Differential equations and integrable models: the SU(3) case

Nuclear Physics B 571 PM 2000 583–606

www.elsevier.nlrlocaternpe

Ž .

Differential equations and integrable models: the SU 3 case

Patrick Doreya, Roberto Tateob

a

Department of Mathematical Sciences, UniÕersity of Durham, Durham DH1 3LE, UK

b

UniÕersiteit Õan Amsterdam, Inst. Õoor Theoretische Fysica, 1018 XE Amsterdam, The Netherlands

Received 5 November 1999; accepted 20 December 1999

Abstract

We exhibit a relationship between the massless aŽ2. _{integrable quantum field theory and a} 2

certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schrodinger equation. This forms part of a more general¨

correspondence involving A -related Bethe ansatz systems and third-order differential equations.2

A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the non-linear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators f , f_{12 21} and f . This is checked against₁₅ previous results obtained using the thermodynamic Bethe ansatz. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 03.65.-Ge; 11.15.Tk; 11.25.HF; 11.55.DS

Keywords: Conformal field theory; Bethe ansatz; Ordinary differential equations; Spectral problems

1. Introduction

A curious connection between certain integrable quantum field theories and the

w x

theory of the Schrodinger equation has been the subject of some recent work 1–4 . In

¨

this paper we extend these results by establishing a link between functional relations for

Ž w x.

A -related Bethe ansatz systems see, for example, Refs. 5,62 and third-order differen-tial equations. Most of our analysis concerns a certain specialisation of the model, a

w x

particularly symmetric case that can also be related to the dilute A-model of 7 .

Ž . Ž .

E-mail addresses: [email protected] P. Dorey , [email protected] R. Tateo .

0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.

Ž .

w x

In the cases studied in 1–4 , the most general differential equation was a radial Schrodinger problem with ‘angular momentum’ l and homogeneous potential x

¨

2 M,

Ž .

initially defined on the positive real axis x g 0,` :

d2 _{l l q 1}

Ž

.

2 M

y ₂q_x q ₂ c

Ž

x , E s Ec x , E .

.

Ž

.

Ž

1.1

.

ž

dx x

/

The relevant integrable quantum field theories were the massless twisted sine-Gordon models or, equivalently, the twisted XXZr6-vertex models in their thermodynamic limits, and their reductions. It is worth noting that these models are all related to the Lie

Ž . w x

algebra A . Spectral functions associated with 1.1 satisfy functional relations 8–11 ,1

and these were mapped into functional equations appearing in the context of integrable

w x

quantum field theory in 1–4 . We will follow a similar strategy here, taking a simple third-order ordinary differential equation as our starting-point and showing that the Stokes multipliers and certain spectral functions for this equation together satisfy relations which are essentially the analogues, for the Bethe ansatz systems treated in

w_{6,7 , of the T–Q systems which arise in the context of the integrable quantum field}x

w x

theories related to A1 12,13 . This is the subject of Section 2, while in Section 3 we borrow some other ideas from integrable quantum field theory in order to derive a non-linear integral equation for the spectral functions, an equation which is put to the test in a simple example in Section 4. Duality properties are discussed in Section 5,

Ž .

allowing us to find the equivalent of the angular-momentum term in 1.1 for the third-order equation. Connections with various perturbed conformal field theories are discussed and tested in Section 6. Finally, Section 7 discusses the most general

A -related BA equations that arise in this context, and Section 8 contains our conclu-2

sions.

2. The differential equation

We begin with the following third-order ordinary differential equation:

yXXX

Ž

x , E q P x , E y x , E s 0,

.

Ž

.

Ž

.

Ž

2.1

.

3 M _{Ž .}

and initially restrict ourselves to purely homogeneous ‘potentials’ x , giving P x, E the form

P x , E s x

_Ž

_.

3 My_E.

_Ž

_2.2

_.

Ž .

These are the simplest higher-order generalisations of the l s 0 cases of 1.1 , and so we

w x

expect that some of the properties of that equation, used in the analysis of 4 , will be

w x

preserved. In particular, motivated by the results of 8,9 for second-order equations, we

Ž . Ž .

suppose that 2.1 has a solution y s y x, E such that:

Ž .i y is an entire function of x, E though, due to the branch point in the potentialŽ .

at x s 0, x must in general be considered to live on a suitable cover of the punctured complex plane;

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 585 Ž .ii y, y s dyrdx and y s d yrdx admit, for M ) 1r2, the asymptotic repre-X XX 2 2

sentations 1 _{Mq 1} 1 _{Mq 1} 1 _{Mq 1} X XX y_{M y x} y _{x M y x} y ; x e Mq 1 , y ; ye Mq1 , y ; x e Mq1 , 2.3

Ž

.

as x tends to infinity in the sector 4p

<_{arg x -}< _.

_Ž

_2.4

_.

3 M q 3

Ž .

Furthermore, these asymptotics, or even just the asymptotic of y x, E with x remaining on the positive real axis, characterise y uniquely.

For M ( 1r2, the story is complicated by the appearance of extra terms in the

Ž .

asymptotic 2.3 . The behaviour of the solution which decays as x ™ q` can be more generally found from the formula

x

y_{1 r3 1r3}

y x , E ; P x , E

Ž

.

Ž

.

exp y

ž

_H

P t , E

Ž

.

dt ,

/

Ž

2.5

.

x0

Ž

with the constant x0 being related to the normalisation of the solution. This is the

.

analogue of an approximate WKB solution of a Schrodinger equation. Since to take

¨

M ( 1r2 would bring other technical problems into the treatment to be given below,

from now on, unless otherwise stated, we shall restrict ourselves to M ) 1r2. This

w _{x Ž} w x

range is the analogue of the ‘semiclassical domain’ of 13 see Refs. 1,2,4 for a

.

discussion in the context of differential equations .

Ž .

Given y x, E , bases of solutions to the third-order equation can be constructed just as in the second-order case. For general values of k, define

y x , E s vk_{y v}y_k x , vy_{3 M k} E , 2.6

Ž

.

Ž

.

Ž

.

k with 2p i v s exp

ž

/

.

Ž

2.7

.

3 M q 3 Then y solves_k yXXX x , E q ey_{2 kp i} P x , E y x , E s 0, 2.8

Ž

.

Ž

.

Ž

.

Ž

.

k k Ž .

and so when k is an integer it provides a possibly new solution to the original problem

Ž2.1 . However, since we will shortly need to consider fractional values, we will leave k.

arbitrary for now. It is convenient to define sectors SS ask

2 kp p

SS : arg x y - _.

_Ž

_2.9

_.

k

3 M q 3 3 M q 3

On the cover of the punctured complex plane on which x is defined, the sector SSk

Ž .

abuts the sectors SS_{ky 1} and SS_kq1, and the sector 2.4 is SS_y3r2j SS_y1r2j SS_1r2j S

S3r2. The pattern of dominance and subdominance of solutions is more involved than in the second-order case, since there are now three different behaviours for solutions at

< < yM _Ž Mq 1 _{Ž ..}

large x . In addition to a solution with leading behaviour x exp yx r _{M q 1 as}

< < yM _Ž "p i r3 Mq1 _{Ž ..}

ŽThis is simply a consequence of the fact that the three third roots of y1 are y1, ep i r3

yp i r3_.

and e . Depending on the sector, either one or two of these solutions tend to zero

< <

at large x . We call ‘subdominant’ the solution which tends to zero fastest in a given

Ž .

sector; then, up to a scalar multiple, y is characterised as the unique solution to 2.8k

subdominant inside SS .k

Ž . Ž .

