Factorization of finite temperature graphs in thermal QED

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by  Fernando T. Brandt

Factorization of finite temperature graphs in thermal QED
F. T. Brandta, Ashok Dasb, Olivier Espinosac, J. Frenkela and Silvana Perezd
a Instituto de F´ısica, Universidade de S˜ao Paulo, S˜ao Paulo, BRAZIL
b Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627-0171, USA
c Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, CHILE and
d Departamento de F´ısica, Universidade Federal do Par´a, Bel´em, Par´a 66075-110, BRAZIL
WeextendourpreviousanalysisofgaugeandDiracfieldsinthepresenceofachemicalpotential.
We consider an alternate thermal operator which relates in a simple way the Feynman graphs in
QED at finite temperature and charge density to those at zero temperature but non-zero chemical
potential. Several interesting features of such a factorization are discussed in the context of the
thermal photon and fermion self-energies.
PACSnumbers: 11.10.Wx
6
0
0 In earlier papers [1, 2], we gave a simple derivation of tential, which involved only a single reflection operator
2 aninterestingrelation[3,4,5]betweenfinitetemperature S(E) that changes the energy as E →−E.
Feynmangraphsandthecorrespondingzerotemperature
n
a diagrams in the imaginary as well as in real time formu- In the present note, we propose an alternate thermal
J lations of thermal field theories. We showed that in the operator without time derivatives, but which involves
0 absenceofachemicalpotential,thefinitetemperaturedi- an additional fermionic distribution operatorNˆ(T,µ)(E).
F
3 agrams involving scalar, fermions or gauge fields can be We will show that such a thermal operator representa-
related to the zero temperature graphs througha simple tion naturally leads in QED to a simple factorization of
1
thermal operator. On the other hand, in the presence the thermalamplitudes inthe presenceofa chemicalpo-
v
of a chemical potential µ, the thermal operator is more tential.
7
complicatedasitinvolvesalsotimederivatives. Inathe-
2
2 ory involving fermionic fields with a chemical potential, For briefness, we will discuss here the theory in the
1 wehaveshownthatacompletefactorizationseemstobe imaginary time formalism [6, 7, 8], although everything
0 violated by the presence of an infrared singular contact also holds in the real time formalism. Using this formu-
6 term [2]. As we have pointed out, such a behavior may lation and the mixed space representation, the fermion
0
be due to the simplestpossible choiceof generalizingthe propagator at zero temperature but finite chemical po-
/
h basic thermal operator in the presence of a chemical po- tential can be written in the form
t
-
p
e
h
: 1
v S(0,µ)(τ,E)= θ(τ)A(E)e−(E−µ)τ +θ(−τ)B(E)e(E+µ)τ , (1)
i 2E
X h i
r
a
where E = p~2+m2 and A(E), B(E) are Euclidean distribution function and s is a sign. As shown below,
this prescription enforces the anti-periodicity condition
matrices giveqn by
of the thermal fermion propagator at non-zero chemical
A(E)=iγ0E−~γ·p~+m, B(E)=−iγ0E−~γ·p~+m. (2) potential.
In terms of the operators S(E) and Nˆ(T,µ)(E), our
F
The action of the operator Nˆ(T,µ)(E) on the above time alternate basic fermion operator has the simple form
F
dependent exponential functions is defined as follows:
Oˆ(T,µ)(E)=1−Nˆ(T,µ)(E)[1−S(E)]. (4)
Nˆ(T,µ)(E)es(E±µ)τ = n (E±µ)es(E±µ)τ F F
F F
≡ n±(E)es(E±µ)τ, (3) Then,actingonthepropagatorS(0,µ)(τ,E),thisfermion
F
operatorwillnaturallyreproducethefermionpropagator
where T is the temperature,n denotes the Fermi-Dirac at finite temperature and chemical potential
F
1
S(T,µ)(τ,E)=Oˆ(T,µ)(E)S(0,µ)(τ,E)= A(E) θ(τ)−n−(E) e−(E−µ)τ +B(E) θ(−τ)−n+(E) e(E+µ)τ , (5)
F 2E F F
n (cid:2) (cid:3) (cid:2) (cid:3) o
2
which satisfies, for −1/T < τ < 0, the anti-periodicity ofarbitrarygraphsinQEDatfinitetemperature. Before
condition [9] wepresentageneralproof,letusconsiderfirstsomesim-
ple amplitudes at one loop order. For example, the inte-
S(T,µ)(τ,E)=−S(T,µ)(τ + 1,E). (6) grandof the fermion self-energy(in the Feynmangauge)
T may be written with the help of Eq. (5) in a completely
factorized form as
The abovefactorizationof the thermalfermion propa-
gator is of fundamental importance for the factorization
Σ(T,µ)(E ,E ,τ)=−e2D(T)(τ,E )γ S(T,µ)(τ,E )γ =O(T)(E )Oˆ(T,µ)(E )Σ(0,µ)(E ,E ,τ), (7)
1 2 1 α 2 α B 1 F 2 1 2
whereO(T) denotesthebasicphotonprojectionoperator and n is the Bose-Einstein distribution function.
