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Gauge independence and relativistic dispersion equations in statistical QED
9 September 1999
Physics Letters B 462 Ž1999. 137–143
Gauge independence and relativistic dispersion equations in statistical QED D. Oliva Aguero, H. Perez A. Amezaga Hechavarrıa ¨ ´ Rojas, A. Perez ´ Martınez, ´ ´ ´ Centro de Matematicas y Fısica Teorica,ICIMAF, Calle E No. 309, Ciudad Habana, Cuba ´ ´ ´ Received 15 January 1999; received in revised form 22 July 1999; accepted 26 July 1999 Editor: M. Cveticˇ
Abstract The gauge parameter dependence of the particle spectra in statistical QED is revisited in a non-perturbative way and it is concluded that the electron spectra is gauge-parameter dependent, the physical spectra being obtained in the Landau gauge. We calculate in the one-loop approximation the gauge parameter dependent behavior of the electron dispersion equation at finite temperature and density, and consider the limit of T s 0 and high density afterwards. The curves obtained confirm the predictions of the general results. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Gauge-parameter dependence The present letter has been motivated essentially for two reasons: first to study the statistical version of the problem of the physical propagators in QED investigated by Lavelle et al. w1,2x, and second to clarify some essential points on gauge invariance in statistical QED which are usually not taken into account in the current literature. The problem about the dependence of the fermionic Green functions on the gauge parameter for QED was solved by Landau and Khalatnikov in w3x and simultaneously by Fradkin in w4x These authors proved that the longitudinal part of the gauge field Green function contributes to the electron Green function only as a phase factor Žsee also w5x.. The last result was extended by Fradkin w6x to the statistical quantum electrodynamics case, where the propagators depend also on the four-velocity of the medium.
It is a well established fact that in QFT the electron propagator is gauge invariant on the one-loop mass shell Ž P 2 s m 2 ., see w7x. For finite temperature, this invariance has been also formally shown w8x. This conclusion has been explicitly verified in several limiting cases w9–11x. However, it should be noted that the calculation of the mass operator is carried out precisely to investigate the system perturbatively out of the mass shell, where the condition of gauge parameter independence frequently mentioned Žsee e.g. w7x. is not fulfilled, since the dispersion equation differs in general from the form P 2 s m2 , although it is expected that the gauge independence of the dispersion equation remains valid near the mass shell. But the departure of the spectrum from the P 2 s m2 form in the temperature case w12x excludes the validity of the above mentioned proof at any order of perturbation theory. We must mention at this point that recently Kalashnikov w13x has found
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 8 8 5 - 0
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D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
that the one-particle spectrum is the same in the Feynman and Coulomb gauges. However, we believe that the problem of fermion spectrum gauge dependence in a medium deserves a general proof, which to the extent of our knowledge, in the current literature does not exist beyond the Žnot widely known. paper by Fradkin w6x. Actually a common practice has been to admit gauge independence, to choose the Feynman gauge and to check it perturbatively in some limits. The purpose of the present letter is to discuss in a non-perturbative, gauge independent way the problem of the electron spectrum gauge dependence in the case of finite temperature and density. At first we must recall the problem of the photon gauge invariance in a medium, and after that we conclude that the physical electron spectrum in statistical QED can be obtained Žin a covariant way. only by using the Landau gauge. Despite of the simplifications introduced by the Feynman gauge, the Landau gauge has especial importance since in other gauges the longitudinal part of the photon propagator contributes as a phase factor to the electron Green function Žsee w6x for a general discussion., leading to a gauge dependence of the spectrum through unphysical photon modes. But when the gauge parameter is chosen in a such a way that the longitudinal part of the photon propagator is zero, the unphysical modes are excluded from the spectrum and the gauge behaves as unitary. We recall that in the temperature formalism the fourth imaginary time coordinate is defined as x 4 , where its range of values is restricted to the interval 0 F x 4 F b , its Fourier conjugate k 4 being 2 npb for bosons and Ž2 n q 1.pb for fermions. After analytical continuation to real frequencies k 4 s i v . In statistics Ža medium. as well as in field theory in vacuum, the photon spectrum is gauge parameter independent. In the case of a medium Ž m / 0, T / 0. due to the gauge invariance of the theory w6x we can write the inverse photon Green function, by introducing in the QED Lagrangian the gauge 21a Ž En An . 2 , as y1 Dmn s Ea2dmn y Em En y P Ž x , y . mn q
1
a
Em En .
Ž 1.
y1 Ž < . 1 2 In Fourier components, Dmn k a s Tmn k y Pmn 1 1 Ž q a km kn s 0 where Tmn s dmn y km knrk 2 .. The
Landau gauge in the propagators is taken as a s 0. In a medium Pmn has the general four-dimensional tensor structure, 1 Pmn s Tmn A q Tmn2 B ,
Ž 2.
where Tmn2 s Ž km kn rk 2 y km un r Ž k P u . yum kn r Ž u P k . q um un k 2rŽ k P u. 2 . and um is the four-velocity of the medium. The photon Green function is then 1 Dmn s Tmn A1 q Tmn2 B1 q
a km kn k4
,
Ž 3.
where A1 s 1r Ž k 2 y A . and B1 s BrŽŽ k 2 y A .Ž k 2 y A q BŽ1 y k 2rŽ k P u. 2 .... In the rest frame the polarization operator and the photon green function Žfor a ™ 0. have the tensor structure:
°d ~ž ¢
ijy
Ž s. Pmn s
ki k j k
2
/
Žs. Cs q P44
P4Žis . s Pi4Ž s . sy
ki k4 k2
k i k j k 42
i , js1,2,3 ,
k2 k2
Ž 4.
