Form factors of descendant operators in the Bullough-Dodd model

0.24 MB·English

by  Oleg Alekseev

Form factors of descendant operators in the Bullough-Dodd model
Oleg Alekseev
Landau Institute for Theoretical Physics,
142432 Chernogolovka of Moscow Region, Russia
and
Center for Quantum Spacetime, Sogang University,
Shisu-dong, Mapo-gu, Seoul 121-742 Korea
2 Abstract
1
0 WeproposeafreefieldrepresentationfortheformfactorsofdescendantoperatorsintheBullough-
2 Dodd model. This construction is a particular modification of Lukyanov’stechniquefor solving the
form factors axioms. We prove that the number of proposed solutions in each level subspace of the
t
c chiralsectorscoincidewiththenumberofthecorrespondingdescendantoperatorsintheLagrangian
O
formalism. We check that these form factors possess the cluster factorization property. Besides, we
propose an alternative free field representation which allows us to study analytic properties of the
0
form factors effectively. In particular, we prove that the form factors satisfy non trivial identities
1
known as the “reflection relations”. We show the existence of the reflection invariant basis in the

] levelsubspaces for a generic values of theparameters.
h
t
- 1. Introduction
p
e
h The form factors of local operatorsin two dimensional integrable models can be exactly obtained as the
[ solutions to a system offunctional equations knownas the formfactor axioms [1, 2, 3]. These equations

1 specify the correct analytical properties of the form factors predefined by the scattering matrix of the
v model. In principle,one cansolvethese equationsrecursively. The spaceofsolutionsto the formfactors
8 axioms is supposed to give a faithful representation of the operator content of the model. However, the
1
correctidentificationoftheformfactorsofthespecificoperatorwithinthisspaceofsolutionsisingeneral
8
non trivial problem. In the case of the exponential operators there exist two useful criteria which assist
2
. in the identification. The first one restricts the asymptotic behavior of the form factors at large values
0
of rapidity [4] while the second one is the so called cluster factorization property [5]. But the family of
1
exponentialoperatorsarefar fromexhausting the full set of operatorsin the theory, whichalso contains
2
1 descendantoperators. Thetwocriteriawediscussedarenotsufficienttosolvethe identificationproblem
: for these operators. Therefore, we need an effective approach to study the form factors of descendant
v
operators.
i
X Here we consider the Bullough-Dodd model [6, 7]. A convenient method for solving the form factor
r axioms is the free field representation proposed by Lukyanov [8]. By using this method the solutions to
a
the form factor axioms in the Bullough-Dodd model that correspond to the exponential operators were
found and completely identified [9]. Following the guidelines of [10] we find the free field representation
for the form factorsof descendantoperators. However,the problemof identificationis not solvedin this
case.
The proposed free field representation allows us to study analytic properties of the form factors
efficiently. In particular, we prove that the form factors satisfy the reflection relations [11]. These
relations originate from the similar relations in the Liouville theory and establish connection between
the exponential operators eaϕ with different values of the parameter a [12]. Besides, we prove the
existence of the reflection invariant bases in the Fock spaces.
This paper is organized as follows. In Section 2 we describe the Bullough-Dodd model and specify
its operator contents. In Section 3 we briefly describe Lukyanov’s free field representation for the form
factorsofexponentialoperators. Weintroduceauxiliaryconstructionandfindthefreefieldrepresentation
for the set of solutions to the form factor axioms. The number of these solutions on each level subspace
1
of the Fock module is shown to be coincide with the dimension of the corresponding level subspace in
the Lagrangian formalism. We prove that these solutions satisfy the cluster factorization property and
study their asymptotic behavior. In Section 4 we introduce alternative free field representation which
is used to obtain recurrence relations between the form factors and prove the reflection properties. In
Section 5 we consider an explicit example, namely, level (2,0) descendant operators. Finally, we draw
our conclusions in Section 6.
2. The Bullough Dodd model
The Bullough-Dodd model is defined by the Eucledean action [6, 7]
SBD = d2x 1 (∂νϕ)2+µ(e√2bϕ+2e−√b2ϕ) . (2.1)
15
Z (cid:18) (cid:19)
Here b is the coupling constant and µ is the regularized mass parameter. The spectrum of the model
consistofthe single particleAofmassM whichcanbe expressedintermsofthe parameterµ [11]. This
particle appears as the bound state of itself in the scattering processes
A A A A A. (2.2)
× → → ×
Hereinafter we shall use the following notation
Q=b+b 1, ω =eiπ/3. (2.3)
−
Also, we shall use the light-cone coordinates
∂ ∂
z =x1 x0, z¯=x1+x0, ∂ = , ∂¯= .
