The BMN string at two loops
Per Sundin and Linus Wulff
Universitá di Milano-Bicocca and INFN Sezione di Milano-Bicocca,
Dipartimento de Fisica, Piazza della Scienza 3, I-20126 Milano, Italy
Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.
We calculate the two-loop correction to the dispersion relation for worldsheet modes of the BMN string in for . For the massive modes the result agrees with the exact dispersion relation derived from symmetry considerations with no correction to the interpolating function . For the massless modes in however our result does not match what one expects from the corresponding symmetry based analysis. We also derive the S-matrix for massless modes up to the one-loop order. The scattering phase is given by the massless limit of the Hernández-López phase. In addition we compute a certain massless S-matrix element at two loops and show that it vanishes suggesting that the two-loop phase in the massless sector is zero.
- 1 Introduction
- 2 Green-Schwarz string action
- 3 BMN expansion in
- 4 Two-loop dispersion relation
- 5 Massless S-matrix in
- 6 Conclusion
- A Sunset integrals for massive propagator in
- B Decoupling of massless modes in
- C Sunset integrals for massless propagator in
The point-like BMN string solution plays a special role in the AdS/CFT-correspondence. Expanding around this solution gives a systematic way to compute corrections to the anomalous dimensions of certain long single-trace operators in the CFT from quantum corrections to the energy of the string [Berenstein:2002jq]. From the point of view of the string worldsheet theory one can fix a light-cone gauge adapted to the BMN geodesic to obtain a free theory of 8 massive (and massless in general) bosons and fermions plus an infinite number of interaction terms suppressed by inverse powers of the string tension. One can then carry out perturbative calculations in this (non-relativistic) 2d field theory. Here we will be interested in examples for which the worldsheet theory is known to be classically integrable (and believed to be quantum integrable). In many such examples one can guess the exact S-matrix and dispersion relations for the 2d theory from symmetries with a few extra assumptions. Direct perturbative calculations then give a valuable test of these arguments and the assumptions made.
From the point of view of perturbative calculations it turns out that the simplest examples are strings in for .111The corresponding supergravity backgrounds preserve 8, 16 and 32 supersymmetries respectively. For the discussion of classical integrability see the original papers [Bena:2003wd, Babichenko:2009dk, Sundin:2012gc, Sorokin:2011rr, Cagnazzo:2011at] or, for a general treatment for strings on symmetric spaces, see [Wulff:2015mwa]. The reason for this is that in these models one has only even order interaction vertices in the BMN expansion. In particular there are no cubic interactions which leads to a big reduction in the number of possible Feynman diagrams compared to other cases such as and [Zarembo:2009au, Rughoonauth:2012qd].
For a long time progress in going beyond tree-level was hampered by the fact that it was not clear how to deal with the divergences that show up and how to regularize in a way that preserves the symmetries expected of the answer. Therefore calculations were done either in the so-called Near-Flat-Space limit [Maldacena:2006rv, Klose:2007wq, Klose:2007rz, Puletti:2007hq, Murugan:2012mf, Sundin:2013ypa], using generalized unitarity [Bianchi:2013nra, Engelund:2013fja, Bianchi:2014rfa] or by computing only quantities that were explicitly finite [Rughoonauth:2012qd, Sundin:2012gc, Abbott:2013kka, Sundin:2014sfa]. Thus one could side-step the issue of regularization. Recently however this hurdle was overcome in [Roiban:2014cia] where it was shown how to compute the one-loop correction to the dispersion relation and S-matrix by correctly treating the divergences as wave function renormalization and using a scheme for reducing the integrals that appeared to a smaller subset which could be easily computed. The same ideas were then applied to the string with a mix of NSNS and RR flux and to the calculation of the two-loop dispersion relation for both massive and massless modes in the Near-Flat-Space limit in [Sundin:2014ema].
Here we will extend these techniques to compute for the first time222The two-loop partition function for folded strings in , related to the cusp anomalous dimension on the gauge theory side, has been computed in [Roiban:2007jf, Roiban:2007dq, Roiban:2007ju, Giombi:2009gd, Giombi:2010fa, Giombi:2010zi, Iwashita:2011ha]. the full BMN two-loop dispersion relation. The result takes the following form
For and the massive modes in the result agrees with the expansion of the proposed exact dispersion relation suggested in [Hoare:2013lja, Lloyd:2014bsa] which takes the form
provided that we take , i.e. the interpolating function receives no corrections up to two loops (strictly speaking this is needed only when ). Here measures the amount of NSNS flux of the background and measures the amount of RR flux. The dispersion relation is obtained by setting . To compare this dispersion relation with the one we calculate in the BMN limit one needs to rescale the spin-chain momentum as (the shift by is needed in the comparison because we have defined the massive modes such that their quadratic action is Lorentz-invariant, see footnote 7). For the symmetries are not enough to completely fix the form of the dispersion relation [Hoare:2014kma] but nevertheless we find the same two-loop correction as for the other cases but with replaced by (the same was observed at one loop in [Roiban:2014cia]).
