Large spin expansion of the long-range Baxter equation in the sl(2) sector of N=4 SYM

0.24 MB·English

by  Matteo Beccaria

PreprinttypesetinJHEPstyle-HYPERVERSION
Large spin expansion of the long-range Baxter
8
0 sl
equation in the sector of SYM
0
2 (2) N = 4

n
a
J

4
2
Matteo BeccariaandFrancescaCatino
]
h
DipartimentodiFisica,Universita’delSalento,ViaArnesano,73100Lecce
t
-
p INFN,SezionediLecce
e E-mail: [email protected],
h
[email protected]
[

3
v
1 ABSTRACT: Recently, several multi-loop conjectureshave been proposedfor the spin de-
9 pendenceofanomalousdimensionsoftwist-2and3operatorsinthesl sectorof
9 (2) N =4
SYM. Currently, these conjectures are not proven, although several consistency checks
1
. have been performed on their large spin expansion. In this paper, we show how these
0
1 expansions can be efficiently computed without resorting to any conjecture. To this aim
7
wepresentin fulldetailsamethodtoexpandatlarge spinthesolutionofthelong-range
0
: Baxterequation. Wetreatthetwist-2and3casesattwoloopsandthetwist-3caseatthree
v
i loops. Several subtleties arise whose resolution leads to a simple algorithm computing
X
theexpansion.
r
a
Contents
1. Introduction 1
2. Theone-loopBaxterequationinthesl sector 4
(2)
2.1 GeneralstructureoftheBaxterequation 4
2.2 Groundstatesolutionintwist2and3 5
2.3 Large expansionoftheexact 6
N (cid:13)1
2.4 Large expansionfromtheBaxterequation 6
N
2.4.1 Twist-2 7
2.4.2 Twist-3 8
2.5 Comments 8
3. Thetwo-loopBaxterequationforthegroundstate 9
3.1 Generalstructure 9
3.2 Two-loopconjecturesfor andtheirlarge expansions 11
(cid:13) N
3.2.1 Twist-2 11
3.2.2 Twist-3 12
3.3 Large expansionfromtheBaxterequation: -methodintwist-2 12
N (cid:1)
3.4 Large expansionfromtheBaxterequation: -methodintwist-3 13
N (cid:1)
4. TheimprovedexpansionoftheBaxterequationintwist-2 14
5. ThethreeloopBaxterequation: Twist-3 18
5.1 Threeloopconjecturefor anditslarge expansion 20
(cid:13) N
5.2 Large expansionfromtheBaxterequation: -method 20
N (cid:1)
5.3 Improvedexpansion 21
6. Conclusions 22
1. Introduction
IntegrabilityoffourdimensionalplanarYang-Millstheoriesisaquiterelevantfeaturethat
permitsinprincipletoobtainnon-perturbativeresults[1]. Itfirstappearedasasurprising
fact in theone-loopanalysis ofspecialQCD subsectors[2]. Later,itsuniversality became
moreandmoreclearduetothemanyextensionsathigherorders[3,4,5].
Currently, a deep understanding of its origin is achieved by means of Maldacena
AdS/CFTcorrespondence[6]. Inthemaximal supersymmetric Yang-Millstheory,
N = 4
integrability ofthe superconformalsideis related to integrability of thedual superstring
–1–
theory on [7]. As a major outcome, a deeply motivated higher loop proposal
5
AdS5 (cid:2)S
for the -matrix of the pair SYM/type II on is now available [8, 9, 10,
5
S N = 4 AdS5 (cid:2)S
11,12,13,14,15,16,17]. Inthispaper,weshallworkwithinthisframeworkassumingthe
validityoftheaboveAnsatz,althoughthesubjectisstillunderdevelopment[18].
Werestrictourconsiderationstotheso-called sl sectorof SYMwhich is an
(2) N = 4
invariant subsector closed under perturbative renormalization mixing. It is spanned by
singletraceoperatorsobtainedaslinearcombinationsofthebasicobjects
Tr (1.1)
n1 nL
(D '(cid:1)(cid:1)(cid:1)D '); n1+(cid:1)(cid:1)(cid:1)+nL =N;
where is one of the three complex scalar fields of SYM and is a light-cone
' N = 4 D
covariant derivative. Thenumbers arenon-negativeintegersandtheirsum isthe
fnig N
total spin. Asusual, the number of fieldsis called the twist ofthe operator. Itequals
L '
the classical dimension minus the spin. The subsectorof stateswith fixedspin and twist
isalsoperturbativelyclosedunderrenormalizationmixing.
