Leading exponential finite size corrections for non-diagonal form factors
LLeading exponential finite size corrections for non-diagonalform factors
Zoltán Bajnok, Márton Lájer, Bálint Szépfalvi and István VonaApril 2, 2019
MTA Lendület Holographic QFT Group, Wigner Research Centre for PhysicsKonkoly-Thege Miklós u. 29-33, 1121 Budapest, HungaryandInstitute for Theoretical Physics, Roland Eötvös University,Pázmány sétány 1/A, 1117 Budapest, Hungary
Abstract
We derive the leading exponential finite volume corrections in two dimensional integrable mod-els for non-diagonal form factors in diagonally scattering theories. These formulas are expressed interms of the infinite volume form factors and scattering matrices. If the particles are bound statesthen the leading exponential finite-size corrections ( µ -terms) are related to virtual processes inwhich the particles disintegrate into their constituents. For non-bound state particles the leadingexponential finite-size corrections (F-terms) come from virtual particles traveling around the finiteworld. In these F-terms a specifically regulated infinite volume form factor is integrated for the mo-menta of the virtual particles. The F-term is also present for bound states and the µ -term can beobtained by taking an appropriate residue of the F-term integral. We check our results numericallyin the Lee-Yang and sinh-Gordon models based on newly developed Hamiltonian truncations. Two dimensional integrable quantum field theories are hoped to be exactly soluble. Theoretically,solvability allows us to find exact values for all physical observables including the energy spectrumand correlation functions at any finite size. However, even in integrable theories this very progressivetask has not been completed yet. Integrability has only offered us a systematic way to attack theseproblems so far.The first step of this systematic solution is to solve the theory in infinite volume by completingthe S-matrix and form factor (FF) bootstraps [1, 2, 3, 4]. In infinite volume the powerful crossingsymmetry can be used to derive restrictive functional relations for the scattering matrix and for thematrix elements of local operators, i.e. for form factors. Having solved these functional relations theresulting S-matrix and FFs can be used to describe all the finite size corrections systematically asfollows.At finite size, the leading corrections are polynomial in the inverse of the volume and originatefrom finite volume momentum quantization [5, 6]. Periodicity of the wave function requires that thescattering phase cancels the translational phase when a particle is moved around the cylinder andscattered through all other particles. The leading exponential corrections for bound states (called µ -terms) are related to the fact that in a finite volume bound states can virtually decay into theirconstituents. The next exponential corrections (F-terms) are caused by the polarization of the non-trivial finite volume vacuum [7]. Pairs of virtual particles can appear from the vacuum. These travel1 a r X i v : . [ h e p - t h ] M a r round the world and scatter on the physical particles, then annihilate each other or get absorbed bythe operators, such that this amplitude is described by the infinite volume form factor. There could beany number of virtual particles, which can also scatter on themselves. Thus, for an exact descriptionall these virtual processes have to be quantified and summed up.For the finite volume energy levels the momentum quantization is given by the so-called Bethe-Yang equations [5, 6], which provide the polynomial corrections. Leading exponential corrections forstanding one-particle states were identified in [7] and were later extended for a single moving particle in[8, 9]. The contribution of a single pair of virtual particles was extended for multiparticle states in [10],while the similar contribution with two pairs of virtual particles was analyzed for the vacuum in [11],and for multiparticle states in [12]. Finally, all virtual processes are summed up by the ThermodynamicBethe Ansatz (TBA) equation, which was derived in the simplest case in [6]. This provides the exactfinite volume ground state energy. Excited states can be obtained by careful analytic continuations[13, 14].For finite volume form factors our understanding is much more restricted. As far as polynomialcorrections are concerned one merely has to take into account momentum quantization and the cor-responding change in the normalization of states, which was proved for non-diagonal form factors in[15]. For diagonal form factors extra disconnected terms appear [16], which can be derived by carefullyevaluating the diagonal limit of a non-diagonal form factor [17]. The finite volume one-point functionscan be expressed in terms of the infinite volume connected form factors and the TBA pseudo energies[18, 19] in a way summing up the contributions of virtual processes. This result has been extended byanalytic continuation for diagonal matrix elements in diagonally scattering theories [20, 21, 22]. Theexpansion of these formulae provides the leading exponential corrections for diagonal form factors.For non-diagonal form factors, however, even these leading exponential corrections are not known ingeneral. For the simplest non-diagonal form factor (vacuum-one-particle state) the leading exponential µ -term corrections were obtained in [23], while the F -term correction in [24]. The aim of the presentpaper is to extend these analyses for generic non-diagonal matrix elements in diagonally scattering the-ories. Although the F -term calculation was based on the form factor expansion of the torus two-pointfunction [24], this method is very difficult to generalize even considering the interesting developmentsin [25, 26]. We thus focus on a formal and direct derivation of the cylinder one-point function in thecrossed channel. We test the conjectured results by comparing them to the µ -term corrections and tonumerical data obtained from the combination of the Truncated Conformal Space Approach (TCSA)and mini-superspace approaches newly developed for the sinh-Gordon theory and from TCSA in theLee-Yang