Hexagons and Correlators in the Fishnet Theory
HHexagons and Correlators in the Fishnet Theory
Benjamin Basso a , Jo˜ao Caetano a,b,c , Thiago Fleury a,da Laboratoire de Physique Th´eorique de l’ ´Ecole Normale Sup´erieure, CNRS, Universit´e PSL, SorbonneUniversit´es, Universit´e Pierre et Marie Curie, 24 rue Lhomond, 75005 Paris, France b C. N. Yang Institute for Theoretical Physics, SUNY, Stony Brook, NY 11794-3840, USA c Simons Center for Geometry and Physics, SUNY, Stony Brook, NY 11794-3636, USA d International Institute of Physics, Federal University of Rio Grande do Norte, Campus Universit´ario,Lagoa Nova, Natal, RN 59078-970, Brazil d Instituto de F´ısica Te´orica, UNESP - Univ. Estadual Paulista, ICTP South American Institute forFundamental Research, Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, S˜ao Paulo, SP, Brazil
Abstract
We investigate the hexagon formalism in the planar 4d conformal fishnet theory.This theory arises from N = 4 SYM by a deformation that preserves both conformalsymmetry and integrability. Based on this relation, we obtain the hexagon form factorsfor a large class of states, including the BMN vacuum, some excited states, and theLagrangian density. We apply these form factors to the computation of several corre-lators and match the results with direct Feynman diagrammatic calculations. We alsostudy the renormalisation of the hexagon form factor expansion for a family of diagonalstructure constants and test the procedure at higher orders through comparison witha known universal formula for the Lagrangian insertion. a r X i v : . [ h e p - t h ] D ec ontents Introduction
The conformal fishnet theory [1–3] may well be the simplest interacting CFT in higherdimensions that is integrable in the planar limit. Defined as the extreme limit of a twistedversion [4–7] of the 4d maximally supersymmetric Yang-Mills theory ( N = 4 SYM), thetheory is minimalistic, but still highly nontrivial. It counts only two complex scalar fieldsand a single quartic coupling, L int = g tr φ † φ † φ φ , (1.1)with the fields filling N × N matrices. It depends, in the planar limit N → ∞ , on a singlemarginal coupling g , much like N = 4 SYM, if not that here double-trace deformations mustbe switched on and finely adjusted to maintain criticality [8, 9]. The theory lacks unitaritybut serves nonetheless as a natural stage for a broad family of perfectly meaningful conformalFeynman integrals, the fishnet graphs. These diagrams host one of the first observedmanifestations of integrability in higher dimensions [10] and, although very special, theygive us a hint at the remarkable mathematical structures that underlie Feynman integralsin general, see e.g. [11–20]. They also form an irreducible subset of the conformal integralsneeded to span correlators and amplitudes in general perturbative CFTs, and in N = 4SYM in particular, see e.g. [17, 19, 20].The integrability of the fishnet theory is not as mysterious as in its supersymmetricparent. It traces back to the properties of the quartic coupling and links directly to thedynamics of non-compact conformal spin chains [10, 14, 21]. Fishnet theories, in general,offer a natural setting for discussing the integrability of these non-compact magnets, ina field theoretical language, and expressing their remarkable properties, at the Feynmandiagrammatic level. They are also intimately tied to integrable non-compact sigma models[22], in the graph thermodynamic limit [10], offering new perspectives on the problem oftheir quantization. Last but not least, fishnet theories form a laboratory for experimentingthe techniques put forward for computing correlation functions and scattering amplitudes atfinite coupling in more sophisticated integrable theories, like N = 4 SYM, see e.g. [23–32].In this paper, we will apply one of these techniques - the hexagon factorisation - to thecorrelation functions and Feynman integrals of the fishnet theory. The method was firstdeveloped for computing structure constants in N = 4 SYM [24] and was later on upgradedto encompass higher-point functions [26, 27] and non-planar corrections [29, 30]. Althoughthe hexagon framework has been fairly tested, see e.g. [33–41, 19, 20], it is still far frombeing a well-oiled machinery and remains limited in some of its applications. The problemis partly due to the nature of the approach, which builds on a form-factor decompositionand requires that complicated sums and integrals over all the magnonic states be taken tonon-perturbatively recover the original observable. Progress with the hexagon formalismis also hindered by the need of renormalising the divergences that show up at wrappingorders [42], when the magnons can circulate around a (non-protected) local operator. Todate, no systematic removal of these divergences is known and it is challenging to push thehexagon strategy to higher loops in N = 4 SYM, even for the simplest structure constant,with one non-protected and two half-BPS operators, see [43–45, 41] for the state of the arton the field theory side. 3he fishnet theory appears as an interesting playground to address these issues. Forinstance, the simplest structure constants of the fishnet theory are all about wrappingcorrections, exposing the problem in its minimal form. Moreover, the ingredients enteringthe integrability framework acquire a direct diagrammatic meaning in the fishnet theory,a feature which helps testing their correctness. We will substantially benefit from thisgraphical intuition, in this paper. It will allow us, for instance, to fill a gap in the hexagonapproach and incorporate the “dilaton” (1.1) in its dictionary. Interest in this operatorstems from its relation to the coupling dependence of the Green functions. Its insertion in apair of conjugated operators, for instance, is fixed in terms of the spectral data [46], offeringa mean of testing the ability of the hexagon method at encoding the scaling dimensions ofthe theory.The main outcome of this paper is a proposal for a large class of hexagon form factors ofthe fishnet theory, applicable to a variety of states, including the BMN vacuum, in the SYMterminology. Our formulae can be understood as a projection to the fishnet theory of theconjectures pushed forward for the SYM theory. We will subject them to a series of tests,by means of comparison with diagrammatic computations in the fishnet theory, and willobtain, on the way, a few predictions for a certain class of three-point Feynman integrals.Finally, we will test the hexagons’ aptitude at reproducing the scaling dimension of theBMN vacuum by considering diagonal structure constants with a Lagrangian insertion. Tothis end, we will generalise the renormalisation procedure put forward in [42] and derive,in a particular regime, an all order representation using the Leclair-Mussardo formula [57].We will verify the renormalised expansion so-obtained up to NNLO by a comparison withthe Thermodynamical Bethe Ansatz (TBA) equations.The paper is structured as follows. In Section 2, we briefly recap the ingredients enteringthe hexagon program and detail the approach we shall follow to obtain their counterpartsin the fishnet theory. In Section 3, we perform several classic