научная статья по теме RESURGENCE, OPERATOR PRODUCT EXPANSION, AND REMARKS ON RENORMALONS IN SUPERSYMMETRIC YANG-MILLS THEORY Физика

Текст научной статьи на тему «RESURGENCE, OPERATOR PRODUCT EXPANSION, AND REMARKS ON RENORMALONS IN SUPERSYMMETRIC YANG-MILLS THEORY»

RESURGENCE, OPERATOR PRODUCT EXPANSION, AND REMARKS ON RENORMALONS IN SUPERSYMMETRIC

YANG-MILLS THEORY

M. Shifman*

William I. Fine Theoretical Physics Institute, University of Minnesota 55455, Minneapolis, MN USA

Received October 4, 2014

We discuss similarities and differences between the resurgence program in quantum mechanics and the operator product expansion in strongly coupled Yang-Mills theories. In Ai = 1 super-Yang-Mills theories, renormalons are peculiar and are not quite similar to renormalons in QCD.

Cwitribvtiwi for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S0044451015030076

1. PREAMBLE

This paper is written for Valory Rubakov-60 Festschrift 011 the basis of my talk at CERN in the summer of 2014. I first met Valory around 1980, when he discovered the monopole catalysis of the proton decay, which later became known as the Callan Rubakov effect. There is a beautiful paper of Edward Witten illustrating subtle points in this effect, which appeared shortly after Rubakov s publication. I remember Witten's seminar based 011 this paper delivered during his only visit to the USSR in the early 1980s.

After the Callan Rubakov effect, Rubakov published many inspiring papers and raised two or three generations of bright students. These students, in turn, now inspire new young generations of theoretical physicists all over the world.

2. INTRODUCTION

The notion of resurgence and trans-series associated with it a breakthrough discovery1^ in constructive mathematics in the 1980s mostly associated with the name of Jean Ecalle gradually spread in mathematical and theoretical physics. I was impressed by di-

E-mail: shifman'Qmmn.edu 11 For a pedestrian review understandable to physicists (at least, in part) and an exhaustive list of references, see [1, 2].

verso and numerous applications of these ideas recently discussed by J. Zinn-Justin, M. Berry, U. Jentschura, G. Dunne, M. Beneke, and others. The issues to be discussed below are rather close to resurgence in quantum mechanics, although they go far beyond and are much more complicated, because I discuss strongly coupled field theories, such as quantum chromodynamics (QCD).

In quantum mechanics, the program of resurgence works well, and trans-series of the type

E(f)2) — EpT, regulariied {f]~ 1

EEE

k=1 I p= os

2A'+1

exp

log-

k—instanton

x ^¿^iz ^

regularized PT

can bo derived for all energy eigenvalues (g2 is assumed to be small; the subscript PT stands for perturbation theory).

In weakly coupled field theories, trans-series could be perhaps constructed, although conclusive arguments have not yet been presented. One of my tasks is to explain why resurgence, being conceptually close to the operator product expansion (OPE), does not work in strongly coupled field theories, for instance, in QCD. It is worth noting that OPE existed in QCD from the mid-1970s, and in its general form, the late 1960s. It grew from a formalism that had been suggested by K. Wil-

Fig. 1. V(x) in the anharmonic oscillator problem (2)

son before the advent of QCD. The first part of this paper is devoted to this issue.

In the second part, I focus on a more technical aspect: peculiarities of the factorial divergence of perturbation theory in A/' = 1 super-Yang Mills (SYM). So far rcnornialons in SYM were scarcely discussed. No final conclusion was reached. To a large extent this question remains open.

3. THE SIMPLEST QUANTUM MECHANICAL EXAMPLES

3.1. Anharmonic oscillator

We consider a one-dimensional anharmonic oscillator,

l / 1 2 I 1 2 2 I 2 4 /.1,

H = -P + -UJ X + g X (2)

(see Fig. 1).

For dcfinitcncss, we focus on the ground state energy E0- There exists a well-defined procedure for constructing E0 order by order in perturbation theory, to any finite order,

£o = | (1 + n.92 + f'2.94 + ..•)• (3)

Nevertheless, Eq. (3) does not define the ground state energy. Indeed, the coefficients c/, are factorially divergent at large k [3],

- (— I/.? 1 /.•!. k » 1. (4)

where B = ui?'/3 is the so-called bounce action2^. Hence, the sum in (3) needs a regularization.

2) Equation (4) is slightly simplified. For a more precise for-

mula, see [3].

