S-matrix on effective string and compactified membrane

Expanding Nambu–Goto action near infinitely long string vacuum one can compute scattering amplitudes of 2d massless fields representing transverse string coordinates. As was shown in (Dubovsky et al 2012 J. High Energy Phys. JHEP09(2012)044), the resulting S-matrix is integrable (provided appropriate local counterterms are added), in agreement with known free string spectrum and also with an interpretation of the static-gauge NG action as a TTˉ deformation of a free massless theory. We consider a generalization of this computation to the case of a membrane, expanding its 3d action near an infinite membrane vacuum that has cylindrical R×S1 shape (we refer to such membrane as ‘compactified’). Representing 3d fields as Fourier series in S 1 coordinate we get an effective 2d model in which the massless string modes are coupled to an infinite KK tower of massive 2d modes. We find that the resulting 2d S-matrix is not integrable already at the tree level. We also compute 1-loop scattering amplitude of massless string modes with all compactified membrane modes propagating in the loop. The result is UV finite and is a non-trivial function of the kinematic variables. In the large momentum limit or when the radius of S 1 is taken to infinity we recover the expression for the 1-loop scattering amplitude of the uncompactified R2 membrane. We also consider a 2d model which is the TTˉ deformation to the free theory with the same massless plus infinite massive tower of modes. The corresponding 2d S-matrix is found, as expected, to be integrable. Contribution to the special issue of Journal of Physics A: ‘Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser’


Introduction
The critical first-quantized string is described, in an appropriate gauge, by an effectively Gaussian path integral.This is not so for a membrane which has a highly non-linear and formally nonrenormalizable 3d action.While the existence of a consistent quantum theory of bosonic membranes may be in doubt, this may not be so for the 11d supermembrane or M2 brane [1,2,3].This may be true, in particular, for the supermembrane in the maximally supersymmetric AdS 4 × S 7 (or AdS 7 × S 4 ) background [4] and its orbifold AdS 4 × S 7 /Z k .M-theory in AdS 4 × S 7 /Z k should be dual to the N = 6 supersymmetric 3d U k (N ) × U −k (N ) Chern-Simons matter (ABJM) theory [5].Recent work [6] provided a remarkable evidence that direct semiclassical quantization of the M2 brane in AdS 4 × S 7 /Z k background reproduces the results of large N localization computations [7] of the 1  2 -BPS Wilson loop and instanton contributions to free energy in the ABJM gauge theory.
While the M2 brane action is highly non-linear, expanded near a classical solution with nondegenerate induced 3d metric it can be quantized in a static gauge with the 1-loop result being UV finite (containing no logarithmic divergences) [8,2,9,6].As the 1/N expansion of the localization results on the gauge theory side have the form of an expansion in the inverse of the effective M2 brane tension T 2 = √ 2k π √ N , this suggests that the matching with the 1-loop M2 brane computations [6] should, in fact, extend also to 2-loop and higher orders.
This requires the corresponding quantum M2 brane theory to be UV finite despite its apparent non-renormalizability.This may somehow happen due to high degree of underlying supersymmetry and possibly to other hidden symmetries of the M2 brane theory in AdS 4 × S 7 background that remain to be uncovered.
The M2 brane action in 11d background is formally related to the type IIA string in the corresponding 10d background by a double dimensional reduction [10].Considering M2 brane world volume of topology Σ 2 × S 1 and expanding 3d fields in Fourier modes in S 1 coordinate one gets an effective 2d string action on Σ 2 coupled to an infinite tower of massive 2d fields.Choosing a static gauge in the M2 brane action one gets a static gauge Nambu-Goto action for the massless transverse string modes coupled to a tower of the massive "Kaluza-Klein" 2d modes.This "effective string" 2d action is essentially equivalent to the original M2 brane action and thus may inherit its hidden symmetries.
With this motivation in mind, here we address the question about possible hidden symmetries in the simplest context of a bosonic membrane in flat R 1,D−1 space-time expanded near a R × S 1 cylindrical vacuum which is the analog of the infinite straight string in the bosonic string theory.
We will focus on the resulting 2d S-matrix comparing it with the one found in the string theory limit which corresponds to the case when the radius R of S 1 is sent to zero, i.e. when all massive KK 2d states decouple.In the opposite limit R → ∞ we should recover the S-matrix on plane R 2 membrane.
Let us first recall some results of past work on 2d S-matrix in the infinite bosonic string vacuum.
The tree-level and 1-loop contributions to the scattering of four massless modes representing the D ≡ D − 2 transverse string coordinates of the NG string in the static gauge was computed in [11].The S-matrix was found to be given simply by a scalar (CDD) phase factor thus representing an integrable theory.It was shown in [12] that using this S-matrix in the thermodynamic Bethe Ansatz one reproduces the expected free bosonic string spectrum.This S-matrix was studied at higher loop orders [13] where particular counterterms are required to cancel UV divergences and also to preserve the integrability.The reason why this S-matrix is given simply by a pure phase factor was further elucidated in [14] by observing that the NG action in a static gauge can be viewed as the T T deformation [15] of a theory of free massless bosons.
One may wonder if the integrability property of the NG string may generalize to a 2d theory containing a special infinite set of massive 2d modes.Our aim will be to address this question for the theory obtained from the bosonic membrane action in the static gauge with S 1 R compact direction viewed as an effective 2d theory with a tower of states with masses m 2 n = n 2 R 2 , n = 0, ±1, ....As mentioned above, this spectrum corresponds to the expansion near the cylindrical R×S 1 membrane vacuum.
We will discuss the corresponding tree-level scattering amplitudes and conclude that this model involving massive fields in addition to massless ones is no longer integrable.We will also compute the 1-loop correction to the scattering amplitude of 4 massless particles with all massless and massive modes running in the loop.We will find that as in the examples considered in [6] this 1-loop amplitude is UV finite in analytic regularization as appropriate for a 3d theory.It is also IR finite as all massless modes have only derivative couplings like in the NG string case.The 1-loop amplitude has a non-trivial dependence on the kinematic variables (and the mass scale or the radius R), characteristic of a non-integrable theory.
For comparison, we will also consider a 2d model with the same free massless plus massive tower spectrum but with interactions defined by the T T deformation.At the massless n = 0 level the resulting action will be given by the same NG action but interaction vertices involving massive fields will be different from those in the compactified membrane action (and, in fact, will not follow from any local 3d action compactified on S 1 ).As this is a T T deformation of a free theory, it should be integrable, and we shall verify this by computing tree-level and 1-loop scattering amplitudes.This paper is organized as follows.In section 2 we shall review in detail the tree-level and 1loop computations in the NG model near infinitely long string vacuum, verifying that the S-matrix admits a pure-phase representation.Section 3 will be dedicated to a similar analysis in the compactified membrane theory.We shall compute 1-loop amplitudes for scattering of 4 massless modes using dimensional regularization near d = 2 and Riemann ζ-function regularization to sum over KK modes.We will also present (in section 3.4) the expression for the 1-loop amplitude in the uncompactified membrane model found by starting directly from the 3d action.
The 2d model obtained by T T deformation of the same free massless plus massive spectrum will be studied in Section 4. Some open problems will be mentioned in section 5. Appendices will contain some basic definitions, useful 1-loop integrals, comments on 6-point tree-level amplitudes and details of computations in section 3.3.