For the massless modes in the exact dispersion relation was suggested, based on a similar symmetry argument that gave the massive mode dispersion relation, to have the same form as in (1.2) except with the in the first term replaced by [Lloyd:2014bsa]. Here however the worldsheet calculation is not in agreement (note that the BMN rescaling is in this case without the -shift). This was already noted in the Near-Flat-Space limit in [Sundin:2014ema].333As remarked there this means that the discrepancy is not due to one calculation being done in type IIA and the other in type IIB as the string action is the same for the two cases in the NFS-limit. In fact we have explicitly checked that the correction to the dispersion relation is the same in the type IIB case. Though we find the same form of the dispersion relation to two loops the coefficient of the two-loop correction differs by a factor of . The -factor can be traced to the types of integrals that contribute in this case, they have one massless and two massive modes running in the loops, and are thus quite different from the integrals with three massive modes which contribute to the massive dispersion relation. As speculated in [Sundin:2014ema] this mismatch could be due to a misidentification of the asymptotic states in the two approaches or due to unexpected quantum corrections to the central charges. Unfortunately we will not be able to resolve it here.
In order to gain further insight into the role of the massless modes we also probe the worldsheet S-matrix in the massless sector of the theory of . At tree-level we find that there is no phase contribution and at one loop the dressing phase is simply given by the massless limit of the well known Hernández-López phase [Beisert:2006ib](up to an IR-divergent piece arising from the limit taken). We furthermore find, to the extent the type IIA and type IIB comparison is valid, that our results are consistent with the symmetry based analysis of [Borsato:2014hja].
We then push the analysis to the two-loop level and compute a forward type scattering element of two massless bosons. Since we are using the full BMN-string, where the relevant vertices are fourth, sixth and eight order in transverse fields, a large class of distinct Feynman diagrams contribute. However, once we go on-shell, and focus on the kinematic regime where the sign of the transverse momenta of the scattered particles is opposite, we find that the contribution from each topology vanishes separately. That is, each integral is multiplied with high enough powers of the external momenta to vanish once we go on-shell. Thus the entire two-loop part of the massless S-matrix, for this specific scattering element, is zero. This indicates that there should be no contribution from the phase at this order in perturbation theory.
The outline of the paper is as follows: In section 2 and 3 we write down the string action and perform the BMN expansion. In section 4 we evaluate the two-loop dispersion relation and explain in detail how to regularize the divergent integrals that appear in the computation. Finally, in section 5 we analyze the massless S-matrix up to the two-loop order and compare with the exact solution, to the extent it is known. We end the paper with a summary and outlook. Some details of the dispersion relation calculation are deferred to appendices.
2 Green-Schwarz string action
The Green-Schwarz superstring action can be expanded order by order in fermions as444The sign of the action is due to using opposite conventions for the 2d and 10d metric.
where is the string tension. In this expansion is known to all orders due to the background being maximally supersymmetric [Metsaev:1998it]. In a general type II supergravity background however, the expansion is only known explicitly up to quartic order [Wulff:2013kga]. This is the action we will use for the string in and . Its form is as follows. The purely bosonic terms in the Lagrangian are given by
where we denote the purely bosonic vielbeins by and is the lowest component in the -expansion of the NSNS two-form superfield . The terms quadratic in fermions take the form555These expressions refer to type IIA. To get the type IIB expressions on should replace the -component Majorana spinor by a doublet of -component Majorana-Weyl spinors and the gamma-matrices by ones Finally, instead of the defined in (2.6) one should use the expression appropriate to type IIB (2.3) For more details and definitions of the gamma-matrices see [Wulff:2013kga].
where the Killing spinor derivative is defined as
is the spin connection, is (the bosonic part of) the NSNS three-form field strength and the type IIA RR fields enter the action through the bispinor
Finally the quartic terms in the Lagrangian take the form
Where we have defined two matrices which are quadratic in fermions
while . In addition two new matrices constructed from the background fields contracted with gamma-matrices appear at this order
The first appears in the dilatino equation and the second in the integrability condition for the Killing spinor equation. Due to this fact and will be proportional to the non-supersymmetric (non-coset) fermions in symmetric space backgrounds such as the ones we are interested in.
