Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes
N. E. J. Bjerrum-Bohr, Poul H. Damgaard, Thomas Sondergaard, Pierre Vanhove
aa r X i v : . [ h e p - t h ] M a r Preprint typeset in JHEP style - HYPER VERSION
IPHT-10-030, IHES/P/10/08
Monodromy and Jacobi-like Relations forColor-Ordered Amplitudes
N. E. J. Bjerrum-Bohr, Poul H. Damgaard and Thomas Søndergaard
Niels Bohr International Academy and DISCOVERY Center,The Niels Bohr Institute, Blegdamsvej 17,DK-2100 Copenhagen Ø, Denmark, email: { bjbohr;phdamg;tsonderg } @nbi.dk Pierre Vanhove
Institut des Hautes Etudes Scientifiques, Le Bois-Marie,F-91440 Bures-sur-Yvette, FranceandCEA, DSM, Institut de Physique Th´eorique, IPhT, CNRS, MPPU,URA2306, Saclay, F-91191 Gif-sur-Yvette, France, email: [email protected] A BSTRACT : We discuss monodromy relations between different color-ordered amplitudesin gauge theories. We show that Jacobi-like relations of Bern, Carrasco and Johansson canbe introduced in a manner that is compatible with these monodromy relations. The Jacobi-like relations are not the most general set of equations that satisfy this criterion. Applica-tions to supergravity amplitudes follow straightforwardly through the KLT-relations. Weexplicitly show how the tree-level relations give rise to non-trivial identities at loop level.K
EYWORDS : Amplitudes, Field Theory, String Theory. ontents
1. Introduction 12. Monodromy relations 33. Jacobi-like identities 7
4. String amplitudes 11
5. Monodromy and KLT relations 176. One-loop coefficient relations 19
7. Conclusion 248. Acknowledgments 24A. Evaluation of the five-point integrals 24
1. Introduction
One of the most striking aspects of string theory is the manner in which it reorganizesthe perturbative calculation of amplitudes in the field theory limit. Perhaps the most re-markable example of this is found in the Kawai-Lewellen-Tye (KLT) relations [1] that linkgauge field tree-level amplitudes based on a non-Abelian gauge group to tree-level am-plitudes in perturbative gravity. As it is based on a relationship between closed and open– 1 –trings [2], it immediately yields an even larger class of relations when considered in thecontext of superstring theory: a whole set of relations between supergravity and super-symmetry multiplets at tree level. For a comprehensive discussion, see, e.g. , the reviewby Bern [3]. These relations are puzzling from the point of view of field theory itself,although there are attempts to see their origin at the Lagrangian level [4].Recently, three of the present authors have provided another example of how stringtheory can be used to derive non-trivial amplitude relations that hold even in the fieldtheory limit, although their origin remains mysterious there [5]. The relations were con-jectured earlier by Bern-Carrasco-Johansson [6], and we shall call them BCJ-relations inwhat follows. The peculiar aspect in this case is that these BCJ-relations seemed to followfrom a new principle of Jacobi-like relations among tree-level amplitudes [6], relationsthat hold on-shell for four-point amplitudes [7], but which do not hold off-shell. Neverthe-less, imposing these Jacobi-like relations even above four-point amplitudes yields correctamplitude relations. It was subsequently shown that analogous amplitude relations can bederived for external particles of the full N = 4 hypermultiplet [8], a result that indeed alsofollows directly from the proof using superstring theory [5].To understand the significance of a new set of amplitude relations one needs to con-sider the factorial growth in n for color-ordered n -point amplitudes. For a tree-level n -point amplitude A n with legs in the adjoint representation of, say, SU ( N ) gauge group,one defines the color-ordered n -point amplitude A n (1 , . . . , n ) through A n = g n − X σ ∈ S n / Z n Tr( T a σ (1) · · · T a σ ( n ) ) A n ( σ (1 , . . . , n )) , (1.1)where g Y M is the coupling constant, and the T ’s are group generators of SU ( N ) . The re-lations we shall discuss all concern the color-ordered amplitudes A n (1 , . . . , n ) . Of course,to obtain cross sections, these must be “dressed” with the appropriate color factors andsummed. The shorter the sum, the faster will routines work that do this sum automatically.It is therefore not only of theoretical interest, but also of great practical value to have exactrelations available among the color-order amplitudes. Because of cyclicity of the ampli-tudes, the basis is not of size n ! but of size ( n − Additional non-trivial generic relationsknown before the BCJ-relations were the following. Reflections: A n (1 , . . . , n ) = ( − n A n ( n, n − , . . . , , , (1.2)the photon decoupling relation X σ A n (1 , σ (2 , . . . , n )) , (1.3)and the Kleiss-Kuijf relations [9] A n ( β , . . . , β r , , α , . . . , α s , n ) = ( − r X σ ⊂ OP { α }∪{ β T } A n (1 , σ, n ) , (1.4)– 2 –here the sum runs over the ordered set of permutations that preserves the order withineach set. Transposition on the set { β } means that order is reversed.It was shown in ref. [9, 10] that these relations reduce the basis of amplitudes from ( n − to ( n − The BCJ-relations reduce the basis down to ( n − As follows from theproof based on monodromy [5], no further reduction for arbitrary n will be possible. Afterimposition of the BCJ-relations one has thus reached the minimal basis of amplitudes.In this paper we confront some of the questions that are raised by the apparentlyvalid imposition of Jacobi-like relations among tree-level amplitudes. Given that the BCJ-relations have now been proven based on monodromy [5] a natural question is whetherthe Jacobi-like relations, conversely, follow from the BCJ-relations. Not unexpectedly, wefind that this is not the case. In fact, we find that a huge extension of these Jacobi-likerelations is possible , still leaving invariant the BCJ-relations.The paper is organised as follows. In section 2, we briefly review monodromy re-lations in string theory, and show how they give rise to string theory generalizations ofboth the Kleiss-Kuijf and BCJ-relations. Section 3 contains a discussion of the connec-tion between monodromy and Jacobi-like relations. There are clearly some issues relatedto gauge symmetry, and we choose in section 4 to consider this from the point of view ofstring theory, which automatically imposes a specific gauge choice. In section 5, we turn togravity, and consider the extended Jacobi-like identities in the light of KLT-relations. Allof these issues concern tree-level amplitudes only. In section 6, we explore what these bynow established tree-level identities imply for loop amplitudes. A straightforward way toattack this is through the use of cuts. We illustrate this in the most simple case of one-loopamplitudes in N = 4 super Yang-Mills theory and comment on applications to theorieswith less, or no, supersymmetry. Finally, section 7 contains our conclusions. Some detailsabout hypergeometric functions are relegated to an appendix.