tests of our hexagon formfactors through the computation of correlators, including some with excited states. InSection 4, we discuss more advanced applications to a family of diagonal structure constants,mostly focusing on the Lagrangian insertion and its higher-charge siblings. We conclude inSection 5. The details omitted in the main text are presented in several Appendices. In this paper, we will analyse planar correlators in the fishnet theory using the hexagonfactorisation. The prototype is the three-point function between a conjugate pair of BMNvacua and a third operator. The former are vacuum states in the spin-chain picture andcan be chosen as O = tr φ L , O = tr φ † L , (2.1)where the traces run over the color degrees of freedom; they have minimal dimensions ∆ , given their U (1) charges, i.e., spin-chain lengths L , . The third operator is designed suchas to permit contractions with both operators in the pair. In N = 4 SYM, we can pick yet4nother BMN vacuum, by rotating the fields in (2.1) using an SO (6) transformation, andwork with e.g. φ (cid:48) = φ + φ † + φ i − φ † i ⇒ O = tr ( φ (cid:48) ) L , (2.2)where φ i (cid:54) =1 is a complex scalar field, charged under a different Cartan generator. This choiceunderlies the SYM hexagon framework and the third operator built in this manner is thereservoir in the terminology of [24]. As well known, the structure constant for three BMNoperators is protected in the SYM theory and given to all orders by its tree level expression.In the fishnet theory, it is not possible to take the third operator in the form (2.2), sincethe above mixture is not an eigenstate of the dilatation operator, due to lack of symmetry.In fact, it is generically not possible to have the three operators appearing on an equalfooting, in the fishnet theory, since no BMN vacuum appears in the OPE of O and O ,barring extremal processes. Instead, the operators entering this OPE look like domainwalls of φ and φ † , and the simplest choice of third operator corresponds to O = tr φ † (cid:96) φ (cid:96) , (2.3)where the splitting lengths, a.k.a bridge lengths, (cid:96) ij = (cid:96) ji determine the pairing of fieldsin the BMN pair (2.1), see figure 1, and are such that (cid:96) − (cid:96) = L − L , for chargeconservation.Interestingly, the domain-wall operator (2.3) is protected in the fishnet theory, as longas (cid:96) , (cid:96) (cid:54) = 0; its anomalous dimension γ = 0, in the planar limit. It belongs to a broaderfamily of protected states, which includes, in particular, the Lagrangian density (1.1), asdiscussed in Subsection 2.3. On the contrary, the BMN operators (2.1), which are half-BPS in the SYM theory, receive anomalous dimensions in the fishnet theory, in lack ofsupersymmetry. Their anomalous dimensions are induced by the so-called wheel graphs[1, 58] which feature loops of the second complex scalar φ around the operators, γ , = ∆ , − L , = O ( g L , ) , (2.4)Every wheel costs L , powers of g and thus the RHS above runs in integer powers of g L , .Assembling our three operators together, we obtain the vacuum structure constant C •◦• = (cid:104) tr { φ (cid:96) φ (cid:96) } (0) tr { φ † (cid:96) φ (cid:96) } (1) tr { φ † (cid:96) φ † (cid:96) } ( ∞ ) (cid:105) , (2.5)where, to prepare the ground for the hexagons, we parameterized all the operators in termsof the bridge lengths, with (cid:96) = L − (cid:96) = L − (cid:96) ; the latter count the numbers of (cid:104) φ φ † (cid:105) ’s in each bridge, as shown in figure 1. Similar structure constants were discussedrecently in [8, 18]; see also [54] for a related set-up. The graphs contributing to (2.5) aresimply obtained by bringing together the wheels dressing each BMN operator; the thirdoperator brings nothing in this respect. Altogether, they generate a double expansion ininteger powers of g L and g L , and, accordingly, the structure constant reads C •◦• = (cid:112) L L (1 + O ( g L ) + O ( g L )) , (2.6) Extremal processes are found when the length of the third operator obeys L = ± ( L − L ), a conditionwhich permits the third operator to be a vacuum state. However, the associated structure constant isexpected to vanish if the admixtures of double-trace operators are take into account. O ` O = ⇥ Figure 1:
Wheeled Feynman diagram contributing to a fishnet structure constant. Whiteand grey dots represent the fields φ and φ † in the operators. Black and red lines representpropagators for φ and φ , respectively. The bridge length (cid:96) ij counts the number of blackpropagators along each edge. Cutting along the three edges, as shown here in dashed lines,splits the Feynman diagram into two hexagons and cuts open the wheel. for canonically normalised operators and after removal of the color factor ∼ /N .Traditionally, in the spin-chain picture, the φ ’s are seen as magnons propagating ontop of the lattice defined by the φ ’s [59]. The magnons circulating along the wheels aremade of the same wood but are not attached to a specific operator. They are the so-calledmirror magnons, which live between two locally BMN operators and account for the virtualparticles winding around them [60, 61]. They are classified according to the little groupof the two boundary operators: each magnon is then labelled with a momentum p , or arapidity u = p/
2, for dilatation r∂/∂r = ip ( u ), and a pair of equal spins ( ( a − , ( a − a = 1 , , . . . , for Lorentz rotations ∼ O (4), see Subsection 2.2.For illustration, a magnon inserted between O and O , sitting at respectively 0 and ∞ ,is given, in the fishnet theory, as a plane wave along the radial direction, | φ ( u ) (cid:105) ∞ = φ (0) · ∞ (cid:90) dr r ip ( u ) φ ( r ) · φ † ( ∞ ) , (2.7)dropping the orbital part and associated spin labels, for simplicity. (An analogous pictureis used to add excitations in the background of a null polygonal Wilson loop, in theform of insertions along its edges [62, 63, 53, 64].) A generic Bethe state is obtained byconcatenating magnons, | φ ( u ) (cid:105) ∞ = | φ ( u ) . . . φ ( u n ) (cid:105) ∞ , and can be cast in the form (2.7)by smearing n insertions within a suitable wave function ψ u ( { r i } ). An essential property ofthe Bethe states, which determines their wave functions, is that they diagonalise the quarticinteractions contained inside the bridge. Namely, the bridge ij should be transparent to aBethe state in the associated frame,bridge ij · | φ ( u ) (cid:105) ij = e − E ( u ) (cid:96) ij | φ ( u ) (cid:105) ij , (2.8)up to an overall factor, controlled by the energy of the state, E ( u ) = (cid:80) i E a i ( u i ). Theembedding of the fishnet theory inside N = 4 SYM dictates that E a ( u ) = − log g / ( u + a ) (2.9)6 u u wv v O O O Figure 2:
