Fixing the hexagon

In this section we will constrain the hexagon form factor. For this we impose four properties on them. The first one we consider is symmetry invariance, this will totally fix the tensor structure of one- and two-particle hexagons. It works remarkably similar to the S-matrix bootstrap from Section3.1. When we fix the two-particle hexagon’s matrix structure we note that each block has an undetermined scalar factor. So using the Watson, decoupling and monodromy axioms, we can fix these factors in terms of the phasesσ_uv^∗∗. Thus with all these properties in mind we finally show a solution for the hexagon form factor fornparticles.

Symmetry axiom

The most important property we used to fix thepsu(₁|₁)^⊕2invariant S-matrix was symmetry. Remember that this is the residual symmetry left after the lightcone gauge-fixing. Another way we can also see this fact is from the point of view of the dual CFT2. Let us start by considering the two-point function that preserves the maximal amount of supersymmetry. This is composed by two half-BPS operators at positions 0 and ∞ [96]. One of the operators is the highest weight and the other one is the lowest weight in the same R-charge multiplet. Then the residual symmetry are all generators of isometries that keep this configuration invariant.

Using an analogy to the Wigner’s symmetry procedure in QFT (see [116]) we consider this as the canonical two-point function configuration and the excitations on top of this will transform under the residual symmetry group. This procedure yield us exactly thepsu(1|1)^⊕_c.e.⁴ symmetry for the full spectral problem [33].

We can do the same procedure for three-point functions and in doing so we discover the symmetry group of the hexagon. So we first consider a canonical configuration that preserves as much symmetry as possible for the three-point functions. This is given by the three half-BPS operators in the same R-charge multiplet where now one is the highest weight, other the lowest weight and the remaining one is neither the lowest or highest weight state, it is in the middle of the multiplet [57,96,74]. Now the operators are localized in positions 0, 1 and

∞. To put a generic configuration of three operators in this canonical frame we consider a supertranslation transformation that acts on a operator at position 0 by O_t,k =e^tTkO(0)e^−tTk, (3.77)

wheretis the position ofO_t,k,T_k is is the supertranslation generator andk∈ _Ca free parameter. This generator is made of a specific combination of R-charge and conformal transformations of the dual CFT2and the residual symmetry group is the subalgebra ofpsu(₁|₁)^⊕_c.e.⁴that is invariant with respect to this generator [74].

Let us workout this subalgebra. First define the following supercharges QA =S_A− ⁱ

kϵ_ABQ^B, Q˜A =Q^˜_A−ikϵ_ABS˜^B,

(3.78)

then the invariant subalgebra with respect to T_k can be seen as the diagonal subalgebra ofpsu(1|1)^⊕_c.e.⁴. This is simply

{_QA,QB} ={_Q^˜_A, ˜QB} =0, {_QA, ˜QB} =−ϵAB

i k

C−k²C^† (3.79)

withCbeing the same central charge as in (3.12). An analogous result was also found for theAdS5×S⁵case, there the residual symmetry is the diagonal subalge-bra ofpsu(2|2)^⊕2[57]. Note that the algebra (3.79) is very similar to thepsu(1|1)^⊕2 algebra and we assume that the hexagon form factor is invariant under it, this is the symmetry axiom.

Knowing the symmetry algebra we need to apply it to the hexagon. For simplicity, we will for now restrict ourselves to hexagons with a single excited operator,id est, only a single physical edge has excitations. Configurations of the hexagon with more than one excited operator can be related to these simpler cases by analytic continuation on the rapidities, which we will detail later. We now change the notation of the hexagon from (3.74) and denote then-particle hexagon form factor as

⟨h|_Xa1,u1· · ·_Xan,un⟩_,

where Xa_j,u_j denotes a particle Xa_j with rapidity u_j. As Xa_j we can consider excitations on the L, R or massless multiplets given in (3.35), (3.37), (3.39) and (3.40). The assumption of invariance under (3.79) is then just

⟨h|Q|Xa₁,u₁· · ·Xan,un⟩ =_{0, (3.80)} whereQis any generator of (3.79). Until now the parameterkis free, however due to (3.80) the central charge annihilates all states being them physical or not, even

with its action being diagonal. This constrains the hexagon too much, therefore we obtain a nontrivial result only if we fixk=1 [74].

