Correlation Functions of the Anharmonic Oscillator:Numerical Verification of Two–Loop Corrections to the Large–Order Behavior

Ludovico T. GiorginiNordita, Royal Institute of Technology and Stockholm University, Stockholm 106 91, Sweden

Ulrich D. JentschuraDepartment of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA and

MTA-DE Particle Physics Research Group, P.O. Box 51, H-4001 Debrecen, Hungary

Enrico M. MalatestaArtificial Intelligence Lab, Bocconi University, 20136 Milano, Italy

Giorgio ParisiDipartimento di Fisica, Sapienza Universita di Roma, P.le Aldo Moro 5, 00185 Rome, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, P.le A. Moro 5, 00185 Rome, Italy andInstitute of Nanotechnology (NANOTEC) - CNR, Rome unit, P.le A. Moro 5, 00185 Rome, Italy

Tommaso RizzoInstitute of Complex Systems (ISC) - CNR, Rome unit, P.le A. Moro 5, 00185 Rome, Italy and

Dipartimento di Fisica, Sapienza Universita di Roma, P.le Aldo Moro 5, 00185 Rome, Italy

Jean Zinn-JustinIRFU/CEA, Paris-Saclay, 91191 Gif-sur-Yvette Cedex, France

Recently, the large-order behavior of correlation functions of the O(N)-anharmonic oscillator hasbeen analyzed by us in [L. T. Giorgini et el., Phys. Rev. D 101, 125001 (2020)]. Two-loop correctionsabout the instanton configurations were obtained for the partition function, and the two-point andfour-point functions, and the derivative of the two-point function at zero momentum transfer. Here,we attempt to verify the obtained analytic results against numerical calculations of higher-ordercoefficients for the O(1), O(2), and O(3) oscillators, and demonstrate the drastic improvement ofthe agreement of the large-order asymptotic estimates and perturbation theory upon the inclusionof the two-loop corrections to the large-order behavior.

CONTENTS

I. Introduction 1

II. One–Dimensional Quantum AnharmonicOscillator 2

III. Two– and Three–Dimensional QuantumAnharmonic Oscillator 4A. Overview 4B. Two-point correlator and second derivative 6C. Four-point correlation function 9D. Correlation function with a wigglet insertion 10

IV. Comparison with Two–Loop Corrections forLarge Order 11A. Analytic Formulas 11B. Significance of the Two–Loop Correction 13

V. Conclusions 14

Acknowledgements 15

A. Some useful definitions and identities 151. Matrix elements 15

2. Case N = 3 16

B. Alternative Procedure 16

References 17

I. INTRODUCTION

For a long time, it has been a dream of physics researchto overcome the predictive limits of perturbative quan-tum field theory. Typically, Feynman diagram calcula-tions become more computationally expensive in largeloop orders, to the point where diminishing returns [1]upon the addition of yet another loop limit the predictivepower of perturbation theory, and of Feynman diagramcalculations. In order to overcome these limits, analytictechniques have been developed over the past decades, toanalyze the large-order behavior from the complementarylimit of “infinite-loop-order” Feynman diagrams [2, 3].These techniques are based on various nontrivial obser-vations. The first is that, upon an analytic continuationof the coupling constant of a theory into a physically “un-stable” domain [4] where partition functions acquire animaginary part, one can write dispersion relations which

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relate the behavior of the theory for a coupling constantsmall in absolute magnitude, within in the unstable do-main, to the large-order behavior of perturbation theory(equivalent to the “infinite-order Feynman diagrams”).The imaginary part of the partition functions, and ofthe correlation functions, is related to so-called instan-ton configurations [5–7]. The second observation is thatperturbations about the instanton configurations can bemapped onto corrections to the large-order behavior ofperturbation theory, thus making the theory amenableto a more accurate analysis in the domain of large looporders. The latter perturbative calculations in the in-stanton sector are related to an expansion of perturbativecoefficients in powers of inverse loop orders 1/K, whereK denotes the loop order. The leading term, of course,as is well known, describes the factorial divergence ofperturbation theory in large orders of the coupling con-stant [2, 8–12].

Previously, the calculation of corrections about thelarge-order behavior was reported for the partition func-tion of anharmonic oscillators [13, 14]. Recently [15],corrections to the large-order behavior of perturbationtheory have been obtained for the two-point and four-point functions of the O(N) quartic anharmonic oscilla-tor. Also, the partition function of the O(N) oscillatorwas studied, and results were obtained for the derivativeof the two-point function at zero momentum transfer [15].These results, however, have not been compared yet toan explicit calculation of perturbative coefficients for therespective functions in large orders. In this work, in or-der to make the comparison possible, we derive generalexpressions for the perturbative coefficients of the corre-lation functions at zero momentum transfer of the one-,two- and three-dimensional isotropic anharmonic oscilla-tor.

In this context, it is extremely interesting to investigatethe “rate of convergence of the expansion about infiniteloop order”, i.e., to investigate to which extent the calcu-lation of the corrections of order 1/K to the leading fac-torial asymptotics improves the agreement of low-orderperturbation theory (corresponding to the successive per-turbative evaluation of loops). In this paper, we thus ana-lyze the perturbation series of the one-dimensional O(N)isotropic quantum harmonic oscillator with a quartic per-turbation, which is otherwise referred to as the N -vectormodel. Higher orders of perturbation theory are calcu-lated for the two-point function, the four-point function,and the correlator with a wigglet insertion, and comparedto the results recently reported in Ref. [15]. Our Hamil-tonian is therefore

H = −1

2

∂2

∂~q 2+

1

2~q 2 +

g

4~q 4 , ~q ≡

N∑i=1

qi ei , (1)

where g is the coupling constant. The rest of the paperis organized as follows. In Sec. II, we start by analyzingthe simple N = 1 case, whereas in Sec. III, we discussthe general N -dimensional case. In Our results are com-

pared with those of Ref. [15] and Sec. IV. Conclusionsare reserved for Sec. V.

II. ONE–DIMENSIONAL QUANTUMANHARMONIC OSCILLATOR

We start with a discussion of the method of calcu-lation for the perturbative corrections to the correla-tion functions exposed in the previous section. Thecase with a trivial internal symmetry group is the eas-iest (N = 1). Since when N = 1 the unperturbedHamiltonian is non-degenerate, we can use standardnon-degenerate Rayleigh-Schrodinger perturbative the-ory techniques. We can write the perturbative expan-sions as follows,

H(0) = − 1

2

∂2

∂q2+

1

2q2 , δH =

g

4q4 , (2a)

|En〉 = |E(0)n 〉+ |E(1)

n 〉+ |E(2)n 〉+ . . . , (2b)

|ψn〉 = |ψ(0)n 〉+ |ψ(1)

n 〉+ |ψ(2)n 〉+ . . . . (2c)

The unperturbed problem, with the unperturbed Hamil-

tonian H(0), is solved by the unperturbed states |ψ(0)n 〉,

H(0) |ψ(0)n 〉 = E(0)

n |ψ(0)n 〉 , (3)

leading to the unperturbed energy eigenvalues E(0)n . The

perturbative corrections E(K=1,2,3,... )n are each propor-

tional to gK and describe a perturbation series in g. Onthe basis of a well-known recursive scheme, we can calcu-late higher-order (in K) perturbations to the wave func-tions, as follows:

g0 : (H(0) − E(0)n ) |ψ(0)

n 〉 = 0 , (4a)

g1 : (H(0) − E(0)n ) |ψ(1)

n 〉 = (E(1)n − δH) |ψ(0)

n 〉 = 0 ,(4b)

g2 : (H(0) − E(0)n ) |ψ(2)

n 〉 = (E(1)n − δH) |ψ(1)

n 〉+ E(2)

n |ψ(0)n 〉 , (4c)

g3 : (H(0) − E(0)n ) |ψ(3)

n 〉 = (E(1)n − δH) |ψ(2)

n 〉+ E(2)

n |ψ(1)n 〉+ E(3)

n |ψ(0)n 〉 , (4d)

gK : (H(0) − E(0)n ) |ψ(K)

n 〉 = (E(1)n − δH) |ψ(K−1)

n 〉+ E(2)

n |ψ(K−2)n 〉+ E(3)

n |ψ(K−3)n 〉+ . . .+ E(K)

n |ψ(0n 〉 .(4e)

This recursive algorithm allows us to calculate the Kth-

order perturbation |ψ(K)n 〉 to the wave function. Note

that the calculation of theKth-order energy perturbation

E(K)n only requires the wave function to order K − 1,

E(K)n = 〈ψ(0)

n |δH|ψ(K−1)n 〉 . (5)

The wave function perturbations |ψ(K)n 〉 are orthogonal

to the unperturbed state |ψ(0)n 〉,

〈ψ(0)n |ψ(1)

n 〉 = 〈ψ(0)n |ψ(2)

n 〉 = 〈ψ(0)n |ψ(3)

n 〉 = . . . = 0 . (6)

3

In order to solve the recursive scheme given by Eq. (4),

for a given unperturbed reference state |ψ(0)n 〉, we can

define the reduced Green function of the unperturbedproblem,

T =

(1

H(0) − E(0)n

)′, (7)

where the inverse is taken over the Hilbert space of un-

perturbed states orthogonal to the reference state |ψ(0)n 〉.

