





Gell-MannLow Function in QCD

I. M. Suslov
Kapitza Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 117973 Russia
e-mail: suslov@kapitza.ras.ru



2
The Gell-MannLow function in QCD (g) (g = g /162, where g is the coupling constant in the Lagrangian)
is shown to behave in the strong-coupling region as g, where 13 and ~ 105.




Recently, using an algorithm proposed in [1, 2] for 1 a a 1 a
summing divergent series of perturbation theory, we L =  --- F F ( )2

4   2
------ A
determined the Gell-MannLow function of 4 theory
[1, 2] and QED [3]. Here, this algorithm is applied to + D^
+ c
(  ),
QCD, for which previous attempts provided ambiguous f f a a g f abcb A (3)
results [4]. f
Fa = a a
 + g f abc Ab Ac ,
1. Information about all terms of the perturbation   A A 
series can be obtained by interpolating its first terms with D
^ = i a
( ig ATa
 ),
the Lipatov asymptotic behavior [5]. The first four
terms in the expansion of the Gell-MannLow function where ,
Aa a
f, and are gluon, quark, and ghost fields,
for QCD are known in the MS scheme [6]: respectively; Ta and f abc are the generators of the fun-
damental representation and structure constants of the
Lee algebra, respectively; is the gauge parameter and
g
( )
= gN
= g2
+ g3
+ g4
+ ...,
N 2 3 4 (1) subscript f specifies the type of quarks.
N
= 0 2. The asymptotic behavior in perturbation theory
2
g = g /162, was discussed for YangMills fields [79] and QCD
[10, 11], but the results are not sufficiently general.
1
where Below, this deficiency will be partially compensated.
The pre-exponential factor of the most general func-
2 38 tional integral for QCD involves M gluon, 2L ghost, and
 = 11  --- N ,  = 102  ------ N ,
2 3 f 3 3 f 2K quark fields, i.e.,

Z = DADDD D A(x )... A(x )

2857 5033 325 MLK 1 M
 = ------------  ------------ N + --------- N2,
4 2 18 f 54 f (4)
 (y )(y )...(y )(y ) z
( )
1 1 L L 1
 z
( )... z
( ) z
( ) exp (S{ A, , , , }),
149753 1 K K
 = ------------------ + 3564 3
( ) (2)
5 6 where vector indices immaterial for the further consid-
eration are omitted. The substitution A B/ g
reduces the Euclidean action to the form
1078361 6508
 --------------------- + ------------ 3
( ) N
162 27 f S{ A, , , , }

(5)
S{B} 4
50065 6472 + d
2 1093 3 ------------- x Q
^ + D^

+ --------------- + ------------ 3
( ) N + ------------ N . f f
162 81 f 729 f g2 f


1 Our view on the renormalon contributions was formulated in [3].
Here, Nf is the number of types of quarks and
g is the The existence of renormalon singularities in QCD was neither
coupling constant in the QCD Lagrangian proven nor disproven, and we shall assume that they are absent.


0327


328

+
and the integration over the fermion fields yields expansion term NFNg2N 2 or FNg2N when determin-
ing it from any vertex. The expansion of the function
Z = (1/g)M DAB(x )...B(x )
MLK 1 M
has the same form [5]. Since g = g2 /162, the coeffi-
 G(y , y )...G(y , y ( , z ( , z (6)
)G z )...G
~ z ) cients of series (1) have the asymptotic behavior
1 1 L L 1 1 K K