2. Monodromy relations
In this section we will briefly recall how to derive monodromy relations for amplitudesthrough string theory. The color-ordered amplitudes on the disc are given by [2] A n ( a , . . . , a n ) = Z n Y i =1 dz i | z ab z ac z bc | dz a dz b dz c n − Y i =1 H ( x a i +1 − x a i ) Y ≤ i 4) = Z dx x α ′ k · k (1 − x ) α ′ k · k , (2.6) A (1 , , , 4) = Z ∞ dx x α ′ k · k ( x − α ′ k · k , (2.7) A (2 , , , 4) = Z −∞ dx ( − x ) α ′ k · k (1 − x ) α ′ k · k . (2.8)We indicate the contour integration from 1 to + ∞ in fig. 1. Figure 1: Thecontour of integration from1 to + ∞ . Under the assumption that α ′ k i · k j is complex and has a negative real part, we are allowedto deform the region of integration so that instead of integrating between from 1 to + ∞ – 4 –n the real axis we integrate either on a contour slightly above or below the real axis. Bya deformation of each of the contours, one can convert the expression into an integrationfrom −∞ to 1. One needs to include the appropriate phases each time x passes through y = 0 or y = 1 (when rotating the contours), ( x − y ) α = ( y − x ) α × ( e + iπ α for clockwise rotation ,e − iπ α for counterclockwise rotation . One can thus deform the integration region in two equivalent ways I + and I − , see fig. 2. e − iα ′ π ( k · k ) e − iα ′ πk · ( k + k ) e iα ′ π k · ( k + k ) e iα ′ π ( k · k ) Figure 2: The contours I + and I − . We have I + = I − = A (1 , , , . If now I + is multiplied by e iα ′ πk · ( k + k ) and I − by e − iα ′ πk · ( k + k ) we get for the contours as illustrated in fig. 3. We thus have I + e iα ′ πk · ( k + k ) − I − e − iα ′ πk · ( k + k ) = 2 i A (1 , , , 4) sin(2 α ′ π k · ( k + k )) . How-ever, the contour obtained after subtracting these two contours can also be interpreted as infig. 4. This is equal to − i A (1 , , , 4) sin(2 α ′ πk · k ) . In this way we arrive at the fol-lowing monodromy relation: sin(2 πα ′ k · k ) A (1 , , , 4) = sin(2 πα ′ k · k ) A (1 , , , where we have used momentum conservation and the on-shell condition. For other exter-nal states of higher spin, the integrals change appropriately to restore the identities (in-cluding sign factors for the fermionic statistics of half-integer spins). e iα ′ π ( k · k ) No phase No phase e − iα ′ π ( k · k ) Figure 3: The contours I + and I − after multiplying with phases e iα ′ πk · ( k + k ) and e − iα ′ πk · ( k + k ) . – 5 – sin( − α ′ π ( k · k )) Figure 4: Another interpretation of the two contours. By deforming the contour of integration of A (2 , , , one finds in an equivalentfashion: sin(2 πα ′ k · k ) A (1 , , , 4) = sin(2 πα ′ k · k ) A (2 , , , . This implies thatall the amplitudes can be related to the A (1 , , , A (1 , , , 4) = sin(2 πα ′ k · k )sin(2 πα ′ k · k ) A (1 , , , , A (1 , , , 2) = A (2 , , , 4) = sin(2 πα ′ k · k )sin(2 πα ′ k · k ) A (1 , , , . (2.9)Taking the limit α ′ → , we get the following relations between the field theory amplitudes A (1 , , , 4) = k · k k · k A (1 , , , ,A (1 , , , 2) = A (2 , , , 4) = k · k k · k A (1 , , , . (2.10)The string theory relations can immediately be checked to hold based on the explicit stringamplitude expression. In the low energy limit, the corresponding relations (2.10) coincidewith those of ref. [6].As shown in ref. [5], one has the following n -point amplitude relations: A n ( β , . . . , β r , , α , . . . , α s , n ) = ( − r × ℜ e hY ≤ i 5) + S k ,k A (1 , , , , , S k ,k + k A (1 , , , , − S k ,k A (1 , , , , 5) + S k ,k A (1 , , , , , S k ,k + k A (1 , , , , − S k ,k A (1 , , , , 5) + S k ,k A (1 , , , , , S k ,k + k A (1 , , , , − S k ,k A (1 , , , , 5) + S k ,k A (1 , , , , . (2.13)Here we have used the notation S p,q ≡ sin(2 α ′ π p · q ) . There are of course various waysof writing these monodromy relations, but they reduce to just four independent equations.One can immediately verify these relations from the explicit form of the tree amplitudesin string theory given by [14–17]. In the field theory limit they reduce to relations that areequivalent to those discussed in ref. [6]. 3. Jacobi-like identities The field theory limit of the monodromy relations were originally conjectured on the basisof an observation for the four-point gluon amplitudes [6]. We start by briefly reviewingthe argument. At four points, the photon decoupling identity reads A (1 , , , 4) + A (2 , , , 4) + A (2 , , , 4) = 0 . (3.1)It holds independently of polarization and external on-shell momenta. The natural waythis identity can be satisfied is through A (1 , , , 4) + A (2 , , , 4) + A (2 , , , 4) = χ ( s + t + u ) = 0 , (3.2)with χ being a common factor .In the amplitude A (1 , , , both pairs of legs (1,2) and (1,4) are adjacent, and weshould thus treat the s and t factors on the same footing. The contribution of this colorordering to eq. (3.2) must therefore be A (1 , , , 4) = − χ ( s + t ) = χu . (3.3)Likewise, one is led to A (2 , , , 4) = χt, A (2 , , , 4) = χs . (3.4) We will discuss the explicit expression for χ in the case of vector particles in section 4. – 7 –liminating χ one obtains tA (1 , , , 4) = uA (2 , , , , sA (1 , , , 4) = uA (2 , , , ,sA (2 , , , 4) = tA (2 , , , . (3.5)These