IL NUOVO CIMENTO VOL. 110 A, N. 3 Marzo 1997 NOTE BREVI
Remark on the ansatz for IR behavior of gluon propagator in QCD
TRANHUUPHAT(*) and NGUYEN TUANANH(**)
International Centre for Theoretical Physics - P.O. Box 586, Trieste 34100, Italy (***)
(ricevuto il 3 Gennaio 1997; approvato il 27 Marzo 1997)
Summary. — It is shown that, in the IR region if the gluon propagator of QCD behaves like D(k) 4m2d(k), then the minimum of effective potential would
correspond to the vanishing value of m in the global color symmetry model (GCS) of QCD. The possibility to maintain the usefulness of this model is also suggested. PACS 12.38 – Quantum chromodynamics.
1. – It is well known that confinement and dynamical chiral symmetry breaking
(DCSB) are two crucial problems of quantum chromodynamics (QCD). They challenge all considerations of nonperturbative effects at large distances. There are actually two main approaches dealing with the infrared structure of QCD. The first one is the lattice QCD [1] which is not in the scope of this paper. The most fruitful method is to use the Schwinger-Dyson equation (SDE) for quark propagator S(k) [2]. The quark propagator provides us with both DCSB and confinement information. It is commonly accepted that confinement is characterized by the fact that S(k) has no pole on the timelike real axis of the complex-k2plane.
The main problem that all of current studies of SDE deal with is to make some ansatz for the gluon propagator D(k) in order confinement and DCSB at large distance to occur. At present two forms of D(k) in the infrared (IR) region are usually accepted, a) D(k) 4 M k4 , (1) b) D(k) 4m2d(k) . (2)
(*) Permanent address: Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam.
(**) Address: Hanoi National University; High College of Physics, Institute of Physics, Hanoi, Vietnam.
(***) E-mail: phatHicpt.trieste.it
TRAN HUU PHATandNGUYEN TUAN ANH 338
The form (1) has been considered for the first time in the pioneering paper of Pagels [3] and, subsequently, in many other publications [4-6].
The form (2) is treated to be the regularization of (1) and it leads to the desired quark propagator [7] and, as a consequence, is the main issue for a lot of considera-tions [8, 9]. In a series of papers [9] Cahill and his coworkers developed successfully the so-called global color symmetry model approximate to QCD, in which the IR behavior (2) of gluon propagator is the main ingredient. Their primary important result is to find out the quark propagator S(k), as the solution of SDE, for the gluon propagator D(k) modelling infrared slavery (2) and asymptotic freedom, namely,
A(s) 4
.
/
´
2 1 1 (112m2/s)1 /2 2 for s Gm2/4 , for s Dm2/4 , (3) B(s) 4.
/
´
(m22 4 s)1 /2 4 mD3 s[ ln (s/L2) ]1.8 for s Dm2/6 , for s Dm2/6 , (4) where S21(k) 4ikOA(k)1B(k) , s 4k2D 0 , m DB 0.42m . (5)In order to calculate the numerical value of the physically observed quantities such as the pion decay constant fp, the mass of mesons etc., the scale parameter m is set to
0.9 GeV.
2. – Now m should be considered to be the variational parameter, whose value is
defined by the minimum of effective potential.
It is well known that the effective potential in the two-loop approximation for GCS model reads V[S , D] 42i
d 4p ( 2 p)4tr [ ln S0 21(p) S(p) 2S 021(p) S(p) 11]1 (6) 1 i 2Cf(N) d4p ( 2 p)4 d4k ( 2 p)4tr [gmS(p) gnS(k) ] g mn D(p 2k) .Inserting (5) into (6) gives
V[A , B , D] 42inf
d4p ( 2 p)4lny
A(p) 1 B2(p) p2A(p)z
2 (7) 24 inf d4p ( 2 p)4g
1 2 p2A(p) p2 A 2 (p) 1B2(p)h
1 2 infCf(N) d4p ( 2 p)4 d4p ( 2 p)4D(p 2k)3 3 B(p) B(k) 2A(p) A(k) pQk [p2A2 (p) 1B2(p) ] [k2A2 (k) 1B2(k) ] .REMARK ON THE ANSATZ FORIRBEHAVIOR OF GLUON PROPAGATOR INQCD 339 Now the expressions (2), (3) and (4) for D(k), A(k) and B(k), respectively, are substitute into (7) and we arrive at the final result as follows
V(m2 ) 42nf
ds ( 4 p)2s lny
A(s) 1 B2(s) s2A(s)z
2 (8) 2 4 nf ds ( 4 p)2sg
1 2 sA(s) sA2(s) 1B2(s)h
2 2 nfCf(N) ds ( 4 p)2s sA2 (s) 2B2(s) [sA2 (s) 1B2(s) ]2 Q 3 16m 2 , which yields: for s Gm2/6, T(m2) 4 def V(m2) LQCD4 ( 4 p)2 nf 4u
m 2 LQCD2v
2 1 36g
ln 3 2 37 36h
; for s Dm2/6 , the m2-dependence of V(m2) is shown in fig. 1.It is clear that the minimum of effective potential V(m2) corresponds to m 40. This implies that the ansatz (2) for gluon propagator is not compatible with the approximation made for GCS model.
The preceding result is unchanged if the vacuum energy density is normalized as the difference between perturbative and nonperturbative vacuum energy density.
From the stability of the vacuum configuration defined by SDE and the minimum of effective potential at m 40 it follows that, in general, the IR behavior of gluon
TRAN HUU PHATandNGUYEN TUAN ANH 340
propagator given by
D(k) 4mF(k; n)
would yield the small value for m at the minimum of effective potential if
F(k ; n) Kd(k) as n KQ ( or 0) .
3. – The result obtained in the preceding paragraph indicates that ansatz (2) is not
consistent with the abelianization made for GCS model [10]. As we know, this model is probably the simplest approximation to QCD which involves the confinement. Now we like to point out that the usefulness of GCS model could be maintained if, instead of (2), another IR behavior of the gluon propagator would be assumed like
G(k) 4md(k)1nu(L2
2 k2) (9)
where m , n are two variational parameters, u is the step function and L is an UV cutoff.
It is known that (8) was first derived by [1] assuming the existence of constant condensate field which leads to the violation of local gauge invariance. However, if we disregard the mechanism yielding (8); instead, we adopte (8) a priori as an ansatz, we could consistently establish the GCS model, which is the simplest generalization of the Nambu-Jona-Lasanio model, including confinement.
* * *
We would like to sincerely thank Profs. M. A. VIRASORO, L. BERTOCCHI and S. RANDJBAR-DEAMI and the International Centre for Theoretical Physics for the hospitality extended to us. The support of Vietnam National Programme for Fundamental Research and the World Laboratory is acknowledged with thanks.
R E F E R E N C E S
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