Kawai-Lewellen-Tye Relations Research Articles

One-Loop Double Copy Relation in String Theory

We discuss relations between closed and open string amplitudes at one loop. While at tree level these relations are known as Kawai-Lewellen-Tye (KLT) and/or double copy relations, here we investigate how such relations are manifested at one loop. Although, we find examples of one-loop closed string amplitudes that can strikingly be written as sum over squares of one-loop open string amplitudes, generically the one-loop closed string amplitudes assume a form reminiscent of the one-loop double copy structure of gravitational amplitudes involving a loop momentum. This double copy structure represents the one-loop generalization of the KLT relations. Published by the American Physical Society 2024

KLT factorization of nonrelativistic string amplitudes

We continue our study of the Kawai-Lewellen-Tye (KLT) factorization of winding string amplitudes in [1]. In a toroidal compactification, amplitudes for winding closed string states factorize into products of amplitudes for open strings ending on an array of D-branes localized in the compactified directions; the specific D-brane configuration is determined by the closed string data. In this paper, we study a zero Regge slope limit of the KLT relations between winding string amplitudes. Such a limit of string theory requires a critically tuned Kalb-Ramond field in a compact direction, and leads to a self-contained corner called nonrelativistic string theory. This theory is unitary, ultraviolet complete, and its string spectrum and spacetime S-matrix satisfy nonrelativistic symmetry. Moreover, the asymptotic closed string states in nonrelativistic string theory necessarily carry nonzero windings. First, starting with relativistic string theory, we construct a KLT factorization of amplitudes for winding closed strings in the presence of a critical Kalb-Ramond field. Then, in the zero Regge limit, we uncover a KLT relation for amplitudes in nonrelativistic string theory. Finally, we show how such a relation can be reproduced from first principles in a purely nonrelativistic string theory setting. We will also discuss connections to the amplitudes of string theory in the discrete light cone quantization (DLCQ), a method that is relevant for Matrix theory.

The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes

We study the Kawai-Lewellen-Tye (KLT) relations for quantum field theory by reformulating it as an isomorphism between two Lie algebras. We also show how explicit formulas for KLT relations arise when studying rational functions on ${\mathcal M}_{0,n}$, and prove identities that allow for arbitrary rational functions to be expanded in any given basis. Via the Cachazo-He-Yuan formulas for, these identities also lead to new formulas for gauge and gravity tree amplitudes, including formulas for so-called Bern-Carrasco-Johansson numerators, in the case of non-linear sigma model and maximal-helicity-violating Yang-Mills amplitudes.

Higher derivative quartic vertex of f(R) gravity in light-cone gauge

In the recent studies of four-dimensional Einstein–Hilbert action, quite a few interesting results such as the Kawai–Lewellen–Tye (KLT) relations, MHV-Lagrangian and quadratic forms have been reported. These results naturally raise an important question: Whether these results are valid for the modified theories of gravity in 4-dimensions? In this work, we consider R+αR2 gravity and derive the complete quartic interaction vertex in light-cone gauge. We then discuss the implications of the results for KLT relations, and computing MHV amplitudes.

New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level.

In this Letter, we extend the tree-level Kawai-Lewellen-Tye (KLT) and Bern-Carrasco-Johansson (BCJ) amplitude relations to loop integrands of gauge theory and gravity. By rearranging the propagators of gauge and gravity loop integrands, we propose the first manifestly gauge- and diffeomorphism-invariant formulation of their double-copy relations. The one-loop KLT formula expresses gravity integrands in terms of more basic gauge invariant building blocks for gauge-theory amplitudes, dubbed partial integrands. The latter obey a one-loop analogue of the BCJ relations, and both KLT and BCJ relations are universal to bosons and fermions in any number of spacetime dimensions and independent on the amount of supersymmetry. Also, one-loop integrands of Einstein-Yang-Mills theory are related to partial integrands of pure gauge theories.