are of course just the monodromy relations eq. (2.10). To proceed further, onecan parameterize the three subamplitudes in terms of their possible pole structures andunspecified numerators A (1 , , , 4) = n s s + n t t , (3.6) A (2 , , , 4) = − n u u − n s s , (3.7) A (2 , , , 4) = − n t t + n u u . (3.8)It follows from (3.5) that n u − n s + n t = 0 . This resembles the Jacobi identity for theassociated color factors. Bern, Carrasco and Johansson [6] took as hypothesis that thiscan be extended iteratively for general n -point amplitudes. This is equivalent to assumingthat one can choose a parametrization in which Jacobi relations for numerator factors canbe imposed in one-to-one correspondence with the genuine Jacobi identities for the colorfactors. Imposing this hypothesis gets quite involved as n grows, but it can be carriedthrough systematically; for details see ref. [6]. This leads to the BCJ-relations [6]. Thesame principle can be used to generate relations for scalar and fermionic matter in the ad-joint representation [8]. We of course now understand that this is because the monodromyrelations hold for the full N = 4 supermultiplet in four dimensions [5].Since the BCJ-relations have been proven [5], one would like to understand the mean-ing of these Jacobi-like identities for the numerators. In the four-point case the identitiesare exact, but only on-shell [7]. Even if the theory in question had only three-point vertices(which it does not) so that all n -point tree-level amplitudes for n ≥ could be constructedby gluing three-point vertices on a four-point function (thus having at least one leg off-shell on all four-point sub-diagrams), this would represent a puzzle. How can this startingpoint then lead to correct amplitude identities? To see what is going on it suffices to focus on the 5-point case. We will simply deriveexactly what follows directly from the field theory BCJ-relations when expressed in termsof the pertinent set of poles for each color-ordered amplitude. We use the parametrization A (1 , , , , 5) = n s s + n s s + n s s + n s s + n s s , (3.9) A (1 , , , , 5) = n s s + n s s + n s s + n s s + n s s , (3.10) A (1 , , , , 5) = n s s − n s s + n s s − n s s + n s s , (3.11)– 8 – (1 , , , , 5) = n s s + n s s − n s s + n s s − n s s , (3.12) A (1 , , , , 5) = n s s − n s s − n s s − n s s − n s s , (3.13) A (1 , , , , 5) = n s s − n s s − n s s − n s s − n s s . (3.14)This can be easily illustrated by diagrams involving only anti-symmetric three-vertices.However, since the coefficients n i may depend on the kinematic variables (and thus cancelpoles) there is no assumption of only three-vertices here. The listed subamplitudes arerelated through the monodromy relations in the field limit of (2.13), i.e. , s + s ) A (1 , , , , − s A (1 , , , , 5) + s A (1 , , , , , (3.15) s + s ) A (1 , , , , − s A (1 , , , , 5) + s A (1 , , , , , (3.16) s + s ) A (1 , , , , − s A (1 , , , , 5) + s A (1 , , , , , (3.17) s + s ) A (1 , , , , − s A (1 , , , , 5) + s A (1 , , , , . (3.18)Plugging the expressions for the amplitudes in terms of the n i ’s into (3.15)–(3.18) weimmediately obtain:1. From (3.15) n − n + n s − n − n + n s − n − n + n s − n − n + n s , (3.19)2. From (3.16) n − n + n s − n − n + n s − n − n + n s − n − n + n s , (3.20)3. From (3.17) n − n + n s + n − n + n s − n − n + n s + n − n + n s , (3.21)4. From (3.18) n − n + n s − n − n + n s − n − n + n s − n − n + n s . (3.22)We thus see that the BCJ-relations can be written as kind of extended Jacobi identitieswhen expressed in terms of the numerators. Let us simplify the notation a bit by denotingthe nine numerator combinations as X ≡ n − n + n , X ≡ n − n + n , X ≡ n − n + n ,X ≡ n − n + n , X ≡ n − n + n , X ≡ n − n + n ,X ≡ n − n + n , X ≡ n − n + n , X ≡ n − n + n . (3.23)– 9 –ur four equations then take the form X s − X s − X s − X s , (3.24) X s − X s − X s − X s , (3.25) X s + X s − X s + X s , (3.26) X s − X s − X s − X s . (3.27)These four equations describe the general constraints on the numerator factors dictated bythe monodromy relations at five points. As long as these equations are satisfied we havenumerator identities leading to eq. (3.15)–(3.18). Of course, the simplest solution is to putall X i = 0 , but this is clearly not the most general solution. To make the amount of freedom one has in the above parametrization of subamplitudesmore clear, let us write the most general solution by means of five arbitrary functions f , f , f , f and f X ≡ s f , X ≡ s f , X ≡ s f , X ≡ s f , X ≡ s f , (3.28) i.e. from eq. (3.24)–(3.27) X ≡ s f , X ≡ s f , X ≡ s f ,X ≡ s f , X ≡ s f , X = s ( f − f + f ) ,X = s ( f − f + f ) , X = s ( f − f − f ) , X = s ( f − f − f ) . (3.29)Note that we have used the canonical set of kinematic variables (generalized Mandelstamvariables for the 5-point case) s , s , s , s , s in our definition of the f i . The s ij occuring in the expression for X , X , X and X are related to this canonical set by s = s − s − s , s = s − s − s ,s = s − s − s , s = s − s − s . (3.30)The freedom we have to generalize the solution, i.e. eq. (3.29), is not just relatedto gauge degrees or