><tr><td>&Xi;<sup>-</sup></td><td>1315</td><td>-1</td><td>-1/2</td><td>1/2</td><td>0.06</td></tr><tr><td>&Delta;<sup>++</sup></td><td>1232</td><td>+2</td><td>+3/2</td><td>3/2</td><td>3.47</td></tr><tr><td>&Delta;<sup>+</sup></td><td>1232</td><td>+1</td><td>+1/2</td><td>3/2</td><td>1.73</td></tr><tr><td>&Delta;<sup>0</sup></td><td>1232</td><td>0</td><td>-1/2</td><td>3/2</td><td>0.06</td></tr><tr><td>&Delta;<sup>-</sup></td><td>1232</td><td>-1</td><td>-3/2</td><td>3/2</td><td>-1.69</td></tr></tbody></table>

the other hand, the $\Delta$ baryon couplings are less constrained due to our limited knowledge about their interactions. From available literature, they are anticipated to exhibit an attractive potential of approximately 2/3 to 1 times the potential of nucleons [847, 848]. We analyse the scenarios with $x_{\sigma\Delta} = x_{\omega\Delta} = 1.0$ and $x_{\sigma\Delta} = x_{\omega\Delta} = 1.2$, keeping $x_{\rho\Delta} = 1.0$, that generates, respectively, potentials $U_{\Delta} = -66.25$ MeV (equal to the nucleon potential) and $-79.50$ MeV.

Furthermore, investigating the impact of strong magnetic fields on baryon resonances is of special interest due to the possibility of them having large electric charge and additional spin projections. The effects of Landau levels in dense stellar matter containing $\Delta$ baryons were initially explored in neutron-star contexts and subsequently in $\Delta$-admixed hypernuclear stellar matter [839]. Different properties of baryons considered in this study are shown in Tab. 33.1.

To model hadronic dense matter under magnetic fields, it is necessary to propose a Lagrangian density that describes the particle interactions with each other and with the external electromagnetic field. To address the strong force interaction, we employ two relativistic models to have two distinct frameworks for comparison. The first model is a nonlinear extension of the Walecka model, where baryon interactions are mediated by mesons like $\sigma$, $\omega$, $\rho$, and $\phi$ within the mean field approximation. Specifically, we adopt the L3$\omega\rho$ parametrization [849], which includes an additional $\omega\rho$ interaction crucial for predicting the symmetry energy slope, neutron-star radii, and tidal deformabilities. The $\phi$ meson, with hidden strangeness, interacts exclusively with hyperons, addressing the hyperon puzzle and enabling the reproduction of higher maximum masses for neutron stars. The second model utilized is the chiral mean-field (CMF) model, based on a nonlinear realization of the chiral sigma model. Baryon masses are dynamically generated by the medium, allowing for chiral symmetry restoration at high temperatures or densities, consistent with lattice QCD results. Here, we focus on the hadronic version developed in Ref. [850], excluding phase transitions to quark matter. An additional $\omega\rho$ interaction is introduced to enhance agreement with symmetry energy slope, neutron-star radii, and tidal deformability data, while a higher-order $\omega^4$ interaction is included to model more massive neutron stars [851, 852].

In order to study modifications introduced by an external magnetic field $B$ on fermions, we modify the calculation of thermodynamical quantities of each (hadronic and leptonic) particle species with non-zero electric charge $q$ according to
<figure><img src="image_96.png" /><figcaption>Figure 33.1: Particle composition of neutron-star matter with $\Delta$s, with $B=0$ (top panels) and magnetic field $B = 3 \times 10^{18}$ G (bottom panels), when considering (solid lines) or disregarding (dashed lines) the effects of the AMMs. Reproduced from [841].</figcaption></figure>

the general procedure of taking

$$ \sum_{\text{spin}} \int d^3 k \rightarrow \frac{|q|B}{(2\pi)^2} \sum_{\text{spin}} \sum_n \int dk_z, \qquad (33.1) $$

where $k$ is the momentum, $z$ is the local direction of the magnetic field and $n$ is the discretized orbital angular momentum that the charged particle acquires in the plane transverse to $B$. The sum in $n$, at zero temperature, goes until a maximum (integer) corresponding to Landau level $\nu$ for which $k_z^2 \ge 0$, i.e.,

$$ \nu \le \nu_{\text{maxb}}(s) = \left[ \frac{(E_{F b}^* + s\kappa_b B)^2 - M_b^{*2}}{2|q_b|B} \right]. \qquad (33.2) $$

where $\nu = n + \frac{1}{2} - \frac{s}{2}|q|$ depends on spin and electric charge, and the total effective energy of a particle with effective mass $M_b^*$ is given by

$$ E_{F b}^{*2} = k_{F,b}^2(\nu, s) + \left( \sqrt{M_b^{*2} + 2\nu|q_b|B - s\kappa_b B} \right)^2. \qquad (33.3) $$

The number and scalar densities are defined for electrically neutral particles as

$$ n_b = \frac{1}{2\pi^2} \sum_s \left\{ \frac{k_{F b}^3(s)}{3} - \frac{s\kappa_b B}{2} \left[ (M_b^* - s\kappa_b B) k_{F b}(s) \times E_{F b}^{*2} \left( \arcsin \left( \frac{M_b^* - s\kappa_b B}{E_{F b}^*} \right) - \frac{\pi}{2} \right) \right] \right\}, \qquad (33.4) $$

and

$$ n_{sb} = \frac{M_b^*}{4\pi^2} \sum_s \left[ E_{F b}^* k_{F b}(s) - (M_b^* - s\kappa_b B)^2 \ln \left| \frac{k_{F b}(s) + E_{F b}^*}{M_b^* - s\kappa_b B} \right| \right], \qquad (33.5) $$
respectively, and, for charged particles as

$$n_b = \frac{|q_b|B}{2\pi^2} \sum_{\nu,s} k_{Fb}(\nu, s) \qquad (33.6)$$

and

$$n_{sb} = \frac{|q_b| B M_b^*}{2\pi^2} \sum_{s,\nu} \frac{\sqrt{M_b^{*2} + 2\nu|q_b|B - s\kappa_b B}}{\sqrt{M_b^{*2} + 2\nu|q_b|B}} \times \ln \left| \frac{k_{Fb}(\nu, s) + E_{Fb}^*}{\sqrt{M_b^{*2} + 2\nu|q_b|B - s\kappa_b B}} \right| \qquad (33.7)$$

The expressions for energy density and pressure are different for each model and can be obtained from the energy-momentum tensor for matter in the usual manner.

In Fig. 33.1, we present the particle composition in neutron-star matter containing $\Delta$ baryons under the influence of strong magnetic fields, considering both scenarios with and without anomalous magnetic moment (AMM) corrections. The enforcement of charge conservation and chemical equilibrium results in diverse behavior among particles, depending on the sign and strength of their AMMs. Overall, the inclusion of AMMs generally leads to an increase in charged particle populations, whereas the populations of neutral particles tend to decrease.

### 33.3. Macrophysics

In addressing the influence of magnetic field effects on the mechanical equilibrium of compact stars, several considerations arise due to deviations from ideal conditions induced by $B$. The TOV equations of relativistic hydrostatic equilibrium are derived imposing isotropic matter and spherical symmetry, both assumptions broken by extreme magnetic fields. Various approaches have been suggested in the literature to address the effects of magnetic fields on compact stars. These include disregarding the anisotropy altogether [839, 840], employing perturbative solutions applicable only to weak magnetic fields [853], which might not influence matter properties significantly, or assuming chaotic distributions of magnetic fields [854].

Also, magnetars exhibit $B = 10^{15}$ G in the star surface, escalating to approximately $10^{18}$ G in its core. Several studies try to delineate the magnetic field profile interpolating these extremes, typically involving the *ad hoc* proposition of profiles for the magnetic field within the TOV framework. Typically, these ansatz are some variation on the parameterization introduced by Ref. [855] in the late 90s. However, [856] highlighted that this kind of magnetic field profiles are physically inaccurate because they do not adhere to Maxwell's equations, not preserving the cobdition $\nabla \cdot \mathbf{B} = 0$. Assuming such profiles in a spherically symmetric star implies a purely monopolar distribution of the magnetic vector field.

Hence, the usual TOV equations are unsuitable for describing magnetars due to their assumption of spherical symmetry, which does not hold under strong magnetic fields. These fields cause highly deformed stellar shapes, requiring the stellar structure to be determined by solving the Einstein equations together with the Maxwell equations. To compute the effect of the strong magnetic fields on the structure of the magnetars, then, we solve the Einstein-Maxwell equations within the numerical library LORENE 15 using a multi-domain spectral method.

To explore thoroughly effects of Landau quantization and AMMs on the star macroscopic properties, one must solve these equations with a magnetic field-dependent EoS. Refs. [857, 858] used this approach with magnetic field-dependent quark EoS but found that the maximum neutron star mass is minimally affected by magnetic fields. Therefore, in this

<footnote n="15">15 http://www.lorene.obspm.fr</footnote>
<figure><img src="image_97.png" /><figcaption>**Figure 33.2:** Magnetic field distribution inside a neutron star of mass $1.8 M_{\odot}$ and central magnetic field of $B = 5 \times 10^{17}$ G. Solid, dashed, dashed-dotted and dotted are $l = 0, 2, 4, 6$. Reproduced from [841].</figcaption></figure>

section of the study, we assume a non-magnetic matter contribution to the EoS, incorporating the magnetic field only through the pure electromagnetic field. However, these effects are relevant for the microscopic properties of matter, as discussed in the previous section.

The numerical routine employed in the solution of the Einstein-Maxwell equations decompose the magnetic field norm in terms of spherical harmonics,

$$B(r, \theta) \approx \sum B_l(r) Y_l^0(\theta). \qquad (33.8)$$

In Fig. 33.2, we plot the first four even multipoles ($l = 0, 2, 4, 6$) as function of coordinate radius for both the EoS models and coupling strengths. We also plot the profile of the dominant monopolar, spherically symmetric, term ($l = 0$) inside the star in Fig. 33.3.

In Fig. 33.4, we show the mass radius relations as a function of equatorial radius for a given stellar central magnetic field. The differences between the mass-radius curves for the same B case (the same line dash pattern) arise from variations in the non-magnetic EoS, while differences between the same EoS with different magnetic fields (the same line color) result from the pure electromagnetic field contributions. The Lorentz force from the electromagnetic field affects the low-density part of the EoS, which is why the maximum mass of very massive stars remains unchanged with increasing magnetic field strength, while the mass and radius of less massive stars increase significantly. We observe that as the radius of the neutron star remains fixed, increasing the strength of the central magnetic field raises both the central baryon and energy densities. This occurs because greater matter pressure is needed to counterbalance the Lorentz force.
<figure><img src="image_98.png" /><figcaption>**Figure 33.3:** Magnetic field distribution inside a neutron star of mass $1.8M_{\odot}$ and central magnetic field of $B = 5 \times 10^{17}$ G. Solid, dashed and dotted are the $l = 0$ term at the $\theta = 0, \pi/4, \pi/2$ orientations. Reproduced from [841].</figcaption></figure>

## 33.4. Conclusion

This research aimed to shed light on the intricate interplay between strong magnetic fields, exotic particles like hyperons and baryon resonances, and the overall properties of magnetars. By investigating the microphysics stellar hadronic matter under extreme magnetic conditions, we highlighted significant alterations in the energy spectra and particle population distributions. These effects are crucial for understanding the composition and behavior of neutron star interiors under magnetic fields that can exceed $10^{18}$ G.

At the macroscopic level, we addressed the challenges posed by strong magnetic fields to the conventional models of compact stars. The spherical symmetry assumptions underlying traditional models like the TOV equations are no longer valid under such conditions, necessitating the solution of the Einstein and Maxwell equations together. This approach reveals how magnetic fields reshape the internal structure and macroscopic properties of magnetars, influencing their magnetic field profiles and mass-radius relations.

In conclusion, the comprehensive analysis presented in this paper enhances our understanding of magnetars as natural laboratories for testing the limits of matter under extreme magnetic fields. Future research directions may explore additional complexities introduced by Landau quantization and AMMs, providing further insights into the interplay between magnetic fields, exotic particles, and the overall properties of neutron stars.
<figure><img src="image_99.png" /><figcaption>**Figure 33.4:** Stellar mass as a function of equatorial radius for different compositions and interaction strengths, for central magnetic fields $B = 0$ (solid lines), $B = 5 \times 10^{17}$ G (dashed lines), and $B = 10^{18}$ G (dotted lines). Reproduced from [841].</figcaption></figure>

## 34. Hyperons in Magnetized Neutron Stars

Anomalous X-ray pulsars and soft γ-ray repeaters are described as magnetized neutron stars with a surface magnetic field of $\sim 10^{14} - 10^{15}$ G [859, 860, 861]. This type of compact stars are identified as magnetars, that is, neutron stars with magnetic fields much larger than the canonical surface dipole magnetic fields $B \sim 10^{12} - 10^{13}$ G of the bulk of the pulsar population [862, 863]. Early works on dense matter in magnetars found out that the equation of state (EoS) of dense nuclear matter will be modified by magnetic fields larger than $B/B_c^e = 10^5$, with $B_c^e = 4.414 \times 10^{13}$ G defined as the critical magnetic field at which the electron cyclotron energy is equal to the electron mass [864, 865, 866, 867, 868, 869, 870].

The study of the effects of very strong magnetic fields upon the EoS of matter made of hyperons was initiated in Ref. [871] and has been addressed later on by several groups [872, 873, 874, 875, 876, 877, 878, 879, 880, 881] within different relativistic mean field (RMF) theories, where baryons interact through the exchange of mesons, providing a covariant description of the EoS.

Within these approaches, the introduction of magnetic fields induces some important modifications in the lagrangian density. Under the assumption of minimal coupling of electromagnetic fields, the covariant derivative becomes $D_\mu = \partial_\mu + iqA_\mu$, where $A_\mu$ is the electromagnetic vector potential and $q$ is the charge of the particle. Also, the coupling of the particles to the electromagnetic field tensor via the baryon anomalous magnetic moments could be considered, as done in Refs. [872, 879]. Moreover, strong magnetic fields modify the orbits of the charged particles (leptons and charged baryons) in the direction perpendicular to the field becoming Landau quantized. Therefore, the single-particle energies of the charged baryons and leptons as well as the scalar and vector densities have to be modified accordingly, where
the integration in the orthogonal to the B-field plane must be replaced by a summation over the Landau levels up to a maximum value (for a review see Ref. [881] and references therein).

With all these modifications, the EoS for hyperonic matter together with the global properties of neutron stars, such as mass and radius, can be obtained under strong magnetic fields. In order to illustrate the behaviour of hyperonic matter in magnetized neutron stars, we consider the approach of Ref. [875]. In Ref. [875] two slightly different parametrizations for baryonic matter within the RMF approach were developed, the FSU2R and FSU2H. The FSU2R was fitted to nuclear matter and finite nuclei properties while fulfilling some restrictions on high-density matter deduced from heavy-ion collisions as well as reproducing $2M_\odot$ observations and radii below 13 km for pure nucleonic matter, whereas the FSU2H emerged by slightly refitting the parameters of the FSU2R so that FSU2H fulfills the $2M_\odot$ limit with hyperons. These parametrizations were improved in Refs. [882, 883] and extended to finite temperature in Refs. [884, 885].

The effects of strong magnetic fields were introduced in Ref. [875] by considering the following density dependent magnetic field profile

$$B(n) = B_s + B_c \{1 - \exp[-\beta(n/n_0)^\gamma]\}, \quad (34.1)$$

with $n$ the baryonic density, following Ref. [864]. The surface magnetic field value of $B_s = 10^{15}$ G was taken, which is consistent with the surface magnetic fields of observed magnetars [859, 860, 861] and a strong core magnetic field value of $B_c = 2 \times 10^{18}$ G. The parameter $\beta$ determines the density where the magnetic field saturates, whereas $\gamma$ gives the steepness of the transition from the surface to the core field. Changes of these parameters were allowed, although the magnetic field configuration with $\beta = 0.0065$ and $\gamma = 3.5$ was the preferred one, as the magnetic field has practically saturated to its maximum value at the densities inside the core ($n \sim 5 - 6n_0$, with $n_0$ the nuclear saturation density). With this magnetic field profile, the Tolman-Oppenheimer-Volkov (TOV) system was solved in order to determine the mass and radius of magnetized neutron stars.

In Fig. 34.1 the particle fractions for the FSU2H parametrization are shown for neutron star matter as functions of the baryon density. The upper panel displays the fractions in the absence of magnetic field, whereas the other panels include the density dependent magnetic field profile previously described for three different values for $\beta$ and $\gamma$. Landau oscillations are seen in the charged particle fractions when a magnetic field is applied, reflecting the successive filling of the Landau levels. For a given density, the smaller the magnetic field, the more Landau level can be accommodated and, hence, the more oscillations observed. As density increases, the magnetic field increases for all cases, and this leads to the need of only one Landau level to fit in the population of the charged particles. Thus, the oscillations smoothen out and disappear with increasing density. Also, it is observed that the magnetic field affects the charged particles mostly, increasing their population with respect the $B=0$ case. At low and intermediate densities up to $n \sim 4n_0$, an increase in the occupation of negatively charged electrons and muons is clearly seen, which delays the appearance of the negatively charged hyperons. Therefore, it can be concluded that hyperonic magnetars re-leptonize and de-hyperonize with respect to zero-field stars. At the same time, the proton abundance increases substantially, which might facilitate direct Urca processes and affect the cooling evolution of a magnetized neutron star.

With regards to the mass and radius of highly-magnetized neutron stars that contain hyperons, in Fig. 34.2 the mass and radius are displayed for the FSU2R parametrization (left panel) and the FSU2H one (right panel), both considering only nucleons as well as nucleons and hyperons. Whereas the solid lines correspond to vanishing magnetic field, the dashed lines include the effects of the magnetic field configuration with $\beta = 0.0065$ and $\gamma = 3.5$. The thin black lines indicate nucleonic neutron stars, denoted by nuc, and the thick red lines correspond to the hyperonic stars, indicated by
<figure><img src="image_100.png" /><figcaption>**Figure 34.1:** Particle fractions as functions of the baryonic density for the FSU2H without magnetic field (upper panel) and including different magnetic field configurations (second, third and fourth lower panels), as described in the text. Figure taken from [875].</figcaption></figure>

*hyp*. The inclusion of magnetic field produces stars with larger maximum masses, due to the increase in the total pressure. This enhancement is larger for the hyperonic stars than for the nucleonic ones, which is essentially due to the additional effect of de-hyperonization that takes place in the presence of a magnetic field. The reduction of hyperons is responsible for enhancing the value of the pressure, since the Fermi contributions of the other species are larger than in the *B* = 0 case. It is observed, however, that large magnetic fields do not lead to large changes in the mass (or the radius).
<figure><img src="image_101.png" /><figcaption>**Figure 34.2:** Mass-radius relationship of neutron stars for FSU2R (left panel) and FSU2H (right panel), with (thick red lines) or without (thin black lines) hyperons, and without (solid lines) or with (dashed lines) a magnetic field with the configuration described in the text. The two shaded bands indicate the masses $M = 1.97 \pm 0.04M_{\odot}$ in the pulsar PSR J1614-2230 (grey band) [886] and $M = 2.01 \pm 0.04M_{\odot}$ in the pulsar PSR J0348+0432 (turquoise band) [887]. Figure taken from [875].</figcaption></figure>

We note that the work of Ref. [875] as well as early works on hyperonic matter under magnetic fields [872, 873, 874] assumed ad hoc profiles of the magnetic field within the TOV system (that results from solving the Einstein equations for a spherically non-rotating neutron star) without solving Maxwell’s equations. However, it is nowadays clear the need of solving the coupled Einstein-Maxwell field equations in order to determine stable anisotropic magnetic-star configurations [888, 878, 879, 880]. Nevertheless, the general conclusions presented here regarding the behaviour of hyperons under strong magnetic fields are expected to remain unchanged.