S-matrix on Nambu-Goto string and its integrability
Let us start with a review of the perturbative S-matrix of the Nambu-Goto string action in the long string vacuum [11].Fixing static gauge one may study scattering of the 2d massless fields representing the transverse string coordinates.One finds that there is no particle production or annihilation at tree-level, which is compatible with the classical integrability of this 2d model.The loop-corrected S-matrix is given by a pure phase provided appropriate counterterms are added, which is consistent with known free string spectrum [12]. 1elow we shall describe the computation of the 1-loop NG string S-matrix in some detail to prepare for the analysis of the membrane case in the next section.
Let us start with a brane with 1 time and d − 1 space world-volume directions moving in a D-dimensional target space-time.We will be interested in the cases of the string with d = 2 and the membrane with d = 3.The corresponding Dirac [18] or Nambu-Goto action is where α, β = 0, 1, . . .d − 1 and µ, ν = 0, 1, . . ., D − 1.We shall consider the expansion of (2.1) near an infinite flat string or membrane vacuum so that we can fix the static gauge X α = σ α .We shall label the remaining transverse coordinates as X j so that the induced metric is Then expanded in derivatives of X j the action (2.1) takes the form so that Notice that in the string case when d = 2 only J 1 and J 2 are independent invariants; for example, Below we shall sometimes keep the values of the coefficients c n in (2.4),(2.7)arbitrary to emphasize simplifications that happen when they take their "Nambu-Goto" values.

Tree-level 4-point amplitude
Starting with (2.3) one can compute the scattering amplitudes of the massless fields X j .From the expression of L 4 in (2.4) we deduce that there are two types of Feynman diagrams contributing to the four-point amplitude, The solid line here represents contraction of momenta, while the dashed lines correspond to a contraction of indices of the fields.Let us label the two incoming particles with indices i, j and momenta p 1 , p 2 and the two outgoing particles with indices k, l and momenta p 3 , p 4 .The amplitude takes the following schematic form (we omit the standard momentum conservation delta-function, (A.6),(A.7)) where A is the annihilation, B the transmission and C the reflection parts.Their tree-level expressions following from (2.4) are 2 Writing these in terms of the Mandelstam variables (see Appendix A) taking into account that here all particles are massless, i.e. s + t + u = 0, we get For the values of the coefficients in (2.4) this becomes Specifying to d = 2 we may use that in this case the kinematical constraints imply that for massless particles stu = 0 or tu = 0 if we assume that s = 0 (see (A.2)) 3 stu = 0 → tu = 0 , s = 0 . (2.12) 3) where T ≡ T1 is the string tension the factors of T −1 or effective appear in the quartic, etc., interaction vertices and thus in the corresponding scattering amplitudes.We will not always explicitly include them below as they can be restored in the final expressions.
3 For example, we may use a Lorentz transformation to go to the center of mass frame where p1 = − p2 = 0, i.e. (p 1 , i) With this choice of the kinematics we get A (0) = 0 in (2.11).Also, B (0) = 0 for u = 0 and C (0) = 0 for t = 0.For the latter case the tree-level S-matrix (2.8) contains only the B (0) δ ik δ jl term which is proportional to the unit operator in the S-matrix (cf.(A.4),(A.5)).Being proportional to the identity, the tree-level S-matrix automatically satisfies the Yang-Baxter equation (A.9),(A.10).