− ∂z ∂z¯
The integrability of the model implies that the scattering processes are purely elastic [13]. Therefore,
the n-particle S-matrix factorizes into the n(n 1)/2 two-particle scattering amplitudes [14]
−
tanh1(θ+ 2iπ)tanh1(θ 2iπ)tanh1(θ 2iπb)
S(θ)= 2 3 2 − 3bQ 2 − 3Q . (2.4)
tanh1(θ 2iπ)tanh1(θ+ 2iπ)tanh1(θ+ 2iπb)
2 − 3 2 3Qb 2 3Q
The S-matrix is invariant under the weak-strong coupling constant duality b b 1. A simple pole of
−
→
the scattering matrix located at θ =2πi/3 corresponds to the bound state of the particle A itself in the
scattering process (2.2).
Letusconsidertheoperatorcontentsofthemodel. ThespaceoflocaloperatorsoftheBullogh-Dodd
model consists of the exponential operators
V (x)=eaϕ(x) (2.5)
a
and their descendants, i.e. the linear combinations of the fields
∂n1ϕ...∂nrϕ∂¯n¯1ϕ...∂¯n¯sϕeaϕ(x). (2.6)
Here the pair of integers (n,n¯) given by
r s
n= n , n¯ = n¯ (2.7)
i j
i=1 j=1
X X
iscalledthelevelofdescendantoperator. Thenumbersnandn¯ separatelyarecalledchiralandantichiral
level correspondingly. Any exponential operator is characterized by its scaling dimension ∆ in the
a
ultraviolet regime which is related to the parameter a. The scaling dimension of the corresponding
descendant operator at the level (n,n¯) is given by ∆ +n+n¯, while its spin is n n¯.
a
−
Let us consider the radial quantization picture at some point in the Eucledean plane, e.g. x =0. In
the vicinity of this point the field ϕ(x) can be expanded as
a a¯
ϕ(x)=Q iPlogzz¯+ mz−m+ mz¯−m. (2.8)
− im im
m=0 m=0
X6 X6
2
HeretheoperatorsQ,P,a anda¯ formaHeisenbergalgebrawiththefollowingcommutationrelations
m m
[P,Q]= i, [a ,a ]=mδ , [a¯ ,a¯ ]=mδ . (2.9)
m n m+n,0 m n m+n,0
−
IntheradialquantizationpicturetheexponentialoperatorV (0)correspondstothehighestweightvector
a
a defined by the relations
rad
| i
a a =a¯ a =0 (m>0), Pa =aa , a =eaQ vac (2.10)
m rad m rad rad rad rad rad
| i | i | i | i | i | i
Let be the Fock module spanned on the vectors generated by the elements a (m < 0) acting on
a m
the hFighest weight vector a . Similarly, the ¯ is a Fock module spanned on−the vectors generated
rad a
| i F
by the elements a¯ (m <0) acting on the same highest weight vector. Evidently, the modules and
m
¯ are isomorphic.−The space of states is given by the tensor product ¯ of chiral and antFichiral
a a
F F ⊗F
components. Up to some numerical factor the descendant operators (2.6) correspond to the vectors
a ...a a¯ ...a¯ a (0<n ... n ,0<n¯ ... n¯ ). (2.11)
−n1 −nr −n¯1 −n¯s| irad 1 ≤ ≤ r 1 ≤ ≤ s
Each module admits a natural gradation into the level subspaces = , where each level
Fa Fa ⊕∞n=0Fa,n
subspace is spanned on the vectors (2.11) with n¯ = 0 and n = n. The dimensions of these
a,n i
F
subspaces are given by the following generating function
P
∞ ∞ 1
qndim = . (2.12)
Fa,n 1 qm
n=0 m=1 −
X Y
The dimension of the level (n,n¯) subspace if given by the product dim dim ¯ .