the freedom to absorb contact terms. It can be seen as the trivialfreedom to add a “zero” to the subamplitude and forcing it into a parametrization of theform eq. (3.9)–(3.14).As a simple example, imagine that we add g − g to eq. (3.9), with g being anarbitrary function. We can then absorb the g ’s in n and n , i.e. A (1 , , , , 5) = ( n + s s g ) s s + n s s + ( n − s s g ) s s + n s s + n s s . (3.31)– 10 –n no other amplitude than A (1 , , , , does n appear, however, n appears in eq. (3.12)so we add g − g to the amplitude, and absorb in the following way: A (1 , , , , 5) = ( n − s s g ) s s + n s s − ( n − s s g ) s s + n s s − n s s . (3.32)We have thereby redefined n , n and n n → n + s s g , (3.33) n → n − s s g , (3.34) n → n − s s g , (3.35)which changes X , X and X X = s f → s ( f − s g ) ≡ s f ′ , (3.36) X = s f → s ( f − ( s + s + s ) g ) = s ( f − s g ) ≡ s f ′ , (3.37) X = s f → s ( f − s g ) ≡ s f ′ , (3.38)and we now have X = s f ′ , X = s f ′ , X = s f ′ ,X = s f , X = s f , X = s ( f ′ − f ′ + f ) ,X = s ( f ′ − f ′ + f ) , X = s ( f ′ − f ′ − f ) , X = s ( f ′ − f ′ − f ) . (3.39)This trivial addition of zeros to the amplitudes illustrates the fact that we can find manydifferent representations of the numerators, all of which are perfectly consistent with themonodromy relations. The freedom is that of general reparametrizations of the amplitudeand not just gauge symmetry. 4. String amplitudes Let us consider tree-level open string amplitudes in superstring theory. We have alreadygiven the needed formulas in section 2. We first focus on the color-ordered four-pointamplitude for vector particles A σ = Z D σ dz | z | α ′ k · k | − z | α ′ k · k ˜ F ( z ) , (4.1)where the domain of integration D σ for each color ordering are given by D = { ≤ z ≤ } , D = { ≤ z } , D = { z ≤ } . Expanding the function ˜ F in (2.5)leads to ˜ F ( y ) = a y + b y − , (4.2) This can be derived with a very tedious expansion [18] of the expression in eq. (2.5). The simplicity ofthe expansion appears naturally in the pure spinor formalism [19,20]. The tilde on F n indicates that we havefixed the three conformal points in the expression. – 11 –here a and b are expressed in terms of the polarizations and the momenta. Their ex-pressions are particularly long but there is a relation between the two coefficients s b − t a = α ′ t m ··· m F m m F m m F m m F m m , (4.3)where F i are the field-strengths corresponding to the external legs. The tensor t is con-tracting the Lorentz indices as defined in appendix 9.A of [2] (it is common to define χ = t m ··· m F m m F m m F m m F m m / ( stu ) ). The quantity a and b are not gauge in-variant but the combination in (4.3) is gauge invariant.For the four-point color-ordered amplitudes we find A (1 , , , 4) = Φ , ( α ′ s, α ′ t ) (cid:18) − a α ′ s + b α ′ t (cid:19) , (4.4) A (1 , , , 4) = Φ , ( α ′ u, α ′ t ) (cid:18) − a + b α ′ u − b α ′ t (cid:19) , (4.5) A (2 , , , 4) = Φ , ( α ′ s, α ′ u ) (cid:18) a α ′ s + a + b α ′ u (cid:19) , (4.6)where we introduced the hypergeometric functions Φ , ( α ′ s, α ′ t ) ≡ F ( − α ′ s, α ′ t ; 1 − α ′ s ; 1) = Γ(1 − α ′ s )Γ(1 − α ′ t )Γ(1 + α ′ u ) . (4.7)In the convention of BCJ [6], n s = − a /α ′ , n t = − b /α ′ , n u = − ( a + b ) /α ′ , (4.8)we immediately obtain the exact relation n u = n t − n s . Let us now consider the five point amplitude. Having fixed the position vertex operatorsat positions z = 0 , z = 1 and z = ∞ , the integrand takes the compact form [20] A σ = Z D σ dz dz Y i 3) = 0 , J (4 , , 3) = 0 . (4.12)In the amplitude (4.9) we have made explicit the poles C , and C , and C , and C , .This freedom corresponds to local monodromy transformations exchanging the posi-tion of neighboring vertex operators. There are as well global monodromy transformationsgiven by moving vertex operators from one side of the line to the other side which are notcaptured by these local transformations.The 12 color-ordered five-point amplitudes are given by specifying the range of inte-gration over z and z over the following domains of integrations D σ D = { ≤ z ≤ z ≤ } ,D = { ≤ z ≤ z ≤ } ,D = { ≤ z ≤ ≤ z } ,D = { ≤ z ≤ ≤ z } ,D = { ≤ ≤ z ≤ z } ,D = { ≤ ≤ z ≤ z } ,D = { z ≤ ≤ z ≤ } ,D = { z ≤ ≤ z ≤ } ,D = { z ≤ z ≤ } ,D = { z ≤ z ≤ } ,D = { z ≤ ≤ ≤ z } ,D = { z ≤ ≤ ≤ z } . (4.13)We now use the result for I ( a, b, c, d, e ) which is given in the appendix A. The integralsare explicitly evaluated in appendix A. We here quote the field theory results. In the fieldtheory limit α ′ → we get A (1 , , , , 5) = As s + B − Gs s s + Cs s + E + Gs s s + D − Gs s s , (4.14) We have ( n − / such domains corresponding to the different ( n − color-ordered amplitudesdivided by 2 by reflection. – 13 – (1 , , , , 5) = A − E − Fs s − D − Gs s s + − Fs s − D − Cs s + B − Ds s , (4.15) A (1 , , , , 5) = A − Cs s + B − Ds s − Cs s + F + B − Ds s − D − Gs s s , (4.16) A (1 , , , , 5) = A − E − Gs s s − B − Gs s s − − Fs s − E + Gs s s − B − Ds s , (4.17) A (1 , , , , 5) = D − C + A − E − Fs s + D − Gs s s + B − E + Gs s s + D − Cs s + B − Gs s s , (4.18) A (1 , , , , 5) = D − C + A − F − B − Gs s s − B − Ds s − B − E + Gs s s − F + B − Ds