A study of form factors in relativistic mixed-flux AdS 3

We study the two-particle form-factors for the relativistic limit of the integrable S -matrix of the mixed-flux AdS 3 × S 3 × T 4 string theory. The S -matrix theory was formally constructed in two distinct ways by two different teams. We focus on the massive theory built up by Frolov, Polvara and Sfondrini, and derive expressions for the minimal solutions to the axioms, in both integral and manifestly meromorphic form, and then proceed to apply the off-shell Bethe ansatz method of Babujian et al . We obtain the integral formulas for the two-particle complete form-factors and check the axioms at this particle number.


Introduction
The integrable system underlying string theory in AdS 3 × S 3 × M 4 [1], the manifold M 4 either being T 4 or S 3 × S 1 [2] -see the reviews [3] -is similar but arguably richer than the AdS 5 counterpart [4].An incomplete list of references is given by [5,6].In AdS 3 massless modes are present [7] and the machinery of massless integrability is called into play [8,9].The possibility of having a mixture of RR and NS-NS flux is also there -in [1] the case of equal radius for the S 3 × S 3 × S 1 is discussed and in the generic radius case is found in [5], with the mixed flux case featured in [10,11,12].The paper [12] in particular describes the full moduli space, and explains the h parameter which connects with the complete moduli dependence.Recently, the Quantum Spectral Curve (QSC) has been written down for purely RR background AdS 3 × S 3 × T 4 in [13], see also [14].A new series of articles [15], see also [16], has revisited the integrability programme in AdS 3 × S 3 × T 4 , has proposed new dressing phases and has formulated the Thermodynamic Bethe Ansatz (TBA).Further recent work can be found in [17].
In [18,19] a change of variables was exhibited which shows the complete difference form of the Smatrix of massless AdS 3 particles.This form coincides with the BMN limit for particles with equal worldsheet chirality [9].These new ('pseudo-relativistic') massless variables prefigured in [20] then led to the massive counterpart being introduced in [15], by virtue of which a part of the massive S-matrix (not the whole of it) also acquires a difference form.
The single-particle mixed-flux dispersion relation is [21] given by k ∈ N being the WZW level, m the mass and h the coupling constant.In [19] the massive mixed-flux S-matrix was subjected to a particular construction leading to a particular massless relativistic S-matrix, for a critical theory whose spectrum is encoded in a massless TBA a la Zamolodchikov.The paper [22] then obtained the same S-matrix from a different limit performed in a way to retain massive excitations, and extended the study to all the different possible masses and bound states.The paper [23] provides an in-depth study of the kinematical and analytic structure underlying the theory and its S-matrices, and a proposal for the complete mixed-flux dressing factor.More very recent work can be found in [24], as this area is shown to be extremely vibrant.
In this paper we wish to initiate the study of relativistic form factors for this massive theory.The form factor programme is very much at the core of AdS 5 /CF T 4 in the shape of the hexagon approach [25].
This approach has been generalised to AdS 3 [26].The traditional relativistic form factor programme was initially pursued in [27], where it was extended to non-difference form.In [28] the standard (pre-hexagon) approach was applied to massless AdS 3 in the BMN-limit, which is a genuinely relativistic model [9] see also [19,15,29] -by following the off-shell Bethe Ansatz technique of [30].We now intend to do the same for the simplest form factors of this deformed system at hand.