a,n a,n¯
F · F
3. Free Field representation for form factors
3.1. Form factors of exponential operators
Let us briefly describe Lukyanov’s free field representation for the form factors of exponential operators
in the Bullough-Dodd model [9]. First, we consider a pair of operatorsΛ+(θ), Λ (θ) and define the one
−
and two-point trace functions as follows
Λσ(θ) =1, Λσ′(θ′)Λσ(θ) =[R(θ θ′)]σ′σ, σ′,σ = , (3.1)
hh ii hh ii − ±
where R(θ) is the two-point minimal form-factor given by [15]
∞ dtcosh6t sinh3tQb sinh3Qtb iθt
R(θ)=exp 4 cosh(t ) . (3.2)
− t sinhtsinh t − π
(cid:16) Z0 2 (cid:17)
The multi-point functions can be calculated by means of the Wick’s theorem. Let us define the normal
ordering procedure :...: by the following relation
ΛσN(θ )...Λσ1(θ )=:ΛσN(θ )...Λσ1(θ ): Λσj(θ )Λσi(θ ) . (3.3)
N 1 N 1 j i
hh ii
1 i<j N
≤Y≤
Hereinafter we shall use the notation
iπ iπ
Λ0(θ)=:Λ+(θ )Λ (θ+ ):. (3.4)
−
− 3 3
The Lukyanov’s generators are given by
T(θ)=ρ(eiπpΛ+(θ)+e iπpΛ (θ)+hΛ0(θ)), (3.5)
− −
where
π(b b 1)
−
h=2sin − ,
6Q
4√2a b+b 1 1
−
p= − , (3.6)
6Q − 2
ρ= sinπ3 exp 2 ∞ dtcosh6t sinh3tQb sinh3Qtb .
ssin23πQbsin32Qπb (cid:16) Z0 t sinhtcosh2t (cid:17)
3
The form factors of exponential operators are given by the multi-point trace functions, namely
vacV (0)θ ,...,θ eaϕ f (θ ,...,θ )= eaϕ T(θ )...T(θ ) , (3.7)
a 1 N a 1 N N 1
h | | i≡h i h ihh ii
where eaϕ is the vacuum expectation value of the corresponding exponential operator [11]. The
h i
functions f (θ ,...,θ ) are analytic functions in the variables θ with complicated analytic structure.
a 1 N i
Taking into account (3.1) we immediately conclude that these functions can be represented as follows
f (θ ,...,θ )=ρNJ (eθ1...,eθN) R(θ θ ). (3.8)
a 1 N N,a i j
−
i i<j N
≤Y≤
HerethefunctionsJN,a(x1,...,xN)aresymmetricrationalfunctionsinthevariablesxi =eθi. Theirpole
structure is governed by the form factor axioms. The only poles located at relative rapidity difference
θ =iπandθ =2πi/3willbereferredtoasthekinematicalandtheboundstatepolescorrespondingly.
ij ij
3.2. Form factors of descendant operators
We argue that the form factors of descendant operators possess the free field representationthat is very
similar to Lukyanov’s one (3.7). The main idea is to modify the generators T(θ). Let us consider a
commutative algebra = generated by the elements α with n > 0. Each level subspace
is spanned on theAelem⊕en∞nt=s0Anα such that n = n. W{e −alns}o introduce another copy ¯ of the
An −ni i A
algebra generated by the elements α¯ . The canonical homomorphism between these algebras is
n
defined bAy the following relationQ: for a{ny−h} lePt us define h¯ ¯ according to the rule α α¯ .
n n
The element g =hh¯ will be referred to as th∈eAlevel (n,n¯) descen∈daAnt if h and h¯ ¯ −. → −
′ n ′ n¯
∈A ∈A
Let us define a bracket on the algebra by
A
∞ ∞ ∞
αkm , αlm = k !δ (3.9)
m m m km,lm
− − !
m=1 m=1 m=1
Y Y Y
and consider the currents
∞
a(z)=exp α zm ,
m
−
(cid:18)m=1 (cid:19)
X
∞
b(z)=exp α ( z)m , (3.10)
m
− − −
(cid:18) m=1 (cid:19)
X
∞
c(z)=exp (ω−m ( 1)mωm)α mzm .