# 35. Massive, magnetized neutron stars

## 35.1. Introduction

Compact stars, particularly neutron stars (NSs) and white dwarfs are incredibly rich objects from a theoretical perspective, integrating various fields of physics. One of the most fundamental results of theoretical astrophysics involving compact stars is the Chandrasekhar limit [889], which defines the maximum mass of a white dwarf. This concept synthesizes astrophysics, gravity and quantum statistics in one single formula. However, despite over five decades since the detection of the first NS, no analogous limit has been established for NSs due to the extreme densities at their cores, which can far exceed the nuclear saturation density. Understanding the behavior of matter at these high densities remains a significant challenge.

NSs serve as unique laboratories in the universe where such extreme densities are found. The exact behavior of matter under these conditions is still unknown, which multiple candidate equation(s) of state (EOS) proposed for the NS interior. These EOS are constrained by both nulcear physics and astrophysical obervations, making the study of NS structure vital for unraveling the mysteries of supranuclear matter.

Recent observations have made the question of the maximum mass of an NS extremely interesting. Specifically, the
discovery of “mass gap” candidates through gravitational waves (GW) and other means has intensified interest in this topic. The lower mass gap, an observational range roughly between $2.5 - 5M_\odot$, was previously void of observed objects. However, events like GW190814 [890], which detected the merger of a $22.2 - 24.3M_\odot$ black hole with an object precisely in this mass gap ($2.5 - 2.67M_\odot$), challenged this notion. Other GW observations, such as GW200210_092254 [891] and recent pulsar timing results [892] further support the existence of mass gap objects.

Thus, the above-mentioned observations suggest that the mass gap is not a theoretical inevitability, prompting the investigation of whether these objects are massive NSs or light black holes. Thus, the critical question arises: How massive can a NS be?

In this section, we explore the possibility that massive NSs could populate the mass gap. It turns out that pure EOS effects might not suffice to support mass gap NSs due to softening effects at high densities. Therefore, we investigate additional physics, specifically the roles of magnetic fields and anisotropy, in enhancing the masses of NSs.

Magnetic fields of compact stars represent another rich area of study with many open questions. It has been shown that strong magnetic fields can significantly affect the structure of NSs. For example, in white dwarfs, they might even be responsible for violation of the Chandrasekhar limit [893]. Observationally, NSs are known to have significant magnetic fields, with typical surface fields ranging from $10^8 - 10^{13}$ G. Magnetars, a subset of NSs with extreme magnetic fields, constitute about 10% of the NS population, with surface fields up to $10^{15}$ G and potentially higher central fields [894, 895]. These magnetic fields modify the structure of NSs through magnetic pressure and stress. Additionally, for high enough fields, the magnetic fields can alter the microphysics of the system via Landau quantization. However, this requires quite high fields, higher than $3 \times 10^{18}$ G [896] for hyperonic matter. We find that magnetized NSs with strong fields ($\simeq 10^{18}$ in the core) can serve as candidates to explain mass gap objects. We put bounds on their magnetic to gravitational energy ratio ($E_{mag}/E_{grav}$) to ensure the physicality of our results.

Another recent constraint on the NS EOS, coming from GW observations, is that of the tidal deformability, which measures how NS matter deforms in an external gravitational field. The tidal deformability is essentially the ratio of the quadrupole moment ($Q_{ij}$) developed in an external field to that external field $\epsilon_{ij}$. Thus, $Q_{ij} = -\lambda\epsilon_{ij}$, where $\lambda$ is the tidal deformability of the star. In dimensionless form, $\Lambda = \lambda/M^5 = (2/3)k_2C^{-5}$, where $C = M/R$ is the compactness of the star. Tidal deformability can further be linked to the dimensionless second love number $k_2$ arising from multipole expansion as shown in previous work [897]. Constraints on $\Lambda$ for a $1.4M_\odot$ star have been computed from GW events GW170817 ($\Lambda_{1.4} < 580$ and $\Lambda_{1.4} < 800$), and GW190814 ($458 < \Lambda_{1.4} < 889$) [898, 899, 890]. However, these constraints are not all independent. In fact the constraint from GW190814 was obtained by re-weighting the spectral EOS distribution from GW170817 with the probability that the maximum mass aligns with the mass of the secondary object observed in GW190814. Thus, the constraint obtained from GW190814 ended up favoring stiffer EOS as compared to the constraints from GW170817. Hence, tidal deformability is an evolving constraint. We require more such events to put better constraints on this quantity. We have, in the current article, ensured that our results are in line with the current constraints.

Massive, magnetized NSs have many interesting implications, including the potential to emit continuous GWs (CGWs). They also could be pulsar candidates emitting electromagnetic radiation. This paper aims to summarize the key features of these NSs and their observational implications, contributing to the broader understanding of compact star physics and the nature of mass gap objects.

This section is organized as follows. In subsection 35.2, we set up the formalism required to describe the structure of
our massive NSs. In subsection 35.3, we describe our results and their possible implications. We explore the plausible detection of massive, magnetized NSs through CGWs in subsection 35.4. We end by summarizing and providing an outlook based on our results in subsection 35.5.

## 35.2. Massive neutron star structure

We obtain the structure of massive NSs by solving the general relativistic, hydrostatic equilibrium equations, i.e., the Tolman-Oppenheimer-Volkoff (TOV) equations [900]. The isotropic TOV equations are modified due to the inclusion of magnetic fields and anisotropic effects (both from the magnetic field and general matter effects). We follow the same modification of the TOV equations outlined in previous work by our group [901, 902], given by

$$ \frac{dm}{dr} = 4\pi r^2 \left(\rho + \frac{B^2}{8\pi}\right), \quad \frac{dp_r}{dr} = \begin{cases} \frac{-(\rho+p_r)\left(\frac{4\pi r^3}{r(r-2m)}\left(\frac{p_r - \frac{B^2}{8\pi}}{\frac{d\rho}{dp_r}}\right)\right) + \frac{2}{r}\Delta}{\left(1 - \frac{d}{d\rho}\left(\frac{B^2}{8\pi}\right)\left(\frac{d\rho}{dp_r}\right)\right)} & \text{for RO} \\ \frac{-(\rho+p_r + \frac{B^2}{4\pi})\left(\frac{4\pi r^3}{r(r-2m)}\left(\frac{p_r + \frac{B^2}{8\pi}}{\frac{d\rho}{dp_r}}\right)\right) + \frac{2}{r}\Delta}{\left(1 + \frac{d}{d\rho}\left(\frac{B^2}{8\pi}\right)\left(\frac{d\rho}{dp_r}\right)\right)} & \text{for TO.} \end{cases} \quad (35.1) $$

Here, $m$ denotes the mass, $\rho$ the density, and $B$ the magnitude of the magnetic field at a given radial distance $r$ within the star. Additionally, we have considered two different orientations for our magnetic field - radially oriented (RO) and transversely oriented (TO). These can be considered as one-dimensional analogues of poloidal and toroidal geometries, respectively. Due to the presence of anisotropy, there is a difference between the pressure along the radial direction, $p_r$, and the pressure along the transverse direction, $p_t$. This difference is expressed in the effective anisotropy factor $\Delta$ defined as

$$ \Delta = p_t - p_r + \frac{B^2}{4\pi} \text{ for RO; } \Delta = p_t - p_r - \frac{B^2}{8\pi} \text{ for TO.} \quad (35.2) $$

### 35.2.1. Modified Bowers-Liang model for anisotropy

In order to close the system of equations defined above, we use the general parametric form first introduced by Bowers and Liang [903]. As done previously in our group [901, 902], we modify the Bowers-Liang form to further include the effects of the magnetic field. The anisotropy factor $\Delta$ is then given by

$$ \Delta = \kappa r^2 \frac{(\rho + p_r) \left( \rho + 3p_r - \frac{B^2}{4\pi} \right)}{1 - \frac{2m}{r}} \quad \text{for RO;} \quad \Delta = \kappa r^2 \frac{(\rho + p_r + \frac{B^2}{4\pi}) \left( \rho + 3p_r + \frac{B^2}{2\pi} \right)}{1 - \frac{2m}{r}} \quad \text{for TO.} \quad (35.3) $$

This model is derived keeping in mind the following key assumptions:

* The anisotropic force must vanish at the center, leading to the anisotropy term vanishing quadratically at the center.

* Anisotropy varies with position inside the star.

* $\Delta$ includes the effects due to local fluid anisotropy as well as the anisotropy due to the magnetic field (both its magnitude and orientation).

We restrict $\kappa$ to the range [-2/3, 2/3] [904]. This ensures the physicality of the solution, i.e., that we do not obtain a positive $dp/dr$.
### 35.2.2. Magnetic field profile

The magnetic field is introduced in the star through a density-dependent profile [905], given by

$$B(\rho) = B_s + B_0 \left[ 1 - \exp\left\{-\eta \left(\frac{\rho}{\rho_0}\right)^{\gamma}\right\} \right]. \qquad (35.4)$$

Through this profile, we obtain the magnitude of the field as a function of the density and, thus, the radius of the star. The parameters here are: $B_s$, the surface field of the star; $B_0$, $\rho_0$, which control the field at the stellar center; $\eta$ and $\gamma$ control how the field decays from the center to the surface. Within our one-dimensional formalism and for the two orientations considered, this profile is consistent with the Maxwell equations [901].

In this work, we choose $B_s$ to be $10^{15}$ G throughout. Our results are not particularly sensitive to this value, as long as $B_s$ and $B_0$ are not comparable. We also restrict ourselves to maximum central fields of $\simeq 10^{18}$ G. At such values, the field is not strong enough the affect the microphysics [896].

### 35.2.3. Equations of state

The final crucial ingredient in our depiction of massive NSs is the EOS. Terrestrial analogs for the matter at supranuclear densities found within NS cores do not exist. There are a number of proposed EOS for high-density matter, encompassing both phenomenological and microscopic approaches. In the current work, we have considered multiple phenomenological EOS. We have considered several relativistic mean field (RMF) EOS: GM1L [906], SWL [907], DD2 [908], DD-ME1 [909], and DD-ME2 [910]. These EOS are chosen because they best satisfy the constraints derived from both NS observations and the properties of nuclear matter at saturation density.

In the RMF approach, matter is modeled at the hadron level (quantum hadrodynamics). Baryon-baryon interactions present in NS matter are mediated by meson fields, which are then set to their mean values. Three such meson fields have been included in this approach: the scalar meson $\sigma$ describing attraction between baryons; the vector meson $\omega$ describing repulsion; and the isovector meson $\rho$ which is required to properly model isospin-asymmetric matter.

At the high densities found within NS cores, another source of uncertainty is the potential presence of exotic particles, i.e., particles that are not energetically favorable at the densities found in atomic nuclei. We have considered in this work, pure nucleonic ($npe\mu$) EOS, as well as hyperon and $\Delta$ admixed ($npe\mu - Y\Delta$) EOS. The inclusion of hyperonic matter is achieved through meson-hyperon couplings based on the SU(3) ESC08 model. The inclusion of $\Delta$ particles is done by including a near-universal meson-$\Delta$ coupling ($x_{i\Delta}$), with $x_{i\Delta} = g_{i\Delta}/g_{iN} = 1.2$. Here, $g$ represents the coupling constants; the subscript $\Delta$ indicates the $\Delta$ particles, $N$ the nucleons, and $i$ represents the mesons mentioned above. All EOS are shown in Fig. 35.1.

## 35.3. Results

One can first investigate the pure EOS effect in massive NSs by examining the stars constructed with $\kappa = 0$ and $B = 0$. As shown in the left panel of Fig. 35.2, all the EOS considered in this study give rise to NSs with maximum-masses larger than $2M_{\odot}$. In the pure nucleonic cases, one obtains masses that are well in the mass gap. However, the presence of hyperon softening reduces the maximum masses to values below $\simeq 2.2M_{\odot}$. Nevertheless, by examining the tidal deformability of these NSs, shown in the right panel of Fig. 35.2, we observe that the $npe\mu - Y\Delta$ EOS are more likely to meet the strictest observational constraints. Thus, the inclusion of exotic particles is well justified from both energetic and observational perspectives.
<figure><img src="image_102.png" /><figcaption>**Figure 35.1:** Different RMF EOS: Here, the upper branch denotes the *npeμ* EOS, while the lower branch represents the hyperon-Δ admixed *npeμ* – YΔ EOS.</figcaption></figure>

<figure><img src="image_103.png" /><figcaption>**Figure 35.2:** Mass-radius curves and tidal deformability for the isotropic cases. The dashed lines represent nucleonic cases, while solid lines represent the *npeμ* – YΔ cases.</figcaption></figure>

We are now thus motivated to investigate massive NSs and potential mass gap candidates by incorporating magnetic fields and/or anisotropy. To reiterate, we control the field values and their decay throughout the star using density-dependent magnetic field controlled by some parameters. However, as it is impossible to probe the magnetic field within a star, the exact magnetic field profile remains an open question.

To investigate these effects, we begin by introducing a magnetic field to the star using two different profiles/variations within the star, as shown in Fig. 35.3, with fixed $\kappa = 0.5$. Profile “1” has a broad variation with $\eta = 0.2, \gamma = 2$, whereas profile “2” has a more shallow variation with $\eta = 0.01, \gamma = 2$. We introduce the field and anisotropy while maintaining the stiffest EOS, namely DDMEX. The resulting mass-radius curves for profile “1”, considering different values of $B_0$ and various field orientations, are depicted on the left side of Fig. 35.4. Similar results for profile “2” are shown on the right side. The corresponding results are presented in Table 35.1.

Some trends emerge. We see that the presence of anisotropic effects itself can enhance the masses of these objects from their isotropic values. However, we must note that in this treatment, the anisotropic and magnetic effects are not strictly independent of each other, as one of the reasons behind anisotropy could be the presence of the magnetic field
<figure><img src="image_104.png" /><figcaption>**Figure 35.3:** Two different magnetic field profiles considered in this work.</figcaption></figure>

<figure><img src="image_105.png" /><figcaption>**Figure 35.4:** Mass-radius relations for anisotropic, magnetized NSs with the DDMEX EOS for two different profiles, with $\kappa = 0.5$.</figcaption></figure>

itself. It is also seen that the presence of a magnetic field does not necessarily increase mass. TO fields increase the mass while RO fields decrease the mass. Thus, the geometry of the field, not just its magnitude, matters. It seems that for addressing the mass gap agenda, it is the TO fields that play an important role.

Additionally, on comparing across profiles, one sees that the variation within the star also ends up playing a significant role. This is shown in Table 35.1. Although profile “1” gives us quite high maximum mass ($M_{max}$), even as high as $\simeq 2.8M_{\odot}$ in the highly TO magnetized case, the $E_{mag}/E_{grav}$ ends up being high enough that it could destabilize the star. On the other hand, for the same central field value, profile “2” gives a slightly lower maximum mass, however with a much better value for $E_{mag}/E_{grav}$.

This difference is further established when we look at the tidal deformability of these cases, as shown for profile “1” in Fig. 35.5 (left) and profile “2” in Fig. 35.5 (right). One can see that the TO magnetized stars of profile “1”, particularly the $\simeq 1.2 \times 10^{18}$ G case mentioned earlier, violates both the limits of tidal deformability set out by GW170817. However, for the similar central values of field in profile “2”, we see that even the strictest constraints of $\Lambda_{1.4}$ are comfortably met by profile “2”.

The presence of anisotropy also has interesting implications for the tidal deformability constraint. As shown in Fig. 35.6, higher degree of anisotropy ($\kappa$) leads to lower values of tidal deformability. Stiffer EOS, which might have been
<table><caption>**Table 35.1:** Numerical values for the physical parameters of magnetized, anisotropic ($\kappa = 0.5$) NSs from the DDMEX EOS.</caption><thead><tr><th colspan="4">Field profile: η = 0.2, γ = 2</th><th colspan="4">Field profile: η = 0.01, γ = 2</th></tr><tr><th>B<sub>c</sub> (10<sup>18</sup>G)</th><th>M<sub>max</sub> (M<sub>☉</sub>)</th><th>R (km)</th><th>E<sub>mag</sub>/E<sub>grav</sub></th><th>B<sub>c</sub> (10<sup>18</sup>G)</th><th>M<sub>max</sub> (M<sub>☉</sub>)</th><th>R (km)</th><th>E<sub>mag</sub>/E<sub>grav</sub></th></tr></thead><tbody><tr><td>0.9 (RO)</td><td>2.34</td><td>12.05</td><td>0.192</td><td>0.92 (RO)</td><td>2.52</td><td>13.32</td><td>0.031</td></tr><tr><td>0.6 (RO)</td><td>2.49</td><td>12.76</td><td>0.057</td><td>0.71 (RO)</td><td>2.56</td><td>13.3</td><td>0.019</td></tr><tr><td>0</td><td>2.62</td><td>13.26</td><td>-</td><td>0</td><td>2.62</td><td>13.26</td><td>-</td></tr><tr><td>0.9 (TO)</td><td>2.75</td><td>13.69</td><td>0.115</td><td>0.95 (TO)</td><td>2.67</td><td>13.22</td><td>0.035</td></tr><tr><td>1.2 (TO)</td><td>2.84</td><td>13.99</td><td>0.192</td><td>1.18 (TO)</td><td>2.69</td><td>12.18</td><td>0.056</td></tr></tbody></table>

<figure><img src="image_106.png" /><figcaption>**Figure 35.5:** Tidal deformability for anisotropic, magnetized NSs with the DDMEX EOS for two different profiles, with $\kappa = 0.5$.</figcaption></figure>

ruled out by purely isotropic analyses, may still be good candidates for the NS EOS if one considers this additional effect. This was also reported in other work, including [911].

