Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990
H I G H T E M P E R A T U R E B E H A V I O U R O F T H E Q C D C O U P L I N G C O N S T A N T R. BAIER a, B. P I R E b a n d D. S C H I F F c
a Fakultiitfur Physik, Universitiit Bielefeld, D-4800 Bielefeld, F R G
b Centre de Physique Thborique ~. Ecole Polytechnique, F-91128 Palaiseau, France
c Laboratoire de Physique Thborique des Hautes Energies z Universitb Paris-Sud Orsay, F-91405 Orsay, France
Received 3 January 1990
We investigate the finite temperature one loop renormalization of the pure QCD coupling constant in the real time formalism.
The temperature dependence of the fl function is derived in a generalized momentum space subtraction scheme. Due to the occurrence of multiple Bose-Einstein distribution functions, we find g2 (T) ~ T - 3 for asymptotic temperatures at fixed momen- tum subtraction scale.
I. Introduction
The question of the high t e m p e r a t u r e b e h a v i o u r of the Q C D coupling constant is a crucial issue for i m p r o v i n g p e r t u r b a t i v e calculations at finite T. It has often been taken for granted [ 1 ] that this T b e h a v i o u r is to be inferred from identifying the r e n o r m a l i z a t i o n scale with T, thus leading to
g 2 ( T ) ~ 1
In (T/AQcD) "
This straightforward extension o f the T = 0 r e n o r m a l i z a t i o n group ( R G ) results is not completely satisfactory.
In ref. [ 2 ], M a t s u m o t o et al. generalize the R G approach at finite T i n the framework of the real time formalism [3]. In this framework, the ultraviolet divergences are the same as at T = 0 b u t in a d d i t i o n to the freedom of choosing the usual m o m e n t u m r e n o r m a l i z a t i o n point, one has to deal with the arbitrariness of the t e m p e r a t u r e at which the r e n o r m a l i z a t i o n p a r a m e t e r s are d e t e r m i n e d . W i t h i n this finite t e m p e r a t u r e R G approach, the be- h a v i o u r o f the Q C D coupling c o n s t a n t with respect to the t e m p e r a t u r e has been investigated by working out the solution of the one loop R G e q u a t i o n s [ 4 - 7 ] .
In this paper, we reanalyze this p r o b l e m in a specific r e n o r m a l i z a t i o n scheme a n d show that c o n t r i b u t i o n s have been overlooked which in fact d o m i n a t e the high T expansion of the Q C D coupling constant. This scheme generalizes at finite T the M O M r e n o r m a l i z a t i o n procedure [ 8 ]. Propagators a n d vertices are r e n o r m a l i z e d at spacelike m o m e n t a a n d at a given temperature. We work in pure Q C D a n d derive the one loop r u n n i n g coupling constant in the F e y n m a n gauge in two M O M r e n o r m a l i z a t i o n schemes attached to the three gluon vertex a n d the g l u o n - g h o s t vertex in the s y m m e t r i c spacelike configuration for external lines p2 = q2 = r 2 = _ M 2 (fig. 1 ).
We find that terms d e p e n d i n g on the second a n d third power of the gluon (or ghost) B o s e - E i n s t e i n statistical d i s t r i b u t i o n are present ~ a n d d o m i n a t e the asymptotic t e m p e r a t u r e b e h a v i o u r o f g 2 ( T ) at fixed M:
g 2 ( T ) ~ ( M / T ) 3
Unit6 Propre du CNRS.
2 Unit6 Asscoci6e au CNRS.
~ In the collinear momentum configuration worked out in ref. [4], these terms do not contribute.
-= + + - d : ~ + ~ ,
b,q,v
c,Go
* + i + ~
Fig. 1. The thermal ( 111 ) three gluon vertex up to one loop. The wavy (dotted) lines represent the gluon (ghost).
T h e a b o v e c o n v e n t i o n a l inverse l o g a r i t h m i c b e h a v i o u r is o f course r e c o v e r e d in the special l i m i t T ~ o o ,
M~ T
fixed.
A n o t h e r interesting result is the c a n c e l l a t i o n o f l o g a r i t h m i c t e r m s (ln M ) in the r e n o r m a l i z a t i o n constants when T ~ oo at fixed M. Such t e r m s have to be absent i f the expected d i m e n s i o n a l r e d u c t i o n o f Q C D at infinite t e m p e r a t u r e is realized [ 9 ]. In this respect, the n 2 a n d n ~ t e r m s play an essential role.
