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A Slavnov-Taylor identity and equality of damping rates for static transverse and longitudinal gluons in hot QCD
19 August 1999
Physics Letters B 461 Ž1999. 131–137
A Slavnov-Taylor identity and equality of damping rates for static transverse and longitudinal gluons in hot QCD M. Dirks, a
a,b,1
, A. Niegawa, ´
a,2
, K. Okano
a,3
Department of Physics, Osaka City UniÕersity, Sumiyoshi-ku, Osaka 558-8585, Japan b Fakultat ¨ f ur ¨ Physik, UniÕersitat ¨ Bielefeld, D-33501 Bielefeld, Germany Received 17 June 1999 Editor: H. Georgi
Abstract A Slavnov-Taylor identity is derived for the gluon polarization tensor in hot QCD. We evaluate its implications for damping of gluonic modes in the plasma. Applying the identity to next to the leading order in hard-thermal-loop resummed perturbation theory, we derive the expected equality of damping rates for static transverse and longitudinal Žsoft. gluons. This is of interest also in view of deviating recent reports of g t Ž p s 0. / g l Ž p s 0. based on a direct calculation of g l Ž p s 0.. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 12.38.Mh; 12.38.-t; 12.38.Bx Keywords: Hot QCD; Slavnov-Taylor identity; Damping rate; Gluon
1. Introduction Much interest is devoted to the physics of a the deconfinement phase of hadronic matter Žquark-gluon plasma, QGP., with both strong experimental and theoretical research going on. A good theoretical understanding of this new phase of matter is urgently required, in particular, in order to identify worthwhile observables of this new phase of matter. Insight can be gained in the framework of ‘Hot QCD’ supplemented with a perturbative approach and im1 Supported by the German Academic Exchange Service ŽDAAD.. E-mail: [email protected]. 2 E-mail: [email protected] 3 E-mail: [email protected]
portant progress has been made during the last decade w1x. In particular thermal effects are known to alter soft modes in an important way, changing dispersion relations, and allowing for Landau damping as well as for new modes of propagation for both fermionic Žquarks. and bosonic Žgluons. quasiparticles w2x. The collective dynamics involved can be consistently described within Hard-Thermal-Loop ŽHTL. resummed perturbation theory w3,4x. In this letter we are interested in damping of collective gluonic excitations as one of the most important characteristics of the plasma dynamics. While on-shell damping is known to be absent to HTL order g 2 T 2 a non zero damping rate is expected to arise at order g Ž g 2 T 2 .. In early work by Braaten and Pisarski w5x a finite, gauge independent
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 2 9 - 1
132
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
result for the damping rate of static transverse gluons g t Ž v s m g , p s 0. has been obtained, as one of the important applications of the HTL-resummation scheme. Note that gauge invariance also follows from a general theorem w6x. Also, in the static regime, g l Ž m g , p s 0. s g t Ž m g , p s 0. has been mentioned in w5x, as is expected since longitudinal Ž l . and transverse Ž t . degrees of freedom can not be distinguished at zero spatial momentum. However, a complete understanding of the damping behavior in the plasma is not yet available. In particular, for non-zero values of the spatial momentum singular results are obtained for transverse and also for longitudinal gluons w7x, which originate from the absence of a chromo-magnetic mass in the HTL resummed gluon propagator. Moreover, also the analyticity of the damping rates around zero momentum has been questioned recently w8x: Expansion for small p of the imaginary part of the gluon polarization tensor has been reported to lead to g t Ž m g ,0. / g l Ž m g ,0. and even to a quadratic divergence in g l Ž m g , p s 0.. In this letter we derive a Slavnov-Taylor ŽST. identity for the gluon polarization tensor in Coulomb gauge. After discussing some general consequences of it, we apply it to the next to leading order contribution to the gluon polarization tensor )P mn , which might be helpful in order to further clarify the physics of damping for gluonic modes in the plasma. From it we easily obtain well known transversality properties as well as new constraints in particular concerning the imaginary part of )P mn related to damping. The expected equality g l Ž m g ,0. s g t Ž m g ,0. can also be derived. We report on these findings in the next two sections and further implications are discussed in the conclusions section. For convenience, the derivation of the ST identity is deferred to the Appendix.
2. Slavnov-Taylor identity and its consequences Within the imaginary-time formalism of hot QCD, ST identities are derived from the BRST invariance of the Lagrangian in the same way as in vacuum theory. For a discussion of dynamical quantities continuation to Minkowski space should be done in the usual way. We briefly summarize the procedure in the Appendix.
Most important to the dynamics of gluonic fields is the identity involving the gluon polarization tensor P mn . In Coulomb gauge it reads: Pn P mn Ž P . s y d nm P 2 y P m Pn q P nm Ž P . P gn Ž P . ,
Ž 1.
with P gn Ž P . related to the Coulomb-ghost self-energy part P g Ž P . through p iP gi Ž P . s P g Ž P . 4 . The derivation of the result Ž1. is explained in the Appendix. Note that it is an exact identity, valid for arbitrary momenta P and to all order in perturbation theory. From Eq. Ž1., other identities are immediately obtained that will be useful for the discussion below. First contracting also the remaining Lorentz index we obtain Pm P mn Ž P . Pn s yPm P nm Ž P . P gn Ž P . s P 2 gmn y Pm Pn P gmP gn q P gmPmn P gn .
Ž 2.
Concerning the imaginary part of P mn Ž P ., since Im P gn Ž P . s 0, we obtain
5
P n Im Pmn Ž P . s yP gn Ž P . Im Pmn Ž P . .
Ž 3.
P m Im Pmn Ž P . P n s P gmP gn Im Pmn .
Ž 4.
Note again that, as Eq. Ž1., also Eqs. Ž2. – Ž4., are general, valid for arbitrary momenta and to all orders in perturbation theory. Eqs. Ž1. – Ž4. contain important informations. At leading and next to leading order several of them have been established through direct calculation. Consider the soft momentum region, P m s Ž v , p . ;
4
Unless otherwise stated, throughout in this letter, Greek letters, m , n , . . . , take 0,1,2,3 while Ratin letters, i, j, . . . , take 1,2,3. 5 This can readily be seen in real-time formalism w4x. The building blocks of perturbation theory, i.e., the propagators and vertices, and then also the self-energy part, are 2=2 matrices in ‘‘thermal space’’. Im P gn Ž P . here is proportional to the Ž1,2.component of Ž P gn Ž P .. i j Ž i, js1,2.. Since, in Coulomb gauge, the ghost propagator-matrix Ž D g Ž P .. i j Ž i, js1,2. is diagonal in thermal space, Ž P gn Ž P .. i j is also diagonal and Ž P gn Ž P ..12 s 0.