The asymptotic 2.3 and the definition 2.6 together imply

1 1 yŽ Mq 1. k Mq 1 X y Ž Mq 1. k Mq 1 Ž Mq 1. k y_{M y} v x y v x y ; vk x e Mq 1 , y ; yek Mq1 , 1 yŽ Mq 1. k Mq 1 XX _{y Ž Mq1. k M y v x} y ; vk x e Mq 1 ,

Ž

2.10

.

< < for x ™ ` with x g SSky 3r2j SSky1r2j SSkq1r2j SSkq3r2.

Ž

2.11

.

Comparing y , y_{k kq1} and y_kq2 in the region SS_kq1r2j SS_kq3r2, where the asymptotics

Ž .

of all three are given by 2.10 , establishes their linear independence. The set

y , y_{k kq1}, y_kq24 therefore forms a basis of solutions to 2.8Ž . Žand, for k integer, to

Ž2.1 . Alternatively, we can examine. .

w

x

W s_{W y , y , y ,}

_Ž

_2.12

_.

k , k , k1 2 3 k1 k2 k3

w x

where the generalised Wronskian W f, g, h is defined to be

X XX f f f X XX Det g g g .

Ž

2.13

.

X XX h h h Ž w x. Ž .

It is a standard result see, for example, Ref. 14 that, for f, g and h solving 2.1 ,

w x

W f, g, h is independent of x, and that f, g and h are linearly independent if and only

w x _{Ž . Ž . Ž .}

if W f, g, h is non-zero. For k , k , k_{1 2 3} s y_{1,0,1 , the asymptotic 2.10 , used in} S

Sy1 r2j SS1r2, shows that

M 2 M

Wy1 ,0 ,1s_8isin

ž

p sin

/ ž

p .

/

Ž

2.14

.

3 M q 3 3 M q 3 It is also the case that

W E s W vy3 M a

E , 2.15

Ž

.

Ž

.

Ž

.

k qa , k qa , k qa1 2 3 k , k , k1 2 3

 4

so W_{k, kq1, kq2} is non-zero for all k, thus confirming the independence of y , y_{k kq1}, y_kq2 .

w x

We now aim to generalise the analysis of 4 to this situation, guided in part by the

w x

treatment of A -related BA systems provided by 6 . Since y , y , y form a basis, we_{2 1 2 3} can write y y SŽ1. _{E y q S}Ž2. _{E y y y s 0 2.16}

Ž

.

Ž

.

Ž

.

0 1 2 3 with W_{0 ,2 ,3} W_{1 ,0 ,3} Ž1. Ž2. S

Ž

E s

.

, S

Ž

E s

.

.

Ž

2.17

.

W1 ,2 ,3 W1 ,2 ,3

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 587

Ž . Ž . Ž1. Ž2.

The coefficient of y in 2.16 is y1 by 2.15 ; S3 and S are Stokes multipliers for

Ž2.1 , and are analytic functions of E. Notice the formal similarity between this equation.

Ž . w x

and Eq. 15 of Ref. 6 .

Ž .

Now suppose that k and k differ by an integer. Then y1 2 k1 and yk2 both solve 2.8

Ž_{with e}y2 kp i_sey2 k1p i_sey2 k2p i_{., and it can be checked by direct substitution that the} function z_{k k}

_Ž

x , E s y y

_.

_k X_k y_yX_{k yk}

_Ž

_2.18

_.

1 2 1 2 1 2 solves zXXX x , E y ey_{2 kp i} P x , E z x , E s 0 . 2.19

Ž

.

Ž

.

Ž

.

Ž

.

k k1 2 k k1 2 Ž .

This is just the equation adjoint to 2.8 ; the observation that the Wronskian of two solutions of a third-order ordinary differential equation satisfies the adjoint equation

w x

dates back at least to Birkhoff 15 . Observe also that if k and k1 2 are shifted by a

Ž .

half-integer, then a solution of the original equation 2.8 results:

XXX _{1 1} y2 kp i ₁ 1

zk q , k q1 2 2

Ž

x , E q e

.

P x , E z

Ž

.

k q , k q1 2 2

Ž

x , E s 0.

.

Ž

2.20

.

2 2

< < _{Ž .}

For k y k - 3, the regions 2.11 for k s k and k s k1 2 1 2 have a non-empty overlap,

Ž .

and an asymptotic for zk k1 2 is easily obtained from 2.10 . In particular, for k s 1,2,3 we have 1 Mq 1 y_{M y2 cosŽp k r3. x} zy_{k r2 , k r2}

Ž

x , E ; 2 isin p kr3 x

.

Ž

.

e Mq 1 , x ™ q` . 2.21

Ž

.

Ž . Ž . Ž

For k s 1, zy_{1 r2,1r2} solves 2.1 , and now from 2.21 we see that it shares up to a

. Ž .

proportionality factor the asymptotic 2.3 . By uniqueness, we deduce

'

zy1 r2 ,1r2

Ž

x , E s i 3 y x , E .

.

Ž

.

Ž

2.22

.

Unfortunately, this argument is not so effective for the other cases. At k s 2, the

Ž .

formula 2.21 shows only that zy_{1 ,1} is not subdominant on the real axis, and this

Ž .

information is not enough to pin the function down. For k s 3, sin p kr3 s 0 and all that can be deduced is that the leading asymptotic of zy_{3 r2,3r2} is subleading to the term over which we have control.

Ž . Ž1. Ž2.

The next step is to manipulate 2.16 in order to eliminate either S or S . We have

yX y y SŽ1. _{E y}X_{y q S}Ž2. _{E y}X _{y y y}X _{y s 0 , 2.23}

Ž

.

Ž

.

Ž

.

1 0 1 1 1 2 1 3 y yXy_SŽ1. E y yXq_SŽ2. E y yXy_{y y}Xs_{0 , 2.24}

Ž

.

Ž

.

Ž

.

1 0 1 1 1 2 1 3 and, subtracting, SŽ2. _{E z s z q z . 2.25}

Ž

.

12 01 13

Ž

.

< <

For the reasons just explained, functions zk k1 2 with k y k s 1 are the most easily1 2

Ž .

handled, so we use the identity y z s y z q z2 13 1 23 12y to rewrite 2.25 as3

SŽ2. _{E y z s y z q z y q z y . 2.26}

Likewise,

SŽ1. _{E y z s z y q y z q z y . 2.27}

Ž

.

1 12 12 0 1 23 01 2

Ž

.

Ž . 15 M r4 21 M r4

Now the result 2.22 can be combined with shifts in E to v E and v E

Ž . Ž .

respectively to rewrite both 2.26 and 2.27 as

T E y

_Ž

_.

_{y1 r4}y_1r4s_yy1r4y5r4q_yy3r4y3r4q_yy5r4y1r4,

_Ž

_2.28

_.

where T E s SŽ1. _v15 M r4 E s SŽ2. _v21 M r4 E . 2.29

Ž

.

Ž

.

Ž

.

Ž

.

As a byproduct, this has established that the two Stokes multipliers SŽ1. _{and S}Ž2. _are

related by an analytic continuation in E.

Ž .

Finally, taking 2.28 at x s 0 yields a functional relation involving E alone. To absorb various phases, it is convenient to set

Qq E s Ey1 r3 M y 0, E , Qq E s Qq vy3 M k E . 2.30

Ž

.