 N f ( )
detQ
^ detD^
( ) exp { S{B}/g2
 } + ..., 11 N  N

= const N + 4N c f
+ ------------------------------ . (8)
N c
6
where G and
G
~ are the Green's functions of the opera-
This result for N
tors
Q
^ and
D
^ , and ellipsis means terms with other pair- c = 2 and Nf =0 agrees with the result
obtained in [7].
ings. It is important that S{B}, G, and
G
~ are indepen-
3. Series (1) is nonalternating, and there exists the
dent of
g . Functional integral (6) is determined by the well-known problem of correct interpretation of the
YangMills action, and the asymptotic behavior of its poorly defined Borel integral. In particular, the principal-
expansion coefficients in
g are calculated by the Lipa- value prescription for it is not necessarily valid [16]. The
tov method [5]. For the saddle-point configuration,
g ~ definition of the gamma function can be rewritten as
N1/2, where N is the order of perturbation theory.
Therefore, each field A(xi) in the pre-exponential factor z
( ) = 
d xz  1, = 1, (9)
xe x

in Eq. (4) provides the factor N1/2, whereas other fields i i
do not give N-dependent factors. The dependence of the i i
Ci
expansion coefficients on N is determined according to
[10]; it differs from the result for the quark correlation where C1, C2, ... are arbitrary contours beginning at the
function only by the factor NM/2. The Nth-order contri- origin and tending to infinity in the right half-plane.
bution to Z The Borel transformation of series (1) yields
MLK has the form

[ 
ZMLK ] g2N = const 162
( ) N
N b  1
g
( ) = dxexx 0 B g
( x),

(7)
( ) i
 M 11 N  N

N + ----- + 4 N c f
+ ------------------------------ i
g2N Ci (10)
2 c 6

for even M (N B z
( ) = B zN, B
N
=
c is the number of colors), and this N N (N + b )
------------------------,
expression should be multiplied by the additional factor 0
N = 0

g N1/2 for odd M. 2 where b0 is an arbitrary parameter. If the Borel trans-
Using the result for the functional integral and form B(z) has singularities in the right half-plane, con-
applying the algebra of factorial series [15], one can tours Ci are no longer equivalent and cannot be reduced
easily obtain the result for any quantity. Let F to the positive semiaxis, as was possible in Eq. (9). For
N g2 N
be
the Nth-order contribution to the vacuum integral (M = this reason, the summation result depends on the choice
L = K = 0). Then, the general term of asymptotic beha- of i and Ci.3 We bypass this problem as follows. For the
power behavior of the Borel transform at infinity, i.e.,
vior (apart from a coefficient) has the form NFNg2N
for when B(z) ~ z, we have
the gluon propagator , FNg2N
for the ghost propagator
g
( ) = , g
G and quark propagator G + g
f, NFN g2 N 1
for the gluon (11)

ghost vertex f + and g
( ) = g , g ,
3 and gluonquark vertex , N2F g2 N 1
3 N

for the three-gluon vertex +
3, and N3FN g2 N 2
for the where the exact relation between and depends on
four-gluon vertex 4. In view of the generalized Ward the chosen
2 i and Ci, but ~ in general case. There-
identities
3 ~ 3G and 4 ~ , the leading contribu-
3 fore, index can be determined and can be esti-
tions to the asymptotic behaviors of 3 and 4 are mated by summing series (1) for negative values of g.
cancelled, and the invariant charge has the general 4. According to the algorithm developed in [1, 2],
2 The term M/2 in the argument of gamma function in Eq. (7) is the resummation of the alternating series with the coef-
related to the number of external fields, 4Nc is half the number of
zero modes, and the term 11(N 3
c  Nf)/6 arises because certain Results for different i and Ci differ by terms proportional to exp(a/g),
zero modes are soft, under more rigorous consideration, and must and such nonperturbative contributions must generally be added
be nontrivially integrated. For the quark correlation function, to the Borel integral. For correctly chosen i and Ci, these contri-
Eq. (6) involves divergences, which were removed in [10, 11] by butions are absorbed by the Borel integral and should not be explic-
the doubtful method [14]. These divergences are absent for M 1. itly taken into account.


