Contents

Imperial-TP-LW-2016-01

ZMP-HH/16-9.

The complete one-loop BMN S-matrix in

Per Sundin and Linus Wulff

II. Institut für Theoretische Physik, Universität Hamburg,

Luruper Chaussee 149, 22761 Hamburg, Germany

Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.

Abstract

We compute the full one-loop 2-particle S-matrix for excitations of the type IIB BMN string. The S-matrix is found to respect the expected symmetries and the phases are consistent with the crossing equations. By analyzing how the relevant integrals scale with the IR regulator we show that scattering of massless bosons is trivial at two loops. Based on our results we argue that the additional S-matrix appearing in the massless sector in the exact solution should trivialize.

## 1 Introduction

String theory in preserving 16 supersymmetries is classically integrable111This is a general property of superstrings in symmetric space backgrounds that preserve some supersymmetry [Wulff:2015mwa]. [Babichenko:2009dk, Sundin:2012gc, Cagnazzo:2012se, Sundin:2013uca, Wulff:2014kja] and is expected to be integrable also at the quantum level. If this is indeed the case it should be possible to determine the spectrum along similar lines as for the string in [Beisert:2010jr]. Here we will consider the type IIB solution supported by Ramond-Ramond three-form flux. An important first step in solving the model is to find the exact S-matrix. Up to phase factors the S-matrix can be completely determined by the symmetries [Borsato:2014hja]. Since the BMN string has both massive and massless excitations the (2-particle) S-matrix splits into three sectors: massive-massive scattering, massive-massless scattering and massless-massless scattering. Each sector comes with phase factors which should be determined by crossing symmetry together with some input from perturbation theory. In the massive sector there are two phases whose all-loop crossing symmetric solution was conjectured in [Borsato:2013hoa] and agrees with perturbative worldsheet calculations at one and two loops [Sundin:2013ypa, Engelund:2013fja, Abbott:2013mpa, Bianchi:2014rfa, Roiban:2014cia].

Here we complete the one-loop analysis of the S-matrix by computing the full one-loop 2-particle S-matrix including mixed and massless sectors. The structure of the S-matrix is found to agree with that of [Borsato:2014hja]222A change in the normalization of the mixed sector S-matrix of that paper is needed for agreement. and the one-loop phases we find in the mixed and massless sectors are shown to respect crossing symmetry. In the mixed sector we show that one-loop scattering is consistent with integrability by verifying that the amplitudes for scattering processes where the particles change their momenta vanish. We are further able to show that the scattering of massless bosons is trivial at two loops, as was argued for a particular process in the type IIA setting in [Sundin:2015uva]. Our perturbative calculations suggest that the extra S-matrix appearing in the massless sector in [Borsato:2014hja] trivializes. The perturbative findings we present in this paper were recently used in [Borsato:2016kbm] to conjecture all-loop expressions for the mixed and massless sector phases together with the corresponding Bethe equations.333For some reservations about the usefulness of said equations see [Abbott:2015pps].

In addition to the S-matrix the exact dispersion relation for the massive and massless modes is also determined by the symmetry analysis of [Borsato:2014hja]. For the massive bosons the dispersion relation agrees with perturbation theory up to two loops, as earlier shown in the type IIA setting in [Sundin:2014ema, Sundin:2015uva]. However, there a discrepancy in the two-loop correction for the massless bosons was found. Here we repeat these calculations for the type IIB string and also include the fermionic modes finding the same result. While our findings are consistent with worldsheet supersymmetry to the two loop order an explanation for the mismatch is still lacking.

The outline of the paper is as follows. In section 2 we describe the Green-Schwarz string action in to quartic order in fermions, its light-cone gauge-fixing and BMN expansion. In section 3 we describe our regularization scheme for the relevant one-loop integrals. Section 4 gives the results of the two-loop two-point function calculations and correction to the dispersion relations. In section 5 we give the form of the S-matrix and section 6 contains a discussion of the mixed and massless sector phases and crossing symmetry. We end with some conclusions and an appendix containing some details of the basic one-loop bubble integrals.

## 2 Green-Schwarz string in AdS3×S3×T4 with RR flux

The Green-Schwarz superstring action can be expanded order by order in fermions as

 S=g∫d2ξ(L(0)+L(2)+…), (2.1)

where we let denote the string tension. In a general type II supergravity background the form of this expansion is known explicitly up to quartic order in fermions [Wulff:2013kga]. This is the action we will use for the string in type IIB . The purely bosonic terms in the Lagrangian are given by

 L(0)=12γijeiaejbηab,γij=√−hhij, (2.2)

where are the (bosonic) vielbeins and we used the fact that the bosonic B-field vanishes in our case. The vielbeins can be read off from the metric which we take to be

for complex transverse coordinates while the metric of is .

