Interpolating the Coulomb phase of little string theory

Journal of High Energy Physics, Dec 2015

Abstract We study up to 8-derivative terms in the Coulomb branch effective action of (1, 1) little string theory, by collecting results of 4-gluon scattering amplitudes from both perturbative 6D super-Yang-Mills theory up to 4-loop order, and tree-level double scaled little string theory (DSLST). In previous work we have matched the 6-derivative term from the 6D gauge theory to DSLST, indicating that this term is protected on the entire Coulomb branch. The 8-derivative term, on the other hand, is unprotected. In this paper we compute the 8-derivative term by interpolating from the two limits, near the origin and near the infinity on the Coulomb branch, numerically from SU(k) SYM and DSLST respectively, for k = 2, 3, 4, 5. We discuss the implication of this result on the UV completion of 6D SYM as well as the strong coupling completion of DSLST. We also comment on analogous interpolating functions in the Coulomb phase of circle-compactified (2, 0) little string theory.

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Interpolating the Coulomb phase of little string theory

JHE Interpolating the Coulomb phase of little string theory Ying-Hsuan Lin 0 1 3 Shu-Heng Shao 0 1 3 Yifan Wang 0 1 2 Xi Yin 0 1 3 0 Cambridge , MA 02139 , U.S.A 1 Cambridge , MA 02138 , U.S.A 2 Center for Theoretical Physics, Massachusetts Institute of Technology 3 Je erson Physical Laboratory, Harvard University We study up to 8-derivative terms in the Coulomb branch e ective action of (1; 1) little string theory, by collecting results of 4-gluon scattering amplitudes from both perturbative 6D super-Yang-Mills theory up to 4-loop order, and tree-level double scaled little string theory (DSLST). In previous work we have matched the 6-derivative term from the 6D gauge theory to DSLST, indicating that this term is protected on the entire Coulomb branch. The 8-derivative term, on the other hand, is unprotected. In this paper we compute the 8-derivative term by interpolating from the two limits, near the origin and near the in nity on the Coulomb branch, numerically from SU(k) SYM and DSLST respectively, for k = 2; 3; 4; 5. We discuss the implication of this result on the UV completion of 6D SYM as well as the strong coupling completion of DSLST. We also comment on analogous interpolating functions in the Coulomb phase of circle-compacti ed (2; 0) little string theory. gauge theory, Superstrings and Heterotic Strings - HJEP12(05) 1 Introduction 2 The Coulomb branch e ective action 3 Perturbative 6D SYM in the Coulomb phase 4 The 0 expansion of little string amplitude 4.1 An interpolating function from weak to strong coupling 5 Discussion 6 Comments on (2; 0) LST and 5D SYM A 6D SYM loop amplitudes contributing to D4F 4 A.1 One-loop A.2 Two-loop A.3 Three-loop A.4 Four-loop B Evaluation of the little string amplitudes the other hand, may be regarded as an expansion near the origin of the Coulomb branch, and describes the strong coupling limit of DSLST. The goal of this paper is to exploit this correspondence, by connecting the two limits of the Coulomb phase of (1; 1) LST. We will inspect the derivative expansion of the Coulomb branch e ective action, focusing on terms of the structure fn(r)D2nF 4, n = 0; 1; 2; etc. Here r stands for the distance from the origin of the Coulomb branch, as measured by the scalar expectation values, and F the eld strength of the U( 1 )k 1 vector multiplets in the Cartan of the SU(k) gauge group. The most convenient way to organize the supersymmetric completion of these higher derivative terms in the e ective action is through the massless superamplitudes they generate [9]. For our purpose, it su ces to focus on the 4-point superamplitudes, which take { 1 { the form1 8(Q)F (s; t; u), where Q is the total supermomentum and s; t; u the Mandelstam variables [10{12]. F (s; t; u) will depend on the color assignment of the Cartan gluons, and depend on r through the W -boson masses. The 4-point superamplitude can be computed in the large r regime by the perturbative double scaled LST [13, 14]. In previous work we have formulated the tree amplitude in the DSLST in terms of an explicit double integration over the cross ratio of four points on the Riemann sphere and over a continuous family of conformal blocks, which is then evaluated numerically. In this paper we will present some higher order terms in the 0-expansion of the DSLST tree amplitude, giving the leading 1=r2 term of the fn(r)D2nF 4 coupling on the Coulomb branch, at large r. In the small r regime, on the other hand, we will perform a perturbative computation in 6D SU(k) SYM. The 4-point amplitude is reduced to 8(Q) times a set of scalar box type integrals, which can be evaluated straightforwardly up to 3-loops. We will present some numerical results for k = 2; 3; 4; 5. Starting at 4-loop order, the 4-point amplitude of Cartan gluons su ers from logarithmic UV divergences. This divergence structure is a bit intricate, as the non-abelian 4-point amplitude already diverges at 3-loop and a 3-loop counter-term of the form D2trF 4 is needed [15, 16]. While this counter-term vanishes when restricted to the Cartan, it gives a nontrivial contribution to the 4-loop amplitude, which has been studied in [16]. In the end, after taking into account suitable 4-loop counter-terms, of the form D4trF 4 and D4tr2F 4, one obtains a 4-loop contribution to f2(r) that involves logarithmic dependence on r, of the form (ln r)2 and ln r. While the nite shifts of the 3-loop and 4-loop counter-terms are not a priori determined in SYM perturbation theory (but should be ultimately xed in the LST), the coe cients of the leading logarithms are unambiguously determined. The results of [16] on the 4-loop divergence of double trace terms then allows for determining certain leading log coe cients, which when combined with 1; 2; 3-loop results produce the rst few terms in the small r expansion of fn(r). The agreement of the r 2F 4 term between a 1-loop computation of 6D SYM and low energy limit of DSLST found in [13], was expected as a consequence of the supersymmetry constraints on the F 4 coupling in the Coulomb branch e ective action [2, 17, 18]. The agreement of r 2D2F 4 term between a 2-loop computation of 6D SYM, the next order 0-expansion of the DSLST amplitude was found in our previous work [14], numerically for k = 2; 3; 4; 5. One anticipates that this agreement should follow from supersymmetry constraints on D2F 4 coupling, namely the function f1(r) should be xed to be the form C1=r2, and the coe cient C1 can then be computed from either small r (SYM) or large r (DSLST). Indeed, the agreement we found in the SU(3) case can be understood in terms of the (sixteen-supercharge) non-renormalization theorem of [18].2 Although the result of [18] 1For comparison, the color-ordered tree-level superamplitude is given by Atree = i 2For SU(2) gauge theory, the D2F 4 term in the Lagrangian is proven to be two-loop exact by [19, 20]. The focus of this