Žs. P44
ss1,2 ,
Ž1. Ž2. where Pmn s Pmn , Pmn s Dmn Ž k <`., C1 s AŽ k, k 4 ., 2 2 P44 s k rk Ž A Ž k ,k 4 . q B Ž k ,k 4 . k 2rk 2 . , C2 s 1rŽ k 2 y A . and D44 s k 4rw k 4 Ž k 2 y P44 .x. The photon spectra in a medium may be obtained y1 either by solving the equation Dmn s 0, in which 1 / 0 factorizes, or from the poles of the four-dia mensional transverse part of Dmn . In the first case we obtain in the rest frame the spectra, which is non-perturbatively gauge-parameter independent, and also gauge invariant. It is,
2
2
k q A Ž k ,k 4 . s 0,
1q
P44 Ž k 2 ,k 4 . k
2
s0 .
Ž 5.
The first equation corresponds to the spatial transverse modes and the second Žafter multiplication by k 2 / 0., to the spatial longitudinal one. These are the three physical modes in a medium, Žwhich in the one-loop approximation and at high temperature have been studied in detail in w10x.. A factor k 4 , accounting for the unphysical modes introduced by the y1 r2 gauge condition, when calculating w Det Dmn x , is removed by the Faddeev-Popov ghost term. The photon spectrum can be also obtained directly from the poles of the Green function Ž3.. We have in Ž3.
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
the non-physical modes k 4 s 0, given by the longitudinal term, which are not present in Ž4. since they were removed by choosing the gauge parameter a ™ 0. In this way the Landau gauge becomes a unitary one. In calculating the exact polarization operator in statistics as well as in quantum field theory, only the transverse part Žthat is, the part of the electron Green function G t depending through the self-energy S from the transverse part of Dmn . of the electron Green function contributes. The property stems from the gauge invariance and gauge parameter independence of Pmn , as can be derived from the Ward identities obtained from the temperature QED effective action G as well as in the vacuum case, which in momentum space reads y1 km Dmn Ž k. s
1
a
kn k 2 ,
Ž 6.
y1 Ž . The substitution of Dmn k s k 2dmn y km kn Ž1 y a1 . y Pmn , leads to the four-dimensional transversality for Pmn . This means that Pmn Ž x, y . is gauge-invariant and gauge-parameter independent w6,14x. Explicitly we have that Pmn Ž x, y . is given by
2
GŽ x yy< a . s G 0 Ž x y y . exp e 2 D l Ž x y y . y D l Ž 0 .
X
X
4
where
Dl Ž x . s y
d3k
a 3
2 Ž 2p . b
where G Ž z , y, y X . s g d Ž z y y . d Ž y y y X . y dS Ž z, y .rd ieAm . Thus, if a gauge parameter dependent GŽ x, y < a . is chosen, cancellations occur in the integrand in such a way that the integral obtained from Ž7. must be the same as that obtained by using GŽ x, y <0. s G t Ž x, y ., where G t Ž x, y . is obviously the exact electron Green function obtained in the Landau gauge Žsee below.. This means that only the physical poles of GŽ x, y . contribute to Pmn , and in consequence, to the transverse part of the exact photon Green function Dmn . Concerning the Fermion spectrum, the gauge parameter dependence is introduced in the calculation of the mass operator S in terms of GŽ x, y < a . and Dmn Ž x, y < a .. Now, for the statistical case, as shown by Fradkin w6x, the gauge parameter dependence is given by a similar formula than in QED. We shall
ÝH k4
k4
eik x
Ž 9.
in Ž8., G 0 is the propagator in the Landau gauge and in our representation coincides with G t ; also, k 4 s 2p n b , n s 0," 1,... The poles of GŽ p, a . are obtained from the poles of the Fourier transform of the right hand term in Ž8. which is the convolution of G 0 Ž x y y . with the exponential a-dependent factor. Due to this convolution the resulting poles are expected to be a-dependent at any order of a perturbative expansion of GŽ p .. We may also present our previous arguments as follows: Let us call S t and S l respectively the transverse and longitudinal terms of the electron mass operator, where e2 2pb
Ý HgmG Ž p q k . k4
4
Ž 7.
4, Ž 8.
S s
s e Tr gmG Ž x , z . Gn Ž z , y, y . G Ž y , x . d xd y ,
H
write it for the one-particle electron Green function as
l
Pmn Ž x , y .
139
km kn
D ld 3 k , Ž 10 . k2 which is obviously nonzero and gauge-parameter dependent. Then, by taking D l Ž k . s ark 2 we observe it contains also the contribution of the unphysical photon modes k 42 q k 2 s 0. Also, by using Ward identities it can be shown w6x that =Gn Ž p q k ,k .
igm pm q m q S t Ž p . G Ž p . s1y
e2
Ý HgmG Ž p q k . 2pb k4
a km k4
d3k .
Ž 11 .
We observe that the poles of GŽ p . are given by the zeros of the term in squared brackets in Ž11. plus the poles introduced by the second term, containing the unphysical photon modes. Thus, the poles of the electron Green function become independent of the unphysical photon modes if the limit a ™ 0 is taken, or in other words, by eliminating the contribution of these modes to S .
