Bled Workshops in Physics Vol. 10, No. 1 p.l

Nambu-Jona-Lasinio model from QCD
B. A. Arbuzov
Skobeltsyn Institute of Nuclear Physics of MSU, 119992 Moscow, Russia
The NJL model [1-3] proves to be effective in description of low-energy hadron physics. The model starts with effective chiral invariant Lagrangian
where ^ is the light quark doublet (u, d). This interaction is non-renormalizable, so one is forced to introduce an ultraviolet cut-off A. Thus we have at least two arbitrary parameters
to be adjusted by comparison with real physics. It comes out that after such adjustment (and similar procedure for the vector sector and for the s-quark terms) we obtain satisfactory description of light mesons and their low-energy interactions.
However, the problem how to calculate the parameters Gt and At from the fundamental QCD was not solved for a long time. The main problem here is to find a method to obtain effective interactions from fundamental gauge interactions, e.g. QCD.
There are also non-local variants of the NJL model, in which one introduces a form-factor F(qt) into the effective interaction of the type (1) instead of a cut-off A. In this case again there was no regular method to obtain this function F and one has to make an arbitrary assumption for the choice.
Our goal is to formulate a regular approach, which allows to obtain a unique solution for the form-factors and other necessary quantities of the effective interactions. In particular we apply this approach to the NJL effective interaction.
The approach is based on the Bogoliubov compensation principle [4,5].
The main principle of the approach is to check if an effective interaction could be generated in a chosen variant of a renormalizable theory.
In previous works [6-12] the Bogoliubov compensation principle was applied to studies of spontaneous generation of effective non-local interactions in renormalizable gauge theories. In view of this one performs an "add and subtract" procedure for the effective interaction with a form-factor. Then one assumes the presence of the effective interaction in the interaction Lagrangian and the same term with the opposite sign is assigned to the newly defined free La-grangian.
(1)
Gi ; Ai ;
The QCD Lagrangian with two light quarks is (u and d)

k=1 1

, Fa Fa	(2)
Let us assume that a non-local NJL interaction is spontaneously generated in this theory. We use the Bogoliubov "add and subtract" procedure to check the assumption. We have
L = Lo + Lint ,
1 ^ ~ ■ ~ -	Gl	, ,.„T.„b„ ■	T . T .A
L° = I	~~	- m0i|ji|j + ^Y (i|jTbY5i|ji|jTby5i|j -
+ ^r (VV^ VV^ +	- \ F^F^ , (3)
1
4
_ pa ra _ ra pa \	/¿\
a i ijV 1 o 1 o / •	(4)
Here the notation e.g. ^ril> 4> 4> 4> means the corresponding non-local vertex in the momentum space
i(2n)4 G i uQ (p) uQ (q) ub (k) ub (t) F(p, q, k, t) 6 (p + q + k +1) , (5)
where F(p, q, k, t) is a form-factor, p, q, k, t are respectively incoming momenta and a, b are isotopic indices of corresponding quarks.
Let us consider expression (3) as the new free Lagrangian L0, whereas expression (4) is the new interaction Lagrangian Lint. The compensation equation demands fully connected four-fermion vertices, following from Lagrangian L0, to be zero. The equation has evidently
1.	a perturbative trivial solution Gi = 0;
2.	but it might also have a non-perturbative non-trivial solution, which we shall look for.
In the first approximation we use the following assumptions.
1.	Loop numbers 0,1, 2. For one-loop case only a trivial solution exists.
2.	Procedure of linearizing over form-factor, which leads to linear integral equations.
3.	Intermediate UV cut-off A, results not depending on the value of this cut-off.
4.	IR cut-off at the lower limit of integration by momentum squared q2 at value m2.
5.	Only the first two terms of the 1 /N expansion (N = 3).
6.	We look for a solution with the following simple dependence on all four variables:
U	i u M + V2 + vl + vj\
F(Pi, V2, Vs, P4) = F I -^—3-1 I .	(6)
Then we come to the following integral equation (see [8])
, ,	3G2 U , x 3 p2 \ (G? + 6Gi G2)N
A2 2
\ 6x log x
1 rx	3 rx
(y2-3p2)F1(y)dy + -
H	2
yFi (y) dy +
H
x2 — 3p' 6~
2
Fi (y)
dy +
yFi (y) dy + x log x
Fi (y) dy +
(logy + ^F! (y)dy+ (2A2-^x)
y log yFi(y) dy +
yFi (y) dy
3
Fi (y) ¿y - 2
log A2
yFi (y) dy + x

