Classical Boltzmann equation and high-temperature QED

0.36 MB·English

by  F. T. Brandt

Classical Boltzmann equation and high-temperature QED
F. T. Brandt, R. B. Ferreira and J. F. Thuorst
Instituto de F´ısica, Universidade de Sa˜o Paulo, S˜ao Paulo, SP 05508-090, Brazil
(Dated: March 3, 2015)
The equivalence between thermal field theory and the Boltzmann transport equation is investi-
gated at higher orders in the context of quantum electrodynamics. We compare the contributions
obtained from the collisionless transport equation with the high temperature limit of the one-loop
thermal Green’s function. Our approach employs the representation of the thermal Green’s func-
tions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly
indicates that the physics described by the leading high temperature limit of quantum electrody-
namics can be obtained from the Boltzmann transport equation. We also present some explicit
examples of this interesting equivalence.
PACSnumbers: 11.10.Wx,11.15.-q
5
1
0
I. INTRODUCTION cause, as we will see, the Boltzmann transport equation
2
formalismyieldsaneffectiveactionwhichisexpressedin

b
terms of the electromagnetic field tensor F .
e This paper is about hard thermal loop (HTL) am- µν
However, it is not known if the same n-photon am-
F plitudes in finite temperature quantum electrodynamics
plitudes can also be obtained from a first principle ap-
(QED).Thishasbeenthesubjectofmanyinvestigations
8 proach, like QED at finite temperature. Our main goal
not only in QED but also in non-Abelian gauge theories
2
in the present work is to investigate this issue using the
and gravity. These HTL amplitudes have loop momenta
imaginarytimeformalism[9–11]. Themainmotivationof
] of the order of the temperature, T, which is considered
h thisinvestigationistoextendtheconceptofHTLampli-
large compared with all the external momenta. In the
t tudesincludingallthecontributionswhicharisefromthe
- case of non-Abelian gauge field theories, it is possible to
p Boltzmanntransportequationapproach. Infact,consid-
sum all the amplitudes in terms of a closed form expres-
e ering that the Boltzmann transport equation formalism
h sion for the effective action [1–3]. Similarly, in gravity,
is a classical limit, it is important to understand how it
[ closed form expressions for the static and the long wave-
relates with the high temperature limit of QED at finite
lenght limits of the effective action have been obtained
4 temperature. As a first step toward this goal, we inves-
[4, 5]. The main motivation for these investigations is
v tigate in the present paper the n-photon amplitudes in
the search for a consistent thermal field theory in the
9
both formalisms.
6 perturbative regime.
In the next section, we review the Boltzmann trans-
4 It is well known that the thermal effective action in
0 QEDhasaleadingtemperaturebehaviorproportionalto port equation approach and derive the effective action
0 T2 whicharisesonlyfromthecontributionofthephoton and the corresponding thermal amplitudes. We obtain
. animplicitexpressionforallthethermalamplitudesand
1 polarization tensor. All the other thermal amplitudes
compute some explicit examples. In Sec. III we consider
0 are either subleading or they vanish as a consequence of
5 the neutrality of the plasma. On the other hand, it is the thermal Green’s functions in QED at finite tempera-
1 ture. We review how the thermal Green’s functions can
also known that the HTL amplitudes can be generated
: be expressed in terms of forward scattering amplitudes.
v using the Boltzmann transport equation. This approach
Then, using the HTL limit, we argue that the leading
i has been used in scalar, non-Abelian gauge theories and
X behavior of each of these forward scattering amplitudes
gravity [6–8].
r agrees with the results obtained in Sec. II. The agree-
a WhenweadopttheBoltzmanntransportequationap-
mentisexplicitlyverifieduptothefourphotonfunction.
proach, in the case of QED, we realize that there is
Finally, in Sec. IV we discuss the main results.
groundforconsideringsomemoregeneralpossibilitiesfor
the thermal amplitudes (although restricted to the semi-
classical limit). First, since the plasma does not need
II. THERMAL AMPLITUDES FROM THE
to be neutral, amplitudes with an odd number of ex-
BOLTZMANN TRANSPORT EQUATION
ternal photons can be generated. Second, amplitudes
which are not simply proportional to a power of T (like
Let us consider the semiclassical description of a QED
the T2 behavior of the two-photon amplitude) are also
plasma in terms of a distribution
generated. (This is the key distinction between the non-
Abelian and Abelian theories, since in the former, each 1
f(x,k,e)= 2θ(k )δ(k2−m2)F(x,k,e), (2.1)
amplitude generated by the Boltzmann transport equa- (2π)3 0
tion formalism is proportional to the same power of T.)
Furthermore, all these contributions are generated from where x = (t,(cid:126)x) and k = (k ,(cid:126)k) are the four-
0
a manifestly gauge invariant effective action, simply be- vectors of position and momenta. The Dirac delta
2
and theta functions impose the conditions k = This shows that we can choose f(x,k,−e)=f(x,−k,e).
√ √
(m/ 1−v2,m(cid:126)v/ 1−v2) for on-shell, positive energy, Therefore we can, alternatively, express Eq. (2.6) as
particles of charge e and mass m.