<figure><img src="image_107.png" /><figcaption>**Figure 35.6:** Change in $\Lambda$ with change of $\kappa$.</figcaption></figure>
### 35.4. Plausible detection through continuous gravitational waves

The detection of NSs is possible in many ways, e.g., by detecting their radio pulses (pulsars), measuring the X-ray counterpart of accreting NSs, and detecting GW from a binary system. However, the detection of isolated NSs remains a challenge due to their faintness and small size. Those isolated rotating NSs, if magnetized, can emit GWs for a very long time (many years) as long as they spin fast with the spin axis misaligned with the magnetic axis, i.e. non-zero obliquity angle. Such GWs are called CGWs because they are emitted over a much longer timescale compared to the transient waves radiated by binary systems. Here, we explore the tantalizing possibility of directly detecting these isolated NSs via CGWs, which would enable the measurement of the magnetic field and provide stronger constraints on the EOS.

The essential source of any system emitting GWs is non-zero, time varying quadrupole moment. If a NS consists of high magnetic field, the magnetic pressure can deform the star. The poloidal and toroidal fields deform the star to oblate and prolate spheroids, respectively. Similarly, rotation can also deform the star, resulting in the oblate spheroid. For non-zero obliquity angle, the system generates a non-zero time-varying quadrupole moment and hence becomes a source of CGWs. The emerging GW signal strength is

$$h(t) = \frac{4G \Omega^2(t) \epsilon I_{xx}}{c^4 d} f(\chi(t), \Omega(t), t) \qquad (35.5)$$

where c is the speed of light, G is Newton's gravitational constant, χ is the obliquity angle, Ω is the angular frequency of the object, and d is the distance between the detector and the source object. The ellipticity is defined as $\epsilon = |I_{zz} - I_{xx}|/I_{xx}$, where $I_{xx}$ and $I_{zz}$ are the principal moments of inertia of the star about the x- and z-axes, respectively. The ellipticity directly depends on the EOS and the magnetic field strength. The function f has a (maximum) value of 0.0110297 for t = 0, d = 10 kpc, and χ = 3° (small angle approximation). To understand the detection viability, we need to calculate the GW signal strength and compare it with the sensitivity of current and future GW detectors in similar frequency ranges. Measuring parameters such as the mass and radius is essential for calculating h.

In the previous sections, for ease of discussion, we neglected the rotation of NSs and focused on one-dimensional calculations. However, for the discussion of CGWs, rotation is indispensable. Therefore, for the present purpose, we perform several simulations solving the Einstein-Maxwell equations using the publicly available XNS code, without restricting the analysis to one dimension. Also unlike the previous sections, we consider here a polytropic EOS of $P = K\rho^{1.95}$ with K being the polytropic constant. Based on the output of the simulation run, we calculate h. However, it is very important to note that the GW signal strength is a function of time as Ω and χ are functions of time. Ω and χ can decrease with time due to angular momentum extraction from a NS due to electromagnetic and gravitational radiation [912]. Here, we will consider a NS having a toroidally dominated internal magnetic field; hence, only gravitational radiation plays an important role in decay. The equations governing the decay are

$$\frac{d(\Omega I_{z'z'})}{dt} = -\frac{2G}{5c^5}(I_{zz} - I_{xx})^2 \Omega^5 f_1(\chi), \quad I_{z'z'} \frac{d\chi}{dt} = -\frac{12G}{5c^5}(I_{zz} - I_{xx})^2 \Omega^4 f_2(\chi), \qquad (35.6)$$

where $f_1(\chi)$ and $f_2(\chi)$ are some complicated functions of χ. Due to the surface poloidal field of the NS, there will be an extra term due to electromagnetic radiation. However, due to the dominance of the central field, the surface field hardly would change the angular momentum extraction (the case would be different if the NS is centrally poloidally dominated). Note, throughout the decays of Ω and χ, one may consider the magnetic field practically constant, as the field decay time scale is at least a thousand times larger than the other two. Hence, it is generally expected that the CGW strength emitted by the NSs just after birth will decrease with time and might become undetected by some detectors.
<figure><img src="image_108.png" /><figcaption>**Figure 35.7:** Dimensionless GW amplitude for NSs before and after $\nu$ and $\chi$ decay along with the sensitivity curves of various detectors for $B_{max} = 1.4 \times 10^{17}$ G, initial $\nu = 500$ Hz, $\chi = 3^\circ$.</figcaption></figure>

<figure><img src="image_109.png" /><figcaption>**Figure 35.8:** GW amplitude for NSs before and after $\nu$ and $\chi$ decay along with the sensitivity curves of various detectors (Left) and SNR for various detectors as a function of integration time for a toroidal magnetic field dominated NS with $B_{max} = 9.5 \times 10^{16}$ G, initial $\chi = 3^\circ$ and $\nu = 200$ Hz (Right). The orange line corresponds to $SNR_{threshold} = 12$.</figcaption></figure>

Here, we show a case in Fig. 35.7 where the NS can be detected initially, but after some years, it becomes undetectable even by Cosmic Explorer.

We may try to increase the detection possibility by calculating the integrated signal-to-noise ratio (SNR) for a year. See other work [912] for details. In some cases, such as in Fig. 35.8, it can be seen that the NS, which was not detectable by any detectors, might come into detection after some months of integrating the signal when it crosses above an SNR threshold limit of detection. Nevertheless, we have to keep in mind that the SNR carries the disadvantage of non-coherent GW search as the rotation rate and, hence, phase of GW changes with time, and hence, the SNR is some factor lower than that of coherent search. Also, in such a long time, the antenna pattern may change with time, and there might be movement of the antenna which may change SNR, making the setup even more challenging.

The decay of GW strength sets a robust upper bound in the detection timescale. Although not a single isolated NS has been detected via CGW so far with LIGO-VIRGO-KAGRA detectors, with the knowledge of the timescales, future detections can be targeted carefully with Cosmic Explorer and Einstein Telescope. If the detection is successful, one can directly estimate the central magnetic field and the EOS property, as the $\epsilon$ is a function of the magnetic field and EOS
type and is easily measurable once the GW is detected. The question remains, however, in their 'active timescale' (as long
as some detectors detect them): what is the probability of detection? We can calculate the number of NSs to be detected
in a galactic volume in 5-year observational timescales by multiplying the birthrate of the NSs (0.002 [913]) in galaxy per
year with the observational timescale (5 yr in this case). This number comes out to be 0.01, much smaller than unity.
To detect a considerable number of NSs in the possible observational time, we may need to strengthen the sensitivity
of detectors and, hence, we will be able to catch more distant sources (at least upto 500kpc). This will dramatically
change the number of detectable NSs as it increases by $d^3$, making the number of NSs to be detected greater than 10 in
observational timescale.

**35.5. Summary and outlook**

The observations of objects with masses in the lower mass gap range raises many interesting questions. The true nature
of these mass gap objects can have many implications to our understanding of compact star physics. One of the most
prominent examples of these objects is the secondary object in GW190814, which has been inferred to have a mass
$2.5 - 2.67 M_{\odot}$. As we report in the present article, we find that the masses of NSs from pure EOS are not high enough
to be in this range, mainly because of the competition from hyperon softening. Hence, we are motivated to turn to the
additional effects of mangetic fields and/or anisotropy.

In this section, we have investigated these objects by probing the possible maximum masses for a NS, under a
combination of different effects including EOS, magnetic field and anisotropy/deformation. We initially consider the
deformation effects to be sufficiently small so as to restrict ourselves to a spherically symmetric framework. On introducing
magnetic fields along with a model anisotropy, we find that there is enhancement of the masses of NSs to mass gap values.
However, this result is very much dependent on not only the magnitude of the magnetic field considered, but also the
geometry/profile that it follows within the star itself. It seems that anisotropy has an essential part to play in all this
as well. The anisotropy considered here, however, is not completely independent of the magnetic field, as we consider
anisotropy to be partly due to the magnetic field effects, along with the matter effects.

Other authors tackled the question of enhancement of maximum masses through inclusion of alternate effects. Specif-
ically for the secondary object in GW190814, the question of it being a quark/hybrid star has been investigated [914].
Some other authors considered modifications to gravity [915], or to the coupling constant arising in the RMF theories
[916, 917]. Another important effect that can significantly effect the eventual masses of NSs is rotation [918, 919].

When considering rotation, an interesting implication is the possibility of CGW emission. These waves arise due to
the non-zero, time-varying quadrupole moment generated by a deformed, rotating, magnetized massive NS. To investigate
these CGWs, simulations were set up using the XNS code with polytropic EOS. However, the plausible detection of the
CGWs is complicated by the fact that there are decays of $\Omega$ and $\chi$ on the timescale at most of a few years. This decay
leads to the decrease in GW amplitude, and leads to a strict upper bound on detection timescales for these objects.
We must consider advanced, highly sensitive future detectors, such as the Cosmic Explorer and the Einstein Telescope,
which will be capable of probing NSs at greater distances. Additionally, integrating the signal can help increase the SNR,
thereby enhancing the possibility of detection. Therefore, the study of massive NSs, facilitated by mass gap observations,
holds significant implications for both the theoretical understanding of compact objects and their potential observational
evidence.
# 36. Strong magnetic fields and the inner crust of neutron stars

## 36.1. Introduction

Neutron stars (NS) are very dense and compact objects, with a typical radius of about 10–15 km and a mass of about $1.4M_{\odot}$ [920]. They are constituted by cold catalysed stellar matter, their composition is strongly asymmetric, very neutron-rich, and their central densities can reach several nuclear saturation densities, $n_{sat} \sim 0.15 \text{ fm}^{-3}$. They are structured in layers, with a surface constituted by an $^{56}$Fe grid, where the pressure is zero. In the outer crust, we have very neutron-rich nuclei embedded in an electron sea. As the density further increases towards the interior of the star, neutrons start dripping out of the nuclei. This defines the transition to the inner crust. In this region of the star, the nucleons can form heavy clusters with different geometrical structures, termed "pasta phases" [921, 922, 923, 924, 925, 926, 927, 928, 929, 930], due to the competition between the Coulomb and the strong forces. Eventually, these structures will melt, and the core of the star starts. In there, purely homogeneous neutral matter exists, and in the very center, hyperons or even deconfined quark matter may occur.

These heavy clusters may form not only in the inner crust of neutron stars, but also in other astrophysical systems, like proto-neutron stars or neutron star mergers, that are under different thermodynamical conditions. In these kind of systems, where the temperatures can reach several MeV [931], other clusters, like deuteron or α-particles, can also form. These light clusters have also been observed in heavy-ion collisions at GANIL [932, 933] or NIMROD [934], in a range of temperatures from ~5 to ~10 MeV.

The appearance of these clusters will modify the neutrino transport, and, therefore, consequences on the dynamical evolution of supernovae and on the cooling of proto-neutron stars are expected [935]. Magnetars, in particular, may have an inner crust even more complex than non-magnetized stars [936, 937, 938, 939] due to the presence of these clusters, as we will see in the course of this paper. In Pons et al [940], it was shown that a fast decay of the magnetic field could be the reason for the absence of stars with periods of rotation higher than 12 s, and this may be related to the existence of an amorphous inner crust, i.e pasta phases. Regarding the macroscopic structure of the star, the explicit inclusion of clusters in the inner crust affects the radius of intermediate mass stars: for $1.4M_{\odot}$, calibrated relativistic mean-field (RMF) models predicted a radius of $13.6 \pm 0.3$ km with a crust thickness of $\Delta R = 1.36 \pm 0.06$ km [941].

In order to obtain the structure of the star, we need the equation of state (EoS) to solve the TOV equations. In the available literature, one can find plenty of models, like the phenomenological ones, whose parameters are fitted to finite nuclei, and are constrained by different observables, like *ab initio* calculations, experiments or observations. The relativistic mean-field models and the non-relativistic Skyrme are examples. A collection of several of these models is gathered in the public, free online repository CompOSE [942, 943]. One example of the several observational constraints available is the simultaneous measurement of the NS mass and radius by the NICER telescope. Their most recent measurements have put the radius and mass of the pulsar PSR J0740+6620 with $12.49_{-0.88}^{+1.28}$ km and $2.073_{-0.069}^{+0.069} M_{\odot}$ [944], and the brightest rotation-powered millisecond pulsar PSR J0437-4715 with $M = 1.418 \pm 0.037 M_{\odot}$ and $R = 11.36_{-0.63}^{+0.95}$ km [945].

Magnetars [946, 947, 948, 949], mainly Soft Gamma Repeaters (SGRs) and Anomalous X-ray Pulsars (AXPs), belong to a kind of neutron stars with very strong magnetic fields at the surface, up to $10^{13} \sim 10^{15}$ G [950, 951], and quite long spin periods, of the order of 2 ~ 20 s. Nowadays, about thirty of such objects have been observed [951]. Moreover, magnetars are good candidates to be a source of continuous gravitational wave emission, and we expect that in the future
it will be possible to detect this type of gravitational waves.

In this section, we will briefly introduce the formalism used to calculate the inner crust EoS under the effect of a strong external magnetic field B within a RMF framework, then we will show some of the results obtained, and finally some conclusions will be drawn.

## 36.2. Theoretical Framework

Here, we describe NS matter within the non-linear Walecka model, where the mesons are responsible for mediating the nuclear force. We include the isoscalar-scalar meson $\sigma$, the isoscalar-vector meson $\omega$ and the isovector-vector meson $\rho$. Electrons are also included to achieve electrical neutrality. An external electromagnetic field, oriented along the z-axis, is considered, $A^\mu = (0, 0, Bx, 0)$. Throughout the results, we define $B^*$ as $B^* = B/B_e^c$, with $B_e^c = 4.414 \times 10^{13}$ G the critical field at which the electron cyclotron energy is equal to the electron mass.

The Lagrangian density of our system is given by

$$ \mathcal{L} = \sum_{i=p,n} \mathcal{L}_i + \mathcal{L}_e + \mathcal{L}_\sigma + \mathcal{L}_\omega + \mathcal{L}_\rho + \mathcal{L}_{\omega\rho} + \mathcal{L}_A. \qquad (36.1) $$

$\mathcal{L}_e$ and $\mathcal{L}_A$ are the electron Lagrangian density and electromagnetic term, given by

$$ \mathcal{L}_e = \bar{\psi}_e [\gamma_\mu (i \partial^\mu + e A^\mu) - m_e] \psi_e, \qquad (36.2) $$

$$ \mathcal{L}_A = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}, \qquad (36.3) $$

with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. The nucleon Lagrangian density is given by

$$ \mathcal{L}_i = \bar{\psi}_i [\gamma_\mu i D^\mu - M^* - \frac{1}{2} \mu_N k_b \sigma_{\mu\nu} F^{\mu\nu}] \psi_i, \qquad (36.4) $$

with

$$ M^* = M - g_\sigma \phi, \qquad (36.5) $$

the nucleon effective mass and

$$ iD^\mu = i\partial^\mu - g_\omega V^\mu - \frac{g_\rho}{2}\tau \cdot \mathbf{b}^\mu - e\frac{1+\tau_3}{2}eA^\mu, \qquad (36.6) $$

The Lagrangian density for the meson fields are given by

$$ \mathcal{L}_\sigma = \frac{1}{2} \left( \partial_\mu \phi \partial^\mu \phi - m_\sigma^2 \phi^2 - \frac{1}{3} \kappa \phi^3 - \frac{1}{12} \lambda \phi^4 \right), \qquad (36.7) $$

$$ \mathcal{L}_\omega = -\frac{1}{4} \Omega_{\mu\nu} \Omega_{\mu\nu} + \frac{1}{2} m_\omega^2 V_\mu V^\mu + \frac{v}{4!} v g_\omega^4 (V_\mu V^\mu)^2, \qquad (36.8) $$

$$ \mathcal{L}_\rho = -\frac{1}{4} \mathbf{B}_{\mu\nu} \cdot \mathbf{B}^{\mu\nu} + \frac{1}{2} m_\rho^2 \mathbf{b}_\mu \cdot \mathbf{b}^\mu, \qquad (36.9) $$

with the tensors written as

$$ \Omega_{\mu\nu} = \partial_{\mu} V_{\nu} - \partial_{\nu} V_{\mu}, \qquad (36.10) $$

$$ \mathbf{B}_{\mu\nu} = \partial_{\mu} \mathbf{b}_{\nu} - \partial_{\nu} \mathbf{b}_{\mu} - g_{\rho} (\mathbf{b}_{\mu} \times \mathbf{b}_{\nu}). \qquad (36.11) $$

The parameters include the couplings of the mesons to the nucleons, $g_\sigma$, $g_\omega$, $g_\rho$, the nucleon and electron masses $M$ and $m_e$, respectively, and the higher-order coupling constants $k$, $\lambda$, and $v$. The electromagnetic coupling constant is given by $e = \sqrt{4\pi/137}$, and $\tau_3 = \pm 1$ is the third component of the Pauli matrices for protons (+1) and neutrons (-1). We also introduce in the model the anomalous magnetic moment (AMM) of the nucleons with $\sigma_{\mu\nu} = \frac{i}{2}[\gamma_\mu, \gamma_\nu]$ and strength $k_b$, with $k_n = -1.91315$ for the neutron and $k_p = 1.79285$ for the proton. $\mu_N$ is the nuclear magneton. We neglect the AMM contribution for the electrons as it is negligible [946].
<table><caption>**Table 36.1:** Symmetric nuclear matter properties at saturation density for the NL3 and NL3ωρ models: binding energy *E*/*A*, saturation density *ρ*₀, normalized nucleon effective mass, *M*/*M*, incompressibility *K*, symmetry energy *E*<sub>sym</sub>, and its slope *L*.</caption><thead><tr><th></th><th><i>E</i>/<i>A</i> (MeV)</th><th><i>ρ</i><sub>0</sub> (fm<sup>-3</sup>)</th><th><i>M</i><sup>*</sup>/<i>M</i></th><th><i>K</i> (MeV)</th><th><i>E</i><sub>sym</sub> (MeV)</th><th><i>L</i> (MeV)</th></tr></thead><tbody><tr><td>NL3</td><td>16.24</td><td>0.148</td><td>0.60</td><td>270</td><td>37.34</td><td>118</td></tr><tr><td>NL3ωρ</td><td>16.24</td><td>0.148</td><td>0.60</td><td>270</td><td>31.66</td><td>55</td></tr></tbody></table>

### 36.2.1. Equation of state

Here, we consider two EoS models, NL3 [952], and NL3ωρ [953, 954, 941], that share the same isoscalar properties. In the NL3ωρ model, an interaction term between the ω and the ρ meson, $\mathcal{L}_{\omega\rho}$, is added to model the density dependence of the symmetry energy, since NL3 has a very large slope of the symmetry energy at saturation, $L = 118$ MeV. NL3ωρ, on the other hand, has $L = 55$ MeV. Some of the symmetric nuclear matter properties at saturation density for these models are shown in Table 36.1. This extra term is given by

$$ \mathcal{L}_{\omega\rho} = \Lambda_{\omega\rho} g_{\omega}^2 g_{\rho}^2 V_{\mu} V^{\mu} \mathbf{b}_{\mu} \cdot \mathbf{b}^{\mu}. $$

