





Gell-MannLow Function in QED

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



The Gell-MannLow function (g) in QED (g is the fine structure constant) is reconstructed. At large g, it
behaves as g with 1 and 1.
PACS numbers: 12.20.Ds; 11.10.Gh


Recently a present author [1, 2] developed a meth- Integration with respect to the fermion fields gives
od of summation divergent perturbation series with
arbitrary coupling constants. With this method, infor- Z , = DAA(x )...A(x )G(y , z )...G(y , z )

mation about all terms of the series is obtained by the M L 1 M 1 1 L L
interpolation of the known first terms with their Lipatov  det(i  m + eA)
asymptotics [3]. In this paper, this method will be used to (2)
reconstruct the Gell-MannLow function in QED.
 1
exp --- 4x(A  A)2
d
+ ...,
Lipatov's method [3] is based on the saddle-point 4

calculation of path integrals near instanton configura-
tions and is being questioned because of the possible where G(x, x') is the Green's function for the Dirac
renormalon contributions [4]. Formally, the asymptotic operator
behavior of perturbation theory is determined by the
singularity nearest to the origin in the Borel plane. (i  + e )G(x, x') = (x  x'), (3)
m A
Whereas the presence of instanton singularities is
beyond question, the existence of renormalon singular- and the ellipsis stands for the terms with other pairings
ities has never been proved, which is acknowledged by of (y ) and z
( ) . Estimations show that the quantity
the most active advocates of this direction [5]. Having i k
been proved in [6], the absence of renormalon singular- eA(x) is large for the saddle-point configuration and
ities in the 4 theory casts doubt on the renorma- the asymptotic form of the determinant at e i can
lon concept as a whole, although similar proofs are be used, because the growth rate is maximal at imagi-
lacking for other field theories. In such a situation, nary e values [9]:
it is possible to assume that the renormalon singulariti-
es are absent. e4 4
ln det(i  + e ) = ----------- d x A2
( )2. (4)
m A

1. The asymptotics of perturbation theory for QED 122
was discussed in the late 1970s [79]; all fundamental
problems were solved by Bogomolny and Fateyev This result is not gauge invariant and is only valid for a
[8, 9], but no results for specific quantities was obtain- specifically chosen gauge; it can be obtained for slowly
ed. Below, we partially fill this gap. varying fields or for configurations with a sufficiently
high symmetry [9]. Taking Eq. (4) into account, a path
The vertex with M photon and 2L electron free lines integral with effective action
is determined by the path integral

4 1 2
Z = ( )... A(x )(y ) z
( )...(y ) S { A} = d x ---
(  )2 4---g2(A )2
 ,
M, DAD DA x
L 1 M 1 1 L
eff
4 A A 3
(5)

 4 1
z
( ) exp  x ---(  )2
d e2
L 4 A A
(1) g = 4
------,


appears in Eq. (2); the asymptotic form of perturbation
-+ (i  m + e A) .
theory for this action can be found by Lipatov's
method. Its structure is determined by the homogeneity


0191


192 SUSLOV

properties of the action [10]; when g2 is used as a cou- High-order coefficients in the expansion of the
pling constant, these properties are the same as in the 4 Gell-MannLow function (g) = 
( g)N coin-

theory, and the general asymptotic term has the form N
N
cide, except for a constant factor, with the coefficients
c SN
0 (N + b)g2N, where S0 is the instanton action. In for the invariant charge [3], which is determined in the
actuality, the expansion is in arbitrary integer (not only electrodynamics by the quantity gD, where D is the
even) g powers, and the general term is c SN/2 photon propagator (M = 2, L = 0). The general asymp-
0 (N/2 + totic term is D
b)gN.1 Taking the value of instanton action into account, N(g)N + 1 ~ NZN(g)N + 1 or NZN  1(g)N ~
one obtains for the Nth-order contribution to the vac- N1/2ZN(g)N, from whence it follows that
uum integral (M = 0, L = 0) [8]:
+