(cid:90)
Letusnowassumethatthecollisionsbetweenthepar-
±J =±eC d4kk f(x,±k,e), (2.10)
ticles are neglectable. Then, when an external electro- µ µ
magnetic field is present, the trajectory followed by the
particles is determined by the equations of motion so that the charge conjugation is equivalent to reversing
the sign of the thermal particle momentum.
mdxµ =kµ; mdkµ =eF kν, (2.2) Letusnowconsiderthegeneralrelationthatmustexist
dτ dτ µν between the current and the thermal effective action of
electromagneticfieldsinteractingwithathermalplasma.
where F = ∂ A −∂ A is the electromagnetic field
µν µ ν ν µ This relation can be simply written as
tensor and τ is the proper time. As a result, the distri-
bution function satisfies the equation (cid:90)
±Γ = d4x±J (A)Aµ(x), (2.11)
transp µ
d ∂f dxµ ∂f dk
f(x,k,e)= + µ =0. (2.3)
dτ ∂xµ dτ ∂kµ dτ where we have made explicit the dependence of ±J on
µ
theexternalfieldA ,whichfollowsfromthesubstitution
Weremarkthattheright-handsideofthepreviousequa- µ
of the solution of Eq. (2.4) into Eq. (2.10).
tion does not vanish whenthe effect of collisions is taken
In the most trivial scenario, when the external field is
intoaccount. UsingEqs. (2.2),weobtaintheBoltzmann
switched off, and the system is in thermal equilibrium,
transport equation
Eq. (2.4) is satisfied by
k·∂f(x,k,e)=−F ·∂ f(x,k,e), (2.4)
k 1
f(x,k,e)=f(0)(k)= 2θ(k )δ(k2−m2)F(0)(k ).
where ∂ is the partial derivative in relation to the four (2π)3 0 0
k
momentum k and we have defined (2.12)
α
The equilibrium distribution F(0)(k ) depends on the
0
F =eF kβ. (2.5) statistics of the particles. In the case of electrons and
α αβ
positrons
ThiscovariantformoftheBoltzmanntransportequation
has been previously obtained in more general cases of 1
F(0)(k )=N (k )= (2.13)
non-Abeliangaugefieldsandalsononcommutativegauge 0 F 0 exp(k /T)+1
0
theories [6, 12–17].
Let us now consider the current which is produced which is the Fermi-Dirac distribution (for our present
when the plasma interacts with the external field. This purposes F(0)(k ) could be any x-independent distribu-
0
canbeexpressedintermsofthedistributionfunctionsfor tion). Then, the partial currents are given by
thenegativeorthepositivechargecarriers(e.g. positrons
or electrons) as eC (cid:90) d3k
±J =±J(0) = ±
(cid:90) µ µ (2π)3 (cid:113)|(cid:126)k|2+m2
±J =±eC d4kk f(x,k,±e). (2.6)
µ µ (cid:18)(cid:113) (cid:19)
N |(cid:126)k|2+m2 k (2.14)
F µ
where C takes into account the degrees of freedom asso-
ciated with the charged particles (C = 2 for an electron
and the corresponding effective actions are
in3+1dimensions). Thetotalcurrentinaplasmawould
be given by eC (cid:90) (cid:90) d3k
±Γ(1) = ± d4x
Jµ =+Jµ+−Jµ. (2.7) transp (2π)3 (cid:113)|(cid:126)k|2+m2
(cid:18)(cid:113) (cid:19)
Ofcourseonecouldthinkofotherpossiblescenariossuch N |(cid:126)k|2+m2 k Aµ(x). (2.15)
F µ
that the net charge of the plasma does not vanish. For
this reason, it is convenient to consider ±J separately.
µ (cid:113)
If we now perform the replacement k → −k in Eq. Introducing the four-vector K = ( 1+m2/|(cid:126)k|2,(cid:126)k/|(cid:126)k|),
(2.4), we obtain
we can write
k·∂f(x,−k,e)=+F ·∂ f(x,−k,e). (2.8)
k CeT3I(1)(m¯)
±Γ(1) = ±
transp (2π)3
On the other hand, if we make e→−e we get
(cid:90) (cid:90)
d4x dΩK Aµ(x), (2.16)
k·∂f(x,k,−e)=+F ·∂ f(x,k,−e). (2.9) µ
k
3
(cid:82)
where dΩ denotes the integration over the directions field, let us look at the details of the integrand in Eq.