− − −
(cid:18)m=1 (cid:19)
X
By using these currents we modify the Lukyanov’s generators as follows
(θ)=ρ eiπpa(eθ)¯b(e−θ)Λ+(θ)+e−iπpb(eθ)a¯(e−θ)Λ−(θ)+hc(eθ)c¯(e−θ)Λ0(θ) . (3.11)
T
For any element g(cid:0) ¯consider the function (cid:1)
∈A⊗A
fg(θ ,...,θ )=( (θ ),..., (θ ) ,g). (3.12)
a 1 N hhT N T 1 ii
It is straightforward to check that this function is a solution to the form factor axioms. Indeed, the
Watson’s theorem and the crossing symmetry condition are evidently satisfied while the kinematical
and the bound state pole conditions are satisfied if the currents (3.10) are subjected to the following
conditions
a(z)b( z)=1, c(z)=a(zω 1)b(zω),
−
−
whicharethecase. Consequently,thefunction(3.12)determinestheformfactorsofanoperatorfromthe
Fock space ( ¯)V (x). The descendant operator which corresponds to this function will be denoted
a
F ⊗F
by Vg(x) that is
a
vacVg(x)θ ,...,θ = eaϕ fg(θ ,...,θ ). (3.13)
h | a | 1 Ni h i a 1 N
4
From expression (3.12) it follows that the solutions fg to the form factor axioms are given by
a
N
fg(θ ,...,θ )=ρNJg (eθ1,...,eθN) R(θ θ ). (3.14)
a 1 N N,a i− j
i<j
Y
Here the functions Jag(x1,...,xN) are symmetric rational functions in the variables xi = eθi with
the kinematical and bound state poles located at rapidity differences θ = iπ and θ = 2iπ/3
ij ij
±
correspondingly. Evidently, the J functions which correspond to the exponential operators are given
by J1 =J . It is straightforwardto get an explicit expression for the function Jg , namely
N,a N,a N,a
JNg,a(x1,...xN)= h#I0e(#I+−#I−)iπpPg(X+|X−|X0)×
I++IX−+I0=I
x x x x x
f iω f iω2 f iω f jω 1 f p . (3.15)
−
× x x x x x
j j k k q
i∈I+,jY∈I−,k∈I0 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(p<Yq)∈I0 (cid:16) (cid:17)
where
h2 1
f(x)=1+ − . (3.16)
x+x 1 1
− −
In expression (3.15) we introduce a set of integers I = 1,...,N and the sum is taken over all
{ }
decompositions of the set I into the three subsets I , σ = +, ,0 , such that I I I = I
σ + 0
and I I = ∅ if σ = σ. With every subset I we associ{ate −the }subset X = x∪i − ∪I . The
σ σ ′ σ σ i σ
′ ∩ 6 { | ∈ }
functions Pg(X Y Z) are polynomials defined by the following relations
| |
Pα−m(X Y Z)=Sm(X) ( 1)mSm(Y)+(ω−m ( 1)mωm)Sm(Z),
| | − − − −
Pα¯−m(X Y Z)=S m(Y) ( 1)mS m(X)+(ω−m ( 1)mωm)S m(Z), (3.17)
| | − − − − − − −
Pg1g2 =Pg1Pg2, Pc1g1+c2g2 =c Pg1 +c Pg2 ( g ,g ¯, c ,c C).
1 2 1 2 1 2
∀ ∈A⊗A ∈
Here we denote the power sums of order m by
N
S (x ,...,x )= xm. (3.18)
m 1 N
i=1
X
In summary, in the framework of the free field representation we obtain the set of solutions fg to the
a
form factor axioms. Besides, in (3.14) and (3.15) we give explicit expressions for these functions. The
formfactors ofexponentialoperatorswhichcorrespondto the case g =1 have been studied indetail [9].
We want to prove that the space of proposed solutions can be bijectively mapped into the space of
descendant operators in the Lagrangian formalism. In other words, we want to prove that for any
descendant operator in the Lagrangianformalism there exist a solution to the form factor axioms given
by certain linear combination of the functions fg.
a
It is a challenging problem to make an identification between the states from the space ¯ and
those ones from ¯. Certain requirements are necessary to identify the form factors oAf a⊗sApecific
F ⊗F
operator among all these functions. In the case of exponential operators the solution to this problem is
known. Itissufficienttoimposerestrictionontheasymptoticbehavioroftheformfactorsatlargevalues
of rapidities [4] and demand the cluster factorization property to be satisfied [5]. However, additional
criteria are required in the case of descendant operators. This problem is not solved in general cases.
Nevertheless, let us consider the analytic properties of the proposed set of functions.
3.3. Cluster factorization property and asymptotic behavior
We considerthe element g =hh¯ suchthat h , ¯h ¯. We areinterestedin the asymptotic behavior
′ ′
∈A ∈A
of the function which corresponds to this element, namely
fhh¯′(θ ,...,θ ,θ +Λ,θ +Λ), (3.19)
a 1 n n+1 N
as Λ . It is convenient to study the asymptotic of this function using the representation (3.14).
→ ∞
Notice that R(θ Λ) 1 as Λ . Besides, is straightforwardto check that in this limit we have
± → →∞
Phh¯′(XeΛ,X′ YeΛ,Y′ ZeΛ,Z′) Ph(XeΛ YeΛ ZeΛ)Ph¯′(X′ Y′ Z′) (3.20)
| | ≃ | | | |
5
Therefore, we get the cluster factorization property for the functions fhh¯′, namely
a
fhh¯′(θ ,...,θ ,θ +Λ,...,θ +Λ) fh(θ +Λ,...,θ +Λ)fh¯′(θ ,...,θ ), (3.21)
a 1 n n+1 N ≃ a n+1 N a 1 n
as Λ + . The cluster factorization property for the form factors of the descendant operators
→ ∞
immediately follows from (3.13). This result is in agreement with those of [16]. The form factors of the
level (n,n¯) descendant operators factorizes into the form factors of the chiral level (n,0) and antichiral
level (0,n¯) descendant operators.
