From the mass gap at large N to the non-Borel-summability in Oð3Þ and Oð4Þ sigma models

Dmytro Volin

Institut de Physique Theorique, CNRS-URA, 2306, C.E.A.-Saclay, Gif-sur-Yvette F-91191, Franceand Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna Str., Kyiv 03143, Ukraine

(Received 26 August 2009; published 10 May 2010)

We give an analytical derivation of the mass gap of the vectorOðNÞ sigma model in two dimensions and

investigate a large-order behavior of the weak-coupling asymptotic expansion for the energy. For

sufficiently large N the series is sign-oscillating, which is expected from the large N solution of the

sigma model. However, for N ¼ 3 and N ¼ 4 the series have positive coefficients.

DOI: 10.1103/PhysRevD.81.105008 PACS numbers: 11.15.Tk, 02.30.Rz

I. INTRODUCTION

The two-dimensional vector OðNÞ sigma model is oftenconsidered as a toy model for the quantum chromodynam-ics. It is asymptotically free and dynamically generates amass scale � (analog of �QCD), although its classical

formulation does not contain any dimensionful parameters.It is widely accepted that the asymptotic states of the sigmamodel are the massive particles in the vector representationof the OðNÞ group. The mass of the particles should beproportional to the only mass scale of the theory:m ¼ c�.

The massm is a physical quantity, while c and� dependon the regularization scheme. The coefficient c cannot bedetermined from the perturbation theory. Luckily, theOðNÞ sigma model can be studied nonperturbatively dueto its integrability. The explicit expression for the coeffi-

cient c in the MS scheme was found in [1,2]:

c ¼�8

e

�1=ðN�2Þ 1

�½1þ 1N�2�

: (1)

We explain the idea of this calculation in the next section.In order to obtain c, one has to solve the integral Eq. (5) inthe weak-coupling regime at the leading and subleadingorder. At the leading order the solution was found by theapplication of the generalized Wiener-Hopf method (for areview of the method, see for example [3]). The subleadingorder was found only numerically, although with a highprecision which allowed one to conjecture the expression(1). In [4] it was mentioned (see Ref. 19 there) that J. Baloghad given an analytical solution at the subleading order.However, to our knowledge, that result had not beenpublished.

The goal of this paper is to give an analytical solution ofthe integral Eq. (5) at subleading and higher orders in arecursive manner. This gives us, in particular, an analyticalderivation of (1). Also, armed with the recursive procedure,we make an estimation for the large-order behavior of thecoefficients of the weak-coupling expansion and studyBorel-summability properties of the model.

Another motivation for this work comes from the AdS/CFT correspondence where the appearance of the Oð6Þsigma model [5] raised the necessity of the solution of

Eq. (5) and its generalization at next-to-subleading order[6].

II. INTEGRAL EQUATION

Let us consider the sigma model in the presence of anexternal field h coupled to a Uð1Þ charge. Such a systemwas initially studied in [7,8]. When the value of h exceedsthe mass gap, a finite density � of equally polarizedparticles is created. At large values of h the free energyof the system can be computed perturbatively due to theasymptotic freedom. Knowing the free energy densityf½h�, we can find the energy density "½�� through theLegendre transform

"½�� ¼ minhðf½h� þ �hÞ: (2)

It is convenient to introduce a running coupling constant�½�� via the relation

1

�þ ð�� 1Þ log� ¼ log

�2���

�MS

�; � ¼ 1

N � 2:

(3)

The perturbative quantum field theory predicts the follow-ing expansion for the energy density:

"½���2

¼ ��

��þ 1

2�2 þ�

X1n¼3

�n�n

�þO

��2MS

�2

�; (4)

where � is evaluated at the scale � ¼ �.The idea of [1,2] was that the energy of the system in the

large volume can be calculated also from the asymptoticBethe ansatz, which explicitly contains the mass m. Theenergy density "½�� can be calculated in the thermody-namic limit, in which the number of particles K and thelength of the system L both go to infinity with fixed � ¼K=L. The comparison of the result of the Bethe ansatz and(4) allows one to find the coefficient c (1).In the thermodynamic limit the Bethe ansatz reduces to

the integral equation for the density of the rapidity distri-bution �½��:

