Little string amplitudes (and the unreasonable effectiveness of 6D SYM)

Journal of High Energy Physics, Dec 2014

We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in six-dimensional super-Yang-Mills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in \( \mathcal{N} \) = 2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the α ′ expansion. We find striking agreements with up to 2-loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a ℤ k invariant point on the Coulomb branch. We comment on the issue of UV divergence at higher loop orders in the gauge theory and discuss the implication of our results.

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Little string amplitudes (and the unreasonable effectiveness of 6D SYM)

Oxford Street 0 Cambridge 0 MA 0 U.S.A. 0 0 Open Access , c The Authors 1 Center for Theoretical Physics, Massachusetts Institute of Technology 2 Jefferson Physical Laboratory, Harvard University We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in sixdimensional super-Yang-Mills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in N = 2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the 0 expansion. We find striking agreements with up to 2-loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a Zk invariant point on the Coulomb branch. We comment on the issue of UV divergence at higher loop orders in the gauge theory and discuss the implication of - effectiveness of 6D 1 Introduction Double scaled little string theory The holographic description SL(2)k/U(1) SU(2)k as a Zk orbifold of U(1)k U(1) 2.6 Identifications among vertex operators in the coset theories Massless string states Spectral flow Zk orbifold SU(2)k/U(1) and N = 2 minimal model Supersymmetric SU(2)3/U(1) and the compact boson k = 2 k = 3 k = 4 k = 5 Non-normalizable, delta function normalizable, and normalizable pri Correlators and amplitudes Winding number conserving correlators Bosonic SL(2)k+2 correlators and Liouville correlators Scattering amplitude in the double scaled little string theory Comparison to 6D super-Yang-Mills amplitudes Structure of perturbative amplitudes Evaluation of color factors and box integrals A Normalizable vertex operators A.1 NS-sector A.2 R-sector B Three-state Potts model There are two interesting maximally supersymmetric quantum field theories in six dimensions: the (2, 0) superconformal field theory and the (1, 1) super-Yang-Mills theory. They are believed to be the low energy limits of (2, 0) or (1, 1) little string theories (LST) [1 While the LST is a strongly coupled string theory that is difficult to get a handle on, it admits a 1-parameter deformation known as the double scaled little string theory (DSLST) [7, 8], the parameter being an effective string coupling. In the weak coupling limit, DSLST can be studied perturbatively. In particular, one can compute the spectrum In this paper, we will focus on the (1, 1) theories. In this case, the LST reduces to 6D SYM at low energies. More precisely, the interactions of massless modes of LST are expected to be described by an effective theory that is 6D SYM deformed by higher dimensional operators (such operators are highly constrained by supersymmetry and we will C.1 Zamolodchikov recurrence formula for conformal blocks C.2 Crossing symmetry D 3-loop UV finiteness of the four-point amplitude for the Cartan gluons 45 gs gY M mW This can be understood by identifying the little string with instanton strings in the SYM. The DSLST, on the other hand, is related to 6D SYM on its Coulomb branch. The relation between the string coupling and the Coulomb branch parameter takes the form Thus one may anticipate a relation between scattering amplitudes of massless modes in the DSLST and those of the Cartan gluons in the 6D SYM of the schematic form ADSLST (gs, 0, E) = ASY M (gY M , mW , E), provided the identifications (1.1) and (1.2). Note however that the weak coupling limit of DSLST corresponds to the regime far from the origin on the Coulomb branch. In the 6D gauge theory, a priori one would expect higher dimensional operators to become important in this limit, and it is not a priori clear whether the gluon scattering amplitudes in the SYM, expanded in 1/m2W , should agree with that of the perturbative amplitudes of massless modes of DSLST. In fact, the 6D SYM diverges at three-loops, and one might be led to think that such an agreement is impossible with just the SYM perturbative amplitudes (without including any higher dimensional operators). However, a more careful inspection indicates that the gauge theory is free of UV divergence at 3-loops. This is because there are only two candidate counter terms that are dimensional 10 operators allowed by sixteen supersymmetries; one of them is a double trace operator known to be 1/4 BPS and finite in 6D SYM according to [9], the other is a single trace operator that is non-BPS but vanishes when restricted to the Cartan subalgebra of the SU(k). (We will explicitly verify the absence of this UV divergence in appendix D.) This suggests that a certain non-renormalization theorem is at play, and one could hope that the 6D SYM by itself already captures some of the dynamics of massless gluons on the Coulomb branch of the (1, 1) little string theory. Remarkably, we find agreement between up to two-loop amplitudes of the massless gluons on the Coulomb branch of the 6D SYM, expanded to leading order in 1/m2W , and the tree level amplitude of the corresponding massless string modes in DSLST, expanded the SU(k) SYM, based on previously known two-loop results derived from unitarity cut methods [1015], and in the DSLST by a worldsheet computation that involves expressing SL(2)/U(1) coset CFT correlators in terms of Liouville correlators, which are then expressed as integrals of Virasoro conformal blocks. The computations on the two sides utilize completely different techniques and the results are rather involved. Nonetheless, we will evaluate the results numerically for k up to 5, and the answers