Contents
Abstract

Integrability is believed to underlie the correspondence with sixteen supercharges. We elucidate the role of massless modes within this integrable framework. Firstly, we find the dressing factors that enter the massless and mixed-mass worldsheet S matrix. Secondly, we derive a set of all-loop Bethe Equations for the closed strings, determine their symmetries and weak-coupling limit. Thirdly, we investigate the underlying Yangian symmetry in the massless sector and show that it fits into the general framework of Yangian integrability. In addition, we compare our S matrix in the near-relativistic limit with recent perturbative worldsheet calculations of Sundin and Wulff.

DMUS-MP-16/13

Imperial-TP-RB-2016-04

NORDITA-2016-79

On the Dressing Factors, Bethe Equations and Yangian Symmetry of Strings on


Riccardo Borsato, Olof Ohlsson Sax, Alessandro Sfondrini,

Bogdan Stefański, jr. and Alessandro Torrielli


1. The Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom

2. Nordita, Stockholm University and KTH Royal Institute of Technology, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

3. Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Str. 27, CH-8093 Zürich, Switzerland

4. Centre for Mathematical Science, City University London, Northampton Square, EC1V 0HB London, UK

5. Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK



r.borsato@imperial.ac.uk, olof.ohlsson.sax@nordita.org, sfondria@itp.phys.ethz.ch, Bogdan.Stefanski.1@city.ac.uk, a.torrielli@surrey.ac.uk


Contents

1 Introduction

Over the last few years integrable methods have been extensively employed in the context of the spectral problem of the correspondence, see for example [David:2008yk, Babichenko:2009dk, OhlssonSax:2011ms, Sundin:2012gc, Cagnazzo:2012se, Borsato:2012ud, Borsato:2013qpa]. Initial progress did not include world-sheet massless modes [Babichenko:2009dk, OhlssonSax:2011ms], for a review see [Sfondrini:2014via].111For some early results on massless modes in the weak- and strong-coupling limits see [Sax:2012jv] and [Lloyd:2013wza, Abbott:2014rca], respectively.. Subsequently, it was shown [Borsato:2014exa, Borsato:2014hja, Lloyd:2014bsa, Borsato:2015mma] that these can be included in a novel integrable all-loop world-sheet S matrix, which was determined, up to dressing phases, for and supported by R-R flux and mixed R-R/NS-NS flux. While the dressing phases are not fixed by the symmetries of the theory, they satisfy crossing equations which were also found.

This progress in exact results was accompanied by a large body of perturbative world-sheet computations and a number of successful comparisons between the two was performed in the massive sector, see for example [Rughoonauth:2012qd, Abbott:2012dd, Beccaria:2012kb, Beccaria:2012pm, Sundin:2013ypa, Sundin:2013uca, Hoare:2013pma, Bianchi:2013nra, Hoare:2013ida, Engelund:2013fja, Abbott:2013ixa, Hoare:2013lja, Sundin:2014sfa, Bianchi:2014rfa, Hernandez:2014eta, Stepanchuk:2014kza, Roiban:2014cia]222Further papers on integrable holographic results include [Abbott:2015mla, Abbott:2014pia, Wulff:2015mwa, Wulff:2014kja, Wulff:2013kga, Chervonyi:2016ajp, Kluson:2015lia, Hernandez:2015nba, Ahn:2014tua, David:2014qta, Kluson:2016dca, Banerjee:2016avv, Prinsloo:2015apa]. In addition, these comparisons have also resulted in two unresolved issues. Firstly, the massless dispersion relation, determined via a (super-)symmetry argument in [Sundin:2014ema], does not agree with a two-loop perturbative computation [Borsato:2014exa, Borsato:2014hja]. Secondly, the massive dressing factor obtained by solving the crossing equations [Borsato:2013hoa] was found to be slightly different from the one calculated using perturbative world-sheet calculations [Beccaria:2012kb, Beccaria:2012pm]. The first issue’s resolution might come about by employing a more symmetric regularisation scheme, or a modified definition of asymptotic states. Given that the all-loop massless dispersion relation follows from a supersymmetric shortening condition, this issue deserves to be better understood. A possible explanation for the second issue was recently proposed in [Abbott:2015pps], where it was argued that a proper incorporation of the wrapping corrections of massless modes is likely to correct the perturbative world-sheet calculations in a way that would make them consistent with crossing.