Using the definition (3.78) and the actions of the supercharges on the states, we find for one-particle hexagons the following result

⟨h|Yu⟩ =1, ⟨h|Zu⟩ =−i, ⟨h|Y^˜u⟩=1, ⟨h|Z^˜u⟩=−i,

⟨h|_χu^˙1⟩=1, ⟨h|χ˜_u^˙1⟩=−i, ⟨h|_χu^˙2⟩ =1, ⟨h|χ˜_u^˙2⟩ =−i, (3.81) with the remaining one-particle form factors vanishing. Note that in this case, the symmetry axiom reduces the number of independent form factors such that we can choose normalizations of the one-particle states that yield (3.81). The normalization factor is the so-called measure and it will play a central role to define finite bridge length corrections to the asymptotic hexagon in Chapter4.

For the two-particle hexagons we note that the action of the supercharges split them into blocks of defined chirality like the S-matrix in Section3.1. This is not surprising since the symmetry algebra is remarkably similar. More than that we find for theLLblock the following structure

⟨h|YuYv⟩=h^LL_uv A^LL_uv,

⟨h|YuZv⟩=−ih^LL_uv B_uv^LL,

⟨h|_Ψ¹_uΨ²_v⟩=−_ihuv^LL_Cuv^LL_,

⟨h|ZuYv⟩=−ih_uv^LLD_uv^LL,

⟨h|ZuZv⟩=h^LL_uv F_uv^LL,

⟨h|_Ψ²_uΨ¹_v⟩ =ih_uv^LLC_uv^LL,

(3.82)

whereh_uv^LLis an undetermined scalar factor. Note that this is basically theLLblock of thepsu(1|1)^⊕2S-matrix up to phases. We find similar structures for the other blocks with their respective scalar factorsh^∗∗_uv. For example in the RRblock we have:

⟨h|Y^˜uY˜v⟩ =h_uv^RR A^RR_uv,

⟨h|Y^˜uZ˜v⟩ =−ih^RR_uv B_uv^RR,

⟨h|_Ψ^˜1_uΨ^˜2_v⟩ =−_ih^RR_{uv Cuv}^RR_,

⟨h|Z^˜uY˜v⟩=−ih^RR_uv D_uv^RR,

⟨h|Z^˜uZ˜v⟩=h^RR_uv F_uv^RR,

⟨h|_Ψ^˜2_uΨ^˜1_v⟩ =ih_uv^RRC_uv^RR.

(3.83)

Here we see theL/R-symmetry explicitly. Now for theLRandRLblocks we have:

⟨h|YuY˜v⟩=h^LR_uv A^LR_uv,

⟨h|YuZ˜v⟩ =−ih^LR_uv C_uv^LR,

⟨h|_Ψ¹_uΨ^˜2_v⟩=−h^LR_uv F_uv^LR,

⟨h|ZuY˜v⟩ =−ih^LR_uv D_uv^LR,

⟨h|ZuZ˜v⟩ =h^LR_uv E_uv^LR,

⟨h|_Ψ²_uΨ^˜1_v⟩ =−h_uv^LR B_uv^LR,

(3.84)

⟨h|Y^˜uYv⟩=h^RL_uv A^RL_uv,

⟨h|Y^˜uZv⟩ =−ih^RL_uv C_uv^RL,

⟨h|_Ψ^˜1_uΨ²_v⟩=−h^RL_uv B_uv^RL,

⟨h|Z^˜uYv⟩ =−ih^RL_uv D_uv^RL,

⟨h|Z^˜uZv⟩ =h^RL_uv E_uv^RL,

⟨h|_Ψ^˜2_uΨ¹_v⟩ =−h_uv^RL F_uv^RL.