model. Our results provide the leading exponential corrections for form factors, which contribute to theleading exponential correction to correlation functions, too. These results can be relevant for variousbranches of physics including finite temperature and finite volume correlation functions in statisticaland solid state systems as well as in lattice gauge theories, where the size of the system is inherentlyfinite and finite size effects are unavoidable. Our results can be useful in the AdS/CFT correspondence,too, where the calculation of correlation functions boils down to the calculation of finite volume formfactors of nonlocal operators [27] or, alternatively, it can be obtained by gluing hexagon [28] andoctagon [29] amplitudes. This gluing procedure is analogous to the calculation of finite size effects ofform factors and requires a regularization procedure [30]. Thus, our systematic method which givesrise to a regulated form factor could be implemented there as well.The paper is organized as follows: In Section 2 we review the exact results for the finite size cor-rections of the energy spectrum. We start this by describing the existing excited state TBA equationsfor the sinh-Gordon and Lee-Yang models and expanding them iteratively to second order. We extractthe µ - and F -term corrections and demonstrate how the µ - terms can be obtained from the F -terms bycalculating appropriate residues. Section 3 deals with the finite size corrections of non-diagonal formfactors. We first review the asymptotic results for polynomial corrections. Assuming the particles arebound states the asymptotic results provide the µ -term corrections, which we derive in a compact form. We note that very similar ideas appeared in an independent investigation by Konik, Mussardo et al., see also footnote5 at the beginning of Section 4. µ -termssubsequently. In Section 4 we check numerically our formulas in the sinh-Gordon and Lee-Yang modelsand conclude in Section 5. Technical details are relegated to the Appendices. In this section we recall how the TBA equations provide an exact description of the energy spectrum.We focus on theories with diagonal scatterings.In the simplest case the theory has a single particle with mass m . Multiparticle scatterings factorizeinto the product of two-particle scatterings, with S-matrix S ( θ ) , which satisfies unitarity and crossingsymmetry S ( − θ ) = S − ( θ ) ; S ( iπ − θ ) = S ( θ ) (2.1)Here θ is the rapidity difference of the particles θ = θ − θ . The simplest non-trivial scattering matrixis S ( θ ) = sinh θ − i sin pπ sinh θ + i sin pπ (2.2)For p > there is no singularity in the physical strip (cid:61) m ( θ ) ∈ [0 , π ] and the scattering matrixcorresponds to the sinh-Gordon theory. However, if p < bound states have to be introduced toexplain the appearing poles. For p = − the scattering matrix S ( θ ) = sinh θ + i sin π sinh θ − i sin π (2.3)satisfies the relation S ( θ + iu ) S ( θ − iu ) = S ( θ ) ; u = π (2.4)which, together with the bound state energy relation m cosh θ = m cosh( θ + iu ) + m cosh( θ − iu ) (2.5)implies that the bound state is the original particle itself. This theory is a consistent scattering theory[31], called the scaling Lee-Yang model. The exact finite volume energy spectrum can be obtained by calculating the continuum limit of anintegrable lattice regularization [32]. A finite volume multiparticle state can be described by the pseudoenergy (cid:15) ( θ |{ θ j } ) and parameters { ¯ θ j } j =1 ,...,N satisfying the non-linear integral equation (cid:15) ( θ |{ θ } ) = mL cosh θ + (cid:88) j log S ( θ − θ j − iπ − (cid:90) ∞−∞ dv π φ ( θ − v ) log(1 + e − (cid:15) ( v |{ θ } ) ) (2.6)where φ ( θ ) = − i∂ θ log S ( θ ) and the particles’ rapidities satisfy the quantization condition Q k ( { ¯ θ } ) = 2 πn k ; Q k ( { θ } ) = − i(cid:15) ( θ k + iπ |{ θ } ) − π ; k = 1 , . . . , N (2.7)Here and from now on we abbreviate the set of rapidities { θ j } j =1 ,...,N as { θ } . Given quantizationnumbers n k , the rapidities { ¯ θ } and the pseudo energy (cid:15) ( θ |{ ¯ θ } ) can be determined, which provide thefinite volume energy of the multiparticle state via E { n } ( L ) = m (cid:88) j cosh ¯ θ j − m (cid:90) ∞−∞ dv π cosh v log(1 + e − (cid:15) ( v |{ ¯ θ } ) ) (2.8)We note that both in(2.6) and (2.8) the terms with the sum can be absorbed into the integral term bychoosing a contour which goes around the singularities of the integrands at v = ¯ θ j + i π . These zero oflogarithm singularities are actually encoded in the quantization conditions (2.7).3 .1.1 Polynomial energy corrections The TBA equations admit a systematic large volume expansion. At leading order, indicated by asuperscript (0) , we drop the exponentially small integral terms and arrive at (cid:15) (0) ( θ |{ θ } ) = mL cosh θ + (cid:88) j log S ( θ − θ j − iπ (2.9)Asymptotic rapidities satisfy the Bethe-Yang equations Q (0) k ( { ¯ θ (0) } ) = 2 πn k ; Q (0) k ( { θ } ) = mL sinh θ k − i (cid:88) j : j (cid:54) = k log S ( θ k,j ) (2.10)where θ k,j = θ k − θ j . This equation has a very transparent meaning. Periodicity of the multiparticlewavefunction requires that, when moving particle k around the circle, the acquired phase – consistingof the translational and the scattering phases – has to be a multiple of π .The energy at leading order is simply the sum of the one-particle energies E (0) { n } ( L ) = m (cid:88) j cosh ¯ θ (0) j (2.11)incorporating all finite volume corrections, which are polynomial in the inverse of the volume. The leading exponential volume correction can be obtained by iterating the exact equations once. Atthis order, denoted by superscript (1) , we have (cid:15) (1) ( θ |{ θ } ) = mL cosh θ + (cid:88) i log S ( θ − θ i − iπ − (cid:90) ∞−∞ dv π φ ( θ − v ) e − (cid:15) (0) ( v |{ θ } ) (2.12)and the quantization conditions get modified as Q (1) k ( { ¯ θ (1) } ) = 2 πn k ; Q (1) k ( { θ } ) = Q (0) k ( { θ } ) + ∂ k Φ( { θ } ) (2.13)where ∂ i ≡ ∂ θ i ≡ ∂∂θ i and Φ( { θ } ) = (cid:90) ∞−∞ dv π (cid:89) j S ( v + i π − θ j ) e − mL cosh v (2.14)The exponentially corrected energy is E (1) { n } ( L ) = m (cid:88) i cosh ¯ θ (1) i − m (cid:90) ∞−∞ dv π cosh v (cid:89) j S ( v + i π − ¯ θ (0) j ) e − mL cosh v (2.15)which can be expressed also in terms of ¯ θ (0) i as in [10]. The integral terms in all formulae above arecalled the F -term corrections. The ground state TBA equation was derived in [6], and careful analytical continuation in the volumelead to the TBA equations of excited states [13]. The same TBA equations can be derived from acontinuum limit of a lattice model as well [33]. The TBA equations are formally the same as in the4inh-Gordon theory except that each particle with rapidity ¯ θ j is represented as a bound state of two’elementary’ particles of rapidities ¯ θ j ± = ¯ θ j ± i ¯ u j . Thus, the pseudo energy equations are (cid:15) ( θ |{ θ ± } ) = mL cosh θ + (cid:88) j,s = ± log S ( θ − θ js − iπ − (cid:90) ∞−∞ dv π φ ( θ − v ) log(1 + e − (cid:15) ( v |{ θ ± } ) ) (2.16)where { θ ± } is the shorthand for { θ j ± } and the quantization conditions are Q k ± ( { ¯ θ ± } ) = 2 πn k ± ; Q k ± ( { θ ± } ) = − i(cid:15) ( θ k ± + iπ |{ θ ± } ) − π ; k = 1 , . . . , N (2.17)It is advantageous to introduce the symmetric and antisymmetric combinations of these equations Q k ( { θ ± } ) = Q k + ( { θ ± } ) + Q k − ( { θ ± } ) ; ¯ Q k ( { θ ± } ) = Q k + ( { θ ± } ) − Q k − ( { θ ± } ) (2.18)The energy formula is also analogous to the sinh-Gordon theory: E { n ± } ( L ) = m (cid:88) js cosh ¯ θ js − m (cid:90) ∞−∞ dv π cosh v log(1 + e − (cid:15) ( v |{ ¯ θ ± } ) ) (2.19) µ -term energy corrections Let us expand the equations as before by dropping the integral terms. We indicate this order by asuperscript ( µ ) on ¯ θ ( µ ) j ± = ¯ θ ( µ ) j ± i ¯ u ( µ ) j as it contains both polynomially and exponentially small volumecorrections. Similarly to the sinh-Gordon case we assign the superscript (0) for polynomial correctionsonly. The pseudo energy at this order is: (cid:15) (0) ( θ |{ θ ± } ) = mL cosh θ + (cid:88) js log S ( θ − θ js − iπ (2.20)while the BY equations read as Q (0) k ± ( { ¯ θ ( µ ) ± } ) = 2 πn k ± with Q (0) k ± ( { θ ± } ) = mL sinh( θ k ± ) − i log S ( θ k ± ,k ∓ ) − i (cid:88) j : j (cid:54) = k,s log S ( θ k ± ,js ) (2.21)Focusing on the imaginary part of the equations we see that in the L → ∞ limit the term imL cosh ¯ θ ( µ ) j sin ¯ u ( µ ) j goes to i ∞ . This can be compensated only by the bound state pole of the scattering matrix S ( θ ) = i Γ θ − iu + S + O ( θ − iu ) , Γ = − √ (2.22)which forces ¯ u j to approach u in the large volume limit. Let us parametrize ¯ u ( µ ) j as ¯ u ( µ ) j = u + δ ¯ u j (2.23)The relation S (2 i ¯ u ( µ ) j ) = Γ δ ¯ u j + . . . together with (2.21) imply that δ ¯ u j is actually exponentially smallin the volume. We can then expand the equations for large volume in δ ¯ u j .At leading order we set δ ¯ u j to be zero, i.e. we keep only the polynomial corrections and take ¯ θ (0) j ± = ¯ θ (0) j ± iu . Using the fusion property of the scattering matrix one can see that Q (0) k ( { ¯ θ (0) ± } ) = Here both ¯ θ j and ¯ u j are real parameters. We could indicate the relevant order by using δ ¯ u ( µ ) j instead of δ ¯ u j , but since we do not go to higher orders in δ ¯ u j we drop its superscript. (0) k ( { ¯ θ (0) } ) . The resulting formulas are exactly the same as the sinh-Gordon equations (2.9-2.11) withquantization numbers n j = n j + + n j − .At the leading non-vanishing order in δ ¯ u k the equation for ¯ Q (0) k ( { ¯ θ } ) determines δ ¯ u k as δ ¯ u k = ( − n k Γ e − mL sin u cosh ¯ θ (0) k (cid:89) j : j (cid:54) = k (cid:118)(cid:117)(cid:117)(cid:116) S (¯ θ (0) k,j + iu ) S (¯ θ (0) k,j − iu ) (2.24)Clearly this expression is at least as small as e − µL with µ = m sin u and that is why we only keptthe polynomial corrections in the θ s, using ¯ θ (0) j here. Alternatively, we could determine δ ¯ u k from theexpansion of the two equations for Q (0) k ± ( { θ ± } ) . By introducing δu k ± ( { θ } ) = Γ e ± im a L sinh( θ k ± iu ) (cid:89) j : j (cid:54) = k S ( θ k,j ± iu ) ± (2.25)the solutions of the Bethe-Yang equations will be δ ¯ u k = δu k + ( { ¯ θ (0) } ) = δu k − ( { ¯ θ (0) } ) (2.26)Using these quantities the Bethe-Yang equations for ¯ θ j at first order in δ ¯ u j , i.e. at order ( µ ) , takesthe form Q ( µ ) j ( { ¯ θ ( µ ) } ) = 2 πn j ; Q ( µ ) j ( { θ } ) = Q (0) j ( { θ } ) + ∂ j (cid:88) k ( δu k + ( { θ } ) + δu k − ( { θ } )) (2.27)where we used that ∂ j δu k ± ( { θ } ) = ± iδu j ± ( { θ } ) ∂ j Q (0) k ± ( { θ ± iu } ) (and the bound state relations (2.4-2.5)).Thus, dropping the integral terms in the TBA equations not only gives the polynomial corrections,but also provides the leading exponential µ -term corrections. This can be seen in the energy formulaas well, which at leading order reads as E ( µ ) { n } ( L ) = m (cid:88) j,s cosh ¯ θ ( µ ) js = m (cid:88) j cosh ¯ θ ( µ ) j − m sin u (cid:88) j cosh ¯ θ (0) j δ ¯ u j (2.28)We note that here ¯ θ ( µ ) j also contains exponentially small corrections coming form the quantizationcondition (2.27), which involves µ -terms. To iterate the integral equations once we use the leading order term in the integrand. These formulasare completely equivalent to (2.12-2.15) except that each rapidity comes in pairs, θ j ± . These equationscontain both the O ( e − mL ) and O ( e − ( µ + m ) L ) corrections. In the following we are only interested inthe O ( e − µL ) and O ( e − mL ) corrections thus we put δu j = 0 in the integrands. At this order, denotedby superscript (1) , we have (cid:15) (1) ( θ |{ θ } ) = mL cosh θ + (cid:88) j log S ( θ − θ j − iπ (cid:88) j φ ( θ − iu − iπ − θ j ) δu j − ( { θ } ) (2.29) − (cid:88) j φ ( θ + iu − iπ − θ j ) δu j + ( { θ } ) − (cid:90) ∞−∞ dv π φ ( θ − v ) (cid:89) j S ( v + i π − θ j ) e − mL cosh( v ) Since the quantization condition ¯ Q k modifies δu k only at order O ( e − ( m + µ ) L ) we focus on Q k . Inaddition to (2.27) we also get an integral term Q (1) k ( { ¯ θ } ) = 2 πn k ± ; Q (1) k ( { θ } ) = Q (0) k ( { θ } ) + ∂ k (cid:88) j ( δu j + ( { θ } ) + δu j − ( { θ } )) + ∂ k Φ( { θ } ) (2.30)6igure 1: Integration contour, which contains both the F - and the µ -terms.The integrand is half the F -term integrand both on the upper and on the