The free energy of hot QED at three and a half loops Page: 3 of 5
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The presence of the large scale T means that the "loop correction" need not be small
compared to the bare propagator. In imaginary time, this is the case for the zero
mode (Matsubara frequency) of the electric propagator at small three momentum
(IpJ ~ eT). (By contrast, static magnetic fields are unscreened in a QED plasma.)
In general there are three ways that one can account for large corrections, such
as Debye screening, in order to restore the perturbative expansion :
(i) Continue with bare Feynman rules and resum by hand dangerous subsets of
diagrams. This procedure is possible in simple cases but has the disadvantage that
one must carefully identify the relevant diagrams, account for symmetry factors and
prevent overcounting; or
(ii) Begin with the non-perturbative skeleton-expansion and truncate down. This
is safe from the point of view of symmetry factors and overcounting but is practical
only in simple cases; or
(iii) Re-organise the bare Lagrangian by adding and subtracting the dominant
non-negligible effects (termed "hard thermal loops" by Braaten and Pisarski ).
This method has the virtue that one continues with Feynman perturbation theory
but with new effective propagators and vertices.
Usually, methods (i) and (ii) are computationally efficient only for static (zero
external energy) Greens functions for which all the power counting analysis can be
done in imaginary-time. For a non-static Greens function one first requires its phys-
ical definition in real time (either by analytic continuation from imaginary time or
through the real-time formalism) and then the method advocated in Ref. is prob-
ably the most efficient.
Recently we computed the order e4 (3-loop) contribution to the free energy den-
sity of massless QED at temperature T, going beyond the e3 term known for many
years . The method used was (i) with dimensional regularisation being extensively
employed to regulate various singularities appearing at intermediate stages of the cal-
culation. From the technical point of view the fourth order calculation required the
evaluation of some complicated overlapping three-loop integrals that did not appear
in a similar three-loop calculation in 04 theory .
Following the order e4 calculation in Ref., the e5 (3Z loop) result was also ob-
tained . This latter term may be viewed as a correction to the three-loop result as
a consequence of Debye screening, just as the e3 term is a similar correction to the
two-loop (order e2) result. Computationally, the order e3 calculation is simpler than
the order e2 calculation because only the zero mode of the photon is involved and
the loop integral along that line becomes three-dimensional. Similarly, the e5 piece
was easier to obtain than the e4 piece, the sum of complicated diagrams factorizing
themselves into a product of simple one-loop integrals. The e5 contribution was also
reconsidered from the point of view of method (ii) in Ref..
_, -, .e:- : ..'Y',. ,, : %" , .T : . ";iY ""' } .sn '. _, r_ r3. . rT'_"ii f.~: .. 1T ; '. S^ '- !^.T
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Parwani, R. R. & Coriano, C. The free energy of hot QED at three and a half loops, article, February 1, 1995; Illinois. (digital.library.unt.edu/ark:/67531/metadc1273101/m1/3/: accessed December 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.