Beside these issues, a more complete understanding of how the massless modes enter the integrable construction was still lacking. While the S-matrix and the crossing equations for processes involving massless modes had been found [Borsato:2014exa, Borsato:2014hja, Lloyd:2014bsa, Borsato:2015mma], the analytic properties of massless modes, and their dressing factors remained to be determined. In this paper we address these outstanding problems and find the minimal solutions to the crossing equations, giving a detailed exposition of the results announced in [Borsato:2016kbm]. On general grounds dressing phases have an expansion in the coupling constant, with the leading and next-to-leading orders on the string theory side conventionally referred to as the Arutyunov-Frolov-Staudacher (AFS) [Arutyunov:2004vx] and Hernández-López (HL) [Hernandez:2006tk] orders. The minimal solution for the massless dressing factor is non-trivial only at HL order, while the mixed-mass dressing factor minimal solution is non-trivial at AFS and HL order. Our solutions have a very natural interpretation as coming from a “massless limit” of the corresponding massive phases. By considering this limit for the non-perturbative Beisert-Eden-Staudacher (BES) phase [Beisert:2006ez] we argue that no natural candidate for homogeneous solutions exists at higher orders, while a homogeneous AFS order term might be natural. Further, we investigate the possibility of massless bound states in the spectrum. We argue that such states are not allowed kinematically and confirm their absence by an explicit analysis of the all-loop S matrix, including our dressing factors. It is worth pointing out that the absence of massless bound states is also a well-known feature of relativistic massless integrable models [Zamolodchikov:1992zr].

We then turn to the derivation of the Bethe equations for the complete closed string spectrum from the S matrix. We impose periodicity and employ the nesting procedure to find the Bethe equations. The structure of the Bethe equations depends on a choice of grading for the underlying super-algebra. We write them in the two inequivalent gradings, showing how these are related through a fermionic duality. We show that these equations reduce to the Bethe equations for the massive modes [Borsato:2013qpa] when no massless excitations are present. We demonstrate that the spectrum has degeneracies that follow from the global symmetry of the theory, as well as from translations along the four directions of the torus. We determine the weak-coupling or spin-chain limit of the Bethe equations and show these latter equations can also be obtained directly from the weakly-coupled limit of the S matrix. Further we write down the Bethe equations for the mixed NS-NS and R-R flux supported background, though in this case solving the crossing equations remains an open problem.

Subsequently, We turn to the Berenstein-Maldacena-Nastase (BMN) limit [Berenstein:2002jq] of the S matrix. In this near-relativistic limit the massless excitations become either left- or right- moving on the worldsheet and the massless S matrix degenerates into two S matrices, depending on whether massless particles of same or different worldsheet chiralities are scattered. For the scattering of same-chirality particles the S matrix is difficult to interpret within a perturbative worldsheet framework. On the other hand the S matrix for scattering particles of opposite worldsheet chiralities has a good perturbative expansion. Both these features are entirely in agreement with the general results for relativistic massless integrable theories [Zamolodchikov:1992zr, Fendley:1993wq, Fendley:1993xa] (see also [Polyakov:1983tt, Polyakov:1984et, Fioravanti:1996rz]). We compare the expansion of the mixed worldsheet chirality S matrix with the recent perturbative world-sheet results [Sundin:2016gqe]. For the most part we find exact agreement. When comparing certain terms that depend on the dressing factors, some of the perturbative calculations suffer from infrared ambiguities making a comparison less well-defined.

Finally, we study the Yangian symmetry underlying the massless-massless S-matrix. We find that the scattering problem in this sector is controlled by a Yangian algebra of the same general type as the massive-massive one. We obtain the specific evaluation representation and the corresponding crossing-symmetry conditions, consistent with the traditional Hopf-algebra framework. We display the Yangian hypercharge generator, and give the rules for its coproduct and charge-conjugation at the zeroth and the first Yangian level. One advantage of small-rank algebras is that it becomes rather elementary to prove a host of determinantal identities. The existence of such identities, and of the associated Yangian central elements, is connected to general principles of integrability. It is however often difficult to get a hold of them in higher-dimensional AdS/CFT situations. In this respect, the massless sector of in particular reveals itself as a favoured playground for testing a variety of exact algebraic methods [Borsato:2013qpa, Pittelli:2014ria, Stromwall:2016dyw].