lower contour.where Φ( { θ } ) is the same as (2.14). The exponentially corrected energy also gets the integral term E (1) { n } ( L ) = m (cid:88) j cosh ¯ θ (1) j − m sin u (cid:88) j cosh ¯ θ (0) j δu j − m (cid:90) ∞−∞ dθ π cosh θ (cid:89) j S ( θ + i π − ¯ θ (0) j ) e − mL cosh θ (2.31)In all the formulas (2.29,2.30,2.31) terms containing δu j are the µ -terms, while the integral termsare the F -terms. Note that the F -terms are universal in the sense that they are the same for boththeories once the corresponding S-matrix is used. It is also very important for our further study topoint out that in the Lee-Yang theory the two corrections are not independent: the µ terms can beobtained as appropriate residues of the F -terms. Indeed, the scattering matrix not only has a pole at θ = 2 iu = i π but also at θ = iπ − iu = i π with opposite residue. This implies that e − (cid:15) (0) ( θ |{ θ } ) haspoles at θ = θ j ± i π with residuesRes θ = θ j ± i π e − (cid:15) (0) ( θ |{ θ } ) = ± iδu j ∓ ( { θ } ) (2.32)We can think of taking the real contour and deforming half of it onto the upper half-plane and theother half to the lower half-plane. Then we can subtract the two residues, which appear with oppositeorientations. As a result we can recover the µ -terms from the F -terms in all the formulas (2.29-2.31).Alternatively, we can choose the contours of integration as shown in Figure 1 and keep only the F -termintegral, which is universal and is the same for both theories. In this section we summarize the results for finite volume form factors. We start by reviewing thedefinition of these quantities together with the available results for the polynomial finite size corrections.We then derive the leading exponential µ - and F -term corrections for general nondiagonal finite volumeform factors. Technical details are presented in Appendices A and B. In Appendix C we also showhow the µ -term correction can be obtained from the F -term corrections.Finite volume form factors are the matrix elements of local operators O ( x, t ) between finite vol-ume energy eigenstates, which can be labeled either by the quantization numbers { n i } or by thecorresponding rapidities { ¯ θ i } : | ¯ θ , . . . , ¯ θ N (cid:105) L = |{ ¯ θ }(cid:105) L ≡ | n , . . . , n N (cid:105) L = |{ n }(cid:105) L (3.1)These rapidities satisfy the exact quantization conditions (2.7) or the related equations for the Lee-Yang theory (2.17). 7ur aim is to express the finite volume form factors in terms of the scattering matrix and the infinite volume elementary form factors defined by (cid:104) |O (0 , | θ , . . . , θ N (cid:105) = F O N ( θ , . . . , θ N ) (3.2)These infinite volume form factors satisfy the monodromy axioms: F N ( θ , . . . , θ N ) = F N ( θ , . . . , θ N , θ − iπ ) = S ( θ i,i +1 ) F N ( θ , . . . , θ i +1 , θ i , . . . , θ N ) (3.3)which together with their known analytic properties allows one to find the relevant physical solutions.Form factors have pole singularities, with either kinematical or dynamical origin. The kinematicalpole is related to disconnected diagrams and appear whenever an outgoing particle coincides with anincoming one. At the level of the elementary form factor this implies that F N +2 ( θ + iπ + (cid:15) , θ − (cid:15) , { θ } ) = i(cid:15) (1 − (cid:89) j S ( θ − θ j )) F N ( { θ } ) + F rN +2 ( θ + iπ, θ, { θ } ) + O ( (cid:15) ) (3.4)where we introduced a specific symmetric evaluation, since the O (1) piece defined this way, that wecall the regulated form factor, will be relevant in the further discussions. The notation { θ } abbreviatesthe ordered set { θ , . . . , θ N } .Dynamical pole singularities are only present for theories in which the scattering matrix has abound state pole. They relate the form factors of elementary particles to those of bound states. Forthe Lee-Yang model they read as F N +2 ( θ + iu + (cid:15) , θ − iu − (cid:15) , { θ } ) = i Γ (cid:15) F N +1 ( θ, { θ } ) + F bN +1 ( θ, { θ } ) + O ( (cid:15) ) (3.5)where the symmetrically evaluated O (1) piece will be used later on. In particular, we will need theexpansion F N ( θ + iu + (cid:15) , θ − iu − (cid:15) , . . . , θ N + iu + (cid:15) N , θ N − iu − (cid:15) N (3.6) (cid:89) j (cid:18) i Γ (cid:15) j (cid:19) (cid:40) F N ( { θ } ) − i Γ − (cid:88) k (cid:15) k F bN,k ( { θ } ) + O ( (cid:15) ) (cid:41) In diagonally scattering theories with a single species the form factors take the form F O N ( θ , . . . , θ N ) = H O N (cid:89) i 112 ) + π gL ; L = (cid:88) n> a − n a n (4.4)This theory has conformal invariance. When the perturbing operators : e ± bϕ : are normal ordered wrt.this theory, they are primary fields of dimension h = ¯ h = − b (8 πg ) − . The mass-gap relation [38] − πµ Γ (cid:16) b πg (cid:17) Γ (cid:16) − b πg (cid:17) = (cid:20) m √ π Γ (cid:18) − p (cid:19) Γ (cid:16) p (cid:17)(cid:21) − p = ( mκ ) − p (4.5)connects the perturbation parameter of the Lagrangian µ to the mass, m , of the sinh-Gordon scatteringparticle, while the scattering parameter p is related to b as p = b πg + b .In order to use the TCSA method, a discrete spectrum needs to be truncated at a given energy cutsuch that the full Hamiltonian can be diagonalized on the truncated space. To ensure this we separatethe zero mode into a mini-Hilbert space with Hamiltonian H mini = 14 πg π + µ (cid:18) L π (cid:19) b πg π ( e bϕ + e − bϕ ) where the volume-dependent coefficient comes from the conformal mapping of the Hamiltonian betweenthe cylinder and the plane H = 2 πL L + ¯ L − 112 + H mini + µ (cid:18) L π (cid:19) b πg πδ P (cid:8) e bϕ (: e b ˆ ϕ : − 1) + e − bϕ (: e − b ˆ ϕ : − (cid:9) (4.6)Here δ P projects to matrix elements which do not change the momentum P = πL ( L − ¯ L ) and ˆ ϕ = ϕ (0 , − ϕ . Technically, we solve numerically the mini-Hilbert space problem first. This can bedone either in the basis of plane waves in a box or using the eigenvectors of the harmonic oscillator. Forsmall volumes even the Liouville reflection factor can be used to get an approximation of the spectrum[38, 39]. We found that using 100 unperturbed vectors we got a reliable spectrum up to 5 digits in therange we were interested in. We kept 6-8 vectors from this zero mode space and calculated the matrixelements of e ± bϕ . By taking the tensor product with the Fock spaces and truncating in the energywith only the zero mode perturbation added we obtained a finite Hilbert space. We then diagonalizedthe full Hamiltonian and calculated the eigenvalues and eigenvectors. These provided the finite sizespectrum and finite volume form factors. 