


JETP Letters, Vol. 71, No. 6, 2000, pp. 217221. Translated from Pis'ma v Zhurnal ksperimental'nooe i Teoreticheskooe Fiziki, Vol. 71, No. 6, 2000, pp. 315321.
Original Russian Text Copyright  2000 by Suslov.


FIELDS, PARTICLES,
NUCLEI


Gell-MannLow Function in the j4 Theory
I. M. Suslov
Kapitza Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 117973 Russia
e-mail: suslov@kapitza.ras.ru
Received November 1, 1999; in final form, February 2, 2000


An algorithm is proposed for the determination of the asymptotics of a sum of a perturbation series from the
given values of its coefficients in the strong-coupling limit. When applied to the 4 theory, the algorithm yields
the (g) g behavior with 1 at large g for the Gell-MannLow function.  2000 MAIK "Nauka/Interpe-
riodica".

PACS numbers: 11.10.-z


Many problems in theoretical physics require and arrived at the conclusion that the Gell-MannLow
advance in the strong-coupling region. The best known function in the 4 theory with the action functional
of them concern the dependence of the effective cou-
pling constant g on the distance scale L; the problems
{ } 1
= ( )2 162
of electrodynamics at ultrashort distances and confine- S d4x --- -----------g4
+ (4)
2 4!

ment are among them. The dependence of g on L in the
renormalizable theories is determined by the equation behaves at large g as 0.9g2, which differs only in the
coefficient from the one-loop result 1.5g2 valid at g
dg/d ln L = (g) (1) 0; similar behavior was obtained for (g) by Kubyshin
[5]. If this result is valid,1 then the 4 theory is self-con-
and generally requires information on the Gell-Mann tradictory. This conclusion seems to be strange from
Low function (g) for arbitrary g [1]. Over many years, the viewpoint of solid-state applications: a reasonable
the problem of reconstruction of the -function seemed model of a disordered system [7, 8], well-defined in the
to be absolutely hopeless because the information on continuous limit, is mathematically reduced to the 4
this function was provided solely by perturbation the- model. Moreover, it was recently proved [9] that there
ory, which allowed the calculation of the first several are no renormalization singularities in the 4 theory;
terms of the expansion and this can be treated as evidence for the self-consis-
tency of the theory.
This paper is aimed at revising the results obtained
(g) = (g)N
N in [4, 5]. We start with the same premises as in [5], i.e.,
with the known first four coefficients of the -function
N = 0 (2) expansion [6, 10]
= g2 g3
 + ... + (g)N + ...,
2 3 N
154.14
= = 0. (g) 3
= ---g2 17
------g3
 + ----------------g4 2338
------------g5
 + ..., (5)
0 1 2 6 8 16

Lipatov [2] proposed a method allowing the calculation and the Lipatov asymptotics with the first-order correc-
of the tion term calculated in [11]:
N asymptotics at large N, which was found to be
factorial for most problems:
1.096 4.7
= ------------- + ...
N7/2 N! 1  ------- . (6)
as N
= caN( N + b) caN Nb  1N! (3) N
N 162

Matching the Lipatov asymptotics (3) with the first The method is different from [4, 5] in that a direct rela-
N
coefficients provides information on all terms of the tion between the (g) asymptotics and the expansion
series and makes it possible to approximately recon- coefficients is used and the interpolation is carried out
in an explicit form.
struct the -function, but this requires a special proce-
dure for the summation of divergent series [3]. Kaza- 1 The authors of [4] do not insist on their statement and emphasize
kov et al. [4] attempted to implement this procedure that it has a tentative character (see also [6]).


0021-3640/00/7106-0217$20.00  2000 MAIK "Nauka/Interperiodica"





218 SUSLOV

1. Let us formulate the problem of reconstruction of ance made for the singularity B(u) ~ (1  u) at the
the -function asymptotics point u = 1. The reexpansion of series (9) gives the fol-
( lowing relation between UN and BN:
g) = , g (7)
g U = B ,
0 0
from the coefficients N of series (2) that grow accord- N
ing to factorial law (3) and are assumed to be numeri- (12)
1
K K  1
cally specified. As in the case of the introduction of crit- U = B --- C ( N 1).