= const  4.886N N 12
---------------- , N . (10)
33/2
N/2 +
N 2
Z (g)N = const -------- N r
------------- (g)N, (6)
N
43 2
The same result is obtained if the invariant charge is
where r = 11 is the number of zero modes including determined through the triple vertex (M = 1, L = 1). In
four translations, a scale transformation, and six four- this case, the dominant contribution to the asymptotic
dimensional rotations (instanton corresponds in sym- expression comes from the amputation of the photon
metry to a rigid body of an irregular shape). line.
In the general case, the functional form of the result 2. The following four terms of the -function expan-
can be found by structural calculations described in sion in the MOM scheme are known [11]
[10] and reduced to dimensional analysis. It is easy to
show that ec ~ N1/4 and Ac(x) ~ N1/2 for the saddle-point 4 202
configuration. To find the dimension of G(x, x'), con- g
( ) = ---g2 4g3 64
------ 3
( )  --------- g4
+ +
sider the Dyson equation 3 3 9
(11)
256 1280
G(x, x') = G (x  x') + ---------  ------------
0 + 186 3
( ) 5
( ) g5 + ... .
(7) 3 3
4
 d yG (x  y)e ( )G(y, x')
0 A y
The series summation procedure should be somewhat
which follows from Eq. (3). In order to clarify the struc- modified as compared to [1, 2], because Lipatov's
ture of the solution, let us consider the scalar analogue asymptotic expression has the form caN(N/2 + b)
of Eq. (7) and assume that the function A(x) is strongly instead of caN(N + b). The Borel transform B(z) is
localized near x = 0; one can then set G(y, x') G(0, x') defined as
in the integral, after which the equation is easily solved:
G(x, x') = G (x  x')  b  1
0
0 g
( ) = dxe x x B(ag x),

4
G 
( x') d yG (x  y)e ( ) 0
0 0 A y (8)
(12)
 -------------------------------------------------------------------------.
4
1 + d yG (y)e ( )
0 A y
B z
( ) = B 
( z)N, B
N
= -----------------------------------,
N N aN(N/2 + b )
N = 0 0
Because eA(x) ~ N1/4 and Eq. (8) is finite in the limit
e , one has G(x, x') ~ N0. It is natural to expect where b0 is an arbitrary parameter. The conformal map-
that this result is general and is not caused by the ping z = u/(1  u) of the Borel transform provides a con-
above assumptions. The Nth-order contribution to the vergent series in u with the coefficients
integral in Eq. (1) has the form
N

33/2
N/2 + +
U = B (1)KCK  1( N 1), U = , (13)
B
const -------- N r M
------------------------ (g)N (9)
N K N  1 0 0
43 2 K = 1

for even M and, with the extra factor eN1/4, for odd whose large-N behavior
M values.

1  1
The direct expansion of Eq. (2) in powers of the last term in U = U , U = ---------------------------------------------- (14)
N N
Eq. (5) is incorrect, because the functional integration will then a () b
( + /2)
0
include the configurations for which result (5) is invalid. The cal-
culation should be carried out by the saddle-point method, which
yields a continuous function of N; the fact that it must be taken at determines the parameters of the asymptotic expression
the integer or half-integer points is an external condition. (g) = g at g .


GELL-MANNLOW FUNCTION IN QED 193

The interpolation is performed for the reduced coef-
ficient function

A A
F N
= ------ = A 1
+ -------------- 2
+ --------------------- + ...,
N as 0 ( )2
N N N
~
 N N
~
 (15)

as = aNNb~(N/2 + b b~
 ),
N



by cutting off the series and choosing the coefficients AK
so that Eq. (15) coincides with the known FN values.

Optimal parametrization corresponds to
b~ = b  1/2 =

5.5 [2], while the parameter
N
~ is used for checking on
the stability of the results and for numerical optimiza-
tion. In contrast to the 4 theory [1, 2], the common coef-
ficient in the asymptotic expression (10) is unknown. Fig. 1.
Technically, this is not a problem because the
parameter
A0 in Eq. (15) is not considered as known but
is found by interpolation. However, this leads to a much
greater uncertainty in the function FN; its first values (in
units of 10-3) F 2 = 63.1, F3 = 7.02, F4 = 0.34, and F5 =
1.23 exhibit only a weak tendency to become a constant,
and the predicted value A0 = lim F changes by several
N
N

orders of magnitude with changing N
~ . At first glance,
no reasonable results can be obtained in such a situa-
tion.

However, the algorithm used for determining the
asymptotic form of (g) is, in a sense, "superstable":
the addition of an arbitrary mth-order polynomial
Pm(N) to BN does not change the coefficients UN at N
m + 2 [2]. This property can be generalized for a wide
class of smooth functions: a change in UN caused by the
replacement BN BN + f(N), where f(N) is an integer
function with rapidly decreasing Taylor-series coeffi-
cients, rapidly decreases with N. Thus, smooth errors
are immaterial even if they are large. In contrast, the
nonsmooth errors lead to a catastrophic effect, which
can be used for optimization: if the interpolation
procedure is not satisfactory, the behavior of UN at
large N cannot be interpreted in terms of a power-law
dependence [2]. Fig. 2.