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
gT; from Eqs. Ž1. and Ž2. we readily rederive two well known results from power counting arguments: 1. At lowest nontrivial order of effective ŽHTL-resummed. perturbation theory w3,4x P mn Ž P . Žs dP mn Ž P .. s O Ž g 2 T 2 . so the left-hand side ŽLHS. of Eq. Ž1. is of O Ž g 3 T 3 .. On the other hand P gn Ž P . s O Ž g 2 T . Žno HTL-contribution in amplitudes with external ghosts w3,4x. and the right-hand side ŽRHS. of Eq. Ž1. is of O Ž g 4 T 3 .. Therefore, to the HTL accuracy, Pn P mn Ž P . , Pn dP mn Ž P . s 0. 2. Similarly, at leading order, the LHS of Eq. Ž2. is of O Ž g 4 T 4 .. In the second line of Eq. Ž2., P gm Ž P . s O Ž g 2 T ., P mn Ž P . s dP mn Ž P . s O Ž g 2 T 2 ., P g Ž P . s O Ž g 3 T 2 .. Therefore, RHSr LHS s O Ž g 2 ., and Pn P nm Ž P . Pm s 0 Ž 5. holds to relative order g 2 . A more precise statement is derived below, cf. Eq. Ž8.. We now proceed exploiting the results Eqs. Ž1. – Ž4. in more detail. We focus on the soft momentum region and the next to leading order contribution to P mn Ž P . from which the leading damping behavior will be deduced. Following standard notations, we write P mn Ž P . to this order dP nm Ž P . q)P nm Ž P . with dP the HTL-contribution and )P the relative order g correction. The diagrams that yield leading contributions to the imaginary part of )P are shown in Fig. 1. In Eqs. Ž1. – Ž4. we need an explicit expression for the ghost contribution. It is sufficient to calculate P gm Ž P . to lowest nontrivial order, which we carry out in Appendix Žcf. Eq. ŽA.17..: g 2 NT i P gm Ž P . , yd m i pˆ , Ž 6. 16 where pˆ i s p irp. It is worth mentioning in passing that P gm Ž v , p s 0. s 0. This is because, for p s 0, the summandrintegrand of Eq. ŽA.15. in Appendix, which is valid for arbitrary ŽEuclidean. four-
133
momentum PE , is odd in the summationrintegration variables K E . First we use Eq. Ž6. in Eq. Ž1.: To next to leading order under consideration we can write dP nm for P nm on the RHS of Eq. Ž1. and using dP nm P n s 0 w3,4x as well as Ž5. we find Pn)P nm Ž P . s
g 2 NT v ) y1 v Dl Ž P . 1, pˆ 16 p p
ž
m
/
Ž 7.
with the inverse plasmon propagator w3,4x, ) y1 Dl
Ž P . s p 2 y dP 00 Ž P . s p 2 q 3m2g 1 y
v 2p
ln
ž
vqp vyp
/
,
where m 2g s N Ž gT . 2r9. Note that the close connection with the inverse propagator is of course apparent in Eq. Ž1.. Also Eq. Ž7. is valid for any value of l, the Coulomb gauge parameter, Eq. ŽA.1. Žin the Appendix.. We have confirmed Eq. Ž7. through direct calculation of the LHS. Incidentally, from Eq. Ž7., Pm)P m i Ž P . e iŽ r . Ž pˆ . s 0, Ž r s 1,2. for any three-vector e iŽ r . Ž p . perpendicular to pˆ and arbitrary Žsoft. P m. Using Eq. Ž6. in Eq. Ž2. we find Pm)P mn Ž P . Pn sy
ž
2
g 2 NTv
/
16
= 1y
1
v2
pˆ idP i j Ž P . pˆ j q O Ž g . ,
Ž 8.
which is of O Ž g 6 T 4 . when P m is soft, thus specifying somewhat the result under point Ž2. above. Finally, making progress towards the discussion of damping in the next subsection, we note that, from Eq. Ž7., the imaginary part of Pm)P mn Ž P . is related to the Landau damping contribution in the HTL-propagator. We have: Im Pm)P mn Ž P . 3p s 32
Fig. 1. Next to leading order contributions to the imaginary part of the gluon polarization tensor. All loop-momenta are soft and HTL-resummed propagators and vertices have to be used.
g 2 NTm2g
v
ž / p
2
u Ž p2 y v 2 . ,
Ž 9.
and therefore for time-like momenta, Im Pm)P mn Ž P . s 0
Ž P 2 ) 0. .
Ž 10 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
134
3. Damping in a gluonic medium Soft gluon damping rates relate to the imaginary part of the gluon polarization tensor P mn , leading damping arises from the relative order g correction ) mn P . The relevant diagrams are shown in Fig. 1 where however the Coulomb-ghost loop does not contribute to Im P mn . HTL resummed propagators and vertices have to be used since the dominant contribution arises from configurations with soft loop momentum. For the following presentation we restrict to strict Coulomb gauge Ž l s 0 in Eq. ŽA.2. in the Appendix.. For gluon damping only the transverse and longitudinal components of )P mn 1 2
rithmic singularity has been extracted for non-zero momentum in both g t and g l w7x. At zero momentum the calculation of g t is well documented in the literature but note that calculating g l requires to expand )P l Ž P . to order O Ž p 2 . – a formidable and subtle task. A recent controversy w8x suggests that approximations should be handled with care and observing general constraints. We are now in a position to show that our results so far leads to g l Ž m g ,0. s g t Ž m g ,0. and thus a consistent calculation necessarily provides it. On the gluon mass-shell, P 2 s v 2 y p 2 ) 0, we obtain from Eq. Ž5. and Eq. Ž10. Im )P
00
Ž P . s yIm )P l Ž P .
ij Pt Ž P . s d H Ž pˆ . P i j Ž P . ,
)
ij dH
)
ij
i
p
s
j
Ž pˆ . s d y pˆ pˆ ,
P l Ž P . s y)P
)
00
Ž P. .
Ž 11 .
will be relevant. In the vicinity of v s v t r l Ž p ., the mass-shell for quasiparticles to HTL-accuracy, the respective inverse propagator component can be written:
žv/
3m 2g v t2 y Ž v t2 y p 2 .
Ž p. s 3 p
m2g
2
p™0
™ 2 mg , 1
v l Ž v l2 y p 2 .
y 3vl
p™0
™
2 p2 mg
g l Ž m g ,0 . s g t Ž m g ,0 . .