B B
If we Fourier transform Eq. (7) in the time variable,
O(T)(E )=1+n (E )[1−S(E )] (8) we readily obtain that
B 1 B 1 1
e2 1+n (E )−n−(E ) n (E )+n−(E )
Σ(T,µ)(E ,E ,p )=− B 1 F 2 − B 1 F 2 γ A(E )γ +
1 2 0 8E E E +E −µ−ip E −E +µ+ip α 2 α
1 2(cid:26)(cid:20) 1 2 0 1 2 0(cid:21)
1+n (E )−n+(E ) n (E )+n+(E )
B 1 F 2 − B 1 F 2 γ B(E )γ , (9)
E +E +µ+ip E −E −µ−ip α 2 α
(cid:20) 1 2 0 1 2 0(cid:21) (cid:27)
which agrees with the result obtained by a direct evalu- Asasecondexample,whichexhibits asubtlety associ-
ation. As shown in [2], apart from a contribution to the ated with the alternate fermion operator in Eq. (4), let
thermal mass of the fermion, the above radiative correc- usconsiderthethermalphotonself-energyatfinitechem-
tionsalsoleadtoafinite renormalizationofthe chemical icalpotential. Using the basicpropertygivenin Eq. (5),
potential which is given by the integrand of this amplitude at one loop order may
also be written in a manifestly factorized form as
e2
µ = 1− µ. (10)
R 16π2
(cid:18) (cid:19)
Π(T,µ)(E ,E ,τ) = e2Tr γ S(T,µ)(τ,E )γ S(T,µ)(−τ,E )
λρ 1 2 λ 1 ρ 2
= Oˆ(T,µh)(E )Oˆ(T,µ)(E )Π(0,µ)(E ,E ,τ)i. (11)
F 1 F 2 λρ 1 2
Using Eq. (1), wesee thatthe factorse±µτ actuallycan- each fermion propagator of energy E a chemical poten-
i
cel in the zero temperature amplitude at finite chemical tial µ . At the end of the calculation, after the thermal
i
potential. However, this would then make the action of operators have acted, one can set µ =µ.
i
the operator Nˆ(T,µ)(E) introduced in Eq. (3) ambigu-
F
ous. In order to define these operators unambiguously, Using this procedure and Fourier transforming Eq.
one may,in the intermediate calculations,associatewith (11) in the time variable, we then readily get the result
3
e2 n−(E )−n−(E ) n+(E )−n+(E )
Π(T,µ)(E ,E ,p )= F 2 F 1 Trγ A(E )γ A(E )+ F 1 F 2 Trγ B(E )γ B(E )+
λρ 1 2 0 8E E E −E +ip λ 1 ρ 2 E −E +ip λ 1 ρ 2
1 2 (cid:20) 2 1 0 1 2 0
1−n+(E )−n−(E ) 1−n+(E )−n−(E )
F 2 F 1 Trγ A(E )γ B(E )+ F 1 F 2 Trγ B(E )γ A(E ) ,
E +E −ip λ 1 ρ 2 E +E +ip λ 1 ρ 2
1 2 0 1 2 0 (cid:21)
(12)
which agrees with the expression for the thermal pho- higher-loopgraph arises when there are internal vertices
ton self-energy obtained by a direct evaluation. In this forwhichthetimecoordinateτ hastobeintegratedover.
