Universality in string interactions

Journal of High Energy Physics, Sep 2016

In this note, we provide evidence for universality in the low-energy expansion of tree-level string interactions. More precisely, in the α′-expansion of tree-level scattering amplitudes, we conjecture that the leading transcendental coefficient at each order in α′ is universal for all perturbative string theories. We have checked this universality up to seven points and trace its origin to the ability to restructure the disk integrals of open bosonic string into those of the superstring. The accompanying kinematic functions have the same low-energy limit and do not introduce any transcendental numbers in their α′-corrections. Universality in the closed-string sector then follows from KLT-relations.

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Universality in string interactions

Received: April Universality in string interactions Yu-tin Huang 0 1 2 5 Oliver Schlotterer 0 1 2 3 Congkao Wen 0 1 2 4 Congkao.Wen@roma 0 1 2 .infn.it 0 1 2 Open Access 0 1 2 c The Authors. 0 1 2 0 Via della Ricerca Scientifica , 00133 Roma , Italy 1 14476 Potsdam , Germany 2 Taipei 10617 , Taiwan, R.O.C 3 Max-Planck-Institut fu ̈r Gravitationsphysik, Albert-Einstein-Institut 4 Dipartimento di Fisica, Universita` di Roma “Tor Vergata” 5 Department of Physics and Astronomy, National Taiwan University In this note, we provide evidence for universality in the low-energy expansion of tree-level string interactions. More precisely, in the α′-expansion of tree-level scattering amplitudes, we conjecture that the leading transcendental coefficient at each order in α′ is universal for all perturbative string theories. We have checked this universality up to seven points and trace its origin to the ability to restructure the disk integrals of open bosonic string into those of the superstring. The accompanying kinematic functions have the same low-energy limit and do not introduce any transcendental numbers in their α′-corrections. Universality in the closed-string sector then follows from KLT-relations. Bosonic Strings; Scattering Amplitudes 1 Introduction 2 Open-string amplitudes 2.1 The bosonic open string Universality and BCJ identities Closed-string amplitudes Universality for closed string theories Universality of UV completion One of the formidable challenges for a theory of quantum gravity is the construction of a gravitational S-matrix which respects unitarity at high energies. Perturbative string theories provide candidate solutions, as its four-point graviton S-matrix is exponentially suppressed in the high-energy limit for fixed-angle scattering [1, 2]. In fact, assuming tree-level causality [3] and unitarity [4] imposes stringent constraints, under which string theories provide the only known analytic solutions so far. Different string theories are understood to be equivalent through a web of strong-weak dualities which relate different orders in the perturbative expansion [5, 6]. At tree level, however, the low-energy description in the form of an effective action with expansion in curvature tensors and covariant derivatives is largely unconstrained by string dualities. More precisely, the coefficients of these higher-dimensional operators are expected to be distinct for different string theories. Thus, if some of these coefficients turn out to be universal, it is then conceivable that such a phenomenon reflects a deeper principle in the theory of quantum gravity beyond the known dualities. At low energies, closed-string theories yield an effective action that augments the Einstein-Hilbert term SEH with higher-dimensional operators. At tree level, type-II superstring theories exhibit the following expansion in the inverse string tension (or cut-off contractions of covariant derivatives and Riemann tensors. The tensor structure of each operator as well as its coefficient furnished by multiple zeta values (MZVs) turally categorized according to their transcendental weight n1 +n2 +. . .