On Higgs boson plus gluon amplitudes at one loop
aa r X i v : . [ h e p - ph ] A ug Prepared for submission to JHEP
IPPP/18/73
On Higgs boson plus gluon amplitudes at one loop
R. Keith Ellis & Satyajit SethIPPP, DurhamE-mail: [email protected], [email protected] . Abstract:
We present analytic results for one-loop Higgs boson + n -gluon amplitudes for n ≤ Keywords:
QCD, Hadron colliders, Higgs boson ontents H + 2 g amplitude 23 Tree level ingredients and cut techniques 4 Q ¯ Q + n gluon amplitudes 43.2 Interference with Higgs amplitudes 43.3 Higgs + 4 gluon amplitude: the coefficient of the scalar pentagon 53.4 Higgs + 5 gluon amplitude: the coefficient of one of the scalar pentagons 6 n = 2 74.2 n = 3 74.3 n = 4 74.4 n = 5 8 γ -matrix traces 11C Reduction of scalar pentagon integrals to boxes 11D Reduction of scalar hexagons integrals to pentagons 12 It is evident that detailed study of the Higgs boson will be a primary focus of the experiments performedat the CERN LHC for at least the next decade. Many calculations of Higgs boson production by gluonfusion are carried out in the Higgs boson effective field theory, valid when the top mass is larger than allother scales in the problem. This approach has the merit that calculations performed in the effectivetheory are easier, since the Born-level matrix element is a tree graph, rather than a one-loop process.However, with increasing statistics the LHC will be able to probe a regime where the effective theoryis no longer valid, yielding valuable information about the intermediaries circulating in the loop thatcouple to the Higgs boson. This is the case in Higgs boson + jet production when the transversemomentum of the Higgs boson or of the jets is large compared to the top quark mass.– 1 –ext-to-leading order (NLO) QCD corrections to Higgs boson plus 1-jet production with full top-quark mass dependence are already known [1, 2]. These calculations use the one-loop Higgs boson + 3parton amplitude as the Born-level cross section, and the one-loop Higgs boson + 4 parton amplitudeas a real radiation correction to the Born-level process. The two loop virtual corrections are calculatedusing an expansion method [1] or sector decomposition [2]. If one were to go further and calculate theNNLO QCD corrections to the Higgs boson + 1 jet process, one of the ingredients would be the Higgsboson + 5 parton amplitudes.The Higgs + 2 jet process via gluon fusion has also been calculated at leading order in the fulltheory [3, 4]. This process constitutes a “background” to the Higgs + 2 jet process occurring viaVector Boson fusion, which also comes accompanied by two jets. The leading order Higgs + 3 jetprocess has been considered in ref. [5]. Phenomenological analyses of Higgs + jets including full masseffects have been performed in refs. [6, 7].Despite this progress the literature does not contain detailed analytic results for Higgs + n -partonamplitudes in the full theory for n ≥
4. Techniques for the analytic calculations based largely onunitarity have been developed over a number of years [8–14]. The purpose of the current paper is toprovide analytic results for the specific case of gluons all having the same helicity. We have undertakenthis work, in order to elucidate patterns which exist for varying values of n . In addition, by calculatingthe all positive helicity processes, which are the simplest, we can assess the feasibility of obtainingsimple analytic forms for all helicities.From a numerical point of view, one loop calculations are a solved problem thanks to tech-niques [15, 16] that allow numerical calculation of the coefficients of the needed loop integrals . Thefull numerical result is obtained by combining these numerical results for the coefficients, with ana-lytic results for the one-loop scalar integrals. Analytic expressions