of Kˆ = (cid:126)k/|(cid:126)k| and we have introduced the integration (2.22). SubstitutingEq. (2.21)intoEq. (2.22)weobtain
operator
(cid:90) (cid:90) 1
±Γ(n) = −e2C d4x d4kAµk kβF
(cid:90) ∞ u4−n 1 m transp µk·∂ αβ
I(n)(m¯)≡ 0 du√u2+m¯2e√u2+m¯2 +1; m¯ ≡ T , ∂ f(n−2)(x,±k,e). (2.23)
(2.17) ∂k
α
(cid:112)
which operates on functions of K = 1+m¯2/u2. (to-
0
Performinganintegrationbypartsinthemomentumin-
gether with the powers of the temperature, I(n)(m¯) give
tegral, yields
the full temperature dependence for the contribution of
(cid:82)
order n). Since dΩKi =0, we finally obtain (cid:90) (cid:90) (cid:18) 1 k ∂α (cid:19)
±Γ(n) = e2C d4x d4kAµ δα − µ
eCT3 (cid:90) ∞ u2 (cid:90) transp µk·∂ (k·∂)2
±Γ(tr1a)nsp = ± 2π2 due√u2+m¯2 +1 d4xA0(x). kβFαβf(n−2)
0
(cid:90) (cid:90)
(2.18) = e2C d4x d4kAµ(cid:0)δαk ∂λ−k ∂α(cid:1)
µ λ µ
Ofcourse,inthecaseofaneutralplasma,thesumofthe 1
kβF f(n−2), (2.24)
two partial contributions vanishes (k·∂)2 αβ
Γ(1) =+Γ(1) +−Γ(1) =0. (2.19) where the derivative of ∂/∂k acting on kβ produces a
transp transp transp α
vanishing contribution as a consequence of the antisym-
As we will see this simple property will remain true for metry of the electromagnetic tensor field. Performing an
all odd powers of the external field (see Eq. (2.25) ). integration by parts in the configuration space integral
In the presence of a general external field, the solu- andusingagaintheantisymmetryoftheelectromagnetic
tion of the Boltzmann transport equation can be quite tensor field we obtain
involved. Here we consider external field configurations
(cid:90) (cid:90) 1
which can be dealt with using a perturbative approach. ±Γ(n) = e2C d4x d4kFµ2ν2 Fµ1ν1
In these cases a formal solution of the Eq. (2.4) can be transp (k·∂)2
written as k k η f(n−2)(x,±k,e) (2.25)
µ1 µ2 ν1ν2
(cid:18) 1 (cid:19)n
f(n)(x,±k,e)= ∓ F ·∂ f(0)(k), (2.20) (from this result one can easily verify that, for odd n,
k·∂ k +Γ(n) +−Γ(n) =0).
transp transp
This form of the effective action encodes a number of
so that each f(n) is of order n in powers of the external
interesting features. First, since Eq. (2.25) is expressed
field and the complete phase space distribution can be
directly in terms of F (notice that f(n−2) is implicitly
expressed as f(x,k,e) = f(0)(k)+f(1)(x,k,e)+.... It µν
dependent on Fµν as a solution of (2.21)), gauge invari-
is convenient to write Eq. (2.20) in a more explicitly
ance is explicitly manifested at any given order. Second,
iterative fashion as
itisstraightforwardtoobtainanyhigherordercontribu-
1 ∂ tions by successive substitutions of Eq. (2.21) into Eq.
f(n)(x,±k,e)=∓e k·∂Fαβkβ∂k f(n−1)(x,±k,e), (2.25). Thesepropertiescanbeconciselysummarizedby
α
expressing the external fields in terms of Fourier compo-
(2.21)
nents so that the effective actions can be written as
where n = 1, 2, .... From these solutions, and using
Eqs. (2.10) and (2.11), we can obtain the corresponding 1 (cid:90) d4p d4p d4p
partial effective actions, defined order by order in terms ±Γ(n) = 1 2 ... n
transp n! (2π)4(2π)4 (2π)4
of the phase space distributions, as follows
A˜µ1(p )...A˜µn(p )
(cid:90) (cid:90) 1 n
±Γ(trna)nsp =±eC d4x d4kAµkµf(n−1)(x,±k,e) ±Πtµr1aµn2s..p.µn(p1,p2,...,pn)
(2.22) (2π)4δ(p1+···+pn), (2.26)
Inprinciple,theeffectiveactioniscompletelydetermined
where ±Πtransp (p ,p ,...,p ) are the thermal ampli-
by this direct procedure starting from the knowledge of µ1µ2...µn 1 2 n
f(0), which can be chosen, for instance, as given by Eqs. tudes associated with the Boltzmann transport equation
(2.12) and (2.13). formalism. Then, from the gauge invariance of ±Γ(n)
transp
The nontrivial effects of the external field interacting under A˜µi → A˜µi +Λ˜pµi, we conclude that the thermal
i
with the plasma will be manifest when we consider the amplitudes must satisfy the Ward identities
second or higher order contributions in Eq. (2.22) (n ≥
2). To derive the explicit dependence on the external pµiΠtransp (p ,p ,...,p )=0. (2.27)
i µ1µ2...µn 1 2 n
4
Notice that in the case of non-Abelian gauge where
theories the corresponding Ward identities
would relate Πtransp (p ,p ,...,p ) with
Πtransp (p ,p ,...,pµ1µ2)...µsno 1tha2t all nthe ampli- F˜µiνi ≡(pµiηνiµ−pνiηµiµ)Aµ. (2.31)
µ1µ2...µn−1 1 2 n−1
tudes would have the same leading high temperature
behavior. On the other hand, in the Abelian theory,
Inserting (2.31) into (2.30) and comparing the resulting
each contribution ±Γ(n) to the effective action is
transp expression with (2.26), we obtain the following result for
individually gauge independent and the corresponding
the two-photon amplitude
amplitudes are not all proportional to the same power
of the temperature. This can be simply understood by
power counting the momentum dependence of successive 2e2CT2I(2)(m¯)(cid:90)
terms obtained from Eq. (2.21). Furthermore, from the ±Πtµrνansp = − (2π)3 dΩ
general form of Eq. (2.25) one can easily see that all
(cid:18) K p +K p p2K K (cid:19)
the amplitudes will be functions of degree zero in the η − µ ν ν µ + µ ν . (2.32)
external momenta p. µν K·p (K·p)2
Asanexample,letusnowcomputetheexplicitformof
thetwo-photonamplitude(thermalself-energy). Making
This particular example of the two-photon function il-
n=2 in Eq. (2.25) one readily obtains
lustrates the general structure of all n-photon functions,
(cid:90) (cid:90) 1 which, as we will see can be expressed in terms of an
±Γ(tr2a)nsp = e2C d4x d4kFµ2ν2(k·∂)2Fµ1ν1 angular integral over the directions of Kˆ ≡(cid:126)k/|(cid:126)k|, the in-
tegrand being a rank n tensor which depends on K as
k k η f(0). (2.28) µ
µ1 µ2 ν1ν2 well as the external momenta. The same structure has
been previously obtained for the leading high temper-
Using Eqs. (2.12) and (2.13)
ature limit of non-Abelian as well as noncommutative
e2CT2I(2)(m¯)(cid:90) gauge theories [16].