paper is the f2(r)D4F 4 term. This is the lowest order in the derivative expansion of the Coulomb branch e ective action where we anticipate a nontrivial interpolating function f2(r) from small r (SYM) to large r (DSLST). Indeed, f2(r) receives all loop perturbative contributions. Collecting numerical results on both sides, we will be able to estimate the interpolating function on the entire Coulomb branch. We will nd that, while the small and large r limits are obviously di erent expansions, when naively extrapolated to the intermediate regime they are not far from one another. In the next section, we describe the general structure of the Coulomb branch e ective action and its relation to 1) massless scalars in 6 dimensions [3]. We denote these massless scalar by i, i = 1; 2; 3; 4, which take values in the U( 1 )k 1 Cartan of the SU(k) gauge group, in the 6D SYM description (which is a priori valid near the origin of the Coulomb branch). We will focus on a Zk-invariant 1-dimensional subspace of the Coulomb moduli space, corresponding to Z 1 + i 2 = r diag(1; e2 i=k; ; e2 i(k 1)=k); 3 = 4 = 0: The large r regime along this 1-dimensional subspace is then described by the perturbative double scaled little string theory [7, 8], with the worldsheet CFT given by R1;5 (SL(2)k=U( 1 )) (SU(2)k=U( 1 )) Zk : The string coupling at the tip of the cigar (target space of SL(2)=U( 1 ) coset CFT) is identi ed with 1=r. The massless degrees of freedom in the Coulomb phase, consisting of k 1 Abelian vector multiplets of the 6D (1; 1) supersymmetry, are governed by a quantum e ective action, that is the U( 1 )k 1 supersymmetric gauge theory action together with an in nite series of higher derivative couplings. We will focus on couplings of the schematic form f ( )D2nF 4 + . Such higher derivative deformations of the Abelian (1; 1) gauge theory { 3 { (2.1) (2.2) are constrained by supersymmetry, though the constraints become weaker with increasing number of derivatives. An illuminating way to organize the higher derivative couplings is through the corresponding supervertex, namely, a set of (super)amplitudes that obey supersymmetry Ward identities with no poles [9]. If we x the scalar vev (say of the form (2.1)), and consider terms of the form D2nF 4 + , then a supersymmetric completion of such a coupling corresponds to a 4-point supervertex of the form where Q is the total supermomentum, de ned by [10{12] 8(Q)F (s; t; u); Q = 4 X qi; i=1 qi = (qiA; qeiB); q iA = i Aa ia; qeiB = eiBb_ ei : _ b (2.3) (2.4) indices, a and b_ on the other hand are SU(2) SU(2) little group indices. i Aa and eiBb_ are 6 dimensional spinor helicity variables, with the null momentum of the i-th particle related a_ b_ = 12 ABCDpiCD. ia and ei_ are a set of 4 Grassby piAB = i Aa Bb i ab, piAB = eiAa_ eiBb_ mannian variables that generate the 24 = 16 states in the supermultiplet of the i-th particle. Corresponding to D2nF 4 coupling, F (s; t; u) would be a function of Mandelstam variables s; t; u of total degree n. For instance, if we x the color structure (choice of Cartan generators), there is a unique supersymmetric completion of the F 4 term, corresponding to the constant term in F (s; t; u). In the SU(2) gauge theory, the massless elds on the Coulomb branch are in a single U( 1 ) gauge multiplet, and thus F (s; t; u) must be symmetric in s; t; u. From this we immediately learn that there is no independent D2F 4 vertex, since s + t + u = 0. This result is also an immediate consequence of the non-renormalization theorem of Paban, Sethi, and Stern [19] which is later extended to the SU(3) case by [18]. In the more general SU(k) theory with k > 3, to the best of our knowledge, there isn't a non-renormalization theorem that determines the D2F 4 completely in terms of the F 4 coupling on the Coulomb branch. In fact, since di erent Cartan generators can be assigned to the 4 external lines of the superamplitude, one can construct nontrivial superamplitudes with F (s; t; u) a linear function of s; t; u. These are the terms computed in [14], from both the SYM at 2-loop and from DSLST. It is likely that by consideration of higher point superamplitudes, and consistency with unitarity, one can derive the supersymmetry constraint on the r-dependence of the f1(r)D2F 4 coupling as in the work of Sethi, but we not will pursue this topic in the current paper. The consideration of superamplitudes allows for an easy classi cation of D2nF 4 couplings for all n. In below we will mostly think in terms of the superamplitudes rather than the terms in the e ective Lagrangian. Now to be precise we will introduce a color label ai 2 Zk for each external line, corresponding to a Cartan gluon in the U( 1 )k 1 that transforms under the Zk cyclic permutation of k NS5-branes by the phase e2 iai=k. The 4-point superamplitude is of course subject to the constraint Pi4=1 ai = 0 (mod k), and { 4 { (2.5) (2.6) (2.7) takes the form where our convention, s = s12 = (p1 + p2)2, t = s14, u = s13 = s t. We also have the following identi cation between the 6D gauge coupling gY M and the little string scale, as seen by matching the tension of the instanton string with the fundamental string of DSLST, and also veri ed in [14]. In this paper we work in units of 0, and so gY2 M = 32 3. Our convention for the Coulomb branch radius parameter r is such that the W -boson corresponding to the D1-brane stretched between the i-th and j-th NS5 brane has mass 8(Q)Fa1a2a3a4 (s; t; u; r); 2 1 0 = In the next two sections, we will study the expansion of the function Fa1a2a3a4 (s; t; u; r) in detail, from perturbative SYM and from DSLST. 3 Perturbative 6D SYM in the Coulomb phase Near the origin of the Coulomb branch, the W -bosons are light compared to the scale set by gY M , and we can compute the 4-point amplitude of Cartan gluons in SYM perturbation theory. A priori, one may expect such a computation to run into two di culties: the loop expansion of the massless scattering amplitude su ers from UV divergence at 4-loop order [16] (while the mixed Cartan gluon and W -boson amplitude diverges at 3-loop [15]), and there may be higher dimensional operators that deform the SYM Lagrangian [21]. The consistency of DSLST [13] combined with non-renormalization theorems of Sethi et al. implies that the SYM Lagrangian at the origin of the Coulomb branch is not deformed by trF 4 terms. The result of [14] further indicates that the 1=4 BPS operator of the form D2tr2F 4 is absent at the origin of the Coulomb branch as well. On the other hand, the 3loop divergence in the non-Abelian sector means that the non-BPS dimension 10 operator D2trF 4 is needed as a counter-term [15]. Likewise, at 4-loop order we will need counterterms of the form D4trF 4 and D4tr2F 4 [16]. It appears that one can proceed with the SYM perturbation theory, and add the appropriate counter-terms whenever a new divergence is encountered at a certain loop order. Of course, the perturbative SYM does not give a prescription for determining the nite part of these counter-terms. Such ambiguities however do