0

-B

Contour of integration

Fig. 2. The perturbative series in the anharmonic oscillator problem is Borel-summable. The y2 series for Eu is sign alternating; f(a) has a singularity on the real negative semi-axis in the Borel parameter complex plane, a is the Borel parameter

In the simplest case under consideration, an appropriate (and exhaustive) regularization is provided by the Borel transformation B,

+ (5)

The Borel transformation introduces 1/kl in each term of series (3), rendering it convergent. Moreover, if the convergent series

°° i

i + EW^/ifl2). (6)

k=l '

which defines the Borel function f(g2). has no singularities on the real positive semi-axis g2 > 0, then we can obtain the ground-state energy E0 starting from the well-defined expression for BE0 and using the Laplace transformation,

oo

E0 = L (BEo) = | f dag-2 exp (-^j /(«). (7) o

This procedure is usually referred to as the Borel summation. Thus, the perturbative expansion in the anharmonic oscillator is Borel-summable because the singularities of /(«) are on the negative real semi-axis. Indeed, we assume that /(«) has a pole at a = —B (see Fig. 2), namely,

ft , ,3

Fig. 3. The same potential with the replacement y2 -¥ -y2, to be denoted as V(x)

Thou the integral (7) is woll-dofinod. At the sanio timo, expanding (8),

x: i

/(«> = £(-i><(£)

0)

k=O

and substituting this series in (7), we immediately arrive at (4).

The fact that the position of the singularity in the u plane is to the left of the origin and that the series is sign-alternating are in one-to-one correspondence with each other.

Exactly fifty years ago, Vainshtcin identified [4] the physical meaning of the factorial growth of coefficients (4) and explained why the underlying singularity in the Borel parameter plane is on the negative semi-axis. Changing the sign of g2 from positive to negative, g2 —¥ —g2, we convert a stable potential V(x) in (2) into an unstable potential V(x) presented in Fig. 3, allowing for the wave function to leak to large distances.

In the leaking potential V, the energy corresponding to the Otli eigenvalue acquires an imaginary part (as do other energy eigenvalues). This imaginary part can be easily determined. Indeed, after the Euclidean time rotation, the potential effectively changes as V(x) —¥ —¥ —V(x), as shown in Fig. 4. Then the so-called bounce trajectory becomes classically accessible3^. The bounce trajectory starts at x = 0, slides to the right, bounces off at x„ = u>/</2.9, and then returns to the point x = 0. The Euclidean action on the bounce trajectory is readily calculable,

.4

D

bounce —

See, e.g., Chapter 7 in [5].

Fig.4. An effective potential in Euclidean time. This potential is a sign reflection of that in Fig. 3, i.e., is —V{x). It vanishes at x = 0 and at x = ±x*, where

where B is defined in Eq. that

(8). In this way, we obtain

T nui B ( B

Ini E0 = — — exp —-2 g2 \ g2

(H)

Now we can calculate the ground-state energy for the original potential in Fig. 1 by using (11) and a dispersion relation in the coupling constant [4],

Eo = ~ [ dg2 Im Eg (g2) =

ir J g + g o

= £ I dz

1

1 + (g2/B).

(12)

The last expression reproduces the series in (3) and (4) with its sign alternation and factorial divergence of the coefficients. Both features are explained by the imaginary part in (11) being proportional to exp(—B/g2).

Summarizing, the perturbative expansion for the anharnionic oscillator is factorially divergent; however, the Borel summability allows finding the closed, well-defined, and exact expressions for the energy eigenvalues. The physical meaning of the factorial divergence, as well as the sign alternation, are fully understood. Now we pass to a more complicated but more interesting lion-Borel-summable case.

3.2. Double-well potential

The double-well problem is described by the Hamil-tonian

H = IP2

1

2 2 U) X

2 4

• 9 •<• .

(13)

H

2 Binst —•—

2 possible

integration contours

Fig. 5. The y2 series in the double-well problem is not sign alternating; f(a) has a singularity on the real negative semi-axis in the Borel parameter complex plane at a = 2Bi„si, where a is the Borel parameter

i.e., the sign of the 0(x2) term is changed, and the point x = 0 becomes unstable. Instead, two stable minima develop at .(;» = ±.(;» with

The shape of the double-well potential is depicted in Fig. 4. Classically, each of the two minima x = = ±u/2\/2g presents a stable solution of the system. Quantum mechanically, zero-point oscillations about the minima occur. Taking the anharmonicity near the minima into account, we generate a porturbativo series for the ground-state energy. This is in the perturbation theory. In fact, the two minima are connected by the tunneling trajectory (instanton) in Euclidean time. The instant on action is

^insl -

UT 12?

(14)

(see, e. g., fC] introduce

In what follows, it will be convenient to

Bjnsl - g Sjnsl —

Ur 12'

(15)

At small g2, the ground-state energy is close to u>/2 plus corrections in g2 and nonporturbativo corrections of the type exp(—c/g2). A crucial distinction from the anharmonic oscillator discussed in Sec. 3.1 is that the g2 series in this case is not sign-alternating (although still factorially divergent), corresponding to a singularity in the Borel function at a real positive value a = 2BingL, i.e., on the integration contour (see Fig. 5). Thus, we have to rethink the Borel summation procedure.

Equation (7) is replaced by

oo

£o = L (BEo) = dag-2 exp (-^j /(«), (16)

where, roughly speaking,

/(«) =

-2 B.

insl

U 2,/i sf

Then, instead of Eq. (4), we obtain ck = kl (2Bin8i.)~k

(17)

(18)

The porturbativo series is not sign alternating, unlike the case of the anharmonic oscillator.

We pause here to take a closer look at the above result

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