The sl sector is very rich and interesting. Even when the twist is kept low, it is a
(2)
non-trivialinfinitedimensionalsectorwithcertainsimilaritieswithanalogousQCDcom-
positeoperatorsappearingindeepinelasticscattering.
Atoneloop,thedilatation operatorcanbeinterpretedastheintegrableHamiltonian
of a spin chain with sites. At each site, the degrees of freedom, associated with ,
n
L D '
transforminthe infinitedimensionalsl representation[19].
s =1=2 (2)
Goingbeyondoneloop,asymptoticall-orderBetheAnsatzequationshavebeenpro-
posed[9]. Importantsuccessfulchecksaredescribedin[20,21,13]. Thankstosupersym-
metry,wrappingproblems(see[22,23]forrecentdevelopments)onlyoccurat loop
L+2
orderintwist- operators[10,11]. Hence,ifweareinterestedinathreeloopanalysis,the
L
twists and3aresafe. Weshallfocusonthesecases.
L=2
ScalingcompositeoperatorsareelementsinthespanofEq.(1.1)whichareeigenvec-
torsofthedilatation operator. Theeigenvaluesare theanomalous dimensions
(cid:13)(N;L;(cid:21))
where is the ’t Hooft planar coupling ( is the number of colors sent to infinity
(cid:21) N
! 1
withconstant )
(cid:21)
2
gYMN
(1.2)
(cid:21) = ;
2
8(cid:25)
At fixed twist , one is interested in the perturbative expansion of that takes
L (cid:13)(N;L;(cid:21))
theform
X1 (1.3)
n
(cid:13) = (cid:13)n(N)(cid:21) :
n=1
Technically, integrability permits to write down Bethe Ansatz equations that compute
(cid:13)
order by order in . The Bethe equations are algebraic equations for the Bethe roots.
(cid:21)
Given a solution, one inserts the Bethe roots in a simple closed expression providing .
(cid:13)
The number of Bethe roots is precisely . So, the above procedure must be repeated at
N
eachgiven . Itisnotobvioushowtofindaparametricexpressionof asafunction
N (cid:13)n(N)
of . Nevertheless,such closed formulae are important in physical applications like, for
N
instance,theBFKLanalysisofpomeronexchange[24].
–2–
Recently,workon twist-2 and 3 operatorshas led to higherorder conjecturesfor the
functions in varioustwistsandingeneralizedsubsectors[22,25,26,27,28]. How-
(cid:13)n(N)
ever,proofsaremissing,atleastbeyondone-loop.
The general way to deduce such conjectures is somewhat deceiving. One starts by
solvingtheBetheAnsatzequationsatvarious . Generally,theBetherootsarenon-trivial
N
algebraic numbers. Nevertheless, turns out to be a rational number to any preci-
(cid:13)n(N)
sion. Givenasufficientlylongsequenceofsuchrationalnumbers,onemakesaninspired
guess about the closed formula for . General arguments from Feynman diagram
(cid:13)n(N)
calculations and deep properties like reciprocity [29, 30, 26] are invoked to constrain the
generalexpression. Inseveralcases,asimpleexpressionisfound.
However,ifitwerenotfortheamountofinspiration,theseconjectureswouldbenoth-
ing but interpolation formulae, although surprisingly simple. It would be very nice to
provetheseformulaewithoutmakingconjectures. Threenaturallinesofinvestigationare
thefollowing:
1. BetheAnsatzequations. Aswementioned,theydealwiththeBetherootsasthebasic
object. This is a bit unnatural if one is ultimately interested in the anomalous di-
mensions which are a much simpler quantity. The Bethe equations can be used to
obtaintheasymptoticdensityofBetheroots,i.e. the limit[13,16,31]. How-
N ! 1
ever, systematiccorrections at large appear to be rather involved and practically
N
restrictedtoafeworders[32].
2. Baxter equation. An alternative approach to the calculation of is based on the
(cid:13)(N)
Baxter approach originally formulated in [33]. The crucial ingredient is the Bax-
ter operator whose eigenvalues obey a relatively simple functional equation.
Q(u)
If is assumed to be a polynomial, then the Baxter equation is equivalent to
Q(u)
the algebraic Bethe Ansatz equations for its roots to be identified with the Bethe
roots. A more general discussion with references to the non-polynomial cases can
be found in [34, 35]. Remarkably, the polynomial turns out to have rational
Q(u)
coefficientsinallcaseswhereaclosedAnsatzfor hasbeenproposed. Accord-
(cid:13)(N)
ingly, the anomalous dimension is a rational combination of derivatives of .
