Form factors of descendant operators: Free field construction and reflection relations

F orm factors of descendan t op erators: F ree field construction and reflection relations Boris F eigin 1 , 2 and Mic hael La shk evic h 1 1 L andau Institute for The or etic al Physics, 142432 Cherno golovka of Mosc ow R e gion, Russia 2 Indep endent University of Mosc ow, 11 Bolshoy Vlasyevsky p er eulok, 11900 2 Mosc ow, Russia De dic a te d to the m emory of A lexey Z amolo dchi kov Abstract The free field represen tation for form factors in the sinh-Gordon model and the sine-Gordon mod el in the breather sector is mo dified to describ e the form factors of descendant op erators, whic h are obtained from th e exp onential ones, e i αϕ , by means of the action of th e Heisenberg algebra associated to the field ϕ ( x ). As a c heck of th e v alidit y of the construction we count the num b ers of op erators defined by the form factors at eac h level in each chiral sector. Another chec k is related to the so called reflection relations, which identif y in the breather sector the descendants of the exp onential fields e i αϕ and e i(2 α 0 − α ) ϕ for generic va lues of α . W e prov e the operators d efi ned by the obtained families of form factors to satisfy such reflection relations. A generalization of the construction for form factors to the kink sector is also prop osed. 1. In tro duction Exact calc ula tion of form facto rs of lo ca l and quasilo ca l op erator s in t wo-dimensional relativistic quantum field theory is known to be reduced to solving a se t of difference e q uation for analytic functions [1–3] called also Ka rowski–W eisz–Smir nov form fa c to r a xioms. One of the techniques for solving these equations is the free field repres ent ation prop osed by Lukyanov [4 ]. It was shown that this r epresentation makes it po ssible to calculate for m factor s o f the exp onential fields e i αϕ in the sine / sinh-Gordo n mo del [5]. But the family of exp onential op er ators is far fro m exhausting the full set of op era tors in the theor y , whic h contains also the descendant op era tors obtained from the exp onential ones by means of the action of the Heisenberg algebr a asso ciated with the field ϕ ( x ). Here we prop ose a construction of the form factors of descendant op erator s in the brea ther sector of the sine-Gor don theory and for the s inh- Gordon theo r y . W e sta rt fro m the proposa l of Babujian a nd Kar owski [6, 7], who express ed the form facto r s o f descendant op erator s in terms of sequences of some auxiliary functions. These sequences must satisfy some conditions to provide form factors of lo ca l oper ators. The main dis tinctio n of our appr oach is that we imp ose muc h mor e r estrictive conditions to these sequences of functions, which makes it p ossible to substantiate the existence of a one-to -one corresp ondence betw een oper ators and sequences of functions. Besides, w e pro po se a n in terpretation of these solutions in terms of an auxiliary commutativ e alge bra and show that at a gener ic v alue of α the dimensions of the level s ubs paces coincide with those for the F o ck mo dules of the Heisenberg algebra . W e go further and, by means of some auxilia ry b osonizatio n pro cedure, prove the existence o f a reflection prop erty , which r elates brea ther for m factor s of desce ndants of the fields e i αϕ and e i(2 α 0 − α ) ϕ (with the v a lue of α 0 known fro m the co nformal field theory ). E a rlier it was conjectured tha t such relations, well known in the Liouville field theory [8], are v alid for the op era tors in the sinh-Gordo n theory [9 , 10]. One may expe ct that they are v alid for the sine- Gordon theory in the sector corr esp onding to the p ertur bed minimal mo del [11, 12]. Surely , this sector includes the brea thers. Notice, that our approach has muc h in commo n with that of [13], though we c oncentrate o ur attention on the br eather sector at generic v alues o f the coupling co ns tants and field pa r ameters. 