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An infrared singularity in the damping rate for longitudinal gluons in hot QCD
23 September 1999
Physics Letters B 463 Ž1999. 117–125
An infrared singularity in the damping rate for longitudinal gluons in hot QCD A. Abada, O. Azi ´ Departement de Physique, Ecole Normale Superieure, BP 92 Vieux Kouba, 16050 Alger, Algeria 1 ´ ´ Received 7 January 1999; received in revised form 19 July 1999; accepted 5 August 1999 Editor: R. Gatto
Abstract We calculate g l Ž0., the damping rate for longitudinal on-shell gluons with zero momentum in hot QCD using the hard-thermal-loop scheme, and find it to be divergent in the infrared. We argue that this result suggests that the scheme may be sensitive in this sector and thus may need to be improved upon. We discuss the validity of our result and emphasize the fact that it depends on the way we perform our calculation. In particular, we do not exclude the possibility that alternative approaches that we briefly comment on may in principle yield infrared-safe results. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 12.38.-t; 12.38.Bx; 12.38.Mh Keywords: Hard thermal loops; Soft gluon damping
Besides their importance regarding the stability of the quark-gluon plasma, gluon damping rates have been crucial in better understanding QCD at high temperature T. In early works, it has been noticed that in this regime, the determination of the dispersion laws for quarks and gluons beyond lowest order using standard perturbation theory is gauge-dependent w1x. This problem has been emphasized in further works in which the gluon damping rates have been calculated to one-loop order in various gauges and schemes and different results have been obtained w2x. It was then realized that the problem was related to the way the expansion in powers of g, the perturbative QCD coupling constant, was performed: at high T Žthe hard scale., when the external momenta are soft, i.e., of magnitude gT, the standard loop expansion is not anymore an expansion in powers of g w3x. Braaten and Pisarski subsequently developed an effective perturbative expansion in the framework of a resummation scheme of the so-called hard thermal loops Žhtl. w4x. Using this scheme, they showed that the transverse-gluon damping rate g t Ž0. with zero momentum was invariant within the Coulomb and covariant gauges and determined it in the strict Coulomb gauge to be finite and positive w5x. Later, a generating-functional formalism in the htl approximation was developed w6x and a relation to the eikonal of a Chern-Simons gauge
1
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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 9 5 9 - 4
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theory was made w7x. From there a hydrodynamic approach showed that the htl approximation was essentially ‘classical’ w8x. However, the htl resummation scheme discusses only the two scales T Žhard. and gT Žsoft. whereas with g and T, one has Žat least. a hierarchy of scales g n T, n an integer, positive if we believe that T is the highest scale in the problem. It may be unnecessary to reorganize perturbation theory taking into account this general structure of scales, but there are indications that the next smaller one g 2 T may be important. Indeed, it is true that in the htl scheme, both on-shell longitudinal and transverse gluons acquire a thermal mass m g of order gT, but static magnetic fields are not screened yet at this scale w9x; they are expected to get so at the hence-called ‘magnetic scale’ g 2 T. Furthermore, the gluon damping rates g Ž p . are to lowest order of magnitude g 2 T and it has been noticed that these damping rates, when calculated in the htl-resummation scheme at soft but nonzero momenta p, exhibit a ln g behavior accompanied with coefficients w9,10x. This logarithmic behavior is not present in the finite expression of g t Ž0., the limit p ™ 0 of g t Ž p ., and no full explanation has been provided yet for this discrepancy. These remarks tend to indicate therefore that the htl scheme may still be sensitive in the ‘infrared’. We think one more direct indication of the infrared sensitivity of the htl-resummation scheme is the observation we made in w11x that, when calculated solely in this scheme, the longitudinal-gluon damping rate at zero momentum g l Ž0. may be potentially infrared-sensitive. Indeed, we have determined in that work the analytic expression to leading order of g l Ž0.. We have obtained for it the following form:
g l Ž 0. s
g 2 Nc T 24p
a lŽ01 . q a lŽ02 . ;
Ž 1.
Ž . w x Nc is the number of colors. aŽ1. l 0 is given in Eq. 3 below; it is a finite number found in 5 to be equal to Ž . 6.63538 and is such that the damping rate g t 0 for transverse gluons with zero momentum is equal to g 2 NcT Ž1. Ž . Ž . a l 0 , see w5x. The other contribution aŽ2. l 0 is given in Eq. 4 and it is this part of g l 0 we remarked it 24p contains terms which are divergent in the infrared w11x. From a physical standpoint and independently of any calculation scheme, there is no difference at zero momentum between longitudinal and transverse gluons w5x. This argument can be used as an important criterion to test the consistency of a given calculational scheme like the htl one. With this in mind, we argued in w11x that the terms contributing to aŽ2. l 0 , in particular the infrared-divergent ones, when put together, may cancel one another. We show in this letter that this is not the case: when all the expressions are treated with care, not only Ž . we find aŽ2. l 0 nonvanishing but the infrared-divergent piece survives too, see Eq. 21 . In the sequel, we present first our calculation and we discuss the result afterwards. All along, we take m g s 1. The longitudinal gluon damping rate in the strict Coulomb gauge is obtained to lowest order by the relation: Im )P l Ž yi v , p . gl Ž p. s E dP l Ž yi v , p . Ev
,
Ž 2.
v s v l Ž p .qi0 q
where dP l Ž)P l . is the longitudinal-gluon htl Žeffective. self-energy and v l Ž p . the on-shell lowest-order energy given for soft momenta by Eqs. Ž8. below. We follow closely the notation of w11x to which this work is a follow-up. Since the denominator in Ž2. is given in a small y p expansion by ErEv dP Žyi v , p .< vs v l Ž p. s y2 p 2 q . . . , to get g l Ž p s 0., we have to expand the imaginary part of the effective self-energy to order p 2 . The rationale behind this expansion goes as follows. To determine the infrared behavior, we introduce an infrared cut-off h , i.e., we replace H0q` dk by Hhq` dk with 0 - h < 1. This means that k is never smaller than h. The cut-off h is fixed for the rest of the calculation and we regard p as always smaller than h Ž p will
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
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eventually go to zero but h stays fixed.. Physically, it may be useful to regard h as representing the magnetic scale, but this is not necessary from a pure technical standpoint. We are therefore always working in the kinematic region 0 F p - h F k. This then allows for the expansion of all functions of q s p y k in powers of small p, functions like 1rQS appearing in the effective vertices and )DŽ Q ., the effective propagators, see w11x for the notation. The expansion of 1rPS doesn’t pose a problem in itself. The subsequent angular integrations and Matsubara sum yield a series in powers of p 2 with a finite radius of convergence that depends of course on h. The first coefficient vanishes as it should and the second one, according to Ž2., yields the damping rate. The Ž2. calculation is reported in w11x and we obtain the result Ž1. where the expressions of aŽ1. l 0 and a l 0 are the following: q`
aŽ1. l0 s 9
q`
dk
Hh
Hy`
dv1
v1
dv2
q`
Hy`
v2
2
d Ž 1 y v 1 y v 2 . k 4 r l1 r l 2 y k 2 Ž k 2 y v 12 . r t1 r l 2 v1
2
q2 Ž k 2 q v 1 v 2 . r t1 r t 2 q
6k3
Ž k 2 y v 12 .