Ž

.

k

Ž

.

Ž

.

Ž

.

Then the relation is

TQq

Qq

s_Qq _Qq q_Qq _Qq q_Qq _Qq _.

_Ž

_2.31

_.

y1 r4 1r4 y1r4 5r4 y3r4 3r4 y5r4 1r4

w x

This is very similar to the equations related to the dilute A model studied in 7,16 . An

XX

Ž . Ž .

equation involving y 0, E can also be derived. First, differentiate 2.25 twice with respect to x:

SŽ2. _{E z}XX _szXX _qzXX _{. 2.32}

Ž

.

12 01 13

Ž

.

Ž . XX XX XX XX XX XX

Using the fact that y , y and y all solve 2.1 , we have y z s y z q z1 2 3 2 13 1 23 12y , and so3

the previous steps can be repeated to find

T E y

_Ž

_.

XX_y yXX s_yXX _yXX q_yXX _yXX q_yXX _yXX _.

_Ž

_2.33

_.

1 r4 1r4 y_{1r4 5r4} y_{3r4 3r4} y_{5r4 1r4}

Again set x s 0, and define

1 XX

y _{1r3 M} y y y_{3 M k}

Q

Ž

E s E

.

2 y

Ž

0, E ,

.

Qk

Ž

E s Q

.

Ž

v E

.

Ž

2.34

.

1

Žthe factor 2 is included for later convenience . Then. TQy Qy s_Qy _Qy q_Qy _Qy q_Qy _Qy _.

_Ž

_2.35

_.

y1 r4 1r4 y1r4 5r4 y3r4 3r4 y5r4 1r4 X Ž . Ž . Ž .

There is no simple relation involving y 0, E alone, but from 2.18 and 2.22 one can deduce X XX XX

'

i 3 y s y y y_{y y ,}

_Ž

_2.36

_.

y1 r2 1r2 y1r2 1r2 X Ž . Ž . XXŽ .

which allows y 0, E to be recovered once y 0, E and y 0, E are known.

3. The non-linear integral equation

"

Ž .

The functions Q E are not single-valued, and to derive an integral equation it is

Ž . XXŽ .

more convenient to work with the functions y 0, E and y 0, E directly. Set

1 XX

q y

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 589

1 1

" "3 M " " " k .3 M " y3 M k

Žso that D ŽE s E. Q ŽE. and Q_k ŽE s v. E D Žv E . These are. .

entire functions of E and can be interpreted as spectral determinants for the third-order

Ž .

equation 2.1 , since their zeroes coincide with the values of E for which the solution y,

Ž

decaying at x ™ q` for all values of E, in addition either vanishes at x s 0 for the

q

. Ž y.

zeroes of D , or has a vanishing second derivative at x s 0 for the zeroes of D .

ŽSee, for example, Ref. 4 for a more detailed discussion of this point in the context ofw x

. "Ž . < <

second-order equations. For M ) 1r2, the functions D E have large- E asymptotics

m " _{< < < <} ln D

Ž

E ; a yE

.

0

Ž

.

, E ™ `, arg yE - p ,

Ž

.

Ž

3.2

.

Ž . Ž . where m s M q 1 r3 M, a s k 3 M,3 , and₀ 1 1 p G 1 q

ž

/ ž

G 1 q

/

sin ` _{a b} 1rb b a a r b k a,b s

_Ž

_.

_H

dx

_Ž

_Ž

x q 1

_.

y_x

_.

s _{p p .} 1 1 0 sin q G 1 q q

ž

/

ž

a b

/

b a 3.3

Ž

.

" Ž .

The growth of ln D E is no larger on the positive real E-axis than elsewhere, so the

orders of Dq

and Dy

as functions of E are both equal to m, and are less than 1 for

M ) 1r2. Invoking the Hadamard factorisation theorem, we can write

` _E " " D

Ž

E s D

.

Ž .

0

_Ł

ž

1 y "

/

.

Ž

3.4

.

E ks1 k " Ž .

The precise values of the constants D 0 are irrelevant for the treatment below, but

 "4

some knowledge of the positions of the zeroes E_k will be crucial. We conjecture that,

"

Ž .

for all M ) 0, all of the zeroes of D E lie on the positive real E-axis. Some numerical evidence in favour of this claim will be presented below.

Ž . Ž .  3 M r4 "4

The generalised T–Q relations 2.31 , 2.35 taken at either E g v E_n or

 y3 M r4 "4 E g v En imply D" vy_{3 M} E" D" v3 M_E"

Ž

n

.

_{. 1}

Ž

n

.

s yv ,

Ž

3.5

.

" y3 M r2 " " 3 M r2 " D

_Ž

v E_n

_.

D

_Ž

v E_n

_.

an equation that can be written in a Bethe-ansatz form as

` " y3 M " ` " y3 M r2 " E_k yv _{En Ek} yv _En . 1 s yv _.

_Ž

_3.6

_.

Ł

" _{3 M} "

Ł

" _{3 M r2} " E yv _{E E} yv _E ks1 k n ks1 k n

This equation is at least not inconsistent with the conjectured reality of the E ’s, sincen

both sides then reduce to pure phases. There are certainly other, complex, solutions to

Ž3.6 , so the reality property should be seen as a way of selecting the particular solution.

relevant to our differential equation, analogous to the selection of the ground state in an integrable model.

w x

A non-linear integral equation, similar to those described in 13,17–19 , can now be obtained for the quantity

D" vy3 M E D" v3 M r2 E

Ž

.

Ž

.

" " 1 d

Ž

E s v

.

" 3 M " y3 M r2 .

Ž

3.7

.

D

_Ž

v _E

_.

_D

_Ž

v _E

_.

w x _{Ž .}

We shall follow a path that completely parallels the treatment given in 13 . By 3.5 ,

"

Ž .  "4 Ž

d E s y1 at the points Ek . The value y1 might also occur at other points; we supplement our previous conjecture with the assumption that none of these points lie on

. Ž .

the positive real axis. The product representation 3.4 implies

` 2 " ln d

_Ž

E s "ip

_.

q

_Ý

_{F ErE}

_Ž

_n

_.

_,

_Ž

_3.8

_.

3 M q 3 _ns1 where 1 y Evy3 M _{1 y Ev}3 M r2

Ž

.

Ž

.

F E s ln

Ž

.

_{3 M} y3 M r2 .

Ž

3.9

.

1 y Ev 1 y Ev

Ž

.

Ž

.

Ž .

The sum over the E in 3.8 can be written as a contour integraln

2 dEX X X

" "

X

ln d

_Ž

E s "ip

_.

q

_H

_{F ErE E ln 1 q d}

_Ž

_.

_E

_Ž

_Ž

_E

_.

_.

_Ž

_3.10

_.

3 M q 3 C2 ip

with the contour C running from q` to 0 above the real axis, winding around 0 and

Ž

returning to q` below the real axis. It is at this point that the conjectures about the

"

Ž . .

locations of the E ’s and of the other zeroes of d_n E q 1 are used. If the new variable u s mln E is introduced, the function F becomes

F e3 Mu rŽ Mq1.

Ž

.