(3.85)

And for the massless block we have:

⟨h|χ^A_u^˙χ_v^B˙⟩ =h^◦◦_uv A^◦◦_uv,

⟨h|χ^A_u^˙χ˜_v^B˙⟩ =−ih^◦◦_uvB^◦◦_uv,

⟨h|T_u^A1˙ T_v^B2˙ ⟩ =ih^◦◦_uvC^◦◦_uv,

⟨h|χ˜_u^A˙χ^B_v^˙⟩ =−ih^◦◦_uvD_uv^◦◦,

⟨h|χ˜_u^A˙χ˜^B_v^˙⟩ =h^◦◦_uv F_uv^◦◦,

⟨h|T_u^A2˙ T_v^B1˙ ⟩ =−ih^◦◦_uvC_uv^◦◦.

(3.86)

Note here that these are blind tosu(2)◦index just like the S-matrix. This similarity of the tensor structure of the hexagon to thepsu(1|1)^⊕2S-matrix is not accidental since this is the only rapidity dependent two-particle tensor invariant under this algebra. Therefore this is the only quantity that could appear here.

Unfortunately the symmetry axiom does not fixesn-particle hexagons, however thispsu(1|1)^⊕2 S-matrix structure hints at an underlying structure for the two-particle hexagon that will allow generalization later. The main argument here is that we can understand the hexagon as a scattering of thepsu(1|1)^⊕2 particles composing the excitations. Indeed, we can write a genericpsu(1|1)^⊕_c.e.⁴excitation as

|_Θ^a_u^a˜⟩=|ξ^a_u⊗ξ^a_u^˜⟩where|ξ_u^a⟩are simply states transforming in theρ_L,ρ_R,ρ◦ and ρ^′_◦ representations ofpsu(1|1)^⊕2. Then the two-particle hexagon can compactly be written as

⟨h|_Θ^a_u^a˜_Θ^b_v^b˜⟩ = (−1)^(Fa+Fa˜)FbK_uK_v|ξ_u^aξ^b_v⟩ ⊗S₍₁₂₎|ξ^a_u^˜ξ_v^b˜⟩, (3.87) where S₍₁₂₎ is the psu(1|1)^⊕2 S-matrix that acts on the right factor composing the excitations andK_u is a contraction operator that acts on |ξ^a_u⟩ and |ξ_u^a˜⟩ and yields⟨h|_Θu^a˜a⟩,id est, it contracts the states and yields a scalar¹⁰. AlsoFa is the fermion number, that is, it is 0 or 1 if|ξ^a_u⟩ is a boson or a fermion, respectively.

Note that in thepsu(1|1)^⊕2 S-matrix we should substituteΣ^∗∗uv → h^∗∗_uv to obtain the two-particle hexagon. We can graphically understand expression (3.87) as exemplified in Figure3.8. One can check then that (3.87) correctly reproduces all the two-particle hexagon form factors and that we can similarly write the mixed mass form factors which are just the massless limits of (3.82) - (3.86). Now we

10This contraction operator was first introduced in the AdS3/CFT2 context. This was not necessary in theAdS5/CFT4case, however here we have fermionic one-particle form factors that do not vanish. Thus we need it to properly account for the statistics of the excitations which is nicely handled by these contraction operators.

Figure 3.8: two-particle hexagon computation. We show here how to compute the form factor⟨h|YuZv⟩. We first denote the excitations on the hexagon as|ξ_u^a⊗ξ^a_u^˜⟩. Then we move the right factors of the tensor product to the right to scatter them.

Here we may have minus factors that we gain when we permute two fermions in the process of moving the particles. After the scattering we contract the states.

In this example we note that the contractions in the first term in the second line yields ⟨h|Yu⟩⟨h|Zv⟩ and the second one we obtain ⟨h|_Ψ¹_u⟩⟨h|_Ψ²_v⟩, thus only the former survive and we obtain the correct two-particle hexagon form factor. All this graphic procedure is encapsulated in (3.87).

move to the next hexagon property.

