Three- and Four-point correlators of excited bosonic twist fields

We compute three- and four-point correlation functions containing excited bosonic twist fields. Our results can be used to determine properties, such as lifetimes and production rates, of massive string excitations localised at D-brane intersections, which could be signatures of a low string scale even if the usual string resonances are inaccessible to the LHC.


Introduction
Orientifold compactifications has been proven to be a very fruitful framework for realistic model building. In those constructions the gauge symmetry does live on the world-volume of lower-dimensional hyperplanes, called D-branes, whereas the chiral matter is localised at the intersection of different D-brane stacks. For recent reviews on D-brane model building and concrete MSSM D-brane realizations, see [1][2][3][4] and references therein.
A particular intriguing subclass of D-brane realizations are string compactifications with a significantly lower string scale [5][6][7]. Such low string scale scenarios with a string scale M s that can be as low as a few TeV provide a potential solution to the hierarchy and cosmological constant problems but may also lead to interesting signatures observable at the LHC. Among those signatures rank experimental signs of anomalous Z ′ physics as well as Kaluza Klein states (see, e.g. ).
Low string scale compactifications also allow for the direct detection of string excitations. In a series of papers [25,[29][30][31]] the authors study tree-level string scattering amplitudes containing at most two chiral fermions. They show that these amplitudes exhibit a universal behaviour independently of the specifics of the compactication, which gives their results a predictive power. The observed poles correspond to the exchanges of Regge excitations of the standard model gauge bosons, whose mass scales like the string mass, and this has enabled the experiments to now limit the string scale to be above 5.08 TeV [39][40][41].
On the other hand at the intersection of two D-brane stacks exists a tower of stringy excitations, dubbed as light stringy states in the following, whose mass scales as M 2 ∼ θM 2 s , where θ denotes the intersection angle. Thus for small intersection angles those states can be significantly lighter than the first Regge excitations of the gauge bosons and are expected to be observed prior to the latter. Therefore it is of utmost interest to study properties, such as decay rates and life time, of these light stringy states.
As recently discussed in [42] the vertex operators of such light stringy states 1 contain excited versions of the bosonic twist fields present in the operators for massless chiral matter states (see e.g. [43,44] for recent work involving these). However, little is known about correlators involving such excited bosonic twist fields 2 , a crucial ingredient in the computation of scattering amplitudes involving light stringy states.
This work is dedicated to filling this gap. For simplicity and reasons of calculability we assume the compactification manifold to be a factorizable six-torus T 6 = T 2 × T 2 × T 2 wrapped by D6-branes. Such models are useful toys, but are still under active development (for original work, see e.g. [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66], for recent work, 1 Here the authors assumed as compactification manifold a factorizable six-torus T 6 = T 2 × T 2 × T 2 . 2 For some preliminary work on excited bosonic twist field correlators, see [45,46]. see e.g. [67,68]). In that case scattering amplitudes split into three separate factorizable parts for each two-tori T 2 , for which one can apply the developed CFT techniques. Using the latter we derive various three-and four-point correlators containing (higher) excited bosonic twist fields, where the latter live on one of three two-tori. More precisely we derive correlators of the type of type where the σ fields are the usual bosonic twist fields while τ and ω denote the excited and double excited bosonic twist fields. The subscript denotes the intersection angle which is measured in units of π and ranges in the interval [0, 1). Since the excited twist fields, τ and ω are not primary conformal fields we will take a detour to determine the correlators of type (1.1) by first evaluating higher point correlators containing solely the primary fields, namely solely bosonic twist fields σ as well as the conformal fields ∂Z and ∂Z. Given those correlators we obtain the correlators displayed in (1.1) by applying various operator product expansions OPE's when performing appropriate limits of the higher point correlators.
This paper is organised as follows: In section 2 we illustrate the method which we use to evaluate the three and four-point correlators containing the excited bosonic twist fields. In section 3, we apply that method to compute the three-point correlation correlators while in section 4 we derive the four-point correlators with one and two independent angles. The appendices A and B contain necessary material for the derivation of the correlators. In appendix C we demonstrate the vanishing of the quantum part of the correlators with only one excited field. In appendices D and E we display the remaining of our results which were omitted in the main text in order to keep the paper in a more readable/compact form.