Form factors
The study of form factors is essential to bring to completion the bootstrap programme and reach a complete solution of an integrable quantum field theory [31,32,33].After having established the exact S-matrix and the spectrum of (bound) states, one proceeds with the form factor analysis [34].Using the information contained in the form factors of the theory, one can then compute the n-point correlation functions.This effectively defines the theory, even in those situation where one does not have access to a Lagrangian.We refer the reader to [31] for a very thorough review of the form factor programme.
In a relativistic theory the n-particle form factor associated with a operator O sitting at the spatial origin is defined as the quantity (2.1) θ i being the rapidity and α i collectively labelling the quantum numbers of the i-th magnon.
The form factors are constrained by a list of axioms -we refer to [31] and [30] for a complete description.
We shall only confine ourselves here to those axioms which will be important in our calculation.We have where we adopt the conventions of [30].
• Periodicity where σ is an appropriate statistical factor.
• Lorentz boost (2.4) where s denotes the spin of O.
We will comment later on on the kinematical singularity and bound state singularity axioms.
The method which we will follow in this paper to obtain the two-particle form factor is the off-shell Bethe ansatz [30].We will describe in detail how this works.Here we provide a short summary of the building blocks which will enter the off-shell Bethe ansatz, and how we will assemble them, to guide with the organisation of the material.This structure follows the decomposition of the S-matrix as [22] S(m 1 , m 2 ; θ) = Φ(m 1 , m 2 ; θ) σ −2 (m 1 , m 2 ; θ) S BB (m 1 , m 2 ; θ) ⊗S BB (m 1 , m 2 ; θ), (2.5) where Φ and σ −2 are scalar factors, and S BB is the S-matrix whose associated R-matrix will be R(θ) (4.1) -with ⊗ being the tensor product of two copies of the fundamental representation to be discussed around (4.23), the hat just helping to keep it separated from ⊗ which denotes the tensor product of the scattering particles.The integers m 1 and m 2 associated with the two particles are quantum numbers which enter all the scattering data, the particle mass being given by Accordingly, the two-particle form factor is build out of: • a block associated with Φ, which will be called F total pre (m 1 , m 2 ; θ) as in "pre-factor"; • a block associated with σ −2 , which will be called F dressing (m 1 , m 2 ; θ) as in "dressing factor" to adopt the nomenclature of [22]; • a block associated with S BB ⊗S BB , made out of two copies of the sub-block to be called -this sub-block will have two indices (each index carrying the fundamental representation) and it will depend specifically on the two scattering rapidities θ 1 and θ 2 to incorporate operators with spin.
To construct these blocks we shall need first to build the minimal solutions and study their singularities.We then present the core of the off-shell Bethe ansatz and proceed to reconstitute the complete list of blocks.The blocks will eventually be combined as to form the complete two-particle form factor.

Minimal two-particle form factor
We can calculate the minimal two-particle form factor by using a classic result by Karowski and Weisz [34].We start from the explicit meromorphic expression for the dressing factor [22] 1 where ψ being the Digamma function ψ(z) = d dz log Γ(z).In the ratio (3.1)only the Gamma functions, of all the parts of (3.2), do not cancel out, and one gets as a net result where The infinite product (3.3)converges to a meromorphic function in the whole complex plane of θ, thanks to the subtle balance of the arguments of the Gamma functions in the numerator and denominator [35].
The dressing factor satisfies We now adopt the standard technique of resorting to the Malmstén representation of the Gamma function: .6)Thanks to the fact that the sum of the arguments of all the Gamma functions at the numerator of (3.3) equals the sum of the arguments at the denominator, we see that most of (3.6) cancels out in the ratio, and only the term containing e −x(z−1) contributes.At this point, the product over ℓ becomes a simple geometric sum.Collecting all such terms, and rescaling the integration variable as x → 2x, we obtain We can clearly see the interesting property of the dressing factor to go into its inverse as we exchange and therefore we can immediately find the minimal two-particle form factor: where By carefully reconstructing backwards a process similar to the one which we followed when going from (3.3) to (3.7), it is possible to find the analytic continuation of (3.9) to the entire complex plane of θ.We summarise the main steps here, while we refer to [28] -section 3.2 in that paper -for a more detailed procedure applied to the case of a completely analogous calculation.It is sufficient to i) rescale the variable x → x 2 ; ii) use two of the three factors (e x − 1) at the denominator (one coming from the sinh x) to reconstruct two geometric sums, the third factor eventually reconstructing the denominator of (3.6); iii) expand the numerator into exponentials and check that the sum of arguments balance between exponentials with the plus (which will end up in the Gamma functions at the numerator) and exponentials with the minus (which will end up in the Gamma functions at the denominator).In this way we have reconstructed the Malmstén representation proceeding backwards, and we find the following result: .11)This product converges for the already mentioned delicate balance of arguments of the Gamma functions [35].We can still see the interesting property of F (m 1 , m 2 ; θ) to go into its inverse as we exchange . For specific values of m 1 , m 2 and k there may be accidental singularities in the third line of (3.11), which does not contain the variable τ = θ 2πi : these cases require to be dealt with separately and may potentially be subject to drastic simplifications.
By the Karowski-Weisz result, combined with the fact that both σ and F are symmetric under the exchange m 1 ↔ m 2 , we conclude that

Singularities of σ
For future purposes, let us list the poles and zeros of σ −2 (m 1 , m 2 ; θ), as can be evinced from (3.3).We list the labels first in a rather pleonastic way -one can in fact streamline the counting.
.., zero of order two, six,.... (3.16)Let us notice for futures scopes that F −1 and σ −1 are also well-defined and meromorphic, with poles and zeros starting from a minimum order one instead of a minimum order two.