This paper is organised as follows. In Section 2 we investigate the analytic structure of the massless modes and solve the crossing equations for the massless and mixed-mass dressing factors. We discuss possible homogeneous solutions of the crossing equations as well as the absence of massless bound states. In Section 3 we determine the Bethe equations for the closed string spectrum, show that these have the expected symmetries and find their weak-coupling limit. We also comment on how the Bethe equations generalise to the background supported by mixed NS-NS and R-R flux. In Section 4 we compare the near-BMN limit of our S-matrix to the perturbative calculations of [Sundin:2016gqe], while in Section 5 we investigate the underlying Yangian symmetry in the massless mode sector. Following the conclusion, we present a number of appendices where some of the more technical results are contained. In Appendix A we give a short review of massless scattering in relativistic integrable systems which we hope might furnish an easy access-point to this classic material.

2 Solving crossing

Symmetries severely constrain the two-body worldsheet S matrix of strings on , determining it up to four independent dressing factors. Scattering of purely massive excitations involves and , while massless-massless and mixed-mass scattering involve and , respectively.333The dressing factors and are related by unitarity. The dressing factors satisfy crossing equations which severely restrict their form. Solutions to the crossing equations for , , have been found some time ago [Borsato:2013hoa] and agree, modulo a small discrepancy discussed in the introduction, with a number of direct calculations [Beccaria:2012kb, Beccaria:2012pm]. In this section we solve the crossing equations for the massless and mixed mass dressing phases and .

2.1 The crossing transformation

Let us start by describing the crossing transformation for massive and massless excitations. Recall that massive excitations have a dispersion relation

(2.1)

It is then useful to introduce Zhukovski variables

(2.2)

so that444In what follows, we will often indicate the arguments of functions as subscripts where convenient, e.g. .

(2.3)
Figure 1: The crossing transformation for massive variables in the planes. Note that, following the conventions of [Borsato:2013hoa], the path crosses the unit circle below the real line. The red and blue zig-zag patterns depict the branch cuts of the massive dressing factors [Borsato:2013hoa].

Massless excitations have a dispersion relation of the form

(2.4)

which, just like the massive one, is -periodic. The massless Zhukovski variables

(2.5)

can be thought of as the limit of equation (2.2), and have the same -periodicity. Finally, as can be seen from the shortening condition (2.3), massless variables satisfy for any , so that it is useful to define

(2.6)

Then their dispersion is simply .

In analogy with relativistic massless particles that have , we may consider a fundamental region for to be and introduce a notion of left- and right-movers on the world-sheet. Right-movers would then have , and lie in the upper-right quadrant of the -plane, while left-movers with would lie in the upper-left quadrant owing to the in equation (2.5), see also Fig. 2. However, it is inconvenient to use a discontinuous map for the massless Zhukovski variables. Taking advantage of the periodicity of (2.5), we therefore define the fundamental region to be

(2.7)

This somewhat obscures the parallel with the relativistic case, but has the advantage that all the discontinuities of the kinematics lie at the boundary of the fundamental region. Left-movers now have and are still mapped to the upper-left quadrant in Fig. 2. Note that all singularities of and hence of lie at the boundaries of our fundamental region.

Under the crossing transformation, energy and momentum must change sign,

(2.8)

In terms of the Zhukovski variables, crossing takes a similar form for massive and massless modes

(2.9)

However, the crossing transformation looks different in the two cases. The physical region for massive modes is , with the imaginary part being positive for and negative for . Then the crossing transformation takes us inside the unit circle, see Fig. 1. If the dressing factor has branch cuts, it is natural to define them on the circle, where changes sign.

For massless modes, again we want to send through the branch cut of the energy in the -plane, see Fig. 2. In the -plane, real momenta live on the upper-half circle owing to the definition (2.6). Crossing takes us to the lower half-circle by crossing the real line, where changes sign. Comparing Fig. 2 with Fig. 1 we can think of the path as coming from the limit of  for in the massive case, when its endpoints tend to the unit circle—which is what happens as we take .

Figure 2: The - and -planes for massless particles. The thick magenta line indicates the domain of real physical momenta. The zig-zag patterns denote where the energy changes sign. Given a point or in corresponding to the real momentum, we depict curves for the crossing transformation. On the -plane, these send and cross a cut of the energy. On the -plane, we have while crossing the real line with .