Hexagon form factor with magnons on the mirror edges and its fishnet counterpart.The quartic interactions are pushed to the boundary and absorbed inside the bridge factors.The hexagon form factor captures the splitting of the magnons’ wave function according tothe pattern of free (red) propagators. for the individual energy of a magnon in the wave | p ( u ) , a (cid:105) , and, as expected, the transportof the state across the bridge results in n × (cid:96) ij powers of the coupling constant.The idea underlying the hexagon factorization is to liberate the mirror magnons byopening up the traces in (2.5) and cutting along the bridges. In the process, every wheel iscut open twice and the end-points so produced are mapped to mirror magnons sitting alongthe edges of two hexagons, see figure 1. The hexagon form factors measure the overlapsbetween the three Bethe states in the three mirror cuts, as shown in figure 2, H ( u , v , w ) = ⊗ (cid:104) φ ( v ) † ⊗ φ ( w ) † | φ ( u ) (cid:105) . (2.10)In the basis of Bethe states, the effect of the bridges boils down to inserting the energyfactors (2.8) and, as a result, the structure constant is given, schematically, as [24] C •◦• /C tree132 = (cid:88) u , v , w e − E ( u ) (cid:96) − E ( v ) (cid:96) − E ( w ) (cid:96) × | H ( u , v , w ) | , (2.11)where each sum runs over a complete basis of states on the associated mirror cut. Thisexpansion is readily seen to reproduce the structure of the perturbative series in (2.6),after taking into account that the number of magnons is conserved, for the processes underconsideration, | u | = | v | + | w | , and that the hexagon form factors are coupling independent,in the fishnet theory, for properly normalised Bethe states.In the following, we derive the expression for H , starting from the conjecture put forwardin the SYM theory. Prior to move to this technical analysis, let us comment on a qualitativeaspect of the hexagons in the fishnet theory. As should be clear from figure 2, all thephysics is pushed to the boundary, where the field theory interactions reside, and only thefree propagators stay inside. The hexagons are seemingly made out of thin air, and, as forthe tree-level pentagon OPE [63, 53, 64] or the tailoring procedure [65], the analysis boilsdown to studying free propagators. (The relation between free propagators and hexagonswill be made more precise in Section 3.) The analysis stays nontrivial, since the propagatorsmust be convoluted with the mirror wave functions ψ in the relevant frames. These wavefunctions are not known in general; constructing them explicitly, using e.g. the Schr¨odinger7 u S ( u , u ) B B A A = u u A B A B Figure 3:
Two-magnon hexagon form factor and its matrix part. A pair of magnons on amirror edge is absorbed by the hexagon. The module of the amplitude is controlled by theabelian factor h ( u , u ). The matrix part accounts for the contraction of the magnons’ leftand right indices. Raising the right indices with the conjugation matrix, we can write it asthe matrix element of the fundamental S matrix S shown in the right panel. equation (2.8), is demanding and evaluating their overlaps (2.10) even more. The hexagonbootstrap bypasses this difficulty by focusing on their asymptotic behaviours, which arecontrolled by the S matrix, but it entails a certain amount of guesswork too. It would beinteresting to place the formalism on firm ground, using “microscopic” methods for buildingthe wave functions. The corresponding problem for null polygonal Wilson loops was solved,for instance, in [53, 64] using the SL (2) Baxter operator and its supersymmetric cousins,and progress was made recently with correlators in the 2d fishnet theory using an SL (2 , C )version of the formalism [16]. A generalisation to SL (4) appears to be needed for thecorrelators of the 4d fishnet theory. The SYM theory has many more fields than the fishnet theory but also many more sym-metries. Its magnons come in more flavours but can all be packed together inside shortirreducible representations of the BMN symmetry group SU (2 | , or, to be precise, of asuitable extension thereof [66]. In particular, the lightest magnons fill a bi-fundamental(16-dimensional) representation, χ A ˙ A ( u ) = χ A ⊗ χ ˙ A ( u ) , (2.12)with χ A ∈ ( ϕ a =1 , | ψ α =1 , ) a quartet of bosonic | fermionic fields and with the rapidity u labelling the energy E ( u ) and momentum p ( u ). Heavier magnons are obtained by binding a fundamental magnons together [67], in the appropriate channel, and fill (4 a ) -dimensionalirreps, with a = 1 , , ... . In the following, we will drop the bound state label, keeping inmind that formulae for bound states entail fusing those for the elementary magnons.Hexagon processes in the SYM theory are also richer than their fishnet counterparts, asthey capture more graphs. In particular, the SYM hexagon can absorb or produce magnons.The simplest form factor quantifies this effect and comes with an ordered set of magnons u = { u , u , . . . } along a single given edge, as shown in figure 3. It can be written formally Note that, in this paper, we work with the anti-clockwise ordering, when drawing magnon sets along h A ˙ A ,A ˙ A ,... ( u ) = (cid:104) h | χ A ˙ A ( u ) , χ A ˙ A ( u ) , . . . (cid:105) ⊗ | (cid:105) ⊗ | (cid:105) , (2.13)where the bra represents the hexagon vertex and the kets the states on its edges. Reshufflingmagnons in a state follows from the action of the S matrix and translates into a constrainton the form factor (2.13). The latter is a universal axiom known as the Watson relation.E.g., for two magnons, it requires that h A ˙ A ,A ˙ A ( u , u ) = S ( u , u ) B ˙ B ,B ˙ B A ˙ A ,A ˙ A h B ˙ B ,B ˙ B ( u , u ) , (2.14)with implicit sums over the B ’s, and with [66, 68, 69] S ( u , u ) B ˙ B ,B ˙ B A ˙ A ,A ˙ A = ( − f S ( u , u ) S ( u , u ) B B A A S ( u , u ) ˙ B ˙ B ˙ A ˙ A , (2.15)the 2-magnon S matrix, with S the abelian factor, S its left/right component, and f = f ˙ A f A + f ˙ B f B a grading factor for the left-right scattering, with f A the fermion numberof χ A , etc.The factorised ansatz put forward in [24] expresses the form factor (2.13) as a square rootof the S matrix, obtained by dropping the right S matrix and mapping the right magnons’components to outgoing particles. More precisely, it casts it into the form h A ˙ A ,... ( u ) = h < ( u ) × M A ˙ A ,... , h < ( u , u ) = (cid:89) i
A generic hexagon form factor. The magnons are distributed on the three mirroredges as in the leftmost panel. In the middle panel, we have the standard representation of thematrix part, obtained by analytically continuing rapidities to the crossed and doubly crossedkinematics. In the rightmost panel, we show an alternative representation where all rapiditiesare set back to the same kinematics using the crossing properties of h and S . This operationflips the orientation of the w lines and makes the cyclic symmetry manifest. The price topay for this re-organisation is a grading of the sums over intermediate states in the loops,represented by the dots. γ : u → u γ starting from the spin chain kinematics. To avoid cluttering our formulae,we shall drop the upper-scripts referring to this mirror move and place ourselves on themirror sheet from the onset. To handle this kinematics properly, we shall adopt the stringworldsheet normalisation and work in the so-called string frame [69].More importantly, the magnons should be more evenly distributed on the top and bottomedges of the hexagon, as in e.g. figure 4, since the magnons to be considered will be chargedw.r.t. the diagonal subgroup. These more generic form factors, h A ˙ A ,... ; A ˙ A ,... ; A ˙ A ,... ( u , v , w ) = (cid:104) h | χ A ˙ A ( u ) . . . (cid:105) ⊗ | χ A ˙ A ( w ) . . . (cid:105) ⊗ | χ A ˙ A ( v ) . . . (cid:105) , (2.19)can be obtained by implementing mirror moves, or crossing transformations [70], on themagnons in (2.16), following the rules spelled out in the appendices of Refs. [24] and [38].Performing these manipulations gives the form factor (2.19) as a S matrix element witharguments u , w − γ , v − γ ; see middle panel in 4. One can massage this expression and obtaina cyclic symmetric representation with all the arguments lying on the same kinematicalsheet. To do so, one simply makes use of the crossing properties of h and S . More precisely,one needs, see [24, 69, 70], h ( u γ , v γ ) = h ( u, v ) , h ( u γ , v ) = 1 h ( v, u ) , (2.20)together with h ( w, v ) h ( w − γ , v − γ ) S ( w − γ , v − γ ) CDAB = C AE S ( v, w ) DEBF C F C ,h ( u, w ) h ( u, w − γ ) S ( u, w − γ ) CDAB = C DE S F CEA ( w, u ) C F B , (2.21)and S ( u, v γ ) CDAB = ( − f D S CDAB ( u, v )( − f B , S ( u γ , v ) CDAB = ( − f C S CDAB ( u, v )( − f A . (2.22)10 ! uu w u u w = ⇥ h ( u, w ) h ( w, u ) Figure 5:
