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ITP-Budapest Report No. 637 Form fa tors of boundary exponential operators in the sinh-Gordon model 8 0 0 ∗ G. Taká s 2 n Theoreti al Physi s Resear h Group of the Hungarian A ademy of S ien es a H-1117 Budapest, Pázmány Péter sétány 1/A J 7 7th January, 2008 ] h t - p e h Abstra t [ Usingthere ently introdu ed boundaryformfa tor bootstrapequations, theform 1 v fa tors of boundary exponential operators in the sinh-Gordon model are onstru ted. 2 Theultraviolets alingdimensionandthenormalizationoftheseoperatorsare he ked 6 against previously known results. The onstru tion presented in this paper an be 9 0 applied to determine form fa tors of relevant primary boundary operators in general . integrable boundary quantum (cid:28)eld theories. 1 0 8 0 1 Introdu tion : v i X The investigation of integrable boundary quantum (cid:28)eld theories started with the seminal r work of Ghoshal and Zamolod hikov [1℄, who set up the boundary R-matrix bootstrap, a whi h makes possible the determination of the re(cid:29)e tion matri es and provides omplete des ription of the theory on the mass shell. For the al ulation of orrelation fun tions, matrix elements of lo al operators between asymptoti states have to be omputed. In a boundary quantum (cid:28)eld theory there are two types of operators, the bulk and the boundary operators, where their names indi ate their lo alization point. The boundary bootstrap program, namely the boundary form fa tor program for al ulating the matrix elements of lo al boundary operators between asymptoti states was initiated in [2℄. The validity of form fa tor solutions was he ked in the ase of the boundary s aling Lee-Yang model al ulating the two-point fun tion using ∗ E-mail: taka selte.hu 1 a spe tral sum and omparing it to the predi tion of onformal perturbation theory. In [3℄ the spe trum of independent form fa tor solutions was ompared to the boundary operator ontent of the ultraviolet boundary onformal (cid:28)eld theory and a omplete agreement was found. Further solutions of the boundary form fa tor axioms were onstru ted and their stru ture was analyzed for the sinh-Gordon theory at the self-dual point in [4℄, and for A 2 the a(cid:30)ne Toda (cid:28)eld theory in [5℄. In the re ent paper [6℄ the validity of the form fa tor solution onje tured for the unique nontrivialboundary primary (cid:28)eld in s aling Lee- Yang model was tested against trun ated onformal spa e and a spe ta ular agreement was found. It is lear from the dis ussion in [3℄ that the most interesting open problem of the boundary form fa tor bootstrap is the identi(cid:28) ation of the operator orresponding to a given solution. For example, in sinh-Gordon theory there are in(cid:28)nitely many form fa tor solutions with minimal growth at large rapidities, whi h an be attributed to the presen e of exponential boundary (cid:28)elds. The task undertaken in this paper is to map the spa e of su h minimal solutions, and make their orresponden e with exponential (cid:28)elds more pre ise. It is shown that spe i(cid:28) solutions