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Gauge invariant values of gluon masses at high temperature
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
GAUGE INVARIANT VALUES OF GLUON MASSES AT HIGH TEMPERATURE Toyoaki FURUSAWA and Keiji KIKKAWA Department of Physics, Osaka University, Toyonaka 560, Japan Received 17 March 1983 Revised manuscript received 16 May 1983
Both electric and magnetic gluon masses in the high temperature QCD plasma are evaluated up to of order g3/2. Gauge covariance is always kept in each step of calculations by taking account of the Ward-Takahashi identity. The magnetic mass is shown to vanish up to O(g3/2 T).
The magnetic mass of the gluon (the inverse of the shielding distance of the static color magnetic field) in the high temperature QCD plasma is a crucial quantity to test the validity of perturbation expansions in finite temperature QCD [1,2]. If the magnetic mass squared of order g3T 2 is nonzero and finite, the perturbation expansion is meaningful to any order as an asymptotic series. If the mass is zero, some non-perturbative methods must be incorporated in high temperature physics. Recently, Kajantie and Kapusta [3,4] reported that the magnetic mass of O(g3/2T) was f'mite in a temporal gauge calculation, while Toimela [5 ] claimed that the magnetic mass was vanishing, but the electric mass of 0(003/2/3 was gauge parameter dependent in a covariant gauge. In these two works, however, the gauge covariance in the calculation was not manifestly shown. The purpose of the present paper is to reinvestigate the problem by keeping the gauge covariance throughout calculations. As is well known, the gauge variance trouble always occurs in any self-consistent method or a partial summation of Feynman diagrams, such as in the Bethe-Salpeter equation and in the BCS superconductor theory [6]. The reason is that gauge invariance is a global concept and the repetition of particular subdiagrams misses inter-subdiagram corrections without which invariance is violated. However, the recovery of gauge invariance is rather easy as far as the self-energy problem is concerned. 218
In our case, to a given propagator, we solve the Ward-Takahashi (W-T) identity and get a set of associated gluon vertices. As a matter of fact the W - T identity does not determine the full vertices. However, since what is needed is the zero momentum transfer vertex in our problem, the W - T identity provides us with enough information. In the following we consider an SU(N) gauge theory in an axial gauge which is specified by a constant vector n u = (n O, n) with nt~-2_ 1. The quark and other matter fields are not included because they are irrelevant to the infrared divergence. The gauge invariance (or variance) of results can be monitored by inspecting the n 2 or n o dependence afterwards. The reason why the axial gauge is adopted is due to the simplicity of the W - T identity. We calculate the electric and magnetic masses up to order g3/2T. Our results disagree with the previous reports [ 3 - 5 ] . We will discuss afterward on these disagreements and show how ours will be obtained by recovering gauge invariance. In the SU(N) gauge theory the gauge is fixed either by imposing nuA ~ = 0 or adding the term (ct/2) X (n~Au) 2 to the lagrangian. In this report we adopt the former. The full gluon propagator can be decomposed into the following SO(3) tensors, ~ , ~ ( k ) = . ~ A ~ + q ~ B ~ + Q C ~ + CDD~,~
(1)
where
0 031-9 ! 63/83/0000--0000/$ 03.00 © 1983 North-Holland
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
where k~ = ku _ (kn)nu '
~(2) = ~g2NT2n 2.
(7)
The color factor ~ab is omitted from the above propagator. The masses are obtained from (5) and (6) as (m(2)~2 = ~g2NT2 ' ~. e l )
+
Auv = 8 ui(Si/ - ninJn2)8/v,
Duv
=
(n2)-l/2(kl~86v
C# v +
-_k uIk lv,
i 8u0k-~v).
(8)
which agree to all previous calculations [2,7]. Before going into self-energy calculations, we point out that the W - T identity in our axial gauge provides us with the gluon vertices in which some of the external lines have zero momenta: I~ubcv(O, q, --q) = -ig.f abc (O/Oq~) [c'/)# v(q)] - 1
Fig. I. Serf-energydiagrams considered in this work.