1 j ₁ p ₁ 3 3 sinh

_ž

₂ u q ip

_/

sinh

_ž

₂ u q i

_/

1 q j 1 q j 1 q j 2 1 q j y_{3 M r2} s_{ln v ,} 1 j ₁ p ₁ 3 3 sinh u y ip sinh u y i



ž

2

/

ž

2

/

0

1 q j 1 q j 1 q j 2 1 q j 3.11

Ž

.

with j s 1rM. Now define

f" u s ln d" e3 Mu rŽ Mq1. , 3.12

Ž

.

Ž

.

Ž

.

" Ž .) " Ž ) .y1 Ž .

use the property d E s_{d E and integrate by parts to recast 3.10 as}

` X X X " " f

Ž

u y

.

_H

_{du R u y u}

Ž

.

_f

Ž

u y i0

.

y` ` 2 X X " X s"ip_{3 M q 3}y_{2 i}

H

_{du R u y u}

Ž

.

_{Im ln 1 q exp f}

Ž

Ž

Ž

u y i0

.

.

.

y` 3.13

Ž

.

with i 3 Mu rŽ Mq1. R u s

Ž

.

E F eu

Ž

.

.

Ž

3.14

.

2p

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 591

Ž . "Ž . Ž .

The term 1 y R ) f u _{on the l.h.s. of 3.13 is easily inverted using Fourier}

transforms. Using

sinh hu q ipt

Ž

.

2 hsin 2tp

Ž

.

iE ln_u s _,

_Ž

_3.15

_.

sinh hu y ipt

Ž

.

cosh 2 hu y cos 2tp

Ž

.

Ž

.

p k sinh

ž

Ž

1 y 2t

.

/

du _{yi ku} 2 hsin 2tp

Ž

.

_{2 h} e s _,

_Ž

_3.16

_.

H

_{2p cosh 2 hu y cos 2tp p k}

Ž

.

Ž

.

sinh

ž

/

2 h we have p p sinh

ž

Ž

1 y j k

.

/

sinh

ž

j k

/

3 3

˜

R k s

_{Ž .}

_p q _p sinh

ž

Ž

1 q j k

.

/

sinh

ž

Ž

1 q j k

.

/

3 3 p k p 2sinh

ž /

cosh

ž

Ž

1 y 2 j k

.

/

6 6 s _{p ,} sinh

ž

Ž

1 q j k

.

/

3 p p sinh

_ž

j k cosh

_/

_ž

k

_/

3 2

˜

1 y R k s

Ž .

p p .

Ž

3.17

.

sinh

ž

Ž

1 q j k cosh

.

/

ž

k

/

3 6

Transforming back to u space and rewriting the imaginary part in terms of values above

"

Ž .

and below the real axis, the functions f u _solve f" u s "ipa y ib eu q w u y uX _{ln 1 q e}f" Ž uX. duX

Ž

.

0

_H

Ž

.

Ž

.

C C1 y w u y uX _{ln 1 q e}y_f"_{Ž u}X_. duX, 3.18

Ž

.

Ž

.

Ž

.

H

C C2

where a s 2r3, the contours CC and C1 C run from y` to q`, just below and just2

above the real u-axis,

p p i ku e sinh k cosh k 1 y 2 j

Ž

.

`

ž

/

ž

/

_{dk 1} 3 6 w u s y

_Ž

_.

_H

_{p p ,} j s _,

_Ž

_3.19

_.

2p M y` cosh

_ž

k sinh k

_/

_ž

j

_/

2 3 M q 1 Ž .

and the constant b s 2sin p0 3 M a0 has been fixed using the asymptotic behaviour

˜

Ž . Ž .

parameter a in 3.18 is analogous to the chemical potential or twist term in the

w x

equations of 13,17–19 .

Ž .

A first consistency check is immediate: in the large u limit of 3.18 the driving term

u

Ž "Ž ..

b e₀ dominates, and so in this limit the functions exp f u _{are y1 at the points} u s u" , or n 1rm 2 " u r m_n E s E s en s

Ž

Ž

_{2 n y 1 "}3

.

prb0

.

Ž

n s 1,2, . . . .

.

Ž

3.20

.

The same limit can be treated directly using a WKB-like approach to the differential

Ž . Ž .

equation 2.1 . Start from 2.5 with x ) x₀ and fix x₀ to be at the inversion point

Ž . Ž 1r3 M. Ž

P x , E s 0 x s E_{0 0} . Now, using analytic continuation see, for example, Section

w _x.

47 in Ref. 20 , the dominant part in the region x - x is₀ x 1 0 y_{1 r3 1r3} < < < < y x , E ; P x , E

Ž

.

Ž

.

exp

ž

_H

P x , E

Ž

.

dx

/

2 x =

'

x 3 ₀ p 1r3 < < cos

ž

_H

P x , E

Ž

.

dx y

/

.

Ž

3.21

.

2 x 3 Ž .

Thus to have y 0, E s 0, requires

x0 1r3 m ₁

3 M q q

'

3

H

Ž

x y_En

.

s_b0

Ž

_En

.

s

Ž

_{2 n y}3

.

p , n s 1,2, . . . ,

Ž

3.22

.

0

where the formula

p p sin

ž

q

/

1 _{a 1rb b a} 1 y x s k a,b 3.23

Ž

.

Ž

.

Ž

.

H

p 0 _sin b Ž . Ž .

was used. The prediction 3.22 agrees perfectly with 3.20 . In Fig. 1 the positions of

q

Ž .

the lowest zeroes of D E are plotted in the range 0.1 - 3 M - 7, and compared with

the WKB-like prediction. Evidence for the reality of the E at 3 M s 1 will be given in_n Section 4; in the meantime, we note that the levels continue smoothly away from that

Ž

point, and the eigenvalues appear to remain real in the range studied. The figure can be

w x

compared with Figs. 1 and 2 of Ref. 4 , which illustrate cases where the spectrum does

. Ž . Ž .

not remain real in the full range displayed. The kernel w u given in 3.19 coincides

with ir2p times the logarithmic derivative of the scalar factor in the Izergin–Korepin

Ž2. w _{x Ž Ž .} w x

S-matrix for the a2 model 21 cf. Eq. 3.21 of Ref. 22 , though note that the

2p

wSmirnov x wthis paper x

w x _.

normalisation of j used by Smirnov in 22 differs from ours: j s j _.

3

Ž .

This is an element of the advertised link between the differential equation 2.1 and the

Ž2. _Ž Ž2.

a2 model, the parameters being related as M s 1rj with j related to the a2

Ž . w x .

coupling g as j s gr 2p y g 22 . When 3 M is an integer the potential is analytic, and the associated scattering theory is diagonal; the same phenomenon was observed in

w x _{Ž .}

the Schrodingerrsine-Gordon case in 1 . The similarity between the relations 2.31 ,

¨

Ž_{2.35 and 3.6 and those arising in the dilute A model 7,16 has already been}. Ž . w x

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 593

q

Ž .

Fig. 1. The positions of the first eight zeroes of D E , plotted on a log scale. Dotted lines show the

WKB-like predictions, and solid lines the results from the non-linear integral equation.

mentioned. Since the aŽ2.

model is conjectured to be the continuum limit of the dilute A

2

Ž w _x.

model see, for example, Ref. 23 , the fact that elements of it emerge here is not a

complete surprise. Nevertheless, it is an encouraging signal that we are on the right track. We will return to this point in Section 6.

4. The linear potential

A simple but non-trivial example occurs when M s 1r3, and is the analogue of the

w x

‘Airy case’ of the second-order problem, discussed in 1,24 . This lies outside the

M ) 1r2 zone treated so far, so we have to assume that the results obtained above

continue to hold as the region of their initial derivation is left. The basic differential equation is

yXXX

Ž

x , E q xy x , E s Ey x , E .