The presence o f c o n t r i b u t i o n s i n v o l v i n g high powers o f statistical d i s t r i b u t i o n s in G r e e n ' s functions at the one l o o p level seems to be c h a r a c t e r i s t i c o f the real t i m e f o r m a l i s m ~2. It m a y r e o p e n the question o f the com- p a r i s o n between this f o r m a l i s m a n d the i m a g i n a r y t i m e f o r m a l i s m [ 12].
Let us now outline the c o n t e n t o f this p a p e r . In section 2, the three gluon vertex is calculated a n d the corre- s p o n d i n g r e n o r m a l i z a t i o n c o n s t a n t Z1 is d e t e r m i n e d at a given t e m p e r a t u r e T. The c o r r e s p o n d i n g r u n n i n g cou- pling c o n s t a n t is d e r i v e d as a function o f T a t fixed M. In section 3, the same p r o c e d u r e is a p p l i e d to the g l u o n - ghost vertex. In section 4, the high T e x p a n s i o n o f the r u n n i n g coupling c o n s t a n t is d e r i v e d at fixed M. T h e e x p a n s i o n in 1 / T is given up to l o g a r i t h m i c t e r m s a n d the In M t e r m s are shown to cancel.
2. Three gluon vertex
We start with the c o m p u t a t i o n o f the three gluon vertex r e n o r m a l i z a t i o n constant at finite t e m p e r a t u r e . The one l o o p d i a g r a m s c o n s i d e r e d are shown in fig. 1 (neglecting c o n t r i b u t i o n s from the q u a r k l o o p ) . T h e y have the s a m e structure as for T = 0. In the real t i m e f o r m a l i s m , however, there is a d o u b l i n g o f the fields (type-1 and -2 ) [ 3 ]. We p e r f o r m the m o m e n t u m s u b t r a c t e d r e n o r m a l i z a t i o n at finite T for the real p a r t o f the ( 1 1 1 ) vertex, as i n d i c a t e d in fig. 1, following the p r e s c r i p t i o n s given in ref. [ 2 ]. T h e r e f o r e only the free ( 11 ) p r o p a g a t o r s are needed, which are for the gluon field (in the F e y n m a n gauge)
,~b ( 1 _2rri6(k2)ne(lko,)) '
( 1 )Au~(k)=6abgu~
k 2 + i ~ a n d for the ghost field,~b (k2~le_2r66(k2)nB(lko[))
( 2 )A g h o s t -~" - -
(~ab
where the indices a, b d e n o t e the c o l o u r degrees o f f r e e d o m ~3. T h e B o s e - E i n s t e i n d i s t r i b u t i o n is
nB(lkol )=[exp(fl[ko[ ) - l ]-', fl=l/T.
( 3 )T h e following c o m m e n t is in order. Inserting e.g. the gluon p r o p a g a t o r ( 1 ) into the three p o i n t d i a g r a m with the gluon loop (fig. 1 ), 2 3 = 8 terms result, which all c o n t r i b u t e to the real p a r t o f the ( 111 ) vertex ~4. One term c o n t a i n s the T = 0 part, which was c a l c u l a t e d by C e l m a s t e r a n d G o n s a l v e s [ 8 ] in the r e n o r m a l i z a t i o n scheme
~2 In the imaginary time formalism at most a single power is present as stressed in refs. [ 10,11 ].
,3 For the T= 0 QCD Feynman rules we follow the conventions and notations of ref. [ 13 ].
,4 Only four of them are taken into account in ref. [ 5 ].
Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990 under consideration. T h e other t e r m s m a y be characterised by the powers o f the Bose-Einstein distribution nB:
terms with one (considered in ref. [ 5 ] ), but also with two and three powers in nB are present. Performing the lengthy, but straightforward calculation for the diagrams in fig. 1 at the euclidean symmetric point (Po = qo = ro = 0,
p2=qa=r2= - M 2 ) ,
we getZ1 = Z ~ =° _ _ -( 4 n ) 2
gaN
-[H_3j~2)_2j~2)
_ _ it(3) 4j~3)] _]_O(g4), 4 0 1 (4) where Z ~ =° is given in ref. [8] and need not be repeated here. The t e m p e r a t u r e dependence is contained in the functions H and j~m), which are functions o fa-tiM.