I l Buy =n -2 6u080v,
(m(2) mag-~2 = 0
(2)
Faabccgo O, q, - q ) = g2(facdfbcd)(o2/Oqc~Oqo) ~tav',
'
X [Cbuv(q)]-1.
(9)
and 8~u
=
60v - non v
kXu = 6 M(k i - ni(k. n)/n 2) . All tensors (A .... , D) are orthogonal to the gauge axis
///a" The diagrams we consider are shown in fig. 1. To def'me gluon masses consider the static limit (k 0 = 0) of correlation functions for the color electric fields (Ea) and magnetic fields (Ba), which can be written in the lowest order of perturbation expansions as
fef> = k2 = ab ab (B i B) ) = eilre/msklkm(ArAs )
(3)
= (k2~ 0 - tq~j)~t(0, k),
(4)
where ~ and qfl are defined in (1). As will be seen later both (3) and (4) are gauge parameter independent. If the renormalization has been done at zero temperature, the masses are given by 2
mel
_
-
[q3(0,0)n2]-I
m2ag
=
M_I(0,0).
(5)
In the second order calculation, where internal lines in fig. 1 are represented by the free gluon propagator and the vertices by the uncorrected perturbational vertices, the one-loop self energy (temperature dependent part) turns out to be
In the skeleton expansion method [8], we use the propagator q) (2) obtained from (6) and the corresponding vertex functions given by (9). The relation (9) and the graphical counting easily show that the temperature dependent part of the full self-energy tensor IIuv (fig. 1) is given by
8II~(0,0)-
g~-fdq o fd3q[exp(-i/3q
0) - 1 ] -1
(0) -I (q)C~u~,(q)], (2) X ( 0 / 0qrj)[(O/Oq,~)ct)uv
(10)
where we have used the inverse of the unperturbed propagator q) (0)-1 v to avoid double counting, while the other is a I~u~lpropagator. If O/3q# operates on c/) (2), the term generates the first diagram in fig. 1, due to Ocb/Oqa = _c-/) [3-'7)-1/0qa ] q), while on c/) (0)-1 ' the second diagram. In (10), if either one or both of a and/3 is the spatial component, the right hand side can be easily shown to vanish because the integrand is a total divergence. The other component 8H00 is non-vanishing and can be obtained by a straightforward integration. The full self-energy part is given by (6) but now ~(2) is replaced by
[~ = [½g2N + (3/27rX½g2N) 3/2] T2n 2.
(11)
(The detailed calculation will be published elsewhere
[91 .) The propagator q)uv = ( ~ (0)-1 + H)u I with k 0 = 0 is given by (1) in which '
+ [~(2)[k~l(knl2ln-2nun,,
(6) 219
Volume 128B, number 3,4
= ilk2,
PIIYSICS LETTERS
qa = 1/(nEk2 + i,),
e=(k2+[j)l[k2"(k "n)2"(k2+D/n2)],
(12)
c~ = _ n o ~ i v Y . (~ ".)" (k 2 + ~/n2)]. The electric and magnetic masses are, therefore, 2 = [½g2N + (3/21r)(lg2N) 3/2] T 2, reel
m2ag = 0, (t3, 14)
which are both gauge parameter independent. Incidentally, we note that in our gauge the vertex corrections due to 8 II /av (2) and (9) are proportional to n u and makes no contribution to (10), because the internal line propagator is orthogonal to n u. This is a special situation in our axial gauge. Our results (13) and (14) do not agree with the preceding reports [ 3 - 5 ] . In the covariant gauge, the W - T identity is not simple and non-trivial vertex corrections will be needed by the gauge invariance. Since Toimela [5] did not include the vertex correction the result can be gauge dependent. The disagreement with Kajantie and Kapusta [3,4] is a little involved. Since our SO(3) tensors are singular in the temporal gauge limit, we cannot simply compare the two calculations term by term. In their gauge the formula (9) must be still true. If one constructs the three-gluon vertex by using (9) and their propagator eq. (2) in ref. [3], one obtains
Pli/( O, q, - q ) ~ 2q18i/- 5h~li - 81iqi
+ (a/aqt) [(G + Fq~/qE)qt
220
25 August 1983
their temporal gauge has a term proportional to q2/q2, which saves the F-term by canceling the factor. We conclude, therefore, that the vertex correction is necessay in their temporal gauge to keep the gauge invariance. Taking account of this vertex correction we recovered the same results as (13) and (14) in their temporal gauge, too. In conclusion, we have calculated the electric and magnetic masses up to O(g3/2T), and have shown that the magnetic mass is vanishing. This implies that the perturbation method does not work in high temperature QCD. As speculated by Polyakov [10] and Gross et al. [2], some non-perturbative method must be incorporated. The authors are grateful to Dr. A. Hosoya and Dr. N. Yamamoto for cooperating discussions.