±Γ(2) = d4x
transp (2π)3 Iterating Eq. (2.21) one more time, Eq. (2.25) yields
(cid:90) K K η
dΩFµ2ν2 µ1 µ2 ν1ν2Fµ1ν1.(2.29)
(K·∂)2 (cid:90) (cid:90) k k η
±Γ(n) = ∓e3C d4x d4kFµ2ν2 µ1 µ2 ν1ν2Fµ1ν1
transp (k·∂)2
ExpressingtheexternalfieldsintermsofFouriercompo-
(cid:18) (cid:19)
nents we obtain 1 ∂
k Fαβ f(n−3)(x,±k,e). (2.33)
βk·∂ ∂kα
e2CT2I(2)(m¯)(cid:90) d4p
±Γ(2) =
transp (2π)3 (2π)4
(cid:90) dΩF˜µ2ν2Kµ1Kµ2ην1ν2F˜µ1ν1,(2.30) Pteegrrfaolr,mthinegraensuilnttceagnrabtieoncabsyt ipnartthseinfotrhmemomentumin-
(K·p)2
±Γ(n) =±e3C(cid:90) d4x(cid:90) d4kFµ1ν1 ∂λ1 Fµ2ν2∂λ2∂λ3Fµ3ν3η T3 f(n−3)(x,±k,e), (2.34)
transp (k·∂)3 (k·∂)3 ν1ν2 µ1µ2µ3ν3λ1λ2λ3
where
T3 = 2k k k k k η −[k k k k k η +µ ↔µ ]+k k k k k η (2.35)
µ1µ2µ3ν3λ1λ2λ3 µ1 µ2 µ3 λ2 λ3 λ1ν3 µ1 µ3 λ1 λ2 λ3 µ2ν3 1 2 µ1 µ2 µ3 λ1 λ2 λ3ν3
It is then straightforward to obtain the three-point am- wheretheexplicitresultforA isgiveninAppendix
µ1µ2µ3
plitudes. Similarlytothepreviouscases,wenowconsider A.
n=3 in the Eq. (2.34) and express the external fields in
Using this iterative approach, we have also computed
terms of Fourier components. Comparing the resulting
the explicit result for the four-photon amplitude. Since
expression with Eq. (2.26) we obtain the result
the calculation is straightforward, but rather involved,
we have made use of computer algebra. The result has
6e3CTI(3)(m¯)(cid:90) beencomparedwiththeoneobtainedusingthehightem-
±Πtransp = ± dΩA .(2.36)
µ1µ2µ3 (2π)3 µ1µ2µ3 perature limit of thermal field theory, as we describe in
5
the next section. an external electromagnetic fields, the external energies
It is clear that, in general, the iterative procedure will in these diagrams have to be analytically continued to
produce a result for the n-photon function which is pro- continuous values, after the Matsubara sums have been
portional to performed. Thisprescriptionextendstheimaginarytime
formalismtoanonequilibriumregimeundertheinfluence
(±e)nCT4−nI(n)(m¯)(cid:90) of external fields.
dΩA (2.37)
(2π)3 µ1µ2...µn
Letusconsidertheone-loopdiagram,showninFig. 1.
Once we compute this basic diagram, the n-point one-
sothateachthermalamplitudehasasimpletemperature
particle-irreducible (1PI) amplitudes can be obtained as
dependence given by the factors T4−nI(n)(m¯).
the sum of certain permutations of the pairs (p , µ ), so
i i
that the bosonic symmetry is satisfied. In the imaginary
timeformalismthecontributionofthisbasicdiagramcan
III. HARD THERMAL LOOPS IN QED
be written as
In this section we consider the QED n-photon ampli-
tudes in the usual thermal field theory setting [9–11]. G =−T (cid:90) d3k (cid:88) f (k ,(cid:126)k) (3.1)
Furthermore, we chose the imaginary time formalism, µ1µ2...µn (2π)3 µ1µ2...µn 0
which, as it will be seen, is the most direct approach k0=iωn
to obtain the thermal amplitudes in the present context.