The fields equations follow from the Euler-Lagrange equations, and in the mean-field approximation, these fields are given by their constant expectation values, $\phi_0$, $V_0$, and $b_0$. In the following, for simplicity, we omit AMM in the equations. The interested reader can consult e.g. Ref. [955] for the equations with that term. The scalar and vector densities for nucleons, and the electron density, are given by

$$ \rho_{s,p} = \frac{q_p B M^*}{2\pi^2} \sum_{\nu=0}^{\nu_{\max}^p} g_s \ln \left| \frac{k_{F,\nu}^p + E_F^p}{\sqrt{M^{*2} + 2\nu q_p B}} \right|, \quad (36.12) $$

$$ \rho_{s,n} = \frac{M^*}{2\pi^2} \left[ E_F^n k_F^n - M^{*2} \ln \left| \frac{k_F^n + E_F^n}{M^*} \right| \right], \quad (36.13) $$

$$ \rho_p = \frac{q_p B}{2\pi^2} \sum_{\nu=0}^{\nu_{\max}^p} g_s k_{F,\nu}^p, \quad (36.14) $$

$$ \rho_n = \frac{k_F^{n3}}{3\pi^2}, \quad (36.15) $$

$$ \rho_e = \frac{|q_e|B}{2\pi^2} \sum_{\nu=0}^{\nu_{\max}^e} g_s k_{F,\nu}^e, \quad (36.16) $$

where $\nu = n + \frac{1}{2} - \frac{1}{2}\frac{q}{|q|}s = 0, 1, \dots, \nu_{\max}$ are the Landau levels (LL) for fermions with electric charge $q$, $q_e = -e$ for electrons and $q_p = e$ for protons. $s$ is the spin quantum number, +1 for spin up cases and -1 for spin down cases. The spin degeneracy factor of the Landau levels, $g_s$, is equal to $g_s = 1$ for $\nu = 0$ and $g_s = 2$ for $\nu > 0$, and $\nu_{\max}$ is the maximum number of LL, for which the square of the Fermi momentum of the particle is still positive, given by

$$ \nu_{\max}^e = \frac{E_F^{e2} - m_e^2}{2|q_e|B}, \quad (36.17) $$

$$ \nu_{\max}^p = \frac{E_F^{p2} - M^{*2}}{2q_p B}. \quad (36.18) $$
$k_{F,\nu}^q$ and $E_F^q$ are the Fermi momenta and energies of the particles, defined as

$$k_{F,\nu}^p = \sqrt{E_F^{p2} - M^{*2} - 2\nu q_p B}, \qquad (36.19)$$

$$k_F^n = \sqrt{E_F^{n2} - M^{*2}}, \qquad (36.20)$$

$$k_{F,\nu}^e = \sqrt{E_F^{e2} - m_e^2 - 2\nu|q_e|B}. \qquad (36.21)$$

The bulk energy density is given by:

$$\mathcal{E} = \mathcal{E}_f + \mathcal{E}_p + \mathcal{E}_n, \qquad (36.22)$$

where

$$\mathcal{E}_f = \frac{m_\omega^2}{2} V_0^2 + \frac{\nu g_v^4}{8} V_0^4 + \frac{m_\rho^2}{2} b_0^2 + \frac{m_\sigma^2}{2} \phi_0^2 + \frac{\kappa}{6} \phi_0^3 \\ + \frac{\lambda}{24} \phi_0^4 + 3 \Lambda_{\omega\rho} g_\rho^2 g_\omega^2 V_0^2 b_0^2, \qquad (36.23)$$

$$\mathcal{E}_n = \frac{1}{4\pi^2} \left[ k_F^n E_F^{n3} - \frac{1}{2} M^* \left( M^* k_F^n E_F^n \right. \right. \\ \left. \left. + M^{*3} \ln \left| \frac{k_F^n + E_F^n}{M^*} \right| \right) \right], \qquad (36.24)$$

$$\mathcal{E}_p = \frac{q_p B}{4\pi^2} \sum_{\nu=0}^{\nu_{\max}} g_s \left[ k_{F,\nu}^p E_F^p + \left( M^{*2} + 2\nu q_p B \right) \right. \\ \left. \cdot \ln \left| \frac{k_{F,\nu}^p + E_F^p}{\sqrt{M^{*2} + 2\nu q_p B}} \right| \right]. \qquad (36.25)$$

The chemical potentials for protons, neutrons, and electrons are given by

$$\mu_p = E_F^p + g_\omega V_0 + \frac{1}{2} g_\rho b_0, \qquad (36.26)$$

$$\mu_n = E_F^n + g_\omega V_0 - \frac{1}{2} g_\rho b_0, \qquad (36.27)$$

$$\mu_e = E_F^e = \sqrt{k_{F,\nu}^{e2} + m_e^2 + 2\nu|q_e|B}. \qquad (36.28)$$

and the pressure is

$$P = \mu_p \rho_p + \mu_n \rho_n - \mathcal{E}. \qquad (36.29)$$

For neutron star matter, the β-equilibrium and charge-neutral conditions are imposed:

$$\mu_n = \mu_p + \mu_e \qquad (36.30)$$

$$\rho_p = \rho_e. \qquad (36.31)$$

### 36.2.2. Pasta structures in the CP and CLD approximations

Here, we consider the coexistence-phase (CP) [956, 955] and the compressible liquid drop (CLD) [957, 958] models to calculate the inner crust structures in β-equilibrium magnetized matter.

In the CP approximation, separated regions of high (heavy clusters) and low (background nucleon gas) densities are considered. Gibbs equilibrium conditions are imposed, together with the charge-neutrality condition:
$$
\begin{align}
\mu_p^I &= \mu_p^{II}, \nonumber \\
\mu_n^I &= \mu_n^{II}, \nonumber \\
P^I &= P^{II}, \nonumber \\
\rho_e &= \rho_p = f \rho_p^I + (1-f) \rho_p^{II}. \tag{36.32}
\end{align}
$$

Here, *I* labels the cluster phase and *II* the gas phase. In this approximation, the finite size effects are taken into account by a surface and a Coulomb terms in the energy density, that is only added after the coexistence phases are achieved.

In the CLD model, the total energy density is minimized, including the surface and Coulomb terms. The equilibrium conditions become

$$
\mu_n^I = \mu_n^{II}, \tag{36.33}
$$

$$
\mu_p^I = \mu_p^{II} - \frac{\mathcal{E}_{surf}}{(1-f)f(\rho_p^I - \rho_p^{II})}, \qquad (36.34)
$$

$$
P^I = P^{II} + \mathcal{E}_{surf} \left[ \frac{3}{2\alpha} \frac{\partial \alpha}{\partial f} + \frac{1}{2\Phi} \frac{\partial \Phi}{\partial f} - \frac{((1-f)\rho_p^I + f\rho_p^{II})}{(1-f)f(\rho_p^I - \rho_p^{II})} \right]. \quad (36.35)
$$

The total energy density of the system is given by

$$
\mathcal{E} = f \mathcal{E}^I + (1-f) \mathcal{E}^{II} + \mathcal{E}_{\text{Coul}} + \mathcal{E}_{\text{surf}} + \mathcal{E}_e, \quad (36.36)
$$

where *f* is the fraction of volume occupied by the dense phase. The surface and Coulomb terms are given by:

$$
\mathcal{E}_{\text{Coul}} = 2\alpha e^2 \pi \Phi R_d^2 (\rho_p^I - \rho_p^{II})^2, \quad (36.37)
$$

$$
\mathcal{E}_{surf} = \frac{\sigma \alpha D}{R_d} \qquad (36.38)
$$

where $\alpha = f$ for droplets, rods and slabs and $\alpha = 1 - f$ for tubes and bubbles, and $R_d$ is the size of the cluster. $\Phi$ is given by

$$
\begin{align}
\Phi &= \left( \frac{2 - D\alpha^{1-2/D}}{D-2} + \alpha \right) \frac{1}{D+2}, \quad D=1,3, \notag \\
\Phi &= \frac{\alpha - 1 - \ln \alpha}{D+2}, \quad D=2. \tag{36.39}
\end{align}
$$

The surface tension parameter $\sigma$ was obtained from a fit to a relativistic Thomas-Fermi calculation [956]. The following relation is obtained, when minimizing the surface and Coulomb terms with respect to $R_d$

$$
\mathcal{E}_{surf} = 2\mathcal{E}_{Coul}, \tag{36.40}
$$

$$
R_d = \left[ \frac{\sigma D}{4\pi e^2 \Phi (\rho_p^I - \rho_p^{II})^2} \right]^{1/3} . \qquad (36.41)
$$

36.3. Results and discussion

Here, we start by addressing the results of the estimation of the crust-core transition density from a dynamical spinodal calculation within the Vlasov formalism [959, 960, 941] for magnetized nuclear matter. For β-equilibration and zero temperature, i.e. NS conditions, this calculation is in very good agreement with more sophisticated calculations, like
<figure><img src="image_110.png" /><figcaption>**Figure 36.1:** Dynamical spinodal regions for NL3ωρ, for a momentum transfer of $k = 75$ MeV, with anomalous magnetic moment for $B = 4.41 \times 10^{16}$ G (left), and $B = 4.41 \times 10^{17}$ G (right). A comparison with the $B = 0$ (black lines) results is also made. The EoS for β-equilibrium matter is also shown (blue). Figure adapted from Ref. [937].</figcaption></figure>

Thomas-Fermi [927]. In this calculation, the instability region is determined from the collective modes of nuclear matter that correspond to small oscillations around equilibrium. Only longitudinal modes are considered, and the boundary of this region is defined by the frequency of these modes to be zero. Inside this (unstable) region, the mode with the largest frequency drives the system to the formation of the instabilities. To calculate the crust-core transition, we cross the EoS with these spinodal surfaces, in the ($\rho_p, \rho_n$) space. In Ref. [941], it was seen that the larger the slope of the symmetry energy, the smaller the spinodal regions. This was then reflected in an anti-correlation between $L$ and the crust-core transition density: the larger the $L$, the smaller the density [941].

In Fig. 36.1, we show the dynamical spinodal regions for the NL3ωρ model, for two different values of the magnetic field, and we also compare with the $B = 0$ result. We observe that the magnetic field is giving rise to alternate bands of homogeneous and non-homogeneous matter, that appear due to the Landau levels. The stronger the magnetic field is, the greater the size of the spinodal region. Also, with the increase of the magnetic field, the number of the bands becomes smaller and the bands become wider. This happens because as the B-field increases, there is a decrease in the number of Landau levels. We also observe that the crust-core transition extends to a larger range of densities, as opposed to what happens at $B = 0$.

In Fig. 36.2, we plot the largest growth rate as a function of the density for NL3ωρ, and three different values of the magnetic field. We observe the appearance of oscillations around the $B = 0$ results, below the $B = 0$ crust-core transition density, that is given when the mode goes to zero, and above this value, we get this alternate regions of clusterized and non-clusterized matter, as already observed in Fig. 36.1. Since the magnetic field is giving rise to larger crust-core transition densities, the correspondent pressures also become larger, and this has a direct influence in the fractional moment of inertia of the crust (see Table I of [937]), that also becomes larger.

In the following, we show the results obtained from a CP and CLD calculations for the NS magnetized inner crust. We start with the CP calculation.

In Fig. 36.3, we show the radii of the Wigner-Seitz (WS) cell (red) and of the nucleus (green) as a function of the density for the same model as above and considering results with (bottom) and without (top) AMM. We consider the highest B-field strength in our calculations to be $B^* = 10^4$, or $B = 4.41 \times 10^{17}$ G. As for comparison, the blue lines
<figure><img src="image_111.png" /><figcaption>**Figure 36.2:** Largest growth rate versus density for NL3ωρ, with a proton fraction of yp = 0.035, with anomalous magnetic moment. Three different values of B are considered: B = 4.41 × 10¹⁵G (red), B = 4.41 × 10¹⁶G (green), and B = 4.41 × 10¹⁷G (blue). A comparison with the B = 0 (black lines) results is also made. Figure adapted from Ref. [937].</figcaption></figure>

<figure><img src="image_112.png" /><figcaption>**Figure 36.3:** Radii of the WS cell (red) and nucleus (green) for β-equilibrium matter using the NL3ωρ parametrization without (left) and with (right) the inclusion of AMM for B = 4.41 × 10¹⁷G. The no-field case is also shown with gray points as a reference. Growth rates obtained with a dynamical spinodal calculation in [937] are plotted with blue lines. Figure adapted from Ref. [955].</figcaption></figure>
represent the maximum growth rates as shown in Fig. 36.2. These results seem to be in agreement with what we previously found with a dynamical spinodal calculation: several disconnected regions of non-homogeneous matter appear above the $B=0$ region. If AMM is considered, these regions are more numerous (the double) and narrower, because the spin polarisation degeneracy is removed.

<figure><img src="image_113.png" /><figcaption>**Figure 36.4:** The proton fraction (left) and the evolution of the pasta phases (right) as a function of the baryonic density for $\beta$-equilibrium matter using the NL3$\omega\rho$ model, considering different magnetic field strengths. The results consider calculations with (dark colors) and without (light colors) AMM. In the case of the shapes, only the results without AMM are shown. Figure adapted from Ref. [955].</figcaption></figure>

In Fig. 36.4, we show the proton fraction and the evolution of the shapes as a function of the density. We observe that the larger the magnetic field, the larger the proton fraction, showing fluctuations due to the opening of new Landau levels. For NL3$\omega\rho$ model, these disconnected pasta regions that appear above the main $B=0$ region contain all types of geometric configurations in their narrow density range. For the NL3 model, that only shows the droplet configuration in the $B=0$ case, the finite magnetic field induces the appearance of all geometric structures in the first region (see Fig. 3 of Ref. [955]), as well as in the narrow disconnected regions.

<figure><img src="image_114.png" /><figcaption>**Figure 36.5:** Baryonic (left) and proton (right) densities of liquid (1, blue) and gas (2, red) phases as function of the total baryon density for the NL3 model in a CP (dashed lines) and CLD (solid line) calculations, with $B^* = 5 \times 10^3$. We also plot the magnetized growth rates divided by a factor $10^2$, $|\omega_{max}|$ (light blue), as well as the densities in the $B=0$ case (black). The green and orange segments indicate, respectively, $\rho_{1\to2}$ and $\rho_{cc}$, both defined in the text. Figure adapted from Ref. [958].</figcaption></figure>
Now we focus on the CLD results. In Fig. 36.5, we show the baryonic and proton densities in the gas and liquid phases as a function of the density. Also shown are the maximum growth rates in a dynamical spinodal calculation (see Fig. 36.2). These results are for the NL3 model. A comparison with a CP calculation from Wang et al [955] is also made, though one can almost not distinguish except for the inset panel in the bottom panel. We see that the crust-core transition density (orange lines, referred as $\rho_{cc}$) gets shifted to higher values with respect to the $B=0$ case (green lines, referred as $\rho_{1\to2}$). As before, in the CP case, this results are in line with the previous studies using the dynamical spinodal calculations [936, 937, 938]. It is interesting to notice that in this extra region that appears due the magnetic field, the proton and baryonic densities of the liquid and gas become very similar.

<figure><img src="image_115.png" /><figcaption>**Figure 36.6:** Baryonic (left) and proton (right) densities of liquid (1, blue) and gas (2, red) phases as function of the total baryon density for the NL3$\omega\rho$ model in a CP (dashed lines) and CLD (solid line) calculations, with $B^* = 5 \times 10^3$. We also plot the magnetized growth rates divided by a factor $10^2$, $|\omega_{max}|$ (light blue), as well as the densities in the $B=0$ case (black). The orange segments indicate the $\rho_{cc}$. Figure adapted from Ref. [958].</figcaption></figure>

In Fig. 36.6, we show the same quantities as above but this time for the NL3$\omega\rho$ model. Unlike for the NL3 model, here we see that with increasing $B$, the crust-core transition decreases and the extra region does not appear, unlike with the CP calculation. In Wang et al [955], an extra region of non-homogeneous matter was found for this model, although smaller than for the NL3 model. However, in this work, the energy criterium, according to which the stable configuration has the lowest free energy, was not applied as in [958], and some non-homogeneous configurations for the larger densities have an energy above homogeneous matter. Notice, however, that the calculation is not self-consistent as it would be, for instance, a Thomas-Fermi calculation that tends smoothly to the homogeneous solution, and, therefore, the crust-core transition should be further analysed. The different behaviour of the two models can be explained by the different behaviour of their symmetry energy: even though the slope of the symmetry energy at saturation is higher for the NL3 model than for NL3$\omega\rho$ (see Tab. 36.1), for densities below $\sim 0.1\text{fm}^{-3}$, the symmetry energy of NL3$\omega\rho$ is higher than the one of NL3, as we can see from Fig. 36.7. This means that NL3$\omega\rho$ (NL3) will have a larger (smaller) proton fraction, that will translate into a smaller (larger) effect of the magnetic field, and therefore a smaller (larger) extension of the crust. The symmetry energy behaviour favours larger proton fractions for NL3$\omega\rho$, and smaller B-field effects, when compared to NL3.

We also need to point out that, even though both these calculations tend to give similar results, they are not self-consistent, since the surface tension is parametrised from a fit to a Thomas-Fermi calculation without magnetic field. This quantity influences the crust-core transition density so in a near future it would be interesting to obtain a calculation
<figure><img src="image_116.png" /><figcaption>**Figure 36.7:** Symmetry energy as a function of the density for the NL3$\omega\rho$ and NL3 models.</figcaption></figure>

for a magnetized surface tension, and to analyze where the crust-core transition occurs.

## 36.4. Conclusions

In this section, the structure of the inner crust of a neutron star in the presence of a strong magnetic field, within a relativistic mean-field framework, using the coexistence phase and compressible liquid drop models for the calculation of the pasta phases, was addressed. These results were compared to a dynamical spinodal method [936, 937, 938]. Two RMF models, NL3 [952] and NL3$\omega\rho$ [953, 954], were considered.

We found that an extended region of clusters appears due to the presence of the magnetic field. This region contains matter in different geometric structures, and, in there, the cluster and gas densities are very close, for both neutrons and protons. This extra region seems to depend on the behaviour of the symmetry energy in the crustal EoS. We found that the transition densities given by the CLD calculation are in good agreement with the CP approximation, and also in agreement with the dynamical spinodal calculation. These heavy clusters are very dependent on the surface tension, therefore a calculation of a magnetized surface tension should be explored in a near future.

# 37. Superconductivity and superfluidity in strong magnetic fields

## 37.1. Introductory remarks

We have observed that extremely large magnetic fields, on the order of $B \sim 10^{18} - 10^{19}$ G, can impact the hydrostatic equilibrium and, consequently, the integral parameters (mass, radius, tidal deformability, etc.) of compact stars. However, much smaller magnetic fields are needed to influence properties governed by the quasiparticle spectrum near the Fermi surface. These properties include the superfluidity and superconductivity of nucleonic fluids (neutrons and protons) and transport phenomena associated with fermionic (primarily electrons, protons, and neutrons) and bosonic degrees of freedom (such as pion and kaon condensates). The changes in these properties due to magnetic fields can significantly affect the macroscopic behavior of magnetars, as discussed in this subsection.

