S-matrix

-10 -5 0 5 10 1.5

2.0 2.5 3.0 3.5

u E˜ Q(u)

(a)

-10 -5 0 5 10

-10 -5 0 5 10

u p˜ Q(u)

(b)

Figure 3.4: Energy and momentum of a mirror massive excitation. We show here the energy (a) and momentum (b) of a massm =2 mirror excitation at coupling h=π²/6. The mirror momentum is non-periodic now and the mirror energy is positive.

one for the mirror theory. Indeed consider E˜(u) = −log

x(u)², (3.51)

˜

p(u) = −h

x(u)− ¹ x(u)

, (3.52)

therefore the region where these satisfy the physical requirements is|_u| ≥ _{2. But} with the care that we are below the cuts, since this is the region where ˜E(u) _is positive as shown in Figure3.5. So the mirror transformation for massive and massless excitations is defined as:

Mirror: x⁺ → ¹

x⁺ for|m| ≥ 1, u ∈ [−2, 2]→u+i0⁻, |u| ≥2 form =0.

This is the mirror kinematics that will be central to define the mirror corrections of the hexagon in Chapter4. Note that for bound states the crossing and mirror transformations are similarly defined.

two--4 -2 0 2 4 0.0

0.5 1.0 1.5 2.0 2.5 3.0

u E˜ o(u)

(a)

-4 -2 0 2 4

-5 0 5

u p˜ o(u)

(b)

Figure 3.5: Energy and momentum of a massless mirror excitation. Here we have the energy (a) and momentum (b) of a massless mirror excitation. The mirror momentum is non-periodic now and the mirror energy is positive in the region where|u| ≥2 and we are below the cuts.

dimensional worldsheet described by the closed string of lengthl. Due to the finite length, the scattering of string modes is not well defined since there are no asymptotically separated states. Therefore we consider thedecompactification limitof l → ∞. In this regime we can properly define the S-matrices and the momentum are quantized by the Bethe equations. Remember thatl corresponds to the R-charge (that is length) of the pseudo-vacuum state in the spin chain language of the dual CFT2, therefore the spectrum is described by the algebraic Bethe ansatz only for asymptotically large chains. Finite length corrections enter through the thermodynamic Bethe ansatz procedure and, for massive states at least, are exponentially suppressed inl⁷[42,43].

Then given the existence of the S-matrix in this decompactification limit let us describe it. We will focus on the description of thepsu(1|1)^⊕2S-matrix, since to find the fullpsu(1|1)^⊕_c.e.⁴ invariant S-matrix of the model we simply take the tensor product of the former. We have four two-dimensional supermultiplets {ρL,ρR,ρ◦,ρ^′_◦}which means that our two-particle state has dimension 64. There-fore thepsu(1|1)^⊕2S-matrix has dimension 64×64, that is, it has 4096 elements.

The number of non-zero components drastically reduces since the S-matrix should satisfy some physical constraints, these are:psu(1|1)^⊕2invariance, braiding uni-tarity, unitarity and crossing invariance [33,38,103]. LetS₍₁₂₎(u,v)be the 2→₂ S-matrix that acts on excitations 1 and 2 with rapidityuandv, respectively. Then

7For massless modes we have that the wrapping corrections to the spectrum enters at the same order of the ABA [44].

braiding unitarity and unitarity are given by

S₍₁₂₎(u,v)S₍₂₁₎(v,u) =_I and S₍₁₂₎(u,v)S₍₁₂₎(u,v)^† =_I, (3.53) respectively. We will not describe here crossing invariance to not enter in some specific details that will not be of use for us, however the reader can consult [33,38,103]. Now the most important constraint ispsu(1|1)^⊕2invariance. For the 2→2 S-matrix it reads as

S₍₁₂₎(u,v)Q₍₁₂₎(u,v) = Q₍₁₂₎(v,u)S₍₁₂₎(u,v), (3.54) whereQ₍₁₂₎(u,v)is a charge ofpsu(1|1)^⊕2 that acts on a two-particle state com-posed with particles of rapidities uandvgiven in (3.14). Note that (3.54) is not exactly a commutation relation since we exchanged the momentum of particles when we act with the S-matrix on the state first. This is due to the fact that the S-matrix exchanges the momentum of the particles in the asymptotic outgoing state.

From the transformation rules (3.29) - (3.32) of psu(₁|₁)^⊕2 we see that the charges map a representation to itself, therefore the 2 → 2 S-matrix splits into blocks of defined chirality like LL → LL, LR → LRandRR → RRfor example.

