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Perturbation Theory with Convergent Series: The Calculation of the 4 - field theory -function (4) The problem of series summation A typical problem: Let F () be given by a formal power series


F ()
k =0

fk k .

Let {fk }k

=1,...,N

be known.

How to calculate the quantity F () (especial ly, for 1)? · The series is asymptotic · The quantity F () is non-analytic at the origin The asymptotics of the fk (k ) is known: A1 ~ fk := C ak k b (k + d) A0 + +O k The Borel summation:


1 k2 (-1)k f k!

.

F () =
0

e B (t) dt ,

-t

B (t) =
k =0

k

· Only finite number of fk 's is known; · Exact analytical properties of F () are hard to obtain;

1


Perturbation Theory with Convergent Series (PTCS) · The method was proposed by V. V. Belokurov, Yu. P. Solovyov and E. T. Shavgulidze (Moscow State University) Consider "the 0-dimensional analogue" gral 1 2 2 I () = e-x - x m dx = exp K1 4 8 0 or (-1)k I () = fk k , fk = k ! 2 4k
k =0

of path inte1 8 (4k )! . (2k )! 1 , 4

Approximate I () with some new integral J (, R) |I () - J (, R)| , The series for J (, R) is convergent

2k 1 4 A2k (R) B (2k )!

J (, R) =
k =0

k 2 k 1 2 A 2k ( R ) B (2k )!

=
k even

k 1 2 A2k (R) B k + 2 (2k )!

k odd k

k 2

-- -
k

R



(-1)k 4k + 1 k! 2

,

where Bk =
2

2k + 1 2
k +

ik Ak (R) = 2

+R -R



e
-

-r

4

e

-ir

dr d

· With PTCS, I () can be calculated precisely, although only finitely many of its PT coefficients are known.
2


PTCS as a metho d for divergent series summation Consider a series fk k
k

Assume |fn | C an nb n! . Then, there exists a function f (t) C ([0, )) s. t. (0) . n! Let the function f (t) satisfy fn = · (-1)k f
(k )

f

(n)

(t) 0 k 0, t [0, );

· allows analytic continuation in Re > 0;
- N · f () - N=01 fn n < C1 aN N b N ! holds uniformly n in , Re > 0;

Then, f () =
0



e

-t

µ(dt) ,

where µ(dt) is a positive Radon measure on [0, ).
N

f ()
n=0

n 1 m A2n (m, R) B (2n)!

n m

,


n Bm =

x m µ(dt) ,
0 +R -R

n

A2n (m, R) = (-1)n 2m () := ~ 1 2
3

2m () 2n d , ~ e
2m

e

-ir -r

dr .


The -function Consider O(n)-symmetric field theory 1 L = µ a µ a 2 The -function


1 2 a a 16 2 + mB + B (a a )2 , 2 4! is (in the M S -scheme)

a = 1, . . . , n

() =
k =2

k k = 1.662 -3.333 +19.974 -175.255 +1898.856 +. . . .

f () = exp(- ()/) = 1-1.6+4.72 -26.33 +2194 -22975 . The asymptotics for the k is known; for n = 2 0.543 7 4.82 1 k k 2 k! 1 - +O . 16 2 k k2 The coefficients A (R = 6) n |A2n (2, 6)| n |A2n (2, 6)| n |A2n (2, 6)| 1 0.995345 4 1526.6842 7 3.5515064·107 2 0.415867 5 42593.555 8 1.0668836613·109 3 51.892553 6 1212765.477 9 3.2743551347·1010 The coefficients Bn/2 Bn/2 n Bn/2 1.16971 9 41374.43266 3.34822 11 976968.175 34.16947 13 1.6115094688·107 845.48256 15 7.21784871286·108 -function 0.4 0.5 6.6 6.6 0.20 0.32 0.05 0.08
4

n 1 3 5 7

The 0.1 0.2 0.3 R 8.4 7.5 6.5 0.01 0.05 0.12 0.01 0.01 0.03

values 0.6 6.6 0.62 0.1

0.7 6.4 0.87 0.4

0.8 6.5 1.2 0.4

0.9 6.4 1.5 0.6

1 6.5 1.9 0.8


The coefficients Bn vs. n 25

20

15 log Bn 10 5 0 0

1

2

3

4 n

5

6

7

8

5