There are two mechanisms of suppression of nucleonic superfluids in strong magnetic fields [961, 962]. The interaction of the magnetic field with the neutron or proton spin induces an imbalance in the number of spin-up and spin-down
particles, which implies that the Cooper pairing will be suppressed because not all spin-up particles will find spin-down "partners" [961]. This so-called *Pauli paramagnetic suppression* acts for both proton and neutron condensates, but is the sole suppression mechanism only in the case of neutrons. The proton condensate is also suppressed due to the Larmor motion of protons in the magnetic field (Landau diamagnetism) [962]. We now discuss these effects on a more quantitative level.

An important threshold is reached when electromagnetic interactions approach the nuclear energy scale for cold neutron stars, typically, between 0.1 and 1 MeV. The first suppression mechanism that acts independent of particle charges arises when the interaction energy between the magnetic field and the nucleon spin is given by $\mu_N B$, where $\mu_N = e\hbar/2m_p$ is the nuclear magneton become of the order of scales indicated above. A simple comparison of these scales shows that the magnetic fields on the order of $10^{16} - 10^{17}$ G would significantly influence 0.1 to 1 MeV-scale physics through the nucleonic spin-B-field interaction.

An additional interaction channel arises for charged particles due to the coupling of the particle charge to the magnetic field. Physically, the suppression becomes effective when the Larmor radius quantifies the winding of the particle trajectory in the magnetic field becomes small enough (once the field is large enough) to be comparable to the condensate coherence length. The latter defines the size of the Cooper pairs in the limit of weak-coupling BCS theory, which is appropriate for neutron and proton fluids in neutron stars. At this threshold the coherence of Cooper pairing is destroyed by the field. To make a simple estimate of the effect, we use the Landau criterion for the critical velocity is given by $v_s \sim \Delta/p_\perp$, where $\Delta$ is the pairing gap and $p_\perp$ is the characteristic momentum in the plane orthogonal to the field. Since this momentum is smaller than the Fermi momentum $p_\perp \le p_F$, we find that the relevant energy scale is given by $v_s p_F \sim \pi(v_s/c)(v/10 \text{ fm})(B/10^{16} \text{ Gauss}) \text{ MeV}$ by equating the particle's Larmor radius $p_F c/eB$ to the coherence length of the condensate, which for neutrons and protons are of the order of $v \simeq 10 \text{ fm}$. In our estimate we used that $v_s/c \le 0.3$. We conclude that the fields needed to suppress the pairing by magnetic fields are significantly smaller than those that would affect their equation of state and structure.

## 37.2. Destruction of S-wave neutron superfluidity by strong magnetic fields

In the case of S-wave pairing the condensate consists of Cooper pairs of neutrons with opposite spins and momenta. When magnetic field is applied, the spin-up and spin-down populations become unequal because of the interaction of the neutron's spin magnetic moment with the magnetic field - there are more neutrons aligned with the magnetic field than neutrons that are anti-aligned. This imbalance has a disruptive effect on S-wave neutron Cooper pairs, as has been first established in the context of ordinary superconductors with magnetic impurities (for a review see [963]). A sufficiently strong magnetic field can completely suppress this pairing as neutrons with spins along the field will not find opposite spin partners to pair. This critical field will be referred to as $H_{c2}^n$ is the analog of the Chandrasekhar-Clogston limit in ordinary superconductors [964].

As the density is lowered there is also a possibility of BCS-BEC crossover from the weakly coupled pairs of neutrons (BCS regime) to the strongly coupled regime (BEC regime). Such a transition in nuclear systems was studied first in the context of proton-neutron pairing where in the vacuum there is a bound state - the deuteron. Naturally enough, one would expect in this case formation of Bose gas of deuterons at very low densities and a smooth transition from the BCS regime to the BEC regime as the density is lowered [965, 966, 967, 968]. However, in this case, the isospin asymmetry - a feature common in the finite and infinite nuclear systems - has a destructive effect, analogous to the spin-polarization, and
<figure><img src="image_117.png" /><figcaption>**Figure 37.1:** Critical unpairing magnetic field $H_{c2}$ for the neutron condensate due to spin alignment, shown for two temperatures. The density is expressed in units of the nuclear saturation density.</figcaption></figure>

the phase diagram is not limited to BCS-BEC crossover but various novel phases may arise [969, 970]. Since two neutrons do not form a bound pair in free space, there is no initial basis for a Bose-Einstein condensate (BEC) of neutron-neutron pairs. However, by applying Noziéres-Schmitt-Rink theory [971], it was possible to identify several signatures in neutron matter that can be viewed as precursors to the BCS-BEC crossover. In the case of zero polarization, similar results were obtained [972, 973, 974, 975].

Ref. [961] have determined the critical field for un-pairing of the neutron condensate to be in the range $B \sim 10^{17}$ G. In Fig. 37.1 we plot values of $H_{c2}$ for the neutron condensate as a function of density, determined based on a phase-shift-equivalent nucleon-nucleon interaction and numerical solutions of the BCS equations in the case of spin-polarized neutron matter [961]. The shape of the curve reflects the corresponding density dependence of the pairing gap and its temperature dependence follows the BCS prediction: it is largest at $T = 0$ and decreases as the pairing gap decreases with increasing temperature. Thus, if the local field in a magnetar crust exceeds the value $H_{c2}^n$, the magnetic field will destroy the condensate. Note that according to Fig. 37.1 the critical field is larger for lower temperatures because (a) the pairing gap increases with temperature and (b) the magnetic field required to produce a certain polarization increases with decreasing temperature.

From a phenomenological perspective, S-wave pairing is important in the crust of magnetars, which extends below the density of half the saturation density of symmetric nuclear matter. At higher densities, the dominant pairing state in neutron matter shifts to the $^3P_2-^3F_2$ channel, where neutrons pair in a total spin-1 state (see [976, 977]). In this situation, the spin-polarizing effect of the magnetic field on the internal structure of spin-1 pairs does not break the pairing [978, 979, 980].

## 37.3. Destruction of S-wave superconductivity of protons in strong magnetic field

BCS superconductors are characterized by at least three distinct length scales: (i) the London penetration depth $\lambda$, (ii) the coherence length $\nu$, and (iii) the interparticle distance $d$. For this discussion, it is assumed that the interparticle distance $d$ is much smaller than the other two scales, indicating that the superconductor is in the weakly coupled regime. The ratio of
<figure><img src="image_118.png" /><figcaption>**Figure 37.2:** The proposed layered structure of a magnetar features a constant magnetic field in the core (represented by the dashed line) [986]. To the right of the point where this field intersects with $H_{c2}$, the proton fluid is non-superconducting; to the left, where $B \le H_{c2}$, it remains superconducting. The crust contains a uniform magnetic field $B$. The density is expressed in units of the nuclear saturation density.</figcaption></figure>

the remaining two scales, $\lambda$ and $v$, defines the type of superconductivity through the Ginzburg-Landau parameter $\kappa = \lambda/v$ (see, e.g., [964]). If $1/\sqrt{2} < \kappa < \infty$, the material is classified as a type-II superconductor; otherwise, it is type-I. In type-II superconductors, the magnetic field is carried by electromagnetic vortices with quantum flux $\phi_0 = \pi/e$ (with $\hbar = c = 1$ assumed hereafter), while in type-I superconductors, the magnetic field forms domain structures [981, 982, 983, 984].

These two scales, $\lambda$ and $v$, also define three distinct magnetic field scales when combined with the flux quantum:

$$H_{c1} \approx \frac{\phi_0}{\lambda^2}, \quad H_{cm} \approx \frac{\phi_0}{v\lambda}, \quad H_{c2} \approx \frac{\phi_0}{v^2} \qquad (37.1)$$

In type-II superconductors, the hierarchy of these fields is $H_{c1} \le H_{cm} \le H_{c2}$ when $\kappa \ge 1$. At and above $H_{c1}$, the formation of a single flux tube (Abrikosov quantum vortex) becomes energetically favorable. The field $H_{cm}$ is the thermodynamic magnetic field, where the energy density equals the difference between the energy densities of the superconducting and normal states. Finally, $H_{c2}$ is the field strength at which superconductivity vanishes, as the density of flux tubes becomes so high that the normal vortex cores overlap.

The Ginzburg-Landau theory of neutron-proton superfluid mixtures was used to compute the upper critical magnetic field in Refs. [962] and is extended to account for coupling between the neutron and proton currents in Ref. [985].

The dependence of the $H_{c2}$ field on density is depicted in Fig.37.2. Several key features are worth mentioning. The $H_{c2}$ field reaches its maximum near the crust-core interface at a density of $n_b = 0.5n_0$, where $n_0$ is the nuclear saturation density. As seen from the figure, a magnetar is likely to have a layered structure, consisting of (a) an inner core devoid of superconductivity, (b) an outer core threaded by flux tubes, and (c) a crust containing a homogeneous magnetic field of magnitude $B$. If the field $B$ exceeds the maximum $H_{c2}$, the flux-carrying region in the outer core vanishes, resulting in the complete destruction of superconductivity within the magnetar.
## 37.4. Astrophysical implications: Precessing magnetars and fast radio bursts (FRBs)

The absence or partial presence of superfluidity in magnetars will significantly impact various both macrophysical and microphysical properties of magnetars. Below, we illustrate the key modifications without the intent to give a complete picture.

The possibility of neutron star precession remains an intriguing unsolved issue, the main obstacle being the fact that the precession is strongly suppressed by the coupling of the crust to the superfluid phases inside the neutron stars, see [987]. The precession in magnetars may not have the same fate if the fields are in the range where the suppression of proton superconductivity is effective [986] and it has been suggested that free precession can account for FRBs [988]. The reason is the following: when protons become unpaired, they can scatter off neutron quasiparticles in the vortex cores, a process that is significantly more effective than the scattering of electrons off magnetized neutron vortices via electromagnetic forces, which is the dominant mechanism in type-I proton superconductors [989, 984]. However, in the more realistic scenario of type-II superconductivity, which would be absent in magnetars due to unpairing, the coupling mechanism is likely more complex due to significant interactions between protonic flux tubes and neutron vortices [990, 991, 992, 993]. It has been proposed that FRBs are associated with magnetars and the subclass of repeated FRBs may originate from the precession of magnetars [994, 995, 996, 997]. The fact that only a small subclass of FRBs shows periodicity might be interpreted as these objects having a favorable magnetic field configuration admitting weak coupling of the neutron superfluid to the crust. If the magnetic field in the core of a magnetar completely unpairs the proton fluid, it will interact with the electron fluid on plasma time scales, which are significantly shorter than hydrodynamical time scales. As a result, the unpaired core of a magnetar can be described as a two-fluid system, consisting of a superfluid neutron condensate and a normal component made up of the proton and electron fluids. The mutual coupling between these components is expected to be weak in this case.

Finally, it is important to note that, in addition to precession, the dynamic coupling between the superfluid and the normal plasma plays a crucial role in understanding the rotational irregularities of magnetars. This includes phenomena such as glitches, anti-glitches, post-glitch relaxation, and oscillations.

## 37.5. Neutrino radiation from magnetars

The suppression of pairing by MeV-scale magnetic fields will also have profound consequences on the thermal evolution of magnetars, because the dominant processes of neutrino radiation will not be suppressed by the Boltzmann factor containing the gap in the quasiparticle spectrum of baryons. At the same time, the processes that are intrinsic to condensates, such as the pair-breaking emission of neutrino-anti-neutrino pairs, will not operate by definition.

Let us start with the direct Urca process: $n \rightarrow p+e+\bar{\nu}_e$ where n stands for neutron, p - proton, e - elections and $\bar{\nu}_e$ for electron antineutrino. It is well known that proton and neutron pairing suppresses the Urca process when nucleons transition into a superconducting or superfluid state. At very low temperatures, emissivity is reduced by a factor of $\exp(-\Delta/T)$ for each participating nucleon, where $\Delta$ represents the relevant pairing gap and T is the temperature. The suppression is evidently reduced when neutron and proton S-wave pairing is terminated by the magnetic field. Since the neutron pairing gap in the P-wave channel is smaller than the proton pairing gap in the S-wave channel, the disruption of proton superconductivity by a MeV-scale magnetic field will significantly impact Urca emissivity. Numerical examples are provided in Ref. [962].

Next, consider pair-formation and breaking (PFB) processes: $[NN] \rightarrow N+N+\nu_f+\bar{\nu}_f$ where N stands for either
neutron or proton, $\nu_f$ and $\bar{\nu}_f$ for neutrino and anti-neutrino of flavor $f$ (for a review see [963]). The rates of neutrino emission through PBF processes follows the relation $\epsilon \propto \Delta^2 T^7$, where $\Delta$ is the pairing gap. As a result, the unpairing of S-wave condensates will effectively eliminate PBF processes in regions where the magnetic field locally surpasses the unpairing thresholds for protons and neutrons. Consequently, the overall neutrino emission rate will asymptotically decrease to the level associated with PBF emission by the P-wave condensate.

Apart from emission rates, the thermal evolution of magnetars depends on the specific heat of the interior matter, i.e. star's thermal inertia. In neutron stars that are fully superconducting or superfluid and have low magnetic fields, the heat capacity is predominantly determined by electrons because of the exponential suppression of specific heat of nucleons. Once magnetic unpair nucleons, their specific heat will contribute to the total budget of the star's thermal inertia. An increase in the specific heat of the interior matter will lead to a longer cooling time scale for a magnetar. For the current state of the simulations of coupled magnetic and thermal evolution see Refs. [998] and references therein.

# 38. Neutron P-wave superfluids in neutron stars: topological surface defects

## 38.1. Introduction

Neutron stars are important subjects which are studied by many researchers in astrophysics, nuclear physics and condensed matter physics (see recent reviews, e.g., Refs. [999, 1000]). Neutron superfluids have been interested since the early stage of researching neutron stars. It is considered that the neutron $^1S_0$ superfluids exist at low density region near the surface of neutron stars. This superfluid is produced by spin-singlet neutron pairings with S-wave by the low-energy scattering process. At higher density, there appears another type of the pairing structure: spin-triplet and P-wave denoted by $^3P_J$. In this pairing, the total angular-momentum $J = 2$ is favored, while the $J = 0$ and $J = 1$ channels are disfavored, because the latter channels are repulsive. Thus, the neutron matter is described as the neutron $^3P_2$ superfluids. The neutron $^3P_2$ pairings exhibit interesting properties induced by the rich structures of the internal degrees of freedom combined by spin and angular momentum. In this section, we review our recent studies on the neutron $^3P_2$ superfluids.

We summarize briefly the history of the researches of the neutron $^1S_0$ and $^3P_2$ superfluids. In the condensed matter physics, there are rapid development of the researches of fermion superfluids. In 1972, $^3$He atom (fermion) superfluids were discovered in experiments [1001, 1002], where it was studied that spin-triplet P-wave Cooper pairs stem from the spin-fluctuation interactions. In 1995, Ru atom (boson) superfluids were discovered in experiments [1003, 1004, 1005]. In 2003, the fermionic condensate was found in an experiment by using $^{40}$K (fermion) atom [1006]. In parallel to the development in the condensed matter physics, the possibility of superfluids has been pursued in nuclear physics. In 1960, Migdal first proposed the existence of neutron $^1S_0$ superfluids at low density [1007]. In 1966, however, Wolf discussed that strong repulsion core makes the interaction to attractive one at high density, and it makes the neutron $^1S_0$ pairings unstable [1008].

As a new type of pairings, in 1968, Tabakin proposed that the neutron $^3P_2$ superfluids can be realized by an attraction in this channel due to the nuclear LS potential at high energy scatterings [1009]. The LS potential exhibits the dependence on both relative angular momentum ($L$) and total spin ($S$) of the scattering two neutrons. After this pioneering work, further studies have been developed by many authors, see, e.g., Refs. [1010, 1011, 1012, 1013, 1014, 1015] as early works. For example, it was presented that the $^3P_2$ channel is coupled to the $^3F_2$, and the mixing effect can enhance the gap size in the neutron $^3P_2$ superfluids leading to a few MeV order [1011, 1012, 1013]. Recent calculations, however, indicate the
pairing size may be smaller due to many-body effects in nuclear matter. The neutron $^3P_2$ pairings are also concerned with the cooling process of neutron stars [1016, 1017, 1018] (see also Refs. [1019, 1020, 1021]).

The P-wave pairings show the topological properties. For example, the Weyl and/or Majorana fermions can exist in the neutron $^3P_2$ superfluids [1022, 1023, 1024]. There are also rich bosonic excitations in the neutron $^3P_2$ superfluids [1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037] which may be relevant to the cooling process. The exotic topological vortices also exhibit many interesting properties such as the spontaneous magnetizations [1014, 1038, 1039, 1040] and vortices with Majorana fermions [1041, 1042, 1042], half-quantized non-Abelian vortices [1043, 1042, 1044, 1045, 1042], solitonic excitations on a vortex [1046], and domain walls [1047]. The neutron $^3P_2$ superfluids also exhibit interesting topological defects at the surface on the neutron stars [1048], as discussed in details in Sec. 38.3. Recently, it was shown that since a singly quantized vortex is split into two half-quantized vortices [1042, 1044], vortex junctions are produced at the interface between the neutron $^1S_0$ superfluid and the neutron $^3P_2$ superfluid, thereby inducing the vortex networks which can explain the power-law of the glitches in neutron stars [1049]. The vortex junctions at the boundary of the superfluids are called the boojums which were discussed originally for $^3$He superfluids by N. D. Mermin, see, e.g., Ref. [1050]. Thus, the $^3P_2$ superfluids are important subjects, not only in the nuclear physics and the astrophysics, but also in the condensed matter physics.

The neutron $^3P_2$ superfluids especially play important roles in magnetars. The magnetars are the special neutron stars with very strong magnetic fields which reach $10^{15}$ Gauss [1051]. This is about hundred times larger than the magnetic fields in neutron stars $10^{13}$ Gauss. The neutron $^3P_2$ superfluids are stable against the strong magnetic fields. This can be intuitively understood: the spin-parallel (spin-one) pairs in the neutron $^3P_2$ superfluids are irrelevant to the Zeeman splitting. This property is sharply contrasted to the fragility of the spin-antiparallel (spin-zero) pairings in the neutron $^1S_0$ superfluids. Therefore, it is expected that the neutron matter in the magnetars is dominated by the neutron $^3P_3$ superfluids.

In the followings, we introduce the basic properties of the neutron $^3P_2$ superfluids in Sec. 38.2. After summarizing the symmetries of the neutron $^3P_2$ superfluids in Sec. 38.2.1, we explain the Bogoliubov-de Gennes (BdG) equation and the Ginzburg-Landau (GL) theory in Sec. 38.2, and present that they give the phase diagram with rich structures. In Sec. 38.3, based on the GL theory, we show our study on the topological surface defects on the boundary of the neutron $^3P_2$ superfluids. We summarize our presentation in Subsec. 38.4. This section is dedicated to the summary of our previous studies performed partly by the author. The relevant references are shown in the text.