General considerations
As already mentioned in the introduction we assume the compactification manifold to be a factorizable six-torus T 6 = T 2 × T 2 × T 2 . In [69][70][71][72] 3 the authors determined the three and four-point bosonic twist field correlators in order to determine the physical Yukawa couplings as well as the Kähler metric for chiral matter 4 . We will perform an analogous derivation for the three-and four-point correlators containing (higher) excited bosonic twist fields.
Let us illustrate the method which we will use to derive the above displayed correlators (1.1). Here we will focus on the three point correlators but the same procedure will apply to the four-point correlators. We start by determining the correlators containing three bosonic twist fields σ and up to two of the bosonic conformal fields ∂Z and ∂Z. More explicitly we derive the four and five point correlators . We have to keep in mind that the conformal fields ∂Z and ∂Z do split into a classical piece ∂Z cl and ∂Z cl , respectively and a quantum piece denoted by ∂Z qu and ∂Z qu Here ∂Z cl and ∂Z cl satisfy the classical equation of motion with the boundary conditions set by the insertions of the bosonic twist fields. Thus the four-and five-point correlators split into a pure quantum part as well as a part which contains the classical piece, dubbed the mixed part in what follows. For the derivation of the pure quantum correlator we will employ the energy momentum tensor method initially used in [76][77][78] in the context of closed string theory on orbifolds and more recently applied to open string theory [45,[70][71][72]79,80]. On the other hand the mixed part requires the knowledge of the classical solutions for ∂Z cl and ∂Z cl for the respective boundary conditions. Those were derived in [71,72] for arbitrary number of bosonic twist field insertions as well as arbitrary intersection angles. Adopting their results for our specific setups in combination with known pure bosonic twist field three-point correlator [46,70,81] allows us to determine the mixed part.
Finally, any correlator is suppressed by the world-sheet instanton e −S cl , which is given by with ∂Z cl and ∂Z cl being again the classical solutions to the equation of motion satisfying the boundary conditions set by the bosonic twist fields. Given those solutions for the two-torus one can easily determine (2.3) where the whole world-sheet instanton contribution is a sum over all possible closed triangles connecting the three intersection points [69][70][71][72] (See also [25]) 5 .
After establishing the complete four-and five-point correlators (2.1), containing the pure quantum and the mixed part as well as the world-sheet contribution, we investigate various limits. Applying the OPE's will then give the three-point correlators containing the excited bosonic twist fields. Even though, the procedure for the derivation of correlators containing exciting twist field correlators was laid out for three-point correlators it generalizes to fourpoint correlators as we will exemplify for four-point correlators containing excited twist fields with one and two independent angles.

Three-point correlators
In this chapter we apply the previously described procedure to obtain three-point bosonic twist field correlators that contain excited bosonic twist fields. As laid out above we take a detour by first deriving higher point correlators containing the bosonic twist fields σ α , σ β and σ γ as well as the conformal fields ∂Z and ∂Z. The insertions of the bosonic twist fields σ lead to the following boundary conditions In the following we assume that the sum of the intersection angles is 1 rather than 2, i.e. α + β + γ = 1. The results for the latter setup can be easily obtained making the replacement α, β, γ → 1 − α, 1 − β, 1 − γ.