The reduced Green function is sometimes denoted as G′

in the literature, but we avoid the notation here becausethe symbol G is already used extensively in other parts ofour considerations, in order to denote perturbative coef-ficients. T has the following matrix elements in the basisof unperturbed eigenstates,

Tm,m′ =

⟨ψ(0)m

∣∣∣∣∣(

1

H(0) − E(0)n

)′∣∣∣∣∣ψ(0)m′

⟩=

δm,m′

E(0)m − E(0)

n

,

(8)assuming that m 6= n and m′ 6= n. Furthermore, we haveTn,n = 0 for m = m′ = n. We can write the perturbationδH = g q4/4 in the unperturbed basis simply by usingthe representation of the position operator in terms ofthe creation and annihilation operators a† and a of theunperturbed Hamiltonian,

q =1√2

(a+ a†

). (9)

We recall that the lowering and raising operators a and

a† act on the unperturbed state |ψ(0)n 〉 as follows,

a|ψ(0)n 〉 =

√n |ψ(0)

n−1〉 , (10a)

a†|ψ(0)n 〉 =

√n+ 1 |ψ(0)

n+1〉 . (10b)

From now on, we will switch to a notation where

|n〉 ≡ |ψn〉 (11)

denotes the (perturbed) eigenstate of the full problem,which includes the quartic term. Averages of a one-dimensional scalar theory can be interpreted as path-integral expressions which in turn can be written as sum-mations over eigenvalues of the corresponding quantumHamiltonian. The two-point function has the transla-tional invariance property

C(2)(t1, t2) = C(2)(0, t2 − t1) ≡ C(2)(t2 − t1) , (12)

where the latter expression defines the correlation func-tion C(2)(t2 − t1) of a single argument. The correlationfunction can be expressed as follows,

C(2)(t) ≡ 〈q(0) q(t)〉 =∑n>0

|〈0|q|n〉|2 e−(En−E0)|t| , (13)

where E0 is the perturbed energy of the ground state (the“vacuum”), and En is the perturbed energy of the state

|n〉. We have for its integrals∫ +∞

−∞dt C(2)(t) = 2

∑n>0

|〈0|q|n〉|2

En − E0,

∼∞∑K=0

[G(2)(1, 1)]K gK , (14)

∫ +∞

−∞dt t2 C(2)(t) = 4

∑n>0

|〈0|q|n〉|2

(En − E0)3

∼∞∑K=0

[G(∂p)(1, 1)]K gK . (15)

Here, the perturbative coefficients [G(2)(N = 1, 1)]K and[G(∂p)(N = 1, 1)]K define an asymptotic series in thecoupling g which exhibits well-known factorial growth forhigher order in K. In the notation, we follow the conven-tions of Ref. [15]; some low-order coefficients [G(2)(1, 1)]Kand [G(∂p)(1, 1)]K are summarized in Table I. A remark is

in order. The expression for the integral∫ +∞−∞ C(2)(t) dt

deceptively looks like the second-order expression forthe energy shift due to a perturbative Hamiltonian pro-portional to q. We should recall, though, that thethe virtual-state eigenfunction |n〉 and the ground-stateeigenfunction |0〉, as well as the energies En and E0, arethe exact eigenfunctions and eigenenergies of the per-turbed anharmonic oscillator and thus contain contribu-tions of arbitrarily higher orders of perturbations propor-tional to gK .

As a side remark, let us briefly consider the limit g → 0,which describes the transition to the unperturbed har-monic oscillator. In this limit, one has

C(2)(t) =∑n>0

|〈0|q|n〉|2 e−(En−E0)|t|

→ C(2)(t) =∑n>0

|〈ψ(0)0 |q|ψ(0)

n 〉|2 e−(E(0)n −E

(0)0 )|t|

= |〈ψ(0)0 |q|ψ

(0)1 〉|2 e−|t| =

1

2e−|t| , (16)

which is the well-known two-point correlation functionfor the unperturbed one-dimensional theory [16].

The connected four-point function is defined as

C(t1, t2, t3, t4) ≡ 〈q(t1) q(t2) q(t3) q(t4)〉− C(2)(t1 − t2)C(2)(t3 − t4)− C(2)(t3 − t2)C(2)(t1 − t4)

− C(2)(t1 − t3)C(2)(t2 − t4) . (17)

For 0 = t1 < t2 < t3 < t4, we can write

C(4)(t2, t2, t4) ≡ 〈q(0)q(t2)q(t3)q(t4)〉= 〈0|q e−(H−E0)∆3 q e−(H−E0)∆2 q e−(H−E0)∆1q|0〉− C(2)(∆1)C(2)(∆3)− C(2)(∆1 + ∆2 + ∆3)C(2)(∆2)

− C(2)(∆1 + ∆2)C(2)(∆2 + ∆3) , (18)

4

where ∆i ≡ ti+1 − ti. We have chosen t1 = 0. For t2, wehave two equivalent regions (t2 can be either positive ornegative), while for t3, we have three equivalent regions.Finally, for t4, one encounters four equivalent regions,bringing the number of equivalent integration regions to4× 3× 2 = 24. The selected integration region 0 = t1 <t2 < t3 < t4 gives rise to the integration measure∫ ∞

0

dt2

∫ ∞t2

dt3

∫ ∞t3

dt4 =

∫ ∞0

d∆1

∫ ∞0

d∆2

∫ ∞0

d∆3 .

(19)The final result is∫ +∞

−∞dt2

∫ +∞

−∞dt3

∫ +∞

−∞dt4 C

(4)(t2, t3, t4)

= 24∑

n,n′,n′′>0

〈0|q|n〉 〈n|q|n′〉 〈n′|q|n′′〉 〈n′′|q|0〉(En − E0) (En′ − E0) (En′′ − E0)

− 24∑

n,n′>0

|〈0|q|n〉|2|〈0|q|n′〉|2 (En + En′ − 2E0)

2(En′ − E0)2(En − E0)2

= 24∑

n,n′,n′′>0

〈0|q|n〉 〈n|q|n′〉 〈n′|q|n′′〉 〈n′′|q|0〉(En − E0) (En′ − E0) (En′′ − E0)

− 24∑

n,n′>0

|〈0|q|n〉|2|〈0|q|n′〉|2

(En′ − E0)2 (En − E0). (20)

A remark is in order, here. The sums over intermedi-ate states in Eq. (20) exclude the ground state (whichhas n = 0). One might wonder, for example, why thepresence of the term −C(2)(∆1)C(2)(∆3) in Eq. (18)does not lead to divergences, when the connected four-point function is integrated over ∆2. The term cancels,though, against the term derived from the expression〈0|q e−(H−E0)∆3 q e−(H−E0)∆2 q e−(H−E0)∆1q|0〉 upon in-sertion of the ground state as the intermediate state inthe exponential e−(H−E0)∆2 . The term with the virtualground state is thus excluded from the sums over inter-mediate states in the representation (20).