Q(u)
This approach leads to the knownexact one-loopformulae. The keyfeature is that
the one-loop Baxter polynomial is known as an analytic function of in this case.
N
Unfortunately,suchaniceresultisnotavailable beyondone-loop.
3. Conformal methods. Theseare reviewed in the recent paper [36]. Leading twist con-
formal primary operatorslie on theunitarity boundand hence are conserved(irre-
ducible) in the free theory. It is possible to cleverly exploit the pattern of breaking
of the irreducibility conditions in the interacting theory. In the end, it is possible
to gain an order of perturbation theory and infer two-loop results from the lowest
ordercalculations.
In this paper, we shall reconsider the Baxter approach. We give up the task of com-
puting the exact Baxter polynomial at more than one loop parametrically in the spin .
N
Instead,wedevelopatechniquetosystematicallyderivethelargespinexpansionof
(cid:13)(N)
–3–
at high orders in the natural logarithmic expansion . This will permit to
n m
(cid:24) log N=N
derive from first principles the expansion of although without knowing its resum-
(cid:13)(N)
mation. Thiscan beregardedas astrongcheckoftheproposedconjecturesat large spin.
It is also a useful result by itself if, for instance, one is interested in proving large spin
propertieslikereciprocity.
The sketch of the paper is the following. First, we illustrate the method in the easy
one-loop case in twist-2 and 3. At this level, we do nothing more than rephrase a very
tricky technique originally devised by G. Korchemsky in [35]. The resulting procedure
will be called -method for brevity. It works well and the desired expansion is system-
(cid:1)
atically obtained in a few lines of calculations, easily implemented on symbolic algebra
packages.
Then, we move to the two-loop case. Here, we find some surprise. The -method
(cid:1)
fails to reproduce the rational part of startingfrom theterms . Curiously,
4
(cid:13)2(N) O(1=N )
thefailureisnotsignaledbyanyapparentpathology. Thelessonisthatthe -methodcan
(cid:1)
bemisleading,althoughitworkswellforthenonrationalpartandforallthelogarithmi-
cally enhanced contributions. The problem is not present in twist-3 as we illustrate and
partiallyexplain.
As a further step, we analyze what is happening in twist-2 and propose a safe im-
proved expansion of the Baxter equation which correctly reproduces the full anomalous
dimension. Finally, we analyze the twist-3 case at three loop to show that again there is
a failure in therational part of . The same solutionadoptedfor thetwist-2, 2-loop case,
(cid:13)
solves completely the problem and appear to be the necessary universal recipe for this
kindofexpansions.
2. The one-loop Baxter equation in thesl sector
(2)
2.1 GeneralstructureoftheBaxterequation
Inthenotationof[37],theone-loopBaxterequationinsl -likesectorsis
(2)
(2.1)
L L
(u+is) Q(u+i)+(u(cid:0)is) Q(u(cid:0)i) =tL(u)Q(u);
where isthetwist, theconformalspin,and apolynomial
L s tL(u)
(2.2)
L L(cid:0)2
tL(u) =2u +qL;2u +(cid:1)(cid:1)(cid:1)+qL;L:
For brevity, in the following, we shall refer to as the transfer matrix with a small
tL(u)
abuseoflanguage.