From this, the damping rates for longitudinal and transverse gluonic modes are obtained as:
g t Ž v t Ž p . , p . s yZt Ž p . Im )P t Ž v t Ž p . , p . p™0
™ y
1 2 mg
Ž 14 .
Finally, using v l r t Ž p s 0. s m g and Eq. Ž13. with Ž14. in Eq. Ž12., we derive that
2
v t Ž v t2 y p 2 .
Ž 13 .
pˆ i pˆ j Im )P i j Ž v ,0 . s 13 Im )P i i Ž v ,0 . s Im )P t Ž v ,0 . .
q i Im )P t r l Ž v s v t r l , p . ,
Zy1 l
pˆ i pˆ j Im )P i j Ž v , p . .
Specializing to p s 0 and using the fact )P i j Ž v , p s 0. A d i j we find
y1 Gy1 t r l , .Zt r l Ž p . Ž v y v t r l .
Zy1 t Ž p. s
2
holds, as is expected on physical grounds. Finally it is interesting to observe what consequences arise if we assume, on physical grounds, that g l Ž m g ,0. s gg Ž m g ,0. to all orders in perturbation theory. From Eqs. Ž11. and Ž12. this assumption leads to
Im )P t Ž m g ,0 . ,
v
lim
)
g l Ž v l Ž p . , p . s qZl Ž p . Im P l Ž v l Ž p . , p . p™0 m g ™ Im )P l Ž m g ,0 . , 2 p2
p™0
Ž 12 .
with m g s v t r l Ž p s 0. the plasma frequency. The direct calculation of g l r t , for general p, has not yet been accomplished, however a leading loga-
Ž 15 .
2
ž / p
Im P
00
s pˆ i pˆ j Im P i j Ž v ,0 . ,
and using this in Eq. Ž4. one finds lim p™0
1 p2
P gm P gn q 2 d n i p i Im Pmn s 0 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
135
The bare gluon propagator D˜ mn Ž PE . reads
4. Conclusions and remarks The Slavnov-Taylor identity for gluon polarization tensor we report on in this letter, Eq. Ž1., allows to obtain various constrains. Applying it to the next to the leading order in HTL-resummed perturbation theory, the identity Eq. Ž7. results, which is relevant to leading-order damping rate of gluonic modes. As has been discussed, for zero three momentum the expected equality g t Ž m g ,0. s g l Ž m g ,0. can be derived with the help of the identity Eq. Ž7.. One of the advantages of the above procedure is that the explicit expansion of )P 00 Ž v l , p . around p ; 0, which has been found to be troublesome, can be avoided in this derivation. Moreover, concerning the direct calculation of g l , the present work indicates that dealing with singularities in intermediate steps of the calculation and necessary changes in the integration variables should be made in a way consistent with the identity Eq. Ž7..
D˜ mn Ž PE . s d m id n j
d i j y pˆ i pˆ j PE2 1
m0 n 0
qd d
p2
V g s yigCa b c p j ,
where p is the outgoing momentum carried by ha . The generating functional reads Z J ,j ,j s D A am Dha D ha exp S q d 4 x Jam Ž x . A am Ž x .
H
HT
dha s
dx 0 d 3 x L Ž x . ' d 4 x L Ž x . ,
ž
H
L Ž x . s y 14 Famn Ž x . Famn Ž x . y qd
mi
ha Ž x . E
i
Dxm, a b
Famn s E mAna y E nA am q gC a b c Dx,ma b Ž
/
HT
1
1 d ha s y = P A a dz , l
2l
2
g 2
C a b ch bhc dz ,
where dz is a Grassmann-number parameter. Using this fact in ŽA.4., we obtain,
HTd
Ž= P A a Ž x . .
4
z BŽ z . Z J , j , j s 0 ,
B Ž z . s Jam Ž z . Dzm, a b
Ž A . hb Ž x . ,
A bm Anc .
1
Ž A.1 .
Here A. ' d a b ErE x m y gC a b c A cm Ž x . and ha and ha are the Coulomb-ghost fields. For notations related to the color space, we follow w9x.
Ž A.4 .
where, the functional integral is to be performed with periodic boundary conditions for all fields Žincluding ghosts w10x., A am Ž x 0 s 0, x . s A am Ž x 0 s 1rT, x ., etc. The action S Žor L ., Eq. ŽA.1., is invariant under the BRST transformation w9x:
Appendix A. Derivation of (1) and (6) We start in the imaginary-time formalism continuing to real energies at the final stage. The QCD action in Coulomb gauge reads
Ž A.2 .
Ž A.3 .
d A am s Damb Ž A . h b dz ,
1rT
,
qj a Ž x . ha Ž x . q ha Ž x . j a Ž x . 4 ,
M.D. likes to thank the German Academic Exchange Office ŽDAAD. for financial and general support of his stay in Osaka. Thanks also to the Faculty of Science, Osaka City University, for kind hospitality.
H0
p4
where PEm s Ž p 0 , p . with p 0 s 2p Tn Ž n s 0 " 1," 2, . . . ., and the bare ghost propagator reads D˜ g Ž p . s 1rp 2 . The form of gluon-ghost vertex can be read off from L 2 gC a b c Ž E j ha .h b A cj ,
Acknowledgements
Ss
ql
PEm PEn
q
l
d
d J jbŽ z.
E ja Ž z .
d
ž /
Ez
d i
d Jai
Ž z.
g d d q Ca b c j a Ž z . . 2 dj b Ž z . dj c Ž z .
Ž A.5 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
136
Computing
d d Jam
we substitute ŽA.7. and ŽA.8. into ŽA.6. and use the form ŽA.2. for D˜ s i to obtain
d
Ž x . dj b Ž y .
d a b d 4 z P˜ nm Ž x y z . E ymDˆ g Ž z y y .
HT
4
= d z B Ž z . ln Z J , j , j
HT
Js j s j s0 ,
s ygC ac d d 4j d 4 z Ž G˜y1 .
by using ŽA.5., we obtain
d a b Exm G˜g Ž x y y . y gC ac d ² A dm Ž x . hc Ž x . h b Ž y . : q da b
1 E
l Ey
i
G˜ m i Ž x y y . s 0 .
Ž A.6 .
Note that the third term of Eq. ŽA.5. does not contribute to ŽA.6.. The ghost propagator G˜g and the gluon propagator G˜ mn in Eq. ŽA.6. are defined, respectively, through
d a b G˜g Ž x y y . s ²ha Ž x . h b Ž y . :
dj b Ž y . dj a Ž x .