Three-point correlators with one excited twist field
Let us start with the three-point correlators containing one excited twist field. Those arise from the four-point correlators where we focus on a setup for which the intersection angles add up to 1, i. e. α + β + γ = 1. As indicated in section 2 the conformal fields ∂Z as well as ∂Z split into a classical and a quantum piece, see eq. (2.2). As we show in appendix C any bosonic twist field correlator containing an odd number of the conformal fields ∂Z qu and ∂Z qu vanishes. Thus the four-point correlators (3.5) consist of a purely mixed part that takes the form The three-point bosonic twist field correlator is given by [46,70,81] On the other hand the classical solutions are given by the asymptotic behaviour close to the insertions of the bosonic twist fields σ, which specifies the solutions up to normalizations that can be evaluated from the global monodromy conditions. This procedure has been applied to three bosonic twist field correlators in [69][70][71] 7 where the authors derived the classical solutions for compactification manifold M, being planar M = R 2 and toroidal M = T 2 . They take the form 8 where we used SL(2, R) invariance to fix the bosonic twist field positions to x 1 = 0, x 2 = 1 and x 3 = ∞. On a plane the vector v a denotes the distance between the two fixed points v a = f 1 − f 2 , where f 1 and f 2 denote the intersection points of two D-brane stacks. However on a two-torus T 2 one has to take into account that the action of the twist operator does not only rotate Z cl but also may shift it by a lattice translation. That results into an infinite number of solutions with v a = f 1 −f 2 + n L a , where In eq. (3.9) I xy = n x m y − n y m x is the intersection number between two D-brane stacks x and y, L x = R 1 n 2 x + (m x T ) 2 the length of the D-brane stack x, with (n x , m x ) being the wrapping numbers and T denoting the complex structure of the two-torus. In the simple setup in which all three D-brane stacks intersect each other exactly once L x coincides with the length L x of the respective D-brane. 6 In case the intersection angles add up to 2 the correlator takes the form . 7 For analogous results in the T-dual IIB framework with magnetized D-branes, see [73]. 8 Here we suppress the x ∞ dependence and assume z < 0.
All correlators are suppressed by world-sheet instanton contributions which can be easily computed applying (2.3) as well as using the classical solutions (3.8). One obtains where v a is given by v a = f 1 − f 2 + n L a . Thus the whole four-point correlator ∂Zσ σ σ takes the form 9 with the world-sheet instanton given in eq. (3.10) and the sum is over all closed triangles, thus over all integers n. On the other hand the four-point correlator ∂Zσ σ σ vanishes. Given the correlator (3.12) we can now derive the three-point correlator containing an excited twist field. Performing the limit z → 0 gives on where we used on the left-hand side the OPE's displayed in eq. (2.4). Reinstating the complete x i dependence gives 10 1 4 x α(1+β) 12 x α(1+γ) 13 x βγ−α 23 e −S cl (3.13) with the world-sheet instanton contribution given by (3.10).

Three-point correlators with two excited twist fields
Three-point correlators containing two excited bosonic twist field (or one double excited bosonic twist field) arise from the five point correlators of type . 9 In the following we will suppress the x ∞ factor, which we will recover when we reinstate the full x i dependence.
10 Any 3-point corrleator of quasi-primary fields takes the form where h i denote the conformal dimension of the field Φ i .
In contrast to the four point correlators discussed in the previous subsection the quantum parts of the correlators (3.14) do not vanish. In order to determine those we define the following functions whose knowledge, together with the three-point correlator (3.7), allows us to derive the 5-point quantum correlator. With the local behaviour (2.4) as well as the form of the functions g, k and m is given by Here we use the fact that any correlation function that contains a conformal field of dimension h at insertion x behaves as x −2h for x → ∞. While the functions g and k are fully determined by requiring the proper local behaviour for the full determination of m we need to impose the monodromy conditions (3.4), that results into the condition 11 which after dividing by ω 1−α,1−β,1−γ (z), using SL(2, R) symmetry to fix the bosonic twist field positions to and performing the limit z → ∞ results into Here g(w) and m(w) take the form Solving (3.26) allows to determine the missing piece of the function m(z, w), C, to Together with the three-point correlator (3.7) we obtain the following five-point quantum correlators 12 Since ∂Z cl = 0 only ∂Z∂Zσ + α σ + β σ + γ gets contribution from the mixed part. Using (3.8) the full five-point correlators take the form where the world-sheet instanton suppression is given in (3.10) and the sum is over all closed triangles formed by the three D-brane intersections.
In order to derive the three-point correlators containing excited bosonic twist fields we take various limits of the five-point correlators, analogously to the derivation performed in the previous subsection. We illustrate this analysis with the example of the five-point correlator ∂Z∂Zσ + α σ + β σ + γ and display the three-point correlators derived from the correlator ∂Z∂Zσ + α σ + β σ + γ in appendix D. From (3.32) we obtain in the limit z → 0, w → 1 After reinstating the x i dependence one gets where again the world-sheet instanton suppression is given in (3.10). Other threepoint correlators can be derived in a similar fashion from (3.33) and we display them in appendix D.