The function H contains the terms proportional to nB.Using the notation o f ref. [ 5 ], it is given by ~5
H = - ~ F o +~Go - 16G2, 2 7 ( 5 )
where e.g.
(" d4k 1
G o = - 4 n 2 M 2 J ~ n B ( I k o l ) ~ ( k 2) [ ( p + k ) 2 + i E ] [ ( k _ q ) 2 + i e ]
1
f f x d x 1
= 2 P
dY 3 e x p ( f l M x ) _ l x 2 ( y 2 + 3 ) _ 1 •
(6)0 0
The other functions Fo, G2 are found in ref. [ 5 ]. The n 2 and n 3 terms are contained in the function J(n m) , i.e.
f d4k 87~ f dx 1
J[m)-(4n)2M2 2 - n n n ' ~ ( I k ° l ) ~ ( k 2 ) 6 ( ( P + k ) 2 ) 6 ( ( k - q ) 2 ) = ~ o ~
[ e x p ( ~ ) - l ] m (7) and8n i x 2 d x 1
j ~ m ) _ 3X/3Y: 0 ~ [ e x p ( ~ ) - l l m ' (8)
where y 2 = ~a 2. These integrals are put into a f o r m relevant for high t e m p e r a t u r e expansion as shown in ref.
[ 14] in which the case m = 1 was investigated.
In order to obtain the renormalized coupling constant gR at finite T i n this renormalization scheme by
Z3/2
gR= --z-g,
(9)we need the gluon-field renormalization constant, which is Z3 = g T = O
g 2N
(47t 2 )-- ( 4 n ) ~ \ ~ S a a - - 3 F o - - F 2 +O(g4), (10)
as calculated in refs. [ 5,8 ], renormalizing the gluon p r o p a g a t o r at p2 = _ M e and t e m p e r a t u r e T.
Applying the renormalization group equations discussed by M a t s u m o t o et al. [2,5] with respect to the tem- perature T and the arbitrary subtraction scale M, one obtains from eq. (9) the running coupling constant as a function o f
T ( for fixed M),
g2( To)
g~t( T) = 1 + [2N/ ( 4n)2]g2 ( To) [-Qg,~e(M/T)
- . Q g l u e ( M / T o ) ] ' (11)~5 We remark that we do not agree with ref. [ 5 ] on the coefficient of Fo.
with
~Qglue (a
=M/T)
-- 292a2 ~ - F o - 3 F 2 -gG07 + 16G2 + 3j}2) _{_ 2j~2) + a a ,11(3) ..k 4T~3)_ 3°3 . For fixed M~ T (=Mo/To) the solution reads(12)
g~( To)
gZR(T)= 1 + [g~(To)/(4zc)2](22/3)Nln T/To ' (13)
with the familiar logarithmic t e m p e r a t u r e dependence [ 1 ].
3. The gluon-ghost vertex
As for T = 0 one m a y p e r f o r m the m o m e n t u m subtraction renormalization for the thermal ( 111 ) gluon-ghost vertex, instead o f the three gluon vertex, i.e. instead o f the definition, eq. (9), the renormalized coupling con- stant gR is then expressed by [ 13 ]
ZI/ZZ3 (14)
g R = Zl g '
where Z3 is the ghost wave function renormalization constant at finite T, given by ref. [ 5 ],
23 = ~T=O ..l_ (__~n)2 Fo + O (g4) , g 2N (15)
with Z3 r=° o f ref. [ 8 ]. In contrast to the case o f the three gluon vertex, the gluon-ghost renormalization constant ZI contains an ambiguity in the finite terms, which is due to the arbitrariness in the definition o f the tensorial structure o f this vertex at the one loop level [ 15 ]. Following ref. [ 15 ] it is expressed by introducing an addi- tional parameter, denoted by b.
The result - in the F e y n m a n gauge - is
g 2N [ F o + ( l + 2 b ) ( G o + ~ j l 2 ) + ~ j ~ 3 ) ) ] + O ( g 4 ) (16) 2, =2~=°(b) - ~
where Z ~ = ° ( b ) is found in ref. [ 15 ]. T h e t e r m s proportional to J lm) are due to n ~ ( m = 2, 3) contributions, which are not taken into account by F u j i m o t o et al. [ 5 ], who f u r t h e r m o r e only discuss the case b = 0.
The additional f r e e d o m in the p a r a m e t e r b m a y be used to adjust it such that the S l a v n o v - T a y l o r identities o f Q C D [ 13 ], namely ZI/Z3 = Z I / Z 3 , are satisfied in this renormalization scheme at a given finite T.