It is also convenient to introduce the massless rapidity

(2.10)

Clearly the physical region is . The energy has cuts for real with , cf. Fig. 3. The crossing transformation takes to itself through the cuts.

2.2 The crossing equations

The S matrix of strings contains four dressing factors. For massive modes, and correspond to the scattering of particle of equal or opposite , respectively. Scattering of two massless modes gives , while mixed-mass scattering gives . They must obey the crossing equations

(2.11)

where

(2.12)

These equations are supplemented by the constraints due to unitarity which requires555Braiding unitarity relates to the massive-massless phase  [Borsato:2014hja]. In fact, owing to that relation, we need only consider .

(2.13)

Introducing666The discussion below is written for any dressing factor and associated phase . It applies equally to each of the four dressing factors , , and .

(2.14)

we will often write when no ambiguity can arise. For physical rapidities the phases have an expansion [Arutyunov:2004vx, Hernandez:2006tk, Beisert:2005wv]

(2.15)

in terms of the local charges of the integrable system [Beisert:2004hm]. Above, the are -dependent coefficients and the local charges are given by

(2.16)

In backgrounds the momentum may enter the expansion (2.15[Beccaria:2012kb]. It is convenient to express in terms of a simpler function

(2.17)

in the case of massive scattering. For mixed-mass scattering one can use (2.6) to reduce the decomposition to two terms, while for massless scattering no decomposition is necessary. The expansion for analogous to (2.15) takes the form

(2.18)

At strong coupling the coefficients have the expansion

(2.19)

The terms are known as AFS [Arutyunov:2004vx] and HL [Hernandez:2006tk] orders, respectively.

Figure 3: The -plane for massless variable.

2.3 Finding

In this subsection we will find the all-loop expression for . The crossing equation for can be decomposed into two auxiliary problems

(2.20)

which we will solve separately.

2.3.1 Finding the massless dressing factor

The crossing equation for involves an auxiliary rapidity . While its dependence on is undetermined, we know that it arises from an invariance and that under crossing we have [Borsato:2014exa, Borsato:2014hja]

(2.21)

Using this, and the fact that  is of difference form in , we can write down a familiar minimal solution for  of difference form [Reshetikhin:1990jn]

(2.22)

It is also straightforward to write homogeneous solutions to the crossing equation. However, as we will argue in Section 4 no such solutions are compatible with perturbative string theory computations of [Sundin:2016gqe] and we are led to take . This trivializes the complete S matrix. In particular the dressing factor , which is consistent with crossing in this limit, since . The massless dressing factor then reduces to

(2.23)

In terms of the crossing equation takes the form

(2.24)

2.3.2 The Riemann-Hilbert problem for

It is useful to express the crossing equation for in terms of . For example (2.24) takes the form

(2.25)

This equation is defined when , see Fig. 2, and involves the value of  on two sheets. If we analytically continue the point  to somewhere very close to the cut of  (and let’s say above it), the equation reads

(2.26)

This is a Riemann-Hilbert problem that can be solved by the Sochocki-Plemelj theorem [Sochocki:1873a, Plemelj:1908a], see also e.g. [Volin:2009uv]. The resulting solution is minimal in the sense that by construction it only allows for the singularities necessary to solve (2.26), and needs to be appropriately anti-symmetrised to account for unitarity (2.13). While it is useful to characterize the solution in this way, it is more convenient to work directly in the -plane, and to make anti-symmetry more manifest from the beginning. To this end, it is helpful to briefly recall some properties of the massive HL phase .

The massless all-loop crossing equation for  has the same form as that of the massive HL phase

(2.27)

where now are interpreted as massive variables. The massive HL phase has the integral representation

(2.28)

valid in the physical region , , with

(2.29)

This definition deviates slightly from the one known in the literature for the choice of the branch-cuts of the logarithms in .777Our choice here ensures that such branch-cuts do not intersect the upper-half disc (lower-half disk) in the -plane in the case of (). As we will see in the next subsection this choice simplifies the analysis of crossing for . In Appendix B we show that (2.28) defines an anti-symmetric function. The expression in (2.28) has discontinuities across the unit circle; defining in the crossed region through analytic continuation, one may check that it satisfies (2.27). Is is also worth mentioning that these expressions can be integrated to an expression involving dilogarithms that has appeared in the literature [Beisert:2006ib, Arutyunov:2006iu].