Js j s j s0
=² A dr Ž z . hc Ž z . h b Ž y . : .
Ž x . d Jbn Ž y .
.
gC ac d ² A dr Ž z . hc Ž z . h b Ž y . :
Ž A.10 .
Ž A.11 .
P˜ nm Ž PE . PEm s yp 2 G˜g Ž PE .
G˜ mn Ž x y y . s D˜ mn Ž x y y .
d nm PE2 y PEn PEm
qP˜ nm Ž PE . 4 P˜ gm Ž PE .
y d 4 ud 4 Õ G˜ mr Ž x y u . P˜ rs
HT
y
= Ž u y Õ . D˜ sn Ž Õ y y . ,
Ž A.7 .
d a b G˜g Ž x y y . s d a b D˜ g Ž x y y .
HT
F .T .
Putting altogether in ŽA.9. and transforming to Fourier space, we finally obtain
Js j s j s0
² A dm Ž x .hc Ž x .h b Ž y .: is the gluon-ghost three-point function. Now, we call for the Schwinger-Dyson equations:
q gC a d e d 4 z
Ž A.9 .
The inverse propagator Ž Gˆy1 . nm is related to P˜ nm through Ž G˜y1 . nm s Ž D˜y1 . nm q P˜ nm . The Fourier transform of ² A dr Ž z .hc Ž z .h b Ž y .: may be written as
p iP˜ gi Ž PE . s P˜ g Ž PE . .
d ln Z d Jam
= d r Ei jmEji D˜ g Ž j y z . q d 4 Ž j y z . d m r
where Ž P˜ gr Ž PE ..a b is related to a ghost self-energy part P˜ g through
,
d a b G˜ mn Ž x y y . s ² A am Ž x . Anb Ž y . : '
Ž xyj .
s yi d a b P˜ gr Ž PE . G˜g Ž PE . ,
d ln Z
'
nm
HT HT
p2
P˜ mn Ž PE . P˜ g Ž PE . ,
Ž A.12 . where use has been made of ŽA.11.. Noting that G˜g Ž PE . s w p 2 q P˜ g Ž PE .xy1 and using P˜ mn s P˜ nm , Eq. ŽA.12. may be ‘solved’ as
ED˜ g Ž x y z . E zi
=² Aie Ž z . hd Ž z . h b Ž y . : ,
PEm
Ž A.8 .
where P˜ rs is the gluon polarization tensor. It can readily be shown that, in the limit g ™ 0, ŽA.6. holds, as it should be. Taking this fact into account,
PEn P˜ mn Ž PE . s y d mn PE2 y PEm PEn q P˜ mn Ž PE . P˜ gn Ž PE . .
Ž A.13 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
137
From this we obtain Eq. Ž1. in the main text after continuation to real energies ip 0 ™ v q i0q according to
1rŽ e z r T y 1.. On summing over k 0 and using nŽ z . , Trz and Hd zr t Ž z ,k .rz s 1rk 2 , we obtain
P m s Ž v , p. ,
P˜ gm Ž PE . , yg 2 NTd m i
PE2 ™ yP 2 s p 2 y v 2 ,
P˜ mn Ž ip 0 , p . ™ Ž yi . P˜ gm Ž ip 0 , p . ™ Ž yi .
dm 0 q dn 0
dm 0
P mn Ž v , p . ,
P gm Ž v , p . .
Ž A.14 .
, yd m i
P gm Ž P .
Finally we compute with soft P to lowest non-trivial order – one-loop contribution – as required for the discussion in the main text. Using Eqs. ŽA.2. and ŽA.3., we have
P˜ gm Ž PE . s yg 2 N T Ý k0
s yg 2 N T Ý k0
d3k
H Ž 2p .
3
Ž p i y k i . ˜im D Ž KE . 2 Ž pyk.
d3k
H Ž 2p .
pi yk i 3
Ž pyk.
ij ˆ = d m jd H Ž k . D˜ Ž K E . q l
2
k i K Em k4
,
Ž A.15 .
ij Ž ˆ. where p i s Ž0, p ., K Em s Ž k 0 , k . and d H k is as in Ž11.. In Eq. ŽA.15., D˜Ž K E . s 1rK E2 for hard p, while, for soft p, the HTL-resummed )D˜Ž K E . is substituted for D˜Ž K E .. After carrying out the renormalization, one can easily see that the leading contribution comes from the first term Žin the square brackets. with soft k region:
P˜ gm Ž PE . , yg 2 N T Ý k0
=
1rT
H0
d3k
Hsoft k Ž 2p .
Ž pi yk i . 3
Ž pyk.
2
ij ˆ d m jd H Ž k.
dt e i k 0t d z r t Ž z ,k . 1 q n Ž z . ey zt ,
H
Ž A.16 . where r t Ž z ,k . is the spectral function of the HTL-resummed transverse gluon propagator and nŽ z . s
g 2 NT 16
p i y Ž p P kˆ . kˆ i
d 3k
Hsoft k Ž 2p .
3
k2 Ž pyk.
pˆ i ,
which is independent of l. Noting that ŽA.17. has only spatial components and is dent of p 0 , we have P gm Ž P . s P˜ gm Ž PE . ŽA.14...
2
Ž A.17 . P˜ gm in indepenŽcf. Eq.