One-loop contribution to the 4-point amplitude
Using the 4-vertex in (2.4) we can build three 1-loop "bubble" diagrams in Figure 1.The total 1-loop amplitude, given by the sum of these three contributions, takes the same form as (2.8) (cf.footnote 2)
Using Feynman parametrization, the s-channel amplitude A (1) s in (2.14) may be written as4 A (1)  s = 1 2i (2.15) where 1 2 is the symmetry factor of the diagram (exchange of two legs in the loop) and (2.17) Here the factor D = D − 2 of the number of transverse fields comes from a diagram with a sum over the target space indices in the loop and A (0) , B (0) , C (0) were defined in (2.9).To simplify the expressions below we will assume that the coefficients c 2 and c 3 take their NG values in (2.4).
It is useful to formally expand N s (2.17) in powers of loop momentum l as ) where Using standard Γ-function relations implying, e.g., that we can rewrite (2.20) as The values of the coefficient functions γ 0 , γ 2 , γ 4 are found from (2. 19) not to depend on d (cf.(2.16)) Setting d = 2 − 2ǫ it is then straightforward to isolate the divergent and finite part of the amplitude in the ǫ → 0 limit: (2.27) Similar expressions are found in the t and u channels, where, e.g.,

A
(1) (2.28) and thus we get (2.27)where now Performing the integrals over the Feynman parameter x gives A (1) A Taking into account that in the string case D = D −2 this is the same expression as found originally in [11].The divergent part in (2.32) is proportional to stu and thus vanishes in d = 2 due to (2.12) (thus the dependence on renormalization scale µ drops out).The same applies to the B (1) and C (1)  amplitudes related to A (1) as in (2.14). 5 Let us note that there is also another 1-loop diagram that could contribute to the 4-point amplitude -the tadpole with 6-vertex from L 6 in (2.6) but it vanishes in dimensional regularization (the vertex contains powers of momenta and the propagator here is massless). 6 The finite part of the 1-loop amplitude is thus given by A f in (2.32) and similar expressions for B (1) f obtained by interchanging s ↔ t and s ↔ u as in (2.14).The first term in A (1) f is 5 As stu = O(d − 2) there may be an "evanescent" finite contribution but it can be cancelled by a local counterterm (cf.footnote 8 and ref. [11]). 6Explicitly, the tadpole contribution found from L6 in (2.6) is , Here the 1 ǫ pole is proportional to the effective "mass" term −iε in the propagator (cf.(2.15)) and thus vanishes for ε → 0.
proportional to D − 24 = D − 26 and hence vanishes in the critical dimension of the bosonic string.The second term is proportional to stu and hence vanishes in d = 2. Thus for s = 0, i.e. tu = 0
The same is true also when D = 1 or D = 3, i.e. when there is just one "transverse" scattering field, so that the amplitude in (2.13) is proportional to For generic dimension D = D − 2 the 1-loop S-matrix (2.36) does not have a CDD phase factor form.However, as expected from the relation of the static-gauge NG action to a T T deformation of a theory of free D scalar fields mentioned above, it should be possible to restore this pure phase structure and consistency with the Yang-Baxter equation by adding contributions of appropriate local counterterms.At quartic order in the fields X i there exist two linearly independent local terms invariant under ISO(1, 1) × SO( D) which are of 6-th order in derivatives (2.38) The corresponding tree-level amplitude is given by (2.8) with The b 1 term does not contribute due to the kinematical constraint (2.12). 8Choosing .40) 7 We use that log(−s − iε) = log s − iπ + O(ε).
8 If we relax this constraint and choose b1 = − D−6 48π 1 ǫ − γ + ln 4π we may cancel the divergent term in (2.32) for any value of stu, including the "evanescent" finite part.Let us note also that the b1 term in (2.38) may be interpreted as originating from the curvature integral evaluated on the induced metric in the long-string vacuum (2.2).It becomes trivial (integral of a total derivative) in d = 2.At the same time, the b2 term in (2.38) is the so called Polchinski-Strominger term [19] originating from the Polyakov term 2) and expanded in derivatives of transverse coordinates (see also [11,20]).
we then get, assuming t = 0, u = −s as in (2.36), Â(1) ≡ A (1) Let us note that in general, the unitarity S † S = 1 of the S-matrix S = 1 + iT implies that 2 ImT = T † T .In the present case this leads to the following relations between the coefficients in (2.8),(2.9)and (2.13) which are indeed satisfied.