This makes many elements in the S-matrix to automatically vanish. Remarkably due to the nontrivial two-particle representations of the particles, modified co-product induced by the off-shell central extension, the entire matrix structure of the 2 → 2 S-matrix is fixed by symmetry alone. The only unknown factors are scalar functions that multiply each block of defined chirality, these are the so-called dressing factors. We will briefly comment on these later. We can compare this with the AdS₅×S⁵case for instance. There we have that the residualpsu(₂|₂)_c.e.also fixes the matrix part of the S-matrix, however we have a single dressing factor (the BES dressing phase) [107]. This is tied to the fact that we have only a single irreducible representation there, but here we have four irreducible representa-tions ofpsu(1|1)^⊕2which implies that we have four dressing factors, at least after identification under theL/R-symmetry.

The first block of the S-matrix that we show is theLLone. Here we follow the

conventions of [74]. We have the following non-zero S-matrix elements:

S₍₁₂₎|_ϕ^B_uϕv^B⟩ = A_uv^LL|_ϕ^B_vϕu^B⟩_,

S₍₁₂₎|ϕ^B_uφ^F_v⟩= B^LL_uv|φ^F_vϕ_u^B⟩+C_uv^LL|ϕ_v^Bφ^F_u⟩, S₍₁₂₎|_φ^F_uϕ^B_v⟩= D_uv^LL|_ϕ^B_vφ^F_u⟩+E_uv^LL|_φ^F_vϕu^B⟩, S₍₁₂₎|φ^F_uφ^F_v⟩ =F_uv^LL|φ_v^Fφ_u^F⟩.

(3.55)

Where each coefficient is given by:

A_uv^LL=1, B_uv^LL =

s x⁻u

x⁺_u

x⁺_u −_x⁺_v x⁻_u −x⁺_v , C_uv^LL =

s xu⁻xv⁺

xu⁺xv⁻

ηu

ηv

x⁻_v −x_v⁺ x⁻u −xv⁺

,

D_uv^LL = s

x⁺v

x⁻_v

x⁻_u −x⁻_v x⁻_u −x⁺_v , E_uv^LL =C_uv^LL,

F_uv^LL =− s

x⁻ux⁺v

x⁺ux⁻v

x⁺_u −x⁻_v x⁻u −x⁺v

.

(3.56)

Note here two things. First, we have an extra sign inF_uv^LLdue to the permutation of fermions. Second, we have here only the matrix part of theLLblock, each factor is then multiplied by the dressing factorΣ^LL_uv.

Another block of the S-matrix is theRRone. The following S-matrix elements are non-zero:

S₍₁₂₎|ϕ_u^Fϕ_v^F⟩= A_uv^RR|ϕ^F_vϕ^F_u⟩,

S₍₁₂₎|ϕ_u^Fφ_v^B⟩ =D_uv^RR|φ^B_vϕ_u^F⟩+E_uv^RR|ϕ_v^Fφ_u^B⟩, S₍₁₂₎|_φu^B_ϕv^F⟩ =B_uv^RR|_ϕv^F_φ^B_u⟩+C_uv^RR|_φ^B_vϕu^F⟩_, S₍₁₂₎|φ_u^Bφ^B_v⟩ = F_uv^RR|φ^B_vφ^B_u⟩.

(3.57)

Which each element being:

A^RR_uv =1, B_uv^RR=

s x⁻u

x⁺_u

x⁺_u −x_v⁺ x⁻_u −x_v⁺, C^RR_uv =

s x⁻_ux⁺_v x⁺ux⁻v

ηu

ηv

x⁻_v −x⁺_v x⁻u −x⁺v

,

D^RR_uv = s

x⁺v

x⁻_v

x⁻_u −x⁻_v x⁻_u −x⁺_v , E_uv^RR =C^RR_uv,

F_uv^RR = s

x⁻_ux⁺_v x⁺ux⁻v

x⁺_u −x⁻_v x⁻u −x⁺v

.

(3.58)

Note here that inF_uv^RRthe minus sign was removed because the statistics of the particles changed. Also note theL/R-symmetry in action since these coefficients are the same as in theLL block. As before each coefficient is multiplied by the

dressing factorΣ^RRuv.