140
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
The zero temperature QED case can be treated by following similar arguments. In the vacuum case, we may cite especially Ref. w1x in which a very interesting discussion of dressed electron physical modes is given in the one-loop approximation for S by using the Coulomb gauge, which is the suitable one for the vacuum case, since there are only two physical Žtransverse. modes. Such dressing was extended in w16x to electrons moving at arbitrary velocities. Thus, although there are general proofs of gauge invariance for gauge boson spectrum w15x, there are not Žwe consider there cannot be. explicit proofs of gauge parameter independence of the fermion collective mode spectrum given by the poles of the Green function, except in very specific cases, as the infrared limit considered by Abrikosov w17x. However, at least in the abelian case, if all calculations are made in the Landau gauge, which is equivalent to choosing the longitudinal part of the photon Green function as equal to zero, the results lead to the physical fermion spectrum. This is more adequate gauge in the case of a medium, where we have three physical modes. Our essential conclusion is then: The physical fields AmŽ x . P, c Ž x . P, which exclude the unphysical longitudinal modes from S Ž x, y ., are obtained from any given fields AmŽ x ., c Ž x . by the gauge transformations AmŽ x . P s AmŽ x . q Em h Ž x ., c Ž x . P s e eh Ž x .c Ž x . in which the condition h Ž x . s yEm AŽ x .m rEm2 q h X Ž x . must hold, where h X Ž x . is an arbitrary function defined on the light cone, i.e., E 2h X Ž x . s 0. Obviously, we get physical fields also by gauge transforming AmŽ x . P, c Ž x . P with regard to the functions h X Ž x .. For non-abelian theories, an expression analogous to Ž8. has not been obtained yet. However, we consider that extensions of Refs. w1,2x, to QCD made in w18,19x, as well as our previous arguments, may be helpful for the consideration of our problem in a medium in the non-abelian case.
for different gauges. We start from the expression for the electron self-energy in the one loop approximation in the temperature formalism Žwe use the shifted momenta pm) s pm y i m e d4, l . w20x
S e Ž p. e2 s
3 Ž 2p . b
Ý Hd 3 k gmG 0 Ž p ) q k . Gn 0 Dmn0 Ž k . , k4
Ž 12 . e2 s
3
Ž 2p . b = gn
Ý Hd 3 k
gm
k4
1 iŽ
2 gm pm) q k q m2e
dmn q Ž a y 1 . km knrk 2 k2
.
.
Ž 13 .
Due to the dependence with regard to the four velocity vector um Žin the rest frame um s Ž0,0,0,i .., we have k P u s ik 4 s yv , and thus the vectors on which the physical quantities may depend, are km and um . The matrix structure of the mass operator is now
S Ž p . s i Ž agr pr q bgr ur q c . s 0 .
Ž 14 .
After some work Žsee w22x., we have, for an arbitrary a asg2
1 p2
Ž p 2 I1 q 4 I3 q 2 I4 q 2 i v I5 .
qa Ž p 2 I1 y I3 y 2 I4 y 2 i v I5 . 4 , bsg2
1 p2
Ž 15 .
y3 i p 2 I2 y 3 v I3 q 2 v I4
y2 i Ž p 2 y v 2 . I5 q 2 p 2v I6 qa y i p 2 I2 q v I3 q 2 v I4 y2 i Ž p 2 y v 2 . I5 q 2 p 2v I6
4,
Ž 16 .
2. Electron self energy
c s Ž 3 q a . i g 2 m e I1 ,
In this section we will perform the computation of the mass operator in the one loop approximation for a general gauge. We will keep the dependence on the gauge parameter to allow a further comparison of the results
where g 2 s e 2r Ž 2p . is the fine structure constant and Ii are integrals defined by expanding the bracket of expression Ž13.; Žsee for details w22x., then these integrals are calculated by performing at first the Matsubara sum over the fourth momentum compo-
2
Ž 17 .
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
nent, and by doing some algebra, subject to the following approximations: Ž1. we drop the terms not depending on the temperature, i.e. we ignore nonthermal electrons; Ž2. we assume that the chemical potential is much larger than the temperature, m 4 T, so that we may approximate the electron distribution function by a step function. From the expressions for the integrals Ii , we carry out the analytic continuation p4 q i m s i v , perform the angular integrations, and finally neglect the mass m2 y m2e , m in the upper limit in the integrals. Note that we do not neglect the mass with regard to momentum. The expressions we obtain read as follows:
(
I1 s y
I2 s y
p
m
p
H0 (k
pi
dk k
m
2
,
AyB
Ž 18 .
(
ž
(k
H0 (k
= y2 q
2
A B
2
2
q m2e
q m 2e
/ ln
AqB AyB
H0 (k
In a medium, the propagation of an electron is described by the dispersion equation ) Sy1 e s Ž i gm pm q m e y S . s 0 .
Ž 25 .
The dispersion equation is thus 2
gm Ž Ž 1 y a . pm y i b dm 4 . y i Ž m e y i c . s 0 .
(Ž 1 y a.
2
2
p 2 y Ž m e y i c . y Ž 1 y a. v q b s 0 .
Ž 27 . p2
AqB
ln
AyB
AqB
ln
y
2
2B
AyB
,
q m 2e
2
A yB
2
p2 y 2B
AqB
ln
AyB
m
H0
ž ( (k q m
dk k 2 v y k 2 q m 2e 2
A y 2 p2 A2 y B 2
1 y 2B
ž
2
q m 2e
b s g 2p Ž 3 q a . ,
AyB
(
2
asg p Ž1ya .
2
Ž 22 .
m2 y 2v
2
/
Ž A yB .
v
2
m y Ž3q2 a .
p
ln
vqp vyp
,
Ž 28 .
2 e
dk k 2 v y k 2 q m 2e
H0 (k
m2
2
/
AqB
ln
In order to obtain analytic representations for the integrals Ii , we neglect the electron mass in the square roots: this is justified because the integrands increase monotonically with regard to k and the upper limit of integration Ž, m . is assumed to be very large. At ultrahigh densities, the electron behaves like a massless fermion quasiparticle. Finally we obtain, for the ultrahigh density limit Ž m 4 m e ., the following expressions for the functions a, b and c:
,
Ž 21 .
I6 s y2 p
3. Dispersion equations
Ž 19 . q m 2e
2 p2 Ž k 2 q p2 q A.
m
Ž 24 .
Bs2 k p .