S-matrix as a pure phase
We conclude that for any D the tree-level (2.11) plus 1-loop (2.41) scattering amplitude is given by (2.8) where (we restore the dependence on the string tension T 1 ≡ T = 1 2πα ′ and assume the d = 2 kinematics with t = 0) (2.43) Thus to 1-loop level there is no particle creation or annihilation, i.e. the scattering is purely elastic.
The S-matrix is related to the amplitude M by (A.6),(A.7).Using (A.8) in the case of a 2d integrable theory with A = C = 0 we conclude that the S is expressed in terms of the transmission amplitude B as (2.44) For massless particle scattering this reduces to (2.45) In the present case of (2.43) we thus get at the tree and 1-loop level (2.46) These are the first terms in the expansion of the pure-phase (unitary) S-matrix [11] (2.47) S = e − i 4T s . (2.47)

S-matrix on compactified membrane
Having reviewed the computation of the world-sheet S-matrix in the NG string theory, let us perform a similar analysis for the bosonic membrane theory.We shall start with the Lagrangian (2.3) with d = 3.In the case of the infinite R 2 membrane in the static gauge the resulting tree-level scattering amplitudes for massless 3d fields will be again given by (2.8), (2.9) where now s, t and u depend on 3d massless momenta so that s + t + u = 0 but there is no extra 2d condition (2.12).
In 3d there is no notion of S-matrix integrability.However, if we assume that the membrane has one compact direction, i.e. its vacuum configuration has topology of R × S 1 , then we may represent the corresponding 3d world-volume theory as an effective string with 2d world sheet coupled to an infinite tower of massive "Kaluza-Klein" 2d modes.We may then ask if the resulting 2d model has an integrable S-matrix once the contribution of the KK modes is included.
As we shall discuss below, allowing massive modes on the external lines one does not get an integrable S-matrix already at the tree level.We will also compute the 1-loop S-matrix of scattering of 4 massless modes generalizing the discussion in the previous section to the case when also an infinite set of the massive KK modes are propagating in the loop.The resulting amplitude will be free of log UV divergences and its finite part will have a rather complicated "non-integrable" dependence on 2d momenta different from the pure-phase structure (2.47) found in the NG string case.

Compactified membrane as effective string coupled to a massive tower
Let us assume that the classical membrane solution has topology of R 1,1 × S 1 .We shall denote the world-volume membrane coordinates as (σ 0 , σ 1 , σ 2 ) and assume that σ 2 is a circle of radius R, i.e. σ 2 ∈ [−πR, +πR).We shall choose a static gauge as (cf.(2.1),(2.2)) with X j (σ 0 , σ 1 , σ 2 ), j = 1, ..., D − 3, being the transverse fluctuation fields.Expanding them in Fourier modes in σ 2 gives we thus get an infinite set of 2d fields X j n (σ 0 , σ 1 ).Integrating over σ 2 in the membrane action (2.1),(2.3)(using ) one gets an effective 2d theory (now σ α = (σ 0 , σ 1 ) and α, β = 0, 1) Thus there are D = D − 3 real massless modes X j 0 and an infinite tower of complex massive modes X j n .T is the "effective string" tension.The quartic interaction term L4 may be written as (cf. where c 2 = 1 2 and c 3 = −1 are the same as in (2.4).Explicitly, The massless X j 0 part here is in the first line and is the same as in the NG action in (2.4).The necessary condition for the integrability of 2d S-matrix is that there cannot be particle production or annihilation, meaning, in particular, that in any process the number of incoming particles should be the same as the number of outgoing particles.Also, there cannot be any particle transmutation.If the two fields X 1 and X 2 have different masses then the processes of the form forbidden.This is because such processes violate macro-causality (see, e.g., [21]).
Let us now show that for the interaction Lagrangian (3.7) amplitudes for such processes are, in fact, non-vanishing already at the tree-level.Thus the theory (3.3) is not classically integrable.

Tree-level 4-point amplitude
Let us consider the scattering of four particles (1 and 2 ingoing and 3 and 4 outgoing) with momenta p r and mode numbers n r .We shall set the compactification radius to one, R = 1, as dependence on it is easy to restore on dimensional grounds.The scattering amplitude will be a generalization of the massless one in (2.8), i.e.

Mij,kl = Âδ
which is multiplied by δ (2) (p 1 + p 2 − p 3 − p 4 ) in the S-matrix.As in the massless case, the factor T −1 of inverse effective tension in (3.3) is implicit here.It follows from (3.7) that there are now five distinct Feynman diagrams, depending on the types of particles on external lines and index contractions (cf.section 2.1) We thus get for the tree-level A-amplitude (cf.(2.9)) (3.9) Using that n 1 + n 2 = n 3 + n 4 and writing this in terms of the Mandelstam variables (A.1) we get (3.10) The amplitudes B and C can be obtained using the crossing relations so that (3.12) When all the external particles are massless we recover the result in (2.9), i.e.Â(0) vanishes for the NG choice of coefficients c 2 and c 3 in (2.4) and tu = 0, ( Â(0) ) 0,0→0,0 = 0. (3.13) The scattering of 3 massless and 1 massive particle is not allowed by conservation of n r .For 2 massless and 2 massive particles we distinguish two cases.If there is massless + massive incoming and massless + massive outgoing states then the kinematic constraint is t = 0 for The following amplitudes are, however, non-zero: The non-zero value of this amplitude indicates that the theory is not integrable: this is an example of particle transmutation when 2 massless particles scatter into 2 massive ones.The non-zero result for Â(0) is found (using again (A.2)) also for amplitudes with 3 and 4 massive particles.Thus the resulting tree-level S-matrix is not integrable for all processes with massive particles on external lines.

