Monodromy and Kawai-Lewellen-Tye Relations for Gravity Amplitudes
aa r X i v : . [ h e p - t h ] M a r Monodromy and Kawai-Lewellen-Tye Relationsfor Gravity Amplitudes
N. E. J. Bjerrum-BohrNiels Bohr International Academy and DISCOVERY Center,The Niels Bohr Institute, Blegdamsvej 17,DK-2100 Copenhagen Ø, Denmark, [email protected]
Pierre Vanhove ∗ Institut des Hautes Etudes Scientifiques, Le Bois-Marie,35 Route de Chartres, F-91440 Bures-sur-Yvette, FranceandCEA, DSM, Institut de Physique Th´eorique, IPhT, CNRS, MPPU,URA2306, Saclay, F-91191 Gif-sur-Yvette, France, [email protected]
Submitted for the 12th Marcel Grossman Meeting 2009August 7, 2018
Abstract
We are still learning intriguing new facets of the string theory motivated Kawai-Lewellen-Tye (KLT) relations linking products of amplitudes in Yang-Mills theo-ries and amplitudes in gravity. This is very clearly displayed in computations of N = 8 supergravity where the perturbative expansion show a vast number of simi-larities to that of N = 4 super-Yang-Mills. We will here investigate how identitiesbased on monodromy relations for Yang-Mills amplitudes can be very useful fororganizing and further streamlining the KLT relations yielding even more compactresults for gravity amplitudes.Keywords: Amplitudes, Quantum gravity, String theory ∗ IPHT-T10/023, IHES/P/10/07. Introduction
The search for a valid construction of quantum gravity has been on for most of theprevious century initiated by Einstein’s formulation of General Relativity in 1916 andthe quantum mechanics revolution in the 1920ties. Physicists today are still huntingthe answers to the ultimate questions, e.g. how was the universe formed and howdoes one comprehend the fabric of space and time? Quantum mechanical correctionsto gravity are crucial for the exact answers but the fundamental concepts of such aquantum theory are unfortunately still very dim. In this paper we will investigate howwe can learn about an ultimate theory of quantum gravity through studying symmetriesin Yang-Mills theories and the links posed between Yang-Mills theories and gravitythrough string theory.The combination of a traditional quantization and the extra symmetry introducedby a super-symmetrization of fundamental interactions appeared for a long while tobe a way out of the troublesome ultraviolet divergences associated with a field theoryfor gravity. The most famous model is possibly is the one of maximal supersymmetry N = 8 supergravity [1, 2]. This theory arises as a low-energy effective descriptionof string theory in four dimensions. Various later arguments based on supersymmetrypoint to a delay in the onset of ultraviolet divergences due to the extra symmetry [3, 4,5, 6] but it has long been the belief that only string theory should be completely free ofUV divergences. However since no explicit ultraviolet divergences have been found sofar in the four-dimensional four-graviton amplitude [7, 8, 9, 10, 11, 12], the effectivefield-theory status of N = 8 supergravity and its relation to string theory have been putinto questions.We know that the on-shell S -matrix elements in string theory depend on the scalarsparameterizing the (classical) moduli space E ( R ) / ( SU (8) / Z ) and that these arecovariant under the discrete U-duality subgroup E ( Z ) [13, 14]. However in super-gravity the S -matrix elements are invariant under the continuous symmetry E ( R ) [15,16, 17, 18]. From the string theory viewpoint the relation between the four-dimensionalPlanck length ℓ and the string scale ℓ s = √ α ′ depend on the (four-dimensional) dila-ton ℓ = α ′ y where y = g s α ′ / ( R · · · R ) and R i are the radii of compactifi-cation. The decoupling limit of string amplitudes goes as ℓ s → , /R i → ∞ and R i /α ′ → ∞ , keeping the four-dimensional Newton’s constant κ = 2 π ℓ fixed.This limit is singular since in this limit some non-perturbative states become masslessand dominate the S -matrix [19, 20]. These non-decoupling results do not imply that2 = 8 supergravity has perturbative ultraviolet problems however because of the lackof concrete data it has become urgent to clarify the status of the ultraviolet behavior of N = 8 supergravity in four dimensions and its relation to string theory.In recent years, by a combination of different inputs from string theory, super-symmetry, unitarity and due to remarkable progress in computational capacity, a hugenumber of amplitudes have been computed [21]. Surprisingly the ultraviolet behaviorof N = 8 supergravity occurs explicitly to be identical to the one of N = 4 super-Yang-Mills at least through four loops [8, 22, 23, 24, 25, 9, 12, 26, 4, 5]. These resultshave made it clear that N = 8 supergravity has a much better perturbative expansionthan power-counting na¨ıvely suggests. It is still an open question if the perturbativeexpansions of the two theories are similar to all loop orders or what in given case willbe the first loop order to have a dissimilarity. These and other aspects are discussedfurther in ref. [27].Motivated by string theory [28] where the massless spectrum of N = 8 supergrav-ity can be factorized as the tensorial product of two copies of N = 4 super-Yang-Millstheories, one can organize N = 8 supergravity tree-level amplitudes according to theKLT relations [28, 8, 29, 21, 30, 31, 32] which we will write schematically in the fol-lowing way M treeGravity ∼ X ij K ij A i L Yang − Mills × A j R