By the analytic continuation ik 4 s yv , we get finally
dk k 2
m
I4 s y2 p
=
(
,
Ž 20 .
I5 s y2 p i
½
A s p 2 y v 2 y m2e q 2 v k 2 q m 2e ,
dk k 2
m
I3 s y2 p
=
with the notation
Ž 26 .
dk k v y k
H0
p
ln
q m2e
AqB
141
,
Ž 23 .
m2 2p
vm ya
p
ln
ln
vqp vyp
vqp vyp
c s y Ž 3 q a . g 2p i m e
m p
ln
,
vqp vyp
Ž 29 . .
Ž 30 .
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
142
Fig. 1. The spectrum of fermionic excitations in an ultrarelativistic plasma for two values of the chemical potential. The dotted curves represent the light cone and the free electron spectrum.
Replacing these expressions in the dispersion equation Ž27., it is easy to obtain the dispersion equations, which constitute our main result,
(
vqs p 2 q m2e 2
qg p
m2 2p
Ž1ya .
q Ž 3 q 2 a . ln
4. Discussion
p
vq vqq p vqy p
.
Ž 31 .
Here, vq is the normal dispersion equation and corresponds to an electron-like collective excitation spectrum. There is also an abnormal dispersion branch Ž vy- 0., which have an essential collective character,
(
vys y p 2 q m2e qg2 p
m2 2p
Ž1ya .
q Ž 3 q 2 a . ln
the limit m s 0. This has been pointed out also by Kalashnikov w13x. However, this is a result obtained perturbatively, which gives back the tree mass shell one-particle spectrum. In w22x the normal and abnormal dispersion curves are plotted for several values of the density. In the case of vq it can be shown that its effective mass grows and its approximation to the light cone with increasing density is faster. For vy ,the curve tends to the light cone faster than for vq, moreover it has a minimum for a finite value of p which decreases with the density. The abnormal branch shows a negative effective mass near p s 0 as a hole quasiparticle. Note that we get two different dispersion equations for particles and holes, unlike other results at finite temperature, such as densities near the electron mass w23x, high temperature but without chemical potential w9x, and others w24x.
We return to the problem of gauge invariance. As shown in Fig. 2, the spectrum is dependent on the gauge parameter, but for prm e ) 1 and near the mass shell, the dispersion curves obtained, for some values of the gauge parameter, approach each other and to the mass shell and become gauge-independent. In this sense this verifies the mass shell P 2 s m2 gauge independence pointed out by several authors w7,9,13x, but it occurs for very large momenta. One must stress at this point that the gauge-dependent scalar b plays an essential role in producing a
p
vy vyq p
. Ž 32 . vyy p The solutions of Ž31. and Ž32. are shown in Fig. 1 together with the light cone and the free electron dispersion law for two different values of the density. As can be observed the behavior of the dispersion curves is analogous to the reported in w21x, although in our case lower densities are employed, which allows us to have a more realistic description for applications in the astrophysical and cosmological contexts. From Ž31. and Ž32. we observe that gauge independence of the spectrum is obtained in
Fig. 2. Dependence of the normal branch with the gauge parameter. The dotted lines correspond to the light cone and the free dispersion curves.
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
departure of the spectrum from the P 2 s m2 mass shell, which manifests especially in the low momentum limit: lim p ™ 0 b s bŽ v , m , a . / 0. Thus, the mass shell gauge invariance of S pointed out in w9x does not work in the present case. However the results of Section 2 indicate that the Landau gauge, despite of introducing algebraic complications, is the appropriate one. The minimum at p 0 / 0 in the abnormal solution, suggests an analogy with a superfluid Žor superconductive, since we are considering charged fermionic particles. behavior, but this conclusion cannot be obtained only from the mass operator. To give a criterium of superconductivity is necessary to analyze the vertex part, which is beyond the scope of the present letter. Acknowledgements It is a pleasure to acknowledge many enlightening discussions with A. Cabo, M. Chaichian, M. RuizAltaba and M. Torres. D.O. thanks especially Professor Virasoro, the ICTP High Energy Group, IAEA and UNESCO for hospitality at the International Centre of Theoretical Physics under The Federation Arrangement Program. References w1x M. Lavelle, D. McMullan, Phys. Lett. B 312 Ž1993. 211. w2x M. Lavelle, D. McMullan, Phys. Lett. B 316 Ž1993. 172.
143
w3x L.D. Landau, I.M. Khalatnikov, Sov. Phys. JETP 2 Ž1955. 69. w4x E.S. Fradkin, Sov. Phys. JETP 2 Ž1955. 258. w5x B. Zumino, J. Math. Phys. 1 Ž1960. 1. w6x E.S. Fradkin, Quantum Field Theory and Hydrodynamics, Proc. of the P.N. Lebedev Physics Inst. No. 29, Consultants Bureau, New York, Ž1967. 1. w7x L.S. Brown, Quantum Field Theory ŽCambridge University Press, Cambridge, U.K., 1996, pp. 521-523.; R. Tarrach, Nucl. Phys. B 183 Ž1981. 384; D. Atkinson, M. Fry, Nucl. Phys. B 156 Ž1979. 301; J.C. Breckenridge et al., Z. Phys. C 65 Ž1995. 155. w8x N.P. Landsman, Phys. Lett. B 172 Ž1986. 46. w9x T. Toimela, Nucl. Phys. B 273 Ž1986. 719. w10x A. Weldon, Phys. Rev. D 26 Ž1982. 2789. w11x P. Elmfors, D. Persson, B. Skagerstam, Nucl. Phys B 464 Ž1996. 153. w12x J. Kapusta, T. Toimela, Phys. Rev. D 39 Ž1989. 3197. w13x O.K. Kalashnikov, Mod. Phys. Lett. A 13 Ž1998. 1719. w14x C. Nash, Relativistic Quantum Fields, Academic Press, London, 1978. w15x R. Kobes, G. Kunsttatter, A. Rebhan, Nucl. Phys. B 355 Ž1991. 1. w16x E. Bagan. M. Lavelle, D. McMullan, Phys. Lett. B370 Ž1996. 128; Phys. Rev. D 56 Ž1997. 3732. w17x A.A. Abrikosov, Zh. Eksp. Teor. Fiz. 30 Ž1956. 96, wSov. Phys. JETP 3 Ž1956. 71x. w18x M. Lavelle, D. McMullan, Phys. Lett. B 436 Ž1998. 339. w19x L. Chen, K. Haller, hep-thr9803250. w20x A. Perez H. Perez ´ Martınez, ´ ´ Rojas, A. Zepeda, Phys. Lett. B 366 Ž1996. 235. w21x J.P. Blaizot, J.Y. Ollitrault, Phys. Rev. D 48 Ž1993. 1390. w22x D. Oliva, H. Perez A. Amezaga, ´ Rojas, A. Perez ´ Martınez, ´ ´ Relativistic Electron Dispersion Equation in Extremely Dense Media hep-phr9804220. w23x E.J. Levinson, Phys. Rev. D 31 Ž1985. 3280. w24x E. Petitgirard, Z Phys. C 54 Ž1992. 673.