1 2. Op erator con te nts of the sine/sinh- Gordon mo del Consider the sine-Gor don mo del S SG [ ϕ ] = Z d 2 x  ( ∂ µ ϕ ) 2 8 π + µ cos β ϕ  . (2.1) W e shall also use the notation β 2 = 2 p p + 1 ≤ 2 , α 0 = 1 β − β 2 = 1 p 2 p ( p + 1) . The spe ctrum of the sine-Gordo n model consists of a kink–a nt ikink (or a soliton–antisoliton) pair of some mass M , which can b e expressed in terms o f to the pa rameter µ [14], and a ser ies of breathers that is no ne mpty for β 2 ≤ 1. The masse s of the brea ther s a re given by the formula m n = 2 M sin π pn 2 , n = 1 , 2 , . . . , pn ≤ 1 . (2.2) Besides, the higher breathers may be cons idered as b ound states of the first breathers of the mass m = m 1 . It mea ns that the form factors w ith r esp ect to the states consisting any n -breathers can b e expressed in terms of the form facto rs with resp ect to the states only containing the 1-breather s . That is why we restrict our considera tion to this 1-brea ther, which will b e just called br eather her e a fter. W e can also consider the sinh-Gor don mo del S ShG [ ϕ ] = Z d 2 x  ( ∂ µ ϕ ) 2 8 π + µ ch ˆ β ϕ  . (2.3) The spectrum of the model consists of the only kind of par ticles, whic h ca n be considered as a n ‘a nalytic contin ua tio n’ of the 1-br eather in the following sense. The ex pressions for the for m factors of every lo cal o p e r ator with resp ect to thes e particle coincide with those with res p ect to the states consisting of the 1-brea ther s after the s ubs titution β → − i ˆ β . Hence, the s inh-Gordon model corresp onds to the region − 1 < p < 0 . The S ma tr ix o f tw o bre a thers is S ( θ ) = th θ +i π p 2 th θ − i π p 2 . (2.4) Consider the o pe rator conten ts of the mo dels. Let us start with the exp onential op erato rs V ( α ) ( x ) = e i αϕ ( x ) . (2.5) Below it will b e mo re conv enien t to use ano ther para meter a = α − α 0 2 α 0 . (2.6) W e shall alwa ys as sume that the parameters a and α are r elated according to (2.6). Since we wan t to use b oth letter s as subscripts, we shall alwa ys use α there in pa rentheses and a without them, e. g. V ( α ) ( x ) ≡ V a ( x ) . The expo nent ial oper ators do not exhaust the oper ator con tent s of the theory . W e hav e to define the descendant o per ators. First of all, recall that at s mall enough distances the field theories (2.1) and (2.3 ) behave lik e a free b oson theory . T ak e any p oint in the Euclidean pla ne, e. g. x = 0, a nd co nsider the radial q ua ntization picture aro und this p oint. Let z = x 1 − x 0 = x 1 + i x 2 , ¯ z = x 1 + x 0 = x 1 − i x 2 , ∂ = ∂ ∂ z , ¯ ∂ = ∂ ∂ ¯ z . The radial quantization means that we consider ra dia l co or dina tes σ , τ such that z = e τ +i σ 2 and co nsider τ as an imaginary time and σ as a space dimension. There is a one-to - one cor resp ondence betw een states |O i rad in this picture and lo ca l o p erators O ( x ) put to the p oint x = 0. This corresp ondence survives the p e rturbation for nearly all fields except some particula r (‘r esonant’) oper ators. The free field ϕ ( x ) can be e xpanded in this picture a s ϕ ( x ) = Q − i P log z ¯ z + X n 6 =0 a n i n z − n + X n 6 =0 ¯ a n i n ¯ z − n . (2.7) The op erator s Q , P , a n , ¯ a n form a Heisenberg algebra with the only nonzer o commutation rela tions [ P , Q ] = − i , [ a m , a n ] = mδ m + n, 0 , [ ¯ a m , ¯ a n ] = mδ m + n, 0 . (2.8) The states | α i rad defined as a n | α i rad = ¯ a n | α i rad = 0 ( n > 0) , P | α i rad = α | α i rad , | α i rad = e i α Q | 0 i rad (2.9) corres p o nd (up to a constant factor) to the o pe r ators V ( α ) ( x ). The descenda nts form a F o c k mo dule of the algebra (2.8) with the highest vector | α i rad . W e may choose the basis 1 a − k 1 . . . a − k s ¯ a − l 1 . . . ¯ a − l t | α i rad ↔ ( − i) s + t Q ( k i − 1)! Q ( l j − 1)! ∂ k 1 ϕ . . . ∂ k s ϕ ¯ ∂ l 1 ϕ . . . ¯ ∂ l t ϕ e i αϕ (2.10) with 0 < k 1 ≤ . . . ≤ k s , 0 < l 1 ≤ . . . ≤ l t . The pa ir of in teg ers ( n, ¯ n ), where n = P k i , ¯ n = P l i , is called the level of the elemen t. The integers n a nd ¯ n are called chiral and antic hiral level corresp onding ly . The descendants only g enerated by the elements a − k are usually c alled chiral de s cendants, while those only gene r ated by the elements ¯ a − l are c a lled antic hiral o nes. L e t F b e the F o ck submo dule o f chiral descendants. The s ubmo dule of ant ichiral descenda nt s will be referred to as ¯ F . Ev ide ntly , the s ubmo dules F and ¯ F ar e isomorphic. The F o ck module spanned on all the vectors (2.10) is the tensor pro duct F ⊗ ¯ F ≃ F ⊗ F . The mo dule F a dmits a natural g radation into the subspace s F n by the chiral lev el n : F = ∞ M n =0 F n , F n = spa n n a − k 1 . . . a − k s | α i rad    s ∈ Z ≥ 0 , s X i =1 k i = n o . The generating function of dimensions of these subspaces (the character) is given by χ ( q ) ≡ ∞ X n =0 q n dim F n = ∞ Y k =1 1 1 − q k . (2.11) 3. F orm factors from free fiel d representat ion 3.1. F orm factors of exp onential op er ators First let us describ e the for m factors of exp onential oper ators. Let | v ac i b e the v a cuum and | θ 1 , . . . , θ N i be the eig e ns tate of the Hamiltonian corr esp onding to N breather s with r apidities θ 1 < . . . < θ N . The form fa ctors of the expo nential can b e written as h θ k +1 , . . . , θ N | V a (0) | θ 1 , . . . , θ k i = G a f a ( θ 1 , . . . , θ k , θ k +1 + i π , . . . , θ N + i π ) . (3.1) Here G a is the v acuum ex pec tation v alue, which is known exactly [15]: G a ≡ h V a (0) i = m α 2 ˇ G a =  m Γ  1+ p 2  Γ  2 − p 2  4 √ π  α 2 exp Z ∞ 0 dt t sh t 2 sh 2 ( a + 1 2 ) t sh t sh pt 2 sh ( p +1) t 2 − (2 a + 1) 2 2 p ( p + 1) e − ( p +1) t ! . (3.2) 1 The states obta ined b y the action of Q m (corresponding to the op erators con taining ϕ m ) can b e obtained as th e m t h deriv atives in α and th us are obtained trivial ly . 3 Using the free field r epresentation [4 ] the ana ly tic functions f a ( θ 1 , . . . , θ N ) are expres sed in terms of tra ce functions of vertex op erator s [5]. Omitting the deta ils let us write the a nswer in the fo r m f a ( θ 1 , . . . , θ N ) = h h T ( θ N ) . . . T ( θ 1 ) i i a . (3.3) Here T ( θ ) is a genera tor of the degenerate deformed Viraso r o a lgebra [16] and h h . . . i i a is a trace function with the prop erty h h X T ( θ ) i i a = h h T ( θ + 2 π i) X i i a ( ∀ X ) . The generato r T ( θ ) can b e written in the form T ( θ ) = i λ ′  e − i π ˆ a Λ + ( θ ) + e i π ˆ a Λ − ( θ )  (3.4) with the constant factor λ ′ =  1 2 sin π p 2  1 / 2 exp  − Z π p 0 dt 2 π t sin t  . The element ˆ a is cent ral so that h h F (ˆ a ) i i a = h h F ( a ) i i ( ∀ F ) . (3.5) The pair trace functions of the vertex op erator s Λ ± are given by h h Λ ± ( θ 2 )Λ ± ( θ 1 ) i i = R ( θ 1 − θ 2 ) , h h Λ ± ( θ 2 )Λ ∓ ( θ 1 ) i i = R − 1 ( θ 1 − θ 2 ∓ i π ) = R ( θ 1 − θ 2 ) sh( θ 1 − θ 2 ) ± i sin π p sh( θ 1 − θ 2 ) , R ( θ ) = exp 4 Z ∞ 0 dt t sh π t 2 sh π pt 2 sh π ( p +1) t 2 sh 2 π t ch( π − i θ ) t ! , (3.6) while the g e neral tra ce functions in the r ight hand side of (3.5) factorize into pair trace functions of the vertex op erato r s Λ ε ( θ ): h h Λ ε N ( θ N ) . . . Λ ε 1 ( θ 1 ) i i = Y 1 ≤ i