not a ect the leading logarithmic dependence on r, and so these leading logs can be computed unambiguously in the framework of SYM perturbation theory at small r. On the other hand, the nite shifts of the counter-terms that cannot be determined by SYM perturbation theory are in principle determined in the full little string theory, and one could hope for extracting such information from the opposite regime, namely the large r limit. Let us begin with the F 4 term in the Coulomb e ective action, or more precisely, its supersymmetric completion, along the 1-dimensional subspace as speci ed in (2.1). The { 5 { k 3 4 5 ` corresponding superamplitude takes the form As was shown in [13], the coe cient C0 is given by where c0 is a constant that is independent of the color assignment. Next consider the D2F 4 term, which can be written as The result of [14] indicates that the corresponding superamplitude takes the form 8(Q) r2 C1;a1a2a3a4 s12 + (1 $ 3) + (2 $ 3) ; i and is two-loop exact. By symmetry of permutation on external lines, C1;a1a2a3a4 is invariant under the permutations (12), (34), as well as (13)(24). Note that there is no 1-loop contribution to f1(r)D2F 4, of order r 4, simply because a 1-loop contribution would come with a C1;a1 a4 factor that is completely symmetric under permutation of a1; ; a4, and thus must be proportional to s+t+u, which is zero. Therefore f1(r) takes the simple form3 with higher numerical precision. We can write the D4F 4 couplings as In [14], the C1 coe cients were computed for k = 2; 3; 4; 5 and color assignment 2 ` 0 . The results are listed here in table 1 Now let us consider the D4F 4 term, which receives contributions from all loop orders. fS;a1a2a3a4 (r)(s2 + t2 + u2)Fa1 Fa4 + fA;a1a2a3a4 (r)s2Fa1 Fa4 (3.6) 3When there is no potential confusion, we will often omit the color indices a1a2a3a4 if a1 = a2 = a3 = a4 = ` + 1. For example, C1 = C1; `+1; `+1; (`+1); (`+1). where S and A stand for symmetric and asymmetric in the Mandelstam variables. fS(r) and fA(r) each admits a small r expansion4 The coe cients CS1=A, CS2=A, CS3=A (which depend on the color factors) are computed from 1, 2, 3-loop amplitudes. The coe cients BS=A and BS0=A come from the 4-loop amplitudes, after canceling the log divergences by 3-loop and 4-loop counter-terms. Note that the appearance of the double log terms is due to nested divergences at 4-loop order. In the UV completed theory, namely the full LST, the divergence of SYM at 4-loop order and higher is re ected as a branch cut in the analytic structure of the function f2(r). The detailed computation and numerical results for the 1, 2, and 3-loop contributions are given in appendix A, for k = 2; 3; 4; 5 and color assignment a1 = a2 = a4 = `+1 with k 2 ` 0. For 3-loop, we need to sum up the scalar integrals represented by the nine diagrams in gure 4. Each of diagrams (e) (f) (g) (i) is in fact UV divergent by itself at linear order in s, t, u, and would potentially contribute to D2F 4. However, these divergences cancel after we sum up these diagrams and the permutations of the external legs, and the remaining parts are quadratic or higher in s, t, u and give nite contributions to D2nF 4 for n 2. The 4-loop divergence can be computed at the origin of the Coulomb branch, as in [16]. After moving away from the origin on the Coulomb branch, in the expansion in external momenta, the logarithmic divergences appear in the form ln( =r), and in the case of nested divergences, (ln( =r))2. After canceling the logarithmic divergences with counter-terms, we are left with logarithmic dependence on r, and the coe cient of the leading log (or double log) is independent of nite shifts of the counter-term. The logarithmic divergence at the origin of the Coulomb branch involves three possible terms, of the form (s2 + t2 + u2)trF 4, (s2 + t2 + u2)(trF 2)2, and s2(trF 2)2 + (2 more). The terms proportional to (s2 + t2 + u2) also contain double pole divergences (in dimensional regularization). To cancel the divergences we need a 3-loop counter-term D2trF 4 (it vanishes when restricted to the Cartan, but is now needed to cancel subdivergences in the 4-loop amplitude) and 4-loop counter-terms of the form D4trF 4 as well as D4(trF 2)2. In the end, one obtain unambiguously the coe cient of and the coe cient of In principle, one can also determine unambiguously the (ln r)2 coe cient of the single trace term proportional to (s2 + t2 + u2)trF 4, but this double pole coe cient has not been evaluated explicitly in [16]. 4The color-ordered one-loop superamplitude is permutation invariant, hence the full amplitude is completely symmetric in s, t, and u, and so CA1 = 0. The log2 divergence is also completely symmetric, as can be seen from (A.50), and hence BA0 = 0. ln r s2(trF 2)2 + (2 more) ; (ln r)2(s2 + t2 + u2)(trF 2)2: (3.8) (3.9) { 7 { The CS1=A, CS2=A, CS3=A and BS=A coe cients for k = 2; 3; 4; 5 and color assignment in the (R,R) sector, of the form [13, 14], Vab_;` = e '2 '2e eip X 2 a eBb_ SASeBV `sl;( 1 A 2 ; `+22;; 12`)+2 V 2`s;u;(2` ; 21 ;2` 12 ); 2 (4.1) the form with ` = 0; 1; ; k 2 labeling the color index of the U(1)k 1 gluons according to their eigenvalues e2 i(`+1)=k with respect to the Zk cyclic permutation of the NS5 branes. spin and eBb_ are the 6D spinor helicity variables as before, and SA; SeB are the left and right elds of the R1;5 part of the worldsheet CFT. There is also an identi cation Vab_;` + Vab_;k 2 ` [13, 14, 22]. It was shown in [14] that the sphere 4-point superamplitude takes A a ADSLST (1`+1; 2`+1; 3 ` 1; 4 ` 1) = 8(Q) Nk;` Z C d2zjzj (`+1)2 s 12 1 k j j z ` (`+1)2 u+ 12 k ; ; ; P ; z)j2 : (4.2) 3; 4; 2 Q { 8 { ; P ; z) is the Liouville 4-point conformal block. See [14] for the precise identi cation of the parameters i , i etc. The evaluation of the conformal block integral and the integration over the cross ratio z are performed numerically, order by order in the 0 expansion.5 For k = 2; 3; 4; 5, the two leading terms in the expansion were given in [14]. We carry out this computation to 0 order, with the order 0n terms corresponding to D2nF 4 coupling in the Coulomb branch e ective action. In the following we normalize the amplitudes by their 00 order terms. 3 1 + 2:10359958(s2 + t2 + u2) + 17:42982502stu + : (4.3) 1 + 6:1080323(s2 + t2 + u2) + 96:795814stu + : 1 1 k = 5; ` = 0; 3 : k = 5; ` = 1; 2 : 2:39679s + 14:9055(s2 + t2 + u2) 14:8295s2 + 302:54stu The omitted terms are of quartic and higher degrees in s; t; u, corresponding to D8F 4 and higher derivative couplings in the e ective action. Note that the DSLST four-point amplitude is invariant under ipping the Zk charges of the vertex operators. In addition when ` + 1 = k=2 (i.e. the vertex operators are identical), the amplitude is invariant under permutation of the Mandelstam variables. 5Since we set 0 = 1, the 0 expansion is an expansion in the Mandelstam variables. { 9 { × × ( ) - ( ) = = = = ( ) = = - ( ) = = HJEP12(05) line is given by the DSLST tree level superamplitude (valid for large r). The lower green line comes from 6D SYM one loop, the middle orange line comes from one and two loops combined, and the upper blue line combines the contributions up to three loops (valid for small r). We interpolate the two ends by a naive extension beyond their regimes of validity. 