on the two sides strikingly agree. While our result may be viewed as a strong check of the duality between DSLST and 6D SYM, it also hints that the higher dimensional operators correcting the 6D SYM are under control. Put differently, the perturbative 6D SYM seems to know a lot more about its UV completion and particularly the instanton strings than one might have naively expected! The paper is organized as the following. After reviewing the construction of DSLST and its spectrum, we will study the tree level four-point amplitude of massless RR vertex operators in six dimensions. In computing such amplitudes, we will need the singular limits of certain correlators in the SL(2)/U(1) coset (cigar) CFT, which are further related to Liouville correlators by Ribault and Teschners relations [16]. This will allow us to express the DSLST amplitudes as integrals of Liouville/Virasoro conformal blocks.1 Order by order in table 1 and the 6D SYM results are summarized in table 3. These amplitudes are then compared with gluon scattering amplitudes of 6D SYM on its Coulomb branch, expanded in 1/m2W to the leading order (if these amplitudes were the full story, higher order terms in the 1/m2W expansion would be mapped to higher genus 1The integration of conformal blocks was circumvented in [17] as the authors of [17] were only concerned blocks will be carried out. double scaled little string amplitudes). The 1-loop and 2-loop amplitudes are perfectly finite and can be obtained using unitarity cut method [1214], and the 3-loop amplitudes have been reduced to scalar integrals as well [18] (as already asserted, they are free of UV divergences when the external lines are restricted to the Cartan subalgebra). An agreement of the 1-loop amplitude at order 1/m2W with the low energy limit of four gluon amplitude involve highly nontrivial group theory factors). While the expressions on both sides are in the three-state Potts model went into the DSLST amplitude computation!) We conclude with a discussion on the implication of these results. Details on the numerical integration of conformal blocks, based on Zamolodchikovs recursion relations [19, 20], are described in appendix C. Double scaled little string theory In section 2.1, we review holographic descriptions of DSLST. In sections 2.2-2.6, we construct the normalizable vertex operators of the supersymmetric quantum numbers satisfying (2.56) subject to the identification (2.62) and (2.64). Our (SL(2)k/U(1) main result is in section 2.7, where we identity the massless bosonic vertex operators of type IIA string theory on R1,5 (SL(2)k/U(1) SU(2)k/U(1))/Zk, which is the T-dual description of IIB (1,1) DSLST. These massless string states correspond to the scalars and gluons in the 6D SYM. The holographic description The physical definition of double scaled little string theory is the decoupled theory on k NS5-branes in type IIA or IIB string theory, spread out on a circle of radius r0 in a transverse plane R in string units. Here one may consider an energy scale of interactions comparable to the string scale (without taking a low energy limit). The configuration or the decoupled theory has a residual global symmetry Zk U(1). In the case of IIB NS5-branes, D1-branes extending between NS5-branes correspond to W-bosons in the six-dimensional SU(k) gauge theory. They become tensionless in the r0 0 limit, where we recover the strongly coupled (1,1) LST. On the other hand, one may directly take the decoupling limit of the string worldsheet theory in the NS5-brane background. Recall that (1,1) LST is described by IIB string theory on the linear dilaton background from the NS5-brane near horizon geometry [3, 21]. T-dual to the SUL((21))k cigar CFT in IIA, similarly for (2,0) DSLST [7, 22, 23] (see also [24]). This leads to the holographic description of DSLST, as IIA (in the (1, 1) case) or type IIB (in the (2, 0) case) string theory with the target space given by [25] in the NSR formalism. The Zk orbifolding acts simultaneously on the two (supersymmetric) coset models. The supersymmetric coset model SL(2)k/U(1) can be constructed from the bosonic 3 and the total bosonic U(1) current at level k (combining a U(1) from the central charge is i The current 2 abcbc gives rise to an SL(2) current algebra at level 2. Altogether, we have a level k SL(2) current for the supersymmetric SL(2)k WZW model. This theory has N = 1 superconformal symmetry, with the supercurrent G = abajb 6 abc as required for critical string theory. SL(2)k/U(1) One starts with the bosonic SL(2)k+2 WZW model, governed by the current algebra ja(z)jb(0) The anti-holomorphic currents will be denoted as ja. The extension to supersymmetric m = 9 + 3(k 2) = 15, Let us summarize the relations between the various currents in this construction. We sum being the total SL(2)k current J a for the supersymmetric SL(2)k WZW model. The J 3 = j3 = JR = i XR, iH = x = follows from (2.5) and (2.7) that the following relations hold among the bosonization scalars: for JR to be well defined in the coset theory. The NS superconformal primaries of the coset model can be constructed starting from the primaries of the supersymmetric SL(2)k WZW model, which can be taken to be the where we have defined G = JR = j j1 ij2. and charge m and m with respect to j3 and j3. The range of j and m, m will be discussed later. The SL(2)/U(1) primary Vjs,ml,m is then obtained by factoring out a U(1) primary,2 js,lm,m = Vjs,ml,m eq k2 (mXm X ), 2As we are focusing on the massless string modes in this paper, we only need to take into account those coset primaries that come directly from current primaries. q k2 X, J3 = + nience. The coset model primary Vjs,ml,m has conformal weights j,m = j(j + 1) + m2 j,m = j(j + 1) + m2 respect to JR. Combining the above two observations, we obtain the R-charges for the coset primary Vjs,ml,m , R(Vjs,ml,m ) = R(Vjs,ml,m ) = Non-normalizable, delta function normalizable, and normalizable priDepending on the value of j, the primary