Illustration of the decoupling of the matrix part for the three-magnon configuration.In the limit where v → u the uv interaction reduces to a permutation, S → − P , and the uw lines can be disentangled up to an overall abelian factor. The relation shown here is equivalentto the unitarity of S after crossing the magnon w . The abelian factor spit out by the matrixpart completes the decoupling of the dynamical factor in (2.23) in the limit v → u . These relations are used, graphically, to flip the orientation of the w lines (as well as to undothe − γ move of the v ’s). Assembling all pieces together, we get the cyclic representation h ( u , v , w ) A ˙ A ,... = h < ( u , u ) h < ( v , v ) h < ( w , w ) h ( u , w ) h ( w , v ) h ( v , u ) × M A ˙ A ,... , (2.23)where the matrix part is illustrated in the right panel of figure 4 on a particular example.The matrix part is easy to spell out for a single magnon on each edge and reads M ( u, v, w ) A ˙ A ,A ˙ A ,A ˙ A = ( − C B ˙ A C B ˙ A C B ˙ A Z B B B A A A , (2.24)with the overall sign ( − = ( − f A + f A + f A ( − f ˙ A f A + f ˙ A f A + f ˙ A f A ( − F + F F , (2.25)where F i = f A i + f ˙ A i . The core of the interaction is obtained by concatenating S matrices, Z B B B A A A = ( − f C + f C + f C S ( u, v ) B C C A S ( v, w ) B C C A S ( w, u ) B C C A , (2.26)with a graded sum over the internal magnons’ flavors C , , . For more magnons, one shoulddress with self-interactions the external legs, as shown in figure 4, scatter the three stackstogether using the mutli-line uplift of the central vertex (2.26) and finally contract left andright movers using the conjugation matrix. One could also remove magnons by sending linesto infinity. E.g., removing w in (2.24), one gets M ( u, v ) A ˙ A ,A ˙ A = ( − f ˙ A f A C B ˙ A C B ˙ A S ( u, v ) B B A A , (2.27)which appears to be the same matrix part as for the 2-body annihilation form factor, seeEq. (2.17). (This well-known relation follows from the fact that the ± γ rotation actstrivially on the matrix part.)The representation (2.23) also makes the kinematical singularities of the hexagon formfactor manifest in the 3 channels. Namely, the form factor has a (simple) pole whenever two Its cyclic symmetry follows from the condition F A + F B + F C = 0 mod 2. h ( u, v ) at u = v .Physically, it represents the situation where a magnon moves far away from the core of thehexagon and decouples. Its residue relates to the measure µ ( u ) normalising the magnonwave function. E.g., decoupling the leftmost particle, for simplicity, by taking v n ∼ u , oneobtains h ( { u , . . . } ; { . . . , v n } ; w ) ∼ i I µ ( u )( v n − u ) × h ( u \{ u } ; v \{ v n } ; w ) , (2.28)where I is a tensor contracting the indices of the decoupled pair of magnons. (Theexplicit expression for I will not be needed but could be read out from Eq. (2.27).) Thefactorisation of the matrix part underlying (2.28) is depicted in figure 5 for the three-magnonconfiguration. The projection to the fishnet theory is done by selecting good scalar components and takingthe weak coupling limit. More precisely, we shall select the SYM magnons carrying maximalcharges under the U (1) R subgroup of SU (2 | D , distribute them along the edges of thehexagon as in figure 2, and finally take the weak coupling limit. This choice of polarisationinsures that the reservoir is transparent to the magnons and reduces to the domain-walloperator (2.3), to leading order at weak coupling. These magnons are transverse, in theterminology of [24], and correspond to φ ( u ) = ϕ ⊗ ϕ ˙1 ( u ) , φ † ( u ) = ϕ ⊗ ϕ ˙2 ( u ) , (2.29)for the elementary ones. Their relatives in the bound-state multiplets form higher repre-sentations of the Lorentz group, see e.g. [71, 26], obtained by attaching derivatives to thescalar fields, e.g., ∂ α ˙ α . . . ∂ α a − , ˙ α a − φ ( u ) = ( −
12 ( a − a − | ϕ ψ α . . . (cid:105)| ϕ ˙1 ψ ˙ α . . . (cid:105) , (2.30)with a the bound state label. They span, for given a , a symmetric traceless representation V a ⊗ ˙ V a of O (4), with spins ( ( a − , ( a − a , and, altogether, they areenough to reconstruct the full 4d massless scalar fields of the fishnet theory. The energy E a ( u ) of a magnon, carrying momentum p a ( u ) = 2 u , is given by the SYM weak couplingformula (2.9).The fishnet S matrix does not depend on our choice of polarisation and follows directlyfrom the scalar component of the SYM S matrix (2.15), after taking the weak coupling limitin the mirror kinematics. As well known, the spin-chain interactions rationalise in the weakcoupling limit, and, as a result, the fishnet S matrix factorises into two copies of the XXX SU (2) R matrix, for the left and right Lorentz indices, respectively, S ab ( u, v ) = S ab ( u, v ) × R ab ( u − v ) ⊗ ˙ R ab ( u − v ) , (2.31)12p to the scalar factor S ab ( u, v ) = a + b + iu − iv a + b − iu + iv (cid:89) k =0 , Γ( k + a − iu )Γ( k + a − b + iu − iv )Γ( k + b + iv )Γ( k + a + iu )Γ( k + a − b − iu + iv )Γ( k + b − iv ) . (2.32)Here, R ab is the standard R matrix [72–74] acting on the tensor product of the a -th and b -th irrep of SU (2), with dimension a and b , respectively, R ab : V a ⊗ V b → V b ⊗ V a , | α, β (cid:105) → R ab ( u − v ) γδαβ | δ, γ (cid:105) , (2.33)with α, ... the multi-spinor indices appropriate for totally symmetric tensors of rank a − b −
1. It can be obtained by fusing the fundamental (spin 1 /