Thesecondcharge isexplicitlyknownandreads
qL;2
(2.3)
qL;2 =(cid:0)(N +Ls)(N +Ls(cid:0)1)+Ls(s(cid:0)1);
where is thespin quantumnumberofthesolution. Weshall be interestedin thepoly-
N
nomialsolution
YN (2.4)
Q(u) = (u(cid:0)ui):
i=1
–4–
Replacing into the Baxter equation, we obtain the BetheAnsatzequations for the
sl spinchain XXXs
(2)
(cid:18)uk +is(cid:19)L YN uk (cid:0)uj (cid:0)i (2.5)
= :
uk (cid:0)is uk (cid:0)uj +i
j=1;j6=k
The one-loop anomalous dimension/energyand quasi-momentum are given in termsof
theBetheroots by
ui
XN 2s i(cid:18) XN uk(cid:0)is (2.6)
(cid:13)1 =k=1u2k+s2; e =k=1uk+is:
TheycanbecalculatedfromtheBaxterpolynomial,completelybypassingtheknowledge
of
fuig
(cid:12)
(cid:13)1 =i (logQ(u))0(cid:12)(cid:12)(cid:12)u=+is; ei(cid:18) = Q(+is): (2.7)
u=(cid:0)is Q((cid:0)is)
In the sl sector, we must set . Also we shall always take even. In this case,
1
(2) s = N
2
the ground statewith minimal anomalous dimension is non degenerateand has
Q(u) =
with automatically vanishing quasi-momentum. This is correct for single-trace
Q((cid:0)u)
composite operators in the gauge theory. Exploiting parity, the anomalous dimension is
simply
(cid:12)
(cid:12) (2.8)
(cid:13)1 =2i (logQ(u))0(cid:12)(cid:12) :
u=+is
2.2 Groundstatesolutionintwist2and3
Thestructureofthegroundstate,i.e. withminimalanomalousdimension,israthersimple
for even and twist 2 or 3. In the twist-2 case, doesnot dependon any unknown
N t2(u)
quantumnumber
(2.9)
2
t2(u) = 2u +q2;2;
1 (2.10)
q2;2 = (cid:0) (cid:0)N(N +1):
2
TheBaxterfunctionistheevenhypergeometricpolynomial[38]
(cid:18) (cid:19)
1 (2.11)
Q(u) = 3F2 (cid:0)N;N +1; (cid:0)iu;1;1;1 :
2
Fromthisexplicitexpressionwecancomputetheone-loopanomalousdimension
(2.12)
(cid:13)1(N) =4S1(N);
where(nested)harmonicsumsaredefinedasusualby
XN (signa)n b XN (signa)n b (2.13)
Sa(N) = jaj ; Sa; = jaj S (n):
n n
n=1 n=1
Inparticular istheharmonicsum
S1
1 1 (2.14)
S1(N) =1+ +(cid:1)(cid:1)(cid:1)+ :
2 N
–5–
The twist-3 case is also simple because again do not depend on any unknown
t3(u)
quantumnumber.
(2.15)
3
t3(u) = 2u +q3;2u;
(cid:18) (cid:19) (cid:18) (cid:19)
3 1 3 (2.16)
q3;2 = (cid:0) (cid:0) N + N + :
4 2 2
TheBaxterfunctionistheevenhypergeometricpolynomial[28,26,22]
(cid:18) (cid:19)
N N 1 1 (2.17)
Q(u) = 4F3 (cid:0) ; +1; +iu; (cid:0)iu;1;1;1;1 ;
2 2 2 2
whichcanbewrittenintermsofWilsonpolynomials(seeAppendixBof[39]).
Fromthisexplicitexpressionwecancomputetheone-loopanomalousdimension
(cid:18) (cid:19)
N (2.18)
(cid:13)1(N) =4S1 :
2
2.3 Large expansionoftheexact
N (cid:13)1
Thelarge expansionof isthatofthefunction forsmallvaluesofthevariable
N (cid:13)1 4S1(1=")
definedas
"
twist-2 1 twist-3 2 (2.19)
" = ; " = :
N N
Theexpansioniswell-knowninanalyticformatallordersandreads
(cid:18)1(cid:19) X1 B2k 2k (2.20)
4S1 =(cid:0)4 log"+4(cid:13)E +2"(cid:0)2 " ;
" k
k=1
where areBernoullinumbersdefinedby
Bk
t X1 Bk k t t2 t4 t6 t8 (cid:16) 10(cid:17) (2.21)
= t =1(cid:0) + (cid:0) + (cid:0) +O t :
t
e (cid:0)1 k! 2 12 720 30240 1209600
k=0
2.4 Large expansionfromtheBaxterequation
N
Now, our aim is to obtain theexpansion Eq. (2.20) directly from the Baxter equation and
withoutexploitingtheknownexactsolution. Letusshowthatthisisquitesimple. Weuse
a trick first introduced by G. Korchemsky [35, 39] with some small modification to easy
theimplementationonsymboliccalculation packages.
First,wedefine and
u=iz
(2.22)
F(z)
Q(iz) =e ; (cid:1)(z) =F(z +1)(cid:0)F(z):
The cases twist-2 and 3 must be treated separately because the calculations differ in the
details.