13 .1.2 Massive boson scheme Alternatively, one can start by perturbing the free boson of mass M . The free massive boson on thecylinder can be considered as a perturbation of the massless one, i.e. H ( M )0 = H + g M L (cid:90) : ϕ ( x, : dx (4.7)This operator can be diagonalized by solving the zero mode harmonic oscillator and applying a Bogoli-ubov transformation to the finite momentum oscillators. Technical details are relegated to AppendixD. As a result the field operator (4.2) is expressed in terms of the new massive creation operators d n as ϕ ( x, 0) = ϕ + (cid:88) n (cid:54) =0 √ Lω n g (cid:0) d n e ik n x + d † n e − ik n x (cid:1) ; ω n = (cid:112) M + k n (4.8)while the Hamiltonian (4.7) becomes H ( M )0 = (cid:88) m ∈ Z ω m d † m d m + ˜ E (cid:48) (4.9)These operators satisfy [ d † m , d n ] = δ n,m and the vacuum energy contribution ˜ E (cid:48) appears due to thedifference between the normal ordering with respect to the mode operators a n or d n .When considering the sinh-Gordon model as a perturbation of the massive boson (e.g. to make aFeynman graph expansion), one may first define the Hamiltonian in infinite volume H ( L →∞ ) = H ( M,L →∞ )0 + gM b ∞ (cid:90) −∞ : cosh ( bϕ ( x )) : M, ∞ dx − gM ∞ (cid:90) −∞ : ϕ ( x ) : M, ∞ dxH ( M,L →∞ )0 = ∞ (cid:90) −∞ (cid:112) M + k d † k d k dk (4.10)Here :: M, ∞ means normal ordering with respect to the modes d k in infinite volume. We first connectthe bare parameter M to the bare coupling µ in the conformal plus zero mode scheme. As a firststep, H ( L →∞ ) is connected to the Hamiltonian on the cylinder. This is achieved by requiring that theperturbation has the same behavior in the UV, i.e. the Hamiltonian density expressed in terms of barefields takes the same form for all volumes: H = H ( M )0 + gM b L (cid:90) : cosh ( bϕ ( x )) : M, ∞ dx − gM L (cid:90) : ϕ ( x ) : M, ∞ dx + ˜ E (4.11)By introducing a UV momentum cutoff in Appendix D we show that : e bϕ (0 , : M, ∞ = e b g ρ ( ML ) : e bϕ (0 , : M,L ; ρ ( x ) = ∞ (cid:90) −∞ du π e x cosh u − (4.12)Note that, similarly to Landau-Ginsburg theories, and as opposed to the sine-Gordon theory, thecoefficient diverges in the limit L → . Now, we bring the zero mode exponentials out of normalordering and obtain the relation µ = gM b πg b (cid:18) e γ E (cid:19) b πg (4.13)14ogether with the vacuum energy contribution ˜ E = − M (cid:90) du u tanh (cid:0) MLu (cid:1) − 12 (1 + u ) tanh (cid:16) ML √ − u (cid:17) (4.14)Following the same line of thought, the normalization of vertex operators can be related in the massiveand massless schemes as well, as: (cid:104) e aϕ (cid:105) ≡ (cid:18) L π (cid:19) a πg (cid:68) (cid:12)(cid:12)(cid:12) : e aϕ ( z =1 , ¯ z =1) : (cid:12)(cid:12)(cid:12) (cid:69) = e − a γE gπ (cid:18) M (cid:19) a gπ (cid:68) (cid:12)(cid:12)(cid:12) : e aϕ (0) : M, ∞ (cid:12)(cid:12)(cid:12) (cid:69) (4.15)where | (cid:105) denotes the interacting ground state.We applied Hamiltonian truncation in this scheme, too. The zero mode problem is again treatedseparately, similarly to [40] and [41]. Comparing the results obtained from the massive scheme to theresults in the massless scheme provided a tool to estimate the numerical error of our approach. Resultsare presented in the next subsection. For the numerical checks, we fixed g = 1 , such that b self-dual = √ π . The UV coupling was fixed to µ = 0 . . The mini Hilbert space was chosen to be diagonalized on the particle in a rigid box basis with states per parity sector. It was sufficient to keep only the states of lowest energy out of these[40, 41]. In the Fock subspace, a conformal cutoff at chiral levels up to (in the finite momentumsector, and ) was used. The dominant cutoff dependence of the results came from this chiralcutoff. This means that the actual computations involved matrices of up to about dimensions.Since the overall momentum is conserved as well as there is a parity Z symmetry present, it sufficesto search the lowest lying eigenpairs of the appropriate sub-Hamiltonians restricted to the differentsymmetry sectors. The above cutoff should be understood separately in each subsector. E m - E m - Figure 3: Theoretical low-lying energy spectrum of sinh-Gordon model at b = 1 . The vacuum energydensity is subtracted. Left: states in the sector of overall momentum (from the bottom up: | vac (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) ). Right: states of overall momentum · πL − (from the bottom up: | (cid:105) , | (cid:105) and | (cid:105) ). Note that we label the states by the corresponding bosonic Bethe Ansatz quantizationnumbers. Bethe-Yang lines are drawn with blue curves. The leading Lüscher correction is depicted byorange curves. The exact TBA result is shown with red curves.The strategy of the computations was to diagonalize the Hamiltonian (4.6) or (4.10) for a numberof volumes, and then plot the volume dependence of the results. A typical spectrum can be seen onFigure 3. 