+nr and constitute a fruitful domain of common interest between high-energy physics and number theory. In fact, for type-II theories, the transcendental weight for each coefficient matches the order for open strings in the type-I theory. The type-I effective action is now an expansion in non-abelian field-strength operators tr(DnF m). In this light, uniform transcendentality for closed strings is inherited from open strings through the Kawai, Lewellen and Tye (KLT) In this letter, we conjecture that the leading transcendental coefficient at each order closed-string theories. We have explicitly verified this up to the seven-point level, and the conjectural all-multiplicity extension is further investigated in a companion paper [8]. This that correspond to the UV completion of tree-level Einstein-Hilbert graviton amplitudes. This is given by the type-II theories. For Heterotic and Bosonic closed string theories, this is augmented by separate terms that correspond to UV completions of amplitudes dilaton or the Tachyon. This remarkable property can be best understood by inspecting the world-sheet correlator of the open-string amplitudes. It was shown in [9] that the n-point tree amplitude of the open superstring can be cast of different color-orderings. These basis integrals exhibit uniform transcendentality upon be cast into the very same basis where — in contrast to the superstring — the augmented Yang-Mills tree amplitude, additional rational functions that contain Tachyon poles. Thus involve rational numbers upon Taylor-expansion, i.e. they do not carry any transcendental have the same leading transcendental pieces as found for the superstring. The same property can be extended to closed strings by utilizing the KLT-relations [7], which assemble closed-string tree amplitudes from products of two open-string trees. The scendentality of the type-II theory. Different double-copies of open bosonic strings and superstrings give rise to three different closed-string theories — bosonic, heterotic and type-II integrals of uniform transcendentality inherited from the open-string constituents. Only specific to open bosonic strings do not introduce any transcendental weight and thereby do not affect the leading-transcendental piece. This completes the argument for univerweight-n coefficient is universal for all perturbative closed-string theories. More over, this representation also allows us to separate a piece of the closed string graviton amplitude that is universal and is given by the type II theories. This paper is organized as follows: in the next section, we give a brief review of the organization of open superstring amplitude as a matrix of disk integrals multiplied by the basis of Yang-Mills amplitude. We will show that through a series of Integration By Parts (IBP) identity, the bosonic string amplitude can be cast into the very same same relations. Finally, we show that the closed string graviton amplitude for all perturbative string theories can be naturally separated into a piece that can be identified as the UV completion of pure tree-level Einstein-Hilbert graviton amplitudes, which is given by that of type II theories. Open-string amplitudes The tree-level amplitude for n gluon-multiplet states in open superstring theory can be conveniently written as [9] where AS and AYM indicate color-ordered amplitudes of the superstring and super YangMills field theory, respectively. The ordering of the two amplitudes does not have to be ( n−2 k−1 k=2 m=1 zkm monodromy relations [11, 12]. weight matches the degree of the accompanying polynomials in sij . Since AYM do not Once undoing the above choice of SL(2) frame, the functions (2.2) can be identified as respectively, and the measure is given by The integral reductions performed in [9] rely on partial-fraction manipulations and integraHowever, as already exploited in a superstring context [9, 19], IBP additionally allows to techniques to gluon amplitudes of the bosonic string yields our main result to be reported in the following. The bosonic open string The tree-amplitude prescription for n-gluon scattering in the bosonic string introduces significantly more rational functions of zij of suitable SL(2) weight than captured by the single cycles in (2.4). In particular, one obtains more terms involving higher order poles as well as multicycle denominators. Still, repeated use of IBP is expected to reduce all of them to the single-cycle form and thereby to the same integral basis as seen in (2.1) and (2.2). For example, at four-points, all double poles can be reduced by using the following identity: 1 − s23 The denominator on the right-hand side signals tachyon exchange specific to the bosonic open superstring, while such double cycle denominators are also present, the OPE among supersymmetric vertex operators guarantees that tachyon poles as in (2.6) are suppressed by numerators 1 − sij . For explicit examples see e.g. [9, 19]. Extending