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�½�� �Z B

�BK½�� �0��½�0�d�0 ¼ mcosh½��; �2 < B2;

K½�� ¼ 1

2�i

d

d�logS0½��;

S0½�� ¼ � �½12 þ i�2���½�þ i�

2���½1þ i�

2���½12 þ �þ i�2��

=c:c:

(5)

The energy density and the density of particles are given by

" ¼ mZ B

�B

d�

2��½�� cosh½��; � ¼

Z B

�B

d�

2��½��: (6)

We see that " depends on � through the parameter B. Tocompare with the expansion (4) we have to consider thelarge �, or equivalently large B, asymptotics of the integralEq. (5).

Prior to giving a large B solution of (5), let us rewrite (5)in terms of the resolvent for the function �½��. For this wefirst notice that the kernel K½�� can be represented as

K½�� ¼ 1

2�i

�DþD2�

1þD�D�1 þD�2�

1þD�1

�1

�; (7)

where D ¼ ei�@� is a shift operator and ð1þD�1Þ�1 �1�D�1 þD�2 � � � � . This representation for the kernelcan be easily derived if one notices the formal equality

�½�i�=2�� ’ ð1=�ÞD2=ð1�D2Þ.The resolvent of �½��, defined by

R½�� ¼Z B

�Bd�0

�½�0��� �0

; (8)

is analytic everywhere in the complex plane except on thesupport ½�B;B� of �½��. The residue of R½�� at infinityequals 2��. The density distribution �½�� can be read fromthe discontinuity of the resolvent on the interval ½�B; B�:

�½�� ¼ � 1

2�iðR½�þ i0� � R½�� i0�Þ: (9)

By substituting (7) and (9) to (5) and then evaluating theintegral using (8) we obtain the following equality:

1�D2�

1þDR½�þ i0� � 1�D�2�

1þD�1R½�� i0�

¼ �2�im cosh½��; �2 < B2: (10)

III. LEADING ORDER SOLUTION FOR THEENERGY DENSITY

In the following we will neglect the terms that giveexponentially suppressed contribution to the value of ".In this approximation we have

"

m’Z B

0e��½�� d�

2�’ eB

Z 0

�1ez=2�½z� dz

4�;

� ¼ Bþ z

2:

(11)

In other words, " receives the main contribution from thevicinity of the branch points �B. Therefore we will con-sider the double scaling limit

B; � ! 1; z ¼ 2ð�� BÞ fixed: (12)

From (9) and (11) we can express the energy density interms of the inverse Laplace transform of the resolvent:

" ¼ meB

4�R½1=2�; R½s� �

Z i1þ0

�i1þ0

dz

2�ieszR½z�: (13)

In the double scaling limit Eq. (10) is simplified:

1�D2�

1þDR½zþ i0� � 1�D�2�

1þD�1R½z� i0� ¼ ��imeBez=2;

z < 0: (14)

The inverse Laplace transform of (14) is derived in theAppendix and is given by:

sin½2��s��eið1�2�Þ�sR½s� i0�cos½�ðs� i0Þ� þ e�ið1�2�Þ�sR½sþ i0�

cos½�ðsþ i0Þ��

¼ m

2ieB�

1

sþ 12 � i0

� 1

sþ 12 þ i0

�; s < 0: (15)

To find the correct solution to (15) we demand the follow-

ing analytical properties for R½s� at each order of 1=Bexpansion:

(i) R½s� is analytic everywhere except on the negativereal axis,

(ii) R½s� has simple poles at s ¼ �n=ð2�Þ and simplezeroes at s ¼ �1=2� n, where n is a positiveinteger,

(iii) R½s� is expanded in negative powers of s at infinity.