Letusdiscusscertainconsequenceswhichfollowfromtheclusterfactorizationproperty. Weconsider
thechiralelementsh andh¯ ¯ . Thecorrespondingdescendantoperatorswillbe denotedbyVh
andVh¯′. Thescaling∈diAmnensions′o∈ftAhen¯seoperatorsintheultravioletlimitaregivenby∆h =∆ +nanad
a a a
∆h¯′ =∆ +n¯, where ∆ is the scaling dimension of the corresponding exponential operator. The spins
a a a
of these descendant operators are the following, sh = n and sh¯′ = n¯. From the cluster factorization
−
property we immediately conclude that the element h corresponds to the chiral level n descendant
operators while the element h¯ corresponds to antichiral level n¯ descendant operators.
′
Let us consider the generic descendant operator Vg related to the element g = hh¯ . The scaling
a ′
dimensionofthisoperatorisgivenby∆ahh¯′ =∆a+n+n¯′whileitsspinisthefollowing,shh¯′ =n−n¯. From
the cluster factorizationproperty we conclude that this operators is givenby certain linear combination
of the level (l,¯l) descendants where l n and ¯l n¯. Therefore, that the cluster factorization property
≤ ≤
do notimpose rigorousconditions onthe descendantoperatorswhichcontribute toa particularfunction
fg.
a
3.4. Descendants counting
In this subsectionwe shallprovethat the number of independent solutions fg to the form factor axioms
a
in each level subspace coincide with the number of the descendant operators in the corresponding level
subspace in the Lagrangianformalism. First, let us prove the following theorem
Theorem 1. Forgeneric values of the parameter a the map (3.12) from the algebra ¯intothe space
A⊗A
of the functions fg is a bijection.
a
Letusprovethatthe differentelements g correspondto thedifferentfunctions fg. First,weconsider
a
the chiral element g = h . Let us prove the linear independence of the set of polynomials (3.17).
∈ A
Indeed, for large enough N the functions
z =S (X) ( 1)mS (Y)+(ω m ( 1)mωm)S (Z) (3.22)
m m m − m
− − − −
are functionally independent. Therefore, the linear independence of the set of polynomials Ph(X Y Z)
reduces to the evident linear independence of the monomials zk1...zk2. | |
1 s
Now, let us consider the asymptotic of the function Jh as the parameter a i . It is easy to
N,a → − ∞
check that
e−iπ(4√26aQ−b+1/b)+i2πJNh,a(x1,...,xN) a i =(a(x1)...a(xN),h)=Ph(X|∅|∅). (3.23)
(cid:12) →−∞
This expression defines a map from the alge(cid:12)bra to the algebra of polynomials in the variables z =
(cid:12) i
Ph(X ∅∅). This map is evidently invertible. ThAe linear independence of the polynomials Ph(X Y Z)
| | | |
wasalreadyproved. Consequently,differentelementsfromthealgebra correspondtodifferentfunctions
A
fh. Now we apply the deformation argument. Since the map from the space of elements h to the
a ∈ A
space of form factors fh is a bijection at one point in the parameter a and the form factors are analytic
a
functions in this parameter this map is a bijections for nearly all values of the parameter a. Hence,
we prove that the form factors fh with different values of h differs. Besides, one can easily prove that
a
dim =dim .
n n
NAowletuscFonsiderthegenericelementg =hh¯ ¯. Takingintoaccounttheclusterfactorization
′
∈A⊗A
property we conclude that the form factors of the different elements differs only if chiral and antichiral
components of these form factors differs. Hence, we proved Theorem 1.
As an immediate consequence of the theorem we have
Proposition 1. For generic values of the parameter a the dimension of the space of the operators Vg
a
withg ¯ is equaltothedimension ofthecorrespondingsubspaceoftheFockspacedim( ¯ ).
n n¯ n n¯
∈A ⊗A F ⊗F
6
3.5. Integrals of motion
The Bullough-Dodd model possesses a set of commuting integrals of motion I of odd integer spin s
s
except the multipliers of 3, i.e.