Physics Letters B 462 Ž1999. 137–143
Gauge independence and relativistic dispersion equations in statistical QED D. Oliva Aguero, H. Perez A. Amezaga Hechavarrıa ¨ ´ Rojas, A. Perez ´ Martınez, ´ ´ ´ Centro de Matematicas y Fısica Teorica,ICIMAF, Calle E No. 309, Ciudad Habana, Cuba ´ ´ ´ Received 15 January 1999; received in revised form 22 July 1999; accepted 26 July 1999 Editor: M. Cveticˇ
Abstract The gauge parameter dependence of the particle spectra in statistical QED is revisited in a non-perturbative way and it is concluded that the electron spectra is gauge-parameter dependent, the physical spectra being obtained in the Landau gauge. We calculate in the one-loop approximation the gauge parameter dependent behavior of the electron dispersion equation at finite temperature and density, and consider the limit of T s 0 and high density afterwards. The curves obtained confirm the predictions of the general results. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Gauge-parameter dependence The present letter has been motivated essentially for two reasons: first to study the statistical version of the problem of the physical propagators in QED investigated by Lavelle et al. w1,2x, and second to clarify some essential points on gauge invariance in statistical QED which are usually not taken into account in the current literature. The problem about the dependence of the fermionic Green functions on the gauge parameter for QED was solved by Landau and Khalatnikov in w3x and simultaneously by Fradkin in w4x These authors proved that the longitudinal part of the gauge field Green function contributes to the electron Green function only as a phase factor Žsee also w5x.. The last result was extended by Fradkin w6x to the statistical quantum electrodynamics case, where the propagators depend also on the four-velocity of the medium.
It is a well established fact that in QFT the electron propagator is gauge invariant on the one-loop mass shell Ž P 2 s m 2 ., see w7x. For finite temperature, this invariance has been also formally shown w8x. This conclusion has been explicitly verified in several limiting cases w9–11x. However, it should be noted that the calculation of the mass operator is carried out precisely to investigate the system perturbatively out of the mass shell, where the condition of gauge parameter independence frequently mentioned Žsee e.g. w7x. is not fulfilled, since the dispersion equation differs in general from the form P 2 s m2 , although it is expected that the gauge independence of the dispersion equation remains valid near the mass shell. But the departure of the spectrum from the P 2 s m2 form in the temperature case w12x excludes the validity of the above mentioned proof at any order of perturbation theory. We must mention at this point that recently Kalashnikov w13x has found
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 8 8 5 - 0
138
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
that the one-particle spectrum is the same in the Feynman and Coulomb gauges. However, we believe that the problem of fermion spectrum gauge dependence in a medium deserves a general proof, which to the extent of our knowledge, in the current literature does not exist beyond the Žnot widely known. paper by Fradkin w6x. Actually a common practice has been to admit gauge independence, to choose the Feynman gauge and to check it perturbatively in some limits. The purpose of the present letter is to discuss in a non-perturbative, gauge independent way the problem of the electron spectrum gauge dependence in the case of finite temperature and density. At first we must recall the problem of the photon gauge invariance in a medium, and after that we conclude that the physical electron spectrum in statistical QED can be obtained Žin a covariant way. only by using the Landau gauge. Despite of the simplifications introduced by the Feynman gauge, the Landau gauge has especial importance since in other gauges the longitudinal part of the photon propagator contributes as a phase factor to the electron Green function Žsee w6x for a general discussion., leading to a gauge dependence of the spectrum through unphysical photon modes. But when the gauge parameter is chosen in a such a way that the longitudinal part of the photon propagator is zero, the unphysical modes are excluded from the spectrum and the gauge behaves as unitary. We recall that in the temperature formalism the fourth imaginary time coordinate is defined as x 4 , where its range of values is restricted to the interval 0 F x 4 F b , its Fourier conjugate k 4 being 2 npb for bosons and Ž2 n q 1.pb for fermions. After analytical continuation to real frequencies k 4 s i v . In statistics Ža medium. as well as in field theory in vacuum, the photon spectrum is gauge parameter independent. In the case of a medium Ž m / 0, T / 0. due to the gauge invariance of the theory w6x we can write the inverse photon Green function, by introducing in the QED Lagrangian the gauge 21a Ž En An . 2 , as y1 Dmn s Ea2dmn y Em En y P Ž x , y . mn q
1
a
Em En .