References w1x T. Hatsuda, Y. Miake, S. Nagamiya, K. Yagi Žeds.., Nucl. Phys. A 638 Ž1998. 1; Proc. 48th Yamada Conference, Quark Matter’97, Tsukuba, December 1-5, 1997; U. Heinz Žed.., rhep-phr9811469rtft98proceedings.html, TFT98: Proc. 5th International Workshop on Thermal Field Theories and their Applications, Regensburg, August 10-14, 1998. w2x V.P. Silin, Zh. Eksp. Teor. Fiz. 38 Ž1960. 1577 wSov. Phys. JETP 11 Ž1960. 1136x; O.K. Kalashnikov, V.V. Klimov, Yad. Fiz. 31 Ž1980. 1357 wSov. J. Nucl. Phys. 31 Ž1980. 699x; V.V. Klimov, Zh. Eksp. Teor. Fiz. 82 Ž1982. 336 wSov. Phys. JETP 55 Ž1982. 199x; H.A. Weldon, Phys. Rev. D 26 Ž1982. 1394; R.D. Pisarski, Physica A 158 Ž1989. 146. w3x R.D. Pisarski, Phys. Rev. Lett. 63 Ž1989. 1129: E. Braaten, R.D. Pisarski, Nucl. Phys. B 337 Ž1990. 569; 339 Ž1990. 310; J. Frenkel, J.C. Taylor, ibid. 334 Ž1990. 199. w4x M. Le Bellac, Thermal Field Theory, Cambridge Univ. Press, Cambridge, 1996. w5x E. Braaten, R.D. Pisarski, Phys. Rev. D 42 Ž1990. 2156. w6x R. Kobes, G. Kunstatter, A. Rebhan, Nucl. Phys. B 355 Ž1991. 1. w7x R.D. Pisarski, Phys. Rev. D 47 Ž1993. 5589; F. Flechsig, A.K. Rebhan, H. Schulz, Phys. Rev. D52 Ž1995. 2994. w8x A. Abada, O. Azi, K. Benchallal, Phys. Lett. B 425 Ž1998. 158; A. Abada, O. Azi, hep-phr9807439 v2, Jan. 1999. w9x C. Itzykson, J.-B. Zuber, Quantum Field Theory, McGrawHill, New York, 1980. w10x H. Hata, T. Kugo, Phys. Rev. D 21 Ž1980. 3333; H. Matsumoto, Y. Nakano, H. Umezawa, ibid. 28 Ž1983. 1931.
Physics Letters B 461 Ž1999. 131–137
A Slavnov-Taylor identity and equality of damping rates for static transverse and longitudinal gluons in hot QCD M. Dirks, a
a,b,1
, A. Niegawa, ´
a,2
, K. Okano
a,3
Department of Physics, Osaka City UniÕersity, Sumiyoshi-ku, Osaka 558-8585, Japan b Fakultat ¨ f ur ¨ Physik, UniÕersitat ¨ Bielefeld, D-33501 Bielefeld, Germany Received 17 June 1999 Editor: H. Georgi
Abstract A Slavnov-Taylor identity is derived for the gluon polarization tensor in hot QCD. We evaluate its implications for damping of gluonic modes in the plasma. Applying the identity to next to the leading order in hard-thermal-loop resummed perturbation theory, we derive the expected equality of damping rates for static transverse and longitudinal Žsoft. gluons. This is of interest also in view of deviating recent reports of g t Ž p s 0. / g l Ž p s 0. based on a direct calculation of g l Ž p s 0.. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 12.38.Mh; 12.38.-t; 12.38.Bx Keywords: Hot QCD; Slavnov-Taylor identity; Damping rate; Gluon
1. Introduction Much interest is devoted to the physics of a the deconfinement phase of hadronic matter Žquark-gluon plasma, QGP., with both strong experimental and theoretical research going on. A good theoretical understanding of this new phase of matter is urgently required, in particular, in order to identify worthwhile observables of this new phase of matter. Insight can be gained in the framework of ‘Hot QCD’ supplemented with a perturbative approach and im1 Supported by the German Academic Exchange Service ŽDAAD.. E-mail: [email protected]. 2 E-mail: [email protected] 3 E-mail: [email protected]
portant progress has been made during the last decade w1x. In particular thermal effects are known to alter soft modes in an important way, changing dispersion relations, and allowing for Landau damping as well as for new modes of propagation for both fermionic Žquarks. and bosonic Žgluons. quasiparticles w2x. The collective dynamics involved can be consistently described within Hard-Thermal-Loop ŽHTL. resummed perturbation theory w3,4x. In this letter we are interested in damping of collective gluonic excitations as one of the most important characteristics of the plasma dynamics. While on-shell damping is known to be absent to HTL order g 2 T 2 a non zero damping rate is expected to arise at order g Ž g 2 T 2 .. In early work by Braaten and Pisarski w5x a finite, gauge independent
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 2 9 - 1
132
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
result for the damping rate of static transverse gluons g t Ž v s m g , p s 0. has been obtained, as one of the important applications of the HTL-resummation scheme. Note that gauge invariance also follows from a general theorem w6x. Also, in the static regime, g l Ž m g , p s 0. s g t Ž m g , p s 0. has been mentioned in w5x, as is expected since longitudinal Ž l . and transverse Ž t . degrees of freedom can not be distinguished at zero spatial momentum. However, a complete understanding of the damping behavior in the plasma is not yet available. In particular, for non-zero values of the spatial momentum singular results are obtained for transverse and also for longitudinal gluons w7x, which originate from the absence of a chromo-magnetic mass in the HTL resummed gluon propagator. Moreover, also the analyticity of the damping rates around zero momentum has been questioned recently w8x: Expansion for small p of the imaginary part of the gluon polarization tensor has been reported to lead to g t Ž m g ,0. / g l Ž m g ,0. and even to a quadratic divergence in g l Ž m g , p s 0.. In this letter we derive a Slavnov-Taylor ŽST. identity for the gluon polarization tensor in Coulomb gauge. After discussing some general consequences of it, we apply it to the next to leading order contribution to the gluon polarization tensor )P mn , which might be helpful in order to further clarify the physics of damping for gluonic modes in the plasma. From it we easily obtain well known transversality properties as well as new constraints in particular concerning the imaginary part of )P mn related to damping. The expected equality g l Ž m g ,0. s g t Ž m g ,0. can also be derived. We report on these findings in the next two sections and further implications are discussed in the conclusions section. For convenience, the derivation of the ST identity is deferred to the Appendix.
2. Slavnov-Taylor identity and its consequences Within the imaginary-time formalism of hot QCD, ST identities are derived from the BRST invariance of the Lagrangian in the same way as in vacuum theory. For a discussion of dynamical quantities continuation to Minkowski space should be done in the usual way. We briefly summarize the procedure in the Appendix.
Most important to the dynamics of gluonic fields is the identity involving the gluon polarization tensor P mn . In Coulomb gauge it reads: Pn P mn Ž P . s y d nm P 2 y P m Pn q P nm Ž P . P gn Ž P . ,
Ž 1.
with P gn Ž P . related to the Coulomb-ghost self-energy part P g Ž P . through p iP gi Ž P . s P g Ž P . 4 . The derivation of the result Ž1. is explained in the Appendix. Note that it is an exact identity, valid for arbitrary momenta P and to all order in perturbation theory. From Eq. Ž1., other identities are immediately obtained that will be useful for the discussion below. First contracting also the remaining Lorentz index we obtain Pm P mn Ž P . Pn s yPm P nm Ž P . P gn Ž P . s P 2 gmn y Pm Pn P gmP gn q P gmPmn P gn .
Ž 2.