2.3.3 Deforming the contour for the HL phase

While the massive HL phase satisfies the same crossing equation as , the position of its branch cuts differs from those of the massless kinematics. Massless dressing factors should have branch cuts for real , as illustrated in Fig. 2. Instead (2.28) has a branch-cut on the half-circle, which is exactly where physical massless particles should live. Thankfully, we can analytically extend the integral (2.28) by deforming its integration contour, in such a way as to move the branch cut to the unit segment, as showin in Fig. 4. What is more, as we show in Appendix D.2, the resulting function continues to satisfy the crossing equation that follows from (2.27). Relegating the technical details to Appendix D.1, after the contour shift we find

(2.30)

where888The function does not depend on the choice of sign that enters .

(2.31)

When shifting the contour, we regulate with an -prescription to avoid collision with the branch-cuts of the functions (see Fig. 4). The contributions appearing in the third line of (2.30) do not involve integrals and come from poles that we pick when moving the contours of integration, as depicted in Fig. 4. The representation (2.30) is valid for and similarly for , with no restrictions on the masses of the excitations.

     

Figure 4: Red dashed lines correspond to the branch cuts of the integrand, while red dots to the poles. In this specific example we move the contour of integration (blue line) from the upper semicircle to the real interval and we pick up a pole.

The phase in equation (2.30) has discontinuities on the real line, as we wanted. As proven in Appendix D.2, we therefore have constructed an anti-symmetric function that satisfies the crossing equation for  and is compatible with massless kinematics. Furthermore, this functions still satisfies massive crossing (2.27) too, as long as the crossing path is taken as in Fig. 1. Hence it can be employed in the massive, massless, or mixed-mass cases. A similar result could have been found starting from the dilogarithm expression of [Beisert:2006ib, Arutyunov:2006iu]. As all these functions coincide in the region , , they give rise to the same analytic extension once we move the branch cut to the real line.

2.3.4 Expansion coefficients for with arbitrary masses

It is often useful to represent the dressing factors as a sum over conserved charges, cf. equation (2.15). Such a representation is valid in the physical region, which depends on the particular kinematics of the excitations under consideration. Specifically, for massive kinematics, the expansion is well-defined for , while for massless modes the variable or lie on the upper-half-circle. In this latter case it is worth noting that the expansion is in fact a Fourier series. In Appendix C we show that the coefficients for are the same as the HL coefficients [Hernandez:2006tk], i.e.

(2.32)

This means that, just as the expression (2.30) is valid for massive, massless and mixed-mass kinematics, so is the double-series representation (2.15) with the coefficients as in (2.32).

2.3.5 Minimal solution of the massless crossing equation

The expression (2.30) is well defined in the massless kinematics, and as shown in Appendix D.2, , satisfies the massless crossing equation (2.24) where the crossing path is taken as in Fig. 2. Evaluating (2.30) in the massless kinematics (2.6) we conclude that the minimal solution for is

(2.33)

In Appendix B we show that the above expression for is anti-symmetric under . Therefore, up to possible CDD factors’, we have solved the crossing equation for .

2.4 Solution of homogeneous crossing

As is well known [Castillejo:1955ed], solutions of crossing equations are only defined up to CDD factors, i.e., solutions of the homogeneous crossing equation. There is a huge degeneracy of such solutions in the massless case. Evaluating (2.16) on the massless kinematics, we find . Therefore any finite linear combination of functions of the form

(2.34)

solves the homogeneous crossing equation. While it is very hard to exclude all such possible solutions, they do not appear to have any particular physical significance. Therefore, we shall focus on a particular class of solution of homogeneous crossing that emerge as massless limits of the massive kinematics.

Note that in the massless limit where , the right-hand-side crossing equation for massive modes (2.11) becomes the same as the one of the massless crossing equation (2.11).999Similarly in this limit the crossing equation satisfied by the BES phase (2.36) also reduces to the massless crossing equation. Therefore, one might wonder whether, in the massless limit, the phases and might suggest a natural homogeneous solution, and whether such a solution should be included in . Our minimal solution was constructed out of the massless limit of the HL-order term in the BES phase. Below we investigate the (leading) AFS term and the higher order terms in the BES phase in the massless limit. As we will show, in this limit the higher-order terms become trivial, while the AFS term will provide a hint that a homogeneous term at this order might naturally be expected.

