Inverse of the string theory KLT kernel

Journal of High Energy Physics, Jun 2017

The field theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the α′-corrected bi-adjoint scalar amplitudes that are exact in α′. We find compact expressions in terms of graphs, where the standard Feynman propagators 1/p 2 are replaced by either 1/sin(πα′p 2 /2) or 1/tan(πα′p 2 /2), as determined by a recursive procedure. We demonstrate how the same object can be used to conveniently expand open string partial amplitudes in a BCJ basis.

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Inverse of the string theory KLT kernel

Received: April Inverse of the string theory KLT kernel Sebastian Mizera 0 1 2 3 0 Department of Physics & Astronomy, University of Waterloo 1 Waterloo , ON N2L 2Y5 , Canada 2 Perimeter Institute for Theoretical Physics 3 Waterloo , ON N2L 3G1 , Canada The eld theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the 0-corrected bi-adjoint scalar amplitudes that are exact in 0. We nd compact expressions in terms of graphs, where the standard Feynman propagators 1=p2 are replaced by either 1= sin( kernel; Scattering Amplitudes; Bosonic Strings; Superstrings and Heterotic Strings - 0p2=2) or 1= tan( 0p2=2), as determined by a recuropen string partial amplitudes in a BCJ basis. Contents 1 Introduction 2 3 4 Review of the bi-adjoint scalar theory Diagrammatic rules for m 0 ( j ~) O -diagonal amplitudes Diagonal amplitudes KLT relations Soft limits of closed strings Basis expansion Future directions A Proof of the sign of m 0 ( j ~) Introduction \square" of the Yang-Mills theory [2, 3]. Yang-Mills2 bi-adjoint scalar we can write: bi-adjoint scalar theory. Closed string = Open string2 0-corrected bi-adjoint scalar an interesting structure. We will explore it in this work. 0s34) tan( 0s51) tan( 0s12) sin( 0s345) tan( 0s12) tan( 0s23) tan( 0s45) tan( the amplitudes1 we will study, m 0 (1234j1234) = m 0 (123456j126345) = m 0 (12345j12345) = theory in the in nite tension limit, 0 ! 0. of closed string amplitudes from the open string ones. theory and eld-theory. Review of the bi-adjoint scalar theory to them as amplitudes of an 0-corrected bi-adjoint scalar theory, since they compute physical quantities at aa~ b~b cc~, where f abc and f~a~~bc~ are the structure constants T a n ) Tr(T~a~ ~1 T~a~ ~2 where the sum goes over all the permutations and ~ modulo cyclicity. The partial appropriate propagator, m( j ) = owing through the edge e. In particular, overall sign can be determined using the rules described in [6]. We will introduce an Examples of the amplitudes are as follows: interaction of the form f abcf~a~~bc~ of the two avour groups. decomposition: mfull = ; ~ 2 Sn=Zn m(123j123) = 1; m(1234j1234) = m(12345j13254) = m(12345j14253) = 0; m(1234j1243) = m(12435j14253) = m(12435j13254) = 0: corresponding amplitudes vanish. derivation [6] leads to the eld theory KLT relation, GR = A which we take as a de nition of the operation AfYuMll = P 2Sn=Zn Tr(T a 1 T a 2 YM( ), analogous to (2.1). By m 1 by permutations larly. The sum over repeated indices and ~ respectively. The vectors A and ~ is implied. Here YM( ~) are de ned simiand ~ range over sets of 3)! permutations forming a BCJ basis, in which case the (n 3)! matrix KLT GR is a pure gravity amorderings to be 2 f(12345); (12435)g and ~ 2 f(13254); (14253)g. We can then use the GR = " YM(12345)#| "1=s23s45 YM(12435) 1=s24s35 YM(14253) = s23s45 A YM(13254) + s24s35 A YM(14253): as long as the matrix is invertible. relations are linking di erent theories, the kernel stays the same. YM( ) = m( j ~) m 1( ~j ) A 3)! matrix and Here, A YM( ) on the left hand side is a vector of size p, m( j ~) is an p 3)! matrix, YM( ) is a vector of size (n combination of (n 3)! Yang-Mills partial amplitudes. The latter forms a basis. For instance, let us expand A YM(12354) in the basis fA YM(13254); A YM(14253)g. We can reuse the matrix from (2.7) to write: YM(12354) = 1=s12s45 1=s23s45 1=s12s35 s12 + s23 YM(13254) #| "1=s23s45 1=s24s35 YM(14253): Here, we have also used two extra bi-adjoint amplitudes: m(12354j12345) = m(12354j12435) = case [21]. studied in [32]. Berends-Giele recursion relations were given in [33]. YM(14253) closed = A In our normalization, m 0 (123j123) = 1; m 0 (1234j1234) = m 0 (12345j13254) = m 0 (12345j14253) = 0; tan( 0s12) tan( 0s23) sin( 0s23) sin( 0s45) m 0 (1234j1243) = m 0 (12435j14253) = sin( 0s24) sin( 0s35) m 0 (12435j13254) = 0: m 0 ( j ~) = = ~ 1 i<j m pai paj . It is straightforward to