Results for the perturbative coefficients obtained fromthe four-point integral

∫ +∞

−∞dt2

∫ +∞

−∞dt3

∫ +∞

−∞dt4 C

(4)(t2, t3, t4)

∼∞∑K=0

[G(4)(1, 1)]K gK (21)

are summarized in Table I.The last correlation function studied here concerns the

two-point function with a wigglet insertion (see Ref. [15]).We have

C(1,2)(t2, t3) = 〈q(0)q(t2)q2(t3)〉− 〈q(0)q(t2)〉〈q2(t3)〉 − 2〈q(0)q(t3)〉〈q(t2)q(t3))〉 , (22)

with the integral finding the perturbative expansion (for

some concrete sample values, see Table I)∫ +∞

−∞dt2

∫ +∞

−∞dt3 C

(1,2)(t2, t3) ∼∞∑K=0

[G(1,2)(1, 1)]K gK .

(23)We obtain in terms of the perturbed oscillator eigen-states,∫ +∞

−∞dt2

∫ +∞

−∞dt3 C

(1,2)(t2, t3)

= 4∑

n,n′>0

〈0|q2|n〉 〈n|q|n′〉 〈n′|q|0〉(En − E0) (En′ − E0)

+ 2∑

n,n′>0

〈0|q|n〉〈n|q2|n′〉〈n′|q|0〉(En − E0) (En′ − E0)

− 2∑n>0

|〈0|q|n〉|2 〈0|q2|0〉(En − E0)2

− 8∑

n,n′>0

|〈0|q|n〉|2|〈0|q|n′〉|2

(En′ − E0) (En′ + En − 2E0)

− 4

(∑n>0

|〈0|q|n〉|2

En − E0

)2

.

The structures encountered for higher-dimensional inter-nal symmetry groups are more complicated and will bediscussed in the following.

III. TWO– AND THREE–DIMENSIONALQUANTUM ANHARMONIC OSCILLATOR

A. Overview

When N > 1, since both the unperturbed Hamiltonianand the perturbation term are radially symmetric, we canuse hyper-spherical coordinates in order to describe theinternal O(N) space of the theory. The Hamiltonian istherefore

H = −1

2

(∂2

∂r2+N − 1

r

∂

∂r−~L2

r2

)+r2

2+g

4r4, (24)

where ~L2 is the angular momentum in N dimensions.The eigenfunctions of the Hamiltonian can be written interms of radial and angular parts, as follows,

ψ(~r) = Rn`(r)Y`1...,`N−2

` (θ1, . . . , θN−1), (25)

where Y`1...,`N−2

` (θ1, . . . , θN−1) are the generalization ofthe spherical harmonics in N dimensions and the eigen-functions of the angular momentum operator

~L2 Y`1...,`N−2

` (θ1, . . . , θN−1)

= `(`+N − 2)Y`1...,`N−2

` (θ1, . . . , θN−1), (26)

5

TABLE I. Sample values are collected for low-order perturbative coefficients of cor-relation function for the scalar theory (N = 1). Results are given as rational numberswhen expressible in compact form.

K [G(2)(N, 1)]K [G(∂p)(N, 1)]K [G(4)(N, 1)]K [G(2,1)(N, 1)]K

0 1 2 0 0

1 − 32

−6 −6 − 32

2 318

1819

992

11

3 − 1683128

− 739796

− 6632

− 8805128

4 13825256

964156128800

277245128

55183128

5 −258.264 −1635.174 − 46658132

−2829.612

. . .

10 3.540 · 106 2.115 · 107 5.003 · 108 9.473 · 107

with the quantum numbers `1 . . . , `N−2 satisfying |`1| ≤|`2| ≤ · · · ≤ |`N−2| ≤ `. Here, the angles θ1, . . . , θN−2

range over [0, π], whereas θN−1 ranges over [0, 2π) (fora more thorough discussion, see Ref. [17]). In N = 2dimensions, the unperturbed normalized radial functions

R(0)n` (r) and Y

`1...,`N−2

` (θ1, . . . , θN−1) are defined as [18]

Y`(θ1) =1√2π

ei`θ1 , (27)

R(0)n` (r) = N (N=2)

n` exp

(−r

2

2

)r|`| L

|`|12 (n−|`|)(r

2), (28)

while in three dimensions, the expressions are as follows(`1 = m takes the role of the magnetic projection),

Y m` (θ1, θ2) = Y`m(θ = θ1, ϕ = θ2) (29a)

= (−1)m

√2`+ 1

4π

(`−m)!

(`+m)!eimθ2 Pml (cos θ1),

R(0)n` (r) = N (N=3)

n` r` exp

(−r

2

2

)L

(`+ 12 )

12 (n−`)(r

2) .

(29b)

Here, the L(α)k (r2) are the generalized Laguerre polyno-

mials of order k, and the Pm` are the associated Legendrepolynomials. For three dimensions (N = 3), we indicatethe relation of the Y m` (θ1, θ2) to the more common nota-tion Y`m(θ = θ1, ϕ = θ2) of the spherical harmonics [19],where θ is the polar angle, and ϕ is the azimuth angle.

The normalization factors are obtained as follows,

N (N=2)n` =

√2[(n− |`|)/2]!√[(n+ |`|)/2]!

,

N (N=3)n` =

(2n+`+2

√π

)1/2[

Γ(n−`

2 + 1)

Γ(n+`

2 + 1)

Γ (n+ `+ 2)

]1/2

.

(30)

Note that the generalized Laguerre polynomial satisfy thefollowing orthogonality relation [20]∫ ∞

0

xα e−xL(α)n (x)L(α)

m (x) dx =(n+ α)!

n!δnm . (31)

It follows that the eigenfunctions are orthogonal,

∞∫0

dr rN−1R(0)n` (r)R

(0)s` (r) =

(N (N)n`

)2

Γ(n+`+N

2

)2 Γ(n−`

2 + 1) δn s ,

(32)with N = 2, 3. The perturbed Schrodinger equation re-duces to an equation for the radial part which reads[

d2

dr2+N − 1

r

d

dr− `(`+N − 2)

r2+ r2 +

g

2r4

]Rnl(r)

= −2EnRnl(r). (33)

The matrix elements of the perturbation δH = g r4/4,can be written in the unperturbed basis evaluating thefollowing integral

〈R(0)n` | r

4 |R(0)sr 〉 = δ` r

∫ ∞0

dr rN+3R(0)n` (r)R

(0)s` (r) . (34)

6

The eigenvalue relation given in Eq. (33) can be writtenin the space of radial functions only,

(H0 + δH)(R(0)n` + δRn`) = (E(0)

n + δEn`) (R(0)n` + δRn`).

(35)The perturbation does not change the angular part, butonly the radial one. So, if we know the unperturbed ra-

dial part R(0)n` (r), then we can compute the perturbation

to the energy δEn` and to the eigenfunction δRn`, forfixed value of `. In fact, for fixed `, the spectrum is notdegenerate, and we can apply standard perturbation the-ory, as we did in the previous section for the case N = 1.

A number of useful formulas for the treatment of theO(N) problem, both in terms of the angular algebra aswell as the perturbative treatment of the radial part, aregiven in Appendices A and B.

