# The force-force correlator at the hard thermal scale of hot QCD

Eamonn Weitz, Jacopo Ghiglieri, Philipp Schicho, Niels Schlusser

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Eamonn Weitz, Jacopo Ghiglieri, Philipp Schicho, Niels Schlusser. The force-force correlator at the hard thermal scale of hot QCD. Strong and Electroweak Matter 2024 (SEWM 2024), Aug 2024, Francfort, Germany. . [hal-04683571](https://hal.science/hal-04683571v1)

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Submitted on 2 Sep 2024

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

Distributed under a Creative Commons Attribution 4.0 International License
# The force-force correlator at the hard thermal scale of hot QCD

Eamonn Weitz, SUBATECH, Nantes Université
with Jacopo Ghiglieri, Philipp Schicho, Niels Schlusser

<figure><img src="image_2.png" /></figure>

## Asymptotic mass

Consider hard parton propagating through quark-gluon plasma.

Forward scattering with medium induces shift in parton's dispersion relation $\Rightarrow m_{\infty}^2$, known as **asymptotic mass**.

In heavy-ion collisions, along with transverse scattering rate, $C(k_{\perp})$, serves as key input for calculations of:

* jet modification (i.e, medium-induced radiation and transverse momentum broadening)

* thermalisation, transport

In hot QCD, take parton momentum $p^{+} = \frac{p^{0}+p^{z}}{2} \gg T$, temperature of plasma $\Rightarrow$ Can integrate out scale $p^{+}$ to write $m_{\infty}^{2}$ [1]

$$m_{\infty}^{2} = g^{2}C_{R}(Z_{g} + Z_{f}) + \mathcal{O}(\frac{1}{p^{+}}) \\ = g^{2}C_{R}(\left[\frac{T^{2}}{6} + \mathcal{O}(g)\right] + \left[\frac{T^{2}}{12} + \mathcal{O}(g^{2})\right]) + \mathcal{O}(\frac{1}{p^{+}}) \quad (1)$$

as sum of gluonic and fermionic condensates.

## Classical Corrections

$Z_g$ given in terms of Wilson lines (representing hard parton) and field strength insertions

$$Z_g = \frac{1}{d_R C_R} \int_0^\infty dx^+ x^+ \text{Tr}\langle U_R(-\infty; x^+) \rangle \quad (2) \\ \times F^{-\perp}(x^+) U_R(x^+; 0) F^{-\perp}(0) U_R(0; -\infty)\rangle$$

Starting at $\mathcal{O}(g)$, $Z_g$ receives contribution from

<figure><img src="image_3.png" /></figure>

with gluon soft $\sim gT$.

Associated Bose-Einstein statistical function

$$n_B(\omega) = \frac{1}{e^{\frac{\omega}{T}} - 1} \approx \frac{T}{\omega} \gg 1$$

$\Rightarrow$ Bose enhancement!

These are **classical corrections**:

* Originate from **zero Matsubara mode**

* For soft gluon exchange, corrections can be computed using Hard Thermal Loop Effective Theory (HTL) but analytically difficult in practice

* Can also have **ultrasoft** gluon $\sim g^2 T$ exchange – these corrections **cannot** be studied perturbatively (Linde problem)

## References

[1] Simon Caron-Huot, 0808.0155

[2] Simon Caron-Huot, 0811.1603

[3] Guy Moore, Niels Schlusser, 2009.06614

[4] Jacopo Ghiglieri, Guy Moore, Philipp Schicho, Niels Schlusser, 2112.01407

[5] Jacopo Ghiglieri, Philipp Schicho, Niels Schlusser, Eamonn Weitz, 2312.11731

[6] Soeren Schlichting, Ismail Soudi, 2111.13731

## Lattice Determination from EQCD

To go from hot QCD to electrostatic QCD (EQCD), integrate out all but zero Matsubara mode. Arrive at **static 3D EFT for Matsubara zero-mode** (dimensional reduction):

* $A^0$ field $\Rightarrow -i\Phi$, adjoint scalar picks up Debye mass, $m_D \sim gT$

* $g_{3d}^2 \sim g^2 T$, $U_R \Rightarrow \tilde{U}_R$

But why is this relevant here? Lightcone analyticity arguments allow us to compute classical corrections to $Z_g$ with EQCD [2] $\Rightarrow$ Can bypass horrible HTL computation and more importantly, gain access to non-perturbative evaluation of classical corrections by studying EQCD on the **lattice**!