Yang − Mills . (1)Here M Gravity , A i L Yang − Mills , A j R Yang − Mills are gravity and color ordered Yang-Millsamplitudes and K ij is a specific function of kinematic invariants needed to ensure thatthe tree-level gravity amplitude has the correct analytic structure.The simple KLT relations between theories of gravity and two gauge theories areobserved directly in on-shell S -matrix elements but have no motivation at the La-grangian level (This is true even if part of the Lagrangian is rearranged as a product ofYang-Mills types of interactions at the two-derivative level [33, 34, 32] or for higherderivative corrections [35]. In the case of pure gravity one needs to take into accountthe contribution from the dilaton in employing the KLT relations.)Because of their high degree of supersymmetry both N = 4 super-Yang-Mills and N = 8 supergravity loop amplitudes are cut constructible in D = 4 − ǫ dimensionsand surprisingly the knowledge of the tree-level amplitudes is enough for reconstruct-ing the full higher-loop amplitudes [36, 9, 10, 15, 12].We will here discuss tree amplitudes from the point of view of the classical N = 8 theory, which can be constructed from the N = 4 super-Yang-Mills tree-level am-plitudes using the KLT relation in (1). (For effective theories of gravity [30] one canalso employ KLT relations in a slightly modified fashion taking into account higherderivative operators introduced through counterterms to ultraviolet divergences.)We will next discuss the construction of tree-level amplitudes in Yang-Mills andgravity from a minimal basis of amplitudes following [37].3 Minimal basis for Yang-Mills and Gravity tree-levelamplitudes
The n -point amplitude in open string theory with U ( N ) gauge group reads A n = ig n − (2 π ) D δ D ( k + · · · + k n ) X ( a ,...,a n ) ∈ S n / Z n tr ( T a · · · T a n ) A n ( a , · · · , a n ) , (2)where D is any number of dimensions obtained by dimensional reduction from dimensions if we consider the bosonic string, or dimensions in the supersymmet-ric case. The field theory amplitudes are obtained by taking the limit α ′ → . Anew series of amplitude identities between different color-ordered amplitudes basedon monodromy for integrations in string theory was derived in [37] (see [38, 39] forrelated discussions). The real part of these relations relates the n -point amplitude withdifferent orderings as A n ( β , . . . , β r , , α , . . . , α s , n ) = ( − r × ℜ e h Y ≤ i 4) = Γ(1 − α ′ s )Γ(1 − α ′ t )Γ(1 − α ′ u ) (cid:16) n s s + n t t (cid:17) , (6) A (1 , , , 4) = Γ(1 − α ′ u )Γ(1 − α ′ t )Γ(1 − α ′ s ) (cid:16) − n u u − n t t (cid:17) , (7) A (2 , , , 4) = Γ(1 − α ′ s )Γ(1 − α ′ u )Γ(1 − α ′ t ) (cid:16) n s s + n u u (cid:17) , (8)where n s , n t and n u depends on the polarizations and the external momenta.The monodromy relations (3) and (4) A (1 , , , 4) = sin(2 πα ′ s )sin(2 πα ′ u ) A (1 , , , , A (2 , , , 4) = sin(2 πα ′ t )sin(2 πα ′ u ) A (1 , , , , (9)imply that the numerator factors satisfy the Jacobi like relation n s = n t + n u . Thegeneralization to higher points gives the new amplitude relations recently conjecturedby Bern et al. in ref. [31]. The string theory monodromy identities for the Kawai-Lewellen-Tye relationship between closed and open string amplitudes give highly sym-metric forms for tree-level amplitudes where the tree-level gravity amplitudes are ex-panded in a basis obtained by the left/right tensorial product of gauge color orderedamplitudes M n = X σ,σ ′ ∈ S n − G σ,σ ′ ( k i · k j ) B Lσ B Rσ ′ . (10)As a direct application of our procedure, we can rewrite the Kawai-Lewellen-Tye rela-tions at four-point level as M = κ α ′ S k ,k S k ,k S k ,k A L (1 , , , A R (1 , , , . (11)The field theory limit of the string amplitude (11), α ′ → gives the symmetric formof the gravity amplitudes of [31] M F T = κ stu (cid:16) n s s + n t t (cid:17) (cid:18) ˜ n s s + ˜ n t t (cid:19) = − κ (cid:18) n s ˜ n s s + n t ˜ n t t + n u ˜ n u u (cid:19) . (12)Here we have made use of the on-shell relation s + t + u = 0 and the four-point Jacobirelation n u = n s − n t . 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