From the knowledge of the eigenfunctions and eigen-values of the Hamiltonian, we can determine the M -pointcorrelation functions of our theory given by [21]

Ci1 i2 ... iM (t1, t2, ..., tM )

≡ 〈φi1(t1)φi2(t2) · · · φiM (tM )〉C≡ Ti1 i2 ... iM C(M)(t2 − t1, ..., tM − t1), (36)

where in the latter form, the expression C(M)(t2 −t1, ..., tM−t1) has M−1 arguments and is defined in anal-ogy to the four-point correlation function from Eq. (18).For example, we have M = 2 for the two-point function,and one argument in C(2)(t2 − t1), while, of course, wehave M = 4 for the four-point function, and three argu-ments in C(4)(t2−t1, t3−t1, t4−t1). Furthermore, the in-dices ij with j = 1, . . . ,M can obtain values 1 ≤ ij ≤ N ,consistent with the structure of the internal symmetrygroup. The designation C indicates that we are con-sidering the connected part of correlation function, andTi1,i2,...,iM is the average value of the product of unit vec-tor components ui1 , ui2 , and so on, taken over the theN -dimensional unit sphere,

Ti1 i2 ... iM = 〈ui1 ui2 . . . uiM 〉SN−1. (37)

with SN−1 being the N − 1-dimensional unit sphere em-bedded in N dimensions; so, SN−1 has a (generalized)surface volume ΩN = 2πN/2/Γ(N/2). We can writean M -point correlation function with arbitrary indicesi1, i2, . . . , iM in terms of the same correlation functionwith fixed indices ı1, ı2, . . . , ıM by multiplying and divid-ing the previous expression by Tı1 ,ı2,...,ıM

Ci1 i2 ... iM (t1, t2, . . . , tM )

≡ Ti1 i2 ... iMTı1 ı2 ...,ıM

Cı1 ı2 ... ıM (t1, t2, . . . , tM ) . (38)

Here, we write the indices of the fixed elementCı1 ,ı2,...,ıM (t1, t2, . . . , tM ) with a hat. Our conventionis that indices with hats are not being summed over,even when they are repeated (in other words, we use theconvention that the Einstein summation convention does

not apply on indices with hats). Comparing Eq. (36) toEq. (38), we can write

C(M)(t2 − t1, ..., tM − t1) =Cı1 ı2 ... ıM (t1, t2, . . . , tM )

Tı1 ,ı2,...,ıM.

(39)This formula allows us to pick one single nonvanishingelement of the correlation function, say, one where allindices are equal,

ı1 = ı2 = . . . ıM , (40)

and to derive a valid expression for any combination ofthe ıj , with j = 1, . . . ,M . It is useful to recall the well-known results [21]

Ti1 i2 = 〈ui1 ui2〉SN−1=δi1 i2N

, (41)

and

Ti1 i2 i3 i4 = 〈ui1 ui2 ui3 ui4〉SN−1

=δi1 i2δi3 i4 + δi1 i3δi2 i4 + δi1 i4δi2 i3

N (N + 2). (42)

In this way, the four-point function written only in termsof the element with all indices fixed to ı1, becomes

Ci1,i2,i3,i4(t1, t2, t3, t4)

≡ δi1 i2δi3 i4 + δi1 i3δi2,i4 + δi1 i4δi2 i33

× Cı1 ı1 ı1 ı1(t1, t2, t3, t4)

≡ δi1 i2δi3 i4 + δi1 i3δi2 i4 + δi1 i4δi2 i3N(N + 2)

×[N(N + 2)

3Cı1 ı1 ı1 ı1(t1, t2, t3, t4)

]≡ δi1 i2δi3 i4 + δi1 i3δi2 i4 + δi1 i4δi2 i3

N(N + 2)

× C(4)(t2 − t1, t3 − t1, t4 − t1) . (43)

So, we have

C(4)(t2 − t1, t3 − t1, t4 − t1) =N(N + 2)

3× Cı1 ı1 ı1 ı1(t1, t2, t3, t4)

≡ N(N + 2)

3C(4)(t2 − t1, t3 − t1, t4 − t1) , (44)

where the latter expression defines the quantity C(4)(t2−t1, t3 − t1, t4 − t1) = Cı1 ı1 ı1 ı1(t1, t2, t3, t4).

B. Two-point correlator and second derivative

We start the discussion from the two-point correlationfunction and its second derivative with respect to themomenta. Changing coordinates and picking up one of

7

TABLE II. Same as Table I, but for the N = 2 theory.

K [G(2)(N, 1)]K [G(∂p)(N, 1)]K [G(4)(N, 1)]K [G(2,1)(N, 1)]K

0 1 2 0 0

1 −2 −8 −6 − 32

2 203

94027

63 50336

3 − 204172

− 911354

− 31516

−108.435

4 142.078 904.550 4179.733 824.861

5 −809.263 −5318.618 −33562.531 −6433.962

. . .

10 1.993 · 107 1.267 · 108 2.114 · 109 3.906 · 108

the possible components (because they all give the samecontribution) we get

Ci1 i2(t) =δi1 i2N

C(2)(t), (45)

with

C(2)(t) ≡ N〈qı1(0)qı1(t)〉 = N Cı1 ı1(t) = N C(2)(t) ,(46)

where C(2)(t) = C(2)ı1 ı1

(t) is equal to any nonvanishingelement within the internal group structure. So, ı1 canassume one of the N possible values. We can thereforechoose for example ı1 = 1, and write

〈qı1=1(0)qı1=1(t)〉 = 〈0|r cos θ1 e−(H−E0)|t| r cos θ1|0〉

=∑~η

|〈0|r cos θ1|n, ~η〉|2 e−(En−E0)|t| . (47)

We have defined |~η〉 = |`, `1, . . . , `N−2〉 as a vector rep-resenting all the angular quantum numbers of the state.The state |n, ~η〉 is characterized by the principal quan-tum number n, and the set of angular quantum numbers~η. For the cases N = 2 and N = 3 under investiga-tion here, the angular integration lead to the followingpicture. First, one observes that the summation over allpossible quantum number summarized in ~η selects thosestates which are coupled to the ground state by a dipoletransition. We define the coordinate system so that (inN = 3) the quantization axis is aligned with the coordi-nate i1 = 1, so that r cos θ1 is the coordinate along thequantization axis. In N = 2, there exists no quantumnumber `1; we only have one angular momentum quan-tum number, `, which can take either positive or negativeinteger values. By contrast, for N = 3, one has two quan-tum numbers, namely, the angular monentum ` and themagnetic projection `1 = m.

For N = 2, there are two nonvanishing contributionsto the sum over intermediate states in Eq. (47), namely,those with ` = ±1. The contributions from ` = −1 and` = +1 are equal to each other and can be taken intoaccount on the basis of an additional multiplicity factor.For N = 3, instead, the only nonvanishing contributionto the sum over intermediate states in Eq. (47) is givenby states with ` = 1. States with ` = 1 are commonlyreferred to as P states in atomic physics [22], and thesecan have three angular momentum projections, namely`1 = m = −1, 0, 1. Of theses, under an appropriate iden-tification of the quantization axis, only the state with`1 = m = 0 contributes, being coupled by the oper-ator r cos θ1. However, as is well known from atomicphysics [22], the final result is independent of the choiceof the quantization axis, provided one sums over m, viz.,sums over ~η.

The two-point correlation function at zero momentumtransfer is obtained by integrating Eq. (46) with respectto time,∫ +∞

−∞C(2)(t) dt = 2

∑n,~η

|〈0|r cos θ1|n, ~η〉|2

En − E0. (48)

Similarly, we get its second derivative at zero momentumtransfer as∫ +∞

−∞t2 C(2)(t) dt = 4

∑n,~η

|〈0|r cos θ1|~η〉|2

(En − E0)3. (49)

As discussed, each matrix element 〈0|r cos(θ)|n, ~η〉 canbe computed using Eq. (A1), in terms of a radial transi-

tion matrix element S00,n`, and an angular element α(N)0,` ,

which depends on the dimension N . After the summa-tion over the angular quantum numbers, one obtains the

8

result∑~η

|〈0|r cos(θ1)|n, ~η〉|2 =∑`

(S00,n`)2(α

(N)0,`

)2

. (50)

where∑` is a sum over a single term ` = 1 for N = 3,

and over ` = ±1 for N = 2.

It is convenient to define the quantity

α(N)0 =

4(α

(2)0,1

)4

for N = 2 ,(α

(3)0,1

)4

for N = 3 .(51)

Calculating the square root of α(N)0 , the previously men-

tioned multiplicity factor two for N = 2 is obtained;it takes care of the two equivalent contributions from` = ±1. We therefore have for the two-point correlationfunction computed at zero momentum,∫ +∞

−∞C(2)(t) dt = 2

√α

(N)0

∑n

S200,n1

En − E0

∼ 1

N

∑K

[G(2)(N, 1)]K gK . (52)

Perturbative coefficients [G(2)(N, 1)]K for N = 2 andN = 3 are given in Tables II and III, respectively. Thesecond derivative of the two-point correlator is given asfollows,∫ +∞

−∞t2 C(2)(t) dt = 4

√α

(N)0

∑n

S200,n1

(En − E0)3.