Also, to fully describe a plasma under the influence of where
1 1 t (k;p ,··· ,p )
f (k ,(cid:126)k)= ··· µ1µ2...µn 1 n . (3.2)
µ1µ2...µn 0 k2−|(cid:126)k|2−m2(k +p )2−((cid:126)k+p(cid:126) )2−m2 (k −p )2−((cid:126)k−p(cid:126) )2−m2
0 0 10 1 0 n0 n
andwehavetakenintoaccounttheminussignassociated where
withthefermionloop. Thefactort (k;p ,··· ,p )
µ1µ2...µn 1 n (cid:90) d3k (cid:90) i∞+δ dk
is a shorthand notation for all the contributions to the ±Gtherm = 0N (k )
numerator which arises, for instance, from the traces of µ1µ2...µn (2π)3 −i∞+δ 2πi F 0
Diracmatricesandcouplingconstantfactors. Thequan- f (±k). (3.6)
µ1µ2...µn
tities ω =(2n+1)πT are the fermionic Matsubara fre-
n
quencies. Using the identity [9–11] This separation in two components is the first necessary
step in order to compare the field theory formalism with
(cid:88)∞ 1 (cid:73) the results obtained in the previous section.
T f(k =iω ) = dk f(k )
0 n 2πi 0 0 Closing the contour of the k0 integration in Eq. (3.6)
n=−∞ C in the right-hand side of the complex plane and taking
(cid:20) (cid:18) (cid:19)(cid:21)
1 1 intoaccountthepolesfromthedenominatorofEq. (3.2)
tanh βk , (3.3)
2 2 0 we obtain
(cid:18)(cid:113) (cid:19)
it is straightforward to show that the temperature de- N ((cid:126)k)2+m2
pendent part of (3.1) can be written as (cid:90) d3k F
±Gtherm = −
(cid:90) d3k (cid:90) i∞+δ dk µ1µ2...µn (2π)3 2(cid:113)((cid:126)k)2+m2
Gtherm = 0N (k )
µ1µ2...µn (2π)3 −i∞+δ 2πi F 0 (cid:88)An (±k,p ,··· ,p ), (3.7)
(cid:104) (cid:105) µ1µ2...µn 1 n
f (k ,(cid:126)k)+f (−k ,(cid:126)k) . cycl.
µ1µ2...µn 0 µ1µ2...µn 0
(3.4) where the quantities An denote on-shell
(cid:113) µ1µ2...µn
Making the change of variables (cid:126)k → −(cid:126)k in the second (k0 = ((cid:126)k)2+m2) forward scattering amplitudes,
(cid:80)
terminsidethebracketandusingtheshorthandnotation as depicted in Fig. 2, and cycl. denotes the sum of
fcoµn1µt2r.i.b.µunt(iokn0,(cid:126)ko)f =thfeµn1µ-2p.h..µonto(kn),dwiaegrcaamnwinritteetrhmesthoefrmtwaol aexllpclayicnlicsopoenr.mutations of the pairs (pi, µi), as we will
components, as At this point, we analytically continue the result
to the Minkowski space-time, so that the amplitudes
Gtherm =+Gtherm +−Gtherm , (3.5) An (k;p ,p ,··· ,p ) can be computed using the
µ1µ2...µn µ1µ2...µn µ1µ2...µn µ1µ2...µn 1 2 n
6
the external photons of momenta p , i=1,...,n, as ad-
i
vanced by Barton [22]. This is clearly in tune with the
.. Boltzmann transport equation approach of the previous
k−pn . section.
Thefullcontributiontothe1PIthermalGreen’sfunc-
k k+p1+p2+p3 tionscannowbeobtainedaddingallthe(n−1)!permu-
tationsofthebasicresultinEq. (3.7). Then,takinginto
(cid:80) (cid:80) (cid:80)
account that = , we can express
perm. cycl. perm.
k+p1 k+p1+p2 the 1PI amplitud(en−s1a)s n
(cid:18)(cid:113) (cid:19)
N ((cid:126)k)2+m2
(cid:90) d3k F
±Πtherm = −
FIG. 1: One-loop amplitude which contribute to the effec- µ1µ2...µn (2π)3 2(cid:113)((cid:126)k)2+m2
tive action. The solid and wavy lines represent fermions and
photons respectively. The external momenta are denoted by (cid:88) An (±k,p ,··· ,p ),
pi, i = 1,...n and k denotes the loop momenta. The sym- µ1µ2...µn 1 n
perm.
metrized amplitude which contributes to the effective action n
isobtainedbyaddingallthepermutationsof(n−1)external (3.10)
photon lines, and dividing the result by (n−1)! .
and the corresponding effective actions are
1 (cid:90) d4p d4p d4p
zero temperature QED Feynman rules. In this way, we ±Γ(n) = 1 2 ... n
obtain from Fig. 2 the following result (an extra minus therm (n−1)! (2π)4(2π)4 (2π)4
sign arises from the analytic continuation) A˜µ1(p )A˜µ2(p )...A˜µn(p )
1 2 n
A1µ1 = −etr(γα1γµ1)kα1 =−4ekµ1 (3.8a) ±Πtµh1eµr2m...µn(p1,p2,...,pn)
Anµ1...µn = −en (trγα1γµ1...γαnγµn) (2π)4δ(p1+···+pn). (3.11)
kα1...(k+sn−1)αn ,(3.8b) (We remark that, due to the symmetry of the basic di-
(p2+2k·p )...(s2 +2k·s ) agram in Fig. 1, the sum of the (n−1)! permutations
1 1 n−1 n−1
of external legs produces completely bosonic symmetric
where amplitudes ±Πtherm (p ,...,p ).)