Ž 1.
y1 Ž < . 1 2 In Fourier components, Dmn k a s Tmn k y Pmn 1 1 Ž q a km kn s 0 where Tmn s dmn y km knrk 2 .. The
Landau gauge in the propagators is taken as a s 0. In a medium Pmn has the general four-dimensional tensor structure, 1 Pmn s Tmn A q Tmn2 B ,
Ž 2.
where Tmn2 s Ž km kn rk 2 y km un r Ž k P u . yum kn r Ž u P k . q um un k 2rŽ k P u. 2 . and um is the four-velocity of the medium. The photon Green function is then 1 Dmn s Tmn A1 q Tmn2 B1 q
a km kn k4
,
Ž 3.
where A1 s 1r Ž k 2 y A . and B1 s BrŽŽ k 2 y A .Ž k 2 y A q BŽ1 y k 2rŽ k P u. 2 .... In the rest frame the polarization operator and the photon green function Žfor a ™ 0. have the tensor structure:
°d ~ž ¢
ijy
Ž s. Pmn s
ki k j k
2
/
Žs. Cs q P44
P4Žis . s Pi4Ž s . sy
ki k4 k2
k i k j k 42
i , js1,2,3 ,
k2 k2
Ž 4.
Žs. P44
ss1,2 ,
Ž1. Ž2. where Pmn s Pmn , Pmn s Dmn Ž k <`., C1 s AŽ k, k 4 ., 2 2 P44 s k rk Ž A Ž k ,k 4 . q B Ž k ,k 4 . k 2rk 2 . , C2 s 1rŽ k 2 y A . and D44 s k 4rw k 4 Ž k 2 y P44 .x. The photon spectra in a medium may be obtained y1 either by solving the equation Dmn s 0, in which 1 / 0 factorizes, or from the poles of the four-dia mensional transverse part of Dmn . In the first case we obtain in the rest frame the spectra, which is non-perturbatively gauge-parameter independent, and also gauge invariant. It is,
2
2
k q A Ž k ,k 4 . s 0,
1q
P44 Ž k 2 ,k 4 . k
2
s0 .
Ž 5.
The first equation corresponds to the spatial transverse modes and the second Žafter multiplication by k 2 / 0., to the spatial longitudinal one. These are the three physical modes in a medium, Žwhich in the one-loop approximation and at high temperature have been studied in detail in w10x.. A factor k 4 , accounting for the unphysical modes introduced by the y1 r2 gauge condition, when calculating w Det Dmn x , is removed by the Faddeev-Popov ghost term. The photon spectrum can be also obtained directly from the poles of the Green function Ž3.. We have in Ž3.
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
the non-physical modes k 4 s 0, given by the longitudinal term, which are not present in Ž4. since they were removed by choosing the gauge parameter a ™ 0. In this way the Landau gauge becomes a unitary one. In calculating the exact polarization operator in statistics as well as in quantum field theory, only the transverse part Žthat is, the part of the electron Green function G t depending through the self-energy S from the transverse part of Dmn . of the electron Green function contributes. The property stems from the gauge invariance and gauge parameter independence of Pmn , as can be derived from the Ward identities obtained from the temperature QED effective action G as well as in the vacuum case, which in momentum space reads y1 km Dmn Ž k. s
1
a
kn k 2 ,
Ž 6.
y1 Ž . The substitution of Dmn k s k 2dmn y km kn Ž1 y a1 . y Pmn , leads to the four-dimensional transversality for Pmn . This means that Pmn Ž x, y . is gauge-invariant and gauge-parameter independent w6,14x. Explicitly we have that Pmn Ž x, y . is given by
2
GŽ x yy< a . s G 0 Ž x y y . exp e 2 D l Ž x y y . y D l Ž 0 .
X
X
4
where
Dl Ž x . s y
d3k
a 3
2 Ž 2p . b
where G Ž z , y, y X . s g d Ž z y y . d Ž y y y X . y dS Ž z, y .rd ieAm . Thus, if a gauge parameter dependent GŽ x, y < a . is chosen, cancellations occur in the integrand in such a way that the integral obtained from Ž7. must be the same as that obtained by using GŽ x, y <0. s G t Ž x, y ., where G t Ž x, y . is obviously the exact electron Green function obtained in the Landau gauge Žsee below.. This means that only the physical poles of GŽ x, y . contribute to Pmn , and in consequence, to the transverse part of the exact photon Green function Dmn . Concerning the Fermion spectrum, the gauge parameter dependence is introduced in the calculation of the mass operator S in terms of GŽ x, y < a . and Dmn Ž x, y < a .. Now, for the statistical case, as shown by Fradkin w6x, the gauge parameter dependence is given by a similar formula than in QED. We shall
ÝH k4
k4
eik x
Ž 9.
in Ž8., G 0 is the propagator in the Landau gauge and in our representation coincides with G t ; also, k 4 s 2p n b , n s 0," 1,... The poles of GŽ p, a . are obtained from the poles of the Fourier transform of the right hand term in Ž8. which is the convolution of G 0 Ž x y y . with the exponential a-dependent factor. Due to this convolution the resulting poles are expected to be a-dependent at any order of a perturbative expansion of GŽ p .. We may also present our previous arguments as follows: Let us call S t and S l respectively the transverse and longitudinal terms of the electron mass operator, where e2 2pb
Ý HgmG Ž p q k . k4
4
Ž 7.
4, Ž 8.
S s
s e Tr gmG Ž x , z . Gn Ž z , y, y . G Ž y , x . d xd y ,
H
write it for the one-particle electron Green function as
l
Pmn Ž x , y .