In a similar way to eqs. ( 1 1 ) and ( 12 ) the T dependence ofgR ( T ) is obtained with 27~2 7
- - ~ o I - - 1 2 a l )
g2ghost ( a ) = ~Sa2-~Fo - ½F2 - ( 1 + 2 b ) ( G o " 11~2) . ± r ~ 3 ) (17) which is in general different from £2glue, eq. ( 1 2 ) , showing the renormalization scheme dependence ofgR ( T ) at fixed M.
4. High temperature behaviour
Let us now examine the high t e m p e r a t u r e limit o f the/~ functions calculated in section 2 and 3 for the three
Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990 gluon vertex and the gluon-ghost vertex respectively. We focus on the large T, fixed momentum M, region. The leading behaviour is obtained from the small a behaviour of the J ~ ' ) (a) functions defined above in eqs. (7),
( 8 ). Since nB (x) ~ x - l for small x, one gets for m > n
oo
j(m) (y) ,~yl--n I d,x x n- 1 ( x Z . ~ _ y 2 ) - (1 + m ) / 2 ( 1 8 )
0
which by rescaling leads to
J~")(y)..~y-"
(19)for small y. The leading behaviour of g2glue (a) is thus a -3, i.e. the running coupling constant behaves as
gZ(V)~M3/r3
(20)for large
T/M.
Let us now discuss the high T expansion o f ~ , e and £2ghost up to logarithmic terms. The asymp- totic expansions of the Fo,2 and Go.2 functions are given in ref. [ 5 ]. Using Haber and Weldon's techniques [ 14 ], we get the corresponding expansion for the j(m) functions. They are8 n ( 3 x / 3 ~ ~ l n a + . . . )
a 2 2 a
8 / ~ ( l n ( a / 2 ~ ) a 2 6a 292 11: ) j ~ 2 , = ~ + 2_7~a+ ~2 in a + . . . .
8Jr ( ~ 3v/3 9 zcxf3 + ) j { 3 ) = ~ \ 4a 3 2a 2f- 2a 31na+... ,
- - ( ~ 3 3 ) 27C2-- 3 7t l 1
8zr [-nV#5 3 In + (21)
J~3) = ~ [~3-a3 -}- 2a 2 lZa 2 2 ~ - ~ - r g In a + . . . . From this, we obtain
~ g l u e =
25/1:26a
3 ~-O(~52 ) (22)and
/2gho,t l +2b zc2 (~52)
- - 2 a 3 +-O . ( 2 3 )
They are equal for the special value of the ghost parameter b = - ~ and therefore yield the same running coupling constant at large T.
We finally comment on the terms proportional to In
M / T
in the high T expansion of the renormalization constants. TheT= 0
parts contain terms InM/It,
where It is the scale in the applied dimensional regularization procedure. As is already known form ref. [ 1 6] these terms combine with InM~ T
to yield a InT/it
dependence in the wave function renormalization constants Z3 and 23, respectively. With the help of the expansions, eq.(2 1 ), we also find it true for the vertex renormalization constants Z~, Zl. However, we stress that for this can- cellation of the In M dependence, the terms proportional to n 2 and n 3 are crucial.
To summarize, the thermal fl function,
fir= T(d/dT)g 2 (T, M)/4~r 2,
behaves for large T and in the investi- gated renormalization scheme (for the three gluon vertex) as25~2N ( T ~ ' g2 T 2
f i r = 4 \ M J 4/r2 + 0 ( ~ ) " (24,
As a c o n s e q u e n c e o f t h e In M t e r m c a n c e l l a t i o n , t h e c o n s t a n t t e r m in this e x p a n s i o n is exactly t h e s a m e as for t h e u s u a l T = 01~ f u n c t i o n :
-~N(g 2/4ze
2)2.Acknowledgement
Partial s u p p o r t o f this w o r k by " P r o j e t s d e C o o p 6 r a t i o n et d ' E c h a n g e " ( P R O C O P E ) is gratefully a c k n o w l e d g e d .
Note added in proof
A f t e r c o m p l e t i o n o f t h i s work, w e r e c e i v e d a p a p e r b y N a k k a g a w a , N i e g a w a a n d Y o k o t a [ 17 ] w h e r e s i m i l a r c o n s i d e r a t i o n s are d e v e l o p e d .
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