The terms quadratic in fermions take the form

 L(2)=ieiaθIγaKijIJDJKjθK,KijIJ=δIJγij−εijσ3IJ, (2.4)

where () are the two 16-component Majorana-Weyl spinors of type IIB and the Killing spinor derivative operator is given by

 DiθI=(δIJ∂i−14δIJωiabγab−12eiaσ1IJP8γ012γa)θJ,P8=12(1+γ012345), (2.5)

where is the spin connection and projects on the supersymmetric fermionic directions. The last term comes from the coupling to the RR three-form flux .

Finally the quartic terms in the Lagrangian take the form (suppressing the -indices)

where

 υ=(1−P8)θ,Uab=12P8γ012γ[aP8γ012γb]−14Rabcdγcd, (2.7)

and

 MαIβJ= MαIβJ+(σ3Mσ3)αIβJ−i2(σ1P8γ012γaθ)αI(θγa)βJ+i4(γabθ)αI(θγaP8γ012γbσ1)βJ, MαIβJ= i2υσ1γ012υδαβδIJ−iθαI(υγ012σ1)βJ+i(σ1γaγ012υ)αI(θγa)βJ. (2.8)

Equipped with this string action the next step is to expand around the point-like BMN string solution given by [Berenstein:2002jq]

 x+=12(t+φ)=τ. (2.9)

At the same time we fix the light-cone gauge and corresponding kappa symmetry gauge

 x+=τ,γ+θI=0. (2.10)

The Virasoro constraints can then be 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 determined by the following conditions on the momentum conjugate to

 p+=−12∂L∂˙x−=1,∂L∂x−′=0. (2.11)

This is often referred to as uniform light-cone gauge.

Rescaling the transverse fields by a factor yields a perturbative expansion in the string tension444The fact that only even orders appear in this expansion for backgrounds greatly simplifies a perturbative treatment.

 L=L2+1gL4+1g2L6+…

where the subscript denotes the number of transverse coordinates. The quadratic Lagrangian takes the form ()

 L2= |∂z|2−|z|2+|∂y|2−|y|2+|∂u1|2+|∂u2|2+i¯χ1L∂−χ1L+i¯χ1R∂+χ1R−¯χ1Lχ1R−¯χ1Rχ1L +i¯χ2L∂−χ2L+i¯χ2R∂+χ2R−¯χ2Lχ2R−¯χ2Rχ2L+i¯χ3L∂−χ3L+i¯χ3R∂+χ3R+i¯χ4L∂−χ4L+i¯χ4R∂+χ4R. (2.12)

The spectrum consists of 4+4 massive and 4+4 massless modes. They are charged under four ’s and the charges are given in table 1. The interaction terms in the Lagrangian are quite complicated and we will not give them here since they are straight-forwardly found by expanding the original Lagrangian.

## 3 One-loop regularization

At one loop one encounters two types of integrals: Bubble integrals

 Brsm1m2(P)=∫d2k(2π)2kr+ks−(k2−m21)((k−P)2−m22), (3.1)

coming from the diagrams in (5.1), and Tadpole integrals

 Trs(P)=∫d2k(2π)2kr+ks−(k−P)2−m2, (3.2)

coming from the diagrams in (5.2). Here is a combination of external momenta. Many of these are power-counting UV divergent,555Due to the presence of massless modes there are also IR divergences. These are dealt with by introducing a small regulator mass for the massless modes which is taken to zero at the end. eg. for and need to be regularized. In [Roiban:2014cia] it was shown that standard dimensional regularization gives the wrong answer. It leads to an S-matrix not compatible with the symmetries due to the introduction of extra rational terms. In the same paper a different regularization scheme, essentially using tensor reduction in strictly two dimensions instead, was introduced which was shown to give a result consistent with the symmetries of the BMN vacuum. The same procedure has been shown to give the right answer also at two loops [Sundin:2014ema] as well as for the spinning GKP string [Bianchi:2015vgw].

The idea is to use the algebraic identity

 1(k−P)2−m22−1k2−m21=k+P−+k−P+−P2+m22−m21(k2−m21)((k−P)2−m22) (3.3)

to reduce all bubble integrals to the (UV) finite integral plus tadpole integrals. Once all divergences are isolated in tadpole integrals these can be evaluated in dimensional regularization without introducing unwanted rational terms. Multiplying the above identity by and integrating we find666We will suppress the subscript on the bubble integrals when there is no risk of confusion.