µ1...µn 1 n
In the previous section we have seen that all the effec-
s ≡p +p +···+p ; n≥2. (3.9)
n−1 1 2 n−1 tiveactions(2.26)haveasimpletemperaturedependence
of the form T4−nI(n)(m¯), as a result of Eq. (2.37). On
We remark that, as a consequence of the on-shell con-
the other hand, the thermal field theory effective actions
dition, all the even powers of m cancel. Also the terms
(3.11)haveamorecomplicatedtemperaturedependence.
proportional to odd powers of m vanish simply because
From a physical point of view, we expect that the Boltz-
they are proportional to the trace of an odd number of
mann transport equation approach is restricted to the
gamma matrices, which is zero when the space-time di-
high temperature limit, when the system approaches a
mension is even (in an odd-dimensional space-time there
classical limit. This indicates that in the thermal field
would be induced Chern-Simons terms proportional to
theory description, we have to consider the limit when
the odd powers of m). This way of expressing the n-
T is high compared with the energy-momentum scale
photon amplitudes in terms of traces of 2n gamma ma-
of the external field. This amounts to consider the so-
trices is convenient in order to carry a computer algebra
called hard thermal loop approximation, which in turn
calculation.
implies that the integration in (3.10) is dominated by
The particular set of forward scattering amplitudes
the region where k (cid:29)p . Denoting the HTL limit of
givenbyEq. (3.8)includestheoneswhicharisefromthe µ iµ
residue of the pole of 1/k2 in Eq. (3.2). It is straightfor- ±Πtµh1e..r.mµn(p1,...,pn)by±Πhµt1l...µn(p1,...,pn),theequiv-
ward to show that the corresponding contributions asso- alence of the two effective actions would then imply that
ciatedwitheachpoleofEq. (3.2)producearesultwhich
1
is given by a cyclic permutation of the pairs (pi, µi). In ±Πhµt1l...µn(p1,...,pn)= n±Πtµr1a..n.µspn(p1,...,pn). (3.12)
this way, the final result can be expressed as the sum
(cid:80) in Eq. (3.7). This representation of one-loop 1PI In QED this is a well-known result for the leading T2
cycl.
diagramsintermsofforwardscatteringamplitudesofon- contribution to the two-photon amplitude. (In the case
shell thermal particles has been derived in various other of non-Abelian gauge theories it can be proved that all
physicalsystemssuchasnon-Abeliangaugetheoriesand the HTL n-gluon functions, which have the same T2 be-
gravity [18–21]. As a result, in all these cases it is nat- havior, can be generated using the Boltzmann transport
ural to think in terms of a physical scenario where on- equation approach [23].) Our main goal here is to inves-
shell thermal particles of momentum k are scattered by tigate the validity of this equivalence also for all other
7
p1 µ1 p2 µ2 p3 µ3 pn µn In the hard thermal loop approximation the denomina-
tors of the forward scattering amplitudes in (3.13) can
be expanded using
. . .
k k+p1 k+p1+p2 k−pn
FIG. 2: Diagram representing the forward scattering ampli-
tude An (k,p ,··· ,p ). A trace over the product of
µ1µ2...µn 1 n
gamma matrices is to be understood.
n-photon amplitudes, which have each a distinct depen- 1 1 1 1 s2
= − n +.... (3.14)
dence on T, in the HTL approximation. s2n+2k·sn |(cid:126)k|2K·sn |(cid:126)k|2(2K·sn)2
Using the explicit expression for the on-shell momen-
(cid:113)
tum k = |(cid:126)k|K , with K = ( 1+m2/|(cid:126)k|2,(cid:126)k/|(cid:126)k|), Eq.
µ µ
(3.10) yields
(cid:18)(cid:113) (cid:19)
N ((cid:126)k)2+m2
1 (cid:90) ∞ F
±Πhtl = − |(cid:126)k|2d|(cid:126)k|
µ1µ2...µn (2π)3 0 2(cid:113)((cid:126)k)2+m2
(cid:90) dΩ (cid:88) An (±|(cid:126)k|K,p ,··· ,p ). When combined with the momentum dependence of the
µ1µ2...µn 1 n numerators, as shown in Eq. (3.8), the resulting expres-
perm. sion for the forward scattering amplitudes in Eq. (3.13)
n
(3.13) can be expressed as
1 1
An (±|(cid:126)k|K,p ,··· ,p )=±|(cid:126)k|An +An ± An + An ±.... (3.15)
µ1µ2...µn 1 n 1µ1µ2...µn 2µ1µ2...µn |(cid:126)k| 3µ1µ2...µn |(cid:126)k|2 4µ1µ2...µn
Letusfirstconsiderthesimplestcase,namelytheone- powers of T. Therefore, a necessary condition to have
photon amplitude. In this case the forward scattering theequivalence(3.12),isthatcontributionsproportional
amplitude is exactly given by to higher powers of T should vanish. For instance, the
term proportional to T3I(1)(m¯), in the two-photon func-
A1µ(±uTK)=∓euT trK/γµ =∓4euTKµ, (3.16) tion, as well as the terms proportional to T3I(1)(m¯)
and T2I(2)(m¯) in the three-photon function, and so on,
whereuT ≡|(cid:126)k|. UsingEq. (3.10)weobtainthefollowing should vanish.