2.4.1 The BES phase

An important ingredient of integrable holography is the non-perturbative BES phase [Beisert:2006ez] which, can be written as a double contour integral over unit circles [Dorey:2007xn]

(2.35)

The resulting phase satisfies the crossing equation

(2.36)

The BES phase (2.35) can be expanded asymptotically using that [Borsato:2013hoa]

(2.37)

where the first term gives the AFS phase and the second one gives the HL one [Vieira:2010kb].

2.4.2 Sub-leading orders

We start by considering the terms arising from the series in (2.37), i.e., these beyond HL order which can be written as

(2.38)

The above expression is ill-defined due to the poles at and . We can regularize this integral by giving a principal-value prescription in e.g. . In fact, to preserve antisymmetry we should sum over the case where the principal value is in and in .101010Equivalently, we may think of taking the radii of the integration circles to be e.g.  and with and small. Anti-symmetry then requires us to sum over this case and the one with . In this way, one recovers for any the expressions of ref. [Beisert:2006ib].

In the massless case we have additional poles on the integration contour. Since , the poles at need to be regularized too. By applying the same prescription, we now find that the integral in e.g.  vanishes as all poles for are on the unit circle. We conclude that the regularization of vanishes when , and therefore we do not expect the sub-leading pieces of the BES phase to play a role in the massless kinematics.

2.4.3 HL order

In the previous sub-section, we discussed how can be deformed to solve crossing for . Recall that and differ from one another by a phase  [Borsato:2013hoa]

(2.39)

Further, is non-trivial precisely at HL order and the corresponding is

(2.40)

where

(2.41)

For massless kinematics, the crossing equation for becomes homogeneous and so one may wondered whether, in this limit, gives rise to a natural non-trivial homogeneous solution at the HL order. In Appendix E we show that by deforming the contour and restricting to massless kinematics, becomes trivial.

2.4.4 AFS order

At the leading order of the strong-coupling expansion (2.37) one recovers the AFS phase, which can be concisely expressed as a series

(2.42)

which is valid for arbitrary masses and . Performing the sum we find

(2.43)

In contrast to and the sub-leading orders of the BES phase, the above expression for the AFS phase does not vanish in the massless limit.111111We could have reached the same conclusion working in terms of the integral obtained from the expansion of (2.37). Furthermore, it is straightforward to check that for two massless excitations under crossing

(2.44)

In fact, as is clear from equation (2.43), taking the massless limit and performing the crossing transformation are two commuting operations. Therefore, while the AFS phase constructed as the limit of the massive AFS phase is non-zero, it satisfies the homogeneous crossing equation. This observation suggests that a homogeneous solution at AFS order may be present in .

2.4.5 Solution of the massless crossing equation

On the basis of the above arguments, we propose the following solution for the massless crossing equation

(2.45)

which differs from the minimal one by the addition of the AFS term that plays the role of a CDD factor.

2.5 Determining the mixed-mass dressing factor

As we have discussed above, the deformation of the integration contour for the HL-order phase can be carried out independently of the mass of the excitations. Further, the AFS-order phase is also well behaved when the mass is varied. As a result, it is straightforward to write down a solution for the mixed-mass crossing equation (2.11). By defining

(2.46)

in terms of (2.302.43) we can check that the crossing equation is satisfied. More specifically, the AFS part of the phase satisfies the crossing equation

(2.47)

in the mixed-mass case, while the HL part satisfies

(2.48)

As for the solution of the homogeneous crossing equation, it is worth noticing that in this case the massless limit of the crossing equation (2.36) does not give our mixed-mass equation. Therefore, it appears that no natural candidate exists for a physically relevant class of solutions to homogeneous crossing.

2.6 Absence of bound states

The massless S matrix is given explicitly in Appendix M of [Borsato:2014hja], supplemented by the dressing factors we just constructed. We are interested in studying its poles, and discussing whether they may be interpreted as arising from bound states. Several entries of the S matrix have a simple pole when

(2.49)

This is the familiar bound-state condition for the scattering of massive particles with , cf. [Dorey:2006dq]. In the massless kinematics (2.6), this reads

(2.50)

It follows that the total energy and momentum of any putative bound state would be

(2.51)

Hence such a bound state would have to be a singlet of the symmetry algebra. Further, solving (2.50) in terms of momenta we find

(2.52)

For in the physical strip (i.e., when ), the only solutions are . Therefore, this putative bound state would have to be a singlet constructed out of two particles with purely imaginary and opposite momenta, and is therefore unphysical. As discussed in Appendix F, this structure is not modified by the dressing factors.