The presence of zeroes in the resolvent can be directly seenfrom Eq. (15). The solution of the correspondent homoge-neous equation should have in addition a zero at s ¼�1=2. The origin of the other stated analytical propertiesis explained in the appendix.The most general solution of Eq. (15) which satisfies the

stated analyticity properties is the following:

R½s� ¼ A�½s��

1

sþ 12

þQ½s��;

A ¼ m

4��e�ð1=2ÞþBþ��½��;

�½s� ¼ 1ffiffiffis

p eð1�2�Þs log½s=e��2�s log½2�� �½2�sþ 1��½sþ 1

2�;

Q½s� ¼ 1

Bs

X1n;m¼0

Qn;m½logB�Bmþnsn

:

(16)

The dependence of Q½s� on B is not a consequence of (15)or analytical properties of the resolvent but can be deducedfrom the considerations of the next section. We see thatQ½s� contains only a finite number of terms at each order of

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1=B expansion and therefore (16) is properly defined. Theleading order should be compared with the expression forG�½i�� of [3].

The leading order of " is given by

" ¼ mA

4��½1=2�eB: (17)

IV. THE PARTICLE DENSITYAND SUBLEADINGCORRECTIONS

First note that if we apply the operator ðD�1=2 þD1=2Þto Eq. (10) we will get

ðD�� �D�Þ � ðD��1=2R½�þ i0� þD1=2��R½�� i0�Þ¼ 0: (18)

The action of the shift operator is understood as analyticalcontinuation. In particular, if �� 1=2< 0 then the ana-

lytical continuation in D��1=2R½�þ i0� includes crossingthe cut of the resolvent.

In the previous section we found the most general solu-tion in the double scaling limit (12). Still we have to fixunknown coefficientsQn;m. For this we consider a different

regime. We take again B ! 1 but now we will be inter-ested in the values of the resolvent R½�� at the distances ofthe order B or larger from the branch points � ¼ �B. Inthis case the shift operator can be expanded in the Taylorseries D ¼ 1þ i�@� � 1

2�2ð@�Þ2 þ � � � .

Since R½�� is an odd function as it follows from (8), it iseasy to check that (18) is perturbatively equivalent to

D��1=2R½�þ i0� þD1=2��R½�� i0� ¼ 0: (19)

For example, if at the leading order (19) is given by X ¼ 0then (18) is given by @�X ¼ 0 integration which gives X ¼const. However, the constant of integration is zero due tothe parity properties of the left-hand side of (19).Solving (19) perturbatively order by order we can ex-

pand the resolvent in the inverse powers of B:

R½�� ¼ X1n;m¼0

Xmþn

k¼0

ffiffiffiffiB

pcn;m;kð�=BÞ�½k�

Bm�nð�2 � B2Þnþ1=2log

��� B

�þ B

�k;

(20)

where �½k� ¼ kmod 2. The perturbative meaning of theexpansion (20) is most easily seen in terms of the variableu ¼ �=B. The solution (20) gives us the value for theparticle density from the residue of the resolvent at infinity:

� ¼ffiffiffiffiB

p2�

�c0;0;0 þ

X1m¼1

c0;m;0 � 2c0;m;1

Bm

�: (21)

If we reexpand (20) in the double scaling limit (12), weshould recover the solution (16) obtained in the previoussection. This condition uniquely fixes all the coefficientscn;m;k and Qn;m.

The expansion (20) in the double scaling limit organizesat each order of 1=B as a 1=z expansion. Therefore weshould compare it with the Laplace transform of the small s

expansion of R½s�. As an illustration, we give here theterms of these expansions which are relevant for calcula-tion of the leading and the subleading orders of � and ":

R ¼ c0;0;0ffiffiffiz

p þ c1;0;0 þ c1;0;1 log½ z4B�

z3=2�

ffiffiffiz

p8B

c0;0;0 þ8c0;1;0 � 3c1;0;0 � 2c1;0;1 þ ð8c0;1;1 þ c1;0;1Þ log½ z

4B�8B

ffiffiffiz

p ;

Z 1

0dse�szR½s� ¼ 2Affiffiffi

zp � A

2� log½e�2 � þ 1þ ð1� 2�Þ logzz3=2

þQ0;0

ffiffiffiz

pB

��2þ 2� log2e� � 1� ð1� 2�Þ logz

z

�:

(22)

V. RESULTS AND DISCUSSIONS

From the results of the previous sections we find theexpressions for � and " at the leading and subleadingorders:

"½B�m2

¼ e2Bþ2��1

16��2��1�½��2

�1þ 1

4B

�;

�½B�m

¼ eðBþ��1=2Þ

4����½�� ffiffiffiffi

Bp

��1�

32 þ ð1� 2�Þ log8B� � log2

�

4B

�:

(23)

Resolving the parametric dependence we obtain exactly �and �2 terms in the expansion (4) if � is defined as

1

�þ ð�� 1Þ log� ¼ log

�

mþ log

��8

e

�� 2�

�½���: (24)

Comparing (24) with (3) we confirm the result (1).Comparing the solutions (16) and (20) one can find the

higher order corrections to the energy density. Up to thefirst four loops they are given by

�3 ¼ 1

2;

�4 ¼ � 1

32ð24ð3Þ�2 þ 8�2 � 42ð3Þ�� 28�þ 21ð3Þ

� 8Þ;�5 ¼ � 1

96ð456ð3Þ�3 þ 24�3 � 918ð3Þ�2 � 60�2

þ 609ð3Þ�� 140�� 105ð3Þ � 24Þ: (25)

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The two-loop result �3 coincides with the field theorycalculations in [6].

The fact that the energy can be expanded in power seriesover the running coupling constant (24) is a nontrivialproperty of the integral Eq. (5). This is a strong check ofthe validity of the bootstrap approach. This hiddenrenormalization-group dynamics of the integral equationwas shown to hold in the case of Gross-Neveu models in[3]. The case of the OðNÞ appeared to be more difficult toprove. We checked the renormalization-group dynamics inthe OðNÞ case at ten first orders of the perturbative expan-sion and then used it to perform calculations at higherorders.

We have found analytical expressions for �n up ton ¼ 26.1 This allowed us to estimate the large n behaviorof �n:

�n ’ �½n�2n�1

an½��: (26)

The dependence of an on n for particular values of � isshown in Fig. 1.

For � ¼ 0 we have an½0� ’ ð�1Þn�1. This result isconsistent with the fact that in the large N limit the IRrenormalon poles in the Borel plane are absent [10,11]. Thelarge-order behavior of the coefficients �n is thereforegoverned by the leading UV renormalon pole leading tothe Borel-summable oscillating behavior (26).

For � ¼ 1 we have an ’ 1:09 and for � ¼ 1=2 we haveanþ1 ’ n�1ð2:09� 0:43ðnmod 2ÞÞ, i.e. the series is non-Borel-summable. The Borel ambiguity is of the order�2=�2 as it should be from the field theory point of view[see (4)].

For arbitrary � � 1 the behavior of the coefficients aninterpolates between those for � ¼ 0, 1=2, and 1. Weestimate the sign oscillation of the coefficients for suffi-ciently large n and �<�c ’ 0:4. For 1 � � � �c all thean are positive. For �> 1 the asymptotic behavior of an is

given by an ’ ��1n�2�n�2, where �1 is positive. This

behavior shows the presence of the singularity in the Borelplane, position of which is proportional to N � 2.The observed non-Borel-summability in theOðNÞ sigma

models for N ¼ 3 and N ¼ 4 can be explained by thepresence of IR renormalons. For N ¼ 4 the observationis supported by the non-Borel-summability of the SUðNÞprincipal chiral field model (PCF) at largeN [12] [since theOð4Þ sigma model can be viewed as an SUð2Þ PCF model].As it follows from [11], one should expect non-Borel-summability at any finite N. Although we cannot showthis explicitly for arbitrary finite N, using the argument ofthe positiveness of an, the observed fact that an þ an�1 >0 can be naturally explained by the presence of IRrenormalons.The additional singularity in the Borel plane at the

position proportional toN � 2 corresponds to the instantonsolution, as one can see by applying the method of [13].The method used in this paper was developed in [14,15]

and with no doubts can be applied to similar systems suchas PCF or Gross-Neveu sigma models.

ACKNOWLEDGMENTS

The author thanks B. Basso, F. David, G. Korchemsky,I. Kostov, D. Serban, and J. Zinn-Justin for many usefuldiscussions. The author is grateful to B. Basso and G.Korchemsky for providing a comprehensive introductioninto the subject. This work has been supported by theEuropean Union through the ENRAGE network(Contract No. MRTN-CT-2004-005616).