4.1 An interpolating function from weak to strong coupling On one hand, perturbative 6D SYM gives a small r expansion of the coe cient f2(r) of each higher derivative term in the Coulomb branch e ective action. On the other hand, perturbative string scattering in DSLST gives an expansion valid at large r. The exact f2(r) is a function that interpolates the two ends. Let us rst consider the D2F 4 term. A non-renormalization theorem by [18] shows that f1(r) is two-loop exact in SU(3) maximal SYM, which means that (3.5) should hold for arbitrary r. Indeed, the result of [14] was that the coe cients of s in the tree-level DSLST superamplitudes exactly match with the C1 obtained from the SYM two-loop superamplitudes (see table 1), for k = 4; 5 as well as k = 3. It is not inconceivable that f1(r) is two-loop exact in 6D SYM for all k, which also implies that all higher genus superamplitudes for the scattering of four Cartan gluons should vanish at 01 order. Next let us consider D4F 4. With the color index assignment a1 = a2 = a3 = a4 = ` + 1 (labeling the Zk charge), the two independent structures are proportional to s2 + t2 + u2 and s2. We will compare the large and small r expansions. On the 6D SYM side, the ra(ln(r= ))b terms after resummation will correct the power of r when one interpolates the function f2(r) to large r. Here the coe cient of (ln(r= ))2 in the small r expansion can be determined by the 4-loop UV divergence at the origin of the Coulomb branch. However, as already mentioned, this computation involves the divergence in the single trace D4trF 4 term, which has not yet been computed in 6D SYM. The scale has absorbed the contribution from the counter term, and is expected to be of order gY M1 in the full LST. Since the actual numerics depends on the precise value of the mass scale , we will not include the ln(r= ) terms in the interpolation function. In each case, the coe cients of 1=r2 are close but not equal between the large and small r expansions. There is no reason for them to be equal, since the large r expansion should be corrected by higher genus contributions of order 1=r2(g+1), and the small r expansion includes one-loop 1=r6 and two-loop 1=r4 terms, and should further be corrected by higher-loop contributions of the form ra(log(r= ))b. For concreteness, we explicitly make the comparison for k = 5, noting that the other cases are qualitatively the same. k = 5; ` = 0 : Large r expansion: and 3 loops) are plotted in solid lines. We interpolate the two ends by a naive extension beyond their regimes of validity. 5 Discussion To summarize our results so far, while the r 2F 4 and r 2D2F 4 terms in the Coulomb branch e ective action are computed exactly by perturbative SYM at one-loop and equal) to the result obtained from captures the large r limit of f2(r). two-loop orders respectively, and match precisely with the corresponding 0-expansion of the tree level amplitude in DSLST, the f2(r)D4F 4 terms involve a set of nontrivial interpolation functions f2(r), that receive a priori all-loop contribution in SYM perturbation theory. We have determined f2(r) in its small r expansion up to 3-loop orders in 6D SYM. Interestingly, the 3-loop contribution that scales like r 2, is numerically close (but not 02 order terms in the tree amplitude of DSLST, which Starting at 4-loop order in the perturbative SYM description, one encounters UV divergences and while the leading log coe cients can be determined unambiguously in perturbation theory, the subleading logs and constant shifts depend on nite parts of 3 and 4-loop counter terms (D2trF 4, D4trF 4, and D4tr2F 4 at the origin of the Coulomb branch), and are a priori undetermined in 6D SYM perturbation theory. In principle, the (1; 1) LST provides an unambiguous UV completion of the perturbative amplitudes of 6D SYM. If one could somehow compute the exact 4-gluon amplitude in DSLST, non-perturbatively in gs, then one should recover all the perturbative SYM loop amplitudes, and x the nite parts of all counter terms. While we do not have the technology for such exact computations on the string theory side, the interpolation results on the Coulomb moduli space so far suggests that, despite the non-renormalizability of the 6D SYM, the naive perturbative expansion is a valid prescription provided that appropriate counter terms are included at each loop order.6 The UV divergences that arise at 4-loop order and higher in the massless amplitudes of 6D SYM in the Coulomb phase, indicate not a trouble with SYM perturbation theory, but rather a feature of the amplitudes and the corresponding couplings in the Coulomb branch e ective action. Namely, the function f2(r), as an analytic function of r on the Coulomb branch, has a branch cut starting from the origin. Where does this branch cut end, in the analytic continuation of Coulomb moduli space? A natural expectation is that perhaps the branch cut goes all the way to r = 1, where the Coulomb phase is described by weakly coupled DSLST. In fact, we generally expect non-analyticity in f2(r) at r = 1, due to the non-convergence of the string perturbation series, and the need for stringy non-perturbative contributions (e.g. D-instanton amplitudes). In fact, due to the identi cation gs we could speculate that non-perturbative string amplitudes of the form exp( 1=gs) contributes to the nite counter terms at the origin of the Coulomb moduli space! 1=r, e r, Going beyond massless amplitudes, the scattering of gluons with W -bosons in 6D SYM may be compared to D-brane scattering amplitudes in DSLST.7 We hope to report on these results in the near future. 6For instance, one could have said that since the 6D SYM theory is expected to be strongly coupled at the scale gY M1 , a UV cuto should be imposed at the scale gY M1 , and there would seem to be no reason to perform the loop integral over momenta above this scale. However, the exact agreement of one-loop and two-loop contributions to the F 4 and D2F 4 terms with DSLST indicates that the naive loop integrals, which happen to be free of UV divergences in these cases, give the correct answer. 