result for the partial thermal one-photon amplitudes
AllthecontributionsproportionaltoT3I(1)(m¯),which
2eT3I(1)(m¯)(cid:90) would contribute to (3.13), come from the first term in
±Πtherm =± dΩK . (3.17)
µ (2π)3 µ Eq. (3.15). From the HTL procedure above described,
onecanseethatAn musthavethehighestdegree
Comparing this result with the left-hand side of Eq. in K, being propo1rtµi1oµn2a..l.µtno
(3.12) (for n = 1) , which can be easily obtained from
Eq. (2.16), we can see that Eq. (3.12) is satisfied pro-
vided that C =2, which is the correct result for fermions
in 3+1 dimensions. Kα1Kα2...Kαn .
Next, let us consider the more interesting cases when (K·p )[K·(p +p )]...[K·(p +p +p )]
1 1 2 1 2 n−1
n ≥ 2. As we have seen in the previous section, (3.18)
the Boltzmann transport equation approach produces Afterperformingthesumoverallthecyclicpermutations
(n ≥ 2)-photon amplitudes which are proportional to and using momentum conservation, it is straightforward
T4−nI(n)(m¯). On the other hand, Eq. (3.15) seems to toshowthattheresultingexpressionvanishes. Morefor-
yield, for each thermal amplitude, when substituted in mally, we have the following necessary condition for the
Eq. (3.13), several other terms proportional to higher equivalence high-temperature QED and the Boltzmann
8
transport equation formalism of the previous section. of Eq. (3.21). Indeed, if we contract it with p we
3µ3
obtainarelationwhichresemblestheLorentzstructureof
(cid:88)
An (K,p ,··· ,p )=0; n≥2. (3.19) thenon-Abeliangaugetheory,withouttheantisymmetric
1µ1µ2...µn 1 n
cycl. colour factor).
We are then left with a leading contribution to the
We remark that this is a property of individual dia- three-photon function which is of order T as in the case
grams which is true at the level of the forward scatter- of the Boltzmann transport equation approach. With
ing amplitudes (one does not need to add the (n−1)! the help of computer algebra, the resulting expression
permutations to cancel these so-called superleading T3 for A3 can be computed in a straightforward way.
contributions). Of course for a neutral plasma the sum After3pµe1rµf2oµr3mingthepermutationsoftheexternalphoton
+Πhtl +−Πhtl would vanish trivially, since the lines, we obtain complete agreement with (3.12), when
µ1µ2...µn µ1µ2...µn
terms which are proportional to any odd power of T are using Eqs. (2.36) and (A1).
also odd in k. However, in the present context, it is Thenextandmoreinterestingn=4-photonordercan
important to check the behavior of the individual com- bedealtwithsimilarlytothepreviouscases,althoughthe
ponents (for both even and odd n), in order to properly calculation becomes much more involved. A straightfor-
compare with the Boltzmann transport equation results ward computer algebra calculation shows that, besides
of the previous section. the cancellation of the contribution proportional to T3,
SinceEq. (3.19)alreadyeliminatestheT3contribution theothercontributionsduetoA4 andA4 also
to the two-photon function, only the second term in Eq. vanish after we add the permut2aµt1i.o..µn4s. Fina3llµy1,...tµh4e re-
(3.15) may contribute to leading order, producing a T2 maining result is also in agreement with (3.12) as in the
contribution. A straightforward calculation yields (p1 = previous cases. This shows that, up to the four-photon
p and p2 =−p) order, wehavefullagreementwiththeBoltzmanntrans-
port equation approach.
(cid:18) K p +K p p2K K (cid:19)
A2 =2e2 η − µ ν ν µ + µ ν . The previous results show that equivalence between
2µ1µ2...µn µν K·p (K·p)2 thermal field theory and the Boltzmann transport equa-
(3.20) tion approach is verified at the level of the forward scat-
Inserting Eq. (3.20) into Eq. (3.13), we obtain tering amplitudes, in the HTL limit, so that it is not
necessary to explicitly perform the angular integrations
2e2T2I(2)(m¯)
±Πhtl = − in Eq. (3.13).
µν (2π)3
(cid:90) (cid:18) K p +K p p2K K (cid:19)
dΩ η − µ ν ν µ + µ ν .(3.21)
µν K·p (K·p)2 IV. DISCUSSION
Comparing Eq. (3.21) with Eq. (2.32) we can see that
In this work we have investigated the equivalence be-
(3.12) is satisfied for C = 2, which is the correct result
tween the high temperature limit of thermal field theory
for fermions in 3+1 dimensions.
and the Boltzmann transport equation. Specifically, we
Up to this point the thermal field theory calculations
have argued that both approaches produce the same n-
havebeenperformedwithoutusinganycomputeralgebra
photonthermalamplitudes. Thisisindicatedbythegen-
tool. If we go further and try to verify Eq. (3.12) for
eral form derived for the amplitudes, as well as from the
n=3,4,... the calculation starts to become much more
explicit calculations up to n = 4. Furthermore, we have
involved. Therefore, to proceed up to at least n = 4 we
considered the more general cases when the plasma may
have employed the FeynCalc computer algebra tool [24].