JH
G2N
Fi (y) dy) I ; p = m0 ; x = p2 ; y = q2 ; (7)
2n2
4N 2tt2 V1 + 2N )
Fi(y) dy
The equation has the following solution decreasing at infinity
F!(z) = C! G«(z|1, 1 1 0, a, b) +C2 G«(z|1, 1 b, a, 1 0, )
1 1
2' 1,
2' 2
+ C3 G^(z|1,0,b, a, I I) ,
b
1 -VI -64u0
1 + a/1 - 64u0
(8)
4	4
where x = p2, y = q2 are respectively external momentum squared and inegra-tion momentum squared,
is a Meijer G-function [13],
(G2 + 6Gi G2)N
P
16n4
nmni „ iai , ... , ap
Gpq ( Z |b1,...,bp

The constants Ct are defined by the boundary conditions
3 G2 P

8n2 2
Fi (y) dy = 0
y Fi (y) dy = 0
y2 Fi(y) dy = 0. (9)
These conditions and the condition A = 0 lead to the cancellation of all terms in equation (7) being proportional to A2 and log A2. So we have the unique solution. The values of the parameter u0 and the ratio of two constants Gt are also fixed
u0 = 1.92•10-8 ~ 2•10-
Gi =^G2.
(10)
y
"X
"X
x
8
We would draw attention to a natural appearance of a small quantity u0 .So G i and G2 are both defined in terms of m0.
Thus we have the unique non-trivial solution of the compensation equation, which contains no additional parameters. It is important that the solution exists only for positive G2 and due to (10) for positive G1 as well.
Now we have the non-trivial solution, which lead to the following effective Lagrangian
+ gs^taA^ - iF^F^
-	^Tby5TjjTf'rby5Tjj - iWiM^
-	-y- (^ViaTH-rVi^ + TKby5yMiHTby5y|aiJ^ •	(ll)
Here g2/4n = as(q2) is the running constant depending on the momentum variable. We need this constant in the low-momenta region. We assume that in this region as (q2) may be approximated by its average value as. The possible range of values of as is from 0.40 up to 0.75.
Thus we come to the effective non-local NJL interaction which we use to obtain the description of low-energy hadron physics [7,8,11]. In this way we obtain expressions for all quantities under study.
Analysis shows that the optimal set of low-energy parameters corresponds to as = 0.67 and m0 = 20.3 MeV. We present a set of calculated parameters for these conditions including the quark condensate,the parameters of the a-meson as well as the parameters of p and ai -mesons:
as = 0.673; m0 = 20.3 MeV;
mn = 135 MeV; mff = 492 MeV; Tff = 574 MeV fn = 93 MeV; m = 295 MeV; < q q >= - (222 MeV)3 ;
(244 MeV)2
Mp = 926.3 MeV(771.1 ± 0.9);	rp = 159.5 MeV(149.2 ± 0.7);
Ma, = 1174.8 MeV( 1230 ±40);	ra, = 350MeV(250 - 600); r(a^ —> an)/rai = 0.23 (0.188 ± 0.043).
G1 = TTTTTTTTTTI9=3.16.
The upper line here is our input, while all other quantities are calculated from these two fundamental parameters. The overall accuracy may be estimated to be on the order of 10 - 15%. The worst accuracy occurs in the value of Mp (20%). It seems that the vectors and the axials need further study.
Important result: average value of as ~ 0.67 agrees with calculated low-energy as [9]. So we have consistent description of low-energy hadron physics with only one dimensional parameter, e.g. m0 or fn.
References
1.	Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961).
2.	T. Eguchi, Phys. Rev. D 14, 2755 (1976).
3.	M. K. Volkov and D.Ebert, Yad. Fiz. 36,1265 (1982).
4.	N. N. Bogoliubov. Physica Suppl. 26,1 (1960).
5.	N. N. Bogoliubov, Quasi-averages in problems of statistical mechanics. Preprint JINR D-781, (Dubna: JINR, 1961).
6.	B. A. Arbuzov, Theor. Math. Phys. 140,1205 (2004).
7.	B. A. Arbuzov, Phys. Atom. Nucl. 69,1588 (2006).
8.	B. A. Arbuzov, M. K. Volkov, I. V. Zaitsev. J. Mod. Phys. A, 21, 5721 (2006).
9.	B.A. Arbuzov, Phys.Lett. B656, 67 (2007).
10.	B.A. Arbuzov, Proc. International Seminar on Contemporary Problems of Elementary Particle Physics, Dubna, January 17-18, 2008, Dubna, JINR, 2008, p. 156.
11.	B. A. Arbuzov, M. K. Volkov, I. V. Zaitsev. J. Mod. Phys. A, 24, 2415 (2009).
12.	B. A. Arbuzov, Eur. Phys. Journal C 61, 51 (2009).
13.	H. Bateman and A. Erdelyi, Higher transcendental functions, Vol. 1. New York, Toronto, London: McGraw-Hill, 1953.