2) R matrix, with a = b = 2in our notations, R ( u ) γδαβ = uu + i δ γα δ δβ + iu + i δ δα δ γβ . (2.34)We spell it out in Appendix A in the symmetric product basis (2.30). Alternatively, we candefine it with no reference to a basis by collecting its eigenvalues, R ab ( u ) = max (cid:88) j =0 ( − j Γ( a + b + iu )Γ( a + b − iu − j )Γ( a + b − iu )Γ( a + b + iu − j ) P a + b − − j , (2.35)where max = ( a + b − | a − b | ) − P a + b − − j the projector on the dim ( a + b − − j ) irrep ⊂ V a ⊗ V b . Its normalisation is such that R ab = 1 in the symmetric channel,corresponding to j = 0, that it reduces to the identity matrix, R ab → I , when u → ∞ andto the permutation operator at u = 0 when b = a . Let us finally recall that it obeys thefunctional (crossing) relation R ab ( u − i ) γδαβ = c ab ( u ) C b βσ R ba ( − u ) σγρα C ρδb , (2.36)where C b is the conjugation matrix defined by C αβ = (cid:15) αβ , C βα = (cid:15) αβ , with (cid:15) = (cid:15) = 1for fundamental spins, and by suitable products thereof for higher b . The crossing factor isgiven by c ab ( u + ) = a + b − (cid:89) j = | a − b | u − − iju + + ij , (2.37)where u ± = u ± i/ Given the S matrix, the next step is to reduce the hexagon form factors. We shall proceedstep-by-step starting with the simplest configurations where all the magnons are elementaryand propagate on the left-hand side of the hexagon, as shown in figure 6. The computationof the corresponding form factor is an immediate application of the general formula givenin the previous subsection. The most complicated component is the matrix part, which is It solves the fusion relations c ab ( u + ) c ab ( u − ) = c a +1 ,b ( u ) c a − ,b ( u ) with the initial conditions c b ( u ) =1 , c b ( u ) = ( u − ib ) / ( u + i ( b − ). u v v m u m u u v v m = 1 m = 2 Figure 6:
Hexagon transition H ( u → v ) with all magnons going to the left. The numbersof incoming and outgoing magnons must match for charge conservation. On the right panel,the SYM domain wall partition functions for the matrix parts when m = 1 and m = 2. Atweak coupling the mirror S matrix is transmission-less and the partition function collapses toa single process where the magnons’ flavors backscatter one each other. represented by the partition function in figure 6. For a single magnon transition u → v , weread out from (2.27), using (2.29), H ( u → v ) = (cid:104) h | φ ( u ) (cid:105) ⊗ | (cid:105) ⊗ | φ † ( v ) (cid:105) = 1 h ( v, u ) (cid:15) (cid:15) S ( u, v ) , (2.38)where, see e.g. appendices in [38], S ( u, v ) = 12 ( A ( u, v ) + B ( u, v )) , (2.39)with A and B parameterising the symmetric and antisymmetric amplitudes of the scalarrestriction of the S matrix. The A amplitude is unitary and fulfills A ( u, v ) A ( v, u ) = 1 atany coupling. This is not a priori the case for the B amplitude, since bosons and fermionscan mix in the antisymmetric channel [66]. However, as well known, this effect is absentto leading order at weak coupling. Moreover, in the mirror kinematics, the weak couplingscattering is transmission-less, and thus B = A + O ( g ) . (2.40)Hence, the hexagon form factor in the fishnet theory is simply given by H ( u → v ) = 1 H ( v, u ) , (2.41)where H ( u, v ) = − A ( u, v ) h ( u, v ) is the scalar hexagon amplitude [24]. The analysis gen-eralises straightforwardly to configurations involving more magnons, as shown in figure 6,thanks to the aforementioned properties of the scalar S matrix. The general formula is fullyfactorised and simply given by H ( u → v ) = H < ( u , u ) H < ( v , v ) H ( v , u ) . (2.42)14 C F K u u w v + + Figure 7:
Processes contributing to the matrix part for the hexagon splitting of a two-magnonwave function, with the magnons arranged as in left panel of figure 4. One of them displays afermion loop represented by the dashed line. (The grading factors drop out since the numberof fermions involved is even.)
Similar simplifications are observed for bound states, although less transparently. In thiscase, the S matrix is more bulky and fermions must be included to represent the derivatives.Nonetheless, the scalar and Lorentz parts are seen to factorise and the final expression is anatural higher spin uplift of (2.42). The abelian part is literally just (2.42) up to H → H ab ,with H ab = − A ab h ab and A ab the bound-state scalar amplitude, while the matrix part has asimilar structure but in terms of R matrices. Putting all factors together, we get H ab ( u → v ) = i f a i f b H < aa ( u , u ) H < bb ( v , v ) H ba ( v , u ) × R ab ( u , v ) R < bb ( v , v ) R < aa ( u , u ) , (2.43)where f a = (cid:80) i ( a i − f b . The indices enter as in the SYM formula, see,e.g., Eq. (2.27), with the dotted indices in the LHS obtained by lowering the outgoing indicesof the R matrices using the conjugation matrix C . E.g, for a single magnon transition, wehave, using multi-spinor indices, H ab ( u → v ) α ˙ α,β ˙ β = i a + b − C a γ ˙ α C b δ ˙ β H ba ( v, u ) R ab ( u − v ) γδαβ . (2.44)The explicit expression for H ab ( u, v ) will be given later on, see Eq. (2.53). The formula forthe transition to the right-hand side of the hexagon follows from turning the picture around,i.e., by exchanging u and v .We proceed with the more complicated situations where the beam of magnons is splitin two, u → v | w . The simplest such process is given by H ( u , u → v | w ) = (cid:104) h | φ ( u ) φ ( u ) (cid:105) ⊗ | φ † ( w ) (cid:105) ⊗ | φ † ( v ) (cid:105) = h ( u , u ) M ( u , u , v, w ) h ( u , w ) h ( u , w ) h ( w, v ) h ( v, u ) h ( v, u ) , (2.45)with M the matrix part depicted in figure 7. Applying the general formula, we find thatthe matrix part M receives three contributions, one for each graph in figure 7 and with the15ast one featuring a fermion loop. They yield M = 12 A u u ( A vw − B vw ) × (cid:20)
18 ( A u v + B u v )( A u v − B u v )( A wu + B wu ) A wu + 18 A u v ( A u v + B u v )( A wu − B wu )( A wu + B wu ) − K u v F u v C wu K wu (cid:21) , (2.46)where to save space we placed the arguments as subscripts, with A, B the scalar amplitudes,
C, F ∼ g the amplitude for creation and annihilation of a pair of fermions, and with K ∼ g the fermion-scalar reflection amplitude. All terms in brackets start at order g , includingthe one with fermions in the loop. Straightforward algebra gives H ( u , u → v | w ) = H ( u , u ) H ( v, w ) H ( u , w ) H ( u , w ) H ( v, u ) H ( v, u ) . (2.47)Remarkably, despite the several internal processes and the fermion loop, the result factorisesand is expressed solely in terms of the basic scalar amplitude. Its structure is suggestingthe general formula H ( u → v | w ) = H < ( u , u ) H < ( v , v ) H < ( w , w ) H ( v , w ) H ( u , w ) H ( v , u ) , (2.48)for a generic distribution of elementary magnons, fulfilling charge conservation, | u | = | v | + | w | . We failed to find a proof of this ansatz, but we tested it extensively with Mathematica.As further evidence for its correctness, we notice that it solves all the bootstrap axioms. Itindeed transforms properly under permutation of the magnons in the states, as a result ofthe Watson relation, H ( u, v ) /H ( v, u ) = S ( u, v ) , (2.49)with S ( u, v ) = S ( u, v ) the scalar S matrix, and it displays decoupling poles wheneverrapidities in bottom and top sets become identical, again, thanks to the correspondingproperty of H ( u, v ), see Eq. (2.55) below. More precisely, one verifies that the decouplingcondition (2.28) is obeyed, with I →