Concerning the imaginary part of P mn Ž P ., since Im P gn Ž P . s 0, we obtain
5
P n Im Pmn Ž P . s yP gn Ž P . Im Pmn Ž P . .
Ž 3.
P m Im Pmn Ž P . P n s P gmP gn Im Pmn .
Ž 4.
Note again that, as Eq. Ž1., also Eqs. Ž2. – Ž4., are general, valid for arbitrary momenta and to all orders in perturbation theory. Eqs. Ž1. – Ž4. contain important informations. At leading and next to leading order several of them have been established through direct calculation. Consider the soft momentum region, P m s Ž v , p . ;
4
Unless otherwise stated, throughout in this letter, Greek letters, m , n , . . . , take 0,1,2,3 while Ratin letters, i, j, . . . , take 1,2,3. 5 This can readily be seen in real-time formalism w4x. The building blocks of perturbation theory, i.e., the propagators and vertices, and then also the self-energy part, are 2=2 matrices in ‘‘thermal space’’. Im P gn Ž P . here is proportional to the Ž1,2.component of Ž P gn Ž P .. i j Ž i, js1,2.. Since, in Coulomb gauge, the ghost propagator-matrix Ž D g Ž P .. i j Ž i, js1,2. is diagonal in thermal space, Ž P gn Ž P .. i j is also diagonal and Ž P gn Ž P ..12 s 0.
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
gT; from Eqs. Ž1. and Ž2. we readily rederive two well known results from power counting arguments: 1. At lowest nontrivial order of effective ŽHTL-resummed. perturbation theory w3,4x P mn Ž P . Žs dP mn Ž P .. s O Ž g 2 T 2 . so the left-hand side ŽLHS. of Eq. Ž1. is of O Ž g 3 T 3 .. On the other hand P gn Ž P . s O Ž g 2 T . Žno HTL-contribution in amplitudes with external ghosts w3,4x. and the right-hand side ŽRHS. of Eq. Ž1. is of O Ž g 4 T 3 .. Therefore, to the HTL accuracy, Pn P mn Ž P . , Pn dP mn Ž P . s 0. 2. Similarly, at leading order, the LHS of Eq. Ž2. is of O Ž g 4 T 4 .. In the second line of Eq. Ž2., P gm Ž P . s O Ž g 2 T ., P mn Ž P . s dP mn Ž P . s O Ž g 2 T 2 ., P g Ž P . s O Ž g 3 T 2 .. Therefore, RHSr LHS s O Ž g 2 ., and Pn P nm Ž P . Pm s 0 Ž 5. holds to relative order g 2 . A more precise statement is derived below, cf. Eq. Ž8.. We now proceed exploiting the results Eqs. Ž1. – Ž4. in more detail. We focus on the soft momentum region and the next to leading order contribution to P mn Ž P . from which the leading damping behavior will be deduced. Following standard notations, we write P mn Ž P . to this order dP nm Ž P . q)P nm Ž P . with dP the HTL-contribution and )P the relative order g correction. The diagrams that yield leading contributions to the imaginary part of )P are shown in Fig. 1. In Eqs. Ž1. – Ž4. we need an explicit expression for the ghost contribution. It is sufficient to calculate P gm Ž P . to lowest nontrivial order, which we carry out in Appendix Žcf. Eq. ŽA.17..: g 2 NT i P gm Ž P . , yd m i pˆ , Ž 6. 16 where pˆ i s p irp. It is worth mentioning in passing that P gm Ž v , p s 0. s 0. This is because, for p s 0, the summandrintegrand of Eq. ŽA.15. in Appendix, which is valid for arbitrary ŽEuclidean. four-
133
momentum PE , is odd in the summationrintegration variables K E . First we use Eq. Ž6. in Eq. Ž1.: To next to leading order under consideration we can write dP nm for P nm on the RHS of Eq. Ž1. and using dP nm P n s 0 w3,4x as well as Ž5. we find Pn)P nm Ž P . s
g 2 NT v ) y1 v Dl Ž P . 1, pˆ 16 p p
ž
m
/
Ž 7.
with the inverse plasmon propagator w3,4x, ) y1 Dl
Ž P . s p 2 y dP 00 Ž P . s p 2 q 3m2g 1 y
v 2p
ln
ž
vqp vyp
/
,
where m 2g s N Ž gT . 2r9. Note that the close connection with the inverse propagator is of course apparent in Eq. Ž1.. Also Eq. Ž7. is valid for any value of l, the Coulomb gauge parameter, Eq. ŽA.1. Žin the Appendix.. We have confirmed Eq. Ž7. through direct calculation of the LHS. Incidentally, from Eq. Ž7., Pm)P m i Ž P . e iŽ r . Ž pˆ . s 0, Ž r s 1,2. for any three-vector e iŽ r . Ž p . perpendicular to pˆ and arbitrary Žsoft. P m. Using Eq. Ž6. in Eq. Ž2. we find Pm)P mn Ž P . Pn sy
ž
2
g 2 NTv
/
16
= 1y
1
v2
pˆ idP i j Ž P . pˆ j q O Ž g . ,
Ž 8.
which is of O Ž g 6 T 4 . when P m is soft, thus specifying somewhat the result under point Ž2. above. Finally, making progress towards the discussion of damping in the next subsection, we note that, from Eq. Ž7., the imaginary part of Pm)P mn Ž P . is related to the Landau damping contribution in the HTL-propagator. We have: Im Pm)P mn Ž P . 3p s 32
Fig. 1. Next to leading order contributions to the imaginary part of the gluon polarization tensor. All loop-momenta are soft and HTL-resummed propagators and vertices have to be used.
g 2 NTm2g
v
ž / p
2
u Ž p2 y v 2 . ,
Ž 9.
and therefore for time-like momenta, Im Pm)P mn Ž P . s 0
Ž P 2 ) 0. .
Ž 10 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
134
3. Damping in a gluonic medium Soft gluon damping rates relate to the imaginary part of the gluon polarization tensor P mn , leading damping arises from the relative order g correction ) mn P . The relevant diagrams are shown in Fig. 1 where however the Coulomb-ghost loop does not contribute to Im P mn . HTL resummed propagators and vertices have to be used since the dominant contribution arises from configurations with soft loop momentum. For the following presentation we restrict to strict Coulomb gauge Ž l s 0 in Eq. ŽA.2. in the Appendix.. For gluon damping only the transverse and longitudinal components of )P mn 1 2
rithmic singularity has been extracted for non-zero momentum in both g t and g l w7x. At zero momentum the calculation of g t is well documented in the literature but note that calculating g l requires to expand )P l Ž P . to order O Ž p 2 . – a formidable and subtle task. A recent controversy w8x suggests that approximations should be handled with care and observing general constraints. We are now in a position to show that our results so far leads to g l Ž m g ,0. s g t Ž m g ,0. and thus a consistent calculation necessarily provides it. On the gluon mass-shell, P 2 s v 2 y p 2 ) 0, we obtain from Eq. Ž5. and Eq. Ž10. Im )P
00
Ž P . s yIm )P l Ž P .
ij Pt Ž P . s d H Ž pˆ . P i j Ž P . ,
)
ij dH
)
ij
i
p
s
j
Ž pˆ . s d y pˆ pˆ ,
P l Ž P . s y)P
)
00
Ž P. .