References [1] A.D. Linde, Phys. Lett. 96B (1980) 289. [2] D. Gross, R. Pisarski and L. Yaffe, Rev. Mod. Phys. 53 (1981) 43. [3] K. Kajantie and J. Kapusta, Phys. Lett. ll0B (1982) 299. [4] K. Kajantie and J. Kapusta, CERN preprint TH-3284 (1982). [5] T. Toimela, Helsinki preprint HU-TFT 82-37 (1982). [6] Y. Nambu, Phys. Rev. 117 (1960) 648. [7] A. Linde, Rep. Prog. Phys. 42 (1979) 389. [8] B.A. Freedman and L.D. McLerran, Phys. Rev. DI6 (1977) 1130. [9] 1~ Furusawa and K. Kikkawa, in preparation; K. Kikkawa, Proc. Workshop on GUTS and early universe (KEK National Laboratory for High Energy Physics, Japan, 1983) KEK Report, to be published. [10] A.M. Polyakov, Phys. Lett. 72B (1978) 477.
PHYSICS LETTERS
25 August 1983
GAUGE INVARIANT VALUES OF GLUON MASSES AT HIGH TEMPERATURE Toyoaki FURUSAWA and Keiji KIKKAWA Department of Physics, Osaka University, Toyonaka 560, Japan Received 17 March 1983 Revised manuscript received 16 May 1983
Both electric and magnetic gluon masses in the high temperature QCD plasma are evaluated up to of order g3/2. Gauge covariance is always kept in each step of calculations by taking account of the Ward-Takahashi identity. The magnetic mass is shown to vanish up to O(g3/2 T).
The magnetic mass of the gluon (the inverse of the shielding distance of the static color magnetic field) in the high temperature QCD plasma is a crucial quantity to test the validity of perturbation expansions in finite temperature QCD [1,2]. If the magnetic mass squared of order g3T 2 is nonzero and finite, the perturbation expansion is meaningful to any order as an asymptotic series. If the mass is zero, some non-perturbative methods must be incorporated in high temperature physics. Recently, Kajantie and Kapusta [3,4] reported that the magnetic mass of O(g3/2T) was f'mite in a temporal gauge calculation, while Toimela [5 ] claimed that the magnetic mass was vanishing, but the electric mass of 0(003/2/3 was gauge parameter dependent in a covariant gauge. In these two works, however, the gauge covariance in the calculation was not manifestly shown. The purpose of the present paper is to reinvestigate the problem by keeping the gauge covariance throughout calculations. As is well known, the gauge variance trouble always occurs in any self-consistent method or a partial summation of Feynman diagrams, such as in the Bethe-Salpeter equation and in the BCS superconductor theory [6]. The reason is that gauge invariance is a global concept and the repetition of particular subdiagrams misses inter-subdiagram corrections without which invariance is violated. However, the recovery of gauge invariance is rather easy as far as the self-energy problem is concerned. 218
In our case, to a given propagator, we solve the Ward-Takahashi (W-T) identity and get a set of associated gluon vertices. As a matter of fact the W - T identity does not determine the full vertices. However, since what is needed is the zero momentum transfer vertex in our problem, the W - T identity provides us with enough information. In the following we consider an SU(N) gauge theory in an axial gauge which is specified by a constant vector n u = (n O, n) with nt~-2_ 1. The quark and other matter fields are not included because they are irrelevant to the infrared divergence. The gauge invariance (or variance) of results can be monitored by inspecting the n 2 or n o dependence afterwards. The reason why the axial gauge is