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Model of nonperturbative running coupling in QCD
Aleksey I. Alekseev
Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia
Abstract
The modified ``physical'' running coupling constant is constructed
on the base of the analytic running coupling constant obtained from
the standard perturbation theory approximation up to the four loop
order. The possibility of providing quark confinement singular infrared
behavior of the running coupling constant and presence of term con­
nected with dynamical gluon mass is studied. In the framework of
the approach fixing the string tension parameter and normalizing for
example at M Ü define completely the running coupling. At that such
quantities as the gluon condensate, the nonperturbative gluon vacuum
energy density, the dynamical gluon mass m g , and the nonperturbative
scale k 0 turn out to be fixed and correspond to another approaches.

The talk is based mainly on:
A.I. Alekseev and B.A. Arbuzov, Yad. Fiz. 61, 314 (1998) [Phys. Atom. Nucl.
61,264. (1998)]; A.I. Alekseev and B.A. Arbuzov, Mod. Phys. Lett. A 13,
1447 (1998); A.I. Alekseev, hep­ph/9808206;
2­loop --- A.I. Alekseev, Phys. Rev. D 61, 114005 (2000);
3­loop --- A.I. Alekseev, Yad. Fiz. 65, 1722 (2002) [Phys. At. Nucl. 65,1678
(2002)];
4­loop --- A.I. Alekseev, J. Phys. G 27, L117 (2001); A.I. Alekseev, Few­Body
Systems 32, 193 (2003).
The idea of the analytic approach:
P.J. Redmond, Phys. Rev. 112, 1404 (1958);
N.N. Bogolubov, A.A. Logunov, and D.V. Shirkov, Zh. '
Eksp. Teor.Fiz. 37,
805 (1959) [Sov. Phys. JETP 10, 574 (1960)].
Application to QCD: D.V. Shirkov and I.L. Solovtsov, JINR Rapid Comm.
2[76]­96, 5 (1996); I.L. Solovtsov and D.V. Shirkov, Teor. Mat. Fiz. 120,
482 (1999) [Theor. Math. Phys. 120, 1210 (1999)]; ... .
1

1 Running coupling models for Q 2 ? 0 from one­loop PT
Matching of the solutions for large and small Q 2 . E.g., B.A. Arbuzov, E.E. Boos,
K.Sh. Turashvili, Z. Phys. C 30, 287 (1986) ) k 0 ' 1 GeV;
fi­function definition, RG­eq. solving;
... .
Logical steps towards our model:
ff (1)
s (Q 2 ) =
4ú
b 0
1
ln(Q 2 =\Lambda 2 )
; (1)
ff (1)
an (Q 2 ) =
4ú
b 0
2
6 4
1
ln(Q 2 =\Lambda 2 )
+
\Lambda 2
\Lambda 2 \Gamma Q 2
3
7 5 ; (2)
ff (1)
an+c (Q 2 ) =
4ú
b 0
2
6 4
1
ln(Q 2 =\Lambda 2 )
+
\Lambda 2
\Lambda 2 \Gamma Q 2 +
c\Lambda 2
Q 2
3
7 5 ; (3)
ff (1) (Q 2 ) =
4ú
b 0
2
6 4
1
ln(Q 2 =\Lambda 2 )
+
\Lambda 2
\Lambda 2 \Gamma Q 2 +
c\Lambda 2
Q 2 +
(1 \Gamma c)\Lambda 2
Q 2 +m 2
g
3
7 5 ; (4)
where
m 2
g =
\Lambda 2
c \Gamma 1
: (5)
2