Tools from the algebraic Bethe ansatz
The minimal form factor is just a building block on the way towards the complete form factor formula.
The method of the off-shell Bethe ansatz [30] requires the use of the Ĉ operator borrowed from the algebraic Bethe ansatz.We do this for two physical spaces (and one auxiliary space).It is important to remark that the integral formulas constructed in [30] for the Sine-Gordon model assume an odd number of particles.We will find here that we can solve the axioms using integral expressions based on the off-shell Bethe ansatz for an even particle number (specifically equal to two) as well.We decided to build integral expressions in this case both i) to prove that they are actually solutions, and ii) to setup the type of building blocks which will be useful in the future in an all-particle generalisation.
The R-matrix associated with the massive-massive scattering is given by [22] 2 the matrices E ij having all zeros except a 1 in row i, column j.For our purposes the state |1⟩ = |ϕ⟩ is a boson and the state |2⟩ = |ψ⟩ is a fermion.We will confine ourselves to massive excitations in this paper.The entries are spelt out in (C.4) of [22], and they read where and Our conventions are such that, if we do not write it explicitly, we always intend while we shall explicitly indicate any other assignment -for instance A(m 2 , m 1 ; θ).We shall also need the S-matrix, which is obtained as where Π is the graded permutation acting on two states: and deg(ϕ Even if the Yang-Baxter equation is solved in the general case, will always assume in this paper that S i = 1, for all i = 0, 1, 2 (with the auxiliary space 0 to appear later on).This is achieved (for the case of non-zero mass which we are concerned with here) by assuming that m ∈ {1, 2, ..., k − 1}, which is sufficient to cover the range of the physical massive spectrum [22].
We construct the monodromy matrix as where the indices denote the space the R-matrix is acting on.The Ĉ operator is obtained by extracting the terms with an E 21 in the auxiliary space.We find having indicated We shall need the action on the pseudovacuum covector: We shall also need

The off-shell Bethe ansatz
The off-shell Bethe ansatz is designed to provide a solution to the permutation axiom of form factors.
We follow closely the conventions of [30] throughout the paper.
The S-matrix of the theory is given as a product of various pieces -each piece will provide a certain contribution to the actual form factor.All these contributions will have to be multiplied together at the end.

Prefactor
There is a contribution from a specific prefactor [22] Φ(m 1 , m with and where The contribution to the complete form factor from each of the individual factors [m] θ can be found using the formulas in appendix C of [30].Namely, every factor where we have set contributes a corresponding term to the complete form factor.The total number of factors [m] θ is always even so the minus sign does not matter, therefore the total contribution from this part is with By the usual Karowski-Weisz theorem, combined with the fact that both Φ and F total pre are symmetric under the exchange m 1 ↔ m 2 , we conclude that

Dressing factor
There is then a contribution from the dressing factor σ −2 (m 1 , m 2 ; θ).This will contribute a term It is clear that we can simply set where F −2 (m 1 , m 2 ; θ 1 , θ 2 ) is our minimal form-factor solution (3.11).We then also have therefore

Matrix part
The matrix part of the complete massive R-matrix is composed of two copies of the psu(1|1) 2 R-matrix (4.1).It is not difficult to convince oneself that this implies a factorised structure: we can solve the permutation axiom for each individual copy of psu(1|1) with a and b in the fundamental two-dimensional representation of psu(1|1) 2 , the complete R-matrix R tot will act on two such states as where R is the 4 × 4 R-matrix (4.1).If we split the total contribution of the matrix part as then the individual functions G ac (θ 1 , θ 2 ) and G bd (θ 1 , θ 2 ) satisfy the permutation axiom with respect to the individual R-matrix R.These functions will now be built as where I 0 is an integral operator with respect to an auxiliary integration variable θ 0 which we will define later on, while the F ab (m 0 , m 1 , m 2 ; θ 0 , θ 1 , θ 2 ) block will come from the off-shell Bethe ansatz.
Let us therefore focus on the individual combinations of states.