adopted is due to the simplicity of the W - T identity. We calculate the electric and magnetic masses up to order g3/2T. Our results disagree with the previous reports [ 3 - 5 ] . We will discuss afterward on these disagreements and show how ours will be obtained by recovering gauge invariance. In the SU(N) gauge theory the gauge is fixed either by imposing nuA ~ = 0 or adding the term (ct/2) X (n~Au) 2 to the lagrangian. In this report we adopt the former. The full gluon propagator can be decomposed into the following SO(3) tensors, ~ , ~ ( k ) = . ~ A ~ + q ~ B ~ + Q C ~ + CDD~,~
(1)
where
0 031-9 ! 63/83/0000--0000/$ 03.00 © 1983 North-Holland
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
where k~ = ku _ (kn)nu '
~(2) = ~g2NT2n 2.
(7)
The color factor ~ab is omitted from the above propagator. The masses are obtained from (5) and (6) as (m(2)~2 = ~g2NT2 ' ~. e l )
+
Auv = 8 ui(Si/ - ninJn2)8/v,
Duv
=
(n2)-l/2(kl~86v
C# v +
-_k uIk lv,
i 8u0k-~v).
(8)
which agree to all previous calculations [2,7]. Before going into self-energy calculations, we point out that the W - T identity in our axial gauge provides us with the gluon vertices in which some of the external lines have zero momenta: I~ubcv(O, q, --q) = -ig.f abc (O/Oq~) [c'/)# v(q)] - 1
Fig. I. Serf-energydiagrams considered in this work.
I l Buy =n -2 6u080v,
(m(2) mag-~2 = 0
(2)
Faabccgo O, q, - q ) = g2(facdfbcd)(o2/Oqc~Oqo) ~tav',
'
X [Cbuv(q)]-1.
(9)
and 8~u
=
60v - non v
kXu = 6 M(k i - ni(k. n)/n 2) . All tensors (A .... , D) are orthogonal to the gauge axis
///a" The diagrams we consider are shown in fig. 1. To def'me gluon masses consider the static limit (k 0 = 0) of correlation functions for the color electric fields (Ea) and magnetic fields (Ba), which can be written in the lowest order of perturbation expansions as
fef> = k2 = ab ab (B i B) ) = eilre/msklkm(ArAs )
(3)
= (k2~ 0 - tq~j)~t(0, k),
(4)
where ~ and qfl are defined in (1). As will be seen later both (3) and (4) are gauge parameter independent. If the renormalization has been done at zero temperature, the masses are given by 2
mel
_
-
[q3(0,0)n2]-I
m2ag
=
M_I(0,0).
(5)
In the second order calculation, where internal lines in fig. 1 are represented by the free gluon propagator and the vertices by the uncorrected perturbational vertices, the one-loop self energy (temperature dependent part) turns out to be
In the skeleton expansion method [8], we use the propagator q) (2) obtained from (6) and the corresponding vertex functions given by (9). The relation (9) and the graphical counting easily show that the temperature dependent part of the full self-energy tensor IIuv (fig. 1) is given by
8II~(0,0)-
g~-fdq o fd3q[exp(-i/3q
0) - 1 ] -1
(0) -I (q)C~u~,(q)], (2) X ( 0 / 0qrj)[(O/Oq,~)ct)uv
(10)
where we have used the inverse of the unperturbed propagator q) (0)-1 v to avoid double counting, while the other is a I~u~lpropagator. If O/3q# operates on c/) (2), the term generates the first diagram in fig. 1, due to Ocb/Oqa = _c-/) [3-'7)-1/0qa ] q), while on c/) (0)-1 ' the second diagram. In (10), if either one or both of a and/3 is the spatial component, the right hand side can be easily shown to vanish because the integrand is a total divergence. The other component 8H00 is non-vanishing and can be obtained by a straightforward integration. The full self-energy part is given by (6) but now ~(2) is replaced by
[~ = [½g2N + (3/27rX½g2N) 3/2] T2n 2.