.

Ž

.

Ž

.

Ž

4.1

.

Ž . Ž .

Setting y x, E s AA x y E , this becomes

AAXXX

Ž

x q x A

.

A x s 0.

Ž

.

Ž

4.2

.

This equation is solvable via a complex-Fourier transform:

1

3 y_{i p xq p}4

AA x s

Ž

.

(

H

e 4 dp,

Ž

4.3

.

2p G

Ž

where the integration path G is represented in Fig. 2. A curious feature of this case is

Ž . .

that the function T E is a constant, equal to 1. Even though the problem is not

Ž . XŽ . XXŽ .

self-adjoint, numerical evidence suggests that all the zeroes of AA x , AA x and AA x

Ž . Ž . Ž .

lie on the negative real axis see Fig. 3 , and so the zeroes of y 0, E ' AA yE , and of

X

Ž . XXŽ .

y 0, E and y 0, E , are positive and real. In the first columns of Tables 1 and 2, the

Ž . XXŽ .

positions of the first ten zeroes of AA yx and AA yx are displayed. Ž .

The approximate positions of the zeroes of AA x can be found from the WKB

formula of the last section. As a check, we rederive them here via a saddle-point

Ž . < Ž .< Ž < Ž .<.

Fig. 3. A search for the zeroes of AA x in the complex x-plane. The function plotted is A x r 1q A x ,

Ž . x Ž . A x s e AA x .

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 595 Table 1 Ž . Zeroes of AA y x Ž . Ž . Ž . k Ek Exact Ek WKB Ek NLIE 1 2.8868281617697677 2.84467 2.886828161769766 2 5.1522519299627660 5.13866 5.152251929962763 3 7.1303732976716514 7.12265 7.130373297671650 4 8.9403621072563961 8.93513 8.940362107256395 5 10.635608688272157 10.6317 10.63560868827213 6 12.245164125544329 12.2421 12.24516412554437 7 13.787063381394688 13.7846 13.78706338139461 8 15.273489957153985 15.2714 15.27348995715393 9 16.713173447810789 16.7114 16.71317344781078 Ž . < <

treatment of 4.3 . The exponent of the integrand for x s y x - 0 has stationary points at

< <1r3 " i2p r3< <1r3

p s i x0 , p s ie" x .

Ž

4.4

.

Deforming the contour G so that it touches the points p , we get"

3 p 3 p 4r 3yip r3 4r3ip r3 <_x< _e qi <x< e yi 4 3 4 3 < < AA y x ; N

_Ž

_.

N e q_{e ,}

_Ž

_4.5

_.

ž

/

where the phases "pr3 are the contributions from the choice of the steepest descent directions, transforming the quadratic terms in the expansion near the saddle points into a pure Gaussian integral

3 ` 3 y <x<2r 3 2t _{< <}y_{1 r3} N Ns

(

H

e 2 dt s x .

Ž

4.6

.

2p y`

Thus for large negative x we have

3 _{4r 3} <_x< _p 8 y_{1r3 3 4r3}

'

< < < < < < AA y x ; 2 x

_Ž

_.

e cos 3 x y _.

_Ž

_4.7

_.

8

ž

3

/

< <1r3

For x ) 0, the dominant saddle point is instead at p s yi x0 , and

3 _{4r 3}

y_{1 r3} y <x<

< <

AA x ; x

Ž

.

e 4 .

Ž

4.8

.

Ž_{This agrees with the general asymptotic 2.5 , since y x, E s A}Ž . Ž . A x y E . The domi-Ž . .

X

Ž . XXŽ . < <

nant behaviours of AA x and AA x for x real and x large are

3 _{4r 3} <_x< ₃ 4r 3 8 4r3 X _{< <} 3

_'

_{< <} X _{< <} y <x< AA y x ; y2 e

Ž

.

cos

Ž

8 3 x

.

, AA

Ž

x ; ye

.

4 ,

Ž

4.9

.

1 ₃ 1 4r 3 ₃ <_x< p 4r 3 3 4r3 3 XX ₈ 3 XX y <_x<

'

< < < < < < < < < < AA

_Ž

y_{x ; 2 x e}

_.

_{cos 3 x} q _{, AA}

_Ž

_{x ; x e}

_.

_{4 .} 8

ž

3

/

4.10

Ž

.

Table 2 XX Ž . Zeroes of AA y x . Ž . Ž . Ž . k Ek Exact Ek WKB Ek NLIE 1 0.8909448213691012 0.85075 0.890944821369104 2 3.6439018898747455 3.66124 3.643901889874749 3 5.8171363674931786 5.82455 5.817136367493172 4 7.7377094704565507 7.74230 7.737709470456556 5 9.5084853382296383 9.51174 9.508485338229636 6 11.174539990553549 11.1770 11.17453999055357 7 12.761111499098718 12.7631 12.76111149909874 8 14.284208133811659 14.2859 14.28420813381167 9 15.754819677279076 15.7562 15.75481967727909

Continuing to differentiate, the general result for the approximate positions of the zeroes

th Ž m ._{Ž .} of the m derivative AA x is 3r4 4 2 m q 1 Ž m . AA

Ž

x s 0:

.

x s y

ž

2 n y

/

p

Ž

n s 1,2, . . . .

.

Ž

4.11

.

'

3 3 3 Ž . Ž .

At m s 0 4.11 reduces to the M s 1r3 WKB prediction 3.22 . In Tables 1 and 2 the

Ž .

results from formula 4.11 are compared with the ‘exact’ result from a numerical

Ž .

treatment of 4.3 , and also against the results of the numerical solution of the non-linear

Ž .

integral equation 3.18 . Clearly the agreement is very good.

5. Duality and a more general chemical potential

Ž .

In this section we shall investigate the effect of a duality transformation on 2.1 ,

w x _{Ž .}

analogous to that studied in 2 for the Schrodinger equation 1.1 . In the Schrodinger

¨

¨

Ž .

case duality maps wavefunctions for confining potentials M ) 0 to wavefunctions for

Ž .

singular potentials y_{1 - M - 0 , in such a way that the theories with M and}

˜

Ž .

M s yMr M q 1 are dual, their respective spectral problems being essentially

equiva-Ž . 2

lent. It also changes the coefficient of the ‘angular momentum’ term l l q 1 rx in

Ž1.1 in a non-trivial way; the same phenomenon here will allow us to guess the. Ž .

corresponding term for the third-order problem 2.1 .

w x

To implement the duality transformation, we begin with a Langer 25 type variable transformation

y x s ez_{u z , z s ln x , 5.1}

Ž

.

Ž .

Ž

.

Ž .

after which 2.1 becomes

uXXX z y uX z q eŽ3 Mq 3. z_yEe3 z _{u z s 0. 5.2}

Ž .

Ž .

Ž

. Ž .

Ž

.

˜

The duality M ™ M is now effected by interchanging the roles of the two exponentials.

ˆ

Substituting z M q 1 z ™ q_ln 1r3

ž

/

M q 1 E

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 597 yields 1 XXX X _{3 z rŽ Mq1. 3 z}

˜

u

_{Ž .}

z y u z q ye

_{Ž .}

_Ž

y_Ee

_.

_{u z s 0,}

_{Ž .}

_Ž

_5.3

_.

2 M q 1

Ž

.