1-loop contribution to the massless 4-point amplitude
Let us now compute the 1-loop contribution to the scattering of 4 massless (n r = 0) modes.
Similarly to the string case in section 2.2 this amplitude is given by the contributions of the 3 bubble diagrams in Figure 1 but now all (massless and massive) modes appear in internal lines.
The amplitude takes the same form as in (2.13) entering the S-matrix together with a δ (2) (p 1 + p 2 − p 3 − p 4 ) factor.
From the properties of the 4-vertex, both legs in the loop have opposite mode numbers (and therefore the same mass).The total s-channel amplitude is given by the sum of contributions of modes with fixed mode number propagating in the loop, where the mode n amplitude has the form (cf. (2.15)) Here we again use dimensional regularization with d = 2 − 2ǫ and implicitly assume the presence of the −iε shift as in (2.15), (2.16) and also the overall tension (coupling) factor of T −2 from (3.3).As in the tree-level amplitude (3.9),(3.10)we also set R = 1 in (3.4),(3.7).As the S 1 radius R enters the Lagrangian (3.3)-(3.7)via n i R , it can be restored by rescaling the mode number or equivalently by rescaling (s, t, u) → R 2 (s, t, u) (implying in particular ∆ s → R 2 ∆ s ) and adding an overall factor in the amplitude.The numerator Ns in (3.19) has the same form as in (2.17), but the tree-level amplitudes A (0) , B (0) , C (0) there should be now replaced with their "massive" counterparts Â(0) , B(0) , Ĉ(0) , as given in (3.9).The expansion of Ns in powers of l has the same form as in (2.18) where now (cf.(2.19)) M µν = ( D − 2)s p 1,µ p 2,ν + p 3,µ p 4,ν , N µνρσ = 4 D p 1,µ p 2,ν p 3,ρ p 4,σ .
In the t-channel (and similarly in the u-channel with t → u) we find that From these expressions we can then compute the coefficients γ k defined as in (2.24) and appearing in the analogs of the expressions (2.23) and (2.27).Remarkably, we find that they do not depend not only on d = 2 − 2ǫ but also on the mode number n and we find the expressions that look the same as in the massless case in (2.26), i.e.
Some details of the calculation of the 1-loop amplitudes at fixed n can be found in Appendix D.
Expanding the analog of (2.23) in ǫ → 0 and summing over the three channels as in (2.14) we get for the singular part of the Â(1) n amplitude: As we have only massless particles on external legs, the stu term vanishes due to the kinematical constraint (2.12) and the divergence is thus proportional to n 2 or the effective mass-squared term.
To sum over n we will apply the Riemann ζ-function regularization.The same regularization was used in the 1-loop membrane computations in [6].It is consistent with the expected absence of the 1-loop logarithmic divergences in the original 3d membrane theory (the absence of 1-loop log UV divergences in a 3d theory should not depend on S 1 compactification).Explicitly, for any positive or zero integer k we will set 10 Thus the sum of the full expression (3.27) is regularized to 0 even without using that stu = 0.
Thus the total 1-loop amplitude is finite and given by the sum over n of the remaining parts of the partial amplitudes (we again choose the kinematics so that t = 0 and thus u = −s) . 10 The existence of these "trivial" zeroes of ζR(z) = ∞ n=1 n −z follows, e.g., from the reflection formula ζR(z) = 2 z π z−1 sin πz 2 Γ(1 − z) ζR(1 − z) after setting z = −2k.This property can be proved also directly by considering the regularized sum ∞ n=1 n 2k e −εn = ∂ 2k ∂ε 2k I(ε) where 1−e −εn and observing that the resulting expansion in ε → 0 contains only odd powers of ε.Indeed, ∂ 2k ∂ε 2k I(ε) changes sign under ε → −ε as 1−e εn + 1 so that and thus there is no finite part after subtracting all poles.
Here the function Q n (s) is defined by (see (D.7)) (3.32)