APPENDIX: ANALYTICAL PROPERTIES OF R½s�1. Expansion at infinity

Since K½�� is analytic on the real axis, from (5) weconclude that �½�� is expanded around � ¼ B as

� ¼ 0 þ ð�� BÞ1 þ ð�� BÞ22 þ � � � : (A1)

This means that expansion of R½z� at z ¼ 0 is given by

R½z� ¼ � log½z��0 þ z

21 þ z2

42 þ � � �

�: (A2)

The inverse Laplace transform of (A2) gives the stated

behavior of R½s� at infinity. The expansion (A2) has a finiteradius of convergency, therefore the inverse Laplace trans-form leads to a divergent asymptotic expansion. This is

coherent with analytical properties of R½s� that wewill nowdiscuss.

2. Zeroes and poles

There is yet another representation of the integralEq. (5). To derive it we have to extend this equation tothe whole real axis. For this we introduce a new function�h½�� (density of holes) which has support complementary

N5 15 25

1

0

1

an

N 5.085 15 25

1

0

1

an

N 4.005 15 25

1

0

1

an

N 4.005 15 25

2

0

2

ann 1 an

N 3.005 15 25

1

0

1

an

N 2.895 15 25

1

0

1

an

FIG. 1. Dependence of an on n for particular values of �.

1The MATHEMATICA code for the calculations is given in [9].

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to ½�B; B�. The function �h½�� is defined to be such thatthe equation

�h½�0� þ �½�0� �Z B

�Bd�00K½�0 � �00��½�00�

¼ mcosh½�0��½B2 � ð�0Þ2� (A3)

is valid for any real �0.Integrating Eq. (A3) with the Cauchy kernel 1

���0 , we get

1�D�2�

1þD�1R½�� þ Rh½�� ¼ T½��; Im½�� _ 0; (A4)

where Rh is the resolvent for �h and T is the resolvent forthe right-hand side of (A3).

The function R½s� is defined by the inverse Laplacetransform [see (13)] and is analytic for Re½s�> 0. We

define the analytical continuation of R½s� to Re½s�< 0using the path that does not cross the ray s < 0. To study

the analytical properties of R½s� in the region Re½s�< 0,Im½s�< 0 we consider Eq. (A4) for Im½��> 0 in thedouble scaling limit and apply the integral

R�i1�0i1�0

dz2�i e

sz.

The result is written as

1� e�4i��s

1þ e�2i�sR½s� ¼ Rh½s� þm

2

eB

sþ 12

; Im½s�< 0;

Rh½s� ¼Z �i1�0

i1�0

dz

2�ieszR½z�: (A5)

Similarly we consider the region Re½s�< 0, Im½s�> 0 and

get the equation

1� e4i��s

1þ e2i�sR½s� ¼ Rh½s� þm

2

eB

sþ 12

; Im½s�> 0:

(A6)

Since Rh½z� is analytic everywhere except on the ray

z > 0 the function Rh½s� is analytic for Re½s�< 0.Therefore, by taking the difference of (A5) and (A6) fors < 0 we get (15).We conclude from (A5) and (A6) for Re½s�< 0, from

analyticity of Rh½s� for Re½s�< 0, and from analyticity of

R½s� for Re½s�> 0 that R½s� is analytic everywhere excepton the negative real axis where it has a cut. From (A5) and

(A6) it also follows that R½s�, on the ray s < 0, must havezeroes at s ¼ n=2� (except for the first one) and may havepoles only at s ¼ n. All the poles should be present in order

to get a correct behavior of R½s� at infinity. This explainsthe stated analytical structure of the resolvent.

Note that the stated analytical properties of R½s� arevalid only in the double scaling limit. The inverseLaplace transform of R½�� at finite B is an entire functionin the s plane. This phenomenon is illustrated by the

function f½s� ¼ 1s � 1

s e�ðs=BÞ. This is an entire function.

However, since we are interested in f½1=2� in the large Blimit the exponentially suppressed term can be neglectedand effectively we get a pole.

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