N K a N  1
ical indices in phase-transition theory, the slow (loga- K = 1
rithmic) corrections to law (7) are considered to be As a result, we arrive at the following simple algorithm:
beyond accuracy. the coefficients BN are calculated from given N [cf.
Treating the sum of series (2) in the Borel sense, we Eq. (8)] and recalculated to UN according to Eq. (12);
use a modified definition of the Borel transform (g): then the UN coefficients at large N are fitted to the power
law UN = UN  1, whose parameters determine and
( b  1
g) = dxexx 0 B(gx), B(g)
= B 
( g)N,
according to Eq. (10). The value is convenient to
N calculate by treating U as a function of b0 and deter-
0 N = 0 (8) mining the slope of linear dependence U
(b0 + ) at
B N
= ------------------------, small b0 + ; this provides an independent estimate for
N (N + b )
0 the index from the root of the U(b0) function.
where b 2. The authors of the majority of works formulated
0 is an arbitrary parameter that is conveniently
used for optimizing the summation procedure [3]. As the algorithm in such a way as to avoid mention of the
was assumed in [3] and proved recently in [9], the Borel coefficients N for intermediate N values. Such an
transform is analytical in the complex g plane with a cut approach is conceptually inconsistent, because a finite
from 1/a to . To analytically continue B(g) from the number of coefficients and their asymptotics can ensure
convergence circle |g| < 1/a to arbitrary complex g val- the construction of a function with any prescribed
ues, a conformal transformation g = f(u) mapping the behavior at infinity.2 The problem can be reasonably
plane with the cut onto the unit circle is used; in this formulated if all N are approximately defined; in this
case, the reexpansion of B(g) in u powers case, the function (g) can be reconstructed within a
certain accuracy. For this reason, the interpolation and
estimation of its accuracy is the necessary step in solv-
B(g) = B (g)N
N g = f (u) ing the problem. Of course, this is possible only on the
N = 0 assumption that
(9) N is a smooth function of N.
The interpolation is convenient to carry out for the
B(u) = U uN
reduced coefficient function
N
N = 0 A A
F N
= ------- = 1 1
+ ------ 2
+ ------ + ... AK
+ ------- + ..., (13)
gives a series convergent at arbitrary g values. We N as N N2 NK
restrict ourselves to the analytic continuation of B(g) to N
the positive semiaxis [which is sufficient for the inte- which varies within the finite limits and has a regular
gration in Eq. (8)] and use a modified conformal trans- expansion in 1/N. Retaining in the series a finite num-
formation g = (u/a)/(1  u) mapping the plane with cut ber of terms and adjusting the coefficients AK to the
(-1/a, ) onto the plane with cut (1, ). This removes known FN values, one obtains the desired interpolation
the g = 1/a singularity to infinity, while the g = sin- formula. Its accuracy is
gularity becomes the nearest to the origin and deter-
mines the following asymptotics for the U A (N  L )(N  L  1)...(N  L)
N coeffi- m + m + 1
F 0
--------------------- 0 0
----------------------------------------------------------------------------, (14)
cients: N m + m + 1 ( )...
N 0 L L  1 L0

U = ------------------------------------------- N  1, N . (10) if the interpolation is performed using m known values
N
a ()(b + )
0 F , F , ..., F
L L + 1 L (m = L  L0 + 1) for m0 known coef-
0 0

This result can easily be obtained by representing the ficients A1, A2, ..., A . Estimate (14) is based on the
m0
expansion coefficients as fact that series (13) is asymptotic [12], so that the error
of its approximation by trincation is of the order of the
du B(u) first omitted term, while the error of interpolating the
U = -------- ------------ (11)
N 2
 iuN+1 2
C A function of a factorial series has the same asymptotics for coef-
ficients (3), but with different c value [8]; the statement formu-
and deforming the contour C enveloping the point u = 0 lated in the text can easily be proved by choosing an appropriate
in such a way that it passes around the cut with allow- linear combination of several functions.


JETP LETTERS Vol. 71 No. 6 2000


GELL-MANNLOW FUNCTION 219

~
UN
(a) b0 = b0 = 10.0 (b) b0 =
5.0

20 40 5.0

1.0
30
1.0
0
10 0.5 20 0.5
0 0.8
0.5 10 1.0
0.9 1.2
0 0
1.1
10 20 30 N 10 20 30 N
1.3 1.4
U 1.5 10 U

1.7 5 1.6
10 2 1.8 20
0
0 1.7
1.9 30
2 5
20 40 1.8
1.0 1.5 b
1.0 1.5 b 0
0


as
Fig. 1. Plots of U
~ = U
N
N (b0 + 2) vs. N at different b0 values for the parametrization of the asymptotics in the form (a) =
N

as
caNNb  1N! and (b) = caN(N + b). The inserts show U
N as a function of b0.