˜

_{Ž .}3 M Mq1 where E s y M q 1 r_{E .}

Now transforming back results in the equation

M M q 2

Ž

.

1 d 1 XXX y3 M rŽ Mq1.

_˜

y q

_˜

y _{y q yx}

_˜

_Ž

y_{E y s 0.}

_.

_˜

_Ž

_5.4

_.

2

ž

2 3

/

dx x x M q 1

Ž

.

As promised, the confining ‘potential’ x3 M _{has been exchanged for a singular potential}

y3 M rŽ Mq1. _Ž y2 y3_.

y_{x , and a new term, proportional to x drdx y x y, has been generated.}

This motivates us to enlarge the set of differential equations under consideration to 1 d 1 XXX _{3 M} y y G y _{y q x}

_Ž

y_{E y s 0}

_.

_Ž

_5.5

_.

2 3

ž

x dx x

/

Ž .

with G a new parameter, analogous to l l q 1 for the Schrodinger equation. Duality

¨

Ž . maps 5.5 to 1 d 1 _˜ XXX _{3 M}

˜

˜

y y G

_˜

y _{y q yx}

_˜

_Ž

y_{E y s 0,}

_.

_˜

_Ž

_5.6

_.

2 3

ž

x dx x

/

where y_{1 1r3 M rŽ Mq1.} y_{1r3 1rŽ Mq1.}

˜ ˜

y x , E,G s M q 1

˜

Ž

.

Ž

.

E x y

Ž

Ž

M q 1 E

.

x , E,G

.

Ž

5.7

.

and 3 M M

Ž

M q 1

.

G y M M q 2

Ž

.

˜

˜

˜

M s y , E s y _{Mq 1} , G s ₂ .

Ž

5.8

.

M q 1 E

_Ž

_{M q 1}

_.

Ž 4. Ž .

Duality therefore maps the 3-parameter family M, E,G of differential equations 5.5 onto itself. The analysis of Section 2 can now be repeated for these generalised

Ž .

problems. It is convenient to write G s g g q 2 and to work mostly with g instead of

Ž .

G. We first enlarge the scope of 2.6 by setting y x , E, g s vky vyk x , vy3 M k E, g ; 5.9

Ž

.

Ž

.

Ž

.

k then y solvesk 1 d 1 XXX y2 kp i y y G y _{y q e P x , E y s 0.}

_Ž

_.

_Ž

_5.10

_.

k

ž

2 3

/

k k dx x x Ž . y2 k1p i y2 k2p i y2 kp i

Next we define z as in 2.18 . If e s_{e ' e , then}

k k1 2 1 d 1 XXX y_{2 kp i} z y_G y _z y_{e P x , E z}

_Ž

_.

s_0,

_Ž

_5.11

_.

k k1 2

ž

_x2 _{dx x}3

/

k k1 2 k k1 2 Ž .1

which is the equation adjoint to 5.10 . We can recover the original problem by shifting

k by a half-integer, and arguing just as before we find that

'

zy1 r2 ,1r2

Ž

x , E, g s i 3 y x , E, g

.

Ž

.

Ž

5.12

.

1

Notice that the operator xy2_{d r dx y x}y3 _{is by itself anti-self-adjoint; this gives some insight as to why it} is a sensible generalisation of the xy2 _{term in the Schrodinger equation.¨}

and then

T E, g y

_Ž

_.

_{y1 r4} y_1r4s_{yy1r4 y5r4}q_{yy3r4 y3r4}q_{yy5r4 y1r4,}

_Ž

_5.13

_.

Ž . Ž1._Ž 15 M r4 _. Ž2._Ž 21 M r4 _. Ž1. Ž2.

where T E, g s S v _{E, g s S} v _{E, g , and S and S are defined as}

Ž . Ž . Ž .

in 2.16 . A non-zero value of G s g g q 2 causes 5.10 to be singular at the origin,

Ž . Ž .

so simply considering 5.13 at x s 0 is not an option. Instead, we expand y x, E, g as

y x , E, g s Dq

E, g x q D0 _{E, g x q D}y

E, g x , 5.14

Ž

.

Ž

.

q

Ž

.

₀

Ž

.

y

Ž

.

 4

where x , x , xq ₀ y forms a basis of solutions defined via behaviour near the origin:

x x , E, g ; xli_{qO x}liq3 _{, i s q,0,y, 5.15}

Ž

.

Ž

.

Ž

.

i

Ž .Ž Ž . Ž ..

with the l ’s the roots of the indicial equation l y 1 l l y 2 y g g q 2 s 0 :i

l s yg ,q l s 1,₀ l s g q 2.y

Ž

5.16

.

Explicitly, the functions Dq

, D0 _{and D}y are

w

x

w

x

w

x

W y, x , x₀ y W y, x , xy q W y, x , xq ₀ q 0 y D s , D s , D s ,

w

x

w

x

w

x

W x , x , xq ₀ y W x , x , xq ₀ y W x , x , xq ₀ y 5.17

Ž

.

w x _{Ž .}3 _{Ž .} X Ž .

with W x , x , x_{q 0 y} s_{2 g q 1 . At g s 0 they reduce to y 0, E , y 0, E and}

XX

Ž .

y 0, E r2 respectively, in agreement with the notation of earlier sections.

Defining Q" E, g s E. Ž gq1.r3 M_D" E, g , Q" s_Q" vy3 M k_{E, g , 5.18}

Ž

.

Ž

.

k

Ž

.

Ž

.

Ž . Ž .

the generalised T–Q relations 2.31 , 2.35 have exactly the same form as before, and

Ž3.6 becomes. ` " y<sub>3 M</sub> " ` " y_{3 M r2} " E_k yv _{En Ek} yv _En . Ž gq1. s yv .

Ž

5.19

.

Ł

" _{3 M} "

Ł

" _{3 M r2} " E yv E E yv E ks1 k n ks1 k n

The arguments of Section 3 can now be repeated essentially verbatim, to discover that, so long as G is such that the conjectures of Section 3 about the zeroes of T and D"

q y

Ž .

remain true, the quantities D and D for the more general differential equation 5.5

Ž .

are again described by the non-linear integral equation 3.18 , but with chemical

2

Ž .

potential term now taking the value a s3 g q 1 .

" 3 M _{Ž .}

At M s 1, v s y_{1, the l.h.s. of 5.19 is 1 and the A -related BA equation}

2

‘collapses’ onto one more closely linked with A . This leads to a rather surprising₁ equivalence between spectral problems for a Schrodinger equation with potential

¨

6 _{Ž .} 2 _{Ž .}

x q l l q 1 rx and the third-order problem 5.5 at M s 1. For these special points,

w x

the quantities in this paper are related to those of 4 as

" _< " y_{3r2 <}Ref . w 4 x

D

_Ž

E, g

_.

_{Ms 1}A_D

_Ž

_{c E,l}

_.

_{Ms3 ,}

< y_{3 r2 <}Ref . w 4 x

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 599

Ž . Ž .

1 2 27 G 7r6 x G 5r3

Ž .

where l q s2 3 g q 1 , c s 4 , and T is one of the ‘fused’ T-operators1

Ž . p

' G 1r3

w x

discussed in Section 4 of Ref. 4 .

6. A link with perturbed conformal field theory

We now return to the relation with the aŽ2.

model, briefly mentioned at the end of

2

w x

Section 3. The analogy with results of 17–19 for the A -related models suggests that1

the quantity 6 ib₀ " " Ž . Ž . u f u u y_f u c s

_H

_{e ln 1 q e}

_Ž

_.