operator Vjs,ml,m can either be non-normalizable, delta function normalizable, or normalizable along the radial direction of the cigar. The non-normalizable primaries correspond to generic real j, whereas the delta function normomentum along the radial direction of the cigar). Note that the conformal weight formula 3This bound is slightly stronger than that imposed by the no-ghost theorem in string theory on Vjs,ml,m = R(j, m, m ; k)Vslj1,m,m , R(j, m, m ; k) = (k)2j+1 (1 2j+1 )(j + m + 1)(j m + 1)(2j 1) k (1 + 2jk+1 )(m j)(m j)(2j + 1) non-negative. If n is negative we simply exchange the role of m and m in the formula for the reflection coefficient. Using this reflection relation we can restrict the non-normalizable upper bound j < k1 to ensure the two point function of Vjs,ml,m is nonsingular [7, 8].3 2 The j3, j3 charges m and m are related to momentum and winding on the cigar (along the circle direction), SL(2) [28, 29]. m m = n Z, m + m = kw kZ, m = m = n + wk So far, for either the non-normalizable or the delta function normalizable primary operators, there are no constraining relations between m, m and j. Importantly, there are also normalizable primary operators at special values of real j [17, 27], corresponding to principal discrete series of SL(2) [29], namely where m0 is given by m0 = (min{|m|, |m |}, min{m, m }, To be more precise, the normalizable vertex operators are the residue of Vj,m,m as j j [27], Vjno,mrm,m = Resjj Vj,m,m . Only delta function normalizable primaries and normalizable primaries are being used to construct the vertex operators of DSLST. The delta function normalizable primaries of the SL(2)/U(1) will lead to a continuum of string modes that propagate down the cigar, whereas the normalizable primaries will give rise to string modes localized at the tip of the cigar, thus effectively living in six dimensions [7, 27]. The scattering amplitudes of the latter is the subject of this paper. The supersymmetric coset model SU(2)k/U(1) can be constructed similarly to the SL(2)k/U(1) model. We will denote the primary operators and charges in the SU(2)k/U(1) model with primes in order to distinguish them from those in the SL(2)k/U(1) model. Let are (in this case there is no distinction between upper and lower indices) ji0(z)jj0 (0) Ji0 = ji0 2 ijk0j 0k. The overall SU(2)k current of the supersymmetric WZW model is given by R-current J R0, G0 = J R0 = k2 j30 + J30 = i j30 = i J R0 = i X0, r k 2 r k 2 XR0, X0 = r k 2 x0 + XR0 = r k 2 The NS superconformal primaries of the SU(2)k/U(1) coset model (which is the same weights and R-charges are j0(j0 + 1) m02 R = k For later application we will recall below two examples of supersymmetric SU(2)k/U(1) coset models where the primary operators and correlators can be easily written down. described as follows. Supersymmetric SU(2)3/U(1) and the compact boson with superconformal currents G0(z) = G0(z) = From (2.22) and (2.23), we can read off the relations between the bosonization scalars with dimension 32 . The coefficient convention for G0(z). The Virasoro primary operators are determined as usual 3n 23 n, w Z. I dz znJ R0(z)(0) = 0, zr+ 12 G0(z)(0) = 0, r > 0, which is always true for the Virasoro primaries (2.30). In the NS-sector, this implies that momentum n and the winding number w, G0+(z)G0(0) 3z3 + 3z3 + 2 T 0(z) + J R0(0) the identity operator. The former ones are Moving on to the R-sector, the superconformal primary condition (2.31) implies that n, w in (2.30): 1 3n + 1 3n 2 1, 2 1, 2 3n + 2 2 2 3n 2 2 corresponding to the R-sector primary operators R : exp i 23 0(z) exp i 23 0(z) , exp i 23 0(z) exp i 23 0(z) , Spectral flow , R = R = 6 , R = R = 2 . another operator O weight and R-charge = j(j + 1) + (m + )2 R = On the other hand, in the SU(2)k/U(1) superconformal coset, the spectral flowed operator R = su, 12 are R-sector vertex operators. Zk orbifold As already mentioned, the worldsheet CFT in the holographic description of DSLST (either type IIA or type IIB case) is The Zk orbifold is inherited from the holographic dual of the (non-doubly-scaled) LST, with worldsheet description stack of NS5-branes. It is a standard fact that the supersymmetric SU(2)k WZW model can be written as the Zk orbifold of the product of a supersymmetric U(1)k WZW model and a supersymmetric coset model SU(2)k/U(1) [32], SU(2)k = Before proceeding to the Zk orbifolding in DSLST, let us recall how (2.41) works. SU(2)k as a Zk orbifold of U(1)k We will write down primary operators of the supersymmetric SU(2)k WZW model in the language of the unorbifolded U(1)k SU(2)k /Zk. By comparing with the general primary operators in SUU((12))k , we will be able to identify the action of the Zk orbifold. the J30 charge, j0,m0 ei0H0 = eiq k2 (m0+0)X0 Vjs0u,m,00 , language of U(1)k U(1) SU(2)k /Zk, X0 is the (holomorphic part of the) compact boson at operator (2.38) in the coset model SU(2)k/U(1), as can be checked by comparing the q k2 X0. In the weights and R-charges on both sides. To complete the identification of vertex operators on the two sides of (2.41), we need to include the antiholomorphic part as well. The vertex operators coming from (2.42) are On the other hand, if we were considering the unorbifolded U(1)k operators would take the form iq k2 (m0+0)X0iq k2 ( m0+0)X 0 Vjs0u,m,(0,0m,00). iq k2 MX0iq k2 M X 0 Vjs0u,m,(0,0m,00), (mod 1) respectively. (2.43) would be reproduced from (2.44) with the identification However, the quantization condition on M and M in the U(1)k are different from that on m0, m 0 of vertex operators in the supersymmetric SU(2)k via this identification. The condition M M Z translates into SL(2)k/U(1) coset model. Hence we will use the same symbol H for the bosonization J Rtot = iH + iH0, Note that the orbifold projection demands that the total Zk charge vanishes, and this is in particular obeyed by (2.45). coset theory, on which the Zk orbifolding acts nontrivially. In preparation for the deformed case, let us introduce some notations in the linear