Watson axiom

Another property that the hexagon should satisfy is the Watson axiom, before we describe it we should first set some notation about the hexagon’s scalar factors that multiply each block just as we did for the dressing factorsΣ^∗∗_uv. At first we have 9 independent functions that are

h_uv^LL, h_uv^RR, h_uv^LR, h^RL_uv, h^∗◦_uv, h^◦∗_uv, and h^◦◦_uv,

with∗= L,R. UsingL/R-symmetry we can reduce the massive part of the scalar factors to only two independent functionsh^••_uvand ˜h^••_uv, where these are related to the previous factors by

h^LL_uv =h_uv^RR=h^••_uv, h^LR_uv =h^˜••_uv,

h^RL_uv = s

x⁻u x⁻v

x⁺_u x⁺_v

1−x_u⁺x_v⁺ 1−x_u⁻x_v⁻h˜^••_uv.

(3.88)

From now one we use the notationh^∗◦_uv=h^•◦_∗,uvandh^◦∗_uv =h^◦•_∗,uvfor the mixed mass terms.

Given these definitions we can properly write the Watson axiom¹¹. This prop-erty tell us that when we exchange two particles in a hexagon form factor we pay a S-matrix. This can be stated as

⟨h|_Xa1,u1· · ·_{Xaj+1,uj+1Xaj,uj}· · ·_Xan,un⟩ =⟨h|S_(j,j+1)(uj,uj+1)|_Xa1,u1· · ·_Xan,un⟩_, (3.89) withS_(j,j+1)(u_j,u_j+1)being thepsu(1|1)^⊕_c.e.⁴S-matrix that acts on particlesjandj+1 only. One can see that the matrix part of the two-particle hexagon automatically satisfies this axiom, however it imposes conditions on the scalar factors. Indeed we have:

h^••_uv

h^••_vu = (_Σ^••_uv)², h˜^••_uv

h˜^••_vu = _Σ^˜••_uv²_, h^◦◦_uv

h^◦◦_vu =−(_Σ^◦◦_uv)²,

h^•◦_∗,uv

h^◦•_∗,vu = _Σ^•◦_∗,uv², h^◦•_∗,uv

h^•◦_∗,vu = _Σ^◦•_∗,uv².

(3.90)

Where the mixed massΣ^•◦∗,uvandΣ^◦•∗,uvare defined in a similar way of the scalar factors. Note that for the massless scalar factor we have an extra minus sign. This is due to the fact that the highest weight particles in the massless supermultiplets are fermionic, so when we exchange them we gain a minus sign. These conditions together with the next axioms allow us to fix these scalar factors in terms of the dressing phases of the S-matrix. The Watson axiom is represented graphically in Figure3.9.

Decoupling axiom

Another condition we impose on the hexagons is the decoupling property. This is heavily inspired and based on decoupling condition of the S-matrix, which we describe now. Consider a scattering event where we have two particles that are the crossing pair of each other, we denote them by|Xu⟩and |X^¯_v2γ⟩. If we take the limitv →u, the composite state of these particles forms|1u⟩which is called singlet. This state scatters trivially with all particles since, as the name suggests,

11The name Watson axiom stems from a similar axiom in the form factor program in two-dimensional integrable quantum field theories, for more on these see [29,59].

Figure 3.9: Watson axiom. Here we represent the Watson axiom, to permute the excitations with rapiditiesuandvwe concatenate apsu(1|1)^⊕_c.e.⁴S-matrixS(u,v) with the hexagon.

it is a singlet¹²ofpsu(1|1)^⊕_c.e.⁴[107,117]. This state has null energy and momenta and carries no R-charge. Since the singlet has trivial scattering with other states it decouples in any scattering event. We apply the same idea now to the hexagon and we have the decoupling property:

Resv→u⟨h|Xa,uX¯_a,v2γXa₁,u₁· · ·Xa_n,u_n⟩ = ¹

µXa(u)⟨h|Xa₁,u₁· · ·Xa_n,u_n⟩_{, (3.91)} where µXa(u) is themeasurefactor associated with particle |Xa⟩. Since we have

⟨h|_∅⟩ =1, the measure is defined as

Resv→u⟨h|Xa,uX¯_a,v^2γ⟩ = ¹

µXa(u)^{. (3.92)}

This object will be central in our discussions of finite length corrections in Chapter 4, so we will abstain on comment it for now. Note that this decoupling property is the one that gave the simple poles to the tree level hexagons discussed previously in Chapter2. We represent the decoupling property in Figure3.10.