not be neutral, so that the amplitudes with odd values
At the three-photon order, there would be in princi-
of n are nonvanishing. This equivalence implies that the
ple contributions from A3, which would give rise to a
2 quantities An in Eq. (3.15) such that p<n should van-
contribution proportional to T2. This would invalidate p
ish. Onlythecontributionwithp=ncanbeinterpreted
theequivalenceoftheBoltzmanntransportequationap-
as arising from the classical limit of finite temperature
proachwiththermalfieldtheory. However,astraightfor-
QED. On the other hand, the p>n contributions repre-
ward calculation yields
sentquantumcorrectionswhicharenotdescribedbythe
A32µ1µ2µ3 = 2Ke3K·pµ33 (cid:18)ηµ1µ2 − Kµ1p1µK2 +·pK1µ2p1µ1 BAobAletlzsimaimnainglanarutgerqeauntihsvpeaoolerrntieceseqauissawwtieeollnll.aksnogwranviitnytahnedcaaslesoofinnothne-
+ (p1)2Kµ1Kµ2(cid:19)−(µ ,p )↔(µ ,p ), particular case of the two-photon amplitude in QED. In
(K·p )2 1 1 3 3 the case of non-Abelian theories it has been useful in or-
1
(3.22) der to derive a closed form for the gauge invariant hard
thermal loop effective action [23]. Our analysis shows
which is manifestly antisymmetric. Therefore, the sum that all the results obtained from the Boltzmann trans-
of its permutations vanish trivially. (It is remarkable portequationcanbeidentifiedwithamoregeneralHTL
how Eq. (3.22) is simply related to the tensor structure limit of the thermal Green’s functions in QED. Conse-
9
quently, the gauge invariant effective action, rather than quencieswhicharesmallcomparedwiththetemperature.
being proportional to a single power of the temperature,
willhaveamoregeneraldependenceonT ascanbeseen
from the form of the thermal amplitudes in Eq. (2.37).
Acknowledgments
We remark that the electron mass is not being neglected
so that the amplitudes with n≥4 are well defined func-
WewouldliketothankCNPqandCAPES(Brazil)for
tions of m/T.
financial support.
Once the equivalence with the Boltzmann transport
equation is established, a full resumation of the thermal
effects in QED, to all orders in the external field, would
require a further investigation, taking into account the Appendix A
gauge invariant effective actions as given by Eq. (2.25).
The resulting effective action would be relevant to de- Here we present the result for the integrand of the
scribe a low density plasma in an external field with fre- three-photon amplitude (2.36):
(cid:34)
A = (cid:88) −Kµ3Kµ2Kµ1p2·p3p12 − 2Kµ3Kµ2Kµ1p1·p3p2·p3 + Kµ3Kµ2p1µ1p2·p3+
µ1µ2µ3 (K·p )2(K·p )2 (K·p )3K·p (K·p )2K·p
perm. 3 1 3 1 3 1
3
2K K p p ·p K K p K·p p 2 K K p p ·p K K p p ·p
µ3 µ2 3µ1 2 3 + µ3 µ1 3µ2 2 1 − µ3 µ1 3µ2 1 2 + µ3 µ1 1µ2 2 3 +
(K·p )3 (K·p )2(K·p )2 (K·p )2K·p (K·p )2K·p
3 3 1 3 1 3 1
2Kµ3Kµ1p3µ2K·p2p1·p3 − Kµ3p1µ1p3µ2K·p2 + Kµ3p2µ1p3µ2 − Kµ3ηµ1µ2p2·p3 −
(K·p )3K·p (K·p )2K·p (K·p )2 K·p 2
3 1 3 1 3 3
2K p p K·p K K p p 2 K K p p ·p K K p p ·p
µ3 3µ1 3µ2 2 + µ2 µ1 2µ3 1 + µ2 µ1 2µ3 1 3 + µ2 µ1 1µ3 2 3 −
(K·p )3 K·p (K·p )2 (K·p )2K·p (K·p )2K·p
3 3 1 3 1 3 1
Kµ2p1µ1p2µ3 − Kµ2p3µ1p2µ3 − Kµ2p2·p3ηµ1µ3 − Kµ1K·p2p12ηµ2µ3 − Kµ1p2µ3p1µ2 +
K·p3K·p1 (K·p3)2 (K·p3)2 K·p3(K·p1)2 K·p3K·p1
Kµ1p1·p2ηµ2µ3 − Kµ1K·p2p3µ2p1µ3 − Kµ1K·p2p1·p3ηµ2µ3 + K·p2p1µ1ηµ2µ3 −.
K·p3K·p1 (K·p3)2K·p1 (K·p3)2K·p1 K·p3K·p1
(cid:35)
p η p η K·p p η K·p p η
2µ1 µ2µ3 + 2µ3 µ1µ2 + 2 3µ1 µ2µ3 + 2 3µ2 µ1µ3 (A1)
K·p3 K·p3 (K·p3)2 (K·p3)2
A similar but much more involved expression has been between the results derived using either the Boltzmann
obtained for the four-photon amplitude, using computer transportequationorthehightemperaturelimitofther-
algebra. In all the cases we have obtained agreement mal field theory.
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