15 inite volume energies In order to compare the TCSA energies to those obtained by solving the integral equation (2.6)-(2.8), itis important to keep in mind that the latter is renormalized fixing both the infinite volume energy andenergy density to . This scheme can be connected to the numerics obtained by TCSA by subtractingthe (exactly known) vacuum energy density of the sinh-Gordon model E = m πp , p = b b + 8 πg (4.16)We note that the TCSA numerics produce reliable results at b = 1 for both the energy levels andthe finite volume form factors. For stronger couplings, e.g. b = 2 , the truncation errors become moresignificant, see fig. 4. Experience suggests the general rule of thumb, using the massive oscillator basis(keeping the same excitation content of the basis) is equivalent to increasing the chiral cutoff of themassless TCSA basis by one. Therefore, for the present work, we mostly consider the case b = 1 . E | v a c > - E | v a c > Figure 4: Volume dependence of the ground state energy: comparison of TCSA data with BY, Lüscherand TBA predictions for b = 1 (left) and b = 2 (right). On these plots, the TBA predictions aresubtracted from each data set. TCSA measurement points are shown with red and brown dots,corresponding to different chiral cutoffs (from bottom up: , , and ). The difference between theBY lines and TBA (blue) and between BY+leading Lüscher and TBA (orange) are also indicated. For b = 1 , TCSA results agree with TBA data to a remarkable precision (the difference is negligible, aswell as the cutoff dependence thereof). For b = 2 truncation errors become more significant.To make the results more transparent, we chose to depict the difference of the quantities of interestfrom some reference data. In Figs. 4-5, the results obtained by numerically solving the TBA system(2.6)-(2.8) are subtracted from each other data sets (in the case of the TCSA points, the energy density(4.16) is also taken into account). Note that we label the states by the corresponding Bethe Ansatzquantization numbers. The Bethe-Yang lines are calculated via (2.9)-(2.11), while the exponentialcorrections follow from (2.12)-(2.15). 16 .0 1.5 2.0 2.5 3.00.000.010.020.030.04 mL E | > E | > - E | > E | > E | > - E | > Figure 5: Volume dependence of the energy of some relevant low-lying states at b = 1 . Again, TBAdata are subtracted from each data set. TCSA data is shown by red dots. The difference betweenthe BY lines and TBA (blue) and between BY+leading Lüscher and TBA (orange) are also indicated.Once again, we see reassuring agreement between the TBA and TCSA data (the difference being closeto , as shown.)From these plots it is clear that in the volume range 1-3 the numerical results can be trusted andthat adding the leading exponential correction to the Bethe-Yang results considerably improved theprecision. We expect similar behaviors for finite volume form factors.17 inite volume form factors In order to check our results in subsection (3.3) we analyze the finite size form factors of the elementaryfield and the exponential operators O k =: e kbϕ : . For both cases the infinite volume form factors havethe form (3.7) with P O k n = det i,j | [ i − j + k ] σ i − j | ; [ k ] = sin kπp sin πp (4.17)where we used the basis of symmetric polynomials defined by n (cid:89) i =1 ( z + x i ) = n (cid:88) k =0 σ k z n − k (4.18)The normalization for the elementary field is given by H ϕ n +1 = (cid:113) Z ( b )2 (cid:16) πpf ( iπ ) (cid:17) n , where Z ( b ) is thewavefunction renormalization constant [42] Z ( b ) = 8 π p gb sin ( πp ) f ( iπ ) (4.19)In the plots regarding the finite volume form factors (Figures 6 and 7), the numerical TCSA data issubtracted. Then the “error” of the polynomial (3.11) approximation (more precisely, its differencefrom TCSA numerics), as calculated from (3.7) and (3.11), is shown by dashed lines. Solid curvesdepict the results of the present paper ((3.21), (3.27)). < vac | ϕ | >< | ϕ | >< | ϕ | > < vac | ϕ | >< | ϕ | > < vac | ϕ | > - - - - - < vac | ϕ | >< | ϕ | > - - - - - Figure 6: Theoretical predictions for various finite volume form factors of the field φ , as compared to nu-merical TCSA data (the latter is subtracted from each data set) at b = 1 . Polynomial (Pozsgay-Takács)results are depicted by dashed curves, while the results containing the first exponential corrections,conjectured by the present article, are shown by continuous curves.18he normalization for the exponentials is given by H kn = (cid:104)O k (cid:105) (cid:16) πpf ( iπ ) (cid:17) n , where (cid:104)O k (cid:105) is given bythe Lukyanov-Zamolodchikov formula [43]: (cid:104)O k (cid:105) = m − k b πg (cid:34) Γ (cid:0) − p (cid:1) Γ (cid:0) p (cid:1) √ π (cid:35) − k b πg (4.20) exp ∞ (cid:90) dtt − sinh (cid:16) kb t πg (cid:17) (cid:16) b t πg (cid:17) sinh t cosh (cid:104)(cid:16) b πg (cid:17) t (cid:105) + k b πg e − t On figure 8 we also present some additional checks for the operators e . bϕ and e bϕ . < vac | e b ϕ | >< | e b ϕ | >< | e b ϕ | > - - - - - < vac | e b ϕ | >< | e b ϕ | >< vac | e b ϕ | > - - - - - < vac | e b ϕ | >< vac | e b ϕ | >< | e b ϕ | > - - - - - - < vac | e b ϕ | >< | e b ϕ | > - - - - - Figure 7: Theoretical predictions for various finite volume form factors of the primary operator e bϕ ,as compared to numerical TCSA data (the latter is subtracted from each data set) at b = 1 . Poly-nomial (Pozsgay-Takács) results are depicted by dashed curves, while the results containing the firstexponential corrections, conjectured by the present article, are shown by continuous curves.As shown on all of the plots including the theoretically predicted leading exponential correctionimproved the data the same way as the similar corrections improved the energy spectrum. This isa very strong support for our conjectured F -term formulae. Let us see the analogous results for thescaling Lee-Yang theory. The scaling Lee-Yang theory is the only relevant perturbation of the conformal Lee-Yang model: S = S LY + λ (cid:90) d z Φ( z, ¯ z ) (4.21)19 vac | e ϕ | >< vac | e ϕ | >< | e ϕ | > - - - - < vac | e ϕ | >< vac | e ϕ | >< | e ϕ | > - - - - - - - Figure 8: Additional checks to some form factors of e . bϕ and e bϕ . Polynomial (Pozsgay-Takács)results are depicted by dashed curves, while the results containing the first exponential corrections,conjectured by the present article, are shown by continuous curves. Again, the TCSA results aresubtracted from each data set.The conformal Lee-Yang model is the simplest non-unitary minimal model with central charge c = − .There are two highest weight representations: one corresponds to the identity operator and the otherone to the perturbing field Φ( z, ¯ z ) with dimension ( − , − ) . The Hamiltonian can be written on theplane as H = 2 πL ( L + ¯ L − c 12 ) + λ (cid:18) L π (cid:19) / πδ P Φ(0 , (4.22)The parameter λ is related to the mass of the scattering particles as m = 2 √ π (Γ( )Γ( ) λ ) Γ( )Γ( ) (4.23) ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▼ ▼▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ● | 〉 ■ | 〉 ◆ | 〉 ▲ | 〉 ▼ | 〉 [ L ] Figure 9: Low lying finite size energy spectrum of the scaling Lee-Yang model obtained from TCSAwith the ground state energy density subtracted. Continuous lines are the results of the Bethe-Yangequations with the various quantization numbers, while discrete data originates from TCSA.20 Y μ μ F μ + F μ + F - - E | > Figure 10: Exponential finite size energy correction for the state labeled by the quantum numbers { , , − } . “ F ” is the F-term correction, while “ µ ” is the leading µ -term correction. “ µ ” contains allthe µ -term corrections summed up, while “ µ + F ” adds the F -term correction to this.The conformal Hilbert space is generated by acting with the negative Virasoro modes on thehighest weight states. On this space L and ¯ L act diagonally and the matrix elements of Φ(0 , canbe calculated exactly. Similarly to the analysis of the sinh-Gordon model, we truncate the Hilbertspace and diagonalize H on this space to obtain the energy spectrum and the form factors of theperturbing operator. The ground state energy density in the TCSA and the TBA formulation aredifferent. To relate the TCSA to the TBA results the ground state energy density (cid:15) = − m √ (4.24)has to be subtracted. The finite size spectrum obtained by the TCSA method after the subtractionlooks like in Figure 9.In order to visualize the various finite size corrections we subtract the numerical TCSA data fromthe theoretical curves as shown on Figure 10. In the domain investigated we compared the TCSA datato the exact TBA result and found that it’s precision was − . Thus there is no visual differencein subtracting TCSA compared to the exact results. For the form factors we do not know the exactresults, therefore we can only subtract the TCSA data and this is the reason why we followed thisapproach here. The Bethe-Yang correction (2.10) contains the polynomial volume corrections. The F term is the leading F -term correction (2.15), while µ is the leading µ -term correction (2.28). The µ correction sums up all the µ -terms by solving (2.21) for the constituents. Then we combine these µ -terms corrections with the leading F -term corrections.Neither the F -term nor the µ -term correction gives a good approximation for volumes − ,however their sum is very close to result of the numerics. The best approximation arises from combiningthe summed up µ -term correction with the leading F -term correction.The best results are demonstrated on figure (11). These suggest that we understand the finite sizecorrection of the energy levels very well, thus we now turn to the investigation of finite volume formfactors.The infinite volume form factors of the perturbing operators are given by (3.7) with (cid:104) Φ (cid:105) = 3 Γ( ) (2 π ) Γ( )Γ( ) ; H n = (cid:32) Γ (cid:112) f ( iπ ) (cid:33) n (4.25)21 >| > - - - - - - E m - | 〉 | 〉 | 〉 Figure 11: Demonstration of the polynomial and exponential volume corrections for the energy. Dashedlines are the difference between the Bethe-Yang energies and TCSA data and solid lines are thedifference of the best µ + F approximation and the TCSA data. The energy cut is . ( μ + F )- BYcut = = < vac | ϕ | > TCSA - BY BY μ μ F μ + F μ + F - - < vac | ϕ | > Figure 12: Left: Cut dependence of the TCSA data. The difference of the TCSA and the BY data atdifferent cuts is marked by dots with various colours and the difference between the µ + F and BYdata is represented with a solid line. Right: Volume corrections of the form factor (cid:104) vac | Φ |{ , − }(cid:105) .Every line shows the absolute value of the difference between the given theoretical curve and the TCSAdata.and with P = 1 , P = σ and for n > P n ( x , . . . , x n ) = σ σ n − det ij | σ i − j +1 | (4.26)where we used the basis of symmetric polynomials.Similarly to the energy spectrum we subtract the TCSA data from the various theoretical curves.Such a result is displayed on Figure 12. The BY curves are the result of (3.11), the leading µ -termcorrection denoted as µ is given by (3.20) and the leading F -term correction, F , by (3.27). Pluggingthe solution of (2.21) into the asymptotic formula (3.13) the form factor µ -terms can be summed up.This correction is denoted by µ . Finally, adding the leading F -term to the µ -terms leads to the bestapproximations.On figure (13) we demonstrate the BY and the best µ + F corrections for various states.In summary, the data presented gives a strong evidence for the correctness of our exponential finitesize corrections. 22 vac | ϕ | vac >< vac | ϕ | > - - < vac | ϕ | >< vac | ϕ | >< vac | ϕ | > - - < | ϕ | >< | ϕ | >< | ϕ | >< vac | ϕ | > - - - - - < | ϕ | >< | ϕ | >< | ϕ | > - Figure 13: Representative figures for the polynomial and exponential corrections. Dashed lines arethe absolute values of the difference between the BY and TCSA data, while solid lines are the samefor the best available exponential corrections. For the (cid:104) vac | Φ | vac (cid:105) diagonal matrix element we showthe F-term only. For (cid:104) vac | Φ |{ }(cid:105) , (cid:104) vac | Φ |{ n, − n }(cid:105) the best available correction is “ µ + F ”, while for (cid:104){ }| Φ |{ n, − n }(cid:105) , (cid:104) vac | Φ |{ , , − }(cid:105) , (cid:104){ m, − m }| Φ |{ n, − n }(cid:105) ; n, m = 1 , , : best available correction is“ µ + F ” 23igure 14: Graphical representation of the form factor F -term Lüscher correction. A virtual particlepair appears from the vacuum and after travelling around the world is absorbed by the operator. In this paper we presented the leading exponentially small volume corrections for non-diagonal formfactors in diagonally scattering theories. In theories with bound states the leading correction is the µ -term, which we derived using the asymptotic finite volume form factor and the assumption thatparticles are composed of their constituents. The F -term is universal in the sense that it is present intheories both with and without bound states, providing the next and leading exponential correction,respectively. We derived the F -term formally and tested the results in various ways.We showed that by