2
Q 1 rt 2 ,
Ž 3.
and: q`
aŽ2. l0 s 9
q
q`
dk
Hh
Hy`
v1 3
ž
1y
ž ž
2 v 12 3
y
v1 6
Ek
k
1q
žž
v1
v 12
qE k k 3 1 q
q
dv1
/
2
k2
/
v1
y
ky
v2
d Ž1 y v1 y v2 .
Ž 1 y 2 v 1 . E k Q 1 rt 2 q
3
k2
dv2
q`
Hy`
v 12 k
r l1 r l 2 q
/
v 12 k2
/
2k 9
/
3
ž
3
1y
v1k
Ž 1 y 2 v1 . Q 1 E k rl2 q
v 12 k
2
/
E k Q 1 E k rt 2 q
v1 6k
6
Q 1 E k2 r l 2
Ž k 2 y v 12 . E k2Q 1
rt 2
Ž k 2 Ž 1 y 3k 2 . q Ž 1 y 4 k 2 . v 1 v 2 y v 12 v 22 . rt1 rt 2
Q 1 rt 2 q
Q 1 rt 2
v1k
v1
k2 9
Ž 1 y 2 k 2 y 4 Ž 1 y k 2 . v 1 q 6 v 12 y 4v 13 . rt1 E k rt 2
q 23 < v 1 < Ž 2 k 2d 1 r l 2 y k 2v 1 Ev 12 d 1 r l 2 q v 1 d 1 r t 2 . .
Ž 4.
The notation is as follows: Q 1 ' Q Ž k 2 y v 12 . where Q is the step function and d 1 ' d Ž k 2 y v 12 .. E k ' ErE k etc., and r l i is a short notation for r l Ž v i ,k ., i s 1,2 and the same for r t i . The spectral densities r t,l are given by w5,12x:
r t ,l Ž v ,k . s z t ,l Ž k . d Ž v y v t ,l Ž k . . y d Ž v q v t ,l Ž k . . q bt ,l Ž v ,k . Q Ž k 2 y v 2 . ,
Ž 5.
an expression in which the residues z t,l Ž k . are given by:
ztŽ k. s
Ž v2yk2 . 2 3v 2 y Ž v 2 y k 2 .
ž
2
;
/
vs v t Ž k .
zlŽ k. s
yŽ v 2 y k 2 . 2 k2 Ž3yv 2 qk2 .
, vs v l Ž k .
Ž 6.
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
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and the cut functions bt,l Ž v ,k . by: 3v Ž k 2 y v 2 .
bt Ž v ,k . s
ž
3
4k
2
3v 2
2
k yv q
2k2
ž
1q
k2yv2 2vk
ln
kqv kyv
2
// ž q
3pv 4k3
2 2
2
Ž k yv .
;
/
y3 v
b l Ž v ,k . s 2k
ž
2
3qk y
3v 2k
ln
kqv
2
3pv
2
.
Ž 7.
/ ž / q
kyv
2k
v t,l Ž p . are the on-shell energies of the gluon to lowest order. For soft gluons we have: 704 91617 v t Ž p . s 1 q 35 p 2 y 359 p 4 q 3000 p 6 y 336875 p8 q . . . ; 3 1 489 v l Ž p . s 1 q 103 p 2 y 280 p 4 q 6000 p 6 q 43120000 p8 q . . . .
Ž 8.
Ž2. w x As we said, aŽ1. l 0 is a finite number found in 5 to be equal to 6.63538. We therefore have to calculate a l 0 . The calculation is mainly a matter of disentangling the infrared-divergent pieces from the finite ones. It very often necessitates an expansion for small k of the residue and cut functions, using their definitions given in Ž6. and Ž7. respectively, and some of their derivatives. Also, care must be taken when handling the delta functions that occur, especially with their first and second derivatives. Finally, extra care must be given to the order in which the integrals over k and v are performed. Let us show explicitly how to carry with one representative term contributing to aŽ2. l 0 . Consider the first term Ž in 4.:
q`
Is9
q`
dk
Hh
Hy`
q`
dv1
v1
q`
Hy` d v
s3
dk
Hh
q`
Hy`
dv2
1y2v 1yv
v2
dŽ 1 y v 1 y v 2 .
v1 3
Ž 1 y 2 v1 . Q1 E k rl2
Q Ž k y v . E k r l Ž 1 y v ,k . .
Ž 9.
We can either apply the derivative ErE k directly to the spectral density r l Ž1 y v ,k . and carry through, or instead write: q`
I s y3
Hh
q`
Hy` d v
dk q`
Hy` d v
q3
1y2v
1y2v 1yv
1yv
d Ž k y v . r l Ž 1 y v ,k .
Q Ž k y v . r l Ž 1 y v ,k .
k™ q` ; ks h
Ž 10 .
the final result is the same 2 . Using the expression of r l given in Ž5., the first term in the above expression has two contributions, i.e., a dd contribution and a dQ contribution. The kinematics for the dd contribution is v s "k with v s 1 y v l Ž k . on the one hand, and v s "k with v s 1 q v l Ž k . on the other. Given the behavior of v l as a function of k, the second kinematics is always forbidden whereas the first one is satisfied only at precisely k s 0. Since h ) 0, the dd contribution is then zero. The kinematics for the dQ contribution is 1 y k F v F 1 q k with v s "k. This is allowed only for v s k and as long as k G 1r2, which means that the q` first term in I is equal to y3H1r2 dk 11yy2kk b l Ž1 y k,k ., a finite number. Note that, since b l Ž1 y k,k . goes to zero as k ™ q`, hard loop momenta are not allowed in any way.
2
This has been checked explicitly.
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We turn to the second term in I. The limit k ™ q` in this term is always zero because in this limit we have z l Ž k . ™ 0 and the same for b l Ž1 y v ,k . for v fixed. We then only have to determine this term at k s h. From the explicit expression of the spectral function r l given in Ž5., here too we have two contributions, i.e., a Qd contribution and a QQ contribution. The second one is zero because we must have from Q Žh y < v <. the conditions yh F v F h and from Q Žh y <1 y v <. the conditions 1 y h F v F 1 q h. These are equivalent to 1 y h F v F h with the condition h G 1r2, which can never be satisfied since h < 1. Now to the Qd contribution. The kinematics are the following: yh F v F h with v s 1 y v l Žh . for the first term and v s 1 q v l Žh . for the second. With the behavior of v l Ž k . for small k given in Eq. Ž8., the second kinematics is Ž . forbidden but the first one is allowed. This contribution is then equal to y3 z l h Ž 1 y 2 v l Ž h . . . If one uses the v l Žh .
expansion: 1
z l Žh . sy
39 13 1 y 35 h 2 q 175 h 4 y 175 h6 q . . . ,
4h 2
Ž 11.
which is obtained from Ž6. and Ž8., one finds z l Žh . 3 9 y3 Ž1 y 2 v l Ž h . . s 2 y . vl Žh . 40 4h Hence one obtains the result: q` 3 9 1y2k Is y y3 dk b Ž 1 y k ,k . . 2 40 1yk l 4h 1r2
Ž 12 .