s − B − Gs s s . (4.19)It is interesting to note that we could use monodromy relations for integrals on the indi-vidual A , B , C etc. terms in (4.9). Thereby one would obtain the same relations as forthe full subamplitudes, but now just for the individual terms. Hence, the OPEs provide uswith expressions for the subamplitudes in which the relations are very explicitly reducedto relations in the pole structure. This can also be checked explicitly for the five-point caseby use of (4.14)–(4.19). In (4.14)–(4.19) we already wrote the amplitudes in terms of double poles. The quantities A to F were naturally put into the double-pole form, but the G term, a single-pole term,was forced into this representation by making a specific choice. Later we will come backto the freedom in absorbing the G terms, but for now we just consider the form givenabove.Comparing with Bern, Carrasco and Johansson’s [6] parametrization ( i.e. (3.9)–(3.14))we identify from (4.14)–(4.19) n = A , n = D − C + A − E − F , n = B − D ,n = B − Gs , n = B − E + Gs , n = A − C ,n = C , n = D − C , n = F + B − D ,n = E + Gs , n = A − E − F , n = D − C + A − F − B − Gs ,n = D − Gs , n = − F , n = A − E − Gs . (4.20)– 14 –he Jacobi-like identities then take the form X = n − n + n = Gs ,X = n − n + n = 0 ,X = n − n + n = 0 ,X = n − n + n = − Gs ,X = n − n + n = 0 ,X = n − n + n = 0 ,X = n − n + n = 0 ,X = n − n + n = − Gs ,X = n − n + n = 0 . (4.21)And from (3.24)–(3.27) it is easy to see that these amplitudes do indeed satisfy the BCJ-relations. Moreover not all X i ’s vanish.Note that the BCJ-relations could also be derived from (4.14)–(4.19) by expressing,for instance, A and B in terms of two subamplitudes and the C to G terms. Using theseexpressions for A and B in the remaining amplitudes leads directly to BCJ-relations (the C to G terms vanish after the substitution). There are many ways of arranging the G terms into the numerators of double poles. Theexpressions given above correspond to just one specific choice. To see this more clearlylet us begin by defining ˜ n i ’s ˜ n = A , ˜ n = D − C + A − E − F , ˜ n = B − D , ˜ n = B , ˜ n = B − E , ˜ n = A − C , ˜ n = C , ˜ n = D − C , ˜ n = F + B − D , ˜ n = E , ˜ n = A − E − F , ˜ n = D − C + A − F − B , ˜ n = D , ˜ n = − F , ˜ n = A − E . (4.22)The amplitudes can then, in all generality, be represented like A (1 , , , , ≡ ˜ n + Gg s s + ˜ n + Gg s s + ˜ n + Gg s s + ˜ n + Gg s s + ˜ n + Gg s s , (4.23)– 15 – (1 , , , , ≡ ˜ n + Gg s s + ˜ n + Gg s s + ˜ n + Gg s s + ˜ n + Gg s s + ˜ n + Gg s s , (4.24) A (1 , , , , ≡ ˜ n + Gg s s − ˜ n + Gg s s + ˜ n + Gg s s − ˜ n + Gg s s + ˜ n + Gg s s , (4.25) A (1 , , , , ≡ ˜ n + Gg s s + ˜ n + Gg s s − ˜ n + Gg s s + ˜ n + Gg s s − ˜ n + Gg s s , (4.26) A (1 , , , , ≡ ˜ n + Gg s s − ˜ n + Gg s s − ˜ n + Gg s s − ˜ n + Gg s s − ˜ n + Gg s s , (4.27) A (1 , , , , ≡ ˜ n + Gg s s − ˜ n + Gg s s − ˜ n + Gg s s − ˜ n + Gg s s − ˜ n + Gg s s , (4.28)where the g i ’s are new parameters representing the fractions of the G terms absorbed intothe specific double poles. Since these expressions must equal (4.14)–(4.19) in order toexpress the actual amplitudes, we get six equations constraining the g i parameters s s s − s s s − s = g s s + g s s + g s s + g s s + g s s , (4.29) s s s − s s s − s = g s s + g s s + g s s + g s s + g s s , (4.30) s = g s s − g s s + g s s − g s s + g s s , (4.31) s = g s s + g s s − g s s + g s s − g s s , (4.32) s s s − s s s − s = g s s − g s s − g s s − g s s − g s s , (4.33) s s s − s s s − s = g s s − g s s − g s s − g s s − g s s . (4.34)Any solution to these equations give a valid distribution of the G terms, i.e. provide uswith a representation of the form (3.9)–(3.14) that satisfy (3.24)–(3.27).– 16 –he representation written out explicitly in (4.14)–(4.19) corresponds to the solution g = 0 , g = 0 , g = 0 ,g = − s , g = s , g = 0 ,g = 0 , g = 0 , g = 0 ,g = s , g = 0 , g = − s ,g = − s , g = 0 , g = − s . (4.35)A numerical check have shown that there do exits solutions for g i such that the nineJacobi identities ( n i − n j + n k = 0 ) are satisfied, and in such a way that four of the g i ’scan be chosen arbitrarily. This correspond to the freedom Bern, Carrasco and Johanssonfind in choosing their α , α , α and α arbitrarily.An example of a (valid) choice of g i ’s which generate n i ’s that satisfy the Jacobiidentities is g = − s , g = − s , g = 0 ,g = − s − s , g = − s , g = 0 ,g = − s , g = − s , g = 0 ,g = − s , g = 0 , g = 0 ,g = − s − s , g = 0 , g = 0 , (4.36)with, e.g. n − n + n = (˜ n − ˜ n + ˜ n ) + G ( g − g + g )= ( C − D + D − C ) + G ( − s − ( − s − s ) − s )= 0 , etc . . . (4.37)From the expansion given by the OPE this might not be the most simple or natural wayof absorbing the G terms into double-poles, but it does show that the assumption of Bern,Carrasco and Johansson is allowed for (at least) the five-point case. 5. Monodromy and KLT relations As a direct application of the monodromy relations in Yang-Mills theory, we can rewritethe Kawai-Lewellen-Tye relations at four-point level in the following manner M = κ α ′ S k ,k S k ,k S k ,k A L (1 , , , A R (1 , , , . (5.1)The field theory