s=6n 1 n=0,1,2... (3.24)
±
Thelocalintegralsofmotionarediagonalizedbytheasymptoticstatesandthecorrespondingeigenvalues
are given by
N
I θ ,...,θ = esθi θ ,...,θ , (3.25)
s 1 N 1 N
| i | i
i=1
X
wherethepropernormalizationoftheintegralsofmotionisassumed. Ontheotherhand,letusconsider
the chiral element α . This element produces a common factor in all terms in (3.15), provided that
m
−
the following relations are satisfied
eiπm = 1, e−i3πm+ei3πm =1. (3.26)
−
The firstrelationshowsthatthe valueofm needtobe odd,while the secondoneshowsthatthe allowed
valuesmarethoseof (3.24). Forthecorrespondingelementsα thecommonfactorinalltermsin(3.15)
s
−
is of the form
Pα s(X X X )=S (X )+S (X )+S (X )=S (X), (3.27)
− + 0 s + s s 0 s
| −| −
where X =X X X . Consequently, we get
+ 0
−∪ ∪
N
fα sg(θ ,...,θ )= esθmfg(θ ,...,θ ), (3.28)
a− 1 N a 1 N
m=1
X
for any g ¯. Comparing (3.25) with (3.28) we conclude that the elements α correspond to the
s
∈A⊗A −
appropriatelynormalizedspin s integrals of motion I while the antichiralelement α¯ correspondto the
s s
integral of motion I , i.e.
s
−
Vα sg(x)=[Vg(x),I ], Vα¯ sg(x)=[Vg(x),I ]. (3.29)
a − a s − a s
−
4. Recurrent relations and reflection property for descendant operators
Inthissectionweshallprovethattheformfactorsofdescendantoperatorssatisfythereflectionrelations.
These relations establish connection between the exponential operators with different values of the
parametera. Namely,uptosomea-dependentfactorthefollowingoperatorsaresupposedtobecoincide
V (x)=R(a) V (x), V (x)=R(a) V (x), (4.1)
a QL−a −a ′ −Q′L+a
where we introduce the notation
1 √2 b
Q = +√2b, Q = + . (4.2)
L √2b ′L b √2
ThefunctionsR(a)andR(a)arethereflectionamplitudes[11]. Thesereflectionrelationsoriginatesfrom
′
the similar relations for exponential operators in the Liouville theory [12]. Indeed, the Bullough-Dodd
model can be treated as two different perturbed Liouville theories with chargesQ or Q depending on
L ′L
which exponent in (2.4) is taken as the perturbing operator. By using recursion relations one can prove
that the J functions of the exponential operators possess the following reflection properties [17]
J (x ,...,x )=J (x ,...,x ), J (x ,...,x )=J (x ,...,x ). (4.3)
N,a 1 N N,QL−a 1 N N,−a 1 N −Q′L+a 1 N
Wearguethatthecorrespondencebetweentheexponentialoperators(4.1)canbeextendedtothewhole
space of descendant operators. More specifically, for any descendant operator of the exponential field
Vg there exist a descendant operator of the exponential field Vg′ such that the form factors of these
a wa
operators coincide. Here we introduce the notation w for the element of the finite group generated
W
by the elements w and w such that
1 2
w a=Q a, w a= Q a. (4.4)
1 L− 2 − ′L−
To prove the reflection property for descendant operators we need to introduce auxiliary construction.
Thisconstructionis analternativefreefieldrepresentationforformfactorsor,tobe morepreciseforthe
functions Jg defined in (3.14).
N,a
7
4.1. The stripped bosonization
From (3.14) it follows that each form factor is proportional to the function Jg up to uniform factor
N,a
which depends on the number of particles only. For the exponential operators, i.e. g = 1, the free
field representation for the functions J was proposed in [17]. Let us briefly recall the construction.
N,a
Thereafter we extend this free field representationto the case of descendant operators.
Let us consider a Heisenberg algebrawith the generatorsd , n Z, n=0. These generatorssatisfy
±n ∈ 6
the following commutation relations
[d ,d ]=0, [d ,d ]=mA δ , (4.5)
±m ±n ±m ∓n ±n m+n,0
where the coefficients A are given by
±n
πn πn π(b b 1)n
A±n =4ω±23 cos 6 cos 3 −cos −3Q− . (4.6)
(cid:16) (cid:17)
Note that
A =A+ =( 1)nA+. (4.7)
−n −n − n
Letaˆbeacentralelementofthealgebra. Besides,let 1 bethevacuumstatesatisfyingtheannihilation
a
| i
conditions for the positive generators d 1 = 0 (n > 0) such that aˆ1 = a1 . The Fock space
±n| ia | ia | ia
ignetnreordautceedtbhyenFeogcaktisvpeaecleemeRntwshdi±−chni(sng>en0er)aftreodmbtyhpeovsaitciuvuemele|1miaenwtsilldbe(nde>no0te)dfrboymDthaLe. vSaimcuiulamrly, w1e.
Da ±n ah |
This vacuum satisfies annihilationconditions for negative modes and 1aˆ= 1a. The Fock spaces we
a a
h | h |
introduced possess decompositions into the level subspaces R = R and L = L .