7At the level of 3-point amplitude of gluon emission by a W -boson, the agreement with the disc 1-point amplitude in DSLST was known in [23]. Comments on (2; 0) LST and 5D SYM In this section we discuss the compacti cation of the (2,0) DSLST to ve dimensions, and constrain the higher derivative terms in the e ective action of the resulting 5D gauge theory on the Coulomb branch. In particular, we will show that the trF 4 coupling at the origin of the Coulomb branch of the circle-compacti ed (2,0) superconformal eld theory is absent. At the perturbative level, or equivalently in the 1=r expansion on the Coulomb branch, the structure of (2; 0) DSLST is very similar to (1; 1) DSLST, di ering only through GSO projection. As far as the massless 4-point amplitude is concerned, at string tree level the only di erence between the (2; 0) and (1; 1) case is the interpretation of the supermomentum delta function 8(Q) in terms of the polarizations of the massless supermultiplets involved. The scalar function of s; t; u that multiplies 8(Q) is identical. An analogous statement holds for the genus one 4-point amplitude as well. In the NSR formalism, this can be seen by noting that the contribution from the (P,P) spin structure vanishes,8 and therefore the IIA and IIB GSO projections yield the same amplitudes, up to reassignment of polarization tensor structure. It is not inconceivable that the massless 4-point amplitudes in (2; 0) and (1; 1) DSLST involve the same scalar function of s; t; u to all order in perturbation theory, though we do not have an argument for this. On the other hand, it appears that the D-instanton amplitudes of massless string scattering will be quite di erent in the two theories, as the BPS D-instanton that is pointlike in the R6 and localized at the tip of the cigar exists only in the (2; 0) DSLST and not in the (1; 1) theory. Such contributions could alter the e ective action near the origin of the Coulomb branch signi cantly, and give rise to entirely di erent low energy dynamics of the (2; 0) and (1; 1) LST at the origin of the Coulomb branch. Nonetheless, in view of the idea that the (2; 0) SCFT, when compacti ed on a circle, is described in the low energy limit as 5D maximally supersymmetric Yang-Mills theory [27] together with an in nite series of higher dimensional operators/counter-terms [28{31], one could ask whether there is a similar interpolation on the Coulomb branch of the (2; 0) LST compacti ed on a circle. In this case, the W -boson comes from D-branes located at the tip of the cigar in the T-dual picture [23]. The parameters in the circle-compacti ed DSLST are the string length `s, the W -boson mass mW which is related to the string coupling9 gs by (6.1) 8It su ces to look at the scattering of the scalars which correspond to (NS,NS) vertex operators. At one loop in the (P,P) sector (here we are following the convention of [24] although historically this had been also referred as the (odd,odd) sector [25]), we need to have three (0; 0)-picture and one ( 1; 1)-picture vertex operators plus one PCO. Hence in the path integral we have a total of 4 insertions of to a vanishing contribution to the total amplitude due to the presence of six zero modes for which leads (and ~ ). and ~ One can also reach the slightly stronger statement that the four point amplitudes in (1; 1) and (2; 0) DSLST agree up to 2-loops following a version of Berkovits' argument in section 3.2 in [26]. 9In this paper, we use gs to denote the string coupling at the tip of the cigar in the IIB picture , not to be confused with the asymptotic string coupling gs1 before taking the decoupling limit of NS5-branes in asymptotically at spacetime in the IIA picture. They are related by gs `sgs1=r [7, 32{34]. mW R gs`s2 ; and the compacti cation radius R, which is related to the 5D gauge coupling g5 by form then we expect R = g 2 From the 5D perspective, the natural mass scale is set by g5 or R, and the two dimensionless parameters are mW R (parameterizing distance from the origin on the Coulomb branch) and R=`s. The 5D gauge theory obtained from compacti cation of (2; 0) SCFT, in its Coulomb phase, is obtained in the limit R=`s ! 1, while holding R and This in particular requires sending gs ! 1 at the same time. If we write the amplitude of massless particles in the compacti ed (2; 0) DSLST in the and the corresponding amplitude in the UV completion of 5D SYM in the form gs!1 lim A(2;0) DSLST E2g2 gs; gsmW 5 ; g52E = A5D GT(g52mW ; g52E): The l.h.s. cannot be captured by DSLST perturbation theory in a straightforward manner. For instance, we can write the D2nF 4 terms in the Coulomb branch quantum e ective Lagrangian in the schematic form 1 n=0 X fn( )D2nF 4; where mW R is the distance parameter on the Coulomb branch, and the subscript n indicates the \number of derivatives". If we assume that the UV completion of the 5D SYM perturbation theory is such that higher dimensional operators are added only when needed as counter-terms,11 then the SYM loop expansion of fn( ) has the structure f0( ) = f1( ) = f2( ) = f ( 1 ) Here the coe cient fn(L) comes from the L-loop 4-point amplitude. Note that the 1loop contribution f1( 1 )= 5 is absent; this is because the 1-loop amplitude involves only 10Note that it is a di erent limit than taking R=`s ! 1 while keeping gs and `s xed, which is the limit of decompacti ed (2,0) DSLST. 11As we will see shortly, while this is expected for the compacti ed (2; 0) SCFT, this is not true for the compacti ed (2; 0) LST. We thank C. Cordova and T. Dumistrescu for a key discussion on this point. (6.2) xed.10 (6.3) (6.4) (6.5) (6.6) (6.7) a single color structure that is invariant under permuting the 4 external lines, and the D2F 4 amplitude would be proportional to s + t + u which vanishes. Note that while the 5D SYM 4-point amplitude is known to have UV divergence at 6-loop order [35], such a divergence vanishes when the external gluons are restricted to the Cartan subalgebra. This is because the counter-term responsible for this divergence is the unique dimension 10 non BPS operator of the form D2trF 4 + [21, 36], which in fact vanishes upon Abelianization (i.e. restricting to the Cartan subalgebra). The 4-point amplitude of Cartan gluons in 5D SYM is expected to diverge rst at 8-loop order, with the counter-term being a non-BPS operator of the form D4tr F 4 + . In the UV completion that is expected to arise from the compacti cation of (2; 0) theory, the D4tr F 4 counter-term should cancel the log divergence, dependence in the Coulomb e ective action, hence the f2(8) ln term in (6.7). Let us focus on the f0( )F 4 coupling for the moment. The argument of [19] and [20] indicates that, at least in the SU(2) case where the Coulomb branch moduli space is just a single R5, f0( ) is a harmonic function on the R5.12 Assuming SO(5) R-symmetry, such a harmonic function must be of the form f0( ) = c + 03 : f ( 1 ) The constant c, if non-vanishing, would correspond to a trF 4 coupling in the non-Abelian SYM at the origin of the Coulomb branch moduli space. In writing (6.7) we have assumed that such coupling is absent in the low energy limit of the compacti ed (2; 0) theory. We will now justify this assumption. The Coulomb phase of the circle-compacti ed A1 (2; 0) LST has a moduli space of vacua R 4 and has size S1. The S1 coming from the compact scalar in the 6D (2; 0) tensor multiplet, (R=`s)2 in units of R.13 In the Coulomb phase of the compacti ed (2; 0) LST, the D2nF 4 couplings come with the coe cients fn(~; R=`s), such that lim R=`s!1 fn(~; R=`s) = fn( ): Here ~ parameterize a point on the R 4 S1 moduli space, and the function fn(~; R=`s) is invariant under SO(4) R-symmetry in 6 dimensions, while the SO(5) is only restored in the R=`s ! 