Four-point correlators
In this chapter we apply the same formalism as above to determine four-point correlators containing excited bosonic twist fields. We will discuss two different scenarios, namely bosonic twist field four-point correlators with one and two independent angles, starting with the case of one independent angle.

Four-point correlators with one independent angle
As for the three-point correlators we derive the four-point correlators containing by computing higher point correlators containing the bosonic twist fields σ + α and σ − α as well as ∂Z and ∂Z. The insertion of the bosonic twist fields σ + ) lead to the following boundary conditions on the world-sheet (4.1) Given those boundary conditions we will determine again the correlators containing one or two excited bosonic twist fields, starting with the correlators containing only one excited bosonic twist field.

Four-point correlators containing one excited twist field
Analogously to the three-point correlator derivation we determine the four-point correlators containing one excited bosonic twist field by first computing the 5-point correlators which get only a contribution from the classical part of ∂Z and ∂Z, respectively. Again the asymptotic behaviour close to the insertion of the bosonic twist fields specify the classical solutions ∂Z cl and ∂Z cl up to normalizations with ω α,β being The normalizations a and b are determined by satisfying the global monodromy conditions. One obtains [70,72] 13 where τ (x) is given by the ratio and with p, q being integers and L x given in eq. (3.9). Together with the four point correlator [45, 70-72, 79, 80, 82] 13 Here we used SL(2, R) invariance to fix the twist field insertions to x 1 = 0, x 2 = x, x 3 = 1 and one obtains for the 5-point correlators (4.2) and (4.3) (4.10) As already indicated both correlators are suppressed by the world-sheet instanton contributions e −S cl which we discuss momentarily. With (2.3) and (4.4) we can compute the classical action in the presence of the four bosonic twist fields compute to [72] (see also [25]) where τ (x) is given by (4.7). The complete world-sheet instanton contribution is the sum over all possible closed polygons connecting the four intersection points, thus we sum over all integers p and q appearing in v a and v b . In the following we will perform the limits z → 0 and z → x in order to derive the four point correlators with one excited bosonic twist field. We will illustrate that with the example ∂Zσ + σ − σ + σ − and display the results arising from the five-point correlator ∂Zσ + σ − σ + σ − in the appendix E.1. From the limit z → 0 one obtains 14 Here we use the fact that an open string four-point correlator is given by where ∆ = 4 i=1 h i and x = x12x34 x13x24 . This expression allows us to reinstate the x i dependence in all the four-point correlators.
Analogously we get for the limit z → x A comparison of (4.14)and (4.15) confirms the following identifications 15 that were already suggested in [42,82] by studying the OPE's of such higher bosonic twist fields. Note that this identification uses the fact that e iπα v a = e iπ(1−α) v a and holds also true for τ − α = τ + 1−α and as we will see later also generalizes to higher excited bosonic twist fields. Let us point out that this identification among bosonic twist fields and its counterpart, the bosonic anti-twist fields, can be also checked by performing the same analysis with the bosonic twist field ordering of the type . The four point-correlator arising from the five-point correlator (4.10) can be derived in an analogous fashion and is displayed in appendix E.1.