taking appropriate residues of the integral for the rapidities of the virtualparticles we can completely recover the µ -term correction. We checked that taking the diagonal limitof the form factors, by sending one rapidity to infinity based on [17] reproduced the diagonal result of[22]. By developing numerical methods to “measure” the finite volume form factors we tested the finitesize corrections in the sinh-Gordon and Lee-Yang models, where we found convincing confirmation ofour formulae.Figure 14 visualizes the physical picture behind the F -term: First a virtual particle-anti-particlepair appears from the finite volume vacuum, then one of them travels around the world and finallyboth are absorbed by the operator. Since the infinite volume form factor with a particle-anti-particlepair is singular we had to regulate the appearing amplitude. Our complicated derivation and checksresulted in the proper definition of this regulated form factor. We found that we had to subtract thekinematical singularity of the form factor in a symmetric way (B.14). This regulated form factor hasvery nice properties. Its phase is the same as the original form factor’s for which we are calculatingthe correction. Its singularities on the upper and lower half planes are related to each other, in such away that the µ -term corrections are correctly reproduced when the residues are taken . In the simplestnon-trivial (vacuum-one-particle) form factor it is real and reproduces our previous results [24], whichwe derived using a finite volume analogue of the LSZ reduction formula of the two-point function.We approached the problem of calculating the partition function and evaluating the asymptoticform factor for bound states through systematic large volume expansions. Clearly this method can beused at higher orders and the resulting finite size form factors give the building blocks of the finitesize or finite temperature correlation functions. These can be used in statistical physical or solidstate systems as well as in the AdS/CFT duality. Although the expansion can be useful for practicalapplications, for obtaining exact results the series has to be summed up. In this respect the integralequation derived recently for diagonal form factors in the sinh-Gordon theory can be useful [44].24 cknowledgments We would like to thank Aron Bodor for the collaboration at an early stage of this work. Furthermore,Janos Balog, Arpad Hegedus, Fedor Smirnov and Gabor Takacs for the useful discussions. M.L.thanks to Robert Konik and Giuseppe Mussardo for the useful discussions regarding the TCSA methodemployed (see also footnote 5). This research was supported by the NKFIH research Grant K116505,the NKFIH Summer Student Internship Program and the Lendulet Program of the Hungarian Academyof Sciences. A Derivation of the form factors’ µ -term correction In this Appendix we calculate the µ -term correction for form factors in the scaling Lee-Yang model.We start with the order (0) form factor (cid:104) |O|{ θ ± }(cid:105) L = F N ( θ , θ − , . . . θ N + , θ N − ) (cid:114)(cid:81) k S ( θ k + ,k − ) ρ N ( { θ ± } ) (cid:81) i In this Appendix we relate the two alternative descriptions of the sinh-Gordon theory based on theperturbation of the massless and the massive free boson theories. First we consider the free massiveboson on the cylinder as a perturbation of the massless one as in eq. (4.6). Let us introduce a new setof creation operators α n as a n = (cid:40) − i √ nα n n > i (cid:112) | n | α †| n | n < a n = (cid:40) − i √ nα − n n > i (cid:112) | n | α †−| n | n < (D.1)We perform a Bogoliubov transformation, which acts on the massless Fock states with a unitaryoperator U = exp (cid:40) − (cid:88) m> χ m (cid:16) α m α − m − α † m α †− m (cid:17)(cid:41) ; e χ n = | k n | ω n (D.2)The creation operators transform according to U α † n U † = d † n ; α n = cosh χ n d n + sinh χ n d †− n (D.3)We would like to emphasize that obtaining the massive vacuum by acting U on the massless groundstate indicates that the massive basis is significantly different from the massless one (from the truncatedspace point of view). The field operator (4.2) is expressed in terms of the new massive creation operators d n as in eq. (4.8). Finally, introducing ϕ = (2 M Lg ) − / (cid:16) d + d † (cid:17) , the Hamiltonian (4.7) becomesthe free massive boson Hamiltonian (4.9). The vacuum energy contribution ˜ E (cid:48) appears due to thedifference between normal ordering with respect to the mode operators α n and d n .Considering the sinh-Gordon model as a perturbation of the massive boson the normal ordering ischosen at infinite volume (4.10), i.e. :: M, ∞ means normal ordering with respect to the modes d k ininfinite volume. Our goal is to connect the bare parameter M to the bare coupling µ in the conformalplus zero mode scheme. As a first step, H ( L →∞ ) needs to be connected to the Hamiltonian on thecylinder. This is achieved by requiring that the perturbation has the same behavior in the UV forboth, i.e. the Hamiltonian density expressed in terms of bare fields takes the same form for all volumes(4.11). Let us assume that we have temporarily introduced an UV momentum cutoff Λ . We use theBCH formula e X + Y = e − [ X,Y ] e X e Y , if [ X, [ X, Y ]] = [ Y, [ X, Y ]] = 0 (D.4)to relate e bϕ (0 , = e b [ ϕ + ,ϕ − ] (Λ) M,L : e bϕ (0 , : M,L = e b [ ϕ + ,ϕ − ] (Λ) M, ∞ : e bϕ (0 , : M, ∞ (D.5)In the relation between the two normal ordered quantities the limit Λ → ∞ can be taken leadingto (4.12). Note that the coefficient diverges in the limit L → . Then, we bring the zero modeexponentials out of normal ordering : e bϕ : M,L = e − b gLM e bϕ and (keeping in mind an UV cutoff again) we can obtain ( ˜ ϕ = ϕ − ϕ ) : e b ˜ ϕ ( x ) : M,L = e b gL (cid:80) q (cid:54) =0 (cid:18) | kq | − ωq (cid:19) : e b ˜ ϕ ( x ) : ,L (D.6)31here the sum in the exponent has an integral representation L (cid:88) q (cid:54) =0 (cid:18) | k q | − ω q (cid:19) = 1 M L + 1 π ln M L π − ρ ( M L ) + γ E π (D.7)Comparing (4.11) with (4.6) we arrive at the relations (4.13) and (4.14). References [1] A. B. Zamolodchikov, A. B. 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