1. Turning the logic around, the ansatz (2.48) appearsas the simplest way of bringing together the left and right form factors, Eq. (2.42) and itsright partner, while preserving the Watson relation and decoupling property. To enforce thelatter requirement we simply added H ( v , w ) in the numerator.At last, we should include the bound states and their matrix degrees of freedom. Herealso it proves easier to bootstrap the answer than to derive it from the SYM partitionfunctions. Drawing inspiration from the structure of the result in the latter theory andassuming a factorised ansatz, one can uniquely determine the missing ingredient, that is,the vertex between the magnons v and w , by imposing the decoupling axiom. More precisely, We should stress that the scaling with the coupling does not imply that the form factor is sub-leading.Indeed, a vanishing result would be in tension with the decoupling property of the fishnet hexagon formfactors. The scaling with the coupling is merely reflecting the implicit normalisation of the external statesin the SYM representation. = u uv u vw w Figure 8:
Binding the left and right interactions in a decoupling friendly way fixes the thirdvertex, here shown as a blob, to be a shifted R matrix. bringing together two R matrices, for the uv and wu scattering, as shown in figure 8, we canthen fix the vw interaction point, denoted R ◦ ( v, w ), by demanding that the latter vertexannihilates the left/right interaction in the right/left decoupling limit. This constraint islinear in R ◦ ( v, w ) and it implies that R ◦ is equal to the R matrix, up to a shift of itsargument and a change of normalisation, R ◦ bc ( v, w ) = c bc ( v − w ) c bc ( v − w − i ) R bc ( v − w − i ) . (2.50)To prove this relation, one simply needs to use the crossing property of the R matrix, seeEq. (2.36), as shown in figure 9. Contrary to the SYM hexagon, here we find that the topvertex is inequivalent to the left and right ones; it goes along with the fact that the fishnethexagon is not cyclic symmetric.Crossing the lines permits to write the final result in the scattering form. E.g., aftercrossing the w ’s, discarding the conjugation of their indices, we can write the core of theinteraction as M ( u , v , w ) | amputated = c ( v , w ) c ( u , w ) × R ( w ++ , v ) R ( u , v ) R ( u , w ++ ) , (2.51)with w ++ = w + i , with implicit bound state labels, and where c ( u , v ) = (cid:89) i,j c a i b j ( u i − v j ) , (2.52)with c ab the crossing factor (2.37). For the sake of clarity, we removed the self-interactionson the external legs – they can be inferred from (2.43) – and the abelian prefactor is givenby (2.48) with the H ’s dressed with bound state indices. In the representation (2.51), themagnons v and w do not appear on an equal footing, but the left decoupling property of thematrix part is manifest, see figure 9. Finally, let us stress that we verified the bound stateansatz (2.51) using Mathematica, for a few magnons and many different choices of boundstate indices, starting from the SYM representation and using the mirror bound state S matrix obtained in [28]. A similar formula would be obtained by crossing the v ’s, making the right decoupling obvious. rossing u u uv ww u uw + iv ! u Figure 9:
Decoupling condition for the three-body matrix part. In the limit v → u the uv interaction reduces to a permutation. After flipping the arrow on the w line, using the crossingproperty of the R matrix, the interactions between u and w are seen to collapse thanks to theunitarity of the R matrix. This is it for the hexagon form factors to be used in this paper. To complete the picture,we quote the expression for the abelian factor H ab ( u, v ) = − A ab ( u, v ) h ab ( u, v ), which followsfrom the weak coupling limit of the fused SYM formula in the mirror kinematics, H ab ( u, v ) = g ( − a − ( a + b + iu − iv )Γ(1 + a − iu )Γ(1 + b − a + iu − iv )Γ(1 + b + iv )( u + a ) Γ( a + iu )Γ( b − a − iu + iv )Γ( b − iv )( v + b ) . (2.53)Its zero at v = u for b = a equips the direct transition (2.44) with the decoupling pole1 H ba ( v, u ) ∼ ( − a − δ ab iµ a ( u )( u − v ) . (2.54)The associated measure reads µ a ( u ) = ag ( u + a / , (2.55)and it is identical to the SYM measure in the mirror kinematics at weak coupling. Onealso verifies the Watson relation, H ab ( u, v ) /H ba ( v, u ) = S ab ( u, v ), with the abelian S matrix(2.32), as it should be. There is an extra ingredient that we need for our investigation. It is associated to theinsertion of magnons on the third operator. It appears natural indeed to enlarge the familyof third operators by considering O → V n,m,n ∗ = tr φ † n ∗ φ † m φ n φ m . (2.56)which includes, in particular, the dilaton, V , , = 1 g L int = tr φ † φ † φ φ . (2.57)18 O O O wu w ! i/ u Figure 10:
Example of a fishnet structure constant with magnons ending on the thirdoperator. We can bring a mirror magnon to this position by continuing its mirror momentumto p ( w ) = 2 w = − i , as shown in the right panel. Owing to the specific ordering of the fields in the trace, the dynamics is frozen and themagnons cannot move in the background of the other fields. In sum, all these operators areprotected.From the integrability viewpoint, operator (2.56) acts as a sink or source for the mirrormagnons. When placed inside a three-point function together with a pair of BMN operators,it leads to the diagram shown in the left panel of figure 10, to leading order at weakcoupling. Importantly, the two sets of magnons, φ m and φ † m , split on two hexagons.Hence, to add the operator (2.56) to our story, we only need to charge the hexagon witha homogeneous reservoir of magnons on the edge associated to the third operator. Theproblem is reminiscent of the charging of the null pentagon Wilson loop [75], used to embedthe non-MHV amplitudes within the pentagon OPE framework in N = 4 SYM. As we shallsee, the outcome is essentially the same.For a unit of charge, we would like to place a single magnon on the edge associated to thethird operator and set its spin-chain momentum p to zero. In this way, we are guaranteedthat the magnon will not generate anomalous dimension. In N = 4 SYM, we could bringthe magnon on the spin-chain edge starting from a neighbouring mirror edge, by using themirror rotation. In the fishnet theory, because of the double scaling limit, the gates to thespin-chain kinematics pinch off at ± i/ u , see figure 10, canbe determined using equations (2.48) and (2.53). We findlim w →− i/ (cid:112) µ ( w ) (cid:112) | ∂ w p ( w ) | H a ( u, w ) = (cid:112) u + a / g ≡ ξ a ( u ) , (2.58)after switching to the spin-chain normalisation. The latter includes the measure µ and theJacobian for the map between rapidity and spin-chain momentum, with p = iE and E themirror energy of the magnon. Note that one would obtain the same result starting from N = 4 SYM, placing a magnon on the relevant edge, and projecting to the fishnet theory.19ore generally, each magnon present on the hexagon gets dressed by a factor thatdepends on its rapidity and representation. Labelling the magnons on the mirror edges asin figure 2, with the third operator at the top, we obtain ξ u /ξ v ξ w , (2.59)where ξ ( u ) = (cid:81) i ξ a i ( u i ), etc. The generalization to the case where we insert m magnons atthe cusp follows from sending m magnons to zero momentum, one after the other, and thedressing factor is obtained by raising (2.59) to the power m . In this section we carry out a battery of tests of our main formulae by comparing theirpredictions for structure constants and correlators with field theoretical calculations. Wewill also obtain a few predictions for a simple class of wheeled 3pt Feynman integrals.