• Two bosons
We start with the pseudovacuum ⟨ϕϕ|.Given that the matrix part of the R-matrix satisfies A(θ 1 − θ 2 ) = 1 for the boson-boson entry, the individual function F ϕϕ (m 0 , m 1 , m 2 ; θ 0 , θ 1 , θ 2 ) can be set equal to 1.This is reproduced by the off-shell Bethe ansatz, because the pseudovacuum simply comes with a coefficient 1 in the construction.In the case of the pseudovacuum, moreover, I 0 is the identity operator, hence we set G ϕϕ (θ 1 , θ 2 ) = 1.In this case the operator could naturally be a boson, and in particular the boost axiom will work for a scalar function and it will certainly be satisfied by our choice.

• Two fermions
Let us continue with the state ⟨ψψ|.The R-matrix entry for two fermions is G(θ 1 − θ 2 ), and the off-shell Bethe ansatz provides us with the appropriate function.Let us set from (4.12) -mindful that the state ⟨ψψ| in the algebraic Bethe ansatz contributes to the form factor of two fermions.This evaluates explicitly to It can be checked that this function satisfies Therefore, if we bear in mind that the S-matrix entry for two fermions is +G(θ 1 − θ 2 ) whereas the R-matrix entry is −G(θ 1 − θ 2 ), we could set which by virtue of (4.29) would satisfy the permutation axiom for all values of θ 0 ∈ C (in this sense one says that the Bethe ansatz is off-shell).In reality, we will not use this expression for the two-fermion form factor, since it turns out not to have good periodicity properties.
Instead, to respect periodicity, it is be better to use the Karowski-Weisz theorem again.We notice that the fermion-fermion entry of the S-matrix is therefore we satisfy the axioms by setting the fermion-fermion contribution to where the operator could naturally be taken to be bosonic.