Finite expression for the amplitude
The total amplitude is thus given by and by similar expressions for B(1) and Ĉ(1) .We shall define the sum over n using again the Riemann ζ-function.Then using (3.28) the terms with only n 2k factors in (3.29) will be zero after the summation.We will also need The remaining non-trivial sums involving Q n (s) can be defined by extracting the part that diverges at large n and define it using (3.28),(3.34).As the expressions in (3.29) contain Q n multiplied by n 2k with k = 0, 1, 2, then in view of (3.31) it is useful to subtract the 1 n 2 and 1 n 4 parts from the large n expansion of Q n and define (for n > 0) Then the sums involving Qn will be finite, i.e. the non-trivial part of the amplitude will be given by finite terms like ∞ n=1 (an We conclude that the 1-loop 4-point massless scattering amplitudes in compactified membrane theory are expressed in terms of complicated (non-polynomial) functions of s that cannot be cancelled by adding local counterterms.This is again an indication of non-integrability of the compactified membrane theory viewed as an effective 2d theory.
One may also obtain an alternative representation for the 1-loop amplitude by starting with the Feynman parameter integral representation (3.19) and first doing the sum over n under the integral.This gives where ∆ s = −x(1 − x)s and γ k,s were given in (3.20) and (3.24), and we defined a function F which is proportional to the Epstein zeta function where we assumed that c > 0 and K ν denotes the modified Bessel function of the second kind. 11his expression provides an analytic continuation of the Epstein zeta function and can be used to regularise the infinite sum for w ≤ 0 (in (3.41) we need w → 0 and w → −1).Using (3.43) we can then take the limit ǫ → 0 and end up with the following finite expression 12 Let us recall that here ∆ s = −x(1 − x)s with x ∈ [0, 1] and s > 0, so that ∆ s < 0 in the physical kinematic region.To define √ ∆ s let us recall that we implicitly assume the iε shift in the propagators in (3.19) as in (2.15),(2.16),i.e.
Similar finite expressions can be found for Â(1) t and Â(1) u using the values of γ-coefficients in (3.25) and (3.26).

Tadpole diagram contribution
In addition to the bubble diagram contributions (cf. Figure 1) leading to (3.19) there is also a possible 1-loop tadpole contribution.It was vanishing in the string case in section 2.2 but may be non-vanishing in the present case of the massive modes propagating in the loop.
There are five different possibilities for the contraction of indices in the 6-point vertex in the action in (2.6), (3.3) that contribute to the one-loop 4-point amplitude A (1) (where particles 1, 2 and 3, 4 have contracted indices).Pictorially, these are given by The first diagram contains an internal loop over the SO( D) index and therefore contributes a factor of D. As a result, the tadpole contribution to the fixed n amplitude A (1) tad,n may be written as This contribution should be combined with the bubble diagram one in (3.37) (where t = 0); as a result the total amplitude does not contain the ζ ′ R (−2)s 2 term.The same applies to B (1) and C (1) amplitudes.

R → ∞ and R → 0 limits
Let us now restore the dependence on the compactification radius R using (3.21) and consider the limits R → ∞ and R → 0, or, equivalently, large and small momenta.For large R or large momenta we should expect to recover the scattering amplitude on uncompactified (i.e.R 2 -shape) membrane, while for small R or small momenta the contributions of massive KK modes should be suppressed and we should get the scattering amplitude of the NG string.
As follows from the form of the action in (3.3),(3.4),(3.7)for R → ∞ we need also to consider n large and introduce as usual a continuous 3rd component of the momentum p 3 = n R thus effectively recovering the case of uncompactified membrane.To take this limit directly in the amplitude 3) as in (3.32) (with T 2 held fixed in the large R limit) and keeping s, t arbitrary (with s + t + u = 0) we thus get for the limit of the total Â(1) amplitude Â(1) (3.53) Assuming that 2d momenta here are restrictions of momenta of massless 3d particles, this is indeed the same expression as for the 1-loop amplitude of scattering of the massless 3d modes that follows directly from the action (2.3) expanded near the uncompactified R 2 membrane vacuum (see (3.59) below), modulo a factor of (2πR) −1 that comes from going from 3d to 2d. 14In the opposite limit of R → 0 the form of the action in (3.4),(3.7)implies that only the modes with n = 0 should contribute, i.e. we should recover the scattering amplitude in the NG string case.Taking this limit directly in (3.44) with ∆ s → R 2 ∆ s we find that the functions in (3.45),(3.46)have the following expansion Adding the contributions of the three channels we get for the R → 0 limit of the total 1-loop amplitudes (assuming t = 0, u = −s and fixing the effective string tension The same expression is found by starting with the first representation (3.37)-(3.39)for the amplitudes in terms of the sums over n where the singular 1 R 2 terms appear in an equivalent form as This singular contribution cancels once one adds the tadpole diagram contribution in (3.51) (with t = 0 and the dependence on R restored).The remaining finite parts of the amplitudes in (3.56) are then indeed the same as in the NG string case in (2.36).

Scattering amplitudes on uncompactified membrane
Let us now consider the S-matrix on uncompactified membrane found by expanding the Dirac action (2.1),(2.3)near the plane R 2 membrane vacuum.The scattering modes are then massless 3d fields representing D = D − 3 transverse membrane coordinates.
The tree-level amplitudes are given by the same expressions (2.11) as in the string case but with the 3d kinematic variables subject only to the s + t + u = 0 constraint.Including the factor of the inverse membrane tension we thus have (cf.footnote 14) The 1-loop amplitudes are given by the same integrals as in (2.15).In particular, the numerator N s in (2.18), (2.19) is the same as in the string case, and so is the expression (2.23) with the coefficients as in (2.24) and (2.25) but now with d = 3.
For d = 3 the integral in (2.23) is UV finite and we find that |s| = i |s| this 1-loop 4-point amplitude contains an imaginary part consistent with unitarity. 16hile this 1-loop S-matrix is finite, the non-renormalizability of the membrane action means that log UV divergences may appear at higher loops.It would be very interesting to find a higher loop generalization of (3.59) that could be the analog of the pure-phase S-matrix (2.47) in the NG string case and, especially, its supermembrane [1] analog that may be better defined.