We begin with the simplest fishnet correlator, the free propagator. Although elementaryon the field theory side, its reconstruction using the hexagon factorisation is instrumental,as it gives a direct access to the hexagon building blocks. More precisely, by embeddingthe propagator inside a four- and five-point function and proceeding with its hexagonalisa-tion [26], we shall be able to perform a direct test of the measure and 2-body form factor.The hexagon processes to be considered are displayed in figure 11, and, in all cases, theinitial and final stages are the charged hexagons described in the previous section.Let us start with the four-point function, which is an adaptation of the integrals consid-ered in [26], see also [15]. It is obtained from the gluing of two hexagons, as shown in theleftmost panel in figure 11, and it involves a complete sum over the 1-magnon eigenstatesalong the middle cut 13. The spectral density to be integrated is ξ a ( u ) µ a ( u ) × geometry , (3.1)where the first factor absorbs the amplitude for production and absorption of the mirrorparticle, on the bottom and top hexagon. The last factor is the geometrical weight forthe dilatation and rotation of the magnon on the edge connecting the two hexagons. Itreads [26] geometry = ρ iu × χ a ( e iφ ) , (3.2)where χ a ( e iφ ) is the SU (2) character in the a -th irrep, i.e., χ a = tr V a ( e iφJ a ) = sin ( aφ )sin φ , (3.3)20ith J a the spin operator on V a . The dilation and rotation parameters, ρ and φ , are givenby ρ = ( zz ) − , e − iφ = (cid:114) zz , (3.4)where z, z are traditional 2d coordinates parameterizing the 4-point cross ratios, zz = x x x x , (1 − z )(1 − z ) = x x x x . (3.5)As described in [26, 27], we should also weight the scalar field insertions on the top andbottom cusps by including the factors (cid:18) | x || x || x | (cid:19) × (cid:18) | x || x || x | (cid:19) = (1 − z )(1 − z ) √ zz x . (3.6)Alternatively, we can omit these extra weights and combine them with the propagator suchas to define a conformally invariant propagator,Propagator = (cid:112) x x x x x x = √ zz (1 − z )(1 − z ) . (3.7)Now, straightforwardly, after using the expression for the measure and ξ factor, see Eqs. (2.55)and (2.58), and picking up the unique residue at u = − ia/
2, we obtain ∞ (cid:88) a =1 (cid:90) du π ξ a ( u ) µ a ( u ) ρ iu tr V a ( e iφJ a ) = ∞ (cid:88) a =1 (cid:90) du π au + a / ρ iu tr V a ( e iφJ a )= ∞ (cid:88) a =1 ρ a χ a ( e iφ ) = Propagator , (3.8)where the last equality is verified as a series expansion of (3.7) around infinity.The ingredients for the five point function read the same but we have one more hexagon,the middle hexagon in the middle picture in figure 11. The magnon trajectory is now cuttwice and we must sum over a complete basis of mirror states both along the zero lengthbridge 13 and 14. At each step the magnon wave function gets stretched and twisted bya dilation and a rotation, determined locally by the surrounding 4pt function. In order toperform the computation, we are going to consider the restriction to the 2d kinematics whereall the points lie in the same plane, since the weight for moving away from the plane hasnot been determined yet. Notice that distances in the plane can be written as x ab = x a,b x a,b and we are going to use this notation below. Only two pairs of cross ratios are needed andthe weights are given by [28] ρ − i − = z ( i − z ( i − = X ,i,i +1 ,i +2 X ,i,i +1 ,i +2 , e − iφ i − = (cid:114) z ( i − z ( i − = X ,i,i +1 ,i +2 /X ,i,i +1 ,i +2 , (3.9)where X ,i,i +1 ,i +2 = x i, x i,i +1 x i +1 ,i +2 x ,i +2 , (3.10)21 u uv uv v Figure 11:
Left and middle panels: Free propagator cut once and twice. We cut the interior ofthe polygon into two and three hexagons, respectively. The dashed middle lines denote bridgesof length zero. Outer bridges / boundaries play no role here. For definiteness, one could givethem arbitrarily large length to emphasize that nothing can leak out of the polygons. Onthe right panel, we give an example of a loop integral that could be hexagonalised by addingbridge lengths - for the horizontal propagators - and further magnons for the vertical ones. and with i = 2 , = ∞ (cid:88) a,b =1 (cid:90) du π dv π ξ a ( u ) µ a ( u ) ξ b ( v ) µ b ( v ) H ba ( v + i , u ) | ρ | iu | ρ | iv F ab , (3.11)where F ab originates from the R matrix in the middle transition, see Eq. (2.44), F ab = tr V a ⊗ V b ( e iφ J a e iφ J b R ab ( u − v )) , (3.12)with the trace taken over the tensor product of the SU (2) modules, of total dimension ab .Using (2.53) and (2.55), we obtain the dynamical part of the integrand ξ a ( u ) µ a ( u ) ξ b ( v ) µ b ( v ) H ba ( v + i , u ) = ab ( − b − Γ( a − iu )Γ( a − b + iu − iv + 0)Γ( b + iv )( a + b − iu + iv )Γ(1 + a + iu )Γ(1 + a − b − iu + iv )Γ(1 + b − iv ) , (3.13)where the i u = v and a = b . Weverify that the net integrand is of order g as needed for a tree-level process. The scalingfollows from, see Eqs. (2.53), (2.55) and (2.58), H ab ( u, v ) = O ( g ) , µ a ( u ) = O ( g ) , ξ a = O (1 /g ) , (3.14)together with the fact that the matrix part is coupling independent. Note also that the ξ factors for production and absorption of the magnon combine nicely with the square rootspresent in the middle transition H ab ( u → v ), see Eqs. (2.53) and (2.58), such as to give a The contour is chosen in a such way that the 5pt integral reduces to the 4pt one in the limit x → x . We evaluate the integral (3.13) by closing the contours of integration in the lower half-planes and summing up the residues. (All the poles are simple; that would not be so if wehad bigger bridge lengths.) We begin by picking up the residues in the lower half u planeand then in the lower half v plane. The former come from the single argument Gammafunction in the numerator and are located at u = − ia/ − ik with k = 0 , , . . . . In principle,we should also worry about the simple poles coming from the matrix part, see Eq. (2.35),at u = v − i a + b − j , j = 1 , . . . , min { a − , b − } , (3.15)to which we can add the pole at u = v − i ( a + b ) /
2, which is visible in (3.13). However, theGamma function of the difference of rapidities in the denominator removes them all, since1Γ(1 + a − b − iu + iv ) → − b + j ) , (3.16)is zero at these points, whenever j (cid:54) b −
1. The next step is to pick up the residues inthe lower half v plane. Here, again, one verifies that they only come from the Gammafunctions in the numerator, and, more specifically, from the Gamma function that dependson the difference of rapidities. Most of these poles are killed by the zeroes coming from thedenominator, such that, in the end, the double integral can be taken at once by extractingthe residues at u = − ia/ v = − ib/ . (3.17)Moreover, b (cid:62) a , as visible from the final expression for the double residue, which is givenby a binomial coefficient. It yields ρ ρ (cid:88) b (cid:62) ρ b − b (cid:88) a =1 ( − ρ ) a − Γ( b )Γ( a )Γ(1 + b − a ) tr V a ⊗ V b ( e iφ J a e iφ J b R ab ( ib − ia )) . (3.18)The sum over a can be viewed as generating the transfer matrices (at a specific point)for a twisted length-one spin chain with spin ( b − / a , we get that the5pt integral reduces to the 4pt one, see Eq. (3.8), ρ ρ (cid:88) b (cid:62) ( ρ (cid:48) ) b − χ b ( e iφ (cid:48) ) = ρ ρ (1 − /z (cid:48) )(1 − /z (cid:48) ) = √ z z √ z z (1 − z + z z )(1 − z + z z ) , (3.19)up to a geometrical redefinition of the cross ratios,( z (cid:48) ) − = z − (1 − z − ) , ( z (cid:48) ) − = z − (1 − z − ) . (3.20) It was observed in [26] by comparing hexagon calculations with perturbation theory in the SYM theorythat it is necessary to dress the mirror bound states with so called Z -markers to obtain an agreement. Thegeneral prescription for dressing the states, which passed all tests so far, was written down in the AppendixA of [28]. In our case, since we deal with transverse scalar excitations, the Z -markers play no role and thedressing trivialises. ¯ O Figure 12:
The two point function of two spiraled states. The red lines correspond to thepropagation of a φ field which is taken to be an excitation over the reference state made outof φ , represented by the dark lines. The Feynman graph corrections wrapping the externaloperators have a configuration of a spiral. Expression (3.19) is then immediately verified to match with the conformal propagator,(3 .