To check this argumentation, a test experiment was
carried out for the 4 theory. The use of complete infor- tory.2 Clearly, the results obtained below should only
mation [i.e., coefficients 25 and parameters A0 and be treated as a zero approximation.
A1 in Eq. (15)] gave = 0.96  0.01 and = 7.4  0.4
[2]; the same procedure without the use of A0 and A1 Following [2], let us approximate UN by the power-
gave = 1.02  0.03 and = 1.7  0.3. Taking into law dependence for a fixed interval 20 N 40 and dif-
account that the uncertainty in the coefficient function ferent b0 and N~ values. The 2 dependence on N~
(estimated through varying N
~ by ~1 near its optimal
2
value) amounts to few percent in the first case and more The difference in the values is not controlled by the estimated error,
but this is quite explainable: the procedure proposed in [2] for
than an order of magnitude in the second, one can con- estimating errors is only justified in the vicinity of the exact
clude that such a stability of the results is quite satisfac- result, where all deviations can be linearized.


194 SUSLOV

minima of 2 at b0 = '/2, ''/2, ..., where the respec-
tive corrections to Eq. (14) vanish, are closest to the
exact value 1 [2].4

Figure 3a shows different estimates for the index
1.0
as a function of N
~ [2]: (1) from the value of eff at the
2 minima corresponding to ' and ''; (2) from the
position of the 2 minimum corresponding to b0 = /2;
(3) from a change in sign of U when processing by
taking the logarithm of UN (solid line in Fig. 2b); and
(4) the same but for processing with a fixed index
(dashed line in Fig. 2b). Figure 3b shows different esti-
mates obtained for : (1) from the U value at the 2
minima corresponding to ' and '' and (2) and (3) from
the slope of the linear portion of the U(b0) dependence
1.0 near the root (upper and lower estimates, respectively).
The discrepancy between different estimates gives a

measure of uncertainty of the results. For N
~ 0.25, the
0.5 results for are consistent with a value slightly smaller
than unity. For N
~ > 0.25, there is a systematic increase
to 1.08, which is not controlled by the estimated error, but the 2
minima are ill-defined and unstable in this case. Similar behavior is
observed for . We take, as the most reliable, values in the
Fig. 3. middle of the chosen
N
~ interval, and accept the conser-
vative estimates for the accuracy including systematic
changes:

= 1.0  0.1, = 1.0  0.3. (16)



It follows from above that even this estimate of error is
not reliable.

It is easy to sum up the series for arbitrary g by cal-
culating the UN coefficients in Eq. (13) for N 30 and
continuing them according to the asymptotic expres-
sion found for U
Fig. 4. N  1. Figure 4 shows the results for

N
~ = 0.2 and b0 = 0. The one-loop law 2g2 matches the
asymptotic dependence
(Fig. 1) enables one to select a set of interpolations g at g ~ 10. At g < 5, (g)
differs only slightly from the one-loop result. Within
(-0.5 N
~ 1.0) for which the power-law behavior of the accuracy adopted, the asymptotic expression for
UN is probable. The typical dependences of 2 and (g) coincides with the upper limit of inequality 0
effective values of U and on b
0 (Fig. 2) indicate that (g) < g, which was derived in [12] from spectral con-
1.3 Indeed, the quantity U reverses its sign [see siderations. For = 1 and = 1, the fine structure con-
Eq. (14)] at b stant in pure electrodynamics increases at small dis-
0 = /2 0.5. At the same b0 value, 2
has a minimum, which corresponds to the fact that the tances L as L2.
leading contribution UN  1 vanishes and the power- This work was supported by INTAS (grant no. 99-
law dependence UN ~ N'  1 prevails, where the index ' 1070) and the Russian Foundation for Basic Research
corresponds to the next correction to the asymptotic (project no. 00-02-17129).
expression for (g) (it is assumed that (g) = g +
4
g' + g'' + ... at large g). The values of In the test examples, minima of 2 are usually observed only for

' '' eff at the and ' [2]. The appearance of additional minima is probably
specific to a small amount of information; it was observed in the
3 For technical reasons, Fig. 2 shows the quantity U~ = U
(b0 + 1). above-mentioned test experiment for the 4 theory.


GELL-MANNLOW FUNCTION IN QED 195

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