## 38.2. Phase diagram of neutron $^3P_2$ superfluids

### 38.2.1. Symmetry

Let us describe some basic properties of the neutron $^3P_2$ pairings in neutron matter. The neutron $^3P_2$ pairs have the total spin one, $s = (s^1, s^2, s^3)$, as a sum of two neutrons. Besides, the neutron $^3P_2$ pairings have also dependence on three-dimensional momentum, $q = (q^1, q^2, q^3)$, transferred between two neutrons. Those properties stem from the LS potential which is dominantly important in the interaction between two neutrons at high-energy scatterings. They induce the tensor-type condensate for the neutron $^3P_2$ superfluids, whose pairing form denoted by $A^{ab}$ is given by

$$A^{ab} \propto \frac{1}{2}(s^a q^b + s^b q^a) - \frac{1}{3}\delta^{ab} s \cdot q, \qquad (38.1)$$
<table><caption>**Table 38.1:** The bulk phases and symmetries of the uniform neutron $^3P_2$ superfluids are shown for the different strengths of magnetic fields (cf. Fig 38.1). *r* is an internal parameter in the pairing matrix, see Eq. (38.3). More detailed classification of phases is found in Ref. [1052].</caption><thead><tr><th>magnetic field</th><th>zero</th><th>weak</th><th>strong</th></tr></thead><tbody><tr><td><i>r</i></td><td>-1</td><td>(-1, -1/2)</td><td>-1/2</td></tr><tr><td>bulk phase</td><td>UN</td><td>D<sub>2</sub>-BN</td><td>D<sub>4</sub>-BN</td></tr><tr><td>symmetry</td><td>O(2)</td><td>D<sub>2</sub></td><td>D<sub>4</sub></td></tr></tbody></table>

as a symmetric and traceless tensor over the indices $a, b, c = 1, 2, 3$. The pairing matrix (38.1) is described essentially by five complex components as independent degrees of freedom. Including the overall global phase, the neutron $^3P_2$ gap can be parametrized by the matrix form. In general, the matrix $A$ in Eq. (38.1) can be expressed by

$$A(\boldsymbol{x}) = \begin{pmatrix} f_1(\boldsymbol{x}) & g_3(\boldsymbol{x}) & g_2(\boldsymbol{x}) \\ g_3(\boldsymbol{x}) & f_2(\boldsymbol{x}) & g_1(\boldsymbol{x}) \\ g_2(\boldsymbol{x}) & g_1(\boldsymbol{x}) & -f_1(\boldsymbol{x}) - f_2(\boldsymbol{x}) \end{pmatrix}, \qquad (38.2)$$

where $f_i(\boldsymbol{x})$ ($i = 1, 2$) and $g_j(\boldsymbol{x})$ ($j = 1, 2, 3$) are complex functions with the three-dimensional coordinate $\boldsymbol{x}$.

When the neutron $^3P_2$ superfluid is uniform in space, the pairing matrix $A$ can be expressed simply by a diagonal form as

$$A = A_0 \begin{pmatrix} r & 0 & 0 \\ 0 & -1-r & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (38.3)$$

Here $A_0$ is a complex scalar including a complex phase, and $r$ indicates the internal parameter in the neutron $^3P_2$ gap which is given by the range of $r$ can be restricted to be $-1 \le r \le -1/2$ without loss of generality. According to the value of $r$, the neutron $^3P_2$ pairing exhibits various phases with different (continuous or discrete) symmetries: the O(2) symmetry for $r = -1$, the D$_2$ symmetry (a discrete symmetry for rectangular form) for $-1 < r < -1/2$ and the D$_4$ symmetry (a discrete symmetry for square prism) for $r = -1/2$. These states are called the UN (uniaxial nematic) phase, the D$_2$-BN (D$_2$ biaxial nematic) phase and the D$_4$-BN (D$_4$ biaxial nematic) phase, respectively. The phases are determined by minimizing the free energy with respect to $r$ at different magnetic fields, see Table 38.1.

In each phase, the symmetry breaking patterns are given as redundant information with the above

$$U(1) \times SO(3)_{L+S} \rightarrow O(2), \qquad (38.4)$$

in the UN phase,

$$U(1) \times SO(3)_{L+S} \rightarrow D_2, \qquad (38.5)$$

in the D$_2$-BN phase and

$$U(1) \times SO(3)_{L+S} \rightarrow D_4, \qquad (38.6)$$

in the D$_4$-BN phase. Here $U(1)$ is the global symmetry and $SO(3)_{L+S}$ is the symmetry for simultaneous rotation in three-dimensional space and spin stemming from the LS potential.
The symmetry breaking patters in Eqs. (38.4), (38.5) and (38.6) are important for the properties of the neutron
$^3P_2$ superfluids. We remind us that, in the symmetry breaking from the group $G$ to the subgroup $H$, the coset space
$G/H$ exhibits nontrivial homotopy groups $\pi_n(G/H)$, see, e.g., Refs. [1022, 1052]. In two-dimensional space, a variety
of homotopy groups in $\pi_1(G/H)$ leads to the rich phenomena of quantum vortices in neutron stars in rotations and/or
magnetic fields.

38.2.2. BdG theory and phase diagram

The neutron $^3P_2$ superfluids are described in terms of the non-relativistic Hamiltonian with the inter-neutron interaction.
The relativistic corrections are supplied by the LS potential which is proportional to $L \cdot S$. Here, the angular momentum
$L$ is coupled to the total spin $S$ in the two neutron pairs, where the total spin is defined by $S = s_1 + s_2$ with $s_i$ the spin
operator for neutrons labeled by $i = 1, 2$. The interaction with the magnetic field ($B$) is introduced through the term
$-\gamma_n s_i \cdot B$, where $\gamma_n = 1.2 \times 10^{-13}$ MeV/T (T for unit of tesla) is the gyromagnetic ratio of a neutron.

The superfluid state can be analyzed by solving the BdG equation for the given Hamiltonian for neutrons. In the BdG equation, the gap function is obtained by the self-consistent solution in the mean-field approximation, see, e.g., Ref. [1022, 1052] as recent studies. This is equivalent to the evaluation of the effective action in the one-loop approximation in the field theory. The BdG equation gives the precise solution of the gap in terms of the fermionic degrees of freedom.

<figure><img src="image_119.png" /><figcaption>**Figure 38.1:** The schematic figure for the phase diagram of the neutron $^3P_2$ superfluids is shown on the plane spanned by temperature ($T$) and magnetic field ($B$) (cf. Table 38.1). More precise information is found in Refs. [1022, 1052]. See the text for the details.</figcaption></figure>

The phase diagram of the neutron $^3P_2$ superfluids is obtained by the BdG equation as shown schematically in Fig. 38.1 (see also Table 38.1). In the figure, $T_c$ is the critical temperature for phase transition from superfluid phase to normal phase. The properties of the phase diagram is explained in the followings. At zero magnetic field, the UN phase is realized on the temperature axis below $T_c$. This phase is, however, unstable against the nonzero magnetic field, and it changes to the $D_2$-BN phase at nonzero but small magnetic fields. At stronger magnetic fields, instead of the $D_2$-BN phase, the
D$_{4}$-BN phase appears as the most stable state. The D$_{4}$-BN phase survives in the strong magnetic fields due the parallel spin in the neutron pairings as explained in Sec. 38.2.1.

More detailed information is explained for the phase diagram in Fig. 38.1. The transitions between the UN phase and the D$_{2}$-BN phase is the second-order transition, while the transitions between the D$_{2}$-BN phase and the D$_{4}$-BN phase exhibit the first-order transition at lower temperatures and the second-order transitions at higher temperatures. It is important to note the existence of the critical end point (CEP) at the meeting point of the lines from the first-order and the second-order phase transitions. The CEP shows the power-law divergences for several physical quantities, such as the heat capacity, the magnetization, and the spin susceptibility [1053]. Interestingly, it leads to the new universality class for powers which has not been known for the other quantum systems.

### 38.2.3. GL theory: bosonic effective theory

Near the critical temperature ($T \lesssim T_c$), the relevant degrees of freedom are given basically in terms of the bosons, i.e., the pairs of fermions (nucleons). In this temperature region, the bosons are regarded as the low-energy effective degrees of freedom. The effective theory for bosons is given by the GL theory as the effective theory of the BdG equation for fermions [1014, 1054, 1055, 1056, 1039, 1057, 1040, 1043, 1058, 1059, 1060, 1048, 1061, 1053, 1052].

Noting the traceless and symmetric $3 \times 3$ tensor, $A$, for the order parameter in Eq. (38.2), whose diagonal form was shown in Eq. (38.3), the GL free energy is given by

$$f_8[A] = f_8^{(0)}[A] + f_2^{(\le 4)}[A] + f_4^{(\le 2)}[A], \qquad (38.7)$$

as a truncated series of the power function of $A$ up to $O(A^8)$, see, e.g., Refs. [1058, 1060, 1053, 1052]. Each term is defined
$$
\begin{align}
f_8^{(0)}[A] &= K^{(0)} \sum_{i,j,\mu=1,2,3} \left( \nabla_j A_{i\mu}^* \nabla_j A_{\mu i} + \nabla_i A_{i\mu}^* \nabla_j A_{\mu j} + \nabla_i A_{j\mu}^* \nabla_j A_{\mu i} \right) \nonumber \\
&\quad + \alpha^{(0)} (\text{tr} A^* A) + \beta^{(0)} \left( (\text{tr} A^* A)^2 - (\text{tr} A^{*2} A^2) \right) \nonumber \\
&\quad + \gamma^{(0)} \left( -3 (\text{tr} A^* A) (\text{tr} A^2) (\text{tr} A^{*2}) + 4 (\text{tr} A^* A)^3 + 6 (\text{tr} A^* A) (\text{tr} A^{*2} A^2) + 12 (\text{tr} A^* A) (\text{tr} A^* A A^* A) \right. \nonumber \\
&\qquad \left. - 6 (\text{tr} A^{*2}) (\text{tr} A^* A^3) - 6 (\text{tr} A^2) (\text{tr} A^{*3} A) - 12 (\text{tr} A^{*3} A^3) + 12 (\text{tr} A^{*2} A^2 A^* A) + 8 (\text{tr} A^* A A^* A A^* A) \right) \nonumber \\
&\quad + \delta^{(0)} \left( (\text{tr} A^{*2})^2 (\text{tr} A^2)^2 + 2 (\text{tr} A^{*2})^2 (\text{tr} A^4) - 8 (\text{tr} A^{*2}) (\text{tr} A^* A A^* A) (\text{tr} A^2) - 8 (\text{tr} A^{*2}) (\text{tr} A^* A)^2 (\text{tr} A^2) \right. \nonumber \\
&\qquad \left. - 32 (\text{tr} A^{*2}) (\text{tr} A^* A) (\text{tr} A^* A^3) - 32 (\text{tr} A^{*2}) (\text{tr} A^* A A^* A^3) - 16 (\text{tr} A^{*2}) (\text{tr} A^* A^2 A^* A^2) \right. \nonumber \\
&\qquad \left. + 2 (\text{tr} A^{*4}) (\text{tr} A^2)^2 + 4 (\text{tr} A^{*4}) (\text{tr} A^4) - 32 (\text{tr} A^{*3} A) (\text{tr} A^* A) (\text{tr} A^2) \right. \nonumber \\
&\qquad \left. - 64 (\text{tr} A^{*3} A) (\text{tr} A^* A^3) - 32 (\text{tr} A^{*3} A A^* A) (\text{tr} A^2) - 64 (\text{tr} A^{*3} A^2 A^* A^2) - 64 (\text{tr} A^{*3} A^3) (\text{tr} A^* A) \right. \nonumber \\
&\qquad \left. - 64 (\text{tr} A^{*2} A A^{*2} A^3) - 64 (\text{tr} A^{*2} A A^* A^2) (\text{tr} A^* A) + 16 (\text{tr} A^{*2} A^2)^2 + 32 (\text{tr} A^{*2} A^2) (\text{tr} A^* A)^2 \right. \nonumber \\
&\qquad \left. + 32 (\text{tr} A^{*2} A^2) (\text{tr} A^* A A^* A) + 64 (\text{tr} A^{*2} A^2 A^{*2} A^2) - 16 (\text{tr} A^{*2} A A^{*2} A) (\text{tr} A^2) + 8 (\text{tr} A^* A)^4 \right. \nonumber \\
&\qquad \left. + 48 (\text{tr} A^* A)^2 (\text{tr} A^* A A^* A) + 192 (\text{tr} A^* A) (\text{tr} A^* A A^{*2} A^2) + 64 (\text{tr} A^* A) (\text{tr} A^* A A^* A A^* A) \right. \nonumber \\
&\qquad \left. - 128 (\text{tr} A^* A A^{*3} A^3) + 64 (\text{tr} A^* A A^{*2} A A^* A^2) + 24 (\text{tr} A^* A A^* A)^2 + 128 (\text{tr} A^* A A^* A A^{*2} A^2) \right. \nonumber \\
&\qquad \left. + 48 (\text{tr} A^* A A^* A A^* A A^* A) \right), \tag{38.8}
\end{align}
$$

$$
f_2^{(\le 4)}[A] = \beta^{(2)} \mathbf{B}^t A^* A \mathbf{B} + \beta^{(4)} |\mathbf{B}|^2 \mathbf{B}^t A^* A \mathbf{B}, \qquad (38.9)
$$

$$
\begin{equation}
\begin{split}
f_4^{(\le 2)}[A] ={}& \gamma^{(2)} \left( -2 |\mathbf{B}|^2 (\text{tr } A^2) (\text{tr } A^{*2}) - 4 |\mathbf{B}|^2 (\text{tr } A^* A)^2 + 4 |\mathbf{B}|^2 (\text{tr } A^* A A^* A) + 8 |\mathbf{B}|^2 (\text{tr } A^{*2} A^2) + \mathbf{B}^t A^2 \mathbf{B} (\text{tr } A^{*2}) \right. \\
& \left. - 8 \mathbf{B}^t A^* A \mathbf{B} (\text{tr } A^* A) + \mathbf{B}^t A^{*2} \mathbf{B} (\text{tr } A^2) + 2 \mathbf{B}^t A A^{*2} \mathbf{A} \mathbf{B} + 2 \mathbf{B}^t A^* A^2 A^* \mathbf{B} - 8 \mathbf{B}^t A^* A A^{*2} \mathbf{A} \mathbf{B} \right. \\
& \left. - 8 \mathbf{B}^t A^{*2} A^2 \mathbf{B} \right),
\end{split}
\tag{38.10}
\end{equation}
$$

with the derivative $\nabla_i$ for the spatial direction $i = 1, 2, 3$ and the external magnetic field $\mathbf{B}$. The coefficients in the GL free energy are obtained as

$$
\begin{align}
K^{(0)} &= \frac{7 \zeta(3) N_F p_F^4}{240 m^2 (\pi T_c)^2}, \quad \alpha^{(0)} = \frac{N_F p_F^2}{3} \log \frac{T}{T_c}, \notag \\
\beta^{(0)} &= \frac{7 \zeta(3) N_F p_F^4}{60 (\pi T_c)^2}, \quad \beta^{(2)} = \frac{7 \zeta(3) N_F p_F^2 \gamma_n^2}{48 (1 + G_0^{(n)})^2 (\pi T_c)^2}, \notag \\
\beta^{(4)} &= - \frac{31 \zeta(5) N_F p_F^2 \gamma_n^4}{768 (1 + G_0^{(n)})^4 (\pi T_c)^4}, \quad \gamma^{(0)} = - \frac{31 \zeta(5) N_F p_F^6}{13440 (\pi T_c)^4}, \notag \\
\gamma^{(2)} &= \frac{31 \zeta(5) N_F p_F^4 \gamma_n^2}{3840 (1 + G_0^{(n)})^2 (\pi T_c)^4}, \quad \delta^{(0)} = \frac{127 \zeta(7) N_F p_F^8}{387072 (\pi T_c)^6}. \tag{38.11}
\end{align}
$$

Here we define $N_F = mp_F/(2\pi^2)$ for the density-of-state of the neutrons on the Fermi surface for the neutron mass $m$
and the Fermi momentum $p_F$. $\zeta(n)$ is the zeta function. The GL free energy (38.7) with the coefficients in Eq. (38.11)
is deduced analytically by the one-loop approximation for the effective action from the BdG equation. In the derivation,
the particle-hole symmetry is supposed as an approximation for simplicity which should be valid at sufficiently high
density. In the above expression, we consider that the neutron magnetic moment $\mu_n$ in vacuum should be replaced to
$\mu_n^* = \gamma_n s / (1 + G_0^{(n)})$ with a neutron spin $s$ and the neutron gyromagnetic ratio $\gamma_n$ due to the modification by the Landau
parameter $G_0^{(n)}$ in the Fermi liquid theory at finite density.

In the GL free energy (38.7), each term has the following meanings. At $\mathcal{O}(A^2)$, the first term ($K^{(0)}$) is the kinetic term and the second term ($\alpha^{(0)}$) indicates the leading interaction. At $\mathcal{O}(A^4)$, the interaction term ($\beta^{(0)}$) exhibits the SO(5)
symmetry as the dynamical symmetry which is absent in the BdG theory. This symmetry appears accidentally up to this order, and it generates the quasi Nambu-Goldstone (qNG) bosons [1062]. The SO(5) symmetry is, however, explicitly broken by the term ($\gamma^{(0)}$) at $O(A^6)$, and hence the qNG bosons disappear finally. Nevertheless, we cannot stop the expansion up to $O(A^6)$, because the GL free energy up to this order has only the local minimum but no global minimum. This problem is solved by the term ($\delta^{(0)}$) at $O(A^8)$ by which the global minimum is realized successfully [1060]. It is important that the terms up to $O(A^8)$ reproduce qualitatively the position of the CEP that is already known in the BdG theory (cf. Fig. 38.1). Thus, the GL free energy up to $O(A^8)$ supplies the minimum set of terms. The complexity of the GL equation for the neutron $^3P_2$ superfluids is qualitatively different from the case of usual S-wave superfluids with second-order phase transition in which usually the terms up the quadratic order are enough.

In Eq. (38.7), the magnetic terms are calculated not only at the leading order (LO) ($\beta^{(2)}$) $O(B^2A^2)$ but also at the next-to-leading orders (NLO) ($\beta^{(4)}$ and $\gamma^{(2)}$) $O(B^4A^2)$ and $O(B^2A^4)$ [1058]. The NLO terms for the magnetic field are necessary in order for checking the convergence of the expansion series in strong magnetic fields such as in magnetars.