19) = (cid:18) | x || x || x | (cid:19) (cid:18) | x || x || x | (cid:19) x , (3.21)after taking into account the aforementioned weights for the scalar insertions at the top andbottom.One could keep going and insert the propagator in higher n point function. The hexagonrepresentation will then involve a sequence of transitions across the various mirror cuts. Weexpect the algebra to be similar to the one carried out here and to reduce to an iteration ofthe geometrical transformation (3.20). One could also consider products of free propagatorsstretching between different cusps of a polygon; the hexagon factorisation would give themin terms of convoluted integrals of products of multi-particle form factors. More ambitiously,one could add loops to the cocktail, of the type shown in figure 11, by dressing each magnonwith the bridge factor e − (cid:96)E a ( u ) , with (cid:96) measuring the number of horizontal propagators alongthe given cut. The resulting representations could be tested using the differential equationsderived from the Yangian symmetry [11, 12], for specific bridge lengths. As a simple and natural generalisation of our set-up, we shall consider spin-chain stateswith φ excitations propagating on top of the BMN vacuum, O ∼ tr φ L → O spiral ∼ (cid:88) n ψ n tr( φ L − N φ N ) , (3.22)where the RHS should be read as a linear superposition of N insertions along the chain.These states are the fishnet counterparts of the states lying in the SU (2) sector of N = 424 u u m v v m O O Figure 13:
In the left figure, the bridge overlap between the spiraled operators is representedin gray. Excitations cannot be contracted with the vacuum operator O so that the onlynontrivial contractions occur in the gray region. On the right figure, we represent the excitationpattern of the hexagon used to compute the bridge overlap. SYM [59] (even though in the fishnet theory only a U (1) subgroup remains). The Feynmangraphs wrapping these operators look like spirals (see figure 12) and for this reason we willrefer to the operators in (3.22) as spiraled states .Distributing magnons on the BMN vacua entering the structure constant (2.5) leadsto graphs of the type shown in figure 13. Hence, contrary to the previous setup, whereall quantum corrections came from virtual particles moving across the bridges, structureconstants for spiraled states receive nontrivial corrections in the form of a perturbativetail in g , before the wrapping corrections ∼ g L , kick in. We will limit ourselves to theasymptotic regime in the following, obtained by neglecting the wrapping corrections. Thisscenario is realised when the bridges connecting the BMN operators to the third operatorare asymptotically thick, i.e. (cid:96) , (cid:96) → ∞ . In these circumstances, the nontrivial part onlycomes from the bridge overlap between the two excited states, as illustrated in figure 13.The spectrum of spiraled states was thoroughly studied in [2]. It is described, asymp-totically, by the double scaling limit of the twisted Beisert-Staudacher equations [7]. Themain outcome of this analysis is that the fishnet limit amounts to performing an infinite(imaginary) boost on the magnons, which pushes them all the way to the mirror kinematics.Therefore, in the end, the magnons sourcing the spirals are just mirror magnons, like theones discussed throughout this paper. The sole difference is that the Bethe ansatz equationssubject them to have imaginary energies and momenta. More precisely, all the Bethe rootsoriginate at the same canonical point p = − i at weak coupling, see Subsection 2.3, and thenspread out along the mirror plane as the coupling increases. (Turning the flow around, onecould say that the Bethe roots are pushed to the spin chain edge, represented by a singlepoint on the mirror sheet, when the coupling is sent to zero.) They admit the expansion u k = − i ∞ (cid:88) j =1 δu ( j ) k g j , (3.23)with an infinite tail of perturbative corrections δu k . The latter are determined iteratively25y solving the Bethe ansatz equations, e iφ j = (cid:18) g u j + 1 / (cid:19) L N (cid:89) k (cid:54) = j S s ( u j , u k ) , (3.24)where S s ( u, v ) = ξ ( v ) ξ ( u ) − S ( u, v ) is the scalar mirror S matrix (2.32), with a = b = 1,in the spin-chain normalisation and where we used that each spiral carries an imaginaryspin-chain momentum p equals to its mirror energy E , e ip = e − E = g u + 1 / . (3.25)Similarly, although the magnons populate different edges of the hexagons, as shown in theright panel of figure 13, the hexagon amplitude takes exactly the same form as before, if notfor the conversion to the spin-chain normalisation. The translation between the string andspin-chain frames boils down to inserting ξ factors, as described in [24], and the hexagonamplitude showed in figure 13 is given by (2.42) up to the replacement H ( u, v ) → H s ( u, v ) = ξ ( v ) ξ ( u ) H ( u, v ) . (3.26)It obeys Watson relation for the spin-chain framed S matrix, H s ( u, v ) /H s ( v, u ) = S s ( u, v ).Asymptotically, the hexagon prescription to compute the structure constant consists inattaching two hexagons together along the bridge 12 and summing over all the ways ofdistributing magnons on both sides of the cut [24]. It yields C •◦• √ L L = N ( u ) N ( v ) (cid:88) α ∪ α = u ,β ∪ β = v e i ( p ( α ) − p ( β )) (cid:96) S s< ( α, α ) S s< ( β, β ) H s (cid:48) ( α, β ) H s (cid:48) ( β, α ) , (3.27)where | α | = | β | for charge conservation. Here, H s (cid:48) ( α, β ) = H s< ( α, α ) H s< ( β, β ) /H s ( β, α ), p is the spin chain momentum defined in (3.25), and the splitting factor is given by S s< ( α, α ) = (cid:89) i ∈ α,j ∈ α,i