1 limit. Note that, importantly, the limit is taken with and so taking R=`s ! 1 requires sending gs ! 1 at the same time. From the 5D perspective, gs of DSLST is related to the vev of a massless scalar eld, whereas R=`s is a rigid parameter (there is no massless graviton propagating in the R1;5 of the DSLST and hence there is no massless 5D scalar associated with the compacti cation radius); in particular, the dependence on gs is constrained by supersymmetry, whereas the dependence on R=`s is not. = R2=(gs`s2) held xed, 12This is consistent with the v4= 3 e ective potential between separate D4 branes moving at a relative velocity [37]. 13To see the size of the S1, we can go back to the NS5-brane picture in type IIA string theory, separated in the transverse R4, with the world volume of the NS5-branes compacti ed on a circle of radius R. A W -boson coming from D2-brane stretched between a pair of the NS5-branes and wrapping the circle has mass mW Rr=(gs1`s3) 2 R=(gs`s) as before. On the other hand, if we are to separate the NS5-branes along the M-theory circle, the M2-brane stretched between the M5-branes and wrapping the compacti cation circle of radius R has mass R=`s2. (6.8) (6.9) At nite R=`s, f0(~; R=`s) is an SO(4)-invariant harmonic function on the R We can write ~ = (~ ; y), where ~ parameterizes the R4 and y is the coordinate on the S1. The harmonic function f0(~; R=`s) is restricted to be of the form f0(~; R=`s) = c(R=`s) + X n2Z f0( 1 )(R=`s) While c may no longer be a constant, it must be a function of the rigid parameter R=`s only. In the limit of large j j, f0 can be expanded as Matching this with the tree level (2; 0) DSLST, we conclude that c(R=`s) argument we also expect that the corrections to the tree level contribution to F 4 coupling in the compacti ed DSLST are entirely non-perturbative in gs. Now, near the origin of Coulomb branch, ( ; y) = (0; 0), f0 can be written as : 3 is generated from 5D SYM by integrating out W -bosons at 1-loop. The second term is non-vanishing at the origin of the Coulomb branch and can be understood in terms of 6D SYM compacti ed on a circle (as in the T-dual (1; 1) LST), with massive Kaluza-Klein modes integrated out at 1-loop. This term vanishes in the R=`s ! 1 limit, and thus the trF 4 coupling is absent in the compacti ed (2; 0) superconformal theory (at the origin of its Coulomb branch). The third term comes from the 1-loop diagram with 6D W -bosons in the loop that also carry nonzero KK momenta, expanded to the second order in the W -boson mass parameter, and gives rise to an SO(5)R breaking dimension 10 BPS operator at the origin of the Coulomb branch of the 5D gauge theory. It should be possible to extend this discussion to higher rank cases as well. A more detailed investigation of the two-parameter interpolation function in the Coulomb phase of compacti ed (2; 0) DSLST, and its interplay with the perturbative structure of 5D SYM, are left to future work. Acknowledgments We would like to thank Chi-Ming Chang for collaboration during the early stage of this project. We are grateful to Ofer Aharony, Lance Dixon, Daniel Ja eris, Cumrun Vafa for useful conversations and correspondences, to Zohar Komargodski and Shiraz Minwalla for key suggestions on the use of non-renormalization theorems in the Coulomb e ective action, to Clay Cordova and Thomas Dumitrescu for important discussions on the compacti cation of (2; 0) theories, and to Travis Max eld and Savdeep Sethi for comments on a preliminary draft. We would like to thank the 7th Taiwan String Workshop at National Taiwan University, Weizmann institute, and Kavli IPMU for their support during the course of this work. The numerical evaluations of loop integrals are performed using FIESTA on the Harvard Odyssey cluster, whereas the conformal block integrations and DSLST amplitudes are computed with Mathematica. S.H.S. is supported by the Kao Fellowship at Harvard University. X.Y. is supported by a Sloan Fellowship and a Simons Investigator Award from the Simons Foundation. Y.W. is supported in part by the U.S. Department of Energy under grant Contract Number DE-SC00012567. This work is also supported by NSF Award PHY-0847457, and by the Fundamental Laws Initiative Fund at Harvard University. 6D SYM loop amplitudes contributing to D4F 4 The term f2(r)D4F 4 receives contribution from all loop orders of the scattering amplitude of four Carton gluons. At each loop order, we need expand the superamplitude to quadratic order in the Mandelstam variables. Each loop order is proportional to the color-ordered four-point tree-level scattering amplitude A tree(1; 2; 3; 4) = i The one-loop amplitude of four Cartan gluons can be written as14 Here I41 loop(s12; s14; mij ) is the scalar box integral ( gure 2)15 I 1 loop(s12; s14; mij ) 4 = Z d6` 1 (2 )6 (`2 +mi2j )((`+p1)2 +mi2j )((`+p1 +p2)2 +mi2j )((` p4)2 +mi2j ) : mij is the mass of the W -boson with gauge indices (ij), and vaj is the polarization vector for the external Cartan gluons. 14The perturbative expansion of the amplitude of massless Cartan gluons takes the form A = gY4 M A 1 loop + gY6 M A 2 loop + + gY2+M2LAL loop + : 15In contrast to the more common convention in the scattering amplitude literature (for example ([16]) where the mostly minus signature is used and s = (p1 + p2)2, here we work in the mostly plus signature and de ne s = (p1 + p2)2. Hence when comparing the two, the Mandelstam variables are the same, but we di er in the de nition of the scalar box integrals by factors of i from Wick rotating d`0 and minus signs from the propagator 1=p2. (A.1) (A.4) (A.5) (A.2) 1 4 1 loop to s2=r6 order. It is straightforward to show that where we have made the following replacements in the integrand: ` pi ` pj ! 1 `2 pi pj = 6 1 192 1 12 2 ` sij ; (` pi)(` pj )(` pk)(` pm) ! (`2)2(sij skm + siksjm + simsjk): Summing up with A1324 1 loop and A1243 1 loop, we obtain the order s2=r6 term for the full one-loop amplitude r6 A 1 loop(1; 2; 3; 4) s2 = va`); A.2 Two-loop The full two-loop amplitude is given by A 2 loop(1; 2; 3; 4) = h s12(A1234 s12s14A + A3421 tree(1; 2; 3; 4) Let us start with the planar contribution, (vam 2 loop;P is the planar scalar two-loop integral ( gure 3(a)), I 2 loop;P (mij; m`i; mj`) = 4 ( ) 4 1 The order s2=r4 terms in A can be computed straightforwardly, I 4 3(`2 + m2 )(`2 + m2 ) ij 2 `i Z 1 `22 +m2 `i 6 6 d `1 d `2 ` 2 1 (`21 +m2 )2 ij # : 2 loop(1; 2; 3; 4) correspond to the s=r4 terms in I ` 2 2 + 4`1 `2 (`22 +m2 )2 `i 3(`21 +m2 )(`22 +m2 ) ij `i Moving on to the non-planar diagram, A1234 = X is the non-planar scalar two-loop integral ( gure 3(b)), I 4 (mij; m`i; mj`) = (2 )6 (2 )6 (`21 +mi2j)((`1 +p1)2 +mi2j)(`22 +m`2i)((`2 +p4)2 +m`2i) ((`2 p1 p2)2 + m`2i)((`1 + `2 p2)2 + mj2`)((`1 + `2)2 + mj2`) As in the planar case, we are interested in the s=r4 term in I computed straightforwardly, (mij; m`i; mj`) " s12 ` 2 2 (`22 + m`2i)2 2`1 `2 + 2`22 s r4 = 1 ! (2 )6 (2 )6 (`21 + mi2j)2(`22 + m`2i)3((`1 + `2)2 + mj2`)2 `22 + m`2i 3(`21 + mi2j)(`22 + m`2i) 3(`21 + mi2j)((`1 + `2)2 + mj2`) + + `21 + `1 `2 `1 `2 +`22 3(`22 +m`2i)((`1 +`2)2 +mj2`) 3(`21 +mi2j)(`22 +m`2i) 3(`22 +m`2i)((`1 +`2)2 +mj2`) 1 + s14 `1 `2 `1 `2 1 : 1 1 2 b ( ) 4 (A.12) (A.13) (A.14) (A.15) !