_{du y}

_H

_{e ln 1 q e}

_Ž

_.

_du

_Ž

_6.1

_.

eff _p2

ž

/

C C1 CC2

should be interpreted as an effective central charge of an underlying conformal field

2

Ž .

theory. For general a s3 g q 1 , this would predict

3 2 c s_{1 y a .}

_Ž

_6.2

_.

eff M q 1 Ž . Ž .

Going further, it is natural to interpret 3.18 and 6.1 as the ultraviolet limit of the following ‘massive’ system:

f u s ip a y ir sinhu q w u y uX _{ln 1 q e}f ŽuX. duX

Ž

.

_H

Ž

.

Ž

.

C C1 y w u y uX _{ln 1 q e}yf ŽuX. duX,

Ž

.

Ž

.

H

C C2 3ir Ž . Ž . f u yf u

ceff

Ž .

r s _p2

ž

_H

sinhu ln 1 q e

Ž

.

du y

_H

sinhu ln 1 q e

Ž

.

du .

/

Ž

6.3

.

C

C1 CC2

This should encode finite-size effects in the massive aŽ2. _{theory, with r s M R, M the}

2 s s

Ž .

mass of the fundamental soliton and R the circumference of the infinite cylinder on

Ž q y

which the theory is living. Notice that there is no need to distinguish f from f any

.

more, since the mapping u ™ yu now has the effect of negating a. There is now a natural scale, which can be related to an operator f perturbing the ultraviolet conformal

w x _{Ž .}

field theory. Standard considerations 26 , based on the u ™ u q i2p M q 1 r3 M

Ž . Ž .

periodicity of f u , suggest that so long as a is not an integer ceff r will have an

6 M rŽ Mq1. _Ž

expansion in powers of r together with an irregular ‘anti-bulk’ term,

irrele-.

vant to the current discussion . This implies for f either the conformal dimensions 3 M

h s h s 1 yf f

Ž

6.4

.

2 M q 2

Ž .

and an expansion of ceff r in which only even powers of the coupling l to the operator f appear, or, alternatively, the conformal dimensions

3 M

h s h s 1 yf f

Ž

6.5

.

M q 1

w x

When a is an integer, the standard considerations of 26 may have to be modified.

Ž . Ž .

Absorbing the term ip a into a shift in f u , 6.3 becomes exactly odd under a

Ž .

negation of u. This forces the shifted f u to be zero at u s 0, even in the far ultraviolet, and so long as the would-be plateau value is non-zero, it splits the plateau region into two pieces, each of half the previous length. As a result, the regular

Ž . 3 M rŽ Mq1. 6 M rŽ Mq1. Ž .

expansion of c_eff r is in powers of r , and not r . Formula 6.4 now describes the situation when both even and odd powers of l appear in the expansion of

Ž .

c , while for even powers only, the correct formula is h s h s 1 y 3 Mr 4 M q 4 ._eff f f

Note though that this plateau-splitting effect does not occur at a s 0, since for this case

Ž .

the plateau value of f is anyway zero, and imposing f 0 s 0 has no effect.

w _{x Ž} w _x. Ž2.

As explained in 22 see also Refs. 28–30 , the a2 model, when appropriately

Ž

quantum-reduced, should correspond to the minimal models MMp, q with p and q

. w x

coprime integers and p - q perturbed by either f , f12 21 or 31,32 f . With f15 12 the

Ž .

perturbing operator, the relation with the parameter j appearing in the kernel 3.19 is

w₂₂x

p 2 j

s _.

_Ž

_6.6

_.

q

Ž

1 q j

.

Since M s 1rj , the ultraviolet effective central charge for a given value of a , as

Ž . predicted by 6.2 , is 3 p 2 c s_{1 y a .}

_Ž

_6.7

_.

eff 2 q Ž . Ž . w x

To recover f21 perturbations one simply has to swap p and q in 6.6 and 6.7 22 ,

w x

while to find f , prq should be replaced by 4 prq 31 .15

For the sine-Gordon model, naturally associated with the f13 perturbing operator

w_{33,34 , it has been observed both analytically and numerically 13,17–19,35,36 that}x w x

reduction is implemented at the level of finite-size effects and the non-linear integral equation via a particular choice of the chemical potential. The similarity between our

w x

equations and those in 17–19 suggests that the same should be true here. To decide

Ž .

which value of a will tune 6.3 onto the ground state of the relevant perturbed minimal model, we demand that the ultraviolet effective central charges match up; the predicted dimensions of the perturbing operators, and a comparison of results at non-zero values of r with those obtained via the thermodynamic Bethe ansatz method, will then provide some non-trivial tests of the proposal.

The effective central charge of the ground state of the theory MM_{p q} is c_effs_{1 y 6rpq.}

Thus to have any chance of matching the vacua of the f -perturbed models, we must12 1

Ž .

set a s 2rp. The required value of h , namely h s 3 pr4 q y , is then matched byf 12 2

Ž6.4 . The value just chosen for a being a non-zero integer if and only if p s 2, 6.4. Ž .

will be the correct formula to use provided the regular parts of the ground state energies of the models MMp q perturbed by f12 expand in even powers of l for p 0 3, and in even and odd powers for p s 2. This ‘prediction’ holds for all of the examples that we

5 2 3 1 4 1

Ž . Ž . Ž . Ž . Ž .

checked. We then compared numerical results for M, a s 4,1 , 3 3, , 2 2, , 3 3, and

Ž_{1,0 against the tables of 27,37 for the A}. w x Ž2. _{ŽYang–Lee , E , E , E and D -related.}

2 8 7 6 4

Ž .

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 601 Table 3

w x

NLIE results versus TBA data from Refs. 27,37–39

Ž .

Model M, a r TBA NLIE

Ž2. _{Ž .} A2 qf12 4,1 0.001 0.399999735051974 0.399999735051971 0.002 0.399998953903823 0.399998953903824 5 2 Ž . E q f8 12 3,3 0.025 0.499926331494289 0.499926331494288 0.05 0.499705463734389 0.499705463734387 3 1 Ž . E q f7 12 2,2 0.02 0.699928050129612 0.699928050129611 0.04 0.699712371203531 0.699712371203531 4 1 Ž . E q f6 12 3,3 0.025 0.857016839032789 0.857016839032790 0.05 0.856639509661813 0.856639509661819 Ž . D q f4 12 1,0 0.01 0.999972507850553 0.999972507850552 0.02 0.999890328583463 0.999890328583464 1 1 Ž . A q f1 21 2,2 0.02 0.499697279140833 0.499697279140832 0.04 0.498957654198721 0.498957654198722 2 1 Ž . A q f2 21 3,3 0.001 0.799999470103948 0.799999470103940 0.002 0.799997907807646 0.799997907807649 1 2 Ž . M q f35 21 5,5 0.1 0.596517064916761 0.596517064916762 0.15 0.592881408017592 0.592881408017593 1 1 Ž . M q f37 15 6,3 0.1 0.709591770021299 0.709591770021299 0.15 0.705031895238354 0.705031895238357

the choice a s 2rq should capture the f21 cases. The conformal weight of h s21 1

Ž3qr4 p y. 2 is matched by 6.4 , provided the swap of p and q in 6.6 is remem-Ž . Ž . Ž .