an be sele ted inside this in(cid:28)nite family, su h that their s aling dimension agrees with the predi tion of onformal (cid:28)eld theory, while their normalization mat hes the available results on the va uum expe tation value of exponential operators. The onstru tion of these solutions an be generalized to determine form fa tors of relevant boundary primary (cid:28)elds in any model where the form fa tors of relevant operators in the bulk theory are known. The outline of the paper is the following. In se tion 2 the ne essary information about boundary sinh-Gordon theory is presented. Se tion 3 des ribes the onstru tion of the form fa tors solutions whi h are onje tured to orrespond to boundary exponential (cid:28)elds. In se tion 4 theirultravioletdimensionand normalizationisevaluatedasa series expansion in the bulk parameter of sinh-Gordon theory, and is shown to be onsistent with known results. Se tion 5 is reserved for the on lusions. 2 Boundary sinh-Gordon theory The sinh-Gordon theory in the bulk is de(cid:28)ned by the Lagrangian density 1 m2 = (∂ Φ)2 (coshbΦ 1) L 2 µ − b2 − It an be onsidered as the analyti ontinuation of the sine-Gordon model for imaginary β = ib oupling . The S-matrix of the model is B B B 2b2 S(θ) = 1+ = ; B = − 2 −2 −2 8π+b2 (cid:18) (cid:19)θ(cid:18) (cid:19)θ (cid:20) (cid:21)θ where sinh 1 (θ+iπx) sinhθ +isinπx (x) = 2 , [x] = (x) (1 x) = θ sinh 1 (θ iπx) θ − θ − θ sinhθ isinπx 2 − − 2 The minimal bulk two-parti le form fa tor belonging to this S-matrix is [7℄ dx x(iπ θ) sinh xB sinh(1 B)x sinh x f(θ) = exp 8 ∞ sin2 − 4 − 2 2 2 N " Z0 x (cid:18) 2π (cid:19) sinh2x # (2.1) where dxsinh xB sinh(1 B)x sinh x = exp 4 ∞ 4 − 2 2 2 N "− Z0 x sinh2x # (2.2) f(θ,B) 1 θ It satis(cid:28)es → as → ∞, and approa hes its asymptoti value exponentially fast. Sinh-Gordon theory an be restri ted to the negative half-linewith the followinga tion 0 1 m2 = ∞ dt dx (∂ Φ)2 (coshbΦ 1) A 2 µ − b2 − (2.3) Z Z (cid:20) (cid:21) −∞ −∞ b ∞ + dtM cosh (Φ(0,t) Φ ) 1 0 0 2 − − Z (cid:18) (cid:18) (cid:19) (cid:19) −∞ whi h maintains integrability [1℄. The orresponding re(cid:29)e tion fa tor depends on two ontinuous parameters and an be written as [8℄ 1 1 B B E 1 F 1 R(θ) = + 1 − − 2 2 4 − 4 2 2 (2.4) (cid:18) (cid:19)θ(cid:18) (cid:19)θ(cid:18) (cid:19)θ(cid:20) (cid:21)θ(cid:20) (cid:21)θ It an be obtained as the analyti ontinuation of the (cid:28)rst breather re(cid:29)e tion fa tor in boundary sine-Gordon model whi h was al ulated by Ghoshal in [9℄. The relation of the E F bootstrap parameters and to the parameters of the Lagrangian is known both from a semi- lassi al al ulation [8, 10℄ and also in an exa t form in the perturbed boundary onformal (cid:28)eld theory framework [11℄. 