Ž 11 .
will be relevant. In the vicinity of v s v t r l Ž p ., the mass-shell for quasiparticles to HTL-accuracy, the respective inverse propagator component can be written:
žv/
3m 2g v t2 y Ž v t2 y p 2 .
Ž p. s 3 p
m2g
2
p™0
™ 2 mg , 1
v l Ž v l2 y p 2 .
y 3vl
p™0
™
2 p2 mg
g l Ž m g ,0 . s g t Ž m g ,0 . .
From this, the damping rates for longitudinal and transverse gluonic modes are obtained as:
g t Ž v t Ž p . , p . s yZt Ž p . Im )P t Ž v t Ž p . , p . p™0
™ y
1 2 mg
Ž 14 .
Finally, using v l r t Ž p s 0. s m g and Eq. Ž13. with Ž14. in Eq. Ž12., we derive that
2
v t Ž v t2 y p 2 .
Ž 13 .
pˆ i pˆ j Im )P i j Ž v ,0 . s 13 Im )P i i Ž v ,0 . s Im )P t Ž v ,0 . .
q i Im )P t r l Ž v s v t r l , p . ,
Zy1 l
pˆ i pˆ j Im )P i j Ž v , p . .
Specializing to p s 0 and using the fact )P i j Ž v , p s 0. A d i j we find
y1 Gy1 t r l , .Zt r l Ž p . Ž v y v t r l .
Zy1 t Ž p. s
2
holds, as is expected on physical grounds. Finally it is interesting to observe what consequences arise if we assume, on physical grounds, that g l Ž m g ,0. s gg Ž m g ,0. to all orders in perturbation theory. From Eqs. Ž11. and Ž12. this assumption leads to
Im )P t Ž m g ,0 . ,
v
lim
)
g l Ž v l Ž p . , p . s qZl Ž p . Im P l Ž v l Ž p . , p . p™0 m g ™ Im )P l Ž m g ,0 . , 2 p2
p™0
Ž 12 .
with m g s v t r l Ž p s 0. the plasma frequency. The direct calculation of g l r t , for general p, has not yet been accomplished, however a leading loga-
Ž 15 .
2
ž / p
Im P
00
s pˆ i pˆ j Im P i j Ž v ,0 . ,
and using this in Eq. Ž4. one finds lim p™0
1 p2
P gm P gn q 2 d n i p i Im Pmn s 0 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
135
The bare gluon propagator D˜ mn Ž PE . reads
4. Conclusions and remarks The Slavnov-Taylor identity for gluon polarization tensor we report on in this letter, Eq. Ž1., allows to obtain various constrains. Applying it to the next to the leading order in HTL-resummed perturbation theory, the identity Eq. Ž7. results, which is relevant to leading-order damping rate of gluonic modes. As has been discussed, for zero three momentum the expected equality g t Ž m g ,0. s g l Ž m g ,0. can be derived with the help of the identity Eq. Ž7.. One of the advantages of the above procedure is that the explicit expansion of )P 00 Ž v l , p . around p ; 0, which has been found to be troublesome, can be avoided in this derivation. Moreover, concerning the direct calculation of g l , the present work indicates that dealing with singularities in intermediate steps of the calculation and necessary changes in the integration variables should be made in a way consistent with the identity Eq. Ž7..
D˜ mn Ž PE . s d m id n j
d i j y pˆ i pˆ j PE2 1
m0 n 0
qd d
p2
V g s yigCa b c p j ,
where p is the outgoing momentum carried by ha . The generating functional reads Z J ,j ,j s D A am Dha D ha exp S q d 4 x Jam Ž x . A am Ž x .
H
HT
dha s
dx 0 d 3 x L Ž x . ' d 4 x L Ž x . ,
ž
H
L Ž x . s y 14 Famn Ž x . Famn Ž x . y qd
mi
ha Ž x . E
i
Dxm, a b
Famn s E mAna y E nA am q gC a b c Dx,ma b Ž
/
HT
1
1 d ha s y = P A a dz , l
2l
2
g 2
C a b ch bhc dz ,
where dz is a Grassmann-number parameter. Using this fact in ŽA.4., we obtain,
HTd
Ž= P A a Ž x . .
4
z BŽ z . Z J , j , j s 0 ,
B Ž z . s Jam Ž z . Dzm, a b
Ž A . hb Ž x . ,
A bm Anc .
1
Ž A.1 .
Here A. ' d a b ErE x m y gC a b c A cm Ž x . and ha and ha are the Coulomb-ghost fields. For notations related to the color space, we follow w9x.
Ž A.4 .
where, the functional integral is to be performed with periodic boundary conditions for all fields Žincluding ghosts w10x., A am Ž x 0 s 0, x . s A am Ž x 0 s 1rT, x ., etc. The action S Žor L ., Eq. ŽA.1., is invariant under the BRST transformation w9x:
Appendix A. Derivation of (1) and (6) We start in the imaginary-time formalism continuing to real energies at the final stage. The QCD action in Coulomb gauge reads
Ž A.2 .
Ž A.3 .
d A am s Damb Ž A . h b dz ,
1rT
,
qj a Ž x . ha Ž x . q ha Ž x . j a Ž x . 4 ,
M.D. likes to thank the German Academic Exchange Office ŽDAAD. for financial and general support of his stay in Osaka. Thanks also to the Faculty of Science, Osaka City University, for kind hospitality.
H0
p4
where PEm s Ž p 0 , p . with p 0 s 2p Tn Ž n s 0 " 1," 2, . . . ., and the bare ghost propagator reads D˜ g Ž p . s 1rp 2 . The form of gluon-ghost vertex can be read off from L 2 gC a b c Ž E j ha .h b A cj ,
Acknowledgements
Ss
ql
PEm PEn
q
l
d
d J jbŽ z.
