Abstract
In this chapter we shift our focus to three point functions. The first and second section are a review of two computational methods: the Tailoring procedure, which provides the weak-coupling limit of the three point function using a spin-chain construction, and the Hexagon proposal, which provides us the all-loop three point function. The last section is devoted to the rewriting of the hexagon proposal in a language more resembling the algebraic Bethe ansatz by means of Zamolodchikov-Faddeev operators.
I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler.
Albert Einstein The Special and the General Theory-A Clear Explanation that Anyone Can understand [1]
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Notes
- 1.
These lengths \(l_{ij}\) are usually called bridges and count the number of Wick contractions between operator \(\mathcal {O}_i\) and \(\mathcal {O}_j\). We must impose this positivity condition because the operator mixing between single-trace operators and double-trace operators is suppressed by a color factor \(\frac{1}{N_c}\) (and therefore does not need to be considered) when the bridge lengths are non-vanishing. These operators are called non-extremal, in contraposition with extremal operators, characterized by having one vanishing bridge length.
- 2.
We will sometimes use a and \(\alpha \) instead of A when we want to differentiate between bosonic an fermionic indices.
- 3.
The explicit expression for this vertex has not been explicitly computed previously, being that the aim of this chapter.
- 4.
Actually we should write \(\otimes ^6 \left| 0 \right\rangle \) and specify in which of the vacuums we apply our operators. However, as we are going to work mostly with the canonical hexagon, all operators will be applied to the same edge. Hence, instead of writing the whole vacuum hexagon, we are going to write only the edge in which all operators will act and drop the label on the operators to alleviate notation.
- 5.
One way to formally do this breaking is to introduce two non-dynamical fields \(\chi _A\) and \(\chi _{\dot{A}}\) with a coupling with the original operators given by \(\exp \left[ \chi _A (u) \otimes \chi _{\dot{A}} (u) \otimes Z_{A \dot{A}} (u) \right] \).
- 6.
Of which we are only going to conserve the two edges on which the operators are going to act, \(\left| 0 \right\rangle _L \otimes \left| 0 \right\rangle _R \), to alleviate notation.
- 7.
Note that, as \(p(u^{-2\gamma })=-p(u)\), the interpretation of this vertex as a boundary operator becomes very appealing.
- 8.
A couples of reasons for that statement are that equal rapidities do not yield proper Bethe wavefunctions [24] and that strings (in the thermodynamical limit the solutions of the Bethe equation cluster around lines called strings) with two equal rapidities have no weight in equilibrium problems [25]. Appart from that, ZF operators with equal rapidities behave like fermionic operators if the S-matrix behaves like \(S_{AB}^{CD} (u,u)=-\delta ^C_ A \delta ^D_B\), which is our case.
- 9.
The operators they propose do not exactly form a ZF algebra because the commutation relations with the annihilation operators \(E_i (z)\) are no exactly the correct ones, as they have an extra Cartan generating function. Note that their definition of Cartan generating function is different from ours, being the equivalence between notations \(H_i (u)=k_{i+1}^+ (u) [k_i^+ (u)]^{-1}\). However the algebra generated by \(\left( H^{-1}_i(u) E_i (u) , F_j (v) \right) \) actually form a ZF algebra. The same happens with the algebra generated by \(\left( E_i (u) , F_j (v) H^{-1}_j(v) \right) \).
- 10.
Note that this coproduct only works for the canonical S-matrix. A coproduct for the Beisert S-matrix would need more structure.
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Nieto, J.M. (2018). Tailoring and Hexagon Form Factors. In: Spinning Strings and Correlation Functions in the AdS/CFT Correspondence. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-96020-3_7
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