case, the only effect of the chemical potential is to yield At finite temperature, τ as well as the external times τ
i
a correction to the thermal mass of the photon [7]. In lieintheinterval[0,1/T]. Ontheotherhand,inthezero
consequence of the symmetry of thermal QED [2], the temperature graphs, the internal time needs to be inte-
above radiative corrections do not renormalize the bare gratedovertheinterval[−∞,∞]. Sincethebasicthermal
zero chemical potential of the photon. operatorsare independent of the time coordinates,these
The complete factorization in the above examples, may be taken out of the time integral. Then, using the
whichoccursinconsequenceofthebasicrelation(5),can procedure outlined after Eq. (11), the essential step in
beimmediatelyextendedtoanyone-loopgraph. Thedif- proving the factorization of an arbitrary graph consists
ficultyinestablishingsuchafactorizationforanarbitrary in showing that the function
0 ∞
V = dτ + dτ S(0,µ1)(τ −τ,E )γ S(0,µ2)(τ −τ ,E )D(τ −τ ,E ) (13)
α 1 1 α 2 2 3 3
" −∞ 1 #
Z ZT
which appears in the basic electron-photon vertex, is annihilated by the thermal operator
O(T,µ)(E ,E ,E )=Oˆ(T,µ1)(E )Oˆ(T,µ2)(E )O(T)(E ). (14)
1 2 3 F 1 F 2 B 3
To show this, we evaluate the τ-integrals in Eq. (13) and note that the result may be written in the form
V = 1+e−(E1+µ1+E2−µ2+E3)/TS(E )S(E )S(E )
α 1 2 3
h A(E )γ B(E ) e−(E1−µ1)τ1 e−(E2+iµ2)τ2 e−E3τ3
1 α 2
. (15)
E −µ +E +µ +E 2E 2E 2E
(cid:20) 1 1 2 2 3 (cid:18) 1 2 3 (cid:19)(cid:21)
Using the relation (3) together with the identities thermal diagram can be factorized in the form
I d3k V
Γ(T) = i (2π)3δ3(k,p)O(T,µ)γ(0,µ), (17)
n±(E) n (E ) N (2π)3 v N
e−(E±µ)/T = F ; e−E3/T = B 3 , (16) Z i=1 v=1
1−n±(E) 1+n (E ) Y Y
F B 3 wherewedenotetheinternalandexternalthreemomenta
of the graph generically by k and p, respectively, and
δ3(k,p)enforcesthethree-momentumconservationatthe
it is now straightforward to show that the thermal op- v
vertexv. The thermaloperatorfor the graphis givenby
erator given in Eq. (14) annihilates the quantity V in
α
Eq. (15). This establishes that for the product of prop-
agators in the basic electron-photon vertex, which are IF I
integrated over the common time τ, we can extend the O(T,µ) = OˆF(T,µi)(Ei) OB(T)(Ej), (18)
range of integration to the interval [−∞,∞]. Yi=1 j=YIF+1
With the helpofthis property,onecanproveby using with IF, I being respectively the number of internal
a procedure similar to the one given in [1] that in the fermion propagators and the total number of propa-
presence of a chemical potential, an arbitrary N-point gators. Furthermore, γ(0,µ) represents the integrand
N
4
of the zero temperature graph at finite chemical po- by following in a straightforward way the analysis pre-
tential, which involves, apart from a product of pho- sented in [2], to a non-Abelian gauge theory. We would
ton propagators, also a product of fermion propaga- like to mention that a rather similar approach has been
tors S(0,µi)(τ ,E ). We note from (1) that, since recently proposed in the context of complex scalar fields
i i
S(0,µi)(τ ,E )=exp(τ µ )S(0,0)(τ ,E ),we canmoreover atfinitetemperatureandchargedensity[10]. Thefactor-
i i i i i i
directlyrelateγ(0,µ) to theintegrandofthe zerotemper- izationpropertyiscalculationallyquiteusefulandallows
N
ature and chemical potential graph γ(0,0). As we have us to study in a direct and transparent way many ques-
N tions of interest at finite temperature.
pointed out, the limit µ → µ is assumed to be taken
i
Acknowledgment
only after the action of the thermal operator.
Thus,wehaveobtainedathermaloperatorrepresenta- ThisworkwassupportedinpartbytheUSDOEGrant
tion for QED at finite temperature and chemical poten- number DE-FG 02-91ER40685, by MCT/CNPq as well
tial, which leads to a simple factorizationof the thermal as by FAPESP, Brazil and by CONICYT, Chile under
amplitudes. Thisinterestingresultcanbealsoextended, grant Fondecyt 1030363and 7040057 (Int. Coop.).
[1] F. T. Brandt, A. Das, O. Espinosa, J. Frenkel and S. [7] M.L.Bellac, Thermal Field Theory(CambridgeUniver-
Perez, Phys. Rev.D 72, 085006 (2005). sity Press, Cambridge, England, 1996).
[2] F. T. Brandt, A. Das, O. Espinosa, J. Frenkel and S. [8] A. Das, Finite Temperature Field Theory (World Scien-
Perez,“ThermalOperatorRepresentationofFiniteTem- tific, NY,1997).
peratureGraphs II”, submited to Phys. Rev.D. [9] R. Kubo,J. Phys.Soc. Japan 12, 570 (1957);
[3] O.EspinosaandE.Stockmeyer,Phys.Rev.D69,065004 P. C. Martin and J. S. Schwinger, Phys. Rev. 115, 1342
(2004). (1959).
[4] O.Espinosa, Phys.Rev. D71, 065009 (2005). [10] M. Inui, H. Kohyama and A. Ni´egawa, “Comment on
[5] J. P. Blaizot and U. Reinosa, Nucl. Phys. A 764, 393 “Thermal Operator Representation of Finite Tempera-
(2006) [arXiv:hep-ph/0406109]. ture Graphs””, arXiv:hep-ph/0601092.
[6] J. I. Kapusta, Finite Temperature Field Theory (Cam-
bridge UniversityPress, Cambridge, England, 1989).