bered. This time a is never an integer, and the use of 6.4 is justified by the regular parts of the ground state energies of the f21 perturbations always being in even powers

1 1 2 1 1 2

Ž . Ž . Ž . Ž .

of the coupling l. For M, a s 2 2, , 3 3, and 5 5, the results from the A and1

w x w x

A -related TBA equations 27 and the M2 M35 model 38,39 were reproduced within our

2 _{Ž . Ž .}

numerical accuracy . Finally, replacing prq by 4 prq in 6.2 q ) 2 p , at a s 1rp Ž .

the models MMp q perturbed by f15 are recovered. This time it is 6.5 which predicts the correct value for h , as expected given that ff 15 perturbations expand in both even and odd powers of l. TBA equations for a number of f -perturbed models have been₁₅

w x _{Ž .}

proposed in 31,38,40 , but so far we have only compared 3.18 with the TBA for the

w x

f_{15 perturbation of the MM37} model given in 38 . A selection of our numerical results for all of the cases just mentioned is presented in Table 3, together with thermodynamic

w x

Bethe ansatz data taken from Refs. 27,37–39 . To facilitate the comparison, we took

r s M R throughout, with M1 1 the mass of the fundamental particle in the reduced scattering theory. For AŽ2. _{and E this is the first breather in the unreduced theory, and}

2 8

Ž . Ž .

M1 is equal to 2cos 5pr12 M and 2cos 3pr10 M respectively. In all of the others s

models in the table, M is equal to M . The results strongly support the claim that the1 s

Ž .

system 6.3 encodes the ground state energies of f , f12 21 and f15 perturbations of minimal models.

2

Ž . w x Ž . 2

In previously-studied examples, equations similar in form to those for the ground

Ž w _x.

state have been found to describe excited states see, for example, Refs. 13,35,41–43 . We expect that the same will be possible here, but we will leave investigation of this point for future work. Finally, we remark that it would be interesting to derive a non-linear integral equation for the aŽ2. _{model directly from finite lattice BA equations.}

2

w x

We understand that progress is currently being made in this direction 44 .

7. General A -related BA equations2

In this section, we discuss the effect of adding a term proportional to xy₃

to the

Ž .

differential equation 5.5 . The equation becomes 1 d 1 XXX y y G y _{y q P x , E, L y s 0,}

_Ž

_.

_Ž

_7.1

_.

2 3

ž

x dx x

/

where L 3 M P x , E, L s x

_Ž

_.

y_{E q .}

_Ž

_7.2

_.

3 x

Duality acts on M and G as before, and transforms L as

L

˜

L ™ L s ₃.

Ž

7.3

.

M q 1

Ž

.

˜

Ž .

The relation 5.7 , apart from the appearance of L and L as arguments of y and y

˜

respectively, is unchanged. The earlier treatment can be generalised by defining

y ' y x , E, g , L s vk_{y v}y_k x , vy_{3 M k} E, g , vy_{3 Ž Mq1. k} L , 7.4

Ž

.

Ž

.

Ž

.

k k so that 1 d 1 XXX y2 kp i y y G y _{y q e P x , E, L y s 0.}

_Ž

_.

_Ž

_7.5

_.

k

ž

_x2 _{dx x}3

/

k k Ž .

If we also define zk k1 2 as in 2.18 , then, for k and k differing by an integer,1 2

1 d 1 XXX y2 kp i z y_G y _z y_{e P x , E, L z}

_Ž

_.

s_0,

_Ž

_7.6

_.

k k1 2

ž

2 3

/

k k1 2 k k1 2 dx x x

Ž_{with e}y2 k1p i_sey2 k2p i_{' e}y2 kp i _{. which is the adjoint to 7.5 . Again we can recoverŽ .} the original problem by shifting k by a half-integer. As before, we find that

'

Ž . Ž . Ž Ž ..

zy1 r2,1r2 x, E, g, L s i 3 y x, E, g, L and cf. 2.28

P. Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583–606 603

The presence of a non-vanishing L has however introduced an extra complication. To see this explicitly, shift k by "1r4 to get

T"y y_{0 " 1r2}s_{y y0 " 3r2}q_{y y1 y1r2}q_yy1y1r2,

_Ž

_7.8

_.

"

Ž . 3 M r4 .

with T s T Ev , g,. iL . If this equation is rewritten in terms of the function

Ž .

y x, E, g, L , both signs of L appear: for k integer or half-integer we have

2 k

k y_k y_{3 M k}

y s v y x vk

_Ž

, Ev , g , y1

Ž

.

L .

_.

Ž

7.9

.

Ž .

Notice that the same does not happen for the argument g or G , which is why this

Ž .

problem did not arise before. From an analytic point of view, y x, E, g, L and

Ž .

y x, E, g,y L are just two distinct points of the same function, but in the derivation of

the non-linear integral equation it is only the analyticity in E that is used. To proceed, it is best to consider L to be held fixed once and for all, and to treat the pair of functions

k _Ž yk y3 M k _. k _Ž yk y3 M k _.

Õ s v y vk x, v E, g, L and Õ s v y vk x, v E, g,y L independently.

Ž .

Then 7.8 becomes

"

T Õ Õ0 " 1r2s Õ Õ0 " 3r2q Õ Õ1 y1r2q Õy1 1r2Õ .

Ž

7.10

.

w x

This equation is very reminiscent of those given in 6 for the A -lattice model. There2

Ž .

remains an x-dependence in 7.10 which can be eliminated, once again, by expanding

Õ s Dq E, g , L x q D0 _{E, g , L x q D}y E, g , L x , 7.11

Ž

.

q

Ž

.

0

Ž

.

y

Ž

.

q ₀ y Õ s D

Ž

E, g , L x q D

.

q

Ž

E, g , L x q D

.

0

Ž

E, g , L x ,

.

y

Ž

7.12

.

 4  4

where x , x , xq ₀ y and x , x , xq ₀ y are alternative bases defined via the behaviour near the origin x _{x , E, g , L ; x}l_i q_{O x}liq3 _{, i s q,0,y, 7.13}

Ž

.

Ž

.

Ž

.

i li l q3i x_i

_Ž

_{x , E, g , L ; x q O x}

_.

_Ž

_.

_{, i s q,0,y,}

_Ž

_7.14

_.

and the l ’s and l ’s are respectively solutions of the indicial equations_{i i}

l y 1 l l y 2 y g g q 2 q_{L s 0,}

Ž

.

Ž

Ž

.

Ž

.

.

l y 1 l l y 2 y g g q 2 y_{L s 0. 7.15}

Ž

.

_Ž

Ž

.

Ž

.

_.

Ž

.

If the labelling is chosen consistently with that of section Section 5, so that the l’s and

Ž .

the l’s reduce to the quantities in 5.16 when L s 0, then

" " " " " " " " " T Q Q s_{Q Q} q_{Q Q} q_{Q Q ,}

_Ž

_7.16

_.

0 " 1r2 0 " 3r2 1 y_1r2 y_{1 1r2} with Q" E, g , L s EŽ l"y1 .r3 M D" E, g , L , Q"s_Q" vy3 M k E, g , L , 7.17

Ž

.

Ž

.

k

Ž

.

Ž

.

" _{Ž l"}y_{1.r3 M} " " " y_{3 M k} Q

_Ž

E, g , L s E

_.

D

_Ž

E, g , L ,

_.

Q s Q_k

_Ž

v _{E, g , L .}

_.

_Ž

_7.18

_.