Da ⊕∞n=0Da,n Da ⊕∞n=0Da,n
Let us consider the exponential operators
d
λ (z)=exp ±nz n, λ0(z)=:λ+(zω 1)λ (zω):. (4.8)
± − − −
n
n=0
X6
Here we use the notation :...: for the normal ordering procedure which is defined by the relations
λ (z )λ (z )=:λ (z )λ (z ):,
± ′ ± ′ ± ′ ± ′
z z (4.9)
λ−(z′)λ+(z)=λ+(z)λ−(z′)=f ω f ω2 :λ+(z)λ−(z′):,
z z
′ ′
(cid:16) (cid:17) (cid:16) (cid:17)
Let us define the current
t(z)=eiπpλ+(z)+e iπpλ (z)+hλ0(z), (4.10)
− −
which looks much the same as Lukyanov’s one (3.5). The functions J which determine the form
N,a
factors of exponential operators are given by the matrix elements of the currents t(x ), namely
i
J (x ,...,x )= 1t(x )...t(x )1 . (4.11)
N,a 1 N a 1 N a
h | | i
Inordertoobtainthefreefieldrepresentationforthe functionsJg weneedanadditionalconstruction.
N,a
Let us consider two representations of the algebra in the Heisenberg algebra, π and π , defined as
R L
A
follows
πR(α−n)= d+nA−+nd−n, πL(α−n)= d+−nA−+nd−−n (n>0). (4.12)
It is easy to get the following commutation relations:
[πR(α n),λ±(z)]=( )n+1znλ±(z),
− ±
[π (α ),λ (z)]= ( )n+1z nλ (z), (4.13)
L n ± − ±
− − ∓
[π (α ),π (α )]= m(A+) 1(1+( 1)m)δ
R −m L −n − m − − m−n,0
We shall also use the notation
h = 1π (h), h¯ =π (h)1 . (4.14)
a a R a L a
h | h | | i | i
8
Consider the chiral element h . Taking into account the commutation relations (4.13) we easily
∈ A
obtain that the J functions of chiral and antichiral descendant operators are given by the following
matrix elements
Jh (x ,...,x )= ht (x )...t (x )1 , Jh¯ (x ,...,x )= 1t (x )...t (x )h¯ . (4.15)
N,a 1 N ah | 1 1 N N | ia N,a 1 N h | 1 1 N N | ia
Now,letusconsiderthegenericelementg =hh¯. Wedenotethecorrespondingmatrixelementasfollows
J˜Nhh¯,a′(x1,...,xN)=ahh|t1(x1),...,tN(xN)|h¯′ia. (4.16)
Notice that this matrix element do not coincide with the function Jhh¯ (X). However, these functions
N,a
can be related to each other. Indeed, by pushing the element π (h) and π (h) from the definition of
R L
the vectors h and h¯ in the expression (4.16) in the opposite directions we get that this function is
given by cerathain| linea|r′cioambination of the functions Jhh¯′, i.e.
N,a
J˜hh¯′(x ,...x )= Jhih¯′i(x ,...,x ). (4.17)
N,a 1 N N,a 1 N
i
X
If in the l.h.s. of this expression we consider the elements from the level subspaces h and h
n ′ n¯
∈A ∈A
then in the r.h.s the sum is taken over the J functions corresponding to the elements h and
i ∈ Ani
h such that n n and n¯ n¯.
′i ∈TAhen¯′ivectors h ain≤d h¯ wiill≤be referred to as the ‘physical’ ones since their matrix elements can
a ′ a
h | | i
be expressed as a linear combination of J functions. These vectors form a subspaces in the spaces
R and L. The ‘physical’ subspaces can be decomposed into the direct sum over the level subspaces
Da Da
R,phys = R,phys and L,phys = L,phys. Evidently that
Da ⊕∞n=0Da,n Da ⊕∞n=0Da,n
dim R,phys =dim L,phys =dim . (4.18)
Da,n Da,n Fn
Further we shall use an alternative definition of the ‘physical’ subspaces. Namely, these subspaces can
be defined as kernels of the operators D defined as follows
n
Dn =ω23nd+n +ω−23nd−n, (n6=0). (4.19)
Itiseasytochecktheconsistencyofthesetwodefinitions. Indeed,theoperatorsD satisfythefollowing
n
commutation relations
[D ,π (h)]=0, [D ,π (h)]=0, [D ,D ]=0. (4.20)
n R n L n m
Besides, one can show that for any state v R such that v D , n > 0, there exist the element
h suchthat v = 1π (h). Similaralyh,|fo∈r aDnaystate v ah |L −sunch thatD v =0, n>0, there
∈A ah | ah | R | ia ∈Da n| ia
exist the element h such that v =π (h)1 .