Four-point correlators containing two excited twist fields
Analogously to the three-point correlators containing two excited bosonic twist fields the four-point correlators containing two excited bosonic twist fields arise from sixpoint correlators containing two conformal fields ∂Z and ∂Z. More specifically, they arise from the six-point correlators In contrast to the five-point correlators considered in the previous subsection (4.1.1) the pure quantum part of the correlators 4.17 does not vanish. We will determine it applying the same procedure performed in section 3.2 for the derivation of the pure quantum five-point correlators.
Let us define the following three functions that will be determined momentarily using local behaviour of the bosonic twist fields as well as global monodromies. Together with the knowledge of the four-point corre- ) the functions g(z, w), k(z, w) and m(z, w) allow us to derive the requested six-point correlators.
Using the local behaviour of the bosonic twist fields (2.4) as well as (3.18) we obtain for the functions g(z, w), k(z, w) and m(z, w) where we use various symmetries of the functions under the exchanges of the x i 's.
Here ω α,β is given by eq. (4.5) and P takes the form Focusing on the derivation of g(z, w) and k(z, w) the constraints (4.25) lead after using SL(2, R) symmetry to fix x 1 = 0, x 2 = x, x 3 = 1 and x 4 = x ∞ , dividing by ω α,α (w) and taking the limit w → ∞ to where g(z) and k(z) are given by Solving the system of equations (4.27) by using the integrals displayed in appendix B we obtain for A(x) (4.30) Together with (4.21) and (4.22) this determines the functions g(z, w) and k(z, w) at the positions (z, w, 0, x, 1, ∞). Note that g(z, w) and k(z, w) are completely independent on the choice of ξ, as it should be. Thus for the function g(z, w), A compensates for the ξ dependence in P .
In an analogous way one can determine C(x) 17 by solving the system of equations (4.26) and obtain for A(x) again (4.29) while for C(x) one gets (4.31) 16 The setup with four bosonic twist field insertions has two independent world-sheet contours which we choose without loss of generality to be 0 to x and x to 1. 17 Here C(x) = −C(0,x,1,x∞) x∞ The knowledge of the four-point correlator (4.8) then allows us to determine the pure quantum part of the six-point correlators ∂Z∂Zσ After a few manipulations one obtains Adding the "mixed " part that contains the classical solutions ∂Z cl and ∂Z cl , given by (4.4) as well as the world-sheet instanton contributions (4.11) we obtain for the three different six-point correlators: with S cl being the world-sheet instanton contribution given in equation (4.11).
Given these six-point correlators we can derive the various four-point correlators containing higher excited twist fields by taking the appropriate limits as we will examplify momentarily. Taking the limit z → 0 and w → x of (4.35) and using the OPE's (2.4) one gets which after reinstating all x i dependence gives (recall x = x 12 x 34 x 13 x 24 ) 18 If we take the limit z → x and w → 0 of (4.35) we get the four-point correlator Comparing this correlator to (4.38) confirms again the identifications that were already suggested in chapter 4.1. Looking at other limits of the six-point correlators (4.35), (4.36) and (4.37) that give us four-point correlators containing excited bosonic twist fields we can generalize this identification to the higher excited twist fields ω − and ω − . Thus we have

Four-point correlators with two independent angles
In this section we perform the same analysis as above, however for the more generalized setup of two independent angles. The bosonic twist field insertions σ + α (x 1 ), σ − α (x 2 ), σ + β (x 3 ) and σ − β (x 4 ) lead to the following boundary conditions Analogously to section 4.1 we will determine the correlators containing one or two excited bosonic twist fields via deriving five-and six-point correlators. Let us start with the derivation of the four-point correlators containing one excited bosonic twist field. 18 Here we use the fact that an open string four-point correlator is given by where ∆ = 4 i=1 h i and x = x12x34 x13x24 . This expression allows us to reinstate the x i dependence in all the four-point correlators.