dilaton theory. j0(j0 + 1) eq k2 jeiH+i0H0 j0,m0 = eq k2 jeiH eiq k2 (m0+0)X0 Vjs0u,m,00 . Now we will consider the deformation from (the internal part of) the worldsheet theory of LST to that of DSLST [7, 8], namely We would like to see how the vertex operator (2.51) is deformed. The weight and the R-charge (determined from (2.50)) of the vertex operator (2.51) are with the quantum numbers satisfying In the deformed theory, (2.51) maps to the following spectral flowed operator in which indeed has the same weight and R-charge (by (2.37) and (2.38)). In other words, as well as the constraints (2.46) and (2.47), will survive the orbifold. operators in the deformed theory SU(2)k /Zk can be written as Combining the holomorphic and antiholomorphic part, we see that a class of vertex Identifications among vertex operators in the coset theories both the SL(2)/U(1) and SU(2)/U(1) coset theories. These identifications can be traced back to the ones in the bosonic coset theories. Let us start with the bosonic SU(2)kbos /U(1) at level kbos, with primary operators Vjb0,oms 0 . The quantum number j0 lies in the range 0 j0 kb2os , whereas m0, unlike in the bosonic SU(2)kbos WZW model, is a priori unconstrained. There is the following identification among primaries labeled by different quantum numbers [17, 33]:5 Vjb0,oms 0 = V kbb2ooss j0, kb2os +m0 statement that m0m 0 (mod 1) is the charge with respect to the residual Zkbos action on kbos the coset theory. j0,m0 ei0H0 = Vjs0u,m,00 eiq k2 (0+m0)X0 = Vjb0,oms 0 eiq k 22 m0x0 ei0H0 . kmb0o2s only applies for |m0| j0 which labels the primaries obtained directly from factoring out U(1). Within this range, the identification is only Vjb0,oms 0 = Vjs0u,m,00 ei0q kk2 XR0+im0 k(2k2) XR0 Vjs0u,m,00 = V ks2u2,0 j0,m0 k 22 eiq kk2 XR0 = V ksu2,01 SU(2)k/U(1) is not invariant under a shift on m0 alone. Rather, we have to the Zk symmetry. We can write the above identifications in a more compact form, Similarly, primary operators in the bosonic SL(2)k+2/U(1) are subject the following identification6 [17, 34] from which we obtain the identification for the supersymmetric SL(2)k/U(1) theory 2 j0,m0 k 22 = V ksu2,0+1 Vjb,mos = V kb2o2sj, k+22 +m, Vjs,ml, = V ks2l,2+1j,m k+22 = V ksl,21 Note the sign difference in 1 and m k+2 when compared with SU(2)k/U(1). Once 2 Massless string states Now we discuss the construction of physical vertex operators in type IIA string theory on SU(2)k /Zk, which is the T-dual description of IIB (1, 1) DSLST [7, 8]. U(1) We will focus on the explicit description of massless string modes in the R1,5, localized at the tip of the cigar. We will further restrict our attention to bosonic string modes, and discuss the (NS,NS) sector and (R,R) sector separately. the highest weight state of Dj to the lowest weight state of D+kbos j2 2 6We are considering here coset primaries that come directly from the lowest weight (resp. highest weight) . The rest of the relations may be thought of as extending the definition of Vjb, mos (see [34] for an explanation of the origin of this identification). Consider the (NS,NS)-sector vertex operators of the form [17, 27] VNS = eeipX Vjs,ml,(,m,)Vj0,m0,m 0 J 3 = in the (NS,NS)-sector. Since VNS has no R1,5 spacetime index, it should be the vertex operator for the six-dimensional scalar fields. The mass shell condition is 2 = 0. the Zk orbifold condition (2.57). The on-shell condition for massless states then reduces to Next let us examine the chiral GSO projection condition, j0(j0 + 1) j(j + 1) FL + R 1 2Z, where FL is the holomorphic worldsheet fermion number of R1,5 (and similar for the anti-holomorphic sector). The total R-charge can be computed using spectral flow (2.37) R = vertex operator of the form (2.66) that obeys GSO projection condition must satisfy (see appendix A.1) normalizable massless vertex operators in these two cases are j = , m = j0 = , m0 = 2 , for ` = 0, 1, , k 2, ( = 1, 0 = 0 or = 0, 0 = 1). Note that m, m0 are both positive in these cases. holomorphic part, To summarize, the normalizable vertex operators from the internal CFT have, in their 2 , `+22 V ` 2s,u,02` , V 2`s,l,`0+22 V 2`s,u2`,1, 2 , `+22 V ` The operators in (2.74) are not all independent, however. Recall that we have the identifications (2.63) and (2.65), and therefore, negative in these cases. with the solutions k 22 . Note that m, m0 are both j = , m = j0 = , m0 = , 2 , `+22 V ` 2s,u,02` = V ksl,20` , k2` V k2` , k2` , su,1 2 2 2 2 , `+22 V ` su,`1 = V ksl,21` , k2` V k2` , k2` . su,0 2 , 2 2 2 2 In particular, the identification flips the sign of m and m0. This will be important when we include the anti-holomorphic part of the vertex operators. Combining with the anti-holomorphic part, normalizability requires in addition that this case m < 0 but m > 0. On the other hand, from the identifications (2.75), some of In the end, there are four inequivalent pairings between the holomorphic and anti-holomorphic quantum numbers that are allowed in the massless vertex operator 0 1 0 1 1 1 1 0 m, m , m0, m0 are all negative for the vertex operators pairing this way (see (2.74)), we can The explicit forms of the four sets of normalizable vertex operators are VNS1,` = eeipX V 2`s,l,(1,1) su,(0,0) , VNS2,` = eeipX V 2`s,l,(0,0) `+22 , `+22 V 2` , 2` , 2 su,(1, `1), VNS3,` = eeipX V 2`s,l,(1`,+202), `+22 V 2` , 2` , 2 su,(0,1`), VNS4,` = eeipX V 2`s,l,(0`,+212), `+22 V 2` , 2` , 2 su,(1,0`), or, equivalently, using the identifications (2.75), we can rewrite them as VNS3,` = eeipX V 2`s,l,`(+202, ,`1+2)2 V `su`,(1`,0), VNS4,` = eeipX V 2`s,l,`(+221,,`0+2)2 V `su`,(0`,1), + + 2 , 2 , 2 2 , 2 , 2 VNSi, ` = VNSi, k2`, i = 1, 2, 3, 4. VNS1,0, VNS2,k2, or equivalently VNS1,k2, VN+S2,0, are in the untwisted sector, i.e. they + Let us compare this with the NS5-brane or six dimensional gauge theory description. We are at the point in the Coulomb branch moduli space where the k NS5-branes are spread on a circle in R and at the same time rotates the circle of spread. The