 P−Br+1,s(P)+P+Br,s+1(P)+(m22−m21−P2)Brs(P)=Trs[m2](P)−Trs[m1](0), (3.4)

where we have indicated explicitly the mass in the tadpole integrals. We also use the very simple fact that for

 Brs(P)=m21Br−1,s−1(P)+Tr−1,s−1[m2](P), (3.5)

which follows by writing and simplifying the integrand. Finally and , which are UV finite, can easily be computed by the following trick. First we note that and are odd functions of as follows by shifting the integration variable. But since is Lorentz invariant they must in fact be of the form

 B10(P)=P+f(P2,m1,m2),B01(P)=P−f(P2,m1,m2), (3.6)

where is some function that must be the same for the two integrals by symmetry. Using this in (3.4) with we can determine the function and finally we find777We have shifted the integration variable in the first tadpole integral.

 B10(P)=P+2P2[(P2+m21−m22)B00(P)+T00[m2](0)−T00[m1](0)], (3.7) B01(P)=P−2P2[(P2+m21−m22)B00(P)+T00[m2](0)−T00[m1](0)]. (3.8)

Using (3.4) and (3.5) we also find (again we have used shifts of the integration variable and Lorentz invariance to simplify the tadpole contributions)

 B30(P)=−1P−[(m22−m21−P2)B20(P)+P+m21B10(P)], (3.9) B03(P)=−1P+[(m22−m21−P2)B02(P)+P−m21B01(P)], (3.10) B20(P)=−1P−[(m22−m21−P2)B10(P)+P+m21B00(P)], (3.11) B02(P)=−1P+[(m22−m21−P2)B01(P)+P−m21B00(P)]. (3.12)

These are all the relations we will need for our one loop calculations.888 These integral identities can also be used to compute the one loop correction to the two-point function for the BMN modes in originally done in [Sundin:2012gc]. In contrast to the case the excitations now have four different masses where controls the relative size of the two -factors and lies in the interval . For the lighter excitations with mass and one finds, where we used dimensional regularization in the last step. If desired one can choose a different regulator for tadpole integrals with different masses in the loop. While not very natural from the point of view of the sigma model, it is possible to make the entire one-loop contribution vanish in this way. The exact solution indicates that the heavy mode from of mass 1 should be a composite state made out of two lighter ones. The way this can happen in the one-loop worldsheet theory is if the bubble contribution to the propagator replaces the pole in the propagator with a branch cut [Zarembo:2009au]. However, in the regularization outlined above we find where the dots denote tadpole contributions. Once we go on-shell, , the entire bubble contribution vanishes so we are lead to the conclusion that in order to see any indication of a composite state we need to go to higher loop order. The integrated result for the basic bubble integral is presented in appendix A.

## 4 Two-point functions and two-loop correction to dispersion relations

To be able to calculate the 2-particle S-matrix we must first compute the off-shell one-loop two-point functions. One finds that only the massive bosons receive a correction which takes the form of wave function renormalization [Roiban:2014cia]

 ⟨¯zz⟩=iZzp2−1+O(g−2),⟨¯yy⟩=iZyp2−1+O(g−2), (4.1)

with the renormalization factors given by

 Zz=1+14π1g(−2ϵ+logπ+γE),Zy=1−14π1g(−2ϵ+logπ+γE), (4.2)

where the contributing tadpole integral has been evaluated in dimensional regularization in dimensions. This wave function renormalization must be taken into account to get a finite S-matrix.

Before we move on to the S-matrix however we will compute the two-loop correction to the on-shell two-point function, i.e. to the dispersion relation. This was done for bosonic modes in [Sundin:2014ema, Sundin:2015uva] (in the type IIA context) and here we will extend that analysis to include also the fermionic modes.

The perturbative result should be compared with the exact dispersion relation fixed by the underlying symmetries. For an excitation of mass the exact dispersion relation was found to take the form [Babichenko:2009dk, OhlssonSax:2011ms, Sax:2012jv, Hoare:2013lja, Lloyd:2014bsa, Borsato:2014hja]

 ε=√m2+4h2sin2(p2). (4.3)

By scaling the momenta as , together with the identification , the large coupling expansion should agree with perturbative calculations.

For this calculation we have to sum three distinct classes of Feynman diagrams. The first one is the sunset-type diagrams

giving rise to integrals of the form

 Irstum1m2m3(p)=∫d2kd2l(2π)4kr+ks−lt+lu−(k2−m21)(l2−m22)((p−k−l)2−m23).