H
The other terms in aŽ2. l 0 are worked out in a similar manner, though some necessitate more labor. We now give the corresponding results. To ease the notation, we denote in a compact way: q` q` d v 1 q` d v 2 D'9 dk d Ž1 y v1 y v2 . . Ž 13 . h y` v 1 y` v 2
H
H
H
H
We have:
H
ž ž
DE k k 3 1 q 2 v 12
q
3
k2
1q
/
3
v1 k
2
r l1 r l 2 q y
v 12 k
2
/
2k 9
Ž k 2 Ž 1 y 3k 2 . q Ž 1 y 4 k 2 . v 1 v 2 y v 12 v 22 . rt1 rt 2
Q 1 rt 2 s y
3
40 y
4
27p 2
.
Ž 14 .
It is worth emphasizing the fact that all contributions from k ™ q` vanish, indeed as it should be since only soft contributions are to contribute. The next piece reads: < v1 < v 12 D 4 k 2d 1 r l 2 q 2 v 1 d 1 r t 2 q Ž 1 y 2 v 1 . 1 y 2 e Ž v 1 . E k Q 1 r t 2 3 k
ž
H
dk
q`
s3
H1r2
1yk
/
Ž 2 k b l Ž 1 y k ,k . q bt Ž 1 y k ,k . . .
Ž 15 .
The next pieces are more delicate to handle: they require working out derivatives of delta functions in the presence of other delta functions with different arguments. After that is done, we get: 1
HD 6 3 s 8
ž
2 v1 k 1 y q`
y3
H1r2
v 12 k2 dk
1yk
/
E k Q 1 E k rt 2 q
bt Ž 1 y k ,k . .
v1 k
Ž k 2 y v 12 . E k2Q 1
r t 2 y v 1 E k2
žž
ky
v 12 k
/
Q 1 rt 2
/ Ž 16 .
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
122
The next term is worked out along the same lines as the previous one. We have:
HD
v1k
Q 1 E k2 r l 2 s y
6
9 8h
27 2
q
3 q
80
dk
q`
H 2 1r2
1yk
b l Ž 1 y k ,k . y k E k b l Ž v ,k . < vs1yk .
Ž 17 .
The next term is the most tedious: it necessitates the additional expansion to order h 2 of E k bt Ž v ,k . with v s 1 y v t Ž k .. When this is done and the contributions put together, we get:
H DE
k2 k
Ž 1 y 2 k 2 y 4 Ž 1 y k 2 . v 1 q 6 v 12 y 4v 13 . rt1 E k rt 2
9
125 s
2 2
27p h
1153 y 189p
2
289536 y
2187p 4
.
Ž 18 .
The last term isn’t more difficult. It reads:
HD
y 23 k 2v 12 e Ž v 1 . Ev 12 d 1 r l 2 t s y
3
dk
q`
H 2 1r2
1yk
Ž b l Ž 1 y k ,k . q k E k b l Ž v ,k . < vs1yk . .
Ž 19 .
Putting all these contributions together, we obtain: 3
ž
aŽ2. l0 s y
125 q 27p
8
dk
q`
y3
H1r2
2
1yk
1
/
h
2
21 y
1433 y
80
189p
2
289536 y
2187p 4
Ž 1 y 4 k . b l Ž 1 y k ,k . q k E k b l Ž v ,k . < vs1yk .
Ž 20 .
Note that the integral in the above equation Žand all the other similar ones actually. is finite. We can evaluate it numerically and we get: aŽ2. l0 s
0.09408
h2
y 4.45366 .
Ž 21 .
Ž . Hence, as announced, aŽ2. l 0 is nonzero and quite divergent. Via Eq. 1 , this means g l Ž 0 . is itself infrared-divergent. Indeed, using the value aŽ1. s 6.63538, we get: l0
g l Ž 0. s
g 2 Nc T 24p
ž
0.09408
h2
/
q 2.18172 .
Ž 22 .
We resume our initial discussion. Our calculation of g l Ž 0 . is performed within the framework of the htl-resummation scheme. This latter treats systematically only the two scales T and gT and we have already remarked that the scale g 2 T may play an important role in QCD at high T. This suggests that the scheme, as it is, may be sensitive in the infrared, i.e., when it comes to deal with quantities that, to lowest order, are already of magnitude g n T, n G 2. We think the behavior of g l Ž 0 . we report in this letter is a manifestation of this. One may ask why the result is finite for g t Ž 0 . whereas we find an infrared divergence for g l Ž 0 . . It is pertinent to note in this respect that there is a main difference between the calculation of g l Ž 0 . and that of g t Ž 0 . : in order to get g l Ž 0 . , we need an unavoidable expansion to order p 2 of the imaginary part of the effective self-energy whereas, for g t Ž0., we do not w5,11x. As a matter of fact, we have determined the analytic
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expression of the transverse-gluon damping rate g t Ž p . to order p 2 w13x. The coefficient a t 0 , that of the zeroth Ž . w5x. But a preliminary investigation shows that the order, is finite and as we said, equal to aŽ1. l 0 of Eq. 3 coefficient a t1 , that of order p 2 , has contributing terms which are divergent in the infrared w14x. It seems then that the expansion of the gluon effective self-energies )P in the htl scheme in powers of the soft momentum p is infrared sensitive beyond zeroth order, and that this sensitivity may be responsible for the divergent behavior of g l Ž0.. One may jump to conclude that the htl scheme is robust in the infrared, and that all this simply means that the expansion in powers of the soft momentum of the gluon effective self-energies is not allowed in the first place. In this regard, we stress that the quark-gluon plasma is to be a stable phase of hadronic matter. Therefore, the damping rates g Ž p . have to be positive and be analytic functions of the external soft momenta, i.e., they have to admit a Taylor-series expansion in powers of Žat least very. small p. This is one important criterion to check with the consistency of a given calculational scheme. We have explained in our introductory remarks the basis upon which the expansion is performed, and within that context, it is valid. If the expansion yields infrared-sensitive coefficients as it is the case, it only means that the scheme in which the calculation is performed needs to be cured andror improved upon, not that the principle of the expansion itself is invalid. In other words, an infrared-improved resummation scheme would still allow for the same expansion but would remove the infrared sensitivity of the actual htl scheme. It is useful to remark that the infrared sector is not the first instance in which the htl scheme yields infinities. Indeed, collinear singularities do appear when external light-like momenta are involved w15,16x and, in response to this, the scheme has been subjected to improvement in w15x where an improved action that removes these singularities is proposed. We think the same treatment is necessary in the infrared sector but the situation is more problematic here. Indeed, such an improvement would very likely necessitate the determination of the magnetostatic screening length which is mostly believed to intervene in a nonperturbative way. It may also be that the story does not end at the scale g 2 T and new physics may manifest itself at lower scales g n T with n ) 2, though nothing is suggesting this for the moment. In any case, it is pertinent to note that besides the result we obtain for g l Ž0., what we think is also interesting in this work is that it sets a calculable framework in which the infrared divergences appear explicitly and, by the same token, where to test an improved htl scheme when found. In the meantime, we can have a very rough estimate of the scale at which the htl scheme starts to be sensitive. If we demand that g l Ž0. be equal to g t Ž0., we may set aŽ2. l 0 equal to zero, which yields h s 0.145 Žin units of m g .. However, one should be prudent in considering this value as an estimation, even rough, of the magnetic mass. Our calculation of g l Ž0. is performed in the imaginary-time formalism and is based on the techniques developed in w4x. In seeking to find an expression for the damping rates, we have followed a sequence of operations which amounts to: Ži. expand the effective self-energies in powers of p; Žii. perform the angular integrals; Žiii. then perform the Matsubara sum. Also, as we previously said, our calculation is valid in the kinematic region p - h F k, where k is the soft loop momentum. How then our result compares to what is already known in the literature? As an illustrative example, consider the estimation:
g t ,l Ž p . ; y
g 2 Nc T 4p
ln g Õt ,l Ž p .