From the decoupling property we obtain functional equations relating distinct hexagon scalar factors. Indeed we obtain the following equations for the massive

12If we want to be more specific, the singlet is a linear combination of crossing pairs of the spectrum with some extra insertions calledZ-markers. Note that crossing symmetry is a QFT concept, there is no analog in the spin chain picture of the integrable model. The role of the singlet and the decoupling condition is to implement crossing symmetry in the S-matrix of the spin chain picture.

Figure 3.10: Decoupling axiom. We show here the decoupling axiom, when the rapidity of a crossing pair is equal we have a pole whose residue is the inverse of measure factor times the hexagon without the crossing pair.

and massless factors

h^••_uvh˜^••_u2γv = s

x⁺u

x⁻u

x⁻_u −x⁻_v x⁺u −_x⁻_v ^, h^◦◦_uvh^◦◦_u2γv = ^xv−xu

1−xuxv.

(3.93)

Note that these equations are very similar to the crossing equations for the dressing phases [40,109]. Therefore the solutions for the scalar factors will be written in terms of these phases. Another hint that this is the case is that these scalar factors have nontrivial monodromy just likeΣ^∗∗_uv. Now we move to the last hexagon axiom we will impose.

Monodromy axiom

We defined the hexagon form factor for a single excited operator, that is we put excitations only in a single physical edge. However we can construct a gen-eral hexagon configuration from analytic continuations of the one we defined previously. We define a jump of an excitation to a neighbor edge in the clockwise direction as a mirror transformation on the excitation [57,74]. Therefore to jump one edge we do a mirror transformation (u→u^γ), two edges a crossing transfor-mation (u → u^2γ), and so on. Similarly we can do counterclockwise moves by the inverse transformations. Therefore we can move excitations from the other operators by analytical continuations in the rapidity and put all excitations on a single edge. An example of this procedure is shown in Figure3.11.

With this we impose the monodromy axiom. This is given by

⟨h|Xa₁,u₁· · ·Xan,unXa,u⟩ =⟨h|X^¯_a,u^6γXa₁,u₁· · ·Xan,un⟩, (3.94)

Figure 3.11: Analytic continuations in the hexagon. This is an example of how can we move excitations of the general excited hexagon to obtain a simpler hexagon with excitations only in a single edge using the rules described here. Note that there is not an unique way of doing these moves, we could do it in another way which would yield the same result.

where ¯Xa,v denotes the crossing pair of particleXa,v. We can understand this as the invariance of the hexagon if we take an excitation and move it clockwise back to the same edge we started it. This is represented in Figure3.12. Just as the decoupling and Watson axioms this imposes functional constraints in the scalar factors. Consider for example the massive scalar factors, the monodromy axiom yields

h^••_uv = s

x⁻v

x⁺_v

x⁻_u −x⁺_v

x⁻_u −x⁻_v h˜^••_v6γu. (3.95) We can obtain similar constraints for the remaining scalar factors. This is not the only constraint coming from analytical continuations. We have that the labels of the operators are not special, therefore we can cyclically rotate the hexagon and obtain the same form factor. This gives us the following constraint for the massive scalar factors

h^••_uv=h^••_u2γv^2γ. (3.96) With similar relations for the other scalar factors. This cyclic invariance is shown in Figure3.12. Combining the Watson, decoupling and monodromy axioms we can completely fix the scalar factors in terms of the dressing phases as we will show below.

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