3 BMN expansion in
The string action simplifies in the cases we are considering since all background fields are constant. The backgrounds we consider are supported by the following combinations of fluxes (see [Wulff:2014kja] for conventions)
where is supported by a combination of NSNS- and RR-flux parameterized by satisfying
Note that for and we are taking the type IIA solutions but we could of course also have used the IIB solutions obtained by T-duality along a torus direction. The and radius are both set to one in these conventions. From (2.6), (2.3) and (2.9) we find
where we have defined the following projection operators
the index indicating the dimension of the space they project on, i.e. the number of supersymmetries preserved in each case.
The -metric is taken to be
where the transverse coordinates are grouped together into two complex coordinates in , one in and one real coordinate, , in . Similarly the -metric is
In these coordinates the NSNS two-form appearing in eq. (2.2), which is only non-zero for , takes the form
Plugging the form of the background fields into the action the next step is to expand around the BMN-solution given by [Berenstein:2002jq]. At the same time we fix the light-cone gauge and corresponding kappa symmetry gauge
The Virasoro constraints are then used to solve for in terms of the other fields. In this gauge the worldsheet metric defined in (2.2) takes the form , where denotes higher order corrections to be determined from the conditions on the momentum conjugate to
Rescaling all transverse coordinates with a factor yields a perturbative expansion in the string tension666The fact that only even orders appear is a special feature of the backgrounds that make them particularly suited to a perturbative treatment.
where the subscript denotes the number of transverse coordinates in each term. The quadratic Lagrangian takes the form777Here . For with mixed flux we have performed a field redefinition of the massive modes and and a similar redefinition for the fermions which puts the quadratic action into a Lorentz-invariant form.
where unprimed indices run over massive modes and primed indices run over massless modes and except for with mixed flux in which case . The spectrum can be summarized as follows
Here all coordinates are complex except and , originating from the transverse directions of and respectively.
4 Two-loop dispersion relation
We now turn to the problem of determining the two-loop correction to the two-point function, i.e. the correction to the dispersion relation. There are three different topologies of Feynman diagrams that appear. The first, and by far the most complicated, are the sunset diagrams
which lead to the following loop integrals
where the masses of the virtual particles are either all the same, , or two the same and one different . Generically the integrals are (power counting) UV-divergent and sometimes IR-divergent when massless particles are involved and must be regularized. Our procedure for regularizing and computing the relevant integrals is described in the next section.
The second type of Feynman topologies are the four-vertex bubble-tadpoles
which lead to a combination of a tadpole integral888Note that can be expressed in terms of (a sum of) by shifting the loop variable. In the following we will assume that this is done and simply write .
and a bubble integral
where both the bubble and tadpole have .
Finally we have the double-tadpoles built out of a six-vertex
which lead to a product of two tadpole integrals (4.4) and are the simplest to evaluate (although a bit cumbersome since the sixth order Lagrangian contains many terms).
4.1 Regularization procedure
Our regularization scheme is similar to the one found to work at one loop in [Roiban:2014cia]. It is based on reducing the integrals that appear, via algebraic identities on the integrand and shifts of the loop variable, to less divergent, or finite, integrals plus tadpole-type integrals. In this way it turns out to be possible to push all UV-divergences into tadpole-type integrals which can then be easily regularized. In and there are also massless modes present which lead to potential IR-divergences. These turn out to be simpler to deal with and we do this simply by introducing a small regulator mass for these modes which is sent to zero at the end. IR-divergences turn out to cancel within each class of diagrams independently unlike the UV-divergences. In [Roiban:2014cia] it was explained how to reduce the bubble integrals so here we will focus on the sunset integrals in (4.2).
The first step is to use the simple identity
on the integrand. This leads to a sunset integral with a lower degree of divergence plus an integral with one less propagator, which, by shifting the corresponding loop momentum , leads to a product of two one-loop tadpole integrals (4.4). By repeating this process the sunset integrals we have to compute are reduced to the following ones
Integrals with are still (power-counting) UV-divergent of course. Note that it is enough to compute the first and third of these as the second and fourth differ only by replacing a -index by a -index and vice versa. To evaluate these it is useful to start by considering the one-loop bubble integrals defined in (4.5). Following [Roiban:2014cia] we use the algebraic identity
on the integrand to derive the recursion relation999Here we have used shifts of the loop variable in the tadpole integrals that appear and the fact that for by Lorentz-invariance.
