

 12 Aug 1994

 THE INFRARED SENSITIVITY OF NONABELIAN DEBYE SCREENING

\Lambda

A. K. REBHAN DESY, Gruppe Theorie, Notkestr. 85, D-22603 Hamburg, Germany

In memory of Tanguy Altherr

ABSTRACT It is shown that a perturbative treatment of nonabelian Debye screening at high temperature suffers from infrared problems already at the next-to-leading order, which is given by ring-resummed one-loop diagrams. Superficial infrared power counting would let one expect a sensitivity to the magnetic mass scale only at higher orders, but the form of Debye screening depends on the analytic structure of the correlation functions, which is strongly sensitive to the existence of screening of static magnetic fields.

1. Introduction

High-temperature gauge theories1 become increasingly complicated as the infrared regime is approached. At the momentum scale gT , where g is the coupling and T the temperature, the spectrum of the theory begins to deviate considerably from the one of the free theory. Gauge bosons as well as fermions acquire new, collective degrees of freedom described by the gauge-invariant (nonlocal) effective action generated by the so-called "hard thermal loops" (HTL)2;3, the high-temperature limit of all one-loop Feynman diagrams proportional to T 2. The resulting leadingorder dispersion laws of the quasi-particle excitations can be understood in classical terms4;5, but already at next-to-leading order the dispersion laws receive contributions from all orders of the conventional perturbation series. In order to restore perturbation theory it is necessary to resum all HTL corrections3, and in this way some corrections to the HTL dispersion laws have been obtained6;7;8. However, in nonabelian theories, the restitution of perturbation theory is not complete. Static chromomagnetic fields are not screened at the HTL level, and the self-interactions of these lead to a breakdown of perturbation theory at a certain loop order. In particular, the magnetic screening mass, which is expected to arise at the order g2T , is not calculable in perturbation theory9;10; HTL resummation does not help in this respect.

\Lambda Invited talk at the Workshop on Quantum Infrared Physics, 6-10 June 1994, American University of Paris, France.

1

On the other hand, HTL resummation should allow one to calculate corrections to the classical chromoelectric (Debye) screening mass m0, which is of the order gT , to wit,

m20 = e

2T 2

3 (1) with e2 = (N + Nf =2)g2 for color group SU(N ) and Nf flavors. Upon HTL resummation, perturbation theory is organised in powers of g rather than g2, and the relative order g correction is determined by resummed one-loop diagrams. According to superficial infrared power-counting1, problems coming from the perturbatively vanishing magnetic mass would be expected to set in at two-loop order.

In this note, I shall demonstrate, however, that already the next-to-leading order term of the resummed perturbation theory is crucially sensitive to the magnetic mass scale.

2. The electrostatic gluon propagator at resummed one-loop order

In momentum space, the `electrostatic' gluon propagator is given by

DL(k) = 1k2 + \Pi

00(k0 = 0; k) (2)

and its Fourier transform \Phi (r) determines the chromoelectric field induced by a single external (conserved) source J through

hEi(x)i = @@x

i \Phi (jxj); (3)

where we have suppressed all color indices. Despite the nonabelian nature, this is an exact result of linear response theory11, since with only one direction in color space introduced by the single source J , the gauge potentials all point in the same direction, which eliminates the nonabelian commutator terms trivially.

At high temperature, the leading contribution to \Pi 00(k0 = 0; k) equals the constant m20 of Eq. (1), and \Phi (r) / e\Gamma m0r=r. However, beyond leading order \Pi 00 is a gauge fixing dependent quanitity. At one-loop resummed order one obtains12;13 in covariant gauges with gauge parameter ff

\Pi 00(0; k) = m20 + gp3N

0m2

0

2ss "

m20 \Gamma k2

m0k arctan

k m0 +

ff \Gamma 2

2 # + O(g

2); (4)

where N 0 = N=(1 + Nf2N ). Computing the Fourier transform of DL(k) with the thus corrected gluon self-energy, one finds a surprising behavior14: \Phi (r) has the form of a Yukawa potential multiplied by an oscillating function which approaches a negative value asymptotically. Its details are gauge parameter dependent except for the screening mass which characterises the exponential decay for very large distances, which is still given by m0.

2

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Execution stack:
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Dictionary stack:
--dict:1100/1123(ro)(G)-- --dict:0/20(G)-- --dict:74/200(L)-- --dict:120/250(L)-- --dict:43/200(L)-- --dict:101/150(L)--
Current allocation mode is local
Last OS error: 2