a L a
∈A | i | i
4.2. Analytic properties of the Jg functions
N,a
The formfactors ofexponentialoperatorsareknownto satisfy recurrencerelations[17]. One canobtain
the similar relations for the form factors of descendant operators. Indeed, the function Jg (z,X) is
N+1,a
ananalyticfunctioninthevariablez dependingontheparametersX = x ,...,x . Thisfunctionhas
1 N
{ }
simple kinematical and bound state poles at points prescribed by the form factor axioms. The residues
at these poles can be evaluated explicitly as shown in [17]. One can separate the kinematical and the
bound state poles contribution from the regular part and obtain
N
x
JNg+1,a(z,X)=JN(∞+)1g,a(z,X)+ z+nx Kn(X)JNg 1,a(Xˆn)+
n −
n=1
X
N x ω2 N x ω 2
+ n B+(X)Jg (x ω,Xˆ ) n − B (X)Jg (x ω 1,Xˆ ), (4.21)
z x ω2 n N,a n n − z x ω 2 n− N,a n − n
n=1 − n n=1 − n −
X X
9
Here we introduce the notation Xˆ =X x and the functions K (X) and B (X) are given by
n \{ n} n n±
(h2 3)(h2 1) x x x x
Kn(X)= i − − f nω2 f nω f nω−2 f nω−1 ,
− 2√3 (cid:18)iY6=n (cid:16)xi (cid:17) (cid:16)xi (cid:17)−iY6=n (cid:16)xi (cid:17) (cid:16)xi (cid:17)(cid:19) (4.22)
h(h2 1) x
Bn±(X)=−i √−3 f xniω±1 .
iY6=n (cid:16) (cid:17)
Thederivationoftheexpansion(4.21)repeatsthemainfeaturesofthoseonefortheexponentialoperators
and we omit it here. The regular part is given by the function J(∞)g (z,X). This function is regular
N+1,a
everywhere except the points z = 0 and z = . Since the sum over residues is of the order O(z 1) as
−
∞
z the asymptotic behavior of the Jg function as a function of z is governed by the regular part
→ ∞
J( )g. Further we shall also use another expansion of the Jg function in the vicinity of z = 0 which is
∞
given by
N x 1
Jg (z,X)=J(0)g (z,X) −n K (X)Jg (Xˆ )
N+1,a N+1,a −n=1z−1+x−n1 n N−1,a n −
X
N x 1ω 2 N x 1ω2
−n − B+(X)Jg (x ω,Xˆ )+ −n B (X)Jg (x ω 1,Xˆ ). (4.23)
−nX=1z−1−x−n1ω−2 n N,a n n nX=1z−1−x−n1ω2 n− N,a n − n
In this case the asymptotic behavior is governedby the function J(0)g (z,X). From expressions (4.21)
N+1,a
and (4.23) we easily obtain that the regular parts of these expansions are related as follows
J(0)g (z,X) J(∞)g (z,X)=Dg (X), (4.24)
N+1,a − N+1,a N,a
where the function Dg (X) is given by
N,a
N N
Dg (X)= x ω2B+(X)Jg (x ω;Xˆ) x ω 2B (X)Jg (x ω 1;Xˆ )+
N,a n n N,a n − n − n− N,a n − n
n=1 n=1
X X
N
+ K (X)Jg (Xˆ ). (4.25)
n N 1,a n
−
n=1
X
The expansions (4.21) and (4.23) allows us to calculate form factors recursively. The recursionrelations
for exponential operators, i.e. g = 1, immediately follows from these expansion since the regular parts
J(0) and J( ) can be easily calculated [17]. In Section 5 we use these expansions to obtain recurrence
∞
relations for the chiral level 2 descendant operators.
4.3. Reflection relations for descendant operators
In this subsection we shall prove that the form factors of descendant operators satisfy the reflection
relations. More precisely, we prove this statement for the matrix elements (4.16). However, from (4.17)
it follows that if the reflection relations hold for J˜functions these relations hold for J functions as well.
Theorem 2. For generic values of the parameter a there exist a representation r (w) of the group
a
W
on the algebra such that for any elements h,h the following relation holds
′
A ∈A
J˜Nhh¯,a′(x1,...,xN)=J˜N(r,aw(aw)h)(r−a(w)h′)(x1,...,xN). (4.26)
The proofs of this theorem is very similar to those of [10]. The main idea is that all form factors
can be obtained as a coefficients of the large rapidity expansion of the form factors of the exponential
operators [18], [19].
First, let us introduce auxiliary current
s(z)=:λ+(zω−3/2)λ−(zω3/2):,
10