Ž 23 .
of the soft-gluon damping rates made in the paper by Pisarski in w9,10x; Õt,l Ž p . are the group velocities. This calculation is different in many respects from the one we carry. In particular, the result Ž23., as explained in that paper, is obtained in the kinematic region where the loop momentum k is restricted to the ‘‘ very soft’’ region Ž0 F k - h . while p is just soft, of order m g . Then, it is obvious why our result is different from that of the estimation Ž23. above: our result is carried in a different kinematic region, i.e., 0 F p - h F k; it does not take account of the ‘‘ very soft’’ loop momenta 0 F k - h and cannot be carried to the region p ) h. On the other hand, the result Ž23. takes account of only the contributions from the ‘‘ very soft’’ region 0 F k - h with p ) h ;
124
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
it cannot be carried to the region p - h , in particular to the point p ™ 0 for it will give zero Žusing the expressions of the group velocities at very soft momenta.. The same comparison can be carried with the other results in the literature. The point to emphasize is that our result is not in contradiction in any way with what is previously known because it is valid in a different kinematic region. It is also important to realize that our result is not at all the final word regarding the issue of the infrared sensitivity of the htl-scheme. This is simply because the sequence of operations we have followed in our work is not unique. Indeed, it is certainly important to be able to carry the calculation in the order: Matsubara sum first, angular integrations second and only third the expansion in powers of p. Provided that the calculation in this order is technically feasible, it is important to see whether in this order, the infrared divergences persist. This is not obvious from the outset because we are not assured of the commutation of the different operations involved. However, from a calculational standpoint, one has to realize that matters are not straightforward if we delay the expansion in powers of p. This is because we have to deal with expressions quite complicated and involved. It is true that the Matsubara sum can easily be performed if the effective propagators and terms appearing in the effective vertices are replaced by their respective spectral representations, but this will bring in more than one energy denominator, which would compromise the straightforward extraction of the imaginary part of the effective self-energy. Also, the advantages of the techniques developed by Braaten and Pisarski will likely be lost. More importantly, the subsequent angular integrations will be very difficult, if not impossible, to handle in a compact way before any expansion is made. Equally important is the real-time approach. It will avoid the Matsubara sum and the analytic continuation to real energies altogether. In this regard, work is being considered in this direction and preliminary results seem to indicate that the infrared problem is present there too w17x. Nevertheless, it is quite possible that these alternative approaches will eventually demonstrate that the one we adopted in our calculation is not the ‘correct’ way of approaching the physical damping rates. In any case, the issue is certainly not closed and our result is only a first step forward. Finally, we have limited ourselves thus far to discussing the gluonic sector only, but certainly quarks play an important role too in the structure of hot QCD. They themselves acquire a thermal mass m f to order gT and their damping rates start also at the scale g 2 T, which means they are not a priori shunned from infrared sensitivity.
Acknowledgements We deeply thank Asmaa Abada for all her help and encouragements. O.A. would like to thank ICTP for their kind hospitality where some of the later part of this work was carried. She had the opportunity there to carry constructive discussions with Samina S. Masood. This is a revised version of the work and we must thank the referee for his substantive remarks that reshaped the emphasis in our discussion. Between the two versions, O.A. has been able to attend a workshop on thermal field theory at CERN and she benefited from stimulating discussions with many people, most particularly with P. Aurenche, U. Heinz, E. Iancu, A. Rebhan, D. Schiff and I. Zahed. O.A. would also like to thank all the people in Annecy for their kind hospitality and most useful suggestions: Gelis, Petitgirard, Peigne´ and Zaraket. Finally, special thanks to Patrick Aurenche and Anton Rebhan for their warm encouragements.
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w2x A representative set of works is D.J. Gross, R.D. Pisarski, L.G. Yaffe, Rev. Mod. Phys. 53 Ž1981. 43; K. Kajantie, J. Kapusta, Ann. Phys. ŽNY. 160 Ž1985. 477; T.H. Hansson, I. Zahed, Phys. Rev. Lett. 58 Ž1987. 2397; Nucl. Phys. B 292 Ž1987. 725; U. Heinz, K. Kajantie, T. Toimela, Phys. Lett. B 183 Ž1987. 96; Ann. Phys. ŽNY. 176 Ž1987. 218; R. Kobes, G. Kunstatter, Phys. Rev. Lett. 61 Ž1988. 392; U. Kraemmer, M. Kreuzer, A. Rebhan, Ann. Phys. ŽNY. 201 Ž1990. 223; a more exhaustive list of works is given in Ref. 1 of E. Braaten, R.D. Pisarski, Phys. Rev. D 42 Ž1990. R2156. w3x R.D. Pisarski, Phys. Rev. Lett. 63 Ž1989. 1129. w4x E. Braaten, R.D. Pisarski, Nucl. Phys. B 337 Ž1990. 569; Phys. Rev. Lett. 64 Ž1990. 1338; Nucl. Phys. B 339 Ž1990. 310; J. Frenkel, J.C. Taylor, Nucl. Phys. B 334 Ž1990. 199; Ref. w3x; Physica A 158 Ž1989. 246; Nucl. Phys. A 498 Ž1989. 423c; The htl scheme is well discussed now in M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996. w5x E. Braaten, R.D. Pisarski, Phys. Rev. D 42 Ž1990. R2156. w6x J.C. Taylor, S.M.H. Wong, Nucl. Phys. B 346 Ž1990. 115; E. Braaten, R.D. Pisarski, Phys. Rev. D 45 Ž1992. R1827. w7x R. Efraty, V.P. Nair, Phys. Rev. Lett. 68 Ž1992. 2891. w8x P.F. Kelly, Q. Liu, C. Lucchesi, C. Manuel, Phys. Rev. Lett. 72 Ž1994. 