limit of the string amplitude (5.1), α ′ → gives the symmetric form ofthe gravity amplitudes of [6] M = 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) . (5.2)– 17 –ere we have made use of the on-shell relation s + t + u = 0 and the four-point Jacobirelation n u = n s − n t .At five point order Bern, Carrasco and Johansson [6] showed that if the subamplitudesare parameterized by numerators like in eqs. (3.9)–(3.14), and we assume the numeratorssatisfy the Jacobi-like identities, then the KLT relation − iM (1 , , , , 5) = s s A (1 , , , , e A (2 , , , , s s A (1 , , , , e A (3 , , , , , (5.3)implies the following form of M − iM (1 , , , , 5) = n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s + n ˜ n s s . (5.4)If we instead use the more general solution for A and e A , i.e. X ≡ s f , X ≡ s f , X ≡ s f ,X ≡ s f , X ≡ s f , X = s ( f − f + f ) ,X = s ( f − f + f ) , X = s ( f − f − f ) , X = s ( f − f − f ) , (5.5)and e X ≡ s g , e X ≡ s g , e X ≡ s g , e X ≡ s g , e X ≡ s g , e X = s ( g − g + g ) , e X = s ( g − g + g ) , e X = s ( g − g − g ) , e X = s ( g − g − g ) . (5.6)Here X = n ′ − n ′ + n ′ and e X = ˜ n ′ − ˜ n ′ + ˜ n ′ , see eq. (3.23), and we obtain − iM (1 , , , , 5) = n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s + n ′ ˜ n ′ s s − (cid:2) f g + f g + f g + f g + f g + f ( g − g ) + g ( f − f )+ f ( g − g ) + g ( f − f ) − f g − g f (cid:3) . (5.7)– 18 –his representation of the gravity is of course guaranteed to be exact due to the KLT-construction. We obtain the simple factorized form (5.4) only when we choose f g + f g + f g + f g + f g + f ( g − g ) + g ( f − f )+ f ( g − g ) + g ( f − f ) − f g − g f = 0 . (5.8)This is evidently satisfied when the numerators fulfill the simple Jacobi-like relations.However, more general parameterizations are consistent with this equation as well. Forinstance, eq. (4.14)–(4.19) implies f = G, f = − G, and f = f = f = 0 , (5.9)and using the same parametrization for e A , eq. (5.8) is seen to be satisfied: f g + f g + f g + g f = G + G − G − G = 0 . (5.10)Again, the freedom in choosing different representations of the KLT-relations arise fromthe freedom to pick parameterizations of the gauge invariant amplitudes in terms of dif-ferent pole structures. These pole structures are not gauge invariant by themselves and wesee that this arbitrariness in the gauge theory is inherited in the gravity amplitude. 6. One-loop coefficient relations We end this paper with an obvious application of the monodromy relations in the fieldtheory limit. We illustrate how these relations can imply relations between coefficientsof integrals in one-loop gluon amplitudes. For simplicity we will focus on amplitudes in N = 4 super Yang-Mills, but it will be evident that most of the considerations here willapply also to the case of less supersymmetric or even non-supersymmetric amplitudes. Our starting point will be the one-loop gluon amplitudes which can be color decom-posed [21] as follows A − loopn = g n [ n/ X c =1 X σ ∈ S n /S n ; c Gr n ; c ( σ ) A n ; c ( σ ) . (6.1)Here [ x ] is the largest integer less than or equal to x . The leading color factor is Gr n ;1 ( σ ) = N c Tr( T a σ (1) · · · T a σ ( n ) ) , (6.2)and the subleading color factors ( c > ) are Gr n ; c ( σ ) = Tr( T a σ (1) · · · T a σ ( c − )Tr( T a σ ( c ) · · · T a σ ( n ) ) . (6.3)– 19 – n here denotes the set of all permutations of n objects. S n ; c is the subset leaving Gr n ; c invariant.It is sufficient to consider the subamplitude A n ;1 which is leading in color counting,since the remaining A n ; c subamplitudes with c > can be obtained as a sum over differentpermutations of A n ;1 [21, 22].In N = 4 super Yang-Mills theory we can always write the one-loop gluon amplitude(using a Passarino-Veltman reduction [23]) as a linear combination of scalar box integralswith rational coefficients [22, 24]. For the leading subamplitude the expression becomes A n ;1 = X (cid:16)b bI m + b cI m e + b dI m h + b gI m + b f I m (cid:17) . (6.4)Here the sum runs over color-ordered box diagrams, and the integrals (defined in dimen-sional regularization) are given by I = − i (4 π ) − ǫ Z d − ǫ l (2 π ) − ǫ l ( l − K ) ( l − K − K ) ( l + K ) . (6.5)The external momenta K i are given by the sum of momenta of consecutive external legs,and all momenta are taken to be outgoing. The labels m, m, m and m refer to thenumber of “massive” corners, i.e. the number of K i = 0 . This is equivalent to the numberof corners with more than one external gluon. The m case is separated into adjacentmassive corners I m h ( h for hard), and diagonally opposite massive corners I m e ( e foreasy).Since the scalar box integrals are all known explicitly [24], calculation of one-loopamplitudes is reduced to finding the coefficients. From that general setting the existenceof relations between coefficients of different one-loop amplitudes is surprising. The indi-cation of such structures does not appear until we introduce unitarity cuts [22,25]. Workingin complex momenta it is possible to do quadruple cuts and derive formulas for generalcoefficients [26] b a α = 12 X S,J n J A tree1 A tree2 A tree3 A