Define q¯q potential as
V (r) = \Gamma
4
3
Z d n q
(2ú) 3 exp (iqr)
4úff(q 2 )
q 2
fi fi fi fi n=3
: (6)
At large distances
V (r)j r!1 ' a 2 r; (7)
a 2 = oe --- string tension,
(3=2)oe = (4ú=b 0 )\Lambda 2
1 ; \Lambda 2
1 = c\Lambda 2 : (8)
At a ' 0:42 GeV (L.D. Soloviev, Phys. Rev. D 58, 035005 (1998); Phys. Rev.
D 61, 015009 (2000)), b 0 = 9 one has \Lambda 1 ' 435 MeV.
At small distances
\DeltaV (r)j r!0 ' oe 0 r; (3=2)oe 0 = (4ú=b 0 )(c \Gamma 1)\Lambda 2 ; (9)
(V. Zakharov's possibility). Let us mention the well known elegant and eco­
nomical Richardson model with UV as (1) and IR as (3)
ff (1)
Rich (Q 2 ) =
4ú
b 0
1
ln(1 +Q 2 =\Lambda 2
1 )
; (10)
3

multiplicatively improved model (A. Nesterenko)
N ff (1)
an (Q 2 ) =
4ú
b 0
Q 2 \Gamma 1
Q 2 ln(Q 2 =\Lambda 2 )
; (11)
and ``frozen'' in the IR coupling model by Yu. Simonov
ff (1)
Sim (Q 2 ) =
4ú
b 0
1
ln((M 2
B +Q 2 )=\Lambda 2 )
(12)
(MB ¸ 1 GeV).
NPT contributions of the models tend to zero in the UV but insufficiently fast.
Introducing the term corresponding to the dynamical gluon mass (J.M. Corn­
wall, Phys. Rev. D 26, 1453 (1982)) we come to the model (4) where the
coefficients, defined according to the principle of minimality of NPT contribu­
tions in the PT region (A.A.,B.A.), decrease at large Q 2 fast enough.
4

From the Running Coupling to the Analytic Running Coupling
The behavior of the QCD running coupling ff s (Q 2 ) is defined by the renormal­
ization group equation
Q 2 @ff s (Q 2 )
@Q 2 = fi(ff s ) = fi 0 ff 2
s + fi 1 ff 3
s + fi 2 ff 4
s + fi 3 ff 5
s + O(ff 6
s ); (13)
where the coefficients
fi 0 = \Gamma 1
4ú
b 0 ; b 0 = 11 \Gamma 2
3
n f ;
fi 1 = \Gamma 1
8ú 2 b 1 ; b 1 = 51 \Gamma 19
3
n f ;
fi 2 = \Gamma 1
128ú 3 b 2 ; b 2 = 2857 \Gamma 5033
9
n f +
325
27
n 2
f ;
fi 3 = \Gamma
1
256ú 4 b 3 ; b 3 =
149753
6
+ 3564i 3
\Gamma
0
@ 1078361
162
+
6508
27
i 3
1
A n f +
0
@ 50065
162
+
6472
81
i 3
1
A n 2
f +
1093
729
n 3
f :
Here n f is the number of active quark flavors and i is the Riemann zeta­
function, i 3 = i(3) = 1:202056903::: .
5

The integration of the RG equation yields
1
ff s (Q 2 )
+
fi 1
fi 0
ln ff s (Q 2 ) +
1
fi 2
0
`
fi 0 fi 2 \Gamma fi 2
1
'
ff s (Q 2 ) +
1
2fi 3
0
`
fi 3
1 \Gamma 2fi 0 fi 1 fi 2
+fi 2
0 fi 3
'
ff 2
s (Q 2 ) +O(ff 3
s (Q 2 )) = \Gammafi 0 ln(Q 2 =\Lambda 2 ) + ¯
C: (14)
The integration constant is represented here as a combination of two constants
\Lambda and ¯
C. It is convenient to introduce constant C,
C = ln(\Gammafi 0 ) + (fi 0 =fi 1 ) ¯
C:
Iteratively solving the equation for ff s (Q 2 ) at L = ln(Q 2 =\Lambda 2 ) !1 and invert­
ing the result one obtains (slight generalization of K.G. Chetyrkin, B.A. Kniehl,
and M. Steinhauser, Phys. Rev. Lett. 79, 2184 (1997))
ff s (Q 2 ) = \Gamma
1
fi 0 L
8 ? !
? : 1 +
fi 1
fi 2
0 L
(ln L + C) +
fi 2
1
fi 4
0 L 2
Ÿ
(ln L + C) 2 \Gamma (ln L +C)
\Gamma1 +
fi 0 fi 2
fi 2
1
3
5 +
fi 3
1
fi 6
0 L 3
2
4 (ln L + C) 3 \Gamma 5
2
(ln L + C) 2 \Gamma
0
@ 2 \Gamma 3fi 0 fi 2
fi 2
1
1
A (ln L +C)
+
1
2
\Gamma fi 2
0 fi 3
2fi 3
1
3
7 5 + O
0
@ 1
L 4
1
A
9 ? =
? ; : (15)
6