We also have the mixedLRand RLblocks. For theLRscattering we have the coefficients

S₍₁₂₎|_ϕu^B_ϕv^F⟩=C_uv^LR|_ϕv^F_ϕu^B⟩,

S₍₁₂₎|ϕ_u^Bφ_v^B⟩ =A^LR_uv|φ^B_vϕ_u^B⟩+B_uv^LR|ϕ_v^Fφ^F_u⟩, S₍₁₂₎|φ^F_uϕ_v^F⟩ =E_uv^LR|ϕ_v^Fφ_u^F⟩+F_uv^LR|φ^B_vϕ_u^B⟩, S₍₁₂₎|_φ^F_uφ^B_v⟩ =D_uv^LR|_φ^B_vφu^F⟩_.

(3.59)

Now for theRLwe have

S₍₁₂₎|ϕ_u^Fϕ_v^B⟩=D^RL_uv|ϕ^B_vϕ_u^F⟩,

S₍₁₂₎|_ϕu^F_φ^F_v⟩ =E^RL_uv|_φv^F_ϕu^F⟩+F_uv^RL|_ϕ^B_vφ^B_u⟩, S₍₁₂₎|φ^B_uϕ_v^B⟩ =A^RL_uv|ϕ^B_vφ_u^B⟩+B_uv^RL|φ^F_vϕ_u^F⟩, S₍₁₂₎|φ^B_uφ_v^F⟩ =C_uv^RL|φ_v^Fφ_u^B⟩.

(3.60)

So for theLRblock the S-matrix elements are simply

A^LR_uv = s

x⁻_u x⁺u

1−x_u⁺x_v⁻ 1−xu⁻xv⁻

,

B_uv^LR = s

x⁻uxv⁻

x⁺_ux_v⁺ 2i

h

ηuηv

1−x⁻_ux⁻_v , C_uv^LR =_1,

D_uv^LR = s

x⁻_u x⁻_v x⁺u x⁺v

1−x⁺_ux_v⁺ 1−x⁻uxv⁻

,

E_uv^LR =− s

x⁻v

x⁺_v

1−x_u⁻x⁺_v 1−x_u⁻x⁻_v , F_uv^LR =−B_uv^LR.

(3.61)

And the coefficients in theRLblock are related to theLRones by braiding unitarity, therefore

A_uv^RL = s

x⁺u

x⁻_u

1−x⁺_u x⁻_v 1−x⁺_u x⁺_v , B_uv^RL = ²ⁱ

h

ηuηv

1−x⁺_ux⁺_v , C_uv^RL =

s x⁺u x⁺v

x⁻u x⁻v

1−x⁻_u x⁻_v 1−x⁺u x⁺v

,

D_uv^RL =1, E^RL_uv =−

s xv⁺

xv⁻

1−x⁻_ux⁺_v 1−_x⁺_ux⁺_v ^, F_uv^RL =−B_uv^RL.

(3.62)

Again each block is multiplied by its dressing factor, with these being Σ^LR_uv and Σ^RL_uv.

Now for the massless-massless scattering we consider it as the massless limit

of theLLorRRblocks. Remember that this limit is defined as:

x⁺(u) →x(u) and x⁻(u) →1/x(u). (3.63) There is one small caveat in taking this limit. We have to consider the change in the excitations’ statistics that may happen when going fromρL toρ◦orρ^′_◦. Indeed theρ◦andρ◦scattering is exactly the same as the LLblock in (3.56), however for theρ^′_◦andρ^′_◦ we have

S₍₁₂₎|_ϕ^F_◦,uϕ◦^F_,v⟩ =−A^LL_uv|_ϕ◦^F_,vϕ◦^F_,u⟩,

S₍₁₂₎|ϕ^F_◦,uφ^B_◦,v⟩ =B_uv^LL|φ_◦^B_,vϕ^F_◦,u⟩+C_uv^LL|ϕ_◦^F_,vφ^B_◦,u⟩, S₍₁₂₎|φ^B_◦,uϕ_◦^F_,v⟩ =D_uv^LL|ϕ_◦^F_,vφ^B_◦,u⟩+E_uv^LL|φ^B_◦,vϕ_◦^F_,u⟩, S₍₁₂₎|_φ^B_◦,uφ^B_◦,v⟩=−F_uv^LL|_φ^B_◦,vφ◦^B_,u⟩_,

(3.64)

note the highlighted minus signs incyandue to the change in statistics. Also note that the we need to take the massless limit in the coefficients. For example, this limit for theLLcoefficients is just

A_uv^LL=1, B_uv^LL = ^xu−xv

1−xuxv, C_uv^LL = ^η◦,u

η◦,v

1−_xv² 1−xuxv,

D_uv^LL = ^xv−_xu 1−xuxv, E_uv^LL=C_uv^LL, F_uv^LL =1.