tree4 . (6.6)Here α represent a specific ordering of external legs, J the spin of a particle (running inthe loop) in the N = 4 multiplet, n J the number of particles in the multiplet with spin J and S is the set of the two solutions to the on-shell conditions S = { l | l = 0 , ( l − K ) = 0 , ( l − K − K ) = 0 , ( l + K ) = 0 } . (6.7)It turns out that for many amplitudes eq. (6.6) simplifies significantly. The helicityconfiguration often kills the sum over non-gluonic states and one of the S solutions. Thesecoefficients are therefore only given by a single term of four tree-level gluon amplitudesmultiplied together. Monodromy relations on these tree amplitudes then leads to relationsamong coefficients for one-loop amplitudes. Most interesting is probably the possibilityof relating coefficient for split-helicity loop amplitudes to mixed-helicity loop amplitudes.For some reviews of the work at tree and loop level involving helicity amplitudes forgluons, see e.g. refs. [27–29]. – 20 – .2 Six-point examples In the following section we give two explicit examples of how the monodromy relations, incombination with unitarity cuts, can be used to obtain relations between scalar box integralcoefficients of different one-loop amplitudes. These should be sufficient to get the idea formore general one-loop amplitudes. Let us begin by considering the b c coefficient to the A (1 + , − , − , + , + , + ) one-loopamplitude, i.e. the coefficient to the I m e integral for a specific ordering of the legs. Herewe choose the one illustrated in fig. 5. Note that with this helicity configuration fig. 5 isthe only diagram that contributes to b c . Any other assignment of helicities to the loop-legsmakes at least one of the corners vanish. In addition, only gluons can run in the loop forthis helicity configuration – fermions and scalars would make the two corners with equalhelicity vanish. − − + + + + +++ + −− −− l l l l Figure 5: Two-mass (easy) cut diagram. Since the four corners are just given by the appropriate (on-shell) tree-level ampli-tudes, we can use the four-point monodromy relations to flip the legs around. One of theadvantages of the monodromy relations is that we can always keep two of the legs fixed.This is important here since we do not want to change the position of legs in the loop. Thediagram in fig. 5, which we denote D m e , is therefore related to the diagram of same type,but with legs 1 and 2 interchanged, through D m e = s ( − l )1 s l D m e . (6.8)The helicity configuration (+ + − ) of the two three-point corners is only consistent withone of the S solutions [26], and the coefficient is simply given by b c = D m e / . The same– 21 –s of course true in the case of leg 1 and 2 interchanged, which imply that b c = s ( − l )1 s l b c ′ , (6.9)where b c ′ is the coefficient to the I m e scalar box integral for the one-loop amplitude A (2 − , + , − , + , + , + ) . This is a very simple relation between coefficients for split-helicity and mixed-helicity loop amplitudes.For completeness, we show how to solve for the loop-momenta and express the frac-tion in front of b c ′ solely in terms of external momenta. For this we will be using the spinorhelicity formalism. From momentum conservation and on-shell conditions we have l = l − p − p , ( l − p − p ) = 0 ,l = l − p = l − p − p − p , ( l − p − p − p ) = 0 ,l = l − p − p = l + p , ( l + p ) = 0 , (6.10)and in terms of spinor products s ( − l )1 s l = s ( − l )1 s ( − l )2 = h l i [ l h l i [ l . (6.11)Since the three-point corners have helicity configuration (+ + − ) we must take the holo-morphic spinors at these corners to be proportional and hence having vanishing h•i prod-uct (remember, we are working with complex momenta, so the [ • ] product can be non-vanishing). In particular we get h l i = 0 = ⇒ | l i = α | i . (6.12)The proportionality factor α can be obtained from ( l − p − p ) = 0 = ⇒ l · ( p + p ) = ( p + p ) , (6.13)and since l · ( p + p ) = h l | | l ] = α h | | l ] , α = ( p + p ) h | | l ] . (6.14)To express the anti-holomorphic spinor of l we use ( l − ( p + p + p )) = 0 = ⇒ l · ( p + p + p ) = ( p + p + p ) , (6.15)and l · ( p + p + p ) = h l | | l ] = α h | | l ] , (6.16)from which follows ( p + p ) h | | l ] = h | | l ]( p + p + p ) ⇐⇒ (cid:2) ( p + p ) h | (1 + 2 + 3) − ( p + p + p ) h | (1 + 2) | {z } ≡ [ γ | (cid:3) | l ] = 0 , (6.17)– 22 – .e. | l ] = β | γ ] . We are not interested in the proportionality factor β since it cancels outfrom eq. (6.11) anyway. Using these expressions for the spinors of l , we get, after a bit ofrewriting, s ( − l )1 s l = − h ih ih ih i . (6.18) Let us now consider a one-mass box integral coefficient. As in the example above we justuse the A (1 + , − , − , + , + , + ) one-loop amplitude to illustrate the idea. The diagramis given in fig. 6, which we denote as D m . Again this helicity configuration kills all otherdiagrams and allow only gluons to run in the loop. − − + + + + ++ + + −− −− l l l l Figure 6: One-mass cut diagram. This time we can use the five-point monodromy relations to connect a diagram ofmixed helicity to two diagrams of split helicities D m = ( s + s ( − l )1 ) D m + s ( − l )1 D m s l , (6.19)with obvious notation for the different diagrams. Like above, the coefficients related tothese diagrams only consist of these single terms, and we can therefore equally well writeit as b b = ( s + s ( − l )1 ) b b + s ( − l )1 b b s l , (6.20)where we have a one-mass integral coefficient belonging to the mixed-helicity amplitude A (2 − , + , − , + , + , + ) related to one-mass coefficients of the split-helicity ampli-tudes A (1 + , − , − , + , + , + ) and A (6 + , − , − , + , + , + ) .Using very similar methods as for the two-mass case we could again express the kine-matic invariants in terms of external momenta. However, this is not our focus here.