the integral reduction along the lines of (2.6) to arbitrary multiplicity leads us to conjecture the following structure for the n-gluon tree in bosonic string theory: In comparison to the superstring result (2.1), the kinematic factors AYM(. . .) are replaced of sij and multilinear in the polarizations ej entering via (ei · ej ) and (ei · kj ), and crucially Yang-Mills tree amplitudes, and is therefore identical to that of the superstring, i.e. B(1,. . ., n; α′) = AYM(1,. . ., n) + X(2α′)kBk(1,. . ., n) . k=1 The simplest instances of the subleading terms occur at the three-point level and signal the F 3 interaction specific to the bosonic string, B1(1,2,3) = (e1 ·k2)(e2 ·k3)(e3 ·k1) , Bk≥2(1,2,3) = 0 . The higher-point case requires integral reductions as in (2.6), and the resulting geometric series yield non-zero Bk(. . .) for any value of k. A. Four-points. In the case of n=4, we find s212(1 − s12) the Bk(1, 2, 3, 4)’s from the second line of (2.10). At five points, after partial fractional manipulations, we require following two identities to reduce all the integrals to a single-cycle basis (2.4): s23 − 1 s13Z(1, 3, 2, 4, 5) 1 − s51 1 − s51 s23 − 1 notebook containing the full expression is attached. To arrive at (2.7) at six points, we find that after partial fractions, besides the single-cycle basis we encounter integrals of following forms, (z23z34z42) (z15z56z61) 2 , We have checked that indeed all above integrals can be reduced to single-cycle integral via IBP, for instance, (z23z34z42) (z15z56z61) Note that all identities of (2.6), (2.11) and (2.13) can alternatively be derived by imposing linearized gauge invariance under ej → kj , and the same is believed to hold for the integral reduction at arbitrary multiplicity. The analogous seven-point checks to arrive at (2.7) have been performed as well. Universality and BCJ identities must agrees with the superstring amplitude. This leads to the conclusion that the leadingare universal in open-string theories. Although the kinematic factors Bk(. . .) in (2.8) differ from AYM(. . .) in tensor structure and mass dimension, we will now argue that they obey the same KK- and BCJ-relations [15– 17]. The universal monodromy relations [11, 12] among bosonic-string subamplitudes have inserting (2.7) into the lowest-transcendentality pieces of the monodromy relations and identifying B0(. . .) ≡ AYM(. . .) yields 0 = Bk(1, 2, . . . , n) + Bk(2, 1, 3, . . . , n) + Bk(2, 3, 1, . . . , n) + . . . + Bk(2, 3, . . . , n − 1, 1, n) 0 = s12Bk(2, 1, 3, . . . , n) + (s12 +s13)Bk(2, 3, 1, 4, . . . , n) + . . . + (s12 +s13 +. . .+s1,n−1)Bk(2, 3, . . . , n − 1, 1, n) ploited in [20] to derive BCJ-relations for subamplitudes of the F 3 operators as well as the supersymmetrized D2F 4 + F 5. Moreover, an all-order argument for single-trace gluon amplitudes of the heterotic string has been given in [21]. By the same reasoning, (2.14) (Mj1 Mj2 . . . Mjp )στ Bk(1, 2τ , . . . , (n−2)τ , n−1, n) . in (2.15) are inevitable to verify permutation invariance of the world-sheet integrand for supergravity trees [29]. Closed-string amplitudes Closed-string amplitudes at tree level can be obtained from squares of open-string ampliable basis w.r.t. Q [22, 24]. The selection rules were identified in [25] with the single-valued projection of MZVs [26, 27]. A representation of the massless closed-superstring tree MnS which manifests the effect of these cancellations has been firstly given in [22]: X A˜YM(1, 2σ, . . . , (n−2)σ, n, n−1)(S0)σρ The polarizations of the type-II supergravity multiplets stem from tensor products of the gauge-multiplet polarizations in A˜YM and AYM. The matrix S0 has entries of order F (α′) = 1 + ζ2P2 + ζ3M3 + ζ22P4 + ζ5M5 + ζ2ζ3P2M3 + ζ23P6 + sij with rational coefficients. Once the MZVs are expressed in terms of their conjectural basis over Q, only one independent matrix Pw or Mw occurs at each weight w along with seen in (3.2) recycling information from lower weights. The rational prefactors 51 , 265 , . . . the terms in (3.2) [22], along with the matrix commutators [M5, M3], . . . can be understood from the coproduct of MZVs which can only be rigorously defined for their motivic versions. A description of motivic MZVs which lead to a rewriting of (3.2) with unit rational coefficients can be G(α′) = 1 + 2ζ3M3 + 2ζ5M5 + 2ζ32M3M3 + 2ζ7M7 + 2ζ3ζ5{M3, M5} spirations in the KLT-formula. Moreover, the properties of the Mw matrices additionally These selection rules have been