Black hole formation and classicalization in ultra-Planckian [formula omitted] scattering

We establish a connection between the ultra-Planckian scattering amplitudes in field and string theory and unitarization by black hole formation in these scattering processes. Using as a guideline an explicit microscopic theory in which the black hole represents a bound-state of many soft gravitons at the quantum critical point, we were able to identify and compute a set of perturbative amplitudes relevant for black hole formation. These are the tree-level N-graviton scattering S-matrix elements in a kinematical regime (called classicalization limit) where the two incoming ultra-Planckian gravitons produce a large number N of soft gravitons. We compute these amplitudes by using the Kawai–Lewellen–Tye relations, as well as scattering equations and string theory techniques. We discover that this limit reveals the key features of the microscopic corpuscular black hole N-portrait. In particular, the perturbative suppression factor of a N-graviton final state, derived from the amplitude, matches the non-perturbative black hole entropy when N reaches the quantum criticality value, whereas final states with different value of N are either suppressed or excluded by non-perturbative corpuscular physics. Thus we identify the microscopic reason behind the black hole dominance over other final states including non-black hole classical object. In the parameterization of the classicalization limit the scattering equations can be solved exactly allowing us to obtain closed expressions for the high-energy limit of the open and closed superstring tree-level scattering amplitudes for a generic number N of external legs. We demonstrate matching and complementarity between the string theory and field theory in different large-s and large-N regimes.

Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator

We revisit the tree-level closed superstring amplitude and identify its α′-expansion as series with single-valued multiple zeta values as coefficients. The latter represent a subclass of multiple zeta values originating from single-valued multiple polylogarithms at unity. Moreover, the α′-expansion of the closed superstring amplitude can be cast into the same algebraic form as the open superstring amplitude: the closed superstring amplitude is essentially the single-valued version of the open superstring amplitude. This fact points to a deeper connection between gauge and gravity amplitudes than is implied by Kawai–Lewellen–Tye relations. Furthermore, we argue that the Deligne associator carries the relevant information on the closed superstring amplitude. In particular, we give an explicit representation of the Deligne associator in terms of Gamma functions modulo squares of commutators of the underlying Lie algebra. This form of the associator can be interpreted as the four-point closed superstring amplitude.

Closed string amplitudes as single-valued open string amplitudes

We show that the single trace heterotic N-point tree-level gauge amplitude ANHET can be obtained from the corresponding type I amplitude ANI by the single-valued (sv) projection: ANHET=sv(ANI). This projection maps multiple zeta values to single-valued multiple zeta values. The latter represent a subclass of multiple zeta values originating from single-valued multiple polylogarithms at unity. Similar relations between open and closed string amplitudes or amplitudes of different string vacua can be established. As a consequence the α′-expansion of a closed string amplitude is dictated by that of the corresponding open string amplitude. The combination of single-valued projections, Kawai–Lewellen–Tye relations and Mellin correspondence reveal a unity of all tree-level open and closed superstring amplitudes together with the maximally supersymmetric Yang–Mills and supergravity theories.

Remarks on the identities of gluon tree amplitudes

Recently, Bjerrum-Bohr, Damgaard, Feng, and Sondergaard derived a set of new interesting quadratic identities of the Yang-Mills (YM) tree scattering amplitudes, besides Bern-Carrasco-Johansson (BCJ) identities. Here we comment that these quadratic identities of YM amplitudes actually follow directly from the KLT (Kawai-Lewellen-Tye) relation for graviton-dilaton-axion scattering amplitudes (in four-dimensional spacetime). This clarifies their physical origin and also provides a simpler version of the new identities. We also comment that the recently discovered BCJ identities of YM helicity amplitudes, at least for the maximal helicity-violating case, can be verified by using (repeatedly) the Schouten identity. We also point out additional quadratic identities that can be written down from the KLT relations.