E ja Ž z .
d
ž /
Ez
d i
d Jai
Ž z.
g d d q Ca b c j a Ž z . . 2 dj b Ž z . dj c Ž z .
Ž A.5 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
136
Computing
d d Jam
we substitute ŽA.7. and ŽA.8. into ŽA.6. and use the form ŽA.2. for D˜ s i to obtain
d
Ž x . dj b Ž y .
d a b d 4 z P˜ nm Ž x y z . E ymDˆ g Ž z y y .
HT
4
= d z B Ž z . ln Z J , j , j
HT
Js j s j s0 ,
s ygC ac d d 4j d 4 z Ž G˜y1 .
by using ŽA.5., we obtain
d a b Exm G˜g Ž x y y . y gC ac d ² A dm Ž x . hc Ž x . h b Ž y . : q da b
1 E
l Ey
i
G˜ m i Ž x y y . s 0 .
Ž A.6 .
Note that the third term of Eq. ŽA.5. does not contribute to ŽA.6.. The ghost propagator G˜g and the gluon propagator G˜ mn in Eq. ŽA.6. are defined, respectively, through
d a b G˜g Ž x y y . s ²ha Ž x . h b Ž y . :
dj b Ž y . dj a Ž x .
Js j s j s0
=² A dr Ž z . hc Ž z . h b Ž y . : .
Ž x . d Jbn Ž y .
.
gC ac d ² A dr Ž z . hc Ž z . h b Ž y . :
Ž A.10 .
Ž A.11 .
P˜ nm Ž PE . PEm s yp 2 G˜g Ž PE .
G˜ mn Ž x y y . s D˜ mn Ž x y y .
d nm PE2 y PEn PEm
qP˜ nm Ž PE . 4 P˜ gm Ž PE .
y d 4 ud 4 Õ G˜ mr Ž x y u . P˜ rs
HT
y
= Ž u y Õ . D˜ sn Ž Õ y y . ,
Ž A.7 .
d a b G˜g Ž x y y . s d a b D˜ g Ž x y y .
HT
F .T .
Putting altogether in ŽA.9. and transforming to Fourier space, we finally obtain
Js j s j s0
² A dm Ž x .hc Ž x .h b Ž y .: is the gluon-ghost three-point function. Now, we call for the Schwinger-Dyson equations:
q gC a d e d 4 z
Ž A.9 .
The inverse propagator Ž Gˆy1 . nm is related to P˜ nm through Ž G˜y1 . nm s Ž D˜y1 . nm q P˜ nm . The Fourier transform of ² A dr Ž z .hc Ž z .h b Ž y .: may be written as
p iP˜ gi Ž PE . s P˜ g Ž PE . .
d ln Z d Jam
= d r Ei jmEji D˜ g Ž j y z . q d 4 Ž j y z . d m r
where Ž P˜ gr Ž PE ..a b is related to a ghost self-energy part P˜ g through
,
d a b G˜ mn Ž x y y . s ² A am Ž x . Anb Ž y . : '
Ž xyj .
s yi d a b P˜ gr Ž PE . G˜g Ž PE . ,
d ln Z
'
nm
HT HT
p2
P˜ mn Ž PE . P˜ g Ž PE . ,
Ž A.12 . where use has been made of ŽA.11.. Noting that G˜g Ž PE . s w p 2 q P˜ g Ž PE .xy1 and using P˜ mn s P˜ nm , Eq. ŽA.12. may be ‘solved’ as
ED˜ g Ž x y z . E zi
=² Aie Ž z . hd Ž z . h b Ž y . : ,
PEm
Ž A.8 .
where P˜ rs is the gluon polarization tensor. It can readily be shown that, in the limit g ™ 0, ŽA.6. holds, as it should be. Taking this fact into account,
PEn P˜ mn Ž PE . s y d mn PE2 y PEm PEn q P˜ mn Ž PE . P˜ gn Ž PE . .
Ž A.13 .
M. Dirks, et al.r Physics Letters B 461 (1999) 131–137
137
From this we obtain Eq. Ž1. in the main text after continuation to real energies ip 0 ™ v q i0q according to
1rŽ e z r T y 1.. On summing over k 0 and using nŽ z . , Trz and Hd zr t Ž z ,k .rz s 1rk 2 , we obtain
P m s Ž v , p. ,
P˜ gm Ž PE . , yg 2 NTd m i
PE2 ™ yP 2 s p 2 y v 2 ,
P˜ mn Ž ip 0 , p . ™ Ž yi . P˜ gm Ž ip 0 , p . ™ Ž yi .
dm 0 q dn 0
dm 0
P mn Ž v , p . ,
P gm Ž v , p . .
Ž A.14 .
, yd m i
P gm Ž P .
Finally we compute with soft P to lowest non-trivial order – one-loop contribution – as required for the discussion in the main text. Using Eqs. ŽA.2. and ŽA.3., we have
P˜ gm Ž PE . s yg 2 N T Ý k0
s yg 2 N T Ý k0
d3k
H Ž 2p .
3
Ž p i y k i . ˜im D Ž KE . 2 Ž pyk.
d3k
H Ž 2p .
pi yk i 3
Ž pyk.
ij ˆ = d m jd H Ž k . D˜ Ž K E . q l
2
k i K Em k4
,
Ž A.15 .
ij Ž ˆ. where p i s Ž0, p ., K Em s Ž k 0 , k . and d H k is as in Ž11.. In Eq. ŽA.15., D˜Ž K E . s 1rK E2 for hard p, while, for soft p, the HTL-resummed )D˜Ž K E . is substituted for D˜Ž K E .. After carrying out the renormalization, one can easily see that the leading contribution comes from the first term Žin the square brackets. with soft k region:
P˜ gm Ž PE . , yg 2 N T Ý k0
=
1rT
H0
d3k
Hsoft k Ž 2p .
Ž pi yk i . 3
Ž pyk.
2
ij ˆ d m jd H Ž k.
dt e i k 0t d z r t Ž z ,k . 1 q n Ž z . ey zt ,
H
Ž A.16 . where r t Ž z ,k . is the spectral function of the HTL-resummed transverse gluon propagator and nŽ z . s
g 2 NT 16
p i y Ž p P kˆ . kˆ i
d 3k
Hsoft k Ž 2p .
3
k2 Ž pyk.
pˆ i ,
which is independent of l. Noting that ŽA.17. has only spatial components and is dent of p 0 , we have P gm Ž P . s P˜ gm Ž PE . ŽA.14...
2
Ž A.17 . P˜ gm in indepenŽcf. Eq.
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