center of mass mode decouples, are scalars that are linear combinations of collective coordinates of the NS5-brane in the R2 that contains the circle of spread, while Zej are scalars associated with the remaining is a center of mass mode and decouples, thus absent from the DSLST spectrum. The is absent from the spectrum. So indeed there is only a single massless complex scalar Z0 that is uncharged under the Zk symmetry. In the (R,R)-sector, we consider the vertex operators [17, 27] VR = a,a e 2 2 eipX SaSea Vj,m,m the U(1)k1 gauge bosons. where Sa, Sea are the spin fields in the R1,5, and a and a are the indices in the 4 and 4 of in the 15. In the massless case, VR will give the vertex operators for the field strength of The on-shell condition for the vertex operator (2.80) is 8 = 0. (2.81) j0(j0 + 1) j(j + 1) Its straightforward to derive that (see appendix A.2) physical vertex operators surviving 1 the GSO projection FL + R 2 2Z must have the following combinations of spectral flow = 2 , 0 = 2 , and FL = 2 , 2 . Let us first consider the case = 12 , 0 = 2 . We must have j = j0 and m = m0 1. The 1 solutions for the normalizable states are or, equivalently, The two are related by the reflection (2.86), Note that m, m0 are both negative in this case. solutions for the normalizable states are Similarly, in the case = 2 , 0 = 12 , we must have j = j0 and m = m0 + 1. The 1 Note that m, m0 are both positive in this case. are in fact identified by (2.63) and (2.65), 2 , `+22 V ` su,1/2 = V ksl,21`/,2k2` V k2` , k2` . su,1/2 2 , 2` 2 2 2 Combining with the anti-holomorphic part, as before, normalizability of the vertex , 0 = 21 in the holomorphic sector with = 2 , 0 = 12 in the anti-holomorphic sector. In the end, the gauge boson 1 . The fact that m and m must take the vertex operators are VR,` = a,a e 2 2 eipX SaSea V ` 2 , `+22 , `+22 V ` su,(1/2,1/2), ` = 0, 1, , k 2, j = , m = ` = 0, 1, , k 2 j = , m = ` = 0, 1, , k 2, j0 = , m0 = 2 , j0 = , m0 = and Q = b + b1 = 5/2. k = 4, ` = 0, 2 : ADSLST 78.96 144.6 2 s12 + O(02s2). 4 ` , k = 4, ` = 0, 2 : ADSLST = 1.831. due to the symmetry in s12, s13, and s14, so ADSLST is zero. k = 5 The relevant operators in the supersymmetric SU(2)5/U(1) coset model for the gauge boson vertex operators (2.87) and (2.88) are Combining with the correlators in the R1,5 and SL(2)4/U(1) parts, we obtain the following scattering amplitude, Z dP ADSLST 1 + 1 z + 1 1 z C 3, 4, 2 iP |F (1, 2, 3, 4; P ; z)|2, V0s,u0,,01/2,1/2(z, z) = V 32 , 32 , 2 3 V 1s,u1,1,/12,1/2(z, z) = V1s,u,1,1/12,1/2(z, z), = = 2 2 2 V1s,u1,,11/2,1/2(z, z) = V 12 , 12 , 2 su,1/2,1/2(z, z), = = 1 V 32s,u32,1,/322,1/2(z, z) = V0,0,0 su,1/2,1/2(z, z) , R = R = , R = R = , R = R = 10 , R = R = 10 These four operators with dimension 3/40 can be realized as a free boson 0 and the V0s,u0,,01/2,1/2(z, z) = V 32 , 32 , 2 su,1/2,1/2(z, z) = exp i 3 11The three-state Potts model is reviewed in appendix B. 0(z) exp i i 0(z) exp i i 0(z) exp i = hV 3 3 3 = |z| 10 |1 z| 10 (z, z)V 3 3 3 (z, z)V 1 1 1 su,1/2,1/2 su,1/2,1/2 (, )i = hV1,1,1 (0, 0)V1,1,1 (z, z)V1,1,1 su,1/2,1/2 (1, 1)V1,1,1 su,1/2,1/2 (, )i (z, z)V0,0,0 (0, 0)V0,0,0 su,1/2,1/2 (1, 1)V0,0,0 su,1/2,1/2 (, )i su,1/2,1/2 su,1/2,1/2 (, )i and Q = b + b1 = 1/ 5 + 5. The amplitude is numerically computed to be k = 5, ` = 0, 3 : ADSLST 110.4 264.5 iP k = 5, ` = 0, 3 : ADSLST = 2.397. (z, z) = V1,1,1 su,1/2,1/2 (z, z) = V 1 (z, z) = V0,0,0 su,1/2,1/2 su,1/2,1/2 1 1 = |z| 30 |1 z| 30 GPotts (z, z). ` = 0, 3 ` = 1, 2 ADSLST Z dP ADSLST Z dP 20 s13+ 175 GPotts (z, z) iP , The four-point functions of the SU(2)5/U(1) coset model for each value of `, which takes three possible values, 0, 1, 2, 3, are then 27.92 + 0 s12 51.28 60.11s12 78.96 144.6s12 157.9 + 0 s12 110.4 264.5s12 220.7 304.6s12 expansion, i.e. the coefficient of s12 divided by the s12-independent term in the full amplitudes. These ratios exactly match with the ratios of the 6D SU(k) SYM amplitudes listed in the last appendix B. The amplitude is numerically computed to be k = 5, ` = 1, 2 : ADSLST = 1.380. The overall normalization of the amplitudes (which depends on k and `) have not been lently, the expansion in the Mandelstam variables, of a particular amplitude are computed unambiguously. These exactly match with the 1-loop and 2-loop 6D SU(k) SYM ampliamplitudes to higher order, and they should be compared to higher than 2-loop amplitudes on the SYM side. We will comment on this later. Comparison to 6D super-Yang-Mills amplitudes The strong coupling limit of DSLST, namely LST, reduces to 6D (1, 1) SU(k) SYM in the low energy limit. It is conceivable that the full dynamics of the massless degrees of freedom at the origin of the Coulomb branch is described by a Wilsonian effective action, which is that of SU(k) SYM deformed by an infinite set of higher dimensional operators, with a 12A fully supersymmetric regulator is assumed. The DSLST corresponds to a point away from the origin on the Coulomb branch of this theory, at which a Zk U(1) subgroup of the SO(4) R-symmetry is preserved. In the gauge A(E) = X(gY M E)2L+2AL(mW /E). A priori, AL is not quite the same as the L-loop amplitude. As we expand the scalars around their vevs, we obtain couplings that involve positive powers of gY M and non It seems reasonable to assume that the Wilsonian effective Lagrangian. With such vertices, AL generally receives contributions from diagrams of no The perturbative amplitude in DSLST, on the other hand, has the structure (after A(E) = X(gY M mW )2h2Alhst where h labels the genus (which is entirely unrelated to the loop order in the SYM theory). also has an analytic expansion in 1/m2W at large mW , namely While AL(mW /gY M ) is naturally defined by an analytic expansion in mW /gY M (as the theory is free of infrared divergences), the duality with DSLST suggests that AL(mW /gY M ) AL(mW /E) = X and in particular, we expect the tree level DSLST amplitude to agree with the 1/m2W part of the gauge theory amplitude, = X(gY M E)2L+4A(L1). pendix