(3.65)

We can similar define the scattering ofρ◦ and ρ^′_◦ modes. Now to compute the scattering of massless and massive modes we simply consider the massless limit in LL, LRorRLblocks, appropriately taking care of the signs coming from statistics.

The dressing factors are given by solutions of the constraints coming from crossing. Before we show the solutions explicitly we useL/R-symmetry to reduce the number of independent dressing factors. For the massive modes we can write all dressing factors as a function of only two denoted byΣ^••_uvand ˜Σ^••_uv. Indeed we have:

Σ^LL_uv =_Σuv^RR=_Σ^••_uv, Σ^LR_uv =_Σ^˜••_uv,

Σ^RL_uv = s

x⁻_u x⁻_v x⁺u x⁺v

1−x_u⁺x_v⁺ 1−xu⁻xv⁻

Σ˜^••_uv.

(3.66)

The solutions for these and the massless dressingΣ^◦◦uvare⁸[74]

(Σ^••_uv)² = ¹ (σ_uv^••)²

x⁺_u x⁻_v x⁻u x⁺v

2

x⁻_u −x_v⁺ x⁺u −xv⁻

1−x⁻_ux⁺_v 1−x⁺ux⁻v

, Σ˜^••_uv2

= ¹

(σ˜_uv^••)² x⁺_u

x⁻u

2

x⁻_v x⁺v

1−x⁻_u x⁻_v 1−x⁺u x⁺v

1−x⁻_u x⁺_v 1−x⁺u x⁻v

, (_Σ^◦◦_uv)² = ^x

2v

(σ_uv^◦◦)^2,

(3.67)

whereσ_uv^••, ˜σ_uv^••andσ_uv^◦◦ are phases analogous to the BES dressing phase of AdS5× S⁵. To find these, one solves a functional equation coming from crossing symmetry.

Note however that these are not rational functions of the Zhukovsky variables, in fact they possess a more complicated analytic structure consisting of an infinitely sheeted Riemann surface instead of simple square root branch cuts induced by x^±, just as the AdS5×_S⁵ case [111]. However there are some unusual features of these with respect to AdS₅×S⁵, one of them is that the one-loop result in the semiclassical string expansion (the so-called HL term) is not universal but the tree level term remains universal (AFS term) [112]. This is unusual since for AdS₄/CFT3andAdS5/CFT₄both terms are equal, here we need to consider the productσ_uv^••σ˜_uv^•• to obtain the same HL term as in these higher dimensional dualities. Another unusual feature is that the weak coupling expansion in the integrability couplinghstarts atO(h)(at least for the proposal in [109]), contrast this with the first nontrivial term in the BES phase starting atO(h⁶)[111]. This means that nontrivial contributions to the spectrum enters much sooner in the h-expansion for AdS₃×S³×T⁴.

One tentative to find the phases σ^∗∗(u,v) is in [109] for massive dressing phases and [40] for the massless ones, however there are some problems with this proposed solution. One of them is that the dispersion relation for massless modes differs from the semiclassical string result at two-loops. Another one is that some phases, likeσ^••(u,v)for example, do not match the semiclassical string result at higher orders. Another proposal was recently made in [48]. It solved some issues of the previous proposal, however the problems of incompatibility with the semiclassical string result still there. Although we can not specifically pinpoint the origin of the discrepancies, a possible hint of their origin may be the massless modes. A problem for the future could be to clarify what exactly do we mean by the massless modes’ S-matrix and how to properly renormalize the string

8For the dressing phases of the remaining S-matrices the interested reader can consult [74].

perturbation theory results to then make further progress in these matters. Turns out that for our purposes we do not need the full form of these dressing phases, we will use only their analytical properties which any proposed solution must possess.

The reader can worry that we fixed only the 2 →2 S-matrix, what about the fulln→mone? Remember from the Introduction (Chapter1) that in a IQFT the S-matrix satisfies the Yang-Baxter equation and there is no particle production. This means that anyn →nprocess can be written as a sequence of 2→2 scatterings (see Figure1.2). Therefore by fixing the 2 → 2 matrix we fixed the entire S-matrix of the model. Note here the distinction to theO(N)case discussed in the Introduction (Chapter1), there we had that symmetry did not fix the S-matrix entirely. But here our symmetry algebra is so constraining that entirely fixes the S-matrix up to phases and we can check that it satisfies the Yang-Baxter equation.

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