3 Boundary form fa tors in sinh-Gordon theory 3.1 The boundary form fa tor axioms The axioms satis(cid:28)ed by the form fa tors of a lo al boundary operator were derived in [2℄ and are listed here without mu h further explanation. Let us assume that the spe trum m S S(θ) ontains a single s alar parti le of mass , whi h has a two-parti le matrix (using R(θ) the standard rapidity parametrization) and a one-parti le re(cid:29)e tion fa tor o(cid:27) the boundary, satisfying the boundary re(cid:29)e tion fa tor bootstrap onditions of Ghoshal and (t) x = 0 Zamolod hikov [1℄. For a lo al operator O lo alized at the boundary (lo ated at , t and parametrized by the time oordinate ) the form fa tors are de(cid:28)ned as ′ ′ ′ θ ,θ ,...,θ (t) θ ,θ ,...,θ = outh 1 2 m|O | 1 2 niin FmOn(θ1′,θ2′,...,θm′ ;θ1,θ2,...,θn)e−imt(Pcoshθi−Pcoshθj′) 3 ′ ′ ′ θ > θ > ... > θ > 0 θ < θ < ... < θ < 0 in/out 1 2 n 1 2 m for and , using the asymptoti state formalism introdu ed in [12℄. They an be extended analyti ally to other values of rapidities. With the help of the rossing relations derived in [2℄ all form fa tors an be expressed in terms of the elementary form fa tors 0 (0) θ ,θ ,...,θ = F (θ ,θ ,...,θ ) outh |O | 1 2 niin nO 1 2 n whi h an be shown to satisfy the following axioms: I. Permutation: F (θ ,...,θ ,θ ,...,θ ) = S(θ θ )F (θ ,...,θ ,θ ,...,θ ) nO 1 i i+1 n i − i+1 nO 1 i+1 i n II. Re(cid:29)e tion: F (θ ,...,θ ,θ ) = R(θ )F (θ ,...,θ , θ ) nO 1 n−1 n n nO 1 n−1 − n III. Crossing re(cid:29)e tion: F (θ ,θ ,...,θ ) = R(iπ θ )F (2iπ θ ,θ ,...,θ ) nO 1 2 n − 1 nO − 1 2 n IV. Kinemati al singularity n ′ i F (θ+iπ,θ ,θ ,...,θ ) = 1 S(θ θ )S(θ+θ ) F (θ ,...,θ ) − Rθ=eθs′ nO+2 1 n − − i i ! nO 1 n i=1 Y V. Boundary kinemati al singularity n iπ g iπ i F (θ + ,θ ,...,θ ) = 1 S( θ ) F (θ ,...,θ ) − Rθ=e0s nO+1 2 1 n 2 − 2 − i nO 1 n (cid:16) Yi=1 (cid:17) g where is the one-parti le oupling to the boundary ig2 π R(θ) , θ i ∼ 2θ iπ ∼ 2 (3.1) − There are also axioms for singularities orresponding to bound states (bulk and boundary), but they are not needed in the sequel. The general solution of the above axioms an be written as [2℄ n Q (y ,y ...,y ) n 1 2 n F (θ ,θ ,...,θ ) = H r(θ ) f(θ θ )f(θ +θ ) n 1 2 n n i i j i j y (y +y ) − (3.2) i i i j i=1 i<j i<j Y Y Q Q Q n where the are symmetri polynomials of its variables, and the minimal one-parti le form fa tor is given by isinhθ π π r(θ) = u(θ,B) , γ = (E 1) γ = (F 1) ′ (sinhθ isinγ)(sinhθ isinγ ) 2 − 2 − (3.3) ′ − − 4 r(θ) 1 θ 1 with asymptoti s ∼ when → ∞, where dt 1 t iπ t ∞ u(θ) = exp 2cosh cos θ t sinh t − 2 2 − π × (cid:26)Z0 (cid:20) 2 (cid:20)(cid:18) (cid:19) (cid:21)(cid:21) sinh xB +sinh 1 B x +sinh x 4 − 2 2 2 sinh2t (cid:0) (cid:1) (cid:27) and n/2 4sinπB/2 H = n f(iπ) (3.4) (cid:18) (cid:19) is a onvenient normalization fa tor. Using the results of [2, 3℄ it is easy to derive the Q n re ursion relations satis(cid:28)ed by the polynomials : : Q ( y,y) = 0 2 K − Q ( y,y,y ,...,y ) = n+2 1 n − (y2 4cos2γ)(y2 4cos2γ )P (y y ,...,y )Q (y ,...,y ) n > 0 ′ n 1 n n 1 n − − | for : Q (0) = 0 1 B Q (0,y ,...,y ) = n+1 1 n 4cosγcosγ B (y ,...,y )Q (y ,...,y ) n > 0 ′ n 1 n n 1 n for (3.5) where n n 1 πB πB B (y ,...,y ) = y 2sin y +2sin n 1 n 4sin πB i − 2 − i 2 ! (3.6) 2 i=1(cid:18) (cid:19) i=1(cid:18) (cid:19) Y Y and n n 1 P (y y ,...y ) = (y y )(y +y ) (y +y )(y y ) n 1 n i i + i i + | 2(y+ y ) " − − − − − # (3.7) − − Yi=1 Yi=1 with the notations y = ωz +ω 1z 1 + − − (3.8) y = ω−1z +ωz−1 , ω = eiπB2 − z y = z +z 1 y = 2coshθ − with the auxiliary variable de(cid:28)ned as a solution of (i.e. writing z = eθ one has ). The two-point fun tions an be omputed from a spe tral representation: ρ (mt) = 0 (t) (0) 0 AB h |A B | i (3.9) n ∞ 1 = dθ ...dθ f (θ ,...,θ )exp imt coshθ (2π)n 1 n n 1 n − i n=0 Zθ1>...