Within the conventional definition of \Lambda as \Lambda MS one chooses C = 0.
Let us introduce the function a(x) = (b 0 =4ú)ff s (Q 2 ), where x = Q 2 =\Lambda 2 . Then
one can write
a(x) =
1
ln x
\Gamma b
ln(ln x) +C
ln 2 x
+ b 2
2
6 6 4
(ln(ln x) + C) 2
ln 3 x
\Gamma ln(ln x) + C
ln 3 x
+
Ÿ
ln 3 x
3
7 7 5
\Gammab 3
2
6 6 4
(ln(ln x) + C) 3
ln 4 x
\Gamma
5
2
(ln(ln x) +C) 2
ln 4 x
+ (3Ÿ + 1)
ln(ln x) + C
ln 4 x
+
¯
Ÿ
ln 4 x
3
7 7 5 :
(16)
where the coefficients b, Ÿ, and ¯
Ÿ are equal to
b = \Gamma
fi 1
fi 2
0
=
2b 1
b 2
0
;
Ÿ = \Gamma1 +
fi 0 fi 2
fi 2
1
= \Gamma1 +
b 0 b 2
8b 2
1
;
¯
Ÿ =
1
2
\Gamma fi 2
0 fi 3
2fi 3
1
=
1
2
\Gamma b 2
0 b 3
16b 3
1
: (17)
At x ' 1 the perturbative running coupling is singular. The analytic approach
removes all these nonphysical singularities in a regular way.
7

The analytic running coupling is obtained by the integral representation
a an (x) =
1
ú
1
Z
0
doe
x + oe
ae(oe); (18)
where the spectral density ae(oe) = Ima an (\Gammaoe \Gamma i0). According to the analytic
approach to QCD we adopt that Ima an (\Gammaoe \Gamma i0) = Ima(\Gammaoe \Gamma i0), where
a(x) is the perturbative running coupling. It is clear that dispersively­modified
coupling of form (2) has analytical structure which is consistent with causality.
Then one obtains the spectral density
ae(oe) = ae (1) (oe) + \Deltaae (2) (oe) + \Deltaae (3) (oe) + \Deltaae (4) (oe); (19)
where
ae (1) (oe) =
ú
t 2 + ú 2 ; (20)
\Deltaae (2) (oe) = \Gamma b
(t 2 + ú 2 ) 2
Ÿ
2útF 1 (t) \Gamma
`
t 2 \Gamma ú 2
'
F 2 (t)
–
; (21)
\Deltaae (3) (oe) =
b 2
(t 2 + ú 2 ) 3
Ÿ
ú
`
3t 2 \Gamma ú 2
' `
F 2
1 (t) \Gamma F 2
2 (t)
'
\Gamma 2t
`
t 2 \Gamma 3ú 2
'
F 1 (t)F 2 (t)
\Gammaú
`
3t 2 \Gamma ú 2
'
F 1 (t) +t
`
t 2 \Gamma 3ú 2
'
F 2 (t) + úŸ
`
3t 2 \Gamma ú 2
'–
; (22)
8

\Deltaae (4) (oe) = \Gamma b 3
(t 2 + ú 2 ) 4
Ÿ`
t 4 \Gamma 6ú 2 t 2 + ú 4
' `
F 3
2 (t) \Gamma 3F 2
1 (t)F 2 (t)
'
+4út
`
t 2 \Gamma ú 2
' `
F 3
1 (t) \Gamma3F 1 (t)F 2
2 (t)
'
\Gamma 10út
`
t 2 \Gamma ú 2
' `
F 2
1 (t) \Gamma F 2
2 (t)
'
+5
`
t 4 \Gamma 6ú 2 t 2 + ú 4
'
F 1 (t)F 2 (t) + 4ú (1 + 3Ÿ) t
`
t 2 \Gamma ú 2
'
F 1 (t)
\Gamma (1 + 3Ÿ)
`
t 4 \Gamma 6ú 2 t 2 + ú 4
'
F 2 (t) + 4ú¯Ÿt
`
t 2 \Gamma ú 2
'–
: (23)
Here t = ln(oe),
F 1 (t) j 1
2
ln(t 2 + ú 2 ) + C; F 2 (t) j arccos
t
p
t 2 + ú 2
; (24)
For the 1 --- 4­loop cases the spectral density of the analytic running coupling
is shown in Fig. 1 (C = 0; n f = 3).
9