(m + m0 + 1)-order polynomial by the (m + m0)-order ing from the slope at b0 = , one obtains for the
polynomial can be calculated exactly. In the case under asymptotics of the -function
consideration, L0 = 2, L = 5, and m0 = 1, so that FN is (g) 8g g . (15)
determined by the coefficient A6, which can be esti-
mated by factorial-law extrapolation of the found A situation occurring at the alternative ways of
A interpolation is shown in Fig. 1b, where the parametri-
1,
..., A5 values [12].
as
The ambiguity of the interpolation procedure is zation = caN
N (N +b) is used. There is also seen
manifested, in particular, in the possibility to differ- curve flattening, but it is not as distinct as in the preced-
ently parametrize the asymptotics [4, 13], e.g., as ing case. The processing of the curves under the
caN(N + b), caNNb  1N! etc. The asymptotics in the assumption that = 1 yields the b0 dependence of U
(see insert in Fig. 1b) going through zero at b0 = 1.3,
instanton calculations [2] has the form c~a~N Nb~ NN, which corresponds to = 1.3. Hence, the results show
which is very close to the Lipatov parametrization a substantial inconsistency. Curve processing with the
caNNb  1N! obtained from the former by applying the power-law dependence yields an value slightly
Stirling formula, whose accuracy is better than 10% exceeding unity (different for different b0), but in this
even at N = 1. With this respect, the parametrization case U turns to zero at b0 = 0.8 (at this value, the
caNNb - 1N! is "natural," whereas its representation in increase in the curves in Fig. 1b changes to a decrease),
alternative functional forms requires additional leading to the same inconsistency. Correspondingly, the
assumptions [e.g., N b for caN(N + b)]. result for the (g) asymptotics becomes less defined,
(g) 24g and = 0.81.3.
Figure 1a shows the coefficients U
~ N = U
N (b0 + 2) as
(normalized so that they have a finite limit at b Parametrizing the asymptotics as =
N
0 )
for the natural parametrization caNNb  1N!. At large N, 
these coefficients distinctly tend to the constant values caN Nb b~
 1 (N + b~ + 1) and expanding FN in inverse
(excepting the curves for b0 1 and b0 2, for which powers of (N  N0), one obtains a two-parameter (b~ and
the large parameters retard the attainment of asymptot- N0) set of the interpolation formulas. The table presents
ics), which corresponds to the value = 1. The U vs. the results for several such interpolations, for which the
b0 curve goes through zero at b0 = 1.03 (see insert in distinctions in the interpolation curves approximately
Fig. 1a), which gives another estimate = 1.03 demon- fit the error range estimated by Eq. (14) for the natural
strating excellent consistency of the results. Determin- interpolation. With allowance made for the uncertain-


JETP LETTERS Vol. 71 No. 6 2000


220 SUSLOV

FN portion U
~ N 1.1(N  1) (dashed line) corresponding to
1.0
(g) a (g) 1.1g2 dependence close to the results obtained
in [4, 5]. This portion is insensitive to changes in b0 and
0.5 0.9 g2 to the interpolation procedure and can pretend to the
1.06 g1.9 role of the true asymptotics, provided that the results
2 for N > 10 are treated as being due to the interpolation
0 50 100 1
200 2.99 g3/2 errors. But such is actually not the case, because the sta-
N 3 bility of this portion is caused by the presence of a char-
8 g acteristic dip in the reduced coefficient function FN at
N 10 (see insert in Fig. 2). If one models this dip by
100 4 setting F3 = F4 = ... = F10 = 0, then the result U
~ N
1.5(N  1) determined by the first nonvanishing coeffi-
cient F2 (see curve for b0 = ) is obtained at N 10 for
1.5 g2 all b0; this is close to the actual situation.3 Such model-
0 5 10 15 g ing of the dip demonstrates that the one-loop law 1.5g2
for the -function extends up to g ~ 10. More precisely
Fig. 2. Qualitative behavior of the Gell-MannLow function (see footnote 3), the result valid in the interval 1 g
in the 4 theory. Curves 1, 2, 3, and 4 are obtained in [4], [5], 10 is given by the function derived in [4, 5] and yielding
[14], and this work, respectively. The insert shows the coef- the value (g) 90 for g = 10 (see Fig. 2), in accordance
as as with [14]. Asymptotics Eq. (15) matches well with the
ficient function F
N = N/ with the = caN(N + b)
N N indicated value, providing indirect support to the opti-
parametrization convenient for an analysis of Eq. (12); the mistic estimation for the accuracy.
dependence on the interpolation procedure is immaterial on
this scale. Although the available information allows only a
rough estimation for the Gell-MannLow function, one
ties, the index is virtually independent of the particu- can state with assurance that it is nonzero at finite g val-
lar interpolation; systematic deviations occur only for ues and its behavior at g is compatible with the
some "extreme" cases for which the interpolation curve assumption that the 4 theory is self-consistent. The
is partially beyond the error range. The interpolation substitution of Eq. (15) in Eq. (1) yields the g(L) L
dependence with 8 at small L, which is slightly
with b~ = 0 and N0 = 0, which we treated as "natural" modified if the index is other than unity or if logarith-
from computational considerations, stands out as the mic branching is present.
most self-consistent algorithm. Therefore, the corre-
sponding result (15) should be considered as the most The results obtained allow an understanding of why
reliable. For a fixed interpolation, its error is less than the numerical simulations on a lattice indicate that the
0.05 for the index and 10% for 4
, which is an opti- theory is "trivial" (see [15] and references therein):
mistic estimate for the accuracy. The error caused by because of the absence of zeros of the -function, the
the interpolation ambiguity is seen from the table: it can g(L) interaction always decreases with distance; and,
be as great as several tenths for the index, whereas the owing to the extended one-loop law, the behavior is
value can differ from Eq. (15) by a factor of 23. indistinguishable from the trivial in a wide range of
parameters [at g 300, for the most popular charge
3. Let us consider the behavior of the -function at definition when the term with interaction in Eq. (4) has
finite g values. For N < 10, Fig. 1 demonstrates a linear the form g4/4].