D), the 3-loop divergence is absent when the external legs are restricted to the Cartan subalgebra. It turns out that the first UV divergence of the Cartan gluon fourpoint amplitude arises at four-loop.13 In any case, one can ask whether the perturbative to be yes, and in fact the finite 1-loop amplitude in SYM, when expanded to first order in 1/m2W , agrees with the leading low energy term in the tree level DSLST amplitude 13We thank the authors of [40] for pointing this out. The relevant four-loop divergence can be extracted up to an overall multiplicative constant [17]. Since the 6D SYM by itself is UV finite at 2-loop as well, one might suspect that A2(mW /E) is also given entirely by the 2-loop 6D Remarkably, we will find that this is indeed the case. Structure of perturbative amplitudes We will consider four-point amplitudes in SU(k) maximally supersymmetric Yang-Mills theory obtained from unitarity cut methods [1015]. While such amplitudes are mostly studied in four-dimensional gauge theories, where the L-loop result can be expressed in terms of the tree level amplitude together with scalar loop integrals, such formulae admit straightforward generalizations to higher dimensions. It is known that up to 3-loop order dimensions can be extended to D dimensions by simply replacing the relevant scalar loop integrals by the D-dimensional loop integrals [10, 11]. We will express the amplitudes in 6D SYM in terms of 6-dimensional spinor helicity are SU(2) SU(2) little group indices, and A, B are spinor indices of SO(6) or SO(5, 1) Lorentz group. The amplitudes involving various particles in a supermultiplet will be is convenient to express the latter in terms of the supermomenta The color-ordered four-point tree-level superamplitude can be written as tree(1, 2, 3, 4) = s12s23 off. The delta function is the one in Grassmann variables. Explicitly, it can be expanded as 8(X qi) = 4(X qiA)4(X qiB) (4!)2 i,j,k,`,m,n,r,s=1 ABCDqiAqjBqkC q`D EF GH qmE qnF qrGqsH . respect to the SU(2) SU(2) R-symmetry (not to be confused with the little group), or rather, U(1) U(1) Cartan generators of the R-symmetry group, these scalars have transverse R R2 of the 5-brane are linear combinations of these two Cartan generators. In this paper we are interested in the scattering amplitudes of the massless gluons in due to the vanishing color factor. The nonvanishing loop amplitudes of the Cartan gluons contain W -bosons in the loops (as well as possibly massless gluon propagators at two loops and higher). The full 1-loop amplitude in SYM is given by14 [12] (see also [13]) color factor associated with the box diagram. This relation holds in any D, which we now need to replace I41loop(s12, s14) by15 scaled little string theory. written in the form In the last line we have expanded the result in 1/m2W . As explained, it is the order 1/m2W result of the SYM amplitude that will be compared with the genus zero amplitude of double In the end, the 1-loop amplitude of Cartan gluons on the Coulomb branch can be A1loop(1, 2, 3, 4) = i8(X qi) C1234 + C1324 + C1243 summed over the species of W -bosons if k > 2. tree(1, 2, 3, 4) refers to the color-ordered partial amplitude (the full amplitude is obtained by summing over s12, s13, s14 channels), A 1loop(1, 2, 3, 4) and A 2loop(1, 2, 3, 4) are full amplitudes. 15The generalization to massive propagators in the loop is justified by consideration of unitarity cuts. The full 2-loop amplitude is given by the tree-level amplitude multiplied by 2-loop scalar integrals [14] A2loop(1, 2, 3, 4) = s12s23Atree(1, 2, 3, 4) 2loop,P and Aabcd ure 2. Once again, the propagators in the loops will be replaced by the appropriate massive W -boson or massless gluon propagators in the amplitudes on the Coulomb branch of the The 3- and higher-loop amplitudes generally contain logarithmic divergences. It is likely that they still contain nontrivial information that captures the DSLST amplitudes Evaluation of color factors and box integrals need to impose the traceless condition by hand in this case, as the overall U(1) decouples due to the interaction vertices). The external massless gluons will be labeled by vectors ~v1, , ~v4 in the Cartan subalgebra of su(k). The mass of the (ij)-W -boson is mij = r0|i j | = 2r0 sin eter that will be related to the inverse string coupling of DSLST. Expanding around the point in Coulomb branch with Zk symmetry, corresponding to the NS5-branes spreading out on the circle in a transverse R2, it is convenient to take ~va to be Zk charge eigenstates, vaj = (j1)na , j = 1, , k, X na 0 mod k. na of interest are16 for ` = 0, 1, , k 2. As discussed before, the gluon vertex operator VR,` in DSLST has Zk momentum (` + 1). Therefore, we see that in order to compare with the DSLST scattering amplitude n1 = n2 = ` + 1, n3 = n4 = k (` + 1), The 1-loop amplitude, expanded to order 1/m2W , is of the form A1loop = i6=j A(s12, s14) + O(mW4), Plugging in the explicit expression for vai, we can further write A(s12, s14) = A1loop = + O(r04) As was shown in [17], the sum collapses into a curiously simple answer, A1loop = 2 A(s12, s14)min{na, k na} + O(r04). Now consider the 2-loop amplitude. In the planar case, let us label the W -boson running through vertices 1,2 by (ij), the W -boson running through 3,4 by (`m), and the where the scalar loop integral is I2loop,P (mij , m`m, mnr) jnr`mi j`mnri 4 I2loop,P (mij , m`i, mj`) Y (vai va) Y (va` vai), j 4 2 Y (vai va) Y (va` vam) j 1 Z d6`1 d6`2 16We shift n3 and n4 by k for later convenience. 