∼ 1

N

∑K

[G(∂p)(N, 1)]K gK . (53)

Again, low-order perturbative coefficients [G(∂p)(N, 1)]Kfor N = 2 and N = 2 are given in Tables II and III, re-spectively. At this point, all matrix elements have beenreduced to radial integrals of unperturbed O(N) oscilla-tor eigenstates. In the evaluation of perturbative coeffi-cients of the energy levels of a number of anharmonic os-cillators, recursion relations have been found [23, 24]. Forthe quartic O(N) oscillator, we refer to Eqs. (22) and (23)of Ref. [23], for the double-well potential to Eqs. (69)and (70) of Ref. [23], and for more general potentials, toEqs. (12)—(25) of Ref. [24]. We do not attempt to gen-eralize the treatment outlined in Refs. [23, 24] to the cor-relation functions investigated here, because the numberof higher-order perturbative coefficients obtained usingthe methods delineated here is fully sufficient for detect-ing the two-loop corrections to the large-order behaviorof the perturbative coefficients [15]. For possible futureinvestigations, we note that it would be interesting to ex-plore recursion relations for the perturbative correlationfunctions.

Correction to the Large–Order Asymptotics:Ground–State Energy for N = 1, 2, 3

(a) Ground–State Energy for N = 1

(b) Ground–State Energy for N = 2

(c) Ground–State Energy for N = 3

FIG. 1. Coefficients of the perturbative expansion of theground-state of the energy for N = 1, N = 2, N = 3 infunction of the inverse of the order of perturbation (bluedots). These coefficients have been divided by their leadingasymptotic estimate, given in Eq. (85), and have been com-pared with their subleading order estimate. Dashed blacklines include the term b − 1 term at the denominator of the1/(K+ b−1) term in Eq. (87), while solid black lines excludethis term, and follow only the 1/K term given in Eq. (86).

9

TABLE III. Same as Tables I and II, but for the N = 3 theory.

K [G(2)(N, 1)]K [G(∂p)(N, 1)]K [G(4)(N, 1)]K [G(2,1)(N, 1)]K

0 1 2 0 0

1 − 52

−10 −6 − 32

2 24524

144527

1532

30518

3 − 601751152

− 271385864

− 45836

−157.199

4 309.971 2007.796 7184.900 1409.880

5 −2058.802 −13870.761 −67294.482 −12793.097

. . .

10 8.948 · 107 5.986 · 108 7.588 · 109 1.374 · 109

C. Four-point correlation function

Similarly to the previous subsection, we apply the sameideas to the connected four-point function,

Ci1 i2 i3 i4(t1, t2, t3, t4) =1

N(N + 2)

× (δi1 i2 δi3 i4 + δi1 i3 δi2 i4 + δi1 i4 δi2 i3)

× C(4)(t2 − t1, t3 − t1, t4 − t1) , (54)

where, according to Eq. (44),

C(4)(t2, t3, t4) =N(N + 2)

3C(4)(t2, t3, t4)

≡ N(N + 2)

3

[〈qı1(0)qı1(t2)qı1(t3)qı1(t4)〉

− C(2)(−t2)C(2)(t3 − t4)− C(2)(t3 − t2)C(2)(−t4)

− C(2)(−t3)C(2)(t2 − t4)

]. (55)

In our derivation leading to Eq. (44), we had stressed thatthe formula is valid for every value that ı1 can assume.We now select, for convenience, one particular compo-nent with ı1 = 1, which, as we assume, is aligned withthe quantization axis for the states in the internal O(N)symmetry group. Hence, we write the relation

〈qı1=1(0)qı1=1(t2)qı1=1(t3)qı1=1(t4)〉= 〈0|r cos θ1e−(H−E0)∆3 r cos θ1 e−(H−E0)∆2

r cos θ1 e−(H−E0)∆1r cos θ1|0〉+

− C(2)(∆1)C(2)(∆3)− C(2)(∆1 + ∆2)C(2)(∆2 + ∆3)

− C(2)(∆1 + ∆2 + ∆3)C(2)(∆2),

where ∆i ≡ ti+1 − ti. We then have, with x1 = r cos θ1,

∫ +∞

−∞dt2

∫ +∞

−∞dt3

∫ +∞

−∞dt4 C

(4)(t2, t3, t4) =

= 24∑

~η,~η′, ~η′′

〈0|x1|~η〉〈~η|x1|~η′〉〈~η′|x1| ~η′′〉〈 ~η′′|x1|0〉(En − E0) (En′ − E0) (En′′ − E0)

− 24∑~η,~η′

|〈0|x1|~η〉|2 |〈0|x1|~η′〉|2(En + En′ − 2E0)

2(En′ − E0)2(En − E0)2. (56)

Using Eq. (A1), we have

∫ +∞

−∞dt2

∫ +∞

−∞dt3

∫ +∞

−∞dt4 C

(4)(t2, t3, t4)

= 24∑

n,`,n′,`′

1

En′ − E0

[α

(N)0,` α

(N)`,`′

S00,n`Sn`,n′`′

En − E0

]2

− 24∑

n,`,n′,`′

(α

(N)0,`′

)2

S200,n` S

200,n′`′

(En′ − E0)2 (En − E0). (57)

The multiplicities are different if we choose N = 2or N = 3. We can insert them in the definition of thequantity

α(N)` =

4(α

(2)0,1

)4

δ`,0 + 2(α

(2)0,1α

(2)1,2

)2

δ`,2 for N = 2 ,(α

(3)0,1

)4

δ`,0 +(α

(3)0,1α

(3)1,2

)2

δ`,2 for N = 3 ,

(58)which depend on the factors given in Eqs. (A2), (A3).

10

Using the angular selection rules, we finally obtain∫ +∞

−∞dt2

∫ +∞

−∞dt3

∫ +∞

−∞dt4 C

(4)(t2, t3, t4)

= 24∑n′

∑`′=0,2

α(N)`′

En′ − E0

[∑n

S00,n1Sn1,n′`′

En − E0

]2

− 24 α(N)0

∑n,n′

S200,n1 S

200,n′1(En + En′ − 2E0)

2(En′ − E0)2(En − E0)2

∼ 3

N(N + 2)

∑K

[G(4)(N, 1)]K gK . (59)

Results for low-order perturbative coefficients for thefour-point correlation function (N = 2 and N = 3) aregiven in Tables II and III.

D. Correlation function with a wigglet insertion

The last quantity we want to look at, within the con-text of theO(N) theory, is the two-point correlation func-tion with a wigglet insertion,

C(1,2)i1,i2

(t2, t3) =δi1 i2N

C(1,2)(t2, t3) , (60)

where we have defined

C(1,2)(t2, t3) ≡ N C(1,2)ı1 ı1

(t2, t3) = N C(1,2)(t2, t3) . (61)

Here, C(1,2)(t2, t3) = C(1,2)ı1 ı1

(t2, t3) is equal to any nonvan-ishing element within the internal group structure; non-vanishing elements have their first index equal to theirsecond index. One finds

C(1,2)(t2, t3) = 〈qı1(0) qı1(t2) q2ı1(t3)〉

− 〈qı1(0) qı1(t2)〉 〈q2ı1(t3)〉

− 2〈qı1(0) qı1(t3)〉〈qı1(t2) qı1(t3))〉 . (62)

Setting ı1 = 1 for simplicity and introducing the quantityx1 = r cos θ1, for the coordinate along the quantizationaxis, as before, we can perform the integrals with respectto the time variables, obtaining∫ +∞

−∞dt2

∫ +∞

−∞dt3 C

(1,2)(t2, t3)

= 4∑

n,~η,n′,~η′

〈0|(x1)2|n, ~η〉〈n, ~η|x1|n′, ~η′〉〈n′, ~η′|x1|n, 0〉(En − E0) (En′ − E0)

+ 2∑

n,~η,n′,~η′

〈0|x1|n, ~η〉〈n, ~η|(x1)2|n′, ~η′〉〈n′, ~η′|x1|0〉(En − E0) (En′ − E0)

+

−2∑n,~η

|〈0|x1 |n, ~η〉|2〈0| (x1)2 |0〉(En − E0)2

−4

∑n,~η

|〈0|x1 |n, ~η〉|2

(En − E0)