Figure 1: The spectral density of the analytic running coupling up to four loop order
10

Figure 2: The higher loop order corrections for the spectral density
Integrating with the replacement (\Gamma1; +1) ! (\GammaT ; +T ) leads to the rela­
tive error ¸ 1=T , and at large T it is important not to lose the higher loop
11

contributions. In Fig. 2 the higher loop corrections to the spectral density are
shown. We shall obtain another more effective method for precise calculation
of ff an (Q 2 ) which is not connected with the numerical integration.
The analytic running coupling is divided into perturbative component and non­
perturbative one in an explicit form (A.A.).
a an (x) = a pt (x) + a npt
an (x): (25)
The expansion of the last term in the inverse powers of x was obtained
a npt
an (x) =
1
X
n=1
c n
x n
; (26)
where
c n = \Gamma1 + bn [1 + C \Gamma fl \Gamma ln(n)] \Gamma 1
2
b 2 n 2
2
6 41 \Gamma ú 2
6
+ Ÿ
+ (1 +C \Gamma fl \Gamma ln(n)) 2
–
+
1
6
b 3 n 3
2
4 2 +
5
2
Ÿ + ¯
Ÿ \Gamma 2i 3
+ (1 + C \Gamma fl \Gamma ln(n)) 3 + 3(1 + C \Gamma fl \Gamma ln(n))
0
B @1 \Gamma ú 2
6
+ Ÿ
1
C A
3
7 5 :
12

Table 1: The dependence of c n and loop corrections on n for the 1 --- 4­loop cases, n f = 3
n c 1\Gammaloop
n \Delta 2\Gammaloop
n \Delta 3\Gammaloop
n \Delta 4\Gammaloop
n c 2\Gammaloop
n c 3\Gammaloop
n c 4\Gammaloop
n
1 ­1.0 0.33405 0.01608 ­0.07825 ­0.66595 ­0.64987 ­0.72812
2 ­1.0 ­0.42724 0.19624 ­0.37379 ­1.42724 ­1.23101 ­1.60480
3 ­1.0 ­1.60196 ­0.63626 ­1.28115 ­2.60196 ­3.23823 ­4.51937
4 ­1.0 ­3.04517 ­3.48651 ­5.07338 ­4.04517 ­7.53168 ­12.60506
5 ­1.0 ­4.68801 ­9.19185 ­16.30462 ­5.68801 ­14.87987 ­31.18449
Here fl is Euler constant, fl ' 0:5772. We can see that power series (26) is
uniformly convergent at x ? 1 and its convergence radius is equal to unity.
For numerical evaluation of the coefficients c n we choose the MS scheme values
of Ÿ, ¯
Ÿ and assume that C = 0. For n f = 3 the result is given in Table. 1.
13

Figure 3: Relative error of the approximation of a an with a npt
an approximated by only the
first term of the series, as a function of x 1=2 = Q=\Lambda for the 1 --- 4­loop cases at n f = 4.
The dash­dotted line, dotted line, dashed line and solid line correspond to the 1­loop,
2­loop, 3­loop and 4­loop cases, respectively.
14

Figure 4: The analytic running coupling a an and its perturbative component a pt and non­
perturbative component a npt
an
as functions of x = Q 2 =\Lambda 2 for the 1 --- 4­loop order cases.
Here n f = 3.
15

Figure 5: The analytic and perturbative couplings ff an (Q 2 ), ff pt (Q 2 ) for the 1 --- 4­loop
order cases. The normalization conditions are ff (n f =5) (M 2
Z
) = 0.1181, M Z = 91.1882
GeV; ff (n f =5) (m 2
b
) = ff (n f =4) (m 2
b
), m b = 4.3 GeV; ff (n f =4) (m 2
c
) = ff (n f =3) (m 2
c
), m c = 1.3
GeV.
16