329

ficients behaving asymptotically as caN(N + b) pro-
vides the convergent series with the coefficients

N
K
U = B aK 
( 1) CK  1, (12)

N K N  1

K = 1

whose behavior for large N


U = U  1 , U = ---------------------------------------- (13)
N N
a () b
( + )
0

determines the parameters of asymptotic form (11).
The coefficient function is interpolated via the formula

= caNNb~(N + b b~
 )
N

(14)
 A A
1 1
+ -------------- 2
+ --------------------- + ...
N N
~
 (N N~
 )2

by breaking the series and choosing the coefficients AK
from agreement with Eq. (2). The optimal parameter-

ization of the Lipatov asymptotics with
b~ = b  1/2 is

taken [2], and parameter N
~ is used to control the stabil- Fig. 1. Quantities 2, eff, and U~ = U
(b0 + 2) vs. b0 for
ity of results and to optimize the procedure. the optimal interpolation with N
~ = 1.58 and averaging
Similar to QED, the parameter c of the Lipatov
asymptotics is unknown. In the previous paper [3] it was interval 23 N 35. Function U
~ (b
0) for | U
~ | < 10 is
determined in the course of interpolation. In the case under shown schematically. The minima at b0 = 15.4 and 15.9 are
consideration, such procedure gives large uncertainty in the treated as the satellites of the principal minimum at b0 =
results, which is not reduced by optimization. So, interpo- 15.5, because they, together with the latter minimum,
are shifted with varying parameters.
lation was carried out for some predermined c value,
which then varied from 105 to 1.4 Under this variation, the
results change only slightly compared to other un-
certainties. Below we present results obtained for N value in the first (from large b
c = 0) minimum of 2 is clos-
3, N est to the exact index 15, because the leading cor-
f = 0, and c = 105. rection to asymptotic form (13) vanishes at b
Fitting U 0 = '.
N by the power law and consid- Different estimations of index
ering the dependence of 2 on
N
~ , we separate the inter- are close to each other when
N
~ is close to the
val 0.5 N
~ 2.0, where the 2 values are minimal. optimal value
N
~ = 1.58 (Fig. 1) and become inconsistent when
This procedure determines the set of interpolations
consistent with the power behavior of U N
~ go away from its optimal value.
N. The typical
behavior of 2 and effective values U and as func- The result for index cannot immediately be taken
tions of b0 (Fig. 1) indicates that 15. Indeed, U as final. First, the large negative index can imitate an exponen-
determined by Eqs. (13) changes sign at b0 =  15.5. tial. Second, for = 0, 1, 2, ..., the leading contribu-
For this b0 value, 2 has a minimum, because the lead- tion to the asymptotic behavior of UN vanishes due to
ing contribution UN  1 vanishes due to the pole the pole of () [see Eq. (13)], and the observed result
of gamma function in Eq. (13), and we have power can correspond, e.g., to ' rather than
behavior U [2]. In view of these circumstances, we sum a series for
N ~ N'  1 corresponding to the first correc-
tion to the asymptotic behavior of (g) [we assume that n
the function W(g) = g s g
( ) and increase integer
(g) = g +
g' + ... for large g ]. The

' eff parameter ns until the observed index W = + ns
becomes positive. The results (Fig. 2a) conclusively demon-
4 Parameter c is equal to the product of the square of the t'Hooft con- strate that we observe a large negative index rather than an ex-
stant cH in the expression for one-instanton contribution [12, 13] ponential. This index is noninteger because in the case = n
2
( c ~ 105 and 104 for N we would observe the behavior shown in the insert.
H f = 0 and 3, respectively) and the Each point in Fig. 2a is obtained by independent opti-
dimensionless integral of the instanton configuration. The latter
factor can be rather large (characteristic scale is 82). mization in N
~ . The optimal N~ value decreases mono-


330

tonically with increasing ns. Uncertainty in the results
is primarily attributed to the dependence at the lower
bound of averaging interval Nmin N Nmax. The upper
points in Fig. 2a correspond to small Nmin and to 2 ~
106 in the minima. As Nmin increases, decreases mono-
tonically until 2 reaches a value of ~103 (lower points).
With a further increase in Nmin, the pattern of 2 minima
becomes indistinct, and the uncertainty of the results
increases considerably. We admit some further
decrease in until the required values 2 ~ 10 are
reached, and take this into account in the course of error es-

timation. Uncertainty in parameter
is of several
order of magnitude (Fig. 2b), but the most probable
value is ~105, which is consistent with the basic array
of data. Thus, we have