6.048 16.876 34.594 39.883 4.500 12.435 25.327 29.136 + O(r04) + O(r04). A1223l4oop,P 1/m2W expansion for four-gluon scattering in 6D SU(k) SYM. Here we choose the Zk charges for The numbers are in units of s12s23A This gives the color-weighted planar amplitude Qa=1,2 e ikna (j`) sin( na(ij) ) Qa=3,4 sin( na(i`) ) k k 0 m,r=0 8k kX1 Z d6`1 d6`2 Qa=1,2 e ikna (rm) sin( nkam ) Qa=3,4 sin( nkar ) (2)6 (2)6 (`21 + sin2 km )3(`22 + sin2 kr )3((`1 + `2)2 + sin2 (mr) ) + O(r04). k (`21 + sin2 (ikj) )2(`22 + sin2 (ik`) )3((`1 + `2)2 + sin2 (jk`) )2 (2)6 (2)6 (`21 + sin2 km )2(`22 + sin2 kr )2((`1 + `2)2 + sin2 (m+r) )3 k These convergent integrals and sums over color factors can be computed numerically. The results for the planar and non-planar contributions to the two-loop amplitude are given in table 2, and the full two-loop amplitudes, whose expression is given by (4.11), are listed in table 3. We see that the ratios listed in the last column remarkably match with the ratios computed from DSLST that are listed in table 1, to the numerical precision of the conformal block integration. The tree DSLST amplitudes provides all order results gY2 M and first order in 1/m2W of the UV completed 6D gauge theory on its Coulomb branch. While the agreement of 6D A1loop 10.548 29.311 59.922 69.019 1.1720 1.8319 2.3969 1.3804 four-gluon scattering in 6D SU(k) SYM. Here we choose the Zk charges for the external gluons to both in units of s12s23A burning question is whether the SYM 3-loop amplitude, which as discussed is finite when the external lines are restricted to the Cartan subalgebra, agrees with the DSLST at nextreduced to scalar integrals, it is merely a matter of evaluating these scalar integrals to answer the question. We hope to report on the result in the near future. One may also try to carry out the DSLST amplitude computation to higher genus, and compare with the higher order terms in the 1/m2W expansion of the SYM amplitude at each loop order. This is not easy as the relevant genus one four-point function in the cigar coset CFT is not yet known, but would nonetheless be interesting. From the point of view of the Abelian effective action on the Coulomb branch, the 2-loop amplitude of order 1/m2W comes from the 1/4 BPS dimension 10 operator of the of this term in the Coulomb effective action, with respect to higher dimensional non-BPS operator corrections to the non-Abelian SYM theory. If so, then the 3-loop test will be particularly important, and an agreement with the DSLST tree amplitude at the next order In any case, the big question here is, to what extent will the agreement between the massless amplitudes of pure 6D SYM on the Coulomb branch and DSLST hold, and why do they agree? It so happens that the Cartan gluon amplitude becomes divergent at four-loop [40]. Therefore we will definitely see some nontrivial disagreement with the LST known but the UV divergence with external legs in the Cartan subalgebra has yet to be extracted [43].18 A priori, there could be all sorts of higher dimensional operators that enter the Wilsonian effective action of the 6D gauge theory and correct the amplitudes of the SYM theory itself. After all, we do expect the presence of the dimension 10 non17We thank Ofer Aharony for pointing this out. 18We thank the authors of [40] for explaining to us the results of [40, 43, 44]. BPS operator (see for instance [45]) as the counter term that cancels the general 3-loop divergence, even though this operator vanishes when the fields are restricted to the Cartan subalgebra. A systematic investigation of the higher dimensional counter terms and their effect on the Cartan gluon scattering amplitude is left to future work. Finally, let us mention that the W-bosons in the 6D SYM are dual to D1-branes stretched between the NS5-branes. The scatterings of strings with the D1-branes correspond to the scatterings of the Cartan gluons with the W-bosons, and also the scatterings of the D1-branes with themselves are dual to the scatterings of W-bosons. Some aspects of open strings and D-branes in DSLST are studied in [46] (also see [47] for the D-branes to the closed string two-point amplitudes on a disc ending on stretched D1-branes, and compare with the scattering amplitudes of two Cartan gluons and two W-bosons. Acknowledgments We would like to thank Ofer Aharony, Clay Cordova, Lance Dixon, Thomas Dumitrescu, Yu-tin Huang, Daniel Jafferis, Ingo Kirsch, Soo-Jong Rey, David Simmons-Duffin, and Andy Strominger for helpful conversations and correspondences at various stages of this project. We would like to thank the Kavli Institute for Theoretical Physics and Aspen Center for Physics during the course of this work. C.M.C. has been supported in part by a KITP Graduate Fellowship. C.M.C. would like to thank the Physics Department of National Taiwan University for hospitality during the final stage of the work. S.H.S. is supported by the Kao Fellowship and the An Wang Fellowship at Harvard University. X.Y. is supported by a Sloan Fellowship and a Simons Investigator Award from the Simons Foundation. This work is also supported by NSF Award PHY-0847457, and by the Fundamental Laws Initiative Fund at Harvard University. 2(j0 j)(j0 + j + 1) = k(1 2 0 ). Normalizable vertex operators Meanwhile the GSO condition demands that to solve is and therefore, For normalizable vertex operators we have j0 j Z0. |m| > j, j0 |m0|, which implies that This further implies, which is equivalent to This is impossible. j j0 < |m| |m0| |m m0| = | 0|. Assuming j > j0, we can combine the two equalities above to get 2|j0 j| > |2 + 02 1|. 2 + 02 1| < 2(j j0) < 2| 0|. Therefore the only normalizable solutions that survive the GSO projection satisfy The on-shell condition in the R sector is and the GSO condition becomes 4(j0 j)(j0 + j + 1) = k(1 22 202), Since we are looking at half-integer spectral flows, this implies that and therefore, Then the mass-shell condition requires that 22 + 202, | 0|2 4Z + 1. j0 j Z0. Assuming j > j0, we have the following inequality from the normalizability condition which demands This is impossible. |22 + 202 1| < 4(j j0) < 4| 0|, 4| 0| |22 + 202 1| + 8 | 0|2 + 7. Their dimensions are = , X = , Y = 3, = = In addition to the scalar primaries, there are spin 1 primaries Three-state Potts model model. The scalar primary operators in this theory are 1, , X, Y, , , Z, Z. and also spin 3 primaries, is relevant to us, The fusion rules of the primaries are given in [48]. Here, we only present the part that can be written as 4 GPotts (z, z) the conformal block. 