• One boson, one fermion
We can now deal with the form factors that do mix.As commented above (4.8),we can restrict to m ∈ {1, 2, ..., k − 1} so that all the signs are positive.We also call m 0 the mass of the auxiliary particle θ 0 in the algebraic Bethe ansatz.From the action of one single Ĉ operator given in (4.11) we define It is then possible to prove by brute force that and (4.36)where we will often use the notation throughout, where i ̸ = j could be any of 0, 1 or 2.
These two conditions reconstruct the permutation axiom applied to the two individual functions that mix: F ϕψ and F ψϕ .Specifically we can set (4.38) and (4.39) again for any θ 0 ∈ C (off -shell).This shows that (4.35) and (4.36) perfectly reproduce the permutation axiom, since C is the S-matrix entry for ϕψ → ϕψ, D is the S-matrix entry for ψϕ → ϕψ, +E is the S-matrix entry for ψϕ → ψϕ and B is the S-matrix entry for ϕψ → ψϕ.
In order to reproduce the method devised in [30], we now need to assemble a few components.First, The advantage of this definition resides in its transformation properties under periodicity: let us start with i = 1 and shift θ 1 by +2iπ.To begin with, we have checked that braiding unitarity, crossing symmetry and analiticity also hold for σ −1 , and not only for σ −2 , if one halves the powers everywhere accordingly.Likewise we have checked that the the relations satisfied by F −1 (m 1 , m 2 ; θ) can be obtained from those satisfied by F −2 (m 1 , m 2 ; θ) by halving the powers everywhere -analyticity also being preserved (no square roots appearing anywhere).We therefore have Φm0,m1 (θ (4.41) having in addition used the crossing properties of the dressing factor.We have also used the symmetry under exchange of m 1 and m 2 of σ −1 (m 1 , m 2 ; θ) and F −1 (m 1 , m 2 ; θ).The function f m1,m2 (θ) is given by [22] Likewise, if we focus on i = 2 and shift θ 2 by −2iπ (the reason for which will shortly become apparent), we have having used at the very end the properties of the function f m1,m2 (θ) in (4.42).
By brute force computation one can also check that and Finally, we have and also (using the symmetry of σ(m 1 , m 2 ; θ) under m 1 ↔ m 2 and its braiding-unitarity property) We can now assemble all the components and obtain the form factors satisfying the axioms.We define the fermion-boson individual function as and the boson-fermion one as If the operator, of which we are computing the form factor, has spin s, we define to satisfy the boost axiom.This turns out to produce the naively expected sign under periodicity for an operator with half-integer spin s.
To prove periodicity, we shift θ 1 + 2iπ in the boson-fermion formula.In the fermion-boson formula instead, we shift θ 1 + 2iπ but simultaneously shift the integration variable θ 0 + 2πi, so as a net effect this is equivalent to shift θ 2 − 2iπ in the integrand.H 12 is a symmetric function under the exchange of 1 and 2, so that permutation is not spoilt.The integration contour will also be chosen to be symmetric under the exchange of 1 and 2, so that permutation is not spoilt.
If we focus on the strip Im(θ 0 ) ∈ [−2π, 2π], a careful analysis of poles and zeros of the integrand of (4.48) and (4.49) reveals the picture in Fig. 1 and Fig. 2.
We will chose the contour to run through the points θ 1 − iπ − iϵ and θ 2 − iπ − iϵ as shown in the figures.
Periodicity works this way: • for the boson-fermion form-factor, we shift θ 1 + 2iπ.By virtue of the formulas obtained above, the integrand becomes identical to the integrand of the fermion-boson form-factor.• for the fermion-boson form-factor, we shift θ 1 + 2iπ and change integration θ 0 + 2iπ.By virtue of the formulas obtained above, the integrand becomes identical to the integrand of the fermion-boson form-factor.The contour shifts but it encounters no poles as it settles back into its original shape.
To complete all cases, we can go back to the two-fermion form factor.
• Recombining the two copies One ought to tensor together two copies of the integral according to the pattern of states, and the total should then be supplemented by a factor F −2 (m 1 , m 2 ; θ 12 )F total pre (m 1 , m 2 ; θ 12 ), (4.51) so that the complete form factor will be We notice that we seem to find the axioms to be satisfied for any value of m 0 (not necessarily integer).
We have not explored whether this is a genuine degree of freedom in our solution, or whether it will be fixed in some way by the theory.
Let us remark that the kinematical singularity axiom degenerates for the case of two particles, as there are no strict sub-channels3 .
The issue related to the bound state axiom is something which will require a more detailed knowledge of the operator.A naive reading of the bound state form factor axiom implies that the residue at the bound state pole must be proportional to the one point form factor of the operator O with the bound state creation operator.It may be necessary to suitably modify our expressions to account for the bound state singularities [32] -in fact we cannot make any claim about the uniqueness of the formulas which we have so far constructed.We reserve the investigation of bound states for future work.

Convergence
We can compute the asymptotics of the components of our integrals (for generic value of the parameters): if we rescale x ≡ y |Re θ| , (5.1) and subsequently send |Re θ| → ∞, we can expand the integrand first and then perform the integrals.
We obtain However we should point out that this expansion is somewhat dangerous, since the subleading order in |Re θ| −1 of the second line turns out to be multiplied by a divergent integral.The conclusions which we draw are therefore only naive and a more thorough analysis might be required.Numerical experimentation reveals that within the error bars a very good approximation is obtained by the following formula: while numerically the asymptotics of σ −2 are quite well reproduced by the formula we gave above.
By inspecting the integrals we see that the naive asymptotic behaviour is therefore ) , Re θ 0 → +∞, hence we get convergence in the region .4)If we wish to set the spin to s = 1 2 or s = 3 2 for instance, we might consider analytically continue the integrals either in m 1 , m 2 or k to a region of convergence (for instance negative values of m1+m2 k ) -see for instance the discussions in [37].Another option would be to modify the contour -our choice in figures 1 and 2 was the simplest choice compatible with the axioms, but there could be more elaborated choices (see for example [30]) which may produce a final result in a wider region of the parameters.We have not explored these possibilities, and we plan to come back to this issue in future work.
Another worthwhile consideration is that our H 12 function (4.50) is tuned to a standard assignement of statistics -equivalently, to the statistics of the bare states.As shown in [22,19], the coproduct acquires a deformation.This alters the basic statistics [22], and therefore one might need to revisit our assignment of H 12 depending on the operators considered.This in turn might change the convergence properties of the integral, which is extremely sensitive to the exponential e H12 .
The occurrence of divergences in the integrals, both in the off-shell Bethe ansatz method and in the Lukyanov method, are in fact contemplated and seem to be difficult to avoid.It appears that the systematic analytic continuations of the integral representations has to be implemented.In the case of the pure RR massless theory an additional source of divergence is due to the infrared limit entangled with the poor convergence of the series expansion, as discussed in [28] -section 8.1 and particularly section 8.1.2. in that paper.