# : 1 2 1 2 1 a ( ) d ( ) ` 1 g ( ) 3 4 4 3 4 1 2 1 2 1 ` 1 ` 7 b ( ) ` 1 ` 3 ` 6 ` 4 e ( ) ` 5 h ( ) ` 9 `10 ` 2 ` 8 3 4 3 4 4 1 2 1 2 1 c ( ) ` 1 (i) f ( ) ` 5 ` 3 ` 2 ` 1 ` 6 The full three-loop amplitude is given by A 3 loop(1; 2; 3; 4) = 1 (d) 4 A1234 + 2A(1e2)34 + 2A(1f2)34 + 4A(1g2)34 + 1 (h) 2 A1234 + 2A(1i2)34 where we have summed over contributions from individual diagrams in gure 4 and permutations of external legs. The coe cients in front of A1234 combined with the overall 1=4 are the symmetry factors. The numerators for the scalar integrals in gure 4 are given in table 4.16 (x) In below we will listed the contribution from each of the nine graphs, with external lines restricted to Cartan gluons, and with the appropriate W -boson mass assignments in the 16In contrast to the convention in [38] where the external momenta are all outgoing, our external momenta are all ingoing. Furthermore, as mentioned before, the momentum square p2 di ers by a sign due to di erent conventions on the signature, while the Mandelstam variables are the same. Moreover since we consider W -bosons propagating through the loops, the loop momenta `i (not all independent) in the expressions of table 4 are taken to be higher dimensional with their extra components constrained by the mass of the propagating particle. These will be made explicit in the expressions for the full scalar integrals below. ` 4 3 4 3 4 4 (A.16) HJEP12(05) Integral I(x) (a)-(d) (e)-(g) (h) (i) 2 s12 s12(`1 p4)2 s12(`1 + `2)2 s14(`3 + `4)2 + s12`52 + s14`62 W -boson mass square m2 term associated to each (` + p)2 factor in the numerator. We later restore these factors in the explicit expressions for A1234 below. internal propagators. The scalar loop integral will then be expanded in powers of external momenta, or in terms of the Mandelstam variables s; t; u. At order s, while some of the loop integrals are subject to UV divergence, these divergences cancel in the full 3-loop amplitude of Cartan gluons. For the purpose of extracting D4F 4 e ective coupling in the Coulomb e ective action, we will expand the scalar integrals to s2 order. Below we will also list these expanded expressions, which can then be evaluated numerically using FIESTA program. Diagram (a) gives, including color factors, (a) Ia(mij ; mi`; mim; mj`; m`m) X X i;j;`;m i;j Ia(mij ; mi`; m`m; mj`; mim) + 4 X Ia(mij ; 0; mij ; mij ; mij ) Y(vai vaj) Y (vai a=1;2 vaj) Y (vai a=3;4 Y (vai a=1;2 vaj) Y (vam a=3;4 vam) va`) where the scalar integral is Ia(mij ; mi`; mim; mj`; m`m) = s122 Z 1 (`22 +mi2m)((`2 +p4)2 +mi2m)((`2 +p3 +p4)2 +mi2m)(`23 +mi2`)((`3 +p1 +p2)2 +mi2`) 1 ((`1 `3)2 + mj2`)((`2 + `3)2 + m`2m) Before proceeding, let's introduce some shorthand notation, dL A1234 = 2 2 X X i;j Ib(mij ; mi`; mj`; m`m; mim) Ib(mij ; mi`; mj`; mim; m`m) Ib(mij ; 0; mij ; mij ; mij ) 4 Y a=1 Y a=1;2 Y a=1;2 (vai (vai (vai vaj); where Ib(mij; mi`; mj`; m`m; mim) Z Expanding in external momenta, we have r2 Ic(mij ; mim; m`m; mj`; mjm; mi`) Z dL 2 3 1j1 13j3 2j5 23j4 3j2 2 2 : Y a=1;2 (vai j va) Y a=3;4 (va` Expanding in external momenta and extracting the order s2 terms, we have Note that by power counting the loop integral scales like m vaj)(v4m v4`)(v3i v3m) (A.22) vam) 4 Y a=1 (vai vaj); (A.23) (A.24) (A.25) (A.26) Diagram (c) gives X where Ic(mij; mim; m`m; mj`; mjm; mi`) Z (`22 +m`2m)((`2 +p4)2 +m`2m)((`2 +p3 +p4)2 +m`2m)(`23 +mj2`)((`1 +p1 +p2 `2 `3)2 +mi2m) Ic(mij ; mij ; mij ; mij ; 0; 0) + Ic(mij ; 0; mij ; 0; mij ; mij ) 1 1 : Z 1 ((`1 `3)2 + mi2`)((`2 + `3)2 + m2 ) jm Expanding in external momenta, 2 Ic(m1; m2; m3; m4; m5; m6)j s2 = s12 r2 dL 3 3 1j1 2j3 3j4 123j2 13j6 23j5 : (A.27) Id(mij ; mjm; m`m; mi`; mim)(v1i v1)(v2 v2m)(v3` v3m)(v4i ` v4) Id(mim; mij ; m`m; mj`; mjm)(v1m v1i)(v2i j v2)(v3m v3`)(v4` j v ) 4 Id(mij ; mij ; mij ; mij ; 0) (vai j va) 4 Y a=1 Diagram (d) gives X X where Id(mij; mjm; m`m; mi`; mim) Z 1 Y a=1;2 (vai Expanding in external momenta, and after some simpli cation of the loop integrals, Ie(m1; m2; m3; m4; m5; m6)j s2 = r2 s12 Z 3 dL 2 2 2 1j1 2j2 3j3 13j4 12j5 23j6 3s12 j 3s12 12j5 " (2 )6 (2 )6 (2 )6 (`21 + mi2j)((`1 + p1)2 + mi2j)((`3 `1)2 + mj2m)((`3 `1 + p2)2 + m2 ) jm 1 : (`22 + mi2`)((`2 + p4)2 + mi2`)((`2 + `3 Expanding in external momenta 2 Id(m1; m2; m3; m4; m5)j s2 = s12 r2 Z dL 2 2 2 1j1 13j2 2j4 23j3 3j5 2 2 : Ie(mij ; mi`; mim; mjm; mj`; m`m) vaj)(v3i v3`)(v4i v4m) Ie(mij ; mij ; mij ; 0; 0; 0) (vai vaj); Diagram (e) gives X where s12 we have Ie(mij ; mi`; mim; mjm; mj`; m`m) Z 6 6 6 d `1 d `2 d `3 (`1 1 1 1;23( s12 1j1 23j6 13;12s12 + + 3 3 2 2 31j4 23j6 Diagram (f ) gives 12;23s12 + 1;31(s14 1 1 2 2 2;31s12 2j2 13j4 2;12s12 2j2 12j5 1 1 3 3 3;12s12 3j3 12j5 + 2 X If (mij ; mij ; mij ; 0; mij ; 0) Y(vai If (mij ; mj`; mim; mjm; m`m; mi`) Y (vai vaj)(v3` v3j)(v4i v4m) where s12 If (mij; mj`; mim; mjm; m`m; mi`) ((`1 + `2)2 + mj2m)(`23 + mi2`) : Expanding in external momenta, we have (A.33) (A.34) (A.35) (A.36) (A.37) a=1;2 vaj); (`1 a=1;2 (`1 p4)2 + m21 4 a=1 1 4 a=1 (2 )6 (2 )6 (2 )6 (`21 + m21)((`1 + p1)2 + m21)((`1 + p1 + p2)2 + m21)(`22 + m23) If (m1; m2; m3; m4; m5; m6)j s2 = r2 " 3s12 1 1 3s12 + 13j2 1;23(s14 1j1 23j5 Diagram (g) gives s12 Z 3 2 1 1 j + 1;1(s14 + 2s12) 1;2(s14 s12) 13;23s12 13j2 23j5 3 13;13s12 #: 2 13j2 j j 2;13s12 2j3 13j2 + 1;13(3s12 s14) 1j1 13j2 where 2 i;j X 2 X Ig(mij ; mij ; mij ; 0; mij ; 0) Y(vai vaj) Ig(mij ; m`m; mim; mjm; mj`; mi`) Y (vai vaj)(v3` v3m)(v4i v4m); Expanding in external momenta, we have Ig(m1; m2; m3; m4; m5; m6)j s2 = r2 s12 Z 3 3s12 + 1;1(s14 +2s12) + 1;2(s14 s12) + 23j2 + 2;23s12 2j3 23j2 2 2 1j1 2j3 13j5 23j2 3j6 12j4 dL 2 3 13;23s12 13j5 23j2 + `3)2 + mi2j)((`1 `3 + p1)2 + mi2j)((`2 + `3)2 + mj2m)((`2 + `3 + p4)2 + mj2m) p2 1 p3)2 + mi2m)(`23 + mj2`) : Expanding in external momenta, we have Ih(m1; m2; m3; m4; m5; m6)j s2 = " 3 + 312;1 1 2 j 312;2 j r2 13j1 s12s23 Z 3 312;23 23j4 # 2 12;12 : 12j6 dL 2 2 2 2 1j2 2j3 13j1 23j4 12j6 3j5 + " 3s12 1 1 3s12 13j5 1;23( 3s12 + s14) 1j1 23j2 Diagram (h) gives i;j + 2 X vaj) Ih(mij; mi`; m`m; mjm; mj`; mim) where where 1 1 2 1;13s12 1j1 13j5 2 13;13s12 #: 2 13j5 2;13s12 2j3 13j5 (A.38) (A.39) (A.40) v4j); (A.41) (A.42) v4m); (A.43) Diagram (i) gives 2 X Ii(mij; mj`; mi`; mim; mjm; m`m) s12((`1 p4)2 + mi2j) + s14((`1 + `2)2 + mi2`) + 13 (s12 s14)(`22 + mj2`) (`21 + mi2j)((`1 + p1)2 + mi2j)(`22 + mj2`)((`2 + p2)2 + mj2`) 12;1s12 12j3 ! + 2 2 2 1j1 2j2 12j3 3j4 13j5 123j6 j 1 1 3 4 Expanding in external momenta, we have Ii(m1; m2; m3; m4; m5; m6)j s2 = 3 1;2(s12 + s14) r2 2 2 1 3 s12 1j1 +s14 12j3 + (s12 s14) 2 2 2;12(2s12 + s14) 2;3(s12 + s14) 3 12;12s12 2j2 12j3 2 2 3 4 12j3 Note that the above expressions for the scalar loop integrals expanded in external momenta to order s2 do not always exhibit symmetries of the graphs in a manifest way. In the numerical evaluation of the loop integrals, veri cation of these symmetries is a basic and useful consistency check. Results for 6D SYM in the Coulomb phase. To make contact with the consideration of 6D SYM in section 3, we set the mass of the W -boson with gauge indices (ij) to be 3 12j3 + 12;3s12 12j3 3j4 1j1 12j3 : (A.45) ((`1 + `2 + p1 + p2)2 + mi2`)((`1 + `2 p4)2 + mi2`)(`23 + mi2m)((`3 + p4)2 + mi2m) 1 ((`1 + `3)2 + mj2m)((`1 + `2 + `3)2 + m`2m) and the polarization vector for the external Cartan gluons to be mij = 2r sin (i j) k ; vaj = !