invariants are always multiplied by the factor 0, from now on we will set 0 = 1 for the n labels. Then, some more interesting examples become: m 0 (I6j126435) = m 0 (I6j126345) = m 0 (I7j1276345) = sin s12 sin s34 sin s345 sin s12 sin s345 tan s34 sin s12 sin s345 tan s34 tan s45 tan s45 tan s67 tan s712 propagators. This motivates a separate discussion of the two cases. work, using the diagrammatic rules described below. O -diagonal amplitudes We can compute the o -diagonal terms, i.e., 6 m 0 (I6j126345) and m 0 (I7j1276345), we have respectively: Here, the permutations and ~ are drawn with black and red lines respectively. The us all the diagrams that are planar with respect to both and ~ [6]. the norm of the momentum owing through the leg e. In this way, we have introduced an 0se), where se = pe2=2 is Let us now dissect each diagram in turn. Firstly, m 0 (I6j126435) = sin s12 sin s34 sin s345 m 0 (I6j126345) = m 0 (I7j1276345) = sin s12 sin s612 sin s12 sin s612 tan s34 tan s45 consideration is just a product of propagators given by sines. Secondly, sin s12 sin s345 tan s34 tan s45 tan s67 tan s712 : (3.12) sin s12 sin s345 = 2: Here we have used (3.4) twice. draw the permutation on a circle, and then follow the points according to the other w(I6j126345) = the polygons form a loop.3 We have, for example: = 0: (3.14) with zero entries, which are easier to invert. Diagonal amplitudes Without loss of generality we can focus on the identity permutations, i.e., = ~ = In. So = 1; tan s12 tan s23 m 0 (I4jI4) = m 0 (I3jI3) = course, the m 0 ( j ~) amplitudes could still be written in the expansion of sine propagators each other. 0se). Hence, to higher multiplicities. m 0 (I5jI5) = tan s12 tan s34 tan s23 tan s45 tan s45 tan s12 tan s51 tan s23 tan s34 tan s51 poles. There is also a contribution that stays nite on all the factorization channels. It m 0 (I6jI6) = BB + 13 other CC + BB terms A terms A tan s12 tan s34 tan s56 tan s12 As we can see, the presence of the new ve-valent vertex introduced a hierarchy in the 0 ! 0 limit, as expected. There are no new vertices introduced at this stage. contact terms appearing at higher multiplicities. We nd: m 0 (I7jI7) = BB terms A + 41 other CC + BB terms A m 0 (I8jI8) = BB m 0 (I9jI9) = BB + 131 other CC + BB + 428 other CC + BB terms A 3 = Ck 1; terms depending on their leading 0 order. Finally: the diagonal amplitudes m 0 ( j ) can be calculated from a graph expansion, where each conjecture that all the remaining vertices follow this pattern, i.e., where Ck is the k-th Catalan number [34]. KLT relations show how to construct KLT relations from the objects m 0 We have veri ed validity of this construction numerically for n 10 and it remains a ( j ~) introduced in this work. conjecture for higher multiplicities. relations according to closed = A Here, we treat m 0 ( j ~) as an (n the permutations 3)! matrix with columns and rows labelled by and ~. There is no restriction on the permutations we use for the open an invertible matrix. = ~ = (1234). We can use the identity sin( z) = = (z) (1 z) to rewrite the amplitude (3.16), m 0 (I4jI4) = 0(s + t)) 0s) sin( 0u) (1 + 0u) closed = A open(1234) m 01(1234j1234) Aopen(1234) (1+ 0s+ 0t) 0u) (1+ 0u) (1+ 0s+ 0t) which is the Virasoro-Shapiro amplitude [36, 37]. We can also use any other basis for the KLT expansion, for example: or alternatively, closed = A open(1234) BB open(1324) 0t) Aopen(1234) Aopen(1324); closed = A open(1234) BB open(1243) 0s) Aopen(1234) Aopen(1243): C A closed = " open(12345)#| 6 6 open(12435) open(12435) 2 f(13254); (14253)g. Reinterpreted in our new language, the expression reads open(14253) open(14253) sin s24 sin s35 = sin( 0s23) sin( 0s45) Aopen(12345) Aopen(13254) + (3 $ 4); blocks, as in the example above. 2 f(13254); (14253)g and ~ 2 f(12354); (12435)g. We then have: closed = open(13254) open(14253) open(12354) open(12435) open(14253) 0s23) sin( sin( 0(s14 + s45)) 0s45) open(13254) sin( 0s24) Aopen(12435) A sin s24 sin s35 open(12354) open(12435) sin( 0s14) Aopen(12354) ; m 0 ( j ~) = 66 : (4.10) columns are labelled by and the rows by ~ 2 f(123456); (124356); (132456); (134256); (142356); (143256)g 2 f(153462); (154362); (152463); (154263); (152364); (153264)g. After a tedious but straightforward calculation we obtain: to calculate the inverse of the rst block: sin s12 sin s34 sin s345 sin s12 sin s345 tan s34 tan s45 sin s12 sin s345 tan s34 sin s12 sin s34 sin s345 sin s12 sin s35 sin s45 sin s12 sin s45 sin(s34 + s35) sin s12 sin s35 sin(s34 + s45) sin s12 sin s35 sin s45 closed = 0s12) sin( 0s45) Aopen(123456) sin( 0s35) Aopen(153462) 0(s34 + s35)) Aopen(154362) + P(2; 3; 4); over permutations of f2; 3; 4g. Soft limits of closed strings particles a; b 6= 1; n 2 f(1; !a; n 1; a; n)g 2 f(1; !