>θn>0 i=1 ! X X 1 Noti e that the normalization used here di(cid:27)ers from that in the earlier papers [2, 3℄. 5 where time translation invarian e was used and f (θ ,...,θ ) = F (θ ,...,θ ) F (θ ,...,θ ) n 1 n nA 1 n † nB 1 n f n The are symmetri (and also even) inalltheirvariables, therefore the spe tral expansion (3.9) an be written in the following form for the Eu lidean two-point fun tion ρAB(mτ) = ∞ n1!(2π1)n ∞dθ1 ∞dθ2... ∞dθne−mτPicoshθifn(θ1,...,θn) n=0 Z0 Z0 Z0 X The operators of interest an be lassi(cid:28)ed a ording to their s aling dimensions, whi h means that the two-point fun tion must have a power-like short-distan e singularity 1 ρ (mτ) = +... AB τ2∆AB (3.10) ∆ where AB is an exponent determined by the ultraviolet s aling weights of the lo al (cid:28)elds. 3.2 The family of minimal solutions and the umulant expansion In the earlier paper [3℄ it was shown that there exist an in(cid:28)nite family of solutions for Q n whi h the polynomials have the minimum possible degree n(n+1) degQ = n 2 These an be thought to orrespond to the exponential operators eαΦ(t,x=0) of whi h only a ountably in(cid:28)nite number is independent sin e they an all be expanded in terms of powers of the (cid:28)eld: ∞ αk eαΦ(t,0) = Φk(t,0) k! k=1 X Foranyminimalsolutionthe orresponding formfa torhasa (cid:28)nitelimitwhen allrapidities are taken to in(cid:28)nity simultaneously F (θ +λ,θ +λ,...,θ +λ) F˜ (θ ,θ ,...,θ )+O(e λ) n 1 2 n n 1 2 n − → (3.11) whi h means that every multi-parti le ontribution in the spe tral expansion (3.9) individ- ually behaves as τ 2δ − δ 0 with the naive s aling dimension equal to [2, 3℄. 6 0 However, the true s aling dimension in general turns out to be di(cid:27)erent from due to logarithmi orre tions in the individual multi-parti le ontributions, whi h may sum up to give an anomalous dimension. It an be omputed using the umulant expansion of the logarithm of the two-point fun tion [13℄ (for a very ni e dis ussion see also [14℄). Consider the onformal operator produ t expansion Ci (τ) (0) (0) A B ∼ τhA+AhBB hiOi − Xhi h h h i where A and B are the ultraviolet weights of the (cid:28)elds A and B, while the are the i weight of the O . It is obvious that 2∆ = h +h h min AB A B − h h min i i where is the minimum of the weights of the operators O appearing in the expansion. Let us suppose that the limiting fun tion (3.11) satis(cid:28)es an asymptoti fa torization property of the form F˜ (θ ,...,θ ,θ +λ,...,θ +λ) = F˜ (θ ,...,θ )F˜ (θ ,...,θ )+O(e λ) n 1 k k+1 n k 1 k n k k+1 n − − (3.12) n = 0 both for the form fa tors of A and B. The leading term in the spe tral expansions is a onstant given by the va uum expe tation value of the (cid:28)eld, whi h is assumed to be 1 . In the ase of bulk form fa tors (3.12) with