2

− 8∑

n,~η,n′,~η′

|〈0|x1|n, ~η〉|2| 〈0|r cos θ1|n′, ~η′〉|2

(En′ − E0) (En′ + En − 2E0). (63)

Using the radial S matrix elements defined in Eq. (A1),and the radial Q matrix elements defined in Eq. (A5), weobtain∫ +∞

−∞dt2

∫ +∞

−∞dt3 C

(1,2)(t2, t3) =

4(2− δN 3)∑n,n′

∑`=0,2

β(N)0,` α

(N)`,1 α

(N)1,0 Q00,n`Sn`,n′1Sn′1,00

(En − E0)(En′ − E0)

+ 2(3− 2δN 3)∑n,n′

β(N)1,1

(α

(N)1,0

)2

S00,n1Qn1,n′1Sn′1,00

(En − E0)(En′ − E0)

− 2(2− δN 3)∑n

(α

(N)0,1

)2

β(N)0,0

S200,n1Q00,00

(En − E0)2

− 8(2− δN 3)2∑n,n′

(α

(N)0,1

)2 (α

(N)0,1

)2

S200,n1S

200,n′1

(En′ − E0)(En′ + En − 2E0)

− 4(2− δN 3)2

[∑n

(α

(N)0,1

)2 S200,n1

En − E0

]2

. (64)

We define the angular factors

β(N)` ≡

2β

(2)0,`α

(2)`,1α

(2)1,0 (N = 2) ,

β(3)0,`α

(3)`,1α

(3)1,0 (N = 3) ,

(65)

γ(N) ≡

3β(2)1,1

(α

(2)1,0

)2

(N = 2) ,

β(3)1,1

(α

(3)1,0

)2

(N = 3) .(66)

Using the expressions given in Eqs. (A2), (A3), (A6),and (A7) for the angular factors, we finally get∫ +∞

−∞dt2

∫ +∞

−∞dt3 C

(1,2)(t2, t3) =

4∑n,n′

∑`=0,2

β(N)`

Q00,n`Sn`,n′1Sn′1,00

(En − E0)(En′ − E0)

+ 2γ(N)∑n,n′

S00,n1Qn1,n′1Sn′1,00

(En − E0)(En′ − E0)

− 2β(N)0

∑n

Q00,00 S200,n1

(En − E0)2− 4α

(N)0

[∑n

S200,n1

En − E0

]2

− 8α(N)0

∑n,n′

S200,n1S

200,n′1

(En′ − E0)(En′ + En − 2E0)

∼ 1

N

∑K

[G(1,2)(N, 1)]K gK . (67)

Results for perturbative coefficients [G(1,2)(N, 1)]K forthe cases N = 2 and N = 3, for low orders of perturba-tion theory, are given in Tables II and III, respectively.This concludes our discussion of the formalism used forobtaining higher orders of perturbation theory for the

11

correlation functions discussed in this articles. We cannow proceed to the comparison with the analytic large-order estimates, and the subleading corrections, evalu-ated in Ref. [15].

IV. COMPARISON WITH TWO–LOOPCORRECTIONS FOR LARGE ORDER

A. Analytic Formulas

In this section, we briefly review the results obtainedin [15] concerning the large order behaviour of the groundstate energy and of an M -point correlation function G fora D-dimensional field theory with N components, whichcan be traced to the two-loop corrections to the instantonconfigurations that describe the leading factorial growthof the perturbative coefficients.. We refer to the pertur-

bative coefficient of order K as G(M)K . When K is large,

we can express G(M)K to order 1/K as

G(M)K =

c(N,D)

πΓ (K + b)

(1

A

)b (− 1

A

)K[1

−Ad(N,D)

K + b− 1+O(K−2)

], b =

M +N +D − 1

2,

(68)

where A = 4/3 is the action of the φ4 theory evaluatedon the instanton saddle-point multiplied by the couplingconstant and with an inverted sign. For large K, we canreplace K − 1 + b→ K in the denominator of the secondterm and identify the 1/K-correction.

We start with the ground-state energy obtained us-ing the relation in Eq. (48) computing the bracket ofthe perturbative Hamiltonian with the unperturbed andperturbed ground state eigenfunctions. Specifically, oneneeds to examine the relation∫ +∞

−∞C(0)(t) dt ∼

∑K

[G(0)(N, 1)]K gK . (69)

Here, [G(0)N,1]K is the specialization of the general asymp-

totic perturbative coefficient GK of order K given inEq. (68) to the ground-state energy function (M = 0)in one spatial dimension (D = 1) and for an internalO(N) symmetry group,

[G(0)(N, 1)]K =c(0)(N, 1)

πΓ (K + b)

(1

A

)b (− 1

A

)K×[1− Ad(0)(N, 1)

K + b− 1+O(K−2)

], b =

N

2, (70)

where we defined

c(0)(N, 1) = 2π2 8N/2

Γ(N/2), (71a)

d(0)(N, 1) =5

24+

5

2π2− 7ζ(3)

4π2

+

(9

16− 1

2π2+

7ζ(3)

4π2

)N +

7

32N2 .

(71b)

For the two point correlation function given in Eq. (48)one needs to examine the relation∫ +∞

−∞C(2)(t) dt ∼

∑K

[G(2)(N, 1)]K gK . (72)

Here, [G(2)N,1]K is the specialization of the general asymp-

totic perturbative coefficient GK of order K given inEq. (68) to the two-point correlation function (M = 2) inone spatial dimension (D = 1) and for an internal O(N)symmetry group,

[G(2)(N, 1)]K =c(2)(N, 1)

πΓ (K + b)

(1

A

)b (− 1

A

)K×[1− Ad(2)(N, 1)

K + b− 1+O(K−2)

], b = 1 +

N

2,

(73)

where we defined

c(2)(N, 1) = 2π2 8N/2

Γ(N/2), (74a)

d(2)(N, 1) =5

24+

5

2π2− 7ζ(3)

4π2

+

(9

16− 1

2π2+

7ζ(3)

4π2

)N +

7

32N2 .

(74b)

For the second derivative of the two-point correlationfunction at zero momentum, the asymptotic relationshipis given in Eqs. (49),∫ +∞

−∞t2 C(2)(t) dt ∼

∑K

[G(∂p)(N, 1)]K gK , (75)

where

[G(∂p)(N, 1)]K =c(∂p)(N, 1)

πΓ (K + b)

(1

A

)b×(− 1

A

)K [1− Ad(∂p)(N, 1)

K + b− 1+O(K−2)

],

b = 1 +N

2, (76)

12

Two–Loop Correction to Large–Order Asymptotics for N = 1

(a) Two-point correlation function (N = 1) (b) Derivative of two-point correlator (N = 1)

(c) Four-point correlation function (N = 1) (d) Correlator with wigglet insertion (N = 1)

FIG. 2. Same as Fig. 1, but the correlation functions in N = 1.

with

c(∂p)(N, 1) = − π4 8N/2

Γ(N/2), (77a)

d(∂p)(N, 1) =5

24+

4

π4− 21 ζ(3)

π4− 93 ζ(5)

2π4(77b)

+

(− 3

16− 6

π4+

93ζ(5)

2π4

)N +

7

32N2 .

For the four-point correlation function, the relationshipis, from Eqs. (59),

∫ +∞

−∞dt2

∫ +∞

−∞dt3

∫ +∞

−∞dt4 C

(4)(t2, t3, t4)

∼∑K

[G(4)(N, 1)]K gK , (78)

where

[G(4)(N, 1)]K =c(4)(N, 1)

πΓ (K + b)

(1

A

)b (− 1

A

)K×[1− Ad(4)(N, 1)

K + b− 1+O(K−2)

], b = 2 +

N

2,

(79)

with

c(4)(N, 1) = 4π4 8N/2

Γ(N/2), (80a)

d(4)(N, 1) =5

24+

13

π2− 7 ζ(3)

2π2(80b)

+

(9

16− 1

π2+

7ζ(3)

2π2

)N +

7

32N2 .