Integrable T T deformation of infinite tower of 2d fields
The integrability of the NG string model (2.3) has a natural explanation in terms of a T T deformation of a free 2d scalar theory [14,16].While the effective 2d model obtained from compactified membrane was found above to be non-integrable, it is possible to construct an integrable model with the same free-field spectrum by applying a T T deformation to the massless plus massive tower of 2d fields in (3.4).In this case the interaction Lagrangian will be different from (3.7) and will not correspond to a compactification of some local Lorentz invariant theory in 3 dimensions.

T T deformation of free massless 2d field
The T T deformation can be applied to any 2d theory with an energy momentum tensor T αβ .When the seed theory is integrable, its T T deformation will preserves integrability.In general, the "T T " operator is defined as det T µν , i.e.
with ǫ 01 = −ǫ 10 = 1 the antisymmetric Levi-Civita symbol.The corresponding action is (λ is a deformation parameter) where, using that Let us start with the Lagrangian of D free massless fields The first-order (t → 0) deformation of the Lagrangian will then be where we follow the notation in (2.5), i.e.X αβ ≡ ∂ α X j ∂ β X j .The interaction term here has the same form as the first term in the expansion of the NG action in (2.4).In d = 2 there are only two independent invariants J 1 and J 2 among J n ≡ Tr[X n ] (for instance, ).This allows one to obtain the closed form of the deformed Lagrangian as L(λ, J 1 , J 2 ).Using that we find that L(λ, J 1 , J 2 ) satisfying the flow equation in (4.2) is given by It is thus equivalent to the NG Lagrangian in the static gauge (cf.(2.1),(2.2),(2.3)).

T T deformation of a tower of massless and massive 2d fields
Let us now start with D free scalars in 3 dimensions and assume that the third direction is a circle of radius R. The equivalent 2d theory is represented by the tower of 2d fields which is the free part in the compactified membrane action in (3.3), i.e.
Computing the corresponding T T operator in (4.3) we get for the deformed Lagrangian

.10)
Here the n = q = 0 term is the same as in (4.5).Compared to the quartic interaction term in the compactified membrane action in (3.7) here we get only a subsector of (∂X) 4 terms with n 1 = −n 2 and n 3 = −n 4 , no X 2 (∂X) 2 terms and the X 4 term with n 1 = −n 2 and n 3 = −n 4 .
Note that in contrast to (3.7) the interaction term in the 2d Lagrangian (4.9) given by the product of the components of the two 2d stress tensors cannot be obtained from a local 3d action: instead of a single δ n 1 +n 2 ,−n 3 −n 4 it contains δ n 1 ,−n 2 δ n 3 ,−n 4 that may only originate from a double integral over the 3rd S 1 direction.
This difference is important for understanding of why the membrane theory interaction term in (3.7) does not correspond to an integrable theory while the T T one in (4.9) may.Indeed, the structure of (3.7) with only a single δ n 1 +n 2 ,−n 3 −n 4 does not prohibit particle transmutation processes when, e.g., 2 particles with mass m scatter into particle with mass m and m ′ .Such processes are, however, excluded by the presence of δ n 1 ,−n 2 δ n 3 ,−n 4 in (4.10).
Below we shall check that the theory (4.10) is indeed integrable by computing the tree-level and 1-loop contributions to the corresponding scattering amplitude.
Expressing (4.12) in terms of the Mandelstam variables in (A.1) we get (4.13) Here we took into account the delta-symbol constraints on the mode numbers or masses in (4.11).
For the specific coefficients in (4.10) the first two terms in each expression in (4.13) vanish, i.e.
17 We set R = 1 and also ignore the effective coupling λ factor in (4.10).
One can check using (A.1)-(A.3)that in the 2d case Thus the only non-vanishing amplitude is the transmission B one, consistent with the integrability.
For the scattering of massive particles we may use the notation in terms of the rapidities, ω r = p r 2 + n 2 r = n r cosh θ r and p r = n r sinh θ r , so that the amplitude B (0) in (4.15) may be written as We conclude that the tree-level 4-point S-matrix is proportional to the identity and the Yang-Baxter equation is trivially satisfied, in agreement with the expected integrability of the T T deformation.