The minimization of the GL free energy with respect to the order parameter $A$ provides the phase diagram of the neutron $^3P_2$ superfluids as shown, e.g., in Refs. [1058, 1060, 1048, 1053, 1052]. The GL equation up to $O(A^8)$ reproduces the general features of the phase diagram obtained in the BdG equation (cf. Fig. 38.1). At quantitative level, however, we notice that the positions of the first-order phase transition line and the CEP in the GL theory are different from the original ones in the BdG theory. Nevertheless, it should be emphasized that the neutron $^3P_2$ superfluid phases changing from the UN phase to the $D_4$-BN phase through the $D_2$-BN phase can be described by the GL theory successfully. Therefore, the GL theory can be used to understand the properties of the neutron $^3P_2$ superfluids inside neutron stars and magnetars. We comment that the GL free energy (38.7) can be furthermore extended to include the condensate of the neutron $^1S_0$ superfluids allowing the coexistence of the neutron $^1S_0$ and $^3P_2$ phases [1061] for studying a wide range of phases in neutron stars and magnetars.

### 38.3. Topological surface defects on neutron star and magnetar

As introduced in Sec. 38.2.1, there are many interesting topological properties in the neutron $^3P_2$ superfluids. Among them, we focus on the topological defects on the surface of neutron star and magnetars [1048]. This subject is interesting not only in the astrophysics but also in the condensed matter physics such as the $^3$He superfluids and liquid crystals [1063, 1064, 1065]. In this subject, we are interested in the properties of the neutron $^3P_2$ superfluids at the surface, not in bulk, and thus it is important to consider the appropriate boundary condition for the neutron $^3P_2$ faced to the outer space.

At the surface, the directions of the neutron $^3P_2$ superfluids are constrained due to the anisotropy of space. Let us consider for simplicity that the boundary region is regarded approximately as the flat plane by neglecting the curvature [1048]. Such approximation will be valid for the surface of a neutron star, because the scale length of the coherent length of the neutron $^3P_2$ pairings is much smaller than the scale size for curvature of the star surface. Thus, the boundary condition is characterized by the normal vector $n$ whose direction is perpendicular to the surface. Here the positive direction of $n$ is defined to be outward from the inside to the outside.

In our setting for the boundary condition, we suppose that the outer space of the neutron $^3P_2$ superfluids is simply a vacuum rather than the other matter phase, such as the neutron $^1S_0$ superfluids. In reality, we may need to consider that the neutron $^3P_2$ superfluid may be faced to the neutron $^1S_0$ superfluids, because the $^1S_0$ superfluids should be realized at lower density region near the surface of the neutron stars (cf. Ref. [1061]). Thus, the present simple setting should be
regarded as being to the first step to investigate the surface effect. Regardless of the simple approximation, we will show that there are rich phenomena induced by topology on the surface.

We consider the constraint condition for the condensate matrix A at the surface:

$$n^t An = 0, \qquad (38.12)$$

where $n$ is the normal vector perpendicular to the surface. Here $n^t$ is a transpose of $n$. In addition, we define the two-dimensional vector

$$A_n = n^t A, \qquad (38.13)$$

which satisfies the condition $A_n \cdot n = 0$ from Eq. (38.12). Thus, $A_n$ is a two-dimensional vector field confined on the surface. The two-dimensional vector $A_n$ exhibits an interesting behavior on the surface: it induces the topological defects as vortices. Such defects emerge from the property that the boundary condition (38.12) has the symmetry $(S^1 \times S^1)/\mathbb{Z}_2$: the invariance under the one-axis rotation around the normal vector $n$ and the global U(1) phase in the complex matrix A.

<figure><img src="image_120.png" /><figcaption>**Figure 38.2:** The schematic figure for the topological charges is shown for the two-dimensional vector fields on the surface. The panels (a)-(d) indicate +1 charge, and the panels (e)-(h) indicate -1 charge. See the text for the details.</figcaption></figure>

Here we note that, in general, there are two different types of topological surface defects with topological charges $\pm 1$, see Fig. 38.2. The vector-field configurations with topological charges can be deformed smoothly. As a result, the defects with the same charge are not distinguished from the topological point of view, while the configurations with different topological charges cannot be transformed without creating singular points. Thus, the charges of topological defects on the surface should be classified to either +1 or -1. Therefore, it should be expected that the two-dimensional vector $A_n$ should exhibit the topological defects as well.

The solutions of the condensate matrix A under the boundary condition (38.12) can be obtained numerically. Similarly
to Eq. (38.2), we express A by

$$A(x) = \begin{pmatrix} f_1(x) & g_3(x) & g_2(x) \\ g_3(x) & f_2(x) & g_1(x) \\ g_2(x) & g_1(x) & -f_1(x) - f_2(x) \end{pmatrix}, \qquad (38.14)$$

where the one-dimensional axis ($x \ge 0$) is defined as the half line. This half line is directed from the surface of the neutron star ($x = 0$) to the bulk space in center of a neutron star ($x \to \infty$). Because the neutron stars are too large in comparison with the coherence length of the neutron $^3P_2$ superfluids, the large limit of $x$ can be regarded as an infinity. The boundary condition at the infinity is supplied by the bulk phases, i.e., the UN, $D_2$-BN and $D_4$-BN phases, according to the strengths of the magnetic fields, as summarized in Table 38.1. We observe, as a matter of course, that the condensate matrix $A$ are dependent on the distance from the surface. Some examples of the behaviors of the solutions are shown in Ref. [1048]. We notice that the off-diagonal components, $g_1$, $g_2$ and $g_3$, in Eq. (38.14) cannot be eliminated by global transformation in general due to the boundary condition (38.12). Indeed, not only the diagonal components, $f_1$ and $f_2$, but also the off-diagonal components, $g_1$, $g_2$ and $g_3$, play important roles for generating the topological surface defects. They all together contribute to form the two-dimensional vector field $A_n$.

With the above setup, the configurations of $A_n$ can be presented on an “atlas” of the neutron star surface, where the Mercator projection is adopted with the longitude ($\phi$) and the attitude ($\theta$) [1048]. We consider zero, weak and strong strengths of the magnetic fields, in which the corresponding bulk phase is given by the UN phase, the $D_2$-BN phase and the $D_4$-BN phase at the center of neutron star, see Table 38.1. The vector stream lines on the surface are different to each magnetic field (or the bulk phase), as presented in Ref. [1048]. We notice that there exist several topological defects with charges $\pm 1$, regarded as the vortices, induced by $A_n$. It is interesting that the spatial distributions of the topological charge on the surface are changed by tuning the strengths of the magnetic field, as summarized in Table 38.2. Correspondingly, the surface defects have +10 and -8 charges in the case of the UN phase, +12 and -10 charges in the case of the $D_2$-BN phase and +8 and -6 charges in the case of the $D_4$-BN phase. In any case, the total charges are always +2 irrespective to the symmetries in the bulk phases.

The total charge +2 is not accidental, because the total charges of defects of two-dimensional vector fields on closed manifolds should be determined solely by the Euler characteristic of the closed manifold. This is the Poincaré-Hopf theorem known as the hairy ball theorem. Precisely, this theorem is stated as

$$\sum_{p \in M} \mathrm{index}_p v = \chi(M), \qquad (38.15)$$

for the closed manifold $M$ and the vector field $v$ on $M$. In the left-hand side, $p \in M$ is the points on $M$ and $\mathrm{index}_p v$ means the index zero-point of $v$ at $p$, indicating charges $\pm 1$ in our case. Thus, the left-hand side presents the sum of the indices on $M$. In the right-hand side, $\chi(M)$ is the Euler characteristic for the manifold $M$, giving $\chi = 2$ in the case of a ball-like neutron stars and magnetars. Therefore, the total charges of the two-dimensional vector fields on the sphere should be always two. This theorem is applied to the case of the neutron $^3P_2$ superfluids, where the two-dimensional vector field is supplied by $A_n$, and the total charge +2 is assured always. In terms of the dynamics, it is interesting to observe pair-creations and annihilations of the defects on the surface of neutron stars: two pairs of defects and anti-defects are created from zero magnetic field to weak magnetic field, and four pairs of defects and anti-defects are annihilated from weak magnetic field to strong magnetic field. Notice that a pair of defects with +1 and -1 charges should be simultaneously created and/or annihilated to conserve the total charge.
<table><caption>**Table 38.2:** The numbers of the topological surface defects are summarized for different bulk phase at center of neutron stars under different strengths of magnetic fields [1048]. See the text for the details.</caption><thead><tr><th>magnetic field strength</th><th>zero</th><th>weak</th><th>strong</th></tr></thead><tbody><tr><td>bulk phase at center</td><td>UN</td><td>D<sub>2</sub>-BN</td><td>D<sub>4</sub>-BN</td></tr><tr><td>positive charge</td><td>10</td><td>12</td><td>8</td></tr><tr><td>negative charge</td><td>-8</td><td>-10</td><td>-6</td></tr></tbody></table>

Summarizing this section, considering the bulk phase and the boundary condition induces the topological surface defects whose total charge is conserved by the Poincaré-Hopf theorem. This also has impact on the astrophysical research: the correspondence between the distributions of defects on the surface and the bulk phase at the center makes it possible to research the internal bulk phase by observing the defects on its surface. Such topological surface defects may be investigated by observing neutron stars and magnetars.

## 38.4. Summary

In this section, we have reviewed recent progress on the neutron $^3P_2$ superfluids in neutron stars and magnetars focusing on the surface defects. The research of the neutron $^3P_2$ superfluids is still producing new results and phenomena which should be shared not only with researchers in astrophysics and nuclear physics but also with those in condensed matter physics. Especially the topology is an important key word to understand the properties of the neutron $^3P_2$ superfluids. It is expected that neutron stars and magnetars are researched in terms of the topological stars in the coming future.

# 39. Neutron star kicks: rocket effects in strong B-fields

## 39.1. Introduction

Pulsars are rotating neutron stars with high magnetic fields causing observable radio dipole signals. Most of the known pulsars are born in the neighborhood of the galactic plane and move away from it with natal kick velocities which are typically higher than those of their progenitors [1066]. This implies that the birth process of pulsars also produces their high velocities and thus cannot be entirely isotropic [1067]. Two examples for directly observed pulsar motion are the isolated neutron star RX J1856.5-3754 and for the binary system B0950+08. The spatial distribution of observed pulsar kick velocities in galactic coordinates is given in [1068]. Up to now, the mechanisms driving this asymmetry are far from clear. There are several hypotheses, ranging from asymmetric supernova explosions [1069] over neutron star instabilities [1070, 1071] and magnetorotational effects [1072, 1073] to the model of an electromagnetic [1074] or neutrino rocket [1069, 1075, 1076, 423]. For a recent overview, see [1077, 1078] and references therein. It is also not clarified whether the distribution of pulsar kick velocities is bimodal with a lower component of $v \le 100$km/s (20 % of the known objects) and a higher component of $v \ge 500$km/s (80%) as suggested by [1079] or whether it can be explained by a one-component distribution [1068]. Note, that the high velocity tail is possibly underrepresented, due to the fact that pulsars with high velocity move out of the observational area faster than lower ones [1080]. Most of the models are capable of explaining kick velocities of $v \sim 100$ km/s, but it is a nontrivial problem to explain the highest measured pulsar velocities around
1600 km/s.

While there is evidence for the alignment of kick velocity vectors with the rotation axis of pulsars [1081] the suggestion of a correlation between magnetic field and kick velocity [1067, 1082] could not be verified from observations so far.

In this section we show that beamed neutrino radiation from a strongly magnetized strange star in a color superconducting state could lead to an acceleration during the early cooling phase of the neutron star. It has recently been suggested that the typical properties of gamma ray bursts might be explained by the neutrino release off a strongly magnetized quark core star [1083, 1084] since neutrinos can very effectively be converted into $e^+e^-$ pairs and gammas [1085]. Here we show that such a collimated neutrino release (neutrino rocket) can lead to high neutron star kick velocities in qualitative agreement with the observational data. The key ingredient to this mechanism are: (i) the mass defect of the order of $10^{52}$ erg in the formation of a (color superconducting) quark star [1086, 1087], (ii) the anisotropic neutrino transport (beaming) since normal quark matter vortices (short mean free path) aligned with the strong magnetic field are immersed in a superconducting matrix (free neutrino propagation) and (iii) parity nonconservation of neutrino production in the strong magnetic field [1088, 1089] which leads to a sufficiently large net momentum transfer to the star resulting in a pulsar kick. The color superconductivity plays an essential role since those neutrinos generating the propulsion of the star propagate without secondary interaction thus circumventing the no-go theorem [1088, 1090], that there is no parity violation in equilibrium. In the remainder of this contribution, we explain the ingredients of the model and present results of a model calculation [1075].

## 39.2. Neutrino beaming by magnetic vortices and cooling delay

When designing a scenario for the origin of natal pulsar kicks it is natural to make contact to other puzzling phenomena occurring in the realm of a supernova explosion such as gamma ray bursts (GRBs). It has been suspected that GRB characteristics could find an explanation in the phase transition to quark matter in the protoneutron star (see [1083, 1084] and Refs. therein) whereby the phase with all quarks being paired with large gaps in a color-flavor-locking (CFL) state is of particular interest for our concern of the neutrino propagation. Let us, in the following, assume that the bulk of the star is in the CFL phase with a tiny hadronic crust and a surface magnetic field in excess of $B_s \sim 10^{12}$ G. Then, one may expect by flux conservation that the inner magnetic field at the quark core surface can reach values of $B_{in,s} = B_s(n_{in}/n_s)^{2/3} = 10^{12} - 10^{17}$ G, where $n_s \sim 6 \times 10^{-10}$ fm$^{-3}$ and $n_{in} \sim 0.3$ fm$^{-3}$ are the corresponding densities at the core-crust interface. The presence of such high magnetic fields allows creation of a vortex structure along the magnetic axis [1091] as shown in Fig. 39.1.

Due to magnetic flux conservation the number of vortices in the quark star interior is $N_{vo} = \pi B_{in,s} R^2 / (6\Phi_0)$, where $\Phi_0 = 2 \cdot 10^{-7}$ G cm$^2$ is the magnetic flux quantum, cf. [1091, 1092, 1093]. Superconductivity is expelled from the vortex interior, where the matter is in the state of a strongly magnetized quark-gluon plasma. The volume of a vortex can be estimated as $V_{vo} \sim 2\pi\lambda_D^2 R$, where the Debye screening length $\lambda_D$ is of the order of the penetration depth

$$ \lambda_{CFL} \approx (\mu_q/300 \text{ MeV})^{-1} (1 - T/T_c)^{-1/2} \text{ fm}, \qquad (39.1) $$

and for the critical temperature the BCS relation, $T_c = 0.57 \Delta$, can be adopted. The vortex volume has a large neutrino emissivity due to the quark direct Urca (QDU) process [1094]

$$ \epsilon_{\nu}^{QDU} \approx 2.2 \times 10^{26} \alpha_s u Y_e^{1/3} T_9^6 \text{ erg cm}^{-3} \text{ s}^{-1}, \qquad (39.2) $$
<figure><img src="image_121.png" /><figcaption>Figure 39.1: Vortex structure in highly magnetized superconducting quark stars.</figcaption></figure>

with $\alpha_s \sim 1$ being the strong coupling constant, the compression $u = n_b/n_0$ is the factor by which the baryon density exceeds its value at saturation $n_0 = 0.16 \text{ fm}^{-3}$, $Y_e$ is the electron fraction and $T_9$ the temperature in units of $10^9 \text{ K}$. Correspondingly, the neutrino mean free path (MFP) in normal quark matter becomes very small

$$ \lambda_{\nu} = 3.4 \times 10^6 \alpha_s^{-1} u^{-1} Y_e^{-1/3} T_9^{-2} \text{ cm.} \qquad (39.3) $$

Note that the direct Urca processes in quark matter, which is the responsible for the high neutrino production inside the star, remain virtually unaltered in the presence of magnetic fields in this range [865]. In the region between the vortices matter is in the CFL state with pairing gaps $\Delta \sim 100 \text{ MeV}$, so that the direct Urca process is suppressed by the exponential factor $\zeta_{QDU} = \exp(-\Delta/T)$ and the neutrino MFP $\lambda_{\nu}^{CFL} = \zeta_{QDU}^{-1} \lambda_{\nu}$ exceeds the radius of the star for temperatures below $T_{opac}^{CFL} \approx 15 \text{ MeV}$. This prominent difference between the neutrino MFP in normal (vortex core) and in CFL quark matter (matrix) leads to anisotropic neutrino emission from the star within a cone of the temperature-dependent opening angle $\theta_{\nu}(T) \sim \tilde{\lambda}_{\nu}(T)/R$, where $\tilde{\lambda}_{\nu}(T) = \lambda_{\nu}(T) V/(N_{vo} V_{vo})$, see Fig. 39.2.

The cooling evolution $T(t)$ can be described by inverting the solution of

$$ t = - \int_{T_i}^{T(t)} \frac{C_V(T') \, dT'}{L(T')} \qquad (39.4) $$

where $T_i \sim 40 \text{ MeV}$ is the initial temperature of the protoneutron star, $C_V(T) = [1 - N_{vo}V_{vo}/V] C_V^q(T)$ $\zeta_{DU}$ is the specific heat dominated by the normal quark matter contribution, and the luminosity

$$ L(T) = [1 - \cos \theta_{\nu}(T)] L_0(T) \approx L_0(T) \theta_{\nu}(T)^2/2 \qquad (39.5) $$
<figure><img src="image_122.png" /><figcaption>Figure 39.2: Time evolution of neutrino luminosity, temperature and neutrino beaming angle.</figcaption></figure>

takes into account the effect of the neutrino beaming. The isotropic luminosity

$$L_0(T) = (N_{vo}V_{vo}/V) \int dV \epsilon_{\nu}^{QDU}(T) + [1 - (N_{vo}V_{vo}/V)] \int dV \epsilon_{\nu}^{QDU}(T) \zeta_{QDU} \quad (39.6)$$

has contributions from both, the unsuppressed QDU process from the small volume of the vortices and the suppressed contribution from the large CFL matrix volume. Results for the time evolution of neutrino luminosity, temperature and neutrino beaming angle shown in Fig. 39.2 demonstrate the effect of cooling delay due to neutrino beaming in a strong magnetic field.

### 39.3. Pulsar kicks from color superconductivity parity violation

Parity nonconservation in the weak interaction QDU neutrino production process under the conditions of a strong magnetic field in a protoneutron star leads to a violation of reflection symmetry since the neutrino flux is a polar vector while the magnetic field is an axial one. The magnitude of this asymmetry has been estimated as $A_{\nu} \sim 10^{-4} B_{14}$, where $B_{14} = B/10^{14}$ G, see [1088] and [1089] for a similar result. This nonvanishing $A_{\nu}$ leads to a net momentum transfer from the neutrino flux to the star of mass M resulting in a time-dependent kick velocity

$$v(t) = \frac{A_{\nu} c}{Mc^2} \int_{0}^{t} dt' L_{\nu}(T), \quad (39.7)$$

which saturates at the B-dependent asymptotic value

$$v = 3.6 \times B_{14}(M/M_{\odot})^{-1} \text{ km s}^{-1} \quad (39.8)$$
as soon as the beaming ceases. Numerical results are shown in Fig. 39.3 for a typical compact star with $M = 1.4 M_\odot$ and $R = 10$ km. In order to obtain pulsar kicks as large as $10^3$ km s$^{-1}$, magnetic fields of the order of $10^{16} - 10^{17}$ G in the stars interior are required. Such values correspond to our above estimates.