4 GPotts(z, z) F 7 (z)F 2 (z) C0,3 F0(z)F3(z) C3,0 F3(z)F0(z) , 4 GPotts(z, z) F 7 (z)F 2 (z) + C0,3 F0(z)F3(z) + C3,0 F3(z)F0(z) , and the rest are found to be 1.09236, Numerical methods Zamolodchikov recurrence formula for conformal blocks function of four scalars is expressed in terms of the three-point functions and the conformal where z is the cross ratio q is defined by z = n= F (i; |z) = (16q)P 2 z Q42 1 2 (1 z) Q42 1 3 3(q)3Q24(1 +2 +3 +4 )H(i2; |q), K(1 z) K(z) = pt(1 t)(1 zt) The product of (r, s) is taken over and the product of (k, `) is taken over Finally, the function H is determined by the following recurrence relation m,n1 r = m + 1, m + 3, , m 1, s = n + 1, n + 3, , n 1, k = m + 1, m + 2, , m, ` = n + 1, n + 2, , n, faster than the corresponding series in x; this can be seen, for example, for small values of z if one notes q = 1z6 + O(z2). then the matrix equation is H(i2, 1,2 + 2|q) = 1 + qq1RR,211+,,112 q2R2,1 1,2+22,1 H(i2, 1,2 + 2|q) to order qN . We note some caveats in the implementation of this method. For special values of the central charge, for example when c equals the central charge of a minimal model, or therefore certain coefficients appearing in the recurrence relation diverge. Nonetheless, we can deform the value of the central charge from c to c + , and as it must all the poles in cancel. Therefore, with a small and high enough numerical precision (high enough so that the divergences cancel properly on the computer), we can still compute the conformal blocks for these seemingly pathological values of the central charge. Crossing symmetry Consider the four-point function (C.1), and let us define G(1, 2, 3, 4|z) which satisfies the following crossing relations The complete set of transformations are T 2 = S2 = 1, (T S)3 = 1. , (1432) z 1 , (1342) ST S : z z 1 ST : z 1 z , (1243) , (1423) T : z 1 z, (1324). We can divide the complex plane into six fundamental regions: I : Re z 2 , |z 1| 1, II : |z| 1, |z 1| 1, III : Re z 2 , |z| 1, region I by the ST S, T S, T, ST, S transformations, respectively. An integral involving conformal blocks over the entire complex plane can be rewritten as an integral over only region I. This is useful for doing numerical integration because, first, region I is bounded, and second, in this region |q| is bounded above by 0.0658287, which means that the Zamolodchikov recurrence formula (C.6) converges very quickly. amplitude for k = 2, 3, 4, 5 and ` = 0 as d2z|z|s12 |1 z|s13 G(1, 2, 3, 4|z) |1 z|s12 (|z|s13 + |z|s14 )G(1, 3, 2, 4|z) , and for k = 5, ` = 1 as d2z|z| 31 s12 |1 z| 32 s13 G(1, 2, 3, 4|z)GPotts (z) 2 2 (|z|s13 |1 z|s14 + |z|s14 |1 z|s13 )|z| 3 |1 z| 3 G(1, 4, 3, 2|z)GPotts(z) 2 1 |1 z|s12 (|z|s13 + |z|s14 )|z| 3 |1 z| 3 G(1, 3, 2, 4|z)GPotts (z) , 3-loop UV finiteness of the four-point amplitude for the Cartan gluons In this appendix we verify that the 3-loop 4-point amplitudes of the U(1) Cartan gluons (photon) in the SU(k) SYM is UV finite (see also [44]). This is indeed expected both and from the inspection on the possible counter terms mentioned in the introduction. The amplitude is reduced to scalar 3-loop integrals, summarized in figure 2 of [18]. The four potentially logarithmic divergent diagrams are shown in figure 3 (ignoring the signs on vertices for now). We will compute the divergent parts of these diagrams with color factors included and show that they cancel among themselves. Let us start with the SU(2) case where no actual calculation is needed to show the cancellation of UV divergence. The key fact here is that since there is only one species of 6D SYM. The signs for the internal vertices denote the two index structures in the double line notation; plus for the left vertex and minus for the right vertex in figure 4. We label the diagrams following the notation in figure 2 of [18]. The above sign assignments together with the other four of the scattering amplitudes of four Cartan gluons. (k) = i Cartan gluon, the amplitudes are invariant under permutation of all four external legs. It follows that the logarithmic divergent part of the amplitude is proportional to (s12 + s13 + Moving on to the general SU(k) 6D SYM, it suffices to show that the UV divergent part is invariant under cyclic permutations on the external legs 2, 3, 4, from which it again In the double line notation, each 3-point vertex can be written as the difference of two vertices shown in figure 4 with different index structures. Each diagram in figure 3 then is easy to see that the only diagrams that give noncyclic invariant amplitudes are the four The color factors for the four external Cartan gluons will be labeled by ~v1, , ~v4 in the Cartan subalgebra of su(k) as in the previous section. The four diagrams in figure 3 only interested in the divergent part, we have set the external momenta to be zero.The number indicates the propagator should be raised to the corresponding power. can be expressed as19 3 2 1 Ilog UV-divergent scalar integrals I1 log and I2log are defined in figure 5. the vertices. The stands for the finite as well as the cyclic invariant terms. The two Next, we need to sum over all the permutations on the external legs. After taking the symmetry factors for each diagram appropriately, the noncyclic invariant part of the divergent amplitude is proportional to 2N 2s12(v~1v~2)(v~3v~4) 1 Ilog + 1 Ilog 2I1log 2 2 2 3 2 1 Ilog = 0. (D.2) Note that we have grouped (v~1 v~3)(v~2 v~4) + (v~1 v~4)(v~2 v~3) with (v~1 v~2)(v~3 v~4) to the In summary, in this appendix we have showed that the 3-loop 4-point amplitudes for gluons in the Cartan subalgebra is free from divergence and we are then left with a finite amplitude. 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Chi-Ming Chang, Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang. Little string amplitudes (and the unreasonable effectiveness of 6D SYM), Journal of High Energy Physics, 2014, 176, DOI: 10.1007/JHEP12(2014)176