The above conclusions can be further confirmed by considering the residues of the -invariant S matrix of [Borsato:2014hja] when the momenta satisfy equation (2.50). One finds non-vanishing residues from four out of the six S-matrix elements, resulting in a residue matrix, just like in the massive case. However, when imposing the massless condition (2.6), the residue matrix has rank one, again indicating that only one mode could be potentially propagating. This leads us to conclude that any putative bound-state would have to be a singlet and hence unphysical.

Let us now consider the case of one massive and one massless particle. Since massive particles have while massless ones have , it is easy to check that it is impossible to construct a bound state satisfying (2.49) for a pair of particle with complex and conjugate momenta. Interestingly, looking at the total energy and momentum, we find

(2.53)

Therefore, such a two-particle configurations has the same dispersion relation as a single particle in the “mirror” kinematics [Arutyunov:2007tc]. We conclude that there are no physical bound states between massless and massive particles.

3 Bethe equations

In this section we write down the all-loop nested Bethe equations for the full worldsheet theory, including the massless modes. They are found by imposing periodicity of the wave-function on the worldsheet, and their solutions give quantisation conditions for the momenta of the excitations. Our analysis will be restricted to states which carry zero momentum and winding along the . Given the complexity of the S-matrix we need to appeal to the “nesting procedure” [Yang:1967bm, Beisert:2005tm] to diagonalise it, which introduces new auxiliary roots. In Sections 3.1 and 3.2, we write down the Bethe equations in bosonic and fermionic gradings respectively, and discuss some of their properties and symmetries. In Section 3.3 we comment on extra symmetries of the Bethe equations associated to the presence of the massless modes, while in Section 3.4 we present the weak coupling limit of the equations. In Section 3.5 we discuss the Bethe equations for the background supported by mixed NS-NS and R-R three-form flux.

3.1 Bethe equations in the bosonic grading

The derivation of the Bethe equations closely follows the one discussed in [Borsato:2013qpa] for the massive sector. Here we include also the massless excitations, and rather than giving the details of the procedure, which can be found in [Borsato:2016hud], we outline the principal points of the method.

We begin by choosing a maximal set of excitations above the BMN vacuum that scatter diagonally with each other, which will make up the level I of the Bethe equations. There are several possible sets of such excitations, corresponding to different choices of gradings of the symmetry algebra. One possible choice, compatible with that of [Borsato:2013qpa], is

Figure 5: The Left and Right massive modules. We indicate explicitly only the lowering supercharges, corresponding to the arrows pointing downwards. In each module the fermions transform in a doublet of .

The first two excitations are massive and belong to the representations shown in Fig. 5, while the latter two are from the massless module depicted in Fig. 6. Note that we have assumed that the symmetry acting on the massless excitations trivially commutes with the S matrix, as implied by the perturbative computations of [Sundin:2015uva, Sundin:2016gqe]. If this were not true, only one of the massless fermions would appear at level I, and we would have an additional auxiliary root corresponding to the action of the lowering operator of . For completness, we have written down such a set of Bethe equations in Appendix G.3.

For each level-I excitation we introduce a set of momentum-carrying Bethe roots. Since the two massless fermions carry exactly the same charges except for the spin, we will only use a single type of root to describe both at the same time. The Bethe roots and excitation numbers corresponding to each type of level-I excitation are summarised in the table below.

Left massive Right massive Massless
Level-I excitation ,
Bethe root
Excitation number

Since scattering among these excitations is diagonal, it is trivial to write down an eigenstate of the S-matrix and the corresponding Bethe equations as long as no other excitations are present.

Figure 6: The massless module. Dashed lines denote the action of generators.

When we include other excitations, non-diagonal processes arise. For these we use the nesting procedure, see Appendix G, and introduce auxiliary Bethe roots, which correspond to the action of the supercharges on the level-I excitations. Acting once on a level-I state we generate level-II excitations, namely the massive fermions of Fig. 5 and the massless bosons of Fig. 6. As suggested by these figures it is enough to introduce two sets of auxiliary roots

,   ,
Bethe root
Excitation number

Below we will discuss how two additional sets of auxiliary roots can be introduced, so that each type of root corresponds to the action of one supercharge.

The massive momentum-carrying roots satisfy the equations