<figure><img src="image_123.png" /><figcaption>**Figure 39.3:** Time evolution of the pulsar kick velocity for different magnetic fields (right panel) and magnetic field dependence of the final kick velocity (left panel).</figcaption></figure>

We have to comment on the argument, that any initial asymmetry in neutrino emission processes is only a local phenomenon and can not survive at large distances due to strong neutrino scattering inside the neutron star (no go theorem [1088, 1090]). Note, that in our case the neutrinos move along the magnetic axis between the normal quark matter vortices in areas where the quark matter is in a CFL phase and therefore all neutrino interactions are exponentially suppressed with the gap in the energy spectrum. This leads to a strong suppression of scattering processes in this direction. The initial asymmetry in neutrino emission can therefore survive and lead to the observable pulsar kick.

Finally, one can test the predictive power of the neutrino rocket scenario by testing the correlation between the pulsar mass distribution and the kick velocity distribution.

## 39.4. Electromagnetic rocket effect

The electromagnetic rocket effect was introduced in [1095] and has recently been revisited by [1074]. It is based on a displacement $s$ of a strong dipolar magnetic field from the center of the NS (see Fig. 39.4) and the sufficiently fast rotation of the latter. As it was discussed in [1074], observational evidence provided by the NICER (Neutron star Interior Composition ExploreR) X-ray observatory, particularly from investigating the location of hot spots on the surfaces of MSPs [1096, 1097], indicates that the magnetic field diverges from a centered dipole configuration. Based on the results for PSR J0030+0451, [1098] estimated the magnetic field structure and found an off-centered magnetic field consisting of dipole and quadruple components.

Since both the neutrino and electromagnetic rocket mechanisms require strong magnetic fields of at least the young pulsar field strength $B \sim 10^{12}$ Gauss, such effects seem at first glance unlikely for “old” MSPs with low magnetic fields of $B \sim 10^8$ Gauss. However, in systems with mass accretion from a companion star like in the case of LMXBs, the argument of [1099] can be applied that the strong magnetic field of the interior gets buried in the crust of the pulsar [1100, 1101, 1102]. Thus, a small surface magnetic field is compatible with a super-strong magnetic field in the interior. Due to such a magnetic field profile, kick velocities of a few dozens of km s$^{-1}$ can be produced in MSPs, as it has been
<figure><img src="image_124.png" /><figcaption>**Figure 39.4:** Displacement of the strong dipolar magnetic field inside the NS as a condition for the electromagnetic rocket effect. Figure from Ref. [1074].</figcaption></figure>

demonstrated in Ref. [1074] by applying the electromagnetic rocket effect to the case of PSR J0030+0451 with its true transverse velocity not exceeding 17 km/s that is obtained by the electromagnetic rocket effect for a displacement $s = 3.02$ km and the initial spin period 3.52 ms.

## 39.5. Eccentric and isolated MSPs from accretion-induced twin star transition

In concluding this section, we discuss a possible application of neutron star kick mechanisms by electromagnetic or neutrino rocket effects for the explanation of the puzzle why a few millisecond pulsars in low-mass X-ray binaries with periods in the range of 20-50 days have orbits with large eccentricities $e \sim 0.1$, while generally the orbits are rather circular with eccentricities following the Phinney line [1103], see Fig. 39.5.

In Ref. [1104] it has been suggested that an accretion-induced phase transition from nuclear to quark matter in the inner core of the neutron star, transforming it to a hybrid star on the disconnected third family branch [1106] entails a mass defect that is sufficient to cause an orbital reconfiguration that explains the observed large eccentricities $e \sim 0.1$. Earlier, in refs. [1107, 1108] the authors have already discussed that an eccentricity of $\sim 0.11 - 0.15$ will be induced just by instantaneous gravitational mass loss of the NS due to a phase transition in its interior, even without invoking an explicit kick mechanism. However, in their scenario, it may not be possible to explain the “instantaneity” of the transition when the NS spin is included. As has been pointed out by Glendenning et al. in [1109] in the context of their scenario for an accretion-induced phase transition to a hybrid star branch connected with the NS one, the reconfiguration of the mass distribution in the NS interior is accompanied by a change in the moment of inertia. This would entail a “pirouette effect” (a spin-up), which in turn would inhibit the completion of the transition due to the sensible dependence of the density profile inside the star on its spin state caused by centrifugal forces. The authors obtain in their scenario a time scale of $10^5$ years for the completion of the phase transition, which is in stark contrast with the requirement that the mass defect has to occur on a timescale much smaller than the orbital period.

This problem is circumvented when the phase transition leads to a member of the third family of compact stars
<figure><img src="image_125.png" /></figure>

<figure><img src="image_126.png" /><figcaption>**Figure 39.5:** Eccentricity vs. orbital period for millisecond pulsars in binaries with white dwarf companions. Left panel: observations compared with the Phinney line [1103], from Ref. [1104]. Right panel: Monte-Carlo simulation for the post-transition distribution of orbital eccentricities resulting from a transition-triggered mass loss of $\Delta M = 0.016 M_{\odot}$ and a fixed magnitude kick velocity $w$ corresponding to model M2 of Ref. [1105].</figcaption></figure>

(so-called “twin stars”) beyond the conventional white dwarf (WD) and NS classifications, which could emerge as the result of a sufficiently strong phase transition in the NS core [1110, 1111, 1112]. In that case, baryon number and angular momentum can be conserved simultaneously during the transition so that this process can occur almost instantaneously, on a dynamic timescale, which is insignificant compared to the orbital period. This twin transition scenario has been worked out in detail recently in Ref. [1105], where also a Monte-Carlo simulation was performed which resulted in a distribution of post-transition eccentricities as a function of the orbital period which is shown in Fig. 39.5 (right panel), including an additional explicit kick velocity parameter $w$, varied between 0 and 100 km/s.

The evolution of the orbit and the NS spin were followed until the NS reached the phase transition threshold (see Fig. 39.6, left panel), when the phase transition occurred instantaneously (i.e., on a time scale that is much shorter than the orbital period). To investigate the impact of the phase transition on the orbit, the prescriptions of [1113] and [1114] were used, in which the ratio of the post-transition semi-major axis $a_f$ to the pre-transition one ($a_i$) is given by:

$$ \frac{a_f}{a_i} = \frac{1 - \Delta M/M}{1 - 2\Delta M/M - (w/v_{\text{rel}})^2 - 2\cos\theta(w/v_{\text{rel}})}. \qquad (39.9) $$

Here $\Delta M$ is the instantaneous mass defect corresponding to the released gravitational binding energy during the phase transition, $M$ is the total mass of the pre-transition system, $v_{\text{rel}}$ is the relative velocity between the two stars ($v_{\text{rel}} = \sqrt{GM/a_i}$), $w$ is the magnitude of the kick velocity, and $\theta$ is the kick angle between the kick velocity vector and the pre-transition orbital velocity vector. The eccentricity of the post-transition binary system is given by:

$$ e = \sqrt{1 + \frac{2E_{\text{orb,f}}L_{\text{orb,f}}^2}{\mu_f G^2 M_{\text{f,1}}^2 M_{\text{f,2}}^2}}, \qquad (39.10) $$

where $L_{\text{orb,f}} = a_i \mu_f \sqrt{(v_{\text{rel}} + w \cos\theta)^2 + (w \sin\theta \sin\phi)^2}$ is the post-transition orbital angular momentum, with $\phi$ being the kick angle on the plane perpendicular to the pre-transition velocity vector, $\mu_f$ is the post-transition reduced mass, and $E_{\text{orb,f}} = -GM_{\text{f,1}}M_{\text{f,2}}/2a_f$ is the post-transition orbital energy as a function of the post-transition masses $M_{\text{f,1}}$ and $M_{\text{f,2}}$ in the binary.

To investigate the post-transition orbital configurations, a mass defect of $\Delta M = 0.016 M_{\odot}$ obtained for the considered hybrid EOS (see Fig. 39.6, right panel) and secondary kicks with magnitudes $w$ up to 100 km s$^{-1}$ were considered. It
<figure><img src="image_127.png" /><figcaption>**Figure 39.6:** Left panel: Gravitational mass vs. equatorial radius for constant angular momentum values $J = 0, ..., 0.45 J_0$ (dotted lines). The black dashed line highlights the points of maximum stability in the hadronic branches, signaling the onset of a phase transition (PT). The thick magenta curve connects points on the hybrid star branches that can be reached through a collapse conserving both angular momentum and baryon mass. Star markers denote the endpoint trajectories for direct (M2) and delayed (M3) collapse models. Here, the arrows are inclined because gravitational mass, unlike baryonic mass, is not conserved during the PT. Right panel: Baryonic mass ($M_b$) and gravitational mass ($M_g$) as functions of the central energy density ($\varepsilon_c$) for a fixed angular momentum $J$. Vertical gray lines mark the central energy densities at the onset ($\varepsilon_c = 275 \text{ MeV/fm}^3$) and completion ($\varepsilon_c = 595 \text{ MeV/fm}^3$) of the phase transition (PT). Black dotted lines and the blue-shaded region denote unstable configurations. The inset zooms into the PT region, illustrating the mass defect $\Delta M_g = 0.016 M_\odot$ associated with the transition from hadronic to quark matter. From Ref. [1105].</figcaption></figure>

was assumed that the kick lacks a preferential orientation, so that the kick angles $\theta$ and $\phi$ were modeled as uniformly distributed variables. The range of kick velocities is consistent with results for the electromagnetic rocket effect discussed in subsection 39.4.

The results for the model M2 are shown in the right panel of Fig. 39.5 and compare well with the observed effect shown in the left panel of that figure, even without an explicit kick mechanism like electromagnetic or neutrino rocket effect. However, when such rocket effects in the strong magnetic field of a pulsar are considered, the model [1105] for accretion-induced twin star phase transitions in low-mass X-ray binaries allows to explain the disruption of the binary and thus provides a scenario for the origin of isolated MSPs.

## 39.6. Summary and outlook

In this section, we have demonstrated on the example of neutrino and electromagnetic propulsion (rocket) effects for neutron stars that if these mechanisms can serve as the explanation of kick velocities up to 1000 km/s, they require strong magnetic fields in excess of $10^{12}$ G. The neutrino rocket effect we outlined here requires a channeling and thus collimation of the neutrino emission along the magnetic field axis where magnetic vortex lines are formed in color superconducting (CFL phase) quark matter. Perpendicular to the vortex lines, the neutrino propagation is in the diffusion regime because of the frequent scattering on the dense network of these lines while parallel to them the neutrinos are free streaming along the B-field axis because of their long mean free path in CFL quark matter where scattering processes are exponentially suppressed by large pairing gaps. Within the cores of the vortex lines quark matter is normal (non-superconducting) and therefore copious neutrino production by the unsuppressed direct Urca process can take place, in particular at the early (hot) stages of the evolution. The neutrino rocket scenario seems to be most appropriate for NS at their birth in supernova
explosions, where velocities up to 1000 km/s can be reached for young pulsars with high, magnetar-like B-fields.

For the late evolution phases of cold, recycled MSPs in LMXBs, the electromagnetic rocket effect seems to be ap-
propriate which produces moderate kick velocities of ~ 10...100 km/s. Those kick velocities are sufficient to provide a
possible scenario for the origin of both eccentric and isolated MSPs, as has been demonstrated in recent Monte-Carlo
simulations discussed in this section. It has been pointed out that for the accretion induced phase transition scenario of
explaining MSPs with eccentric orbits it is important that a disconnected third family branch of of hybrid stars exists.
This requires that the QCD deconfinement transition is strongly first order and occurs at moderate densities in NS with
typical masses of 1.4 $M_{\odot}$.

Future observations and further theoretical studies have to contribute to a further clarification of the role of the phase
transition to quark matter, color superconducting phases and anomalous neutrino transport in QCD at strong magnetic
fields as they can be met in neutron stars.
# Part VII

## Conclusions

In this work we have presented a comprehensive review of the state-of-the-art in the investigation of the properties of particle systems subject to the influence of strong magnetic fields. The review was motivated by discussions and presentations 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. We have divided the work into five parts, each accounting for the main subject discussed during one of the five days of the workshop. Based on the different frontier topics that are hereby discussed, we can conclude that the field is very active and rapidly evolving. Nevertheless, the present work captures a large enough cross section of the main subjects of current interest, and provides also a very extensive list of references. The prospective for future developments will benefit from this kind of crosstalk exercises where different methods and techniques are used across seemingly different fields but in reality closely related by a driving subject, in this case, the physics of systems of particles in the presence of magnetic fields.
Acknowledgements

This work has been supported by STRONG-2020 "The strong interaction at the frontier of knowledge: fundamental research and applications" which received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 824093. M.K. was supported, in part, by the U.S. Department of Energy grant DE-SC0012447. The material presented in this article is based in part upon work done while C.C. was supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education (ORISE) for the DOE. ORISE is managed by ORAU under contract number DE-SC0014664. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE. CC is currently supported in part by the Netherlands Organisation for Scientific Research (NWO) under the VICI grant VI.C.202.104. S.G. was supported in part by the Office of Science, Office of Nuclear Physics, U.S. Department of Energy under Contract No. DE- FG88ER40388 and in part by a Feodor Lynen Research fellowship of the Alexander von Humboldt foundation. Part of the material presented in this work is based upon work done while S.G. was supported by the Fulbright Visiting Scholar Program which is sponsored by the US Department of State and the German-American Fulbright Commission and the DAAD. J.H. is supported by FWO-Vlaanderen through a Junior Postdoctoral Fellowship. Research by J.H. is also supported by FWO-Vlaanderen project G012222N, and by Vrije Universiteit Brussel through the Strategic Research Program High-Energy Physics. A.M. was financed, in part, by the São Paulo Research Foundation (FAPESP), Brasil, Process Number 2023/08826-7. The work of G.E., E.G.V. and G.M. was funded by the DFG (Collaborative Research Center CRC-TR 211 "Strong-interaction matter under extreme conditions" - project number 315477589 - TRR 211). G.E. acknowledges support by the Hungarian National Research, Development and Innovation Office (Research Grant Hungary). E.G.V. acknowledges support by the Helmholtz Graduate School for Hadron and Ion Research (HGS-HIRe for FAIR). The work of P.B. was funded in part by the UK STFC Consolidated Grant ST/T000988/1. This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), 312032/2023-4 (R.L.S.F.); Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (FAPERGS), Grants Nos. 19/2551-0000690-0 and 19/2551-0001948-3 (R.L.S.F.); The work is also part of the project Instituto Nacional de Ciência e Tecnologia - Física Nuclear e Aplicações (INCT - FNA), Grant No. 464898/2014-5, by Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) under Grant No. SEI-260003/019544/2022 (W.R.T). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 (R.M.N.) and Grant no. 88887.826087/2023-00 (R.P.C). A.B. acknowledges the support from the Alexander von Humboldt foundation. C.V. acknowledges financial support from ANID/FONDECYT under Grant No. 1190192. P.M.L. acknowledges partial support from the Polish National Science Center (NCN) under the Opus grant no. 2022/45/B/ST2/01527. The work of G.K. is a part of the project INCT-FNA proc. No. 464898/2014-5. G.K. is also supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant No. 309262/2019-4 and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Grant No. 2018/25225-9. The work of V.S.T. is also supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant No. 305004/2022-0 and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Grant No. 2019/10889-1. The work of A.A. was supported by UNAM-PAPIIT grant number IG100322 and by Consejo Nacional de Humanidades, Ciencia y Tecnología grant number CF-2023-G-433. The work of L.T. is supported under contract No. PID2022-139427NB-I00 financed by the Spanish MCIN/AEI/10.13039/501100011033/FEDER, UE; by the General-
itat de Catalunya under contract 2021 SGR 00171; by Generalitat Valenciana under contract PROMETEO/2020/023; from the project CEX2020-001058-M (Unidad de Excelencia "María de Maeztu"); from the European Union Horizon 2020 research and innovation programme under the program H2020-INFRAIA-2018-1, grant agreement No. 824093 of the STRONG-2020 project; and from the CRC-TR 211 'Strong-interaction matter under extreme conditions'- project Nr. 315477589 - TRR 211. A.S. is supported by Deutsche Forschungsgemeinschaft (DFG) Grant No. SE 1836/5-3 and the Polish National Science Center (NCN) Grant No. 2023/51/B/ST9/02798. K.D.M. thanks the financial support from the São Paulo State Research Foundation (FAPESP), under Grant No. 2024/01623-6. The work of M.C., D.G.D. and N.N.S. has been partially funded by CONICET (Argentina) under Grant No. PIP 2022 GI-11220210100150CO, by ANPCyT (Argentina) under Grant No. PICT20-01847. The work of D.G.D. has also been funded by the National University of La Plata (Argentina), Project No. X824. S.N. acknowledges support by Ministerio de Ciencia e Innovación and Agencia Estatal de Investigación (Spain), and European Regional Development Fund Grant No. PID2019-105439GC21, by EU Horizon 2020 Grant No. 824093 (STRONG-2020), and by Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital, Generalitat Valenciana, GVA PROMETEO/2021/083. The work was also partially supported by national funds from FCT (Fundação para a Ciência e a Tecnologia, I.P, Portugal) under projects UIDB/04564/2020 and UIDP/04564/2020, with DOI identifiers 10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively, and the project 2022.06460.PTDC with the associated DOI identifier 10.54499/2022.06460.PTDC. H.P. acknowledges the grant 2022.03966.CEECIND (FCT, Portugal) with DOI identifier 10.54499/2022.03966.CEECIND/CP1714/CT0004. L.S. acknowledges the PhD grant 2021.08779.BD (FCT, Portugal) with DOI identifier 10.54499/2021.08779.BD. P.A. is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics and Quantum Horizons Program under Award Number DE-SC0024385. P.A. also acknowledges the hospitality of the Kavli Institute for Theoretical Physics, Santa Barbara, through which the research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. M.E.T-Y acknowledges support by Consejo Nacional de Humanidades, Ciencia y Tecnología grant number A1-S-7655. The work of O.L. is a part of the project INCT-FNA proc. No. 464898/2014-5. It was also supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grants No. 307255/2023-9 and 401565/2023-8 (Universal) and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Grant No. 2022/03575-3 (BPE). The work of M.D. is a part of the project INCT-FNA proc. No. 464898/2014-5. It is also supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant No. 308528/2021-2 and No. 401565/2023-8 (Universal). This study was also financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001 - Project number 88887.687718/2022-0. F.L.B. is a member of INCT-FNA, 464898/2014-5 and he acknowledges partial support also from CNPq-312750/2021-8 and CNPq-407162/2023-2.
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