~b; n 1; n; b)g; where !a and !~b denote the permutations of the remaining n 4 labels. We can arrange the matrix m 0 ( j ~) so that a and b label its (n 4)! blocks. Let us consider the soft limit of the particle n, pn = ! 0. We keep 0 nite. particles in both orderings . Therefore, there are no trivalent vertices involving blocks go as O( 0) is the soft limit. allowed to interact via higher-order vertices, but these give rise to nite contributions. Hence, the diagonal blocks behave as O( 1) in the soft limit. More precisely, near = 0 we have: m 0 (1; !a; n 1; a; nj1; !~b; n 1; n; b) ! m 0 (1; !a; n 1; aj1; !~a; n 1; a) + : : : ; 4)! diagonal blocks on the right hand side becomes a small inverse KLT matrix for n 1 particles. as follows [9]: open(1; !a; n 1; a; n) ! open(1; !a; n 1; a) + : : : ; string amplitude with n 1 particles. Working to leading order and neglecting constant factors, let us now collect all the terms together to obtain the result, ~ is the polarization vector of the massless state. In the last line we amplitudes [39], coinciding with the pure gravity result. Basis expansion Open string partial amplitudes can be expanded in a basis of size (n 3)! [21, 45]. This can way, utilizing the object m 0 ( j ~) introduced in this work. Let us consider an (n 3)! vector A 3)! matrix m 0 ( j ~) with open( ), and additionally (n 3)! + 1 matrix constructed from four blocks: an and ~ ranging in a set forming a BCJ basis, an 3)! transposed vector m 0 = 0; rst equality. Since the sets over which determinant of m 0 ( j ~) is non-vanishing, so we conclude that: and ~ range form a BCJ basis, the which is a basis expansion for the open string amplitude A It is a direct analogue of the expansion (2.8) for the eld-theory amplitudes. Using the open( ) in terms of a BCJ basis. following, we illustrate the expansion (5.1) with a few examples. open(1234)g. This corresponds to taking open(1243) in the one-element basis = f(1243)g, = f(1234)g, and say ~ = f(1324)g. We obtain: A slightly more involved example is: open(1243) = BB open(1324) = BB open(1234): open(1234): open(1234) C A open(1234) open(1234) inverted only once. to string theory. Making the choice f(13254); (14253)g we get: tan s23 7 6 sin s23 sin s45 sin s24 sin s35 open(14253); open(14253) open(14253) open(12354) = 66 = 6 tan s12 sin s12 sin s35 0(s12 + s23)) open(13254) can be graphically summarized in a cartoon: closed is understood as gluing of two open string partial open( ~). The object stitching the two amplitudes together is Here, the closed string amplitude Mn amplitudes, A the KLT kernel, m 01 gluing, with and ~ each ranging over a set of (n 3)! permutations. Similar picture can ( j ~). The sum proceeds over all independent ways of performing the be made to intuitively understand the change of basis relation (5.1). argued that m 0 ( j ~) can be understood as an interesting object on its own right. In fact, present some supporting evidence for this conjecture below. associated to Selberg integrals [49, 50].4 program of light on the relations between the two. 4We thank Matilde Marcolli and Oliver Schlotterer for pointing out these references to us. SUGRA = A open( ) = Z KLT also equivalent to a change 0 ! 0. In addition, it is known [8{11] that open string a string theory KLT of open string amplitudes with a small \twist," Its disk integral representation, up to an overall factor, reads: Z ( ) = vol SL(2; R) z 1; 2 z 2; 3 Note that the two orderings play di erent roles. The permutation gives a disk ordering that is inherited by the open string. The ordering enters the Parke-Taylor factor in SUGRA = ASYM KLT and (6.2) to obtain: ASYM, it is a simple linear algebra exercise to combine it with (6.1) m( j~) = Z( ) or equivalently m 0 ( j ~) = Z KLT Due to the relation (6.4), the 0-corrected bi-adjoint amplitudes m 0 disk orderings. Furthermore, using (6.2), one can show that (5.1) implies, which is distinct from the change of basis for the other permutation [9], Z ( ) = m( j~) m 1(~j") Z ("): from [62], one nds, of (6.8). P ( j ) = in this note. Acknowledgments and Science. 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Sebastian Mizera. Inverse of the string theory KLT kernel, Journal of High Energy Physics, 2017, 1-24, DOI: 10.1007/JHEP06(2017)084