For the two-point correlation function with a wiggletinsertion, the relationship is, from Eqs. (67),

∫ +∞

−∞dt2

∫ +∞

−∞dt3 C

(1,2)(t2, t3)

∼∑K

[G(1,2)(N, 1)]K gK , (81)

where

G(1,2)K =

c(1,2)(N, 1)

πΓ (K + b)

(1

A

)b (− 1

A

)K×[1− Ad(1,2)(N, 1)

K + b− 1+O(K−2)

], b = 2 +

N

2,

(82)

13

Two–Loop Correction to Large–Order Asymptotics for N = 2

(a) Two-point correlation function (N = 2) (b) Derivative of two-point correlator (N = 2)

(c) Four-point correlation function (N = 2) (d) Correlator with wigglet insertion (N = 2)

FIG. 3. Same as Fig. 2, but for N = 2.

with

c(1,2)(N, 1) = 8π2 8N/2

Γ(N/2), (83a)

d(1,2)(N, 1) =35

24+

5

2π2− 7 ζ(3)

4π2(83b)

+

(15

16− 1

2π2+

7ζ(3)

4π2

)N +

7

32N2 .

For reference, the first few perturbative coeffi-cients [G(2)(N, 1)]K , [G(∂p)(N, 1)]K , [G(4)(N, 1)]K , and[G(1,2)(N, 1)]K , for K = 0, 1, 2, 3, 4, 5 and 10, for the in-ternal O(N) symmetry group, with N = 1, 2, 3, are alsogiven in Tables I—III.

B. Significance of the Two–Loop Correction

In Figs. 1—4, we plot the asymptotic expression of thecoefficients of the perturbative expansion of the ground-state energy and of the correlation functions as a func-tion of the inverse of the order of perturbation K. Thesecoefficients have been divided by their leading order ex-pression reported in Eq. (68), that is the expression pro-

portional to

[Q(M)]K =c(N, 1)

πΓ (K + b)

(1

A

)(M+N)/2 (− 1

A

)K.

(84)and they read

[Ξ(M)]K =[G(M)]K[Q(M)]K

. (85)

These coefficients have been compared with their next-to-leading order estimate, i.e., with the multiplicative term

[Ξ(M)]K ≈ µ(M)K = 1−Ad(M,N)

K, (86)

and with

[Ξ(M)]K ≈ ν(M)K = 1−A d(M,N)

K + b− 1

= 1−Ad(M,N)

K+A

(b− 1)d(M,N)

K2+O

(1

K

)3

,

(87)

which carries a (part of the) next-to-next-to-leading1/K2 correction term.

In Figs. 1–4, we observe a good agreement betweenthe asymptotic estimate of the perturbative coefficients

14

Two–Loop Correction to Large–Order Asymptotics for N = 3

(a) Two-point correlation function (N = 3) (b) Derivative of two-point correlator (N = 3)

(c) Four-point correlation function (N = 3) (d) Correlator with wigglet insertion (N = 3)

FIG. 4. Same as Fig. 2, but for N = 3.

of the correlation functions obtained in Ref. [15] with theexplicit higher-order calculations reported here. Indeed,the improvement of the agreement upon the inclusion ofthe next-to-leading order correction is quite remarkable.The calculation of the large-order behavior of the pertur-bative expansion of the correlation functions for N > 1was much more computationally expensive with respectto the case N = 1, therefore it was possible to obtain lessorders of perturbation.

V. CONCLUSIONS

In this article, we have discussed the explicit higher-order calculation of the perturbative expansions of corre-lation functions for the O(N) quartic anharmonic oscilla-tor. We discussed the N = 1 quantum anharmonic oscil-lator in Sec. II. In Sec. III, we discussed the formulationof the perturbative expansion of the correlation functionsof the O(N) quantum anharmonic oscillator, where theinternal symmetry group is assumed to be O(2) or O(3),and general formulas are given which allow us to enter aunified evaluation of the perturbative expansions. Specif-ically, we considered the two-point correlation function inSec. III B, the four-point correlator in Sec. III C, and thecorrelation function with a wigglet insertion in Sec. III D.

The comparison with analytic results together with a re-view of the previously (Ref. [15]) obtained results for thelarge-order behavior of the correlation functions was car-ried out in Sec. IV. The data in Figs. 1—3 underlinethe importance of the next-to-leading order correction tothe large-order factorial growth of the perturbative coef-ficients for the demonstration of the agreement of asymp-totic estimates and explicit perturbative calculations.

Let us take, as an example, the coefficient of order g8

(the “eight-loop coefficient”) for the two-point correlationfunction in the O(3) model. The explicit result is

[G(4)N=3]K=8 ' 3.03 · 106 . (88)

The leading asymptotic term is

[G(4)N=3,1]K=8 ≈ 8.87 · 106 . (89)

With the inclusion of the two-loop (order 1/K) correc-tion, we find the (much better) estimate

[G(4)N=3,1]K=8 ≈ 2.38 · 106 . (90)

which differs from the the exact perturbative coefficientby roughly 20 percent. At order K = 10, we alreadyobserve 93 percent agreement, whereas at order K = 14,the agreement it is slightly better than 97 percent. In

15

some other cases, the agreement is surprisingly good evenat very low orders. For example, for N = 2, the two-looplarge-order estimate of the coefficient of the four-pointcorrelation function agree with the exact perturbativecoefficient at the level of 98 percent, in eight-loop order(K = 8).

The tests presented here are essential to have a goodstarting point from which to extend the calculations tofield theory, i.e., to the case D > 1, where this type ofchecks are not possible anymore. Specifically, the agree-ment between these two different approaches ensures thecorrectness of the method described in [15], which can bethen generalized to obtain the perturbative expression ofthe correlation functions for two and three dimensionalN -vector model where is not possible to use the conven-tional techniques of perturbation theory. We recall that,irrespective of the dimension D, the same dispersion rela-tion relates the large-order growth of the coefficients of acorrelation function with the behaviour at small orders ofperturbation of its imaginary part for negative coupling.

ACKNOWLEDGEMENTS

This work has been supported by the National ScienceFoundation (Grant No. PHY–2110294), by the SwedishResearch Council (Grant No. 638–2013–9243) and by theSimons Foundation (Grant No. 454949).

Appendix A: Some useful definitions and identities

1. Matrix elements

In this section, we will derive the expression for thevarious angular matrix elements denoted as α, and β,which appear in the main text. We start consideringthe matrix element 〈n, ~η|r cos θ1|n′, ~η′〉. If the coordinater cos θ1 is aligned with the quantization axis, then theonly nonvanishing transition matrix element will be ob-tained for all magnetic projections equal to zero. Wecan therfore assume that |n, ~η〉 ≡ |n, `, 0, . . . , 0〉 and|n′, ~η′〉 ≡ |n′, `′, 0, . . . , 0〉. Under these assumptions, wecan write the matrix element as

〈n, ~η|r cos θ1|n′, ~η′〉 = Sn`,n′`′ α(N)`,`′ (A1a)

where the radial part is given as follows,

Sn`,n′`′ ≡∫dr rNRn`(r)Rn′`′(r) (A1b)

α(N)`,`′ ≡

∫dθ1 . . . dθN−1 cos(θ1)J(θ1, . . . , θN−1)

× Y 0...0∗` (θ1, . . . , θN−1)Y 0...0

`′ (θ1, . . . , θN−1) ,(A1c)

and J(θ1, . . . , θN−1) is the Jacobian due to the hyper-spherical change of coordinates.