–6–
2.4.1 Twist-2
WewritetheBaxterequationas
(cid:18) (cid:19)2 (cid:18) (cid:19)2 (cid:18) (cid:19)
2 1 (cid:1)(z) 2 1 (cid:0)(cid:1)(z(cid:0)1) 2 1 2 (2.23)
" z+ e +" z (cid:0) e =1+"+" +2z :
2 2 2
Theanalysisof[35]canberephrasedasthefollowingverysimpleAnsatzfortheasymp-
toticexpansionof
(cid:1)(z)
X1 (2.24)
n
(cid:1)(z) =(cid:0)2 log"+ an(z)" :
n=0
This expansion is derived for sufficiently large positive and is assumed to admit an
z
analyticcontinuationdownto .
z =1=2
Now,theprocedurethatweshallcall -methodgoesin3simplesteps:
(cid:1)
1. ReplaceEq.(2.24)intoEq.(2.23)andmatchthevariousintegerpowersof toobtain
"
thefunctions .
an(z)
2. Given ,getbackthederivativeof ,i.e. .
(cid:1)(z) logQ F(z)
3. Evaluate togettheone-loopanomalousdimension.
0
F (1=2)
Letusseehowallthiscanbeaccomplished. First,step(1). Afterexpansion,weget
(cid:18) (cid:19) (cid:18) (cid:19)
1 2 2 2 1 3 (2.25)
(cid:1)(z) = (cid:0)2 log "(cid:0)2 log z + +"+2z " (cid:0) 2z + " +
2 6
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
1 2 2 4 1 4 3 2 5
(cid:0) 4z +1 12z (cid:0)8z (cid:0)1 " + 240z (cid:0)160z +120z (cid:0)40z +3 " +
16 40
(cid:16) (cid:17)
1 6 5 4 3 2 6
+ 80z (cid:0)144z +120z (cid:0)96z +21z (cid:0)3z " +
12
(cid:16) (cid:17)
1 6 5 4 3 2 7
+ (cid:0)2240z +4032z (cid:0)5040z +3808z (cid:0)1540z +364z (cid:0)33 " +(cid:1)(cid:1)(cid:1) :
112
Toperformstep(2),wefirstnoticethatfromthedefinitionof wededuce
(cid:1)(z)
0 D (2.26)
F (z) = (cid:1)(z):
D
e (cid:0)1
Thelinearoperatoronther.h.s. canbeevaluatedinclosedformwhenactingonmanyspe-
cial . Inparticular,thisistrueforall thatwillappearinthisandlatercalculations.
g(z) g(z)
Thesimplestcaseisthatofpolynomial
g(z)
D n (2.27)
D z =Bn(z);
e (cid:0)1
where isthe -thBernoullipolynomialdefinedbythegeneratingfunction
Bn(z) n
tetz X1 Bk(z) k (2.28)
= t :
t
e (cid:0)1 k!
k=0
–7–
Also,for ,wehavetheveryusefulidentity
g(z) =log z
D where d (2.29)
log z = (z); (z) = log(cid:0)(z):
D
e (cid:0)1 dz
Takingderivatives,wealsoobtain
n+1 n
D 1 ((cid:0)1) (n) (n) d (2.30)
= (z); (z) = (z):
D n n
e (cid:0)1 z (n(cid:0)1)! dz
Goingbacktotheproblemofdetermining ,weimmediatelyfind
0
F (z)
(cid:18) (cid:19) (cid:16) (cid:17)
0 1 1 2 2 1 2 3
F (z) = (cid:0)2 log "(cid:0)2 z+ +"+ 6z (cid:0)6z +1 " (cid:0) (2z (cid:0)1) " +
2 3 2
(cid:16) (cid:17)
1 4 3 2 4 1 3 5
(cid:0)720z +1920z (cid:0)1560z +480z (cid:0)41 " + (2z (cid:0)1) (6z (cid:0)7)" +
240 8
(cid:16) (cid:17)
1 6 5 4 3 2 6
+ 1680z (cid:0)8064z +14280z (cid:0)12096z +5145z (cid:0)1008z +61 " +
252
(cid:16) (cid:16) (cid:17)(cid:17)
1 3 2 7 (2.31)
(cid:0) (2z (cid:0)3)(2z (cid:0)1) 20z (cid:0)36z +17 " +(cid:1)(cid:1)(cid:1) :
16
The last step(3) is trivial. Using to evaluate we immediately
0
(1) = (cid:0)(cid:13)E (cid:13)1 = 2F (1=2)
recoverthecorrectexpansioninfullagreementwithEq.(2.20)!
2.4.2 Twist-3
The same analysis can be repeated in twist-3. The details are quite similar, but slightly
different. Thefinalexpressionofthefunction isnow
(cid:1)(z)
(cid:18) (cid:19) 2 ! 2 !