139
km kn
D ld 3 k , Ž 10 . k2 which is obviously nonzero and gauge-parameter dependent. Then, by taking D l Ž k . s ark 2 we observe it contains also the contribution of the unphysical photon modes k 42 q k 2 s 0. Also, by using Ward identities it can be shown w6x that =Gn Ž p q k ,k .
igm pm q m q S t Ž p . G Ž p . s1y
e2
Ý HgmG Ž p q k . 2pb k4
a km k4
d3k .
Ž 11 .
We observe that the poles of GŽ p . are given by the zeros of the term in squared brackets in Ž11. plus the poles introduced by the second term, containing the unphysical photon modes. Thus, the poles of the electron Green function become independent of the unphysical photon modes if the limit a ™ 0 is taken, or in other words, by eliminating the contribution of these modes to S .
140
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
The zero temperature QED case can be treated by following similar arguments. In the vacuum case, we may cite especially Ref. w1x in which a very interesting discussion of dressed electron physical modes is given in the one-loop approximation for S by using the Coulomb gauge, which is the suitable one for the vacuum case, since there are only two physical Žtransverse. modes. Such dressing was extended in w16x to electrons moving at arbitrary velocities. Thus, although there are general proofs of gauge invariance for gauge boson spectrum w15x, there are not Žwe consider there cannot be. explicit proofs of gauge parameter independence of the fermion collective mode spectrum given by the poles of the Green function, except in very specific cases, as the infrared limit considered by Abrikosov w17x. However, at least in the abelian case, if all calculations are made in the Landau gauge, which is equivalent to choosing the longitudinal part of the photon Green function as equal to zero, the results lead to the physical fermion spectrum. This is more adequate gauge in the case of a medium, where we have three physical modes. Our essential conclusion is then: The physical fields AmŽ x . P, c Ž x . P, which exclude the unphysical longitudinal modes from S Ž x, y ., are obtained from any given fields AmŽ x ., c Ž x . by the gauge transformations AmŽ x . P s AmŽ x . q Em h Ž x ., c Ž x . P s e eh Ž x .c Ž x . in which the condition h Ž x . s yEm AŽ x .m rEm2 q h X Ž x . must hold, where h X Ž x . is an arbitrary function defined on the light cone, i.e., E 2h X Ž x . s 0. Obviously, we get physical fields also by gauge transforming AmŽ x . P, c Ž x . P with regard to the functions h X Ž x .. For non-abelian theories, an expression analogous to Ž8. has not been obtained yet. However, we consider that extensions of Refs. w1,2x, to QCD made in w18,19x, as well as our previous arguments, may be helpful for the consideration of our problem in a medium in the non-abelian case.
for different gauges. We start from the expression for the electron self-energy in the one loop approximation in the temperature formalism Žwe use the shifted momenta pm) s pm y i m e d4, l . w20x
S e Ž p. e2 s
3 Ž 2p . b
Ý Hd 3 k gmG 0 Ž p ) q k . Gn 0 Dmn0 Ž k . , k4
Ž 12 . e2 s
3
Ž 2p . b = gn
Ý Hd 3 k
gm
k4
1 iŽ
2 gm pm) q k q m2e
dmn q Ž a y 1 . km knrk 2 k2
.
.
Ž 13 .
Due to the dependence with regard to the four velocity vector um Žin the rest frame um s Ž0,0,0,i .., we have k P u s ik 4 s yv , and thus the vectors on which the physical quantities may depend, are km and um . The matrix structure of the mass operator is now
S Ž p . s i Ž agr pr q bgr ur q c . s 0 .
Ž 14 .
After some work Žsee w22x., we have, for an arbitrary a asg2
1 p2
Ž p 2 I1 q 4 I3 q 2 I4 q 2 i v I5 .
qa Ž p 2 I1 y I3 y 2 I4 y 2 i v I5 . 4 , bsg2
1 p2
Ž 15 .
y3 i p 2 I2 y 3 v I3 q 2 v I4
y2 i Ž p 2 y v 2 . I5 q 2 p 2v I6 qa y i p 2 I2 q v I3 q 2 v I4 y2 i Ž p 2 y v 2 . I5 q 2 p 2v I6
4,
Ž 16 .
2. Electron self energy
c s Ž 3 q a . i g 2 m e I1 ,
In this section we will perform the computation of the mass operator in the one loop approximation for a general gauge. We will keep the dependence on the gauge parameter to allow a further comparison of the results
where g 2 s e 2r Ž 2p . is the fine structure constant and Ii are integrals defined by expanding the bracket of expression Ž13.; Žsee for details w22x., then these integrals are calculated by performing at first the Matsubara sum over the fourth momentum compo-
2
Ž 17 .
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
nent, and by doing some algebra, subject to the following approximations: Ž1. we drop the terms not depending on the temperature, i.e. we ignore nonthermal electrons; Ž2. we assume that the chemical potential is much larger than the temperature, m 4 T, so that we may approximate the electron distribution function by a step function. From the expressions for the integrals Ii , we carry out the analytic continuation p4 q i m s i v , perform the angular integrations, and finally neglect the mass m2 y m2e , m in the upper limit in the integrals. Note that we do not neglect the mass with regard to momentum. The expressions we obtain read as follows:
(
I1 s y
I2 s y
p
m
p
H0 (k
pi
dk k
m
2
,
AyB
Ž 18 .
(
ž
(k
H0 (k
= y2 q
2
A B
2
2
q m2e
q m 2e
/ ln
AqB AyB
H0 (k
In a medium, the propagation of an electron is described by the dispersion equation ) Sy1 e s Ž i gm pm q m e y S . s 0 .
Ž 25 .
The dispersion equation is thus 2
gm Ž Ž 1 y a . pm y i b dm 4 . y i Ž m e y i c . s 0 .
(Ž 1 y a.
2
2
p 2 y Ž m e y i c . y Ž 1 y a. v q b s 0 .