(j 1)na ; j = 1; ; k; where ! = e2 i=k. For the four Cartan gluon scattering of interest, n1 = n2 = ` + 1; n3 = n4 = k (` + 1) with values ` = 0; 1; ; k The partial amplitudes and full amplitudes for each case are listed in the tables below. The quantity listed is the three-loop contribution to D4F 4 normalized by the one-loop F 4 amplitude A 1 loop(1; 2; 3; 4) s2 r6 s12s14Atree(1; 2; 3; 4) s122 + s213 + s214 r6 k 184320 L=1 Xk1 sin2 L(k`+1) sin2 L(kk ` 1) : (A.49) sin6 L k In the notation of section 3, this quantity is CS3 (s2 + t2 + u2) + CA3s2. (A.44) HJEP12(05) (A.46) (A.47) (A.48) diagram (a) (b) (c) (d) total 3:772838(s2 + t2 + u2) k = 3; ` = 0 : diagram (a) (b) (c) (d) (e) (f) (g) (h) (i) gY M A3 loop=A1 loop 4 14:39876(s2 + t2 + u2) 10:376120s2 5:976425(s2 + t2 + u2) 4:223506s2 5:1697610(s2 + t2 + u2) 3:7469277s2 3:8144749(s2 + t2 + u2) 1:2321663s2 0:56439858(s2 + t2 + u2) + 0:42112441s2 0:68831287(s2 + t2 + u2) + 0:37394094s2 1:0393916(s2 + t2 + u2) 0:73705051s2 0:17584295(s2 + t2 + u2) 0:12991690s2 0:030527986(s2 + t2 + u2) + 0:091583958s2 symmetry factor 16 4 4 8 2 2 1 8 2 total 7:086485(s2 + t2 + u2) 4:505248s2 diagram (a) (b) (c) (d) (e) (f) (g) (h) (i) k = 4; ` = 1 : total 11:619831(s2 + t2 + u2) 8:729678s2 diagram (a) (b) (c) (d) (e) (f) (g) (h) (i) gY M A3 loop=A1 loop 4 17:16058(s2 + t2 + u2) 6:703913(s2 + t2 + u2) 6:131683(s2 + t2 + u2) 4:8779369(s2 + t2 + u2) 0:762594(s2 + t2 + u2) 1:252848(s2 + t2 + u2) 1:2121707(s2 + t2 + u2) 0:21142504(s2 + t2 + u2) 0 total 8:521180(s2 + t2 + u2) symmetry factor 16 4 4 8 2 2 1 8 2 diagram k = 5; ` = 1 : diagram (a) (b) (c) (d) (e) (f) (g) (h) 9:645323(s2 + t2 + u2) 7:922393s2 6:151710(s2 + t2 + u2) 1:9295194s2 1:538466(s2 + t2 + u2) + 0:7303561s2 1:308312(s2 + t2 + u2) + 0:645134s2 1:879007(s2 + t2 + u2) 1:451108s2 0:32813257(s2 + t2 + u2) 0:26805768s2 0:0512934(s2 + t2 + u2) + 0:1579127s2 total 17:38894(s2 + t2 + u2) 13:955903s2 symmetry factor total 12:88988(s2 + t2 + u2) 8:901921s2 A.4 Four-loop dimensions is The result of [16] for the 4-loop 4-point amplitude of maximal SU(k) SYM in D = 6 2 A ( (k2 3 + 25 5) Tr12Tr34s2 + Tr14Tr23t2 + Tr13Tr24u2 + (single trace): When restricted to the Cartan gluons, of charge na 2 Zk (a = 1; 2; 3; 4) with respect to the Zk action, the single trace term is always proportional to (s2 + t2 + u2) P na ( here stands for Kronecker delta modulo k). The coe cient will involve 1= 2 and 1= divergences. These have not been computed explicitly. On the other hand, for the double trace terms, we have (k; na + nb 0; otherwise: 0 mod k; For the amplitude of gluons with Zk charge (n; n; n; n) (n = ` + 1 in our notation), we always have Tr13 = Tr14 = Tr23 = Tr24 = k. Tr12 = Tr34 = 0 for n 6= k=2, and Tr12 = Tr34 = k for n = k=2. In the case k = 4, by comparing ` = 0 with ` = 1, we can separate a contribution from double trace terms only, 4 loop Ak=4;`=1 4 loop Ak=4;`=0 = (stAtree) (4 )12 4 64 (s2 + t2 + u2) e 4 ( 2 2 16 + 36 3 + 1 16 35 18 + 4 3 + 9 4 + 20 5 3 (16 3 + 25 5)s2 (A.51) (A.52) (A.53) 3 3 ) (8 + 18 3)(8 ln r)2 + A ln r + B After subtracting o the 4-loop counter-terms, we expect 4 loop Ak=4;`=1 4 loop Ak=4;`=0 = ( (s(t4A)tr1e2e) 64 (s2 + t2 + u2) + s2 3(16 3 + 25 5)(8 ln r + C) : Here A is a constant that depends on and B; C are constants that depend on nite shifts of the 3-loop D2trF 4 counter-term, nite shifts of the 4-loop D4trF 4 and D4tr2F 4 counter-terms. They cannot be determined from SYM perturbation theory alone. In the n = k=2 cases, all terms are proportional to s2 + t2 + u2, and we cannot separate the double trace terms from the single trace terms at all. In the k = 3 and k = 5 cases, as well as the k = 4; ` = 0 case, since Tr12 = Tr34 = 0, we can determine ( Ak;` 4 loop = (s(t4A)tr1e2e) k3 (s2 + t2 + u2)(unknown) s 2 3(k2 3 + 25 5)(8 ln r + C) : (A.54) ) B Evaluation of the little string amplitudes In this appendix, we discuss some machinery that went into the numerical evaluation of the double scaled little string theory amplitude (4.2). The conformal block can be written F ( i; P jz) = (16q)P 2 z Q42 1 2 (1 Q2 z) 4 1 3 3(q)3Q2 4( 1+ 2+ 3+ 4)H( i; P jq); q(z) = e i (z); (z) = i K(1 K(z) K(z) = 1 Z 1 2 0 where P = Q42 + P 2, z is the cross ratio q is the nome of z, de ned by and 3 is the Jacobi theta function de ned by z = z12z34 ; z14z32 3(p) = 1 X n= 1 pn2 : H satis es Zamolodchikov's recurrence formula [39, 40], which allows one to obtain H as a series expansion in q. Alternatively, we can compute F as a series expansion in z by computing inner products between Virasoro descendants of the external primary states. The resulting expression is manifestly a rational function in c, i, and P . For this reason the latter brute-force method is more advantageous for obtaining simple analytic expressions, although its computational complexity (with respect to the order of the series in q) is much higher than the complexity of the recurrence method. The conformal block written in the form (B.1) converges much faster than a naive series expansion in z, due to the fact that jq(z)j is much smaller than z (note for example that 16jq(z)j jzj and jq(z)j < 1 for all z 2 C). Given an order-N series in z, we can rewrite it in the form of (B.1) by performing a variable transformation and then truncate H to order qN . If we want to integrate z over regions far from the origin, it is crucial that we approximate the conformal block by a truncation of (B.1) instead of a series in z. The Liouville structure constant C( 1; 2 ; 3) is expressed as ratios of the special function , which has an integral representation [41, 42] log (x) = Z 1 dt " 0 Q 2 x 2 e t sinh2( Q 2 sinh b2t sinh 2tb x) 2t # that is is convergent for 0 < Re x < Q. For x lying outside this region, can be analytically continued via the shift formulae b(x + b) = (bx)b1 2bx b(x + 1=b) = (x=b)b 2bx 1 where (1 (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) numerically, the oscillatory behavior of the second term at large t must be taken care of by stripping out an exponential integral function Z 1 dt e( Q2 x)t 4t sinh b2t sinh 2tb = E1(xt0) + 4t sinh b2t sinh 2tb Z 1 dt (e bt + e b e Qt)e( Q2 x)t : (B.8) To obtain the Liouville four-point function, we then integrate over the Liouville momentum P of the intermediate state. This integral is performed by a simple Riemann sum. Finally we are in place to evaluate the integral with respect to the cross ratio z. 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Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang, Xi Yin. Interpolating the Coulomb phase of little string theory, Journal of High Energy Physics, 2015, 1-35, DOI: 10.1007/JHEP12(2015)022