Four-point correlators containing one excited twist field
As in the subsection 4.1.1 the four-point correlators containing one excited bosonic twist field arise from the five-point correlators which has no purely quantum part, but is completely dictated by the classical part of ∂Z and ∂Z. They have been computed in [72] (see also [70]) and take the form with ω α,β being (4.5) and a and b given by Here v a and v b are the same as in the previous subsection 4.1 and we used the following definitions with which I(x) takes the form It is easy to check that the at the limit β → α the classical solution (4.46) simplifies to the one with just one independent angle (4.4). Moreover in the limit x → 1 this solution goes over into the classical solution of three bosonic twist field insertions (3.8) discussed in chapter 3. Together with the four point correlator [25,46,70,72,82] we obtain for the two five point correlators The classical action can be computed in an analogous fashion as previously using (2.3) as well as (4.46) which gives [71] (see also [25]) where τ (x) and ρ are and v a and v b are defined as previously. Again we have to sum over all world-sheet instanton contribution, i.e. sum over all possible closed polygons connecting the four intersection points, which implies a sum over all integers p and q. Given those five-point correlators (4.51) and (4.52) we can derive the desired four-point correlators. We illustrate this by applying this procedure to the five-point correlator (4.51). The limit z → 0 gives which after reinstating the whole x i dependence takes the form where we used the definitions (4.48) as well as (4.49) and the world-sheet instanton contributions are given by (4.53). The other independent four-point correlator arising from the five-point correlator ∂Zσσσσ is displayed in appendix E.2.

Four-point correlators containing two excited twist fields
Four-point correlators containing two excited bosonic twist fields arise from the sixpoint correlators which we derive momentarily, applying the same procedure as for the five-point correlators in section 3.2. Let us start again with the derivation of the pure quantum part, for which we define in a similar fashion to the case with one independent angle .

(4.62)
Applying the OPE's (2.4) the ansatz's for the functions read where ω α,β is given by (4.5) and P takes the form Here we again used various symmetries of the function g(z, w) as well as the singular behavior (3.18). The parameter ξ is not fixed by any symmetry or the singular behavior, but as before in the one independent angle case we expect that g(z, w) itself is independent of ξ. Thus we expect that the ξ dependence of P is compensated by and C ({x i }) will be determined momentarily using the monodromy constraints that read again (4.25) and (4.26). As before we fix the vertex operator positions to x 1 = 0, x 2 = x, x 3 = 1 and x 4 = x ∞ using SL(2, R) symmetry, divide the monodromy constraint (4.25) by ω α,β (w) and the monodromy constraint (4.26) by ω 1−α,1−β (z) and eventually take the limit w → ∞ and z → ∞, respectively. This will lead to the monodromy constraints with Here A, B, C and ω are defined as in the previous subsection (see eq. (4.28) and footnote 17). Solving the system of equations (4.67) and (4.68) gives where we used the definitions of (4.48) and (4.49). Note that as required the functions g(z) and g ′ (z) are indeed independent of the choice of ξ.
Together with the expression for the four point correlator one obtains for the quantum six-point correlators Adding the mixed part containing the classical solutions ∂Z cl and ∂Z cl (4.46) as well as the world-sheet instantons (4.53) Note that taking the limits x → 1 of the six-point functions (4.78), (4.79) and (4.80) as well as use the OPE one obtains the five-point functions (3.32), (3.33) and (3.34), which were evaluated in the previous section, respectively. Given all the six-point correlators we determine the four-point correlators containing the excited bosonic twist fields by taking various limits of the six-point correlators. Performing the limit z → 0, w → x in (4.78) one obtains which after reinstalling all x i dependence we get: where again x = x 12 x 34 x 13 x 24 and the world-sheet instanton contribution is given by (4.53). The remaining four-point correlator functions for two independent angles are displayed in appendix E.2.