3461; Phys. Rev. D 50 Ž1994. 4209; R. Jackiw, Q. Liu, C. Lucchesi, Phys. Rev. D 49 Ž1994. 6787. w9x A.V. Smilga, Phys. Rep. 291 Ž1997. 1; F. Flechsig, A.K. Rebhan, H. Schulz, Phys. Rev. D 52 Ž1995. 2994; A.K. Rebhan, Nucl. Phys. B 430 Ž1994. 319; Phys. Rev. D 48 Ž1993. R3967. w10x R.D. Pisarski, Phys. Rev. D 47 Ž1993. 5589; C.P. Burgess, A.L. Marini, Phys. Rev. D 45 Ž1992. R17; A.K. Rebhan, Phys. Rev. D 46 Ž1992. 482; F. Flechsig, H. Schulz, Phys. Lett. B 349 Ž1995. 504; H. Schulz, Nucl. Phys. B 413 Ž1994. 353; see also E. Braaten, Phys. Rev. Lett. 74 Ž1995. 2164; see also a discussion in real-time formalism in S.S. Masood, M.Q. Haseeb, Astroparticle Physics 3 Ž1995. 405; see in the context of QED and with a different approach J.P. Blaizot, E. Iancu, Phys. Rev. D 55 Ž1997. 973; Phys. Rev. Lett. 76 Ž1996. 3080; see also in the context of scalar QED M.H. Thoma, C.T. Traxler, Phys. Lett. B 378 Ž1996. 233. w11x A. Abada, O. Azi, K. Benchallal, Phys. Lett. B 425 Ž1998. 158. w12x R.D. Pisarski, Physica A 158 Ž1989. 146; Nucl. Phys. B 309 Ž1988. 476. w13x A. Abada, O. Azi, A. Tadji, Damping rate for transverse gluons with finite soft momentum in hot QCD, ENSK–TP–02, hep-phr9710549, unpublished. w14x A. Abada, O. Azi, K. Bouakaz, work in progress. w15x F. Flechsig, A.K. Rebhan, Nucl. Phys. B 464 Ž1996. 279. w16x R. Baier, S. Peigne, ´ D. Schiff, Z. Phys. C 62 Ž1994. 337; P. Aurenche, T. Becherrawy, E. Petitgirard, preprint ENSLAPP-A-452-93. w17x H. Zaraket, private communication.
Physics Letters B 463 Ž1999. 117–125
An infrared singularity in the damping rate for longitudinal gluons in hot QCD A. Abada, O. Azi ´ Departement de Physique, Ecole Normale Superieure, BP 92 Vieux Kouba, 16050 Alger, Algeria 1 ´ ´ Received 7 January 1999; received in revised form 19 July 1999; accepted 5 August 1999 Editor: R. Gatto
Abstract We calculate g l Ž0., the damping rate for longitudinal on-shell gluons with zero momentum in hot QCD using the hard-thermal-loop scheme, and find it to be divergent in the infrared. We argue that this result suggests that the scheme may be sensitive in this sector and thus may need to be improved upon. We discuss the validity of our result and emphasize the fact that it depends on the way we perform our calculation. In particular, we do not exclude the possibility that alternative approaches that we briefly comment on may in principle yield infrared-safe results. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 12.38.-t; 12.38.Bx; 12.38.Mh Keywords: Hard thermal loops; Soft gluon damping
Besides their importance regarding the stability of the quark-gluon plasma, gluon damping rates have been crucial in better understanding QCD at high temperature T. In early works, it has been noticed that in this regime, the determination of the dispersion laws for quarks and gluons beyond lowest order using standard perturbation theory is gauge-dependent w1x. This problem has been emphasized in further works in which the gluon damping rates have been calculated to one-loop order in various gauges and schemes and different results have been obtained w2x. It was then realized that the problem was related to the way the expansion in powers of g, the perturbative QCD coupling constant, was performed: at high T Žthe hard scale., when the external momenta are soft, i.e., of magnitude gT, the standard loop expansion is not anymore an expansion in powers of g w3x. Braaten and Pisarski subsequently developed an effective perturbative expansion in the framework of a resummation scheme of the so-called hard thermal loops Žhtl. w4x. Using this scheme, they showed that the transverse-gluon damping rate g t Ž0. with zero momentum was invariant within the Coulomb and covariant gauges and determined it in the strict Coulomb gauge to be finite and positive w5x. Later, a generating-functional formalism in the htl approximation was developed w6x and a relation to the eikonal of a Chern-Simons gauge
1
E-mail: [email protected]
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 9 5 9 - 4
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theory was made w7x. From there a hydrodynamic approach showed that the htl approximation was essentially ‘classical’ w8x. However, the htl resummation scheme discusses only the two scales T Žhard. and gT Žsoft. whereas with g and T, one has Žat least. a hierarchy of scales g n T, n an integer, positive if we believe that T is the highest scale in the problem. It may be unnecessary to reorganize perturbation theory taking into account this general structure of scales, but there are indications that the next smaller one g 2 T may be important. Indeed, it is true that in the htl scheme, both on-shell longitudinal and transverse gluons acquire a thermal mass m g of order gT, but static magnetic fields are not screened yet at this scale w9x; they are expected to get so at the hence-called ‘magnetic scale’ g 2 T. Furthermore, the gluon damping rates g Ž p . are to lowest order of magnitude g 2 T and it has been noticed that these damping rates, when calculated in the htl-resummation scheme at soft but nonzero momenta p, exhibit a ln g behavior accompanied with coefficients w9,10x. This logarithmic behavior is not present in the finite expression of g t Ž0., the limit p ™ 0 of g t Ž p ., and no full explanation has been provided yet for this discrepancy. These remarks tend to indicate therefore that the htl scheme may still be sensitive in the ‘infrared’. We think one more direct indication of the infrared sensitivity of the htl-resummation scheme is the observation we made in w11x that, when calculated solely in this scheme, the longitudinal-gluon damping rate at zero momentum g l Ž0. may be potentially infrared-sensitive. Indeed, we have determined in that work the analytic expression to leading order of g l Ž0.. We have obtained for it the following form:
g l Ž 0. s
g 2 Nc T 24p
a lŽ01 . q a lŽ02 . ;
Ž 1.