the parti ular normalization given above entails that the operator has a unit va uum expe tation value [15℄. For boundary form fa tors this an be arried over as a reasonable assumption whi h was used previously to normalizetheminimalformfa toroftheboundaryLee-Yangmodelin[2℄. Thisassumption is reasonable from the onsisten y of the arguments in this paper and is also he ked expli itly in subse tion 4.2. Under the above assumptions, the logarithm of the orrelation fun tion an be written as logρAB(mτ) = ∞ n1!(2π1)n ∞dθ1 ∞dθ2... ∞dθne−mτPicoshθicn(θ1,...,θn) (3.13) n=1 Z0 Z0 Z0 X c f n n where the are the umulants of the fun tions de(cid:28)ned re ursively by f (θ ) = c (θ ) , f (θ ,θ ) = c (θ ,θ )+c (θ )c (θ ) 1 1 1 1 2 1 2 2 1 2 1 1 1 2 f (θ ,θ ,θ ) = c (θ ,θ ,θ )+c (θ )c (θ ,θ )+c (θ )c (θ ,θ )+c (θ )c (θ ,θ ) 3 1 2 3 3 1 2 3 1 1 2 2 3 1 2 2 1 3 1 3 2 1 2 +c (θ )c (θ )c (θ ) 1 1 1 2 1 3 ... De(cid:28)ning c˜ (θ ,...,θ ) = lim c (θ +λ,...,θ +λ) n 1 n n 1 n λ (3.14) →∞ 7 c˜ n it is easy to see that the fun tions depend only on the di(cid:27)eren es of the rapidities. From (3.12) it is easy to obtain the following property of the asymptoti umulants c˜ (θ ,...,θ ,θ +λ,...,θ +λ) O(e λ) k = 1,...,n 1 n 1 k k+1 n − ∼ − (3.15) and also note that c (θ +λ,...,θ +λ) = c˜ (θ ,...,θ )+O(e λ) n 1 n n 1 n − (3.16) It an then be shown that ∞ 1 1 ∞ ∞ 2∆ = dθ ... dθ c˜ (0,θ ,...,θ ) AB n!(2π)n 2 n n 2 n (3.17) Xn=1 Z−∞ Z−∞ The derivation of this result is a bit more involved than in the bulk ase where the transla- n = 1 tionalinvarian eof the form fa torinrapidityspa e an be used. Let us examine the ontribution in more detail. The formulae en ountered will also be useful in subse tion 4.2. Consider the integral dθ ∞ c (θ)e mτcoshθ 1 − 2π Z0 c (θ) 1 Due to (3.16), approa hes its limiting value exponentially fast c (θ) = c˜ +O(e αθ) 1 1 − The exponential fa tor has the property 2 e mτcoshθ 1 θ log − ∼ ≪ mτ τ and therefore in the limit of small dθ dθ dθ ∞ c (θ)e mτcoshθ ∞ (c (θ) c˜ )+ ∞ c˜ e mτcoshθ 1 − 1 1 1 − 2π ∼ 2π − 2π (3.18) Z0 Z0 Z0 whi h an be evaluated as c˜ 1 + K (mτ) 0 onstant 2π Using the asymptoti s K (mτ) logmτ +log2 γ 0 E ∼ − − (3.19) γ = 0.577215... E (where is Euler's onstant) the short-distan e exponent is given in this approximation as c˜ 1 2∆ = +... AB 2π (3.20) n > 1 Turning to , the expansion (3.13) an be rewritten as logρAB(mτ) = ∞ n1!(2π1)n21n ∞ dθ1 ∞ dθ2... ∞ dθne−mτPicoshθicn(θ1,...,θn) Xn=1 Z−∞ Z−∞ Z−∞ 8 f c n n using that the fun tions and onsequently also are even in all their arguments. For mτ 1 ≪ , the support of the integrand is on entrated inside the hyper ube θ . log(2/mτ) i | | O(1) and in the interior (ex ept for a transient shell-like region whose thi kness is inde- τ pendent of ) e−mτPicoshθi 1 ∼ c c˜ O(1) n n Using the properties (3.15) and (3.16) an be repla ed by up to terms. One an θ n = 1 1 now repeat the same pro edure in the integral as for , following the line of the derivation used in the bulk ase [14℄. However, some additional are must be taken sin e c 2n 1 n − the fun tions are even, and therefore there are independent