Arrive at EQCD version of Eq. (2): $Z_g^{3d} = -\frac{4T}{d_A} \int_0^\infty dL L \langle FF \rangle$ defined through $\tilde{U}_R$, $D_i\Phi$ and $F_{ij}$

Short-distance behavior: $\langle FF \rangle_{m_D L \ll 1} = \frac{c_0}{L^3} + \frac{c_2 g_{3d}^2}{L^2} + \frac{c_4 g_{3d}^4}{L} + \dots$

To obtain **finite lattice determination**, subtract off LO, UV **power law** divergence and separate into pieces

$$Z_g^{3d}|_{\text{merge}} = -\frac{4T}{d_A} \left\{ \int_0^{L_{\text{min}}} dL L \left[ \langle FF \rangle_{\text{NLO}} - \frac{c_0}{L^3} - \frac{g_{3d}^2 c_2}{L^2} \right] + \int_{L_{\text{min}}}^{L_{\text{max}}} dL L \left[ \langle FF \rangle_{\text{lattice}} - \frac{c_0}{L^3} \right] + \int_{L_{\text{max}}}^{\infty} dL L \left[ \langle FF \rangle_{\text{tail}} - \frac{c_0}{L^3} \right] \right\} \quad (3)$$

**Lattice piece calculated** and **IR tail fitted** in [3, 4]

At short distances, switch to **perturbative NLO EQCD evaluation**, calculated in [4]

$\Rightarrow$ additionally subtract off $\mathcal{O}(g^2)$ logarithmic divergence $\sim c_2$

**Motivation for our work [5],** where we compute $\mathcal{O}(g^2)$ thermal scale $\sim T$ contribution from hot QCD:

* $\mathcal{O}(g^2)$ EQCD logarithmic divergence still needs to be matched against corresponding one from hot QCD to eliminate spurious $L_{\text{min}}$-dependence

* Obtain all $\mathcal{O}(g^2)$ corrections to $Z_g$

## Cutoff-independent results

We find that in Feynman gauge, only need to compute one diagram in hot QCD (left below)
Compare to its EQCD counterpart (right below)

<figure><img src="image_4.png" /></figure>

Isolate K zero-mode contribution

<figure><img src="image_5.png" /></figure>

Probe UV limit by taking $K \gg gT$

Logarithmic divergences cancel $\Rightarrow$ Finite cutoff-independent result for non-perturbative evaluation!

<table><thead><tr><th>T</th><th>Z<sub>g</sub><sup>non-pert class only</sup><br>T<sup>2</sup></th><th>Z<sub>g</sub><sup>non-pert class + finite O(g<sup>2</sup>)</sup><br>T<sup>2</sup></th><th>Z<sub>g</sub><sup>3d</sup><sub>LO</sub><br>T<sup>2</sup></th></tr></thead><tbody><tr><td>250 MeV</td><td>-0.513(138)(45)(7)</td><td>-0.340(138)(45)(7)</td><td>-0.376</td></tr><tr><td>500 MeV</td><td>-0.619(99)(39)(3)</td><td>-0.491(99)(39)(3)</td><td>-0.324</td></tr><tr><td>1 GeV</td><td>-0.462(71)(9)(7)</td><td>-0.356(71)(9)(7)</td><td>-0.305</td></tr><tr><td>100 GeV</td><td>-0.327(16)(5)(2)</td><td>-0.274(16)(5)(2)</td><td>-0.223</td></tr></tbody></table>

Errors from statistical errors in lattice EQCD (first bracket), integration of IR tail (second bracket) and quadrature of lattice data (third bracket)

First column contains non-perturbative evaluation. Second contains in addition finite part (see outlook) of quantum $\mathcal{O}(g^2)$ corrections from $T$ scale $\Rightarrow$ **Larger in magnitude** than $Z_g^{LO}$ for all temperatures!

## Outlook

* Find double-logarithmic divergence in region $k^+ \gtrsim T$, $k^- \lesssim g^2 T$, $k_\perp \sim gT \Rightarrow K^2 \sim g^2 T^2$ $\Rightarrow$ Collinear, hard divergence hints at sensitivity to regions where:

- LPM resummation in necessary

- Scale hierarchy assumed in Eq. (1) is no longer respected

- Outstanding divergence implies our separation of non-perturbative classical corrections and quantum $\mathcal{O}(g^2)$ ones is somewhat ambiguous. Complete $\mathcal{O}(g^2)$ evaluation of collinear and hard contributions needed

* Looking farther forward, want to evaluate impact of non-perturbative corrections on medium-induced radiation, similar to what was done in [6] for $C(k_\perp)$