Ž . w x Nc is the number of colors. aŽ1. l 0 is given in Eq. 3 below; it is a finite number found in 5 to be equal to Ž . 6.63538 and is such that the damping rate g t 0 for transverse gluons with zero momentum is equal to g 2 NcT Ž1. Ž . Ž . a l 0 , see w5x. The other contribution aŽ2. l 0 is given in Eq. 4 and it is this part of g l 0 we remarked it 24p contains terms which are divergent in the infrared w11x. From a physical standpoint and independently of any calculation scheme, there is no difference at zero momentum between longitudinal and transverse gluons w5x. This argument can be used as an important criterion to test the consistency of a given calculational scheme like the htl one. With this in mind, we argued in w11x that the terms contributing to aŽ2. l 0 , in particular the infrared-divergent ones, when put together, may cancel one another. We show in this letter that this is not the case: when all the expressions are treated with care, not only Ž . we find aŽ2. l 0 nonvanishing but the infrared-divergent piece survives too, see Eq. 21 . In the sequel, we present first our calculation and we discuss the result afterwards. All along, we take m g s 1. The longitudinal gluon damping rate in the strict Coulomb gauge is obtained to lowest order by the relation: Im )P l Ž yi v , p . gl Ž p. s E dP l Ž yi v , p . Ev
,
Ž 2.
v s v l Ž p .qi0 q
where dP l Ž)P l . is the longitudinal-gluon htl Žeffective. self-energy and v l Ž p . the on-shell lowest-order energy given for soft momenta by Eqs. Ž8. below. We follow closely the notation of w11x to which this work is a follow-up. Since the denominator in Ž2. is given in a small y p expansion by ErEv dP Žyi v , p .< vs v l Ž p. s y2 p 2 q . . . , to get g l Ž p s 0., we have to expand the imaginary part of the effective self-energy to order p 2 . The rationale behind this expansion goes as follows. To determine the infrared behavior, we introduce an infrared cut-off h , i.e., we replace H0q` dk by Hhq` dk with 0 - h < 1. This means that k is never smaller than h. The cut-off h is fixed for the rest of the calculation and we regard p as always smaller than h Ž p will
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
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eventually go to zero but h stays fixed.. Physically, it may be useful to regard h as representing the magnetic scale, but this is not necessary from a pure technical standpoint. We are therefore always working in the kinematic region 0 F p - h F k. This then allows for the expansion of all functions of q s p y k in powers of small p, functions like 1rQS appearing in the effective vertices and )DŽ Q ., the effective propagators, see w11x for the notation. The expansion of 1rPS doesn’t pose a problem in itself. The subsequent angular integrations and Matsubara sum yield a series in powers of p 2 with a finite radius of convergence that depends of course on h. The first coefficient vanishes as it should and the second one, according to Ž2., yields the damping rate. The Ž2. calculation is reported in w11x and we obtain the result Ž1. where the expressions of aŽ1. l 0 and a l 0 are the following: q`
aŽ1. l0 s 9
q`
dk
Hh
Hy`
dv1
v1
dv2
q`
Hy`
v2
2
d Ž 1 y v 1 y v 2 . k 4 r l1 r l 2 y k 2 Ž k 2 y v 12 . r t1 r l 2 v1
2
q2 Ž k 2 q v 1 v 2 . r t1 r t 2 q
6k3
Ž k 2 y v 12 .
2
Q 1 rt 2 ,
Ž 3.
and: q`
aŽ2. l0 s 9
q
q`
dk
Hh
Hy`
v1 3
ž
1y
ž ž
2 v 12 3
y
v1 6
Ek
k
1q
žž
v1
v 12
qE k k 3 1 q
q
dv1
/
2
k2
/
v1
y
ky
v2
d Ž1 y v1 y v2 .
Ž 1 y 2 v 1 . E k Q 1 rt 2 q
3
k2
dv2
q`
Hy`
v 12 k
r l1 r l 2 q
/
v 12 k2
/
2k 9
/
3
ž
3
1y
v1k
Ž 1 y 2 v1 . Q 1 E k rl2 q
v 12 k
2
/
E k Q 1 E k rt 2 q
v1 6k
6
Q 1 E k2 r l 2
Ž k 2 y v 12 . E k2Q 1
rt 2
Ž k 2 Ž 1 y 3k 2 . q Ž 1 y 4 k 2 . v 1 v 2 y v 12 v 22 . rt1 rt 2
Q 1 rt 2 q
Q 1 rt 2
v1k
v1
k2 9
Ž 1 y 2 k 2 y 4 Ž 1 y k 2 . v 1 q 6 v 12 y 4v 13 . rt1 E k rt 2
q 23 < v 1 < Ž 2 k 2d 1 r l 2 y k 2v 1 Ev 12 d 1 r l 2 q v 1 d 1 r t 2 . .
Ž 4.
The notation is as follows: Q 1 ' Q Ž k 2 y v 12 . where Q is the step function and d 1 ' d Ž k 2 y v 12 .. E k ' ErE k etc., and r l i is a short notation for r l Ž v i ,k ., i s 1,2 and the same for r t i . The spectral densities r t,l are given by w5,12x:
r t ,l Ž v ,k . s z t ,l Ž k . d Ž v y v t ,l Ž k . . y d Ž v q v t ,l Ž k . . q bt ,l Ž v ,k . Q Ž k 2 y v 2 . ,
Ž 5.
an expression in which the residues z t,l Ž k . are given by:
ztŽ k. s
Ž v2yk2 . 2 3v 2 y Ž v 2 y k 2 .
ž
2
;
/
vs v t Ž k .
zlŽ k. s
yŽ v 2 y k 2 . 2 k2 Ž3yv 2 qk2 .
, vs v l Ž k .
Ž 6.
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
120
and the cut functions bt,l Ž v ,k . by: 3v Ž k 2 y v 2 .
bt Ž v ,k . s
ž
3
4k
2
3v 2
2
k yv q
2k2
ž
1q
k2yv2 2vk
ln
kqv kyv
2
// ž q
3pv 4k3
2 2
2
Ž k yv .
;
/
y3 v
b l Ž v ,k . s 2k
ž
2
3qk y
3v 2k
ln
kqv
2
3pv
2
.
Ž 7.
/ ž / q
kyv
2k
v t,l Ž p . are the on-shell energies of the gluon to lowest order. For soft gluons we have: 704 91617 v t Ž p . s 1 q 35 p 2 y 359 p 4 q 3000 p 6 y 336875 p8 q . . . ; 3 1 489 v l Ž p . s 1 q 103 p 2 y 280 p 4 q 6000 p 6 q 43120000 p8 q . . . .
Ž 8.
Ž2. w x As we said, aŽ1. l 0 is a finite number found in 5 to be equal to 6.63538. We therefore have to calculate a l 0 . The calculation is mainly a matter of disentangling the infrared-divergent pieces from the finite ones. It very often necessitates an expansion for small k of the residue and cut functions, using their definitions given in Ž6. and Ž7. respectively, and some of their derivatives. Also, care must be taken when handling the delta functions that occur, especially with their first and second derivatives. Finally, extra care must be given to the order in which the integrals over k and v are performed. Let us show explicitly how to carry with one representative term contributing to aŽ2. l 0 . Consider the first term Ž in 4.:
q`
Is9
q`
dk
Hh
Hy`
q`
dv1
v1
q`
Hy` d v
s3
dk
Hh
q`
Hy`
dv2
1y2v 1yv
v2
dŽ 1 y v 1 y v 2 .
v1 3
Ž 1 y 2 v1 . Q1 E k rl2
Q Ž k y v . E k r l Ž 1 y v ,k . .
Ž 9.
We can either apply the derivative ErE k directly to the spectral density r l Ž1 y v ,k . and carry through, or instead write: q`
I s y3
Hh
q`
Hy` d v
dk q`
Hy` d v
q3
1y2v
1y2v 1yv
1yv
d Ž k y v . r l Ž 1 y v ,k .