asymptoti dire tions in whi h the integrand is non-vanishing: c (θ +λ,θ +σ λ,...,θ +σ λ) c˜ (θ ,σ θ ,...,σ θ ) n 1 2 2 n n n 1 2 2 n n → σ = 1 θ i 1 depending on the hoi es of the signs ± (the sign of is (cid:28)xed as only the relative θ σ θ 2n 1 i i i − signs matter). After a rede(cid:28)nition of integration variables → this results in n 1 1/2 1/2 identi al ontribution whi h an el out − fa tors of . The remaining fa tor an θ 1 be used to map ba k the integration to the half-line where (3.19) an be used. 1/2n In the bulk the fa tor are absent, and translationalinvarian e of the form fa tor of c c˜ n n a spinless operator implies that ≡ . Due to (3.15) only the asymptoti region with all 2n 1 − signs positive ontributes, so there is no fa tor either. Therefore the bulk singularity exponent is twi e the one in the boundary ase (provided that the asymptoti umulants are taken to be identi al). However, if the singularity of the bulk two-point fun tion is taken to be of the form 1 τ4∆AB ∆ thenthisfa tor an els outinthe(cid:28)nalformulafor AB andsothebulk umulantexpansion 2 is identi al to (3.17). The additional fa tor of in the exponent is natural sin e in the bulk the onformal weight re eives two identi al ontributions from the left and the right movers. 3.3 The limiting ase: bulk form fa tor solutions in the sinh- Gordon model Now the task is to resolve the ambiguity in the minimal solution. From [3℄ it is known to originate from the kernel of the re ursion relations, whi h is a homogeneous polynomial of n(n+1)/2 n order at -parti le level. Let us introdu e Q˜ (y ,...,y ) = lim Λ n(n+1)/2Q (Λy ,...,Λy ) n 1 n − n 1 n Λ →∞ Q n(n+1)/2 n whi h gives the terms in whi h are exa tly of degree (all other terms have lower degree). 9 Taking now a solution of the form (3.2), the limiting pro edure (3.11) gives F˜ (θ ,θ ,...,θ ) = lim F (θ +λ,θ +λ,...,θ +λ) n 1 2 n n 1 2 n λ →∞ Q˜ (x ,x ...,x ) n 1 2 n = H f(θ θ ) n i j x (x +x ) − i i i j i<j i<j Y Q Q x = eθi i where . It was observed in˜[3℄ that as a onsequen e of the boundary form fa tor F axioms listed in subse tion 3.1, is always a solution of the bulk form fa tor axioms S built upon the same bulk matrix. For the bulk sinh-Gordon theory, a omplete set of minimalsolutionswas foundby Koubek andMussardo in[16℄. Letus de(cid:28)ne theelementary symmetri polynomials by n n (x+x ) = xn lσ(n)(x ,...,x ) i − l 1 n i=1 l=1 Y X (n) σ 0 l < 0 l > n l ≡ if or The upper index will be omitted in the sequel, as the number of variables will always be lear from the ontext. Let us also denote ωn ω n sin nπB [n] = − − = 2 ω ω 1 sin πB − − 2 Koubek and Mussardo state that the followingfamilyof formfa torsolutionsparametrized k by the real number (k) P (x ,x ...,x ) F˜(k)(θ ,θ ,...,θ ) = H n 1 2 n f(θ θ ) n 1 2 n n (x +x ) i − j i j i<j i<j Y Q (k) P n where the polynomials are given by (k) P = [k] 1 P(k) = [k]detM(n)(k) n > 1 n (n) M (k) = [i j +k]σ (x ,x ...,x ) i,j = 1,...,n 1 ij − 2i−j 1 2 n − orrespond to the bulk exponential operators ekgΦ normalized so that their va uum expe tation value is unity: ekgΦ = 1 (cid:10) (cid:11) 10