Ž 27 . p2
AqB
ln
AyB
AqB
ln
y
2
2B
AyB
,
q m 2e
2
A yB
2
p2 y 2B
AqB
ln
AyB
m
H0
ž ( (k q m
dk k 2 v y k 2 q m 2e 2
A y 2 p2 A2 y B 2
1 y 2B
ž
2
q m 2e
b s g 2p Ž 3 q a . ,
AyB
(
2
asg p Ž1ya .
2
Ž 22 .
m2 y 2v
2
/
Ž A yB .
v
2
m y Ž3q2 a .
p
ln
vqp vyp
,
Ž 28 .
2 e
dk k 2 v y k 2 q m 2e
H0 (k
m2
2
/
AqB
ln
In order to obtain analytic representations for the integrals Ii , we neglect the electron mass in the square roots: this is justified because the integrands increase monotonically with regard to k and the upper limit of integration Ž, m . is assumed to be very large. At ultrahigh densities, the electron behaves like a massless fermion quasiparticle. Finally we obtain, for the ultrahigh density limit Ž m 4 m e ., the following expressions for the functions a, b and c:
,
Ž 21 .
I6 s y2 p
3. Dispersion equations
Ž 19 . q m 2e
2 p2 Ž k 2 q p2 q A.
m
Ž 24 .
Bs2 k p .
By the analytic continuation ik 4 s yv , we get finally
dk k 2
m
I4 s y2 p
=
(
,
Ž 20 .
I5 s y2 p i
½
A s p 2 y v 2 y m2e q 2 v k 2 q m 2e ,
dk k 2
m
I3 s y2 p
=
with the notation
Ž 26 .
dk k v y k
H0
p
ln
q m2e
AqB
141
,
Ž 23 .
m2 2p
vm ya
p
ln
ln
vqp vyp
vqp vyp
c s y Ž 3 q a . g 2p i m e
m p
ln
,
vqp vyp
Ž 29 . .
Ž 30 .
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
142
Fig. 1. The spectrum of fermionic excitations in an ultrarelativistic plasma for two values of the chemical potential. The dotted curves represent the light cone and the free electron spectrum.
Replacing these expressions in the dispersion equation Ž27., it is easy to obtain the dispersion equations, which constitute our main result,
(
vqs p 2 q m2e 2
qg p
m2 2p
Ž1ya .
q Ž 3 q 2 a . ln
4. Discussion
p
vq vqq p vqy p
.
Ž 31 .
Here, vq is the normal dispersion equation and corresponds to an electron-like collective excitation spectrum. There is also an abnormal dispersion branch Ž vy- 0., which have an essential collective character,
(
vys y p 2 q m2e qg2 p
m2 2p
Ž1ya .
q Ž 3 q 2 a . ln
the limit m s 0. This has been pointed out also by Kalashnikov w13x. However, this is a result obtained perturbatively, which gives back the tree mass shell one-particle spectrum. In w22x the normal and abnormal dispersion curves are plotted for several values of the density. In the case of vq it can be shown that its effective mass grows and its approximation to the light cone with increasing density is faster. For vy ,the curve tends to the light cone faster than for vq, moreover it has a minimum for a finite value of p which decreases with the density. The abnormal branch shows a negative effective mass near p s 0 as a hole quasiparticle. Note that we get two different dispersion equations for particles and holes, unlike other results at finite temperature, such as densities near the electron mass w23x, high temperature but without chemical potential w9x, and others w24x.
We return to the problem of gauge invariance. As shown in Fig. 2, the spectrum is dependent on the gauge parameter, but for prm e ) 1 and near the mass shell, the dispersion curves obtained, for some values of the gauge parameter, approach each other and to the mass shell and become gauge-independent. In this sense this verifies the mass shell P 2 s m2 gauge independence pointed out by several authors w7,9,13x, but it occurs for very large momenta. One must stress at this point that the gauge-dependent scalar b plays an essential role in producing a
p
vy vyq p
. Ž 32 . vyy p The solutions of Ž31. and Ž32. are shown in Fig. 1 together with the light cone and the free electron dispersion law for two different values of the density. As can be observed the behavior of the dispersion curves is analogous to the reported in w21x, although in our case lower densities are employed, which allows us to have a more realistic description for applications in the astrophysical and cosmological contexts. From Ž31. and Ž32. we observe that gauge independence of the spectrum is obtained in
Fig. 2. Dependence of the normal branch with the gauge parameter. The dotted lines correspond to the light cone and the free dispersion curves.
D. OliÕa Aguero ¨ et al.r Physics Letters B 462 (1999) 137–143
departure of the spectrum from the P 2 s m2 mass shell, which manifests especially in the low momentum limit: lim p ™ 0 b s bŽ v , m , a . / 0. Thus, the mass shell gauge invariance of S pointed out in w9x does not work in the present case. However the results of Section 2 indicate that the Landau gauge, despite of introducing algebraic complications, is the appropriate one. The minimum at p 0 / 0 in the abnormal solution, suggests an analogy with a superfluid Žor superconductive, since we are considering charged fermionic particles. behavior, but this conclusion cannot be obtained only from the mass operator. To give a criterium of superconductivity is necessary to analyze the vertex part, which is beyond the scope of the present letter. Acknowledgements It is a pleasure to acknowledge many enlightening discussions with A. Cabo, M. Chaichian, M. RuizAltaba and M. Torres. D.O. thanks especially Professor Virasoro, the ICTP High Energy Group, IAEA and UNESCO for hospitality at the International Centre of Theoretical Physics under The Federation Arrangement Program. References w1x M. Lavelle, D. McMullan, Phys. Lett. B 312 Ž1993. 211. w2x M. Lavelle, D. McMullan, Phys. Lett. B 316 Ž1993. 172.
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