[Paper Review] Thermal forward scattering amplitudes in temporal gauges

F. T. Brandt, J. Frenkel+1 more|ArXiv.org|Aug 31, 2000

Numerical methods in inverse problems1 references3 citations

TL;DR

This paper develops a forward scattering amplitude technique to compute the thermal gluon self-energy in temporal gauges, demonstrating that the leading $T^2$ and sub-leading $\ln T$ contributions match those in covariant gauges. It proves the one-loop self-energy is exactly transverse and that prescription poles do not affect high-temperature behavior, extending transversality to all orders via BRS identities.

ABSTRACT

We employ the thermal forward scattering amplitudes technique in order to compute the gluon self-energy in a class of temporal gauges. The leading T^2 and the sub-leading ln(T) contributions are obtained for temperatures high compared with the external momentum. The logarithmic contributions have the same structure as the ultraviolet pole terms which occur at zero temperature (we have recently extended this result to the Coulomb gauge). We also show that the prescription poles, characteristic of temporal gauges, do not modify the leading and sub-leading high-temperature behavior. The one-loop calculation shows that the thermal self-energy is transverse. This result has also been extended to higher orders, using the BRS identities.

Motivation & Objective

• To extend the forward scattering amplitude technique to temporal gauges, which are physical and ghost-free but complicated by extra poles at $q \cdot n = 0$.
• To compute the full tensor structure of the one-loop gluon self-energy in this class of gauges, beyond the static limit previously studied.
• To verify whether the leading $T^2$ and sub-leading $\ln T$ contributions match those in covariant gauges, particularly in structure to ultraviolet poles at zero temperature.
• To investigate whether the prescription poles in temporal gauges affect high-temperature behavior of the self-energy.
• To establish the transversality of the thermal self-energy to all orders using BRS identities.

Proposed method

• Employ the forward scattering amplitude technique by expressing thermal Green functions in terms of on-shell thermal particle scattering amplitudes.
• Use the tensor decomposition $\Pi^{ab}_{\mu\nu} = \delta^{ab}(\Pi_T P^{T}_{\mu\nu} + \Pi_L P^{L}_{\mu\nu} + \Pi_C P^{C}_{\mu\nu} + \Pi_D P^{D}_{\mu\nu})$ to isolate components with different pole structures.
• Apply partial fraction decomposition and momentum shifts ($q \to q - k$) to cancel temporal gauge poles in $\Pi_C$ and $\Pi_D$ before summing over Matsubara frequencies.
• Use the contour integral formula $\sum_n f(i\omega_n) = \frac{1}{2\pi i} \oint_C dq_0 f(q_0) \frac{1}{2} \coth(\frac{1}{2}\beta q_0)$ for bosons to evaluate finite-temperature integrals.
• Apply a prescription for poles at $q \cdot n = 0$ in $\Pi_A$ and $\Pi_B$, and show they do not contribute to leading or sub-leading high-temperature terms.
• Use BRS identities to extend the transversality of the self-energy from one-loop to all orders.

Experimental results

Research questions

• RQ1 Can the forward scattering amplitude technique be successfully applied to a class of temporal gauges, despite their non-covariant structure and extra poles?
• RQ2 Do the leading $T^2$ and sub-leading $\ln T$ contributions in temporal gauges match the structure of ultraviolet pole terms at zero temperature?
• RQ3 Do the prescription poles in temporal gauges affect the high-temperature behavior of the gluon self-energy?
• RQ4 Is the one-loop thermal gluon self-energy exactly transverse in temporal gauges, and does this property extend to all orders?
• RQ5 Can the full tensor structure of the self-energy be consistently computed and expressed in terms of on-shell forward scattering amplitudes in temporal gauges?

Key findings

• The leading $T^2$ and sub-leading $\ln T$ contributions to the gluon self-energy in temporal gauges have the same structure as ultraviolet pole terms at zero temperature, confirming a conjecture from previous work.
• The one-loop thermal gluon self-energy is exactly transverse in temporal gauges, a property not generally true in other gauges except the Feynman gauge at one-loop order.
• The prescription poles characteristic of temporal gauges do not contribute to the leading $T^2$ or sub-leading $\ln T$ high-temperature behavior of the self-energy.
• The components $\Pi_C$ and $\Pi_D$ vanish at one-loop order due to cancellations from partial fractions and momentum shifts, and this result is extended to all orders via BRS identities.
• The full tensor structure of the self-energy is consistently computed and expressed in terms of forward scattering amplitudes of on-shell thermal gluons in temporal gauges.
• The transversality of the self-energy is proven to hold to all orders in perturbation theory, a distinctive feature of temporal gauges not shared by general covariant gauges.