Q Ž k y v . r l Ž 1 y v ,k .
k™ q` ; ks h
Ž 10 .
the final result is the same 2 . Using the expression of r l given in Ž5., the first term in the above expression has two contributions, i.e., a dd contribution and a dQ contribution. The kinematics for the dd contribution is v s "k with v s 1 y v l Ž k . on the one hand, and v s "k with v s 1 q v l Ž k . on the other. Given the behavior of v l as a function of k, the second kinematics is always forbidden whereas the first one is satisfied only at precisely k s 0. Since h ) 0, the dd contribution is then zero. The kinematics for the dQ contribution is 1 y k F v F 1 q k with v s "k. This is allowed only for v s k and as long as k G 1r2, which means that the q` first term in I is equal to y3H1r2 dk 11yy2kk b l Ž1 y k,k ., a finite number. Note that, since b l Ž1 y k,k . goes to zero as k ™ q`, hard loop momenta are not allowed in any way.
2
This has been checked explicitly.
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
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We turn to the second term in I. The limit k ™ q` in this term is always zero because in this limit we have z l Ž k . ™ 0 and the same for b l Ž1 y v ,k . for v fixed. We then only have to determine this term at k s h. From the explicit expression of the spectral function r l given in Ž5., here too we have two contributions, i.e., a Qd contribution and a QQ contribution. The second one is zero because we must have from Q Žh y < v <. the conditions yh F v F h and from Q Žh y <1 y v <. the conditions 1 y h F v F 1 q h. These are equivalent to 1 y h F v F h with the condition h G 1r2, which can never be satisfied since h < 1. Now to the Qd contribution. The kinematics are the following: yh F v F h with v s 1 y v l Žh . for the first term and v s 1 q v l Žh . for the second. With the behavior of v l Ž k . for small k given in Eq. Ž8., the second kinematics is Ž . forbidden but the first one is allowed. This contribution is then equal to y3 z l h Ž 1 y 2 v l Ž h . . . If one uses the v l Žh .
expansion: 1
z l Žh . sy
39 13 1 y 35 h 2 q 175 h 4 y 175 h6 q . . . ,
4h 2
Ž 11.
which is obtained from Ž6. and Ž8., one finds z l Žh . 3 9 y3 Ž1 y 2 v l Ž h . . s 2 y . vl Žh . 40 4h Hence one obtains the result: q` 3 9 1y2k Is y y3 dk b Ž 1 y k ,k . . 2 40 1yk l 4h 1r2
Ž 12 .
H
The other terms in aŽ2. l 0 are worked out in a similar manner, though some necessitate more labor. We now give the corresponding results. To ease the notation, we denote in a compact way: q` q` d v 1 q` d v 2 D'9 dk d Ž1 y v1 y v2 . . Ž 13 . h y` v 1 y` v 2
H
H
H
H
We have:
H
ž ž
DE k k 3 1 q 2 v 12
q
3
k2
1q
/
3
v1 k
2
r l1 r l 2 q y
v 12 k
2
/
2k 9
Ž k 2 Ž 1 y 3k 2 . q Ž 1 y 4 k 2 . v 1 v 2 y v 12 v 22 . rt1 rt 2
Q 1 rt 2 s y
3
40 y
4
27p 2
.
Ž 14 .
It is worth emphasizing the fact that all contributions from k ™ q` vanish, indeed as it should be since only soft contributions are to contribute. The next piece reads: < v1 < v 12 D 4 k 2d 1 r l 2 q 2 v 1 d 1 r t 2 q Ž 1 y 2 v 1 . 1 y 2 e Ž v 1 . E k Q 1 r t 2 3 k
ž
H
dk
q`
s3
H1r2
1yk
/
Ž 2 k b l Ž 1 y k ,k . q bt Ž 1 y k ,k . . .
Ž 15 .
The next pieces are more delicate to handle: they require working out derivatives of delta functions in the presence of other delta functions with different arguments. After that is done, we get: 1
HD 6 3 s 8
ž
2 v1 k 1 y q`
y3
H1r2
v 12 k2 dk
1yk
/
E k Q 1 E k rt 2 q
bt Ž 1 y k ,k . .
v1 k
Ž k 2 y v 12 . E k2Q 1
r t 2 y v 1 E k2
žž
ky
v 12 k
/
Q 1 rt 2
/ Ž 16 .
A. Abada, O. Azi r Physics Letters B 463 (1999) 117–125
122
The next term is worked out along the same lines as the previous one. We have:
HD
v1k
Q 1 E k2 r l 2 s y
6
9 8h
27 2
q
3 q
80
dk
q`
H 2 1r2
1yk
b l Ž 1 y k ,k . y k E k b l Ž v ,k . < vs1yk .
Ž 17 .
The next term is the most tedious: it necessitates the additional expansion to order h 2 of E k bt Ž v ,k . with v s 1 y v t Ž k .. When this is done and the contributions put together, we get:
H DE
k2 k
Ž 1 y 2 k 2 y 4 Ž 1 y k 2 . v 1 q 6 v 12 y 4v 13 . rt1 E k rt 2
9
125 s
2 2
27p h
1153 y 189p
2
289536 y
2187p 4
.
Ž 18 .
The last term isn’t more difficult. It reads:
HD
y 23 k 2v 12 e Ž v 1 . Ev 12 d 1 r l 2 t s y
3
dk
q`
H 2 1r2
1yk
Ž b l Ž 1 y k ,k . q k E k b l Ž v ,k . < vs1yk . .
Ž 19 .
Putting all these contributions together, we obtain: 3
ž
aŽ2. l0 s y
125 q 27p
8
dk
q`
y3
H1r2
2
1yk
1
/
h
2
21 y
1433 y
80
189p
2
289536 y
2187p 4
Ž 1 y 4 k . b l Ž 1 y k ,k . q k E k b l Ž v ,k . < vs1yk .
Ž 20 .
Note that the integral in the above equation Žand all the other similar ones actually. is finite. We can evaluate it numerically and we get: aŽ2. l0 s
0.09408
h2
y 4.45366 .
Ž 21 .
Ž . Hence, as announced, aŽ2. l 0 is nonzero and quite divergent. Via Eq. 1 , this means g l Ž 0 . is itself infrared-divergent. Indeed, using the value aŽ1. s 6.63538, we get: l0
g l Ž 0. s
g 2 Nc T 24p
ž
0.09408
h2
/
q 2.18172 .
Ž 22 .