identified in [25] with the single-valued projection of MZVs, see [26, 27] for further mathematical background. Together with the polarization-dependence from A˜YM, AYM, they encode the tensor contractions of the DnRm operators in the tree-level effective action to the orders seen in (1.1). Given the ubiquitous matrix products with summations over permutations in Universality for closed string theories II closed-string amplitudes are a property of the sphere integrals involving two copies of the integrands in (2.2). Accordingly, the results on the integrals can be exploited in further contexts such as gravitational tree amplitudes MnH or MnB in the heterotic or the closed bosonic string which rest on one or two copies of the bosonic-string integrand in (2.7). The only modification as compared to the superstring (3.1) is an exchange of Clearly, the Einstein-Hilbert interaction can be recovered from (3.6) and (3.7) at leadstructure of (3.6) is expected to capture multitrace interactions in the gauge sector of the In complete analogy to the open bosonic string, it is natural to organize (3.6) and (3.7) While the kinematic factherefore identical in (3.5), (3.6) and (3.7). Hence, we have shown that, at leading transcendentality, gravitational tree-level interactions are universal to the bosonic, heterotic and type-II closed-string theories. Universality of UV completion It is instructive to consider the four-dimensional spinor helicity form of B(1, 2, 3, 4). For different helicity choices, it takes on the following simple form: B(1−, 2−, 3−, 4−) = −u B(1−, 2−, 3−, 4+) = u B(1−, 2−, 3+, 4+) = u − 1 . − 1 For MHV amplitudes we will only need the last form. Thus the MHV amplitude in closed string theories can be written as M4S(1−, 2−, 3+, 4+) = h12i4[34]4 f (s, t, u) with the function f (s, t, u) given as: Super f (s, t, u) = Heterotic f (s, t, u) = Bosonic f (s, t, u) = tree-level Einstein-Hilbert graviton amplitudes. For the heterotic string the extra term Thus we see that the UV completion of the the tree-level Einstein-Hilbert graviton amplitude is in fact unique in all perturbative string theories. The low energy amplitudes that stem from the presence of three-point higher-dimensional operators are completed separately. Note that the uniqueness of the UV completion for Einstein-Hilbert, follows directly from our conjectural form of the bosonic tree function in eq. (2.7), which has been proven up to seven-points. It will be extremely interesting to see whether the additional terms for heterotic and bosonic string graviton amplitudes at higher points can also be written as separate completion for tree-amplitudes of higher dimensional operators. We leave this for future study. In this letter, tree-level amplitudes in all perturbative open- and closed-string theories are versality can be achieved by casting the world-sheet correlators of the bosonic open string matic factors. We have explicitly shown that such a reorganization can be achieved up to seven points, and the conjectural all-multiplicity extension is relegated to future work [8]. Generalizations to closed-string interactions in bosonic, heterotic and type-II theories directly follow from the KLT-relations. These universality results have greatly facilitated the construction of matrix elements for counterterms in half-maximal supergravity [30]. It would be interesting to apply the same organizing principles to massive-state scattering. We expect the same basis of disk integrals to capture tree amplitudes among any combination of massive open-string resonances. Moreover, the structure of (3.5) is believed to apply to closed-string trees among massive resonances upon appropriate replacements of AYM and A˜YM. Finally the conjectural form of eq. (2.7) leads to the interesting conclusion that the UV completion of pure Einstein-Hilbert gravity is in fact unique amongst perturbative string theory. The difference lies in the presence of separate terms in the amplitude that R3. This structure is straight forward at four-points, and it will be interesting to see if the latter persists to higher points. We are grateful to Johannes Bro¨del, Paolo Di Vecchia, Michael Green and Henrik Johansson for inspiring discussions and valuable comments on a draft of the manuscript. 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Yu-tin Huang, Oliver Schlotterer, Congkao Wen. Universality in string interactions, Journal of High Energy Physics, 2016, 155, DOI: 10.1007/JHEP09(2016)155