The retarded electron propagator S_{R}(p_{0},p) at high momentum p>>T was shown by Blaizot and Iancu to be an entire function of complex p_{0}. In this paper a specific form for S_{R}(p_{0},p) is obtained and checked by showing that its temporal Fourier tr

Theelectronthermalpropagatoratp T:Anentirefunctionofp0

H.ArthurWeldon

DepartmentofPhysics,WestVirginiaUniversity,Morgantown,WestVirginia26506-6315

(Dated:September13,2003)TheretardedelectronpropagatorSR(p0,p)athighmomentump TwasshownbyBlaizotandIancutobeanentirefunctionofcomplexp0.Inthispaperaspeci cformforSR(p0,p)isobtainedandcheckedbyshowingthatitstemporalFouriertransformSR(t,p)hasthecorrectbehavioratlarget.Potentialinfraredandcollineardivergencesfromtheemissionofsoftphotonsdonotoccur.

I.

INTRODUCTION

3

HightemperatureQCDis,inonerespect,simpler0thanhightemperatureQED.Inboththeoriesthedamp-02ingratesforchargedparticlesatveryhighmomentum, p T,haveinfraredsingularitiesatthemassshellwhencomputedinperturbationtheoryusingBraaten-epSPisarskiresummationThedivergencecomesfromthe emissionandabsorptionofquasistatic,transversegauge9bosons.InQCDthisapparentinfrareddivergenceinthe2quarkandgluondampingratesiscuto bythenonper- 1turbativemagneticscreeningmassfortransversegluonsvandthisgives nitedampingratesForquarksthe2dampingrateγdeterminesthelocationofthepolein2theretardedpropagatorsR(p0,p):

390squark(p0,p)=1

R

30

/

h

p

-

p

e

:

h

v

i

X

raαT

exp aexp

ζ

4π

exp[ C 1](1.3)

Apropagatorthatisanentirefunctionofp0hasnopoles,nobranchpoints,andnoessentialsingularities.

Thepurposeofthispaperisto ndthefunctionsR(p0,p)whoseFouriertransformhastheasymptoticbehaviorinEq.ThephysicalapproximationswhichleadtoEq.arevalidintheregionp08,9,10,11]anditisonlythisregionthatisinves-≈ptigatedhere.Sec.IIshowsthattheBloch-NordsieckpropagatorsR(t,p)hastheveryunusualpropertyofbe-inganentirefunctionofcomplext.InSec.IIIonlytheleadingtermintheasymptoticexpansionisemployed,viz.

i√αTexp i

p0 pαT .(1.4)ThisapproximationforsR(p0,p)workssurprisinglywell.

ItischeckedbyshowingthattheFouriertransformoverenergiesp0tivationfor≈thepagreesansatz,withEq.Eq.isatgivenlargeint.AppendixThemo-B.ThereacompleteasymptoticexpansionofsR(p0,p)isobtained.The rsttermofthisexpansionisEq.(1.4).Sec.IVshowshowtheabsenceofsingularitiesintheelec-tronpropagatorimplythatinhardscatteringprocesses,therewillbenoinfraredorcollineardivergencesandthequantitativecontributionoftheseregionsisnumericallysmall.Sec.Vdiscussessomefurtherconsequences.

II.

COMMENTONCOMPLEXTIME

ThelogarithmicdependenceontinEq.(1.1)doesnotimplythatthereisabranchcutfornegativet.Thefull