and the same for with the order of the indices switched and . We can now use these relations inside the sunset integrals and we find
When the latter integral does not contribute, however when we need to compute it. To do this we use the fact that to find the dispersion relation we only need to compute the integrals on-shell101010To compute for example the wave function renormalization at two loops one would need to evaluate the sunset integrals off-shell. This would require a more sophisticated approach to the regularization. and for the contributing sunset integrals (4.2) have . This in turn implies the algebraic identity
from which it follows that
where we have shifted in the last term. All the integrals appearing in the sum are in fact zero by Lorentz invariance since they are evaluated at . Putting this together we have the relations
and the same for with . For general masses these recursion relations allow us to solve for integrals of the type in terms of integrals of the type , which turns out to be enough for our purposes. However, in the case when all masses are equal, i.e. , we have the extra symmetry which allows us to solve completely for in terms of and we find
and the same expression with instead of for .
Repeating the same steps for the integrals we find
and using the fact that (note that the last term vanishes unless )
where we have shifted the loop variable in the tadpole-like integral and used the Lorentz symmetry of the measure, this becomes
and the same for with . It turns out that for these expressions to be consistent we must use a scheme where all power-like divergences are set to zero so that for example
and the last term in the first expression drops out. This is of course what we often do, e.g. in dimensional regularization, and was needed also at one loop [Roiban:2014cia]. For general masses these recursion relations turn out to be enough for our purposes while in the special case that all masses are equal we can again use the symmetry in the indices to solve for completely in terms of
Note the appearance of divergences when is odd, coming from the tadpole integral , which were absent in (4.17). Let us recall again that in this derivation of the identities (4.16), (4.20), (4.17) and (4.22) for sunset integrals we have used the fact that the integrals that occur have either
The relations for the former type of integrals are valid off-shell and for the latter only on-shell.
and, in the case of massless external legs,
where we have evaluated the finite sunset integrals by standard means (e.g. Feynman parametrization). The tadpole integral is IR-finite but contains a logarithmic UV-divergence and it can be evaluated for example in dimensional regularization. However as the contribution from this integral cancels out in the final answer we do not need to explicitly evaluate it.
After this discussion of the regularization issues involved we can now turn to the actual computation of the two-loop correction to the dispersion relation. Since some of the expressions involved are very long we have chosen to include the full set of contributing integrals for the bosons in and the massless bosons in in the appendix only. The analysis for the remaining cases is similar and we will skip most of the technical details. We start by analyzing the correction for massive modes.
4.2 Massive modes
The simplest case is that of where all worldsheet excitations have the same mass. Due to the maximal supersymmetry of the background the Green-Schwarz action is known to all orders in fermions and is given by the supercoset model of [Metsaev:1998it, Arutyunov:2009ga]. Since we know in particular the sixth order -terms this allows us to compute the two-loop correction to the two-point functions for the fermions as well. Thus, for we will compute the two-loop correction to the dispersion relation for both bosons and fermions.
We start by discussing the sunset diagrams. The expression for the sunset contributions, before any integral identities are used, is quite lengthy. For completeness it is given, for the bosons, in equation (A.1). After going on-shell and using the identities for sunset integrals derived in the previous section it simplifies dramatically however and the end result is, for bosons and fermions respectively,
Using the integral identities for bubbles and tadpoles the bubble-tadpole contribution becomes (the full off-shell expression for the bosons without using any integral identities is given in (A))
so the bubble-integrals left after using (4.7) cancel out.
Finally the six-vertex tadpole contribution becomes (again the full expression for the bosons is given in (A))
Summing the three contributions gives
in agreement with the proposed exact dispersion relation. It is worth pointing out that this is the first two-loop computation ever performed utilizing the full BMN-string so it is gratifying to see that the final result is manifestly finite and in agreement with what we expect based on symmetries and related arguments.
This theory is more complicated since beside massive worldsheet excitations, with mass (which reduces to in the pure RR case), we now have massless excitations as well. The massless modes correspond to supersymmetries broken by the background and to describe them properly we have to use the full GS-string which currently is only known up to quartic order in fermions. This means that we can only compute the correction to the two-point function for the bosons.
For the sunset diagrams we now have two combinations of virtual particles running in the loop; either all three particles are massive or one is massive and two are massless. The contribution of the latter turn out to exactly cancel, as can be verified from the full expression for their contribution given in (B) by making use of the identities (4.16) and (4.20). Hence, in our regularization scheme, we see that the massless modes completely decouple from the massive dispersion relation, i.e. we could simply put them to zero in the Lagrangian and work with the supercoset sigma-model. This decoupling has already been observed for the two-loop dispersion relation in the NFS-limit and for the one-loop BMN S-matrix in [Sundin:2014ema, Roiban:2014cia].