Conclusions
In this paper we have derived the three-and four-point correlators containing exited bosonic twist fields. More specifically, we computed correlators that consist of regular bosonic twist fields σ as well as excited and doubly excited bosonic twist fields τ and ω, respectively. The knowledge of these correlators is required in the computation of lifetime and decay rates of string excitations that arise at intersections of D-branes. Assuming a low string scale scenario those stringy states can be significantly lighter than the first Regge excitations of the gauge bosons, and thus significantly lighter than the string scale and potentially observable at LHC.
In order to evaluate these correlators, we took a detour and determined five-and six-point correlators containing the conformal fields ∂Z and ∂Z as well as the regular bosonic twist fields σ. Given those five-and six-point correlators we performed various limits to derive the three-and four-point correlators containing higher excited bosonic twist fields.
The detailed computation of amplitudes that contain stringy states localized at the intersection of two D-brane stacks as well as their decay rates and lifetime is relegated to future work. However let us already point out a few phenomenological implications of our results.
Our results reveal that the three-point correlator containing just one excited bosonic twist field τ does not contain a purely quantum part, but is dictated by the classical solution. That implies that the decay rate of the first excited stringy scalar state, localized at the intersection of D-brane stacks, into two massless fermions, depends crucially (not just via world-sheet instanton suppression) on the displacement in the compactification manifold between the stringy massive state and the massless states localized at different D-brane intersections.
On the other hand the displacement in the compactification manifold between different D-brane intersection points is related to the observed Yukawa coupling hierarchies of the massless fermions. This potentially allows one to obtain bounds on the decay rate of light stringy states in terms of observed mass hierarchies. We now have all of the components assembled to do such an analysis.
As already shown in [42], in 2 → 2 processes involving chiral superfields, only the second state in the tower is exchanged, and these processes would be suppressed by the Yukawa couplings -so the bounds on these states will be drastically weaker than those for string Regge excitations. Indeed, from dijet searches for related resonances [39][40][41], since the cross-section is so suppressed we can infer that the bounds on such states will be much less than a TeV. This raises the intriguing prospect that the string scale could be just out of reach of the LHC, but the light stringy states could be hiding in plain sight.

A. OPE's of the twist fields
Here, we provide the OPE's of ∂Z(x) and ∂Z(x) on twisted fields: With these OPE's one can determine the conformal dimension of the respective twist fields. We summarize our findings in table 1. The above OPE's suggest the

Fields conformal dimensions
Fields conformal dimensions following identifications among twist-and anti-twist fields which can be easily generalized to higher excited twist fields.

B. Various integrals
Here we display all necessary integrals needed throughout this work.
C. Vanishing of the correlators ∂Z qu σ α σ β σ γ Here we demonstrate that the quantum part of amplitudes involving a single singlyexcited twist field must vanish. To do this, we simply construct the function where i runs from 0 to L. The functions f (z) andf (z) are holomorphic functions which do exhibit the poles in agreement (2.4). Since ∂Z, ∂Z have unit conformal weight, we furthermore require For L twist field insertions, there are L − 2 independent contours that we can choose for boundary conditions. There are L = 1 contours for L points, but the boundary must form a closed polygon, so one is not independent. There are also and L − 2 differentials which make up f andf in total. We construct the cut differentials where M ≡ α i . Then we can write so the number of differentials is given by the number of independent polynomials p,p which have order L − M − 2, M − 2, respectively. This is determined by counting the poles and allowing one parameter for the overall normalisation; the total number is thus (L − M − 2 + 1) + (M − 2 + 1) = L − 2. These are forced by the boundary conditions to be real parameters. We can then solve the boundary conditions for the quantum amplitudes These are a non-degenerate set of L − 2 equations in L − 2 real parameters, and hence all of the correlators containing just one ∂Z qu are zero.

E.2 Various four-point correlators with two independent angle
Here we display all the different four point correlators that can be derived from the five point correlator (4.52) and the six-point correlators , (4.75), (4.76) and (4.77).