1 z 1 2 z 1 3
(cid:1)(z) = (cid:0)2 log "+log z (cid:0)3log z+ +2log 2+ "+ (cid:0) " + (cid:0) (cid:0) " +
2 2 8 2 24
!
4 3 2
3z z 13z 3z 1 11 1 4
(cid:0) + + + (cid:0) + + " +
16 8 64 64 1024(z (cid:0)1) 256 1024z
!
4 3 2
3z z 3z 3z 1 7 1 5 (2.32)
(cid:0) + (cid:0) + (cid:0) (cid:0) " +
8 4 32 32 512(z (cid:0)1) 640 512z
!
6 5 4 3 2
5z 3z 5z z z 9z 7 7 7 6
(cid:0) (cid:0) + (cid:0) + (cid:0) (cid:0) + " +(cid:1)(cid:1)(cid:1) :
48 16 32 16 8 128 4096(z (cid:0)1) 1536 4096z
Again, each term can be worked out explicitly to give a simple analytic expression for
. Although is quite different than that for twist-2, one checks that again the
0
F (z) (cid:1)(z)
sameandcorrectexpansionof (nowwith )isobtained.
(cid:13)1 " =2=N
2.5 Comments
Beforemovingtothetwo-loopcase,itisconvenienttostopandcommentabouttheabove
procedure.
The -methodinitsoriginalformisbasedonthefollowingobservation. Thel.h.s. of
(cid:1)
theBaxterequationmustbalancethetransfermatrix atlarge . Thisisaccomplished
t(u) N
–8–
forIm byassumingthattheterm isdominantandtreatingthesecond
L
u > 0 (cid:24) (u+i=2)
piece as a perturbation. This seems to be quite reasonable since in the end we want to
compute in wherethesecondpiecevanishes. Ourimplementationstarting
0
F (z) z = 1=2
directly from a logarithmic expansion of follows this simple idea. Notice however
(cid:1)(z)
that the expansion is guaranteed to work for large enough , depending on the order of
z
theexpansion. Intheoriginalwork[35],onehastoanalyticallycontinuedownto
z =1=2
whichappearstobedangerous. Nevertheless,themachineryworkswellatone-loop. We
shallfindsurprisesattwoloops.
Asasecondcomment,wenoticethattheBaxterequationisinvariantunderanytrans-
formation
with (2.33)
Q(u) !h(u)Q(u); h(u) =h(u+i):
In other word, is determined modulo an additive term with periodicity 1 in the
F(z) z
variable. In the above one-loop examples, this term is absent. Heuristically, this can be
explained as follows. Given a polynomial , it seems reasonable to conclude that at
Q(u)
eachorderin , mustgrowatmostpolynomiallywith . Aperiodiccontribu-
1=N logQ(u) u
tion in the variable would instead lead to exponentially growing quantities. Certainly,
z
itwouldbenicetounderstandthispointbetter.
Wenowmovetothemoreinterestingtwo-loopcasewhereweshallshowthepartial
failureofthe -method.
(cid:1)
3. The two-loop Baxterequation for the ground state
3.1 Generalstructure
ThelongrangeBaxterequationforthesl sectorof SYMisdiscussedatthreeloop
(2) N =4
accuracyin[40]. Here,weshallbeinterestedinthetwoloopreductionthatwediscussin
fulldetailsinthecaseofthegroundstate.
Weshallrequirethefollowingdefinitions
0 s 1
u 2(cid:21) (3.1)
x(u) = 1+ 1(cid:0) A;
2
2 u
i (3.2)
u(cid:6) = u(cid:6) ; x(cid:6) =x(u(cid:6)):
2
Also,for ,weintroduce
(cid:27) =(cid:6)1
(cid:12) (cid:18) (cid:19)
(cid:3)(cid:27) = d log Q(u)(cid:12)(cid:12)(cid:12) ; (cid:1)(cid:27)(x) =xL exp (cid:0)(cid:21)(cid:3)(cid:27) : (3.3)
du u=i (cid:27) x
2
Notice that is independent on the spectral parameter . However, it depends on the
(cid:3)(cid:27) u
couplinghiddenintheBaxterfunction .
Q(u)
Thelong-rangeBaxterequationreads
(3.4)
(cid:1)+(x+)Q(u+i)+(cid:1)(cid:0)(x(cid:0)) Q(u(cid:0)i) =tL(u)Q(u);
–9–