The sunset contribution contains the meat of the calculation since they constitute the only class of diagrams that can give a (regularization independent) finite contribution to the amplitude.

The other two types of diagrams are the bubble-tadpole

and six-vertex double tadpoles999Since we do not have the -terms in the Lagrangian we can only compute these contributions for the bosons. For the fermions we will simply assume that all divergences cancel.

and they are both UV-divergent and give rise to -terms that cancel divergences coming from the sunset diagrams.

In order to evaluate the sunset integrals we use a tensor reduction scheme similar to the one described for the one-loop bubble integrals in the previous section. We won’t give the details here but rather refer to [Sundin:2014ema, Sundin:2015uva]. Here we will only state the final result.

For the massive modes we find a correction

 ⟨¯zz⟩(2)=⟨¯yy⟩(2)=⟨¯χ1χ1⟩(2)=⟨¯χ2χ2⟩(2)=−163g2p41I0000111(p)|p2=1=−p4112g2. (4.4)

The fact that the two-loop correction is the same for bosonic and fermionic modes is a strong indication that the worldsheet theory is, as expected, supersymmetric to this loop order and that the regularization employed is consistent with this. It is easy to see that this correction precisely agrees with the exact dispersion relation (4.3).

The situation becomes more interesting in the case of massless modes. The calculation gives

 ⟨¯u1u1⟩(2)=⟨¯u2u2⟩(2)=⟨¯χ3χ3⟩(2)=⟨¯χ4χ4⟩(2)=−4g2p31I0100011(p)|p2=0=−p412π2g2. (4.5)

While this result is again consistent with worldsheet supersymmetry it disagrees with the expansion of the exact dispersion relation (4.3). This mismatch of was already found for the bosonic modes in [Sundin:2014ema] but here we demonstrate for the first time that the same result appears also for the fermionic modes. Although expected this is nevertheless an important consistency check.

## 5 One-loop S-matrix

Here we compute the two body S-matrix up to the one loop level. The S-matrix splits into three sectors according to whether the external particles are both massive, one massive and one massless or both massless. For each sector the contributing diagram topologies are a tree-level contact diagram, one-loop and -channel diagrams

 (5.1)

 (5.2)

When calculating the S-matrix we must take into account the wave function renormalization of the massive bosons (4.2) and the Jacobian from the energy-momentum conservation delta functions and external leg factors which give a factor

 J=141ωqp−ωpq,ωp=√m2p+p2 (5.3)

and similarly for where and denote the masses of the corresponding external particles. We write

 S=1+i\mathbbmT (5.4)

and compute the action of on generic101010To save some space and time we will give only the action on either BF or FB in-states but not both. We will also omit the action on FF in-states for the most part. These additional amplitudes contain no new information and are sometimes tedious to compute. two-particle states.

### 5.1 Massive sector

For scattering up to one loop we find111111Additional S-matrix elements can be obtained from the fact that the theory is symmetric under interchange of fields with their complex conjugates.