Due to the orthogonality relations between the hyper-

spherical harmonics, only few of the α(N)`,`′ terms are non-

zero. For example, if ` = 0, then we have for N = 2 andN = 3 respectively,

α(2)0,`′ =

1

2(δ`′,1 + δ`′,−1) , (A2a)

α(3)0,`′ =

1√3δ`′,1 . (A2b)

When ` = 1, one instead obtains

α(2)1,`′ =

1

2(δ`′,0 + δ`′,2) , (A3a)

α(3)1,`′ =

1√3δ`′,0 +

2√15δ`′,2 . (A3b)

In general, one has

α(2)`,`′ =

1

2(δ`′,`+1 + δ`′,`−1) . (A4)

For the two-point function with a wigglet inser-tion, we also have to consider the matrix element〈n, ~η|(r cos θ1)2|n′, ~η′〉 where |n, ~η〉 ≡ |n, `, 0, . . . , 0〉 and|n′, ~η′〉 ≡ |n′, `′, 0, . . . , 0〉. It can be written as

〈n, ~η|(r cos θ1)2|n′, ~η′〉 = Qn`,n′`′ β(N)`,`′ , (A5a)

where

Qn`,n′`′ ≡∫

dr rN+1Rn`(r)Rn′`′(r) , (A5b)

β(N)`,`′ ≡

∫dθ1 . . . dθN−1 (cos(θ1))2J(θ1, . . . , θN−1)

× Y 0...0∗` (θ1, . . . , θN−1) Y 0...0

`′ (θ1, . . . , θN−1) .(A5c)

Also in this case, because of the orthogonality relationsbetween the hyperspherical harmonics, only few of the

β(N)`,`′ terms are non-zero. For example, if ` = 0, then we

have for N = 2 and N = 3, respectively

β(2)0,`′ =

1

4(2δ`′,0 + δ`′,2 + δ`′,−2) , (A6a)

β(3)0,`′ =

1

3

(δ`′,0 +

2√5δ`′,2

). (A6b)

When ` = 1, instead, one has

β(2)1,`′ =

1

4(δ`′,−1 + 2δ`′,1 + δ`′,3) , (A7a)

β(2)−1,`′ =

1

4(2δ`′,−1 + δ`′,1 + δ`′,−3) (A7b)

β(3)1,`′ =

1

5

(3 δ`′,1 + 2

√3

7δ`′,3

). (A7c)

The above formulas can be used to perform all requiredangular integrals for the correlation functions consideredin our investigations.

16

2. Case N = 3

For N = 3, the coefficients α(N=3)`,`′ and β

(N=3)`,`′ can be

written in terms of the Gaunt coefficients Y ``′`′′

mm′m′′ definedas the integral over three spherical harmonics

Y ``′`′′

mm′m′′ =

∫ 2π

0

dϕ

∫ π

0

dθ sin(θ)

× Y ∗`m(θ, ϕ)Y`′m′(θ, ϕ)Y`′′m′′(θ, ϕ) .

(A8)

By writing cos(θ) and cos(θ)2 in terms of spherical har-monics we get

α(3)`,`′ ≡ 2

√π

3Y `1`

′

000 , (A9)

and

β(3)`,`′ ≡

1

3

(4

√π

5Y `2`

′

000 + 2√

2Y `0`′

000

). (A10)

Alternatively, the integral appearing in Eq. (A8) can beinterpreted using the Wigner-Eckart theorem and can bewritten as the product of the Clebsch-Gordan coefficientcorresponding to the quantum numbers `, `′, `′′, m, m′,and m′′, and the reduced matrix element of the sphericalharmonic tensor Y m

′

`′ , as

∫ 2π

0

dϕ

∫ π

0

dθ sin(θ)Y ∗`m(θ, ϕ)Y`′m′(θ, ϕ)Y`′′m′′(θ, ϕ)

= (−1)`′−`′′+`C

`m`′m′`′′m′′√

2`+ 1〈`||~Y`′ ||`′′〉, (A11)

where the reduced matrix can be expressed using a 3jsymbol with three zero magnetic projections

〈`||~Y`′ ||`′′〉 = (−1)`√

(2`+ 1)(2`′ + 1)(2`′′ + 1)

4π

×

(` `′ `′′

0 0 0

). (A12)

Using this convention the coefficients α(3)`,`′ and β

(3)`,`′ will

read

α(3)`,`′ ≡ (−1)1−`′+` 2

√π

3

C`m10`′m′√2`+ 1

〈`||~Y1||`′〉 , (A13)

and

β(3)`,`′ ≡

1

3

((−1)2−`′+`4

√π

5

C`m20`′m′√2`+ 1

〈`||~Y2||`′〉

+ (−1)−`′+`2√

2C`m00`′m′√

2`+ 1〈`||~Y0||`′〉

). (A14)

These results are in agreement with the formulas ob-tained in Appendix A 1.

Appendix B: Alternative Procedure

The perturbative treatment of the radial part can beaccomplished by a direct mapping of the procedure out-lined in Eqs. (2)—(8) onto a computer algebra system.However, it is useful to delineate an alternative procedureto compute the eigenvalues and eigenfunctions of the per-turbed three dimensional harmonic oscillator. We willdeal with the N = 3 case; the generalization to N = 2 isrelatively straightforward. We adapt to our case the re-sults of Ref. [25], where the computation has been carriedout for a general central field perturbation. The eigen-values and eigenfunctions of our Schrodinger equation

(−~∇2 + r2 +

g

2r4)

Ψn`m(r, θ, ϕ)

= αn`Ψn`m(r, θ, ϕ) , αn` = 2En` , (B1)

where En` can be written as a perturbative series in thecoupling parameter g,

αn` =

∞∑K=0

αn`,K gK , (B2a)

Ψn`m(r, θ, ϕ) = Rn`(r)Y`m(θ, ϕ) , (B2b)

Rn`(r) = Nn` e−r2/2 r`

∞∑K=0

u(K)n` (r2) gK , (B2c)

where Nn` = N (N=3)n` is defined in Eq. (30). Each coef-

ficient uin`(r2) can be expressed as a linear combination

of the eigenfunctions of the unperturbed case (it can beshown that only 4K terms will contribute to the Kthorder perturbation)

u(K)n` (r2) =

qn`+2K∑j=max(qn`−2K,0)

AKj L`+ 1

2j (r2) (B3a)

qn` = αn`,0 − 3− 2` =1

2(n− `). (B3b)

The coefficients αn`,K and AKj can be determined from

Eqs. (32) and (33) of Ref. [25] by setting Bj =δ1j2 . Spe-

cial cases are

A0,j = δj,qn`, AK,qn`

= 0 ∀K > 0 . (B4)

However, there are some typos in the passages of the pa-per and we report here a corrected version of the mainpassages needed to arrive to the result. We use the follow-ing recursive relation of the generalized Laguerre polyno-mials,

rmL`+ 1

2j (r2) =

2m∑n=0

a(m,n, j)L`+ 1

2j+m−n(r2) , (B5)

17

where

a(m,n, j) =(−1)m−n(m!)2

Γ(j +m− n+ `+ 32 )

×min(j+m−n,j)∑k=max(0,j−n)

(j+m−n

k

)Γ(m+ k + `+ 3

2 )

(j − k)!(n− j + k)!(m− j + k)!. (B6)

We can then rewrite Eq. (26) of [25] as

∞∑j=0

(qn` − j)AKj L`+ 1

2j (r2)

=1

8

K−1∑w=0

δ1(K−w)

∞∑j=0

Awj

2(K−w+1)∑i=0

a(K − w + 1, i, j)

× L`+12

j+K−w+1−i(r2)− 1

4

K−1∑w=0

α(K−w)n`

∞∑j=0

AwjL`+ 1

2j (r2)

=1

8

∞∑j=0

A(K−1)j

j+2∑i=j−2

a(2, j + 2− i, j)L`+12

i (r2)

− 1

4

K−1∑w=0

α(K−w)n`

∞∑j=0

AwjL`+ 1

2j (r2) . (B7)

For a specific value of L`+ 1

2s (r2), with the convention

Θ(0) = 1 (B8)

for the Heaviside step function Θ, we get

AKs =1

8(qn` − s)

∞∑j=0

A(K−1)j a(2, j + 2− s, j)

×Θ(2− |s− j|)− 1

4(qn` − s)

K−1∑w=0

α(K−w)n` Aw,s

=1

8(qn` − s)

qn`+2(K−1)∑max(j=qn`−2(K−1),0)

A(K−1)j a(2, j+2−s, j)

− 1

4(qn` − s)

K−1∑w=0

α(K−w)n` Aw,s αn`,K

=1

2

qn`+2(K−1)∑j=max(qn`−2(K−1),0)

A(K−1)ja(2, j + 2− qn`, j) .

(B9)

We can therefore write a recursive relation that will pro-vide an expression for the perturbative coefficients of theeigenvalues and eigenfunctions of the Schrodinger equa-tion. The two-dimensional case can be derived fromthree-dimensional one using the expression for the eigen-functions reported in Eq. (28).

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