Perturbation theory
There is limit where formulas simplify -in particular the dressing factor becomes the identity -which is the k → ∞ limit.It turns out that we can expand our formulas in orders 1  k by relying on the integral representations, expanding the integrand first, and then performing the integral.We assemble here the various pieces: .1)We can also expand F ψϕ and F ϕψ up to order 1 k 2 (their expansion not being very illuminating), and finally the CDD-like pre-factor (in the case of m Let us focus for instance on the fermion-boson and boson-fermion form factors. Assembling all the pieces we find that at the first non-trivial order we can naively perform the final contour integration of figures 1 and 2 if we analytically continue in the spin s to the region 3 2 < s < 5  2 , with the spin acting as a regulator.We then obtain It is amusing to observe that we could tune the -so far unspecified -parameter m 0 to neutralise the divergence.We could for instance choose to send m 0 ∼ (s − 1 2 ) 2 as long with s → 1 2 , and we would probably be safe in the limit (the same with 3  2 ).We can observe that, if we have anywhere a G ϕψ or a G ψϕ , such as in the form factor of the states (ϕ ⊗ϕ) ⊗ (ψ ⊗ϕ) for instance, then the leading order of the form factor is dictated by (6.3) or (6.4), for instance ⟨0|O|(ϕ ⊗ϕ) θ1 ⊗ (ψ ⊗ϕ) θ2 ⟩ = "Φ F −2 G ϕψ G ϕϕ " = (6.3)+ ... (6.6) (with O being fermionic).Similarly for other combinations involving G ϕψ or G ψϕ .

Conclusions
In this paper we have studied the form factors of the mixed flux AdS 3 × S 3 × T 4 scattering theory in the relativistic limit, using the formulation of [22], which uses a distinct limit from the one studied in [19].Here we have focused on the massive sector and derived the minimal solutions to the form factor axioms corresponding to two-particles.We have then constructed the complete set of two particle form factors using the off-shell Bethe ansatz method of [30].
The highlight of our construction is that we can solve the axioms of two-particle form factors, although the integrals for the fermion-boson and boson-fermion case appears to be divergent for certain value of the spin s.We have discussed possibilities for regularising the integral, for instance analytically continuing in the parameter 1 k which appears to behave like a small coupling in perturbation theory.It is also possible that the contour used for the integrals could be modified.Finally, the deformed statistics of the dressed particles might force one to change the exponential function H 12 , upon which the convergence of the integrals depends quite strongly.This will have to be done in accordance with which operator one is considering.
There are a number of directions which one should explore, most importantly whether one can properly cure the divergence.As always, it would be extremely useful to obtain perturbative checks with Feynman diagrams to show that we are on the right track.Generalising to more particles appears in principle standardised by the off-shell Bethe ansatz method, but not too easily implementable practically, as it appears to depend quite significantly on the precise distribution of singularities of the integrand.In particular the bound state axiom and the kinematical singularities would have to be carefully analysed.
It would also be interesting to approach the problem using Lukyanov's method [38,33] and see whether we obtain a different result.We plan to come back to these issues in future work.

Figure 1 :
Figure 1: The locations of the poles of the fermion-boson integrand are denoted with little circular dots with their respective order, in a portion of the complex plane of θ 0 .We have assumed 0 < m 0 < min[m 1 , m 2 ] and k sufficiently large.In red we have indicated the integration countour.

Figure 2 :
Figure 2: The locations of the poles of the boson-fermion integrand are denoted with little circular dots with their respective order, in a portion of the complex plane of θ 0 .We have assumed 0 < m 0 < min[m 1 , m 2 ] and k sufficiently large.In red we have indicated the integration countour.