– 23 – . Conclusion We have reconsidered the BCJ-relations in gauge theories from several points of view.Based on the monodromy proof, we have explored the extent to which Jacobi-like relationsfor residues of poles (and multiple poles) can be derived . We have found that Jacobi-likerelations can be introduced consistently with the constraints of the monodromy relations.But extended Jacobi-like identities are also perfectly consistent with the gauge invariantrelations. We have demonstrated this explicitly from both field and string theoretic angles.We have also considered the implications for gravity amplitudes. Very symmetricforms follows in a simple manner through using the KLT-relations together with the linkposed by monodromy in the gauge theory side. This direction appears worthwhile topursue in the future.As an application of monodromy relations, we have explicitly illustrated how thesetree-level relations give rise to non-trivial identities at loop level. The simplest case is thatof N = 4 super Yang-Mills theory where relations between one-loop box functions aredirectly derivable through quadruple cut techniques. Similar considerations are valid forless supersymmetric or non-supersymmetric amplitudes as well, although in such cases therelations are rather more complicated. There are thus clearly several interesting directionsfor future work that will exploit these relations. 8. Acknowledgments NEJBB and TS would like to acknowledge financial support from the Danish Council forIndependent Research (FNU) and the L´eon Rosenfeld Foundation, respectively. A. Evaluation of the five-point integrals In this appendix we evaluate the five point amplitudes (4.9) for the ordering (1 , , , , .We use the result I ( a, b, c, d, e ) = Z dz Z z dz z a ( z − z ) b (1 − z ) c (1 − z ) d z e (A.1) = Γ( a + 1)Γ( b + 1)Γ( d + 1)Γ( a + b + e + 2)Γ( a + b + 2)Γ( a + b + d + e + 3) × F ( a + 1 , − c, a + b + e + 2; a + b + 2 , a + b + d + e + 3; 1) , that expresses the integral in terms of the hypergeometric function F . We introduce thenotation I ( a, b, c, d, e ) = Γ( α ′ s + a + 1)Γ( α ′ s + b + 1)Γ( α ′ α ′ s + d + 1)Γ( α ′ s + a + b + e + 2)Γ( s , + a + b + 2)Γ( α ′ s , + a + b + d + e + 3) × F ( α ′ s + a +1 , − s − c, α ′ s + a + b + e +2; α ′ s , + a + b +2 , α ′ s , + a + b + d + e +3; 1) , (A.2)– 24 –etting ˆ s i,j = α ′ s i, we have Contribution A The integral is I ( − , , , , − 1) = 1ˆ s , ˆ s , Γ(ˆ s , + 1)Γ(ˆ s , + 1) Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + 1)Γ(ˆ s , + ˆ s , + ˆ s , + ˆ s , + 1) × F (ˆ s , , − ˆ s , , ˆ s , + ˆ s , + ˆ s , ; ˆ s , + ˆ s , + 1 , ˆ s , + ˆ s , + ˆ s , + ˆ s , + 1; 1) , (A.3) Contribution B I (0 , − , − , , 0) = 1ˆ s , ˆ s , Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + 1)Γ(ˆ s , + ˆ s , + 1) h F (ˆ s , + 1 , − ˆ s , , ˆ s , + 1; ˆ s , + ˆ s , + 1 , ˆ s , + ˆ s , + 1; 1) − ˆ s , (ˆ s , + 1)(ˆ s , + ˆ s , + 1)(ˆ s , + ˆ s , + 1) F (ˆ s , +1 , − ˆ s , , ˆ s , +2; ˆ s , + ˆ s , +2 , ˆ s , + ˆ s , +2; 1) i , (A.4) Contribution C I ( − , , , − , 0) = 1ˆ s , Γ(ˆ s , + 2)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + ˆ s , + 3)Γ(ˆ s , + ˆ s , + 3)Γ(ˆ s , + ˆ s , + ˆ s , + ˆ s , + 3) × F ( − ˆ s , , ˆ s , + 2 , ˆ s , + ˆ s , + ˆ s , + 3; ˆ s , + ˆ s , + 3 , ˆ s , + ˆ s , + ˆ s , + ˆ s , + 3; 1) , (A.5) Contribution D I (0 , , − , − , 0) = 1ˆ s , × Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + ˆ s , + 2)Γ(ˆ s , + ˆ s , + 2)Γ(ˆ s , + ˆ s , + ˆ s , + ˆ s , + 2) × F (ˆ s , + 1 , ˆ s , + ˆ s , + ˆ s , + 2 , − ˆ s , ; ˆ s , + ˆ s , + 2 , ˆ s , + ˆ s , + ˆ s , + ˆ s , + 2; 1) , (A.6) Contribution E I (0 , − , , , − 1) = 1ˆ s , ˆ s , Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + 1)Γ(ˆ s , + ˆ s , + ˆ s , + ˆ s , + 1) × F ( − ˆ s , , ˆ s , + 1 , ˆ s , + ˆ s , + ˆ s , ; ˆ s , + ˆ s , + 1 , ˆ s , + ˆ s , + ˆ s , + ˆ s , + 1; 1) , (A.7)– 25 – ontribution F I (0 , , − , , − 1) = Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + ˆ s , + 1)Γ(ˆ s , + ˆ s , + 2)Γ(ˆ s , + ˆ s , + ˆ s , + ˆ s , + 2) × F (ˆ s , + 1 , ˆ s , + ˆ s , + ˆ s , + 1 , − ˆ s , ; ˆ s , + ˆ s , + 2 , ˆ s , + ˆ s , + ˆ s , + ˆ s , + 2; 1) , (A.8) Contribution G I (0 , − , , , 0) = ˆ s , + ˆ s , (ˆ s , − s , ˆ s , × Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + 1)Γ(ˆ s , + ˆ s , + 1)Γ(ˆ s , + ˆ s , + 1) × F ( − ˆ s , , ˆ s , + 1 , ˆ s , + ˆ s , + ˆ s , ; ˆ s , + ˆ s , , ˆ s , + ˆ s , + ˆ s , + ˆ s , + 1; 1) , References [1] H. 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