This work was supported by the INTAS (grant
Table no. 96-0580) and the Russian Foundation for Basic
Research (project no. 00-02-17129).
Interpola- Interpola-
tion with tion with
N0 = 0 b~ = 0 REFERENCES

1. N. N. Bogoliubov and D. V. Shirkov, Introduction to the
b~ = 3.5 24 0.81.3 N0 = 0.5 5.4 1.11.6
Theory of Quantized Fields, 3rd ed. (Nauka, Moscow,
1984, 4th ed.; Wiley, New York, 1980, 3rd ed.).
b~ = 1.5 14 1.01.1 N0 = 0.3 16 0.81.2

3
b~ = 0 8.1 1.0 If one sets F3 = F4 = ... = F10 = , then Eq. (12) yields a linear
dependence with the slope 1.5(1  /F2) for b0 = b  p with inte-
b~ = 1.5 10 1.01.1 N0 = 0.3 4.3 0.91.0 ger p in the interval p + 2 N 10. For = 0.2 (see Fig. 2); the
correct slope 1.1 and a weak dependence on b0 occur in the inter-
b~ = 2.5 2.7 1.51.7 N0 = 0.5 2.3 0.91.0 val 1 < b0 < 10.


JETP LETTERS Vol. 71 No. 6 2000


GELL-MANNLOW FUNCTION 221

2. L. N. Lipatov, Zh. ksp. Teor. Fiz. 72, 411 (1977) [Sov. 10. F. M. Dittes, Yu. A. Kubyshin, and O. V. Tarasov, Teor.
Phys. JETP 45, 216 (1977)]. Mat. Fiz. 37, 66 (1978).
3. J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39, 11. Yu. A. Kubyshin, Teor. Mat. Fiz. 57, 363 (1983).
95 (1977); Phys. Rev. B 21, 3976 (1980).
12. I. M. Suslov, Zh. ksp. Teor. Fiz. 117 (2000) (in press).
4. D. I. Kazakov, O. V. Tarasov, and D. V. Shirkov, Teor.
Mat. Fiz. 38, 15 (1979). 13. V. S. Popov, V. L. Eletskioe, and A. V. Turbiner, Zh. ksp.
5. Yu. A. Kubyshin, Teor. Mat. Fiz. 58, 137 (1984). Teor. Fiz. 74, 445 (1978) [Sov. Phys. JETP 47, 232
6. A. A. Vladimirov and D. V. Shirkov, Usp. Fiz. Nauk (1978)].
129,
407 (1979) [Sov. Phys. Usp. 22, 860 (1979)]. 14. A. N. Sissakian, I. L. Solovtsov, and O. P. Solovtsova,
7. M. V. Sadovskioe, Usp. Fiz. Nauk 133, 223 (1981) [Sov. Phys. Lett. B 321, 381 (1994).
Phys. Usp. 24, 96 (1981)]. 15. A. Agodi, G. Andronico, P. Cea, et al., Mod. Phys. Lett.
8. I. M. Suslov, Usp. Fiz. Nauk 168, 503 (1998) [Phys. Usp. A 12, 1011 (1997); .
41, 441 (1998)].
9. I. M. Suslov, Zh. ksp. Teor. Fiz. 116, 369 (1999) [JETP
89, 197 (1999)]. Translated by R. Tyapaev





JETP LETTERS Vol. 71 No. 6 2000