We resume our initial discussion. Our calculation of g l Ž 0 . is performed within the framework of the htl-resummation scheme. This latter treats systematically only the two scales T and gT and we have already remarked that the scale g 2 T may play an important role in QCD at high T. This suggests that the scheme, as it is, may be sensitive in the infrared, i.e., when it comes to deal with quantities that, to lowest order, are already of magnitude g n T, n G 2. We think the behavior of g l Ž 0 . we report in this letter is a manifestation of this. One may ask why the result is finite for g t Ž 0 . whereas we find an infrared divergence for g l Ž 0 . . It is pertinent to note in this respect that there is a main difference between the calculation of g l Ž 0 . and that of g t Ž 0 . : in order to get g l Ž 0 . , we need an unavoidable expansion to order p 2 of the imaginary part of the effective self-energy whereas, for g t Ž0., we do not w5,11x. As a matter of fact, we have determined the analytic
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expression of the transverse-gluon damping rate g t Ž p . to order p 2 w13x. The coefficient a t 0 , that of the zeroth Ž . w5x. But a preliminary investigation shows that the order, is finite and as we said, equal to aŽ1. l 0 of Eq. 3 coefficient a t1 , that of order p 2 , has contributing terms which are divergent in the infrared w14x. It seems then that the expansion of the gluon effective self-energies )P in the htl scheme in powers of the soft momentum p is infrared sensitive beyond zeroth order, and that this sensitivity may be responsible for the divergent behavior of g l Ž0.. One may jump to conclude that the htl scheme is robust in the infrared, and that all this simply means that the expansion in powers of the soft momentum of the gluon effective self-energies is not allowed in the first place. In this regard, we stress that the quark-gluon plasma is to be a stable phase of hadronic matter. Therefore, the damping rates g Ž p . have to be positive and be analytic functions of the external soft momenta, i.e., they have to admit a Taylor-series expansion in powers of Žat least very. small p. This is one important criterion to check with the consistency of a given calculational scheme. We have explained in our introductory remarks the basis upon which the expansion is performed, and within that context, it is valid. If the expansion yields infrared-sensitive coefficients as it is the case, it only means that the scheme in which the calculation is performed needs to be cured andror improved upon, not that the principle of the expansion itself is invalid. In other words, an infrared-improved resummation scheme would still allow for the same expansion but would remove the infrared sensitivity of the actual htl scheme. It is useful to remark that the infrared sector is not the first instance in which the htl scheme yields infinities. Indeed, collinear singularities do appear when external light-like momenta are involved w15,16x and, in response to this, the scheme has been subjected to improvement in w15x where an improved action that removes these singularities is proposed. We think the same treatment is necessary in the infrared sector but the situation is more problematic here. Indeed, such an improvement would very likely necessitate the determination of the magnetostatic screening length which is mostly believed to intervene in a nonperturbative way. It may also be that the story does not end at the scale g 2 T and new physics may manifest itself at lower scales g n T with n ) 2, though nothing is suggesting this for the moment. In any case, it is pertinent to note that besides the result we obtain for g l Ž0., what we think is also interesting in this work is that it sets a calculable framework in which the infrared divergences appear explicitly and, by the same token, where to test an improved htl scheme when found. In the meantime, we can have a very rough estimate of the scale at which the htl scheme starts to be sensitive. If we demand that g l Ž0. be equal to g t Ž0., we may set aŽ2. l 0 equal to zero, which yields h s 0.145 Žin units of m g .. However, one should be prudent in considering this value as an estimation, even rough, of the magnetic mass. Our calculation of g l Ž0. is performed in the imaginary-time formalism and is based on the techniques developed in w4x. In seeking to find an expression for the damping rates, we have followed a sequence of operations which amounts to: Ži. expand the effective self-energies in powers of p; Žii. perform the angular integrals; Žiii. then perform the Matsubara sum. Also, as we previously said, our calculation is valid in the kinematic region p - h F k, where k is the soft loop momentum. How then our result compares to what is already known in the literature? As an illustrative example, consider the estimation:
g t ,l Ž p . ; y
g 2 Nc T 4p
ln g Õt ,l Ž p .
Ž 23 .
of the soft-gluon damping rates made in the paper by Pisarski in w9,10x; Õt,l Ž p . are the group velocities. This calculation is different in many respects from the one we carry. In particular, the result Ž23., as explained in that paper, is obtained in the kinematic region where the loop momentum k is restricted to the ‘‘ very soft’’ region Ž0 F k - h . while p is just soft, of order m g . Then, it is obvious why our result is different from that of the estimation Ž23. above: our result is carried in a different kinematic region, i.e., 0 F p - h F k; it does not take account of the ‘‘ very soft’’ loop momenta 0 F k - h and cannot be carried to the region p ) h. On the other hand, the result Ž23. takes account of only the contributions from the ‘‘ very soft’’ region 0 F k - h with p ) h ;
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it cannot be carried to the region p - h , in particular to the point p ™ 0 for it will give zero Žusing the expressions of the group velocities at very soft momenta.. The same comparison can be carried with the other results in the literature. The point to emphasize is that our result is not in contradiction in any way with what is previously known because it is valid in a different kinematic region. It is also important to realize that our result is not at all the final word regarding the issue of the infrared sensitivity of the htl-scheme. This is simply because the sequence of operations we have followed in our work is not unique. Indeed, it is certainly important to be able to carry the calculation in the order: Matsubara sum first, angular integrations second and only third the expansion in powers of p. Provided that the calculation in this order is technically feasible, it is important to see whether in this order, the infrared divergences persist. This is not obvious from the outset because we are not assured of the commutation of the different operations involved. However, from a calculational standpoint, one has to realize that matters are not straightforward if we delay the expansion in powers of p. This is because we have to deal with expressions quite complicated and involved. It is true that the Matsubara sum can easily be performed if the effective propagators and terms appearing in the effective vertices are replaced by their respective spectral representations, but this will bring in more than one energy denominator, which would compromise the straightforward extraction of the imaginary part of the effective self-energy. Also, the advantages of the techniques developed by Braaten and Pisarski will likely be lost. More importantly, the subsequent angular integrations will be very difficult, if not impossible, to handle in a compact way before any expansion is made. Equally important is the real-time approach. It will avoid the Matsubara sum and the analytic continuation to real energies altogether. In this regard, work is being considered in this direction and preliminary results seem to indicate that the infrared problem is present there too w17x. Nevertheless, it is quite possible that these alternative approaches will eventually demonstrate that the one we adopted in our calculation is not the ‘correct’ way of approaching the physical damping rates. In any case, the issue is certainly not closed and our result is only a first step forward. Finally, we have limited ourselves thus far to discussing the gluonic sector only, but certainly quarks play an important role too in the structure of hot QCD. They themselves acquire a thermal mass m f to order gT and their damping rates start also at the scale g 2 T, which means they are not a priori shunned from infrared sensitivity.
Acknowledgements We deeply thank Asmaa Abada for all her help and encouragements. O.A. would like to thank ICTP for their kind hospitality where some of the later part of this work was carried. She had the opportunity there to carry constructive discussions with Samina S. Masood. This is a revised version of the work and we must thank the referee for his substantive remarks that reshaped the emphasis in our discussion. Between the two versions, O.A. has been able to attend a workshop on thermal field theory at CERN and she benefited from stimulating discussions with many people, most particularly with P. Aurenche, U. Heinz, E. Iancu, A. Rebhan, D. Schiff and I. Zahed. O.A. would also like to thank all the people in Annecy for their kind hospitality and most useful suggestions: Gelis, Petitgirard, Peigne´ and Zaraket. Finally, special thanks to Patrick Aurenche and Anton Rebhan for their warm encouragements.
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