1-loop scattering amplitude for massless fields
To compare to the results in section 3 for the 1-loop massless scattering amplitude in the compactified membrane theory let us consider the same problem in the case of the T T deformed theory (4.9).We shall thus assume that for the external particles n 1 = n 2 = n 3 = n 4 = 0.The mode number in the loop will be denoted as n.
Let us start with the contribution of the bubble diagrams in Figure 1.The resulting expression for A s amplitude is given by the same expressions as in (3.18),(3.19),(2.18) where instead of (3.22) we get (in the s-channel) This leads to (2.27) where now (cf.(3.24)) For the massless loop mode n = 0 this reduced to the coefficients found in (2.26).
Similarly, in the t-channel we get (the u-channel expressions are found by t ↔ u and p 3 ↔ p 4 ) This gives As a result, the pole part of the total A (1) amplitude reads Compared to the compactified membrane case in (3.27) here we get no s 2 term so this expression vanishes for t = 0 choice of the 2d kinematics even before summation over n, i.e. the amplitude is UV finite for fixed n.
For the remaining finite part of the fixed-n amplitudes we find (setting t = 0, u = −s) 18 For n = 0 this of course reduces to the string scattering amplitudes in (2.36).The total amplitude is given by the sum over n which we may define again using the Riemann ζ-function prescription as in (3.28).Then the Ds 3 and Ds 2 n 2 terms in A (1) and C (1) disappear and the remaining real s 3 terms can be cancelled by a local counterterm as in (2.41).The real contribution to B (1) )) should cancel against the contribution of the tadpole diagrams as in (3.51). 19 Then the final expression for the 1-loop massless amplitude is the same as in the purely massless (string) theory.This is consistent with the origin of (4.10) as a T T deformation of the free (integrable) model: the S-matrix should be given by that of the undeformed theory, dressed by a CDD factor.The CDD factor is only sensitive to the quantum numbers of the external particles, not those of the virtual particles in the loop.Therefore, we should indeed end up with the same amplitude as in the purely massless case.
Let us note that in the case of 1-loop correction to scattering of massive fields in the deformed theory (4.9) one will find UV divergent terms that need to be cancelled by appropriate counterterms [17].The ambiguity in the structure of finite counterterms may be fixed by requiring that the theory should be integrable at the quantum level, i.e. its S-matrix should satisfy the YB equation. 20

Concluding remarks
We have found that the S-matrix of the effective 2d model corresponding to the bosonic membrane action expanded near the cylindrical vacuum is not integrable, so the question of possible hidden symmetries in membrane theory remains open. 18To find the expression for B n,f we used the small n expansion of the function Qn(s) in (3.30),(2.31)implying that Qn(s) + Qn(−s) δn,0 = 2iπs −1 δn,0.
19 The direct computation of the tadpole diagram requires fixing the 6-point vertex in the deformed Lagrangian in (4.9). 20This was demonstrated on an example of a T T deformation of a single massive field in [17] using a momentum cutoff regularization.We have checked that similar result is found using dimensional regularisation.
There are several possible extensions.An interesting problem is to extend the computation of the S-matrix in the bosonic string and membrane theories to the Green-Schwarz superstring 21 and the supermembrane theories.Starting with the GS string in the static gauge one finds [23] that its partition function is UV finite and trivial at 1-loop and also 2-loop orders (the 2-loop log divergences cancel also in AdS 5 × S 5 case [23,24]).It would be important to show that the corresponding 2d scattering amplitudes are also 1-loop and 2-loop finite, despite formal non-renormalizability of the GS theory.
Similarly, in the case of the supermembrane in flat target space it would be interesting to check first that the 2-loop partition function of the cylindrical membrane is UV finite (the 1-loop one is always finite in d = 3).One may then compute the corresponding S-matrix to 2-loop order to see if it is also well defined and have a simple structure.
One may also investigate the T T deformation of the free 2d theory obtained from compactified supermembrane (i.e. containing free superstring modes plus a tower of massive 2d fields).This should produce an integrable model with an infinite set of 2d bosons and fermions, generalizing the static-gauge GS action.
where 1 denotes the identity (corresponding to the free theory) and T encodes the contribution of interactions.The operators 1 and T act on the asymptotic states for the 4-point scattering as X k ( p 3 )X l ( p 4 )|1|X i ( p 1 )X j ( p 2 ) = δ (d−1) ( p 1 − p 3 )δ (d−1) ( p 2 − p 4 ) δ ik δ jl , (A.where it is assumed that all particles are on shell, i.e. one has ω r = p r 2 + m 2 r .Note that M ij,kl has mass dimension 4 − d, so that the right-hand sides of both (A.5) and (A.6) have the same mass dimension 2 − 2d.This is consistent with the expressions for the amplitudes in the main text.Restoring the tension factors both tree-level and one-loop amplitudes M ij,kl have mass dimension 2 in d = 2 and 1 in d = 3.
For c < 0 we define (B.6) by an analytic continuation.
choosing to keep s = 0 (here p is spatial component of momentum, see Appendix (A) for notation).Then by the momentum conservation p3 = − p4 and the energy conservation gives | p1| = | p3| or tu = 0.

Figure 1 :
Figure 1: The three "bubble" diagrams contributing to the 1-loop 4-point amplitude.The arrows denote the flow of momentum: the particles p 1 and p 2 are incoming, while the particles p 3 and p 4 are outgoing.By momentum conservation we have q = p 1 + p 2 − p for the first diagram (s-channel), q = p 1 − p 3 − p for the second diagram (t-channel) and q = p 1 − p 4 − p for the third diagram (u-channel).

. 15 )
The second case is when both incoming (or both outgoing) particles are massless.Then n 1 = n 2 = 0 and n 3 = −n 4 = n and the relation between the Mandelstam variables in (A.2) implies that tu = n 4 .Consequently, for the NG values of c 2 and c 3 in (2