2 The QCD coupling constant model
ff an (Q 2 ) ) ff(Q 2 ),
ff(Q 2 ) = ff an (Q 2 ) +
4ú
b 0
2
6 4
c\Lambda 2
Q 2 \Gamma
d\Lambda 2
Q 2 +m 2
g
3
7 5 ; (27)
with c, d, m g j m \Lambda \Lambda to be fixed. Analogously to the one­loop case we demand
d = c + c 1 ; m 2
\Lambda = \Gammac 2 =(c + c 1 ): (28)
The dynamical gluon mass
m g = m g (\Lambda; n loops ) = \Lambda
v u u u u t
\Gammac 2 \Lambda 2
\Lambda 2
1 + c 1 \Lambda 2
: (29)
for the effective theory with n f = 3 is shown in Fig. 6. At \Lambda = 375 ##V m g '
0.6 GeV.
17

Figure 6: The dynamical gluon mass m g as a function of \Lambda
18

Let us consider three variants of the normalization conditions for ff(Q 2 ; \Lambda; n loops )
and its components.
ffl ff(M 2
Ü ) = 0:323; M Ü = 1:777GeV;
ffl ff(M 2
Ü ) = 0:35; M Ü = 1:777GeV;
ffl ff(M 2
Z ) = 0:1181; M Z = 91:1882GeV:
Parameters \Lambda, m g , c, d are given in Table. 2. In Figs. 7, 8, 9 the dependencies
of ff, ff an and ff pt on Q are given. For the first two cases the loop stability for
the model is higher than that for perturbative case (apart from 1­loop case,
e.g., at Q = 0.7 GeV \Delta pt = 0.11, \Delta an = 0.006, \Delta = 0.06). The last case is
inconsistent with n f = 3.
Superposition of the figures can be elucidatory.
19

Table 2: Parameters \Lambda pt (MeV), \Lambda an (MeV), \Lambda (MeV), dynamical gluon mass m g (MeV) and parameters
of the model c, d for 1 -- 4­loop cases. n f = 3. Normalization conditions are: I. ff(M 2
Ü ) = 0.323, M Ü =
1.777 GeV; II. ff(M 2
Ü ) = 0.35, M Ü = 1.777 GeV; III. ff(M 2
Z ) = 0.1181, MZ = 91.1882 GeV
Norm. I 1­loop 2­loop 3­loop 4­loop
\Lambda pt 204.65 381.89 351.07 344.82
\Lambda an 240.46 599.22 494.71 505.31
\Lambda 204.65 383.43 351.89 345.90
m g 109.01 580.34 416.06 473.66
c 4.5247 1.2890 1.5304 1.5839
d 3.5247 0.6230 0.8806 0.8558
Norm. II 1­loop 2­loop 3­loop 4­loop
\Lambda pt 241.78 431.53 400.74 391.97
\Lambda an 296.73 748.60 612.48 627.67
\Lambda 241.80 435.68 402.87 394.73
m g 161.51 902.79 621.20 715.75
c 3.2413 0.9984 1.1676 1.2162
d 2.2413 0.3324 0.5178 0.4881
Norm. III 1­loop 2­loop 3­loop 4­loop
\Lambda pt 246.98 741.10 640.95 646.72
\Lambda an 247.11 743.24 642.30 648.27
\Lambda 246.98 741.10 640.95 646.72
m g 170.17 i 1562.91 i 1637.59 i 1562.21
c 3.1066 0.3450 0.4613 0.4531
d 2.1066 ­0.3209 ­0.1886 ­0.2750
20

Figure 7: ff pt (Q 2 ), ff an (Q 2 ) # ff(Q 2 ) as functions of Q for 1 --- 4­loop cases. For all of them the normalization
condition is ff(M 2
Ü ) = 0.323, M Ü = 1.777 GeV
21

Figure 8: ff(Q 2 ), ff an (Q 2 ) and ff pt (Q 2 ) as functions of Q for 1 --- 4­loop cases. Normalization for all of
them: ff(M 2
Ü ) = 0.35, M Ü = 1.777 GeV
22