Stringy KLT relations, global symmetries, and E 7(7)-violation

We study consequences of the Kawai-Lewellen-Tye (KLT) relations applied to tree amplitudes in toroidal compactifications of string theory to four dimensions. The closed string tree amplitudes with massless external states respect a global SU(4)xSU(4) symmetry, which is enhanced to the SU(8) R-symmetry of N=8 supergravity in the field theory limit. Our analysis focuses on two aspects: (i) We provide a detailed account of the simplest SU(8)-violating amplitudes. We classify these processes and derive explicit superamplitudes for all local 5- and 6-point operators with SU(4)xSU(4) symmetry at order alpha'^3. Their origin is the dilatonic operator exp(-6 phi) R^4 in the closed-string effective action. (ii) We expand the 6-point closed string tree amplitudes to order alpha'^3 and use two different methods to isolate the SU(8)-singlet contribution from exp(-6 phi) R^4. This allows us to extract the matrix elements of the unique SU(8)-invariant supersymmetrization of R^4. Their single-soft scalar limits are non-vanishing. This demonstrates that the N=8 supergravity candidate counterterm R^4 is incompatible with continuous E_7(7) symmetry. From the soft scalar limits, we reconstruct to quadratic order the SU(8)-invariant function of scalars that multiplies R^4, and show that it satisfies the Laplace eigenvalue equation derived recently from supersymmetry and duality constraints.

KLT and new relations for $ \mathcal{N} = 8 $ SUGRA and $ \mathcal{N} = 4 $ SYM

In this short note, we prove the supersymmetric Kawai-Lewellen-Tye (KLT) relations between N=8 supergravity (SUGRA) and N=4 super Yang-Mills (SYM) tree-level amplitudes in the frame of S-matrix program, especially we do not use string theory or the explicit Lagrangian form of corresponding theories. Our supersymmetric KLT relations naturally unify the non-supersymmetric KLT relations and newly discovered gauge theory identities and produce more identities for amplitudes involving scalars and fermions. We point out also that these newly discovered identities can be used to reduce helicity basis from (n-3)! further down.

SURPRISING SIMPLICITY OF $\mathcal{N}=8$ SUPERGRAVITY

Gravity amplitudes are, via the Kawai–Lewellen–Tye relations intimately linked to products of Yang–Mills amplitudes. Explicitly, this shows up in computations of [Formula: see text] supergravity where the perturbative expansion and ultraviolet behavior of this theory is akin to [Formula: see text] super-Yang–Mills at least through three loops. Full persistence to all loop orders would be truly remarkable and imply finiteness of [Formula: see text] supergravity in four dimensions.

The quintic interaction vertex in light-cone gravity

We consider pure gravity in light-cone gauge and derive the complete quintic interaction vertex. Up to quartic order, the Kawai–Lewellen–Tye (KLT) relations can be made manifest at the level of the Einstein–Hilbert Lagrangian. The quintic interaction vertex represents an essential first step in further extending the off-shell validity of the KLT relations to higher order vertices.

KLT relations from the Einstein–Hilbert Lagrangian

The Kawai–Lewellen–Tye (KLT) relations derived from string theory tell us that perturbative gravity amplitudes are the “square” of the corresponding amplitudes in gauge theory. Starting from the light-cone Lagrangian for pure gravity we make these relations manifest off-shell, for three- and four-graviton vertices, at the level of the action.

Multi-leg one-loop gravity amplitudes from gauge theory

By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the ‘maximally helicity-violating’ amplitudes in N = 8 supergravity, and the ‘all-plus’ helicity amplitudes in gravity with any minimally coupled massless matter content. The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and gauge theory tree amplitudes. For n ⩽ 6, we determine the all-plus amplitudes explicitly from their unitarity cuts. The KLT relations, applied to the cuts, allow us to extend to gravity a previously found ‘dimension-shifting’ relation between (the cuts of) the all-plus amplitudes in gauge theory and the maximally helicity-violating amplitudes in N = 4 super-Yang-Mills theory. The gravitational version of the relation lets us determine the n ⩽ 6 N = 8 supergravity amplitudes from the all-plus gravity amplitudes. We infer the two series of amplitudes for all n from their soft and collinear properties, which can also be derived from gauge theory using the KLT relations.