 Boson-Boson: (5.5) \mathbbmT|z(p)z(q)⟩= (−ℓ(0)1+i2(ℓ(0)1)2+2θ(1)LL)|z(p)z(q)⟩, \mathbbmT|y(p)y(q)⟩= (ℓ(0)1+i2(ℓ(0)1)2+2θ(1)LL)|y(p)y(q)⟩, \mathbbmT|z(p)y(q)⟩= (ℓ(0)2+i2(ℓ(0)2)2+2θ(1)LR)|z(p)y(q)⟩, \mathbbmT|y(p)z(q)⟩= (−ℓ(0)2+i2(ℓ(0)2)2+2θ(1)LR)|y(p)z(q)⟩, \mathbbmT|z(p)¯z(q)⟩= (−ℓ(0)3+i2[(ℓ(0)3)2+2(ℓ(0)7)2]+2θ(1)LR)|z(p)¯z(q)⟩−i(ℓ(0)7)2|¯y(p)y(q)⟩ +(ℓ(0)7−i2ℓ(0)3ℓ(0)7)[|χ1(p)¯χ1(q)⟩+|¯χ2(p)χ2(q)⟩], \mathbbmT|y(p)¯y(q)⟩= (ℓ(0)3+i2[(ℓ(0)3)2+2(ℓ(0)7)2]+2θ(1)LR)|y(p)¯y(q)⟩−i(ℓ(0)7)2|¯z(p)z(q)⟩ +(−ℓ(0)7−i2ℓ(0)3ℓ(0)7)[|¯χ1(p)χ1(q)⟩+|χ2(p)¯χ2(q)⟩], \mathbbmT|z(p)¯y(q)⟩= (ℓ(0)2+i2[(ℓ(0)2)2+2(ℓ(0)8)2]+2θ(1)LL)|z(p)¯y(p)⟩+i(ℓ(0)8)2|¯y(p)z(q)⟩ +(ℓ(0)8+i2ℓ(0)2ℓ(0)8)[|χ1(p)¯χ2(q)⟩−|¯χ2(p)χ1(q)⟩], \mathbbmT|y(p)¯z(q)⟩= (−ℓ(0)2+i2[(ℓ(0)2)2+2(ℓ(0)8)2]+2θ(1)LL)|y(p)¯z(p)⟩+i(ℓ(0)8)2|¯z(p)y(q)⟩ +(ℓ(0)8−i2ℓ(0)2ℓ(0)8)[|¯χ1(p)χ2(q)⟩−|χ2(p)¯χ1(q)⟩], Boson-Fermion: \mathbbmT|z(p)χ1(q)⟩= (−ℓ(0)4+i2[(ℓ(0)4)2+(ℓ(0)8)2]+2θ(1)LL)|z(p)χ1(q)⟩ +(−ℓ(0)8+i2[ℓ(0)4ℓ(0)8+ℓ(0)5ℓ(0)8])|χ1(p)z(q)⟩, \mathbbmT|z(p)¯χ1(q)⟩= (ℓ(0)6+i2[(ℓ(0)6)2+(ℓ(0)7)2]+2θ(1)LR)|z(p)¯χ1(q)⟩ +(ℓ(0)7+i2[ℓ(0)6ℓ(0)7+ℓ(0)7ℓ(0)9])|¯χ2(p)y(q)⟩, \mathbbmT|z(p)χ2(q)⟩= (ℓ(0)6+i2[(ℓ(0)6)2+(ℓ(0)7)2]+2θ(1)LR)|z(p)χ2(q)⟩ +(−ℓ(0)7−i2[ℓ(0)6ℓ(0)7+ℓ(0)7ℓ(0)9])|χ1(p)y(q)⟩, \mathbbmT|z(p)¯χ2(q)⟩= (−ℓ(0)4+i2[(ℓ(0)4)2+(ℓ(0)8)2]+2θ(1)LL)|z(p)¯χ2(q)⟩ +(−ℓ(0)8+i2[ℓ(0)4ℓ(0)8+ℓ(0)5ℓ(0)8])|¯χ2(p)z(q)⟩, \mathbbmT|y(p)χ2(q)⟩= (ℓ(0)4+i2[(ℓ(0)4)2+(ℓ(0)8)2]+2θ(1)LL)|y(p)χ2(q)⟩ +(ℓ(0)8+i2[ℓ(0)4ℓ(0)8+ℓ(0)5ℓ(0)8])|χ2(p)y(q)⟩, \mathbbmT|y(p)¯χ2(q)⟩= (−ℓ(0)6+i2[(ℓ(0)6)2+(ℓ(0)7)2]+2θ(1)LR)|y(p)¯χ2(q)⟩ +(−ℓ(0)7+i2[ℓ(0)6ℓ(0)7+ℓ(0)7ℓ(0)9])|¯χ1(p)z(q)⟩, \mathbbmT|y(p)χ1(q)⟩= (−ℓ(0)6+i2[(ℓ(0)6)2+(ℓ(0)7)2]+2θ(1)LR)|y(p)χ1(q)⟩ +(ℓ(0)7−i2[ℓ(0)6ℓ(0)7+ℓ(0)7ℓ(0)9])|χ2(p)z(q)⟩, \mathbbmT|y(p)¯χ1(q)⟩= (ℓ(0)4+i2[(ℓ(0)4)2+(ℓ(0)8)2]+2θ(1)LL)|y(p)¯χ1(q)⟩ +(ℓ(0)8+i2[ℓ(0)4ℓ(0)8+ℓ(0)5ℓ(0)8])|¯χ1(p)y(q)⟩,

where the superscript denotes tree-level or one-loop contributions and we have used (A.1). The phase factor, , only contributes to diagonal scattering elements. At the one-loop level, where the phase and matrix factors come with different powers of , we have separated the contributions in different expressions.

The tree-level elements are given by

 ℓ(0)1=12g(p+q)2ωqp−ωpq,ℓ(0)2=12gp2−q2ωqp−ωpq,ℓ(0)3=12g(p−q)2ωqp−ωpq, (5.6) ℓ(0)4=12gq(p+q)ωqp−ωpq,ℓ(0)