Figure 9: ff pt (Q 2 ), ff an (Q 2 ) and ff(Q 2 ) as functions of Q for 1 --- 4­loop cases. Normalization at MZ ,
ff(M 2
Z ) = 0.1181, MZ = 91.1882 GeV. Here n f = 3
23

3 Gluon condensate and fixing of the nonperturbative param­
eters
Up to the quadratic approximation in the gluon fields, the gluon condensate
K = lim
x!y
! vac j
ff s
ú
: F a
¯š (x) F a
¯š (y) :j vac ?
ú
3
ú 3
1
Z
0
dk 2 k 2 ff npt (k 2 ) )
3
ú 3
1
Z
0
dk 2 k 2 ff npt
reg (k 2 ): (30)
The following procedure is our definition of the regularized perturbative and
nonperturbative parts of ff(k 2 ):
ff(k 2 ) = ff pt (k 2 ) + ff npt
an (k 2 ) +
4ú
b 0
2
6 4
c\Lambda 2
k 2
\Gamma d\Lambda 2
k 2 + m 2
g
3
7 5 (31)
= ff pt (k 2 ) + ff npt (k 2 ) = ff pt
reg (k 2 ) + ff npt
reg (k 2 ); (32)
where
ff pt
reg (k 2 ) = ff pt (k 2 ) + `(k 2
0 \Gamma k 2 )ff npt
an (k 2 ) (33)
has no power corrections at k 2 !1 and
ff npt
reg (k 2 ) = `(k 2 \Gamma k 2
0 )ff npt
an +
4ú
b 0
` c\Lambda 2
k 2
\Gamma d\Lambda 2
k 2 +m 2
g
'
(34)
24

at k 2 !1 has power corrections coinciding with that of ff npt (k 2 ). One obtains
K (\Lambda; k 0 ) =
12\Lambda 4
ú 2 b 0
8 ? !
? :
1
X
n=1
c n+2
n
k \Gamma2n
\Lambda \Gamma c 2 ln
2
6 4
`
\Lambda 2
1 =\Lambda 2 + c 1
' k 2
\Lambda
\Gammac 2
3
7 5 \Gamma c 1 k 2
\Lambda
9 ? =
? ; : (35)
For one­loop case
K (\Lambda; k 0 ) =
12\Lambda 4
ú 2 b 0
ae
ln
Ÿ`
\Lambda 2
1 =\Lambda 2 \Gamma 1
'
(k 2
\Lambda \Gamma 1)
–
+ k 2
\Lambda
oe
; (36)
k \Lambda = k 0 =\Lambda. The results of the numerical study are given in Fig. 10. Note,
the convexity property exists for multiloop cases. The maximum of gluon
condensate corresponds to the minimum of nonperturbative vacuum energy
density. The condition of its minimality allows one to fix parameters of the
model.
25

Figure 10: Fourth root of the gluon condensate K 1=4 as function of \Lambda at k 0 from 0.5 GeV (lower curves) to
1.5 GeV (upper curves) with 0.2 GeV step for 1 --- 4­loop cases. At that n f = 3. Dashed lines correspond
to k 0 = 0.777, 0.876, 0.895, 0.896 for 1 --- 4­loop cases, respectively. The ``standard'' value of the gluon
condensate K 1=4 = 0.33 GeV is indicated by dots
26

Conclusions
Table 3: \Lambda (MeV), m g (MeV), k 0 (GeV), c, d, ff(M 2
Ü ) for the conventional string tension oe 1=2 =0.42 GeV
and gluon condensate K 1=4 = 0.33 GeV
1­loop 2­loop 3­loop 4­loop
\Lambda 375 379 382 359
k 0 0.777 0.876 0.895 0.896
m g 638 562 527 535
c 1.35 1.32 1.30 1.45
d 0.346 0.650 0.647 0.730
ff(M 2
Ü
) 0.448 0.32 0.34 0.33
A preliminary consideration of the model in the presence of heavy quarks with
matching of the coupling for different n f at corresponding thresholds leads to
the encouraging results.
27