=  13  2,
105
(15)

for Nf = 0. One has = 12  3 and the same most prob-
able value
for N
f =3 (while the total scatter is
=
1107). The stability in the results against a change in
the summation procedure testifies that their uncertainty
is adequately estimated. Some underestimation of the
error is possible due to the nonlinear effects [3] and
Fig. 2. (a) Index in the case when the asymptotics is reached slowly.
W obtained by summing the series for
n
function W(g) = g s (g) vs. n Large uncertainty in _\infty corresponds to comparatively
s for various averaging inter-
vals N small uncertainty in the function itself. The characteristic
min N Nmax: ( ) for Nmin = 22 + ns and Nmax = 35
+ ns and ( , , , , , , ) for sequentially increasing scale where one-loop law

2g2 is matched with asymptotic be-
N
min by one unit; (b) parameter s as a function of ns. havior (11) appears to be g* ~ 2, and
changes by
four orders of magnitude as g* changes by a factor of

two. The sign of
is indeterminate in negative W
region, because error in is large and the factor () in
Eq. (13) is alternating, but this sign is definitely nega-
tive in positive W region (large ns values). Figure 3
shows (solid line) the behavior of function for g < 0
and (dashed line) the analytic continuation to positive g
values, where the behavior is qualitatively the same, but
the sign of asymptotic function (11) can change.5 Nev-
ertheless, the behavior of the effective coupling con-
Fig. 3. stant as a function of the length scale L is rather
definite (Fig. 4). In the one-loop approximation, g(L)
has a pole at L = L 0 = 1/QCD (dashed line in
Fig. 4). For the obtained function (Fig. 3), g(L)
increases near L0 up to ~g* and then either (for > 0)
becomes constant or (for < 0) increases as (lnL)0.07,
which is practically indistinguishable from a constant.

In the weak-coupling region, interaction V(L)
between quarks is described by he modified Coulomb

law /
g2(L) L, and the sharp increase in g(L) near L = L0
testifies to the tendency to confinement. In the strong-
coupling region, the relation between V(L) and g(L) is

5 In particular,



= c

os
for the principal-value interpreta-
Fig. 4. tion of the Borel integral.


331

unknown, but the close in spirit result was obtained by Wil- 2. I. M. Suslov, Zh. ksp. Teor. Fiz. 120, 5 (2001) [JETP
son [17] for the lattice version of QCD: 93, 1 (2001)].
3. I. M. Suslov, Pis'ma Zh. ksp. Teor. Fiz. 74, 211 (2001)
ln 3g2 a
( ) [JETP Lett. 74, 191 (2001)].
V(L) = --------------------- L, g a
( ) 1 (16)
2 4. D. I. Kazakov, in Proceedings of the Conference
a QUARKS-80, Sukhumi, 1980 (INR, Moscow, 1981),
where a is the lattice constant. From the condition that p. 52.
the result is independent of a, the function in the 5. L. N. Lipatov, Zh. ksp. Teor. Fiz. 72, 411 (1977) [Sov.
strong-coupling region is estimated as (g) ~ glng [18], Phys. JETP 45, 216 (1977)].
which is, however, incorrect. The cross size of the 6. T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin,
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QCD is equal to ~a, which is
considerably higher than its actual physical size 7. E. B. Bogomolny and V. A. Fateyev, Phys. Lett. B 71, 93
~1/ (1977).
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( ) Sci. Rev., Sect. A 2, 247 (1980).
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 ~ 20, and, because of (1996).
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This work was supported by INTAS (grant no. 99- 41, 441 (1998)].
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1. I. M. Suslov, Pis'ma Zh. ksp. Teor. Fiz. 71, 315 (2000)
[JETP Lett. 71, 217 (2000)]. Translated by R. Tyapaev