(11)
(The detailed calculation will be published elsewhere
[91 .) The propagator q)uv = ( ~ (0)-1 + H)u I with k 0 = 0 is given by (1) in which '
+ [~(2)[k~l(knl2ln-2nun,,
(6) 219
Volume 128B, number 3,4
= ilk2,
PIIYSICS LETTERS
qa = 1/(nEk2 + i,),
e=(k2+[j)l[k2"(k "n)2"(k2+D/n2)],
(12)
c~ = _ n o ~ i v Y . (~ ".)" (k 2 + ~/n2)]. The electric and magnetic masses are, therefore, 2 = [½g2N + (3/21r)(lg2N) 3/2] T 2, reel
m2ag = 0, (t3, 14)
which are both gauge parameter independent. Incidentally, we note that in our gauge the vertex corrections due to 8 II /av (2) and (9) are proportional to n u and makes no contribution to (10), because the internal line propagator is orthogonal to n u. This is a special situation in our axial gauge. Our results (13) and (14) do not agree with the preceding reports [ 3 - 5 ] . In the covariant gauge, the W - T identity is not simple and non-trivial vertex corrections will be needed by the gauge invariance. Since Toimela [5] did not include the vertex correction the result can be gauge dependent. The disagreement with Kajantie and Kapusta [3,4] is a little involved. Since our SO(3) tensors are singular in the temporal gauge limit, we cannot simply compare the two calculations term by term. In their gauge the formula (9) must be still true. If one constructs the three-gluon vertex by using (9) and their propagator eq. (2) in ref. [3], one obtains
Pli/( O, q, - q ) ~ 2q18i/- 5h~li - 81iqi
+ (a/aqt) [(G + Fq~/qE)qt
220
25 August 1983
their temporal gauge has a term proportional to q2/q2, which saves the F-term by canceling the factor. We conclude, therefore, that the vertex correction is necessay in their temporal gauge to keep the gauge invariance. Taking account of this vertex correction we recovered the same results as (13) and (14) in their temporal gauge, too. In conclusion, we have calculated the electric and magnetic masses up to O(g3/2T), and have shown that the magnetic mass is vanishing. This implies that the perturbation method does not work in high temperature QCD. As speculated by Polyakov [10] and Gross et al. [2], some non-perturbative method must be incorporated. The authors are grateful to Dr. A. Hosoya and Dr. N. Yamamoto for cooperating discussions.
References [1] A.D. Linde, Phys. Lett. 96B (1980) 289. [2] D. Gross, R. Pisarski and L. Yaffe, Rev. Mod. Phys. 53 (1981) 43. [3] K. Kajantie and J. Kapusta, Phys. Lett. ll0B (1982) 299. [4] K. Kajantie and J. Kapusta, CERN preprint TH-3284 (1982). [5] T. Toimela, Helsinki preprint HU-TFT 82-37 (1982). [6] Y. Nambu, Phys. Rev. 117 (1960) 648. [7] A. Linde, Rep. Prog. Phys. 42 (1979) 389. [8] B.A. Freedman and L.D. McLerran, Phys. Rev. DI6 (1977) 1130. [9] 1~ Furusawa and K. Kikkawa, in preparation; K. Kikkawa, Proc. Workshop on GUTS and early universe (KEK National Laboratory for High Energy Physics, Japan, 1983) KEK Report, to be published. [10] A.M. Polyakov, Phys. Lett. 72B (1978) 477.