for scalar one-loop integrals arecompletely known both for IR-finite [18] and divergent[19] cases. The downside of this semi-numericalapproach is that it can lead to instabilities in corners of phase space. These instabilities can be solvedby moving to higher precision calculation, at the cost of increased computer time. Analytic calculationson the other hand are less prone to these instabilities. H + 2 g amplitude The aim of this paper is to calculate one-loop results for Higgs boson + gluon amplitudes. Theseamplitudes contain a quark of mass m circulating in the loop, (dominantly the top quark) and thecoupling of the Higgs boson to the quark is given by − im/v where v ≈
246 GeV is the vacuumexpectation value of the Higgs field. The mass of the Higgs boson is denoted by M h .We will calculate colour-ordered sub-amplitudes for the production of a Higgs boson and n gluonsdefined as follows: A n ( { p i , h i , c i } ) = i g ns π v X { , ,...,n } ′ tr ( t c t c . . . t c n ) A n (1 h g , h g , . . . n h n g ; H ) , (2.1)where the sum with the prime , P { , ,...,n } ′ , is over all ( n − non-cyclic permutations of 1 , , . . . , n and the t matrices are the SU(3) matrices in the fundamental representation normalized such that, tr ( t a t b ) = δ ab . (2.2)Because of Bose symmetry it will be sufficient to calculate one permutation, and the other coloursub-amplitudes can be obtained by exchange. For a review and a complete set of references, see ref. [17]. – 2 – igure 1 . Unitarity approach to calculating the Higgs + 2 gluon amplitude
The unitarity method seeks to calculate this result by sewing together tree-level colour-orderedsub-amplitudes. For the tree graph process, qgg . . . g ¯ q , these are defined as, G nab ( p a , h a , { p i , h i , c i } , p b , h b ) = ig ns X σ ∈ S n ( t c σ (1) t c σ (2) . . . t c σ ( n ) ) ab G tree n ( a q , σ (1) , . . . σ ( n ) , b ¯ q ) , (2.3)where S n is the permutation group on n elements, and G tree n are the tree-level partial amplitudes. Ina similar way we can define the tree-level sub-amplitudes for the production of a Higgs boson andgluons from a massive fermion line, H nab ( p a , h a , { p i , h i , c i } , p b , h b ) = − i g ns v X σ ∈ S n ( t c σ (1) t c σ (2) . . . t c σ (2) ) ab H tree n ( a q , σ (1) , . . . σ ( n ) , H, b ¯ q ) . (2.4)For the case of Higgs + 2 gluons the only non-zero amplitude is when the gluons have the samehelicity. We sketch the calculation of this amplitude, which closely follows the approach of Bern andMorgan [11]. The relevant component tree diagrams can be extracted from Fig. 1. The left-hand sideof the diagram is the colour-ordered amplitude for the qgg ¯ q process with positive helicity gluons whichis given by, G tree2 ( a, + , + , b ) = [1 2] h i ¯ u ( p a ) γ R ( µ + m ) u ( p b )( s a − m ) , γ R = (1 + γ ) / . (2.5)The components of the d -dimensional momenta p a beyond four dimensions are denoted by µ and s ai = ( p a + p i ) . With the normalization defined by Eq. (2.6) the right-hand side of the diagram inFig. 1 is given by H tree0 = m ¯ u ( p b ) u ( p a ) . (2.6)Sewing Eqs. (2.5) and (2.6) together and summing over the polarizations of fermions a and b we getin four dimensions, m [1 2] h i Tr { γ R ( p b + m )( p a + m ) } ( s a − m ) = m [1 2] h i (2 p a · p b + 2 m )( s a − m ) = m [1 2] h i (4 m − M h )( s a − m ) . (2.7)Restoring the propagators which were put on shell and exploiting the linkage between the mass termsand µ , we obtain a result for the amplitude, evaluated on the two particle cut, A (1 + , + ; H ) scut = m [1 2] h i iπ Z d d l (4( m + µ ) − M h )( l − m )(( l + p ) − m )(( l + p ) − m ) . (2.8)– 3 –he symbol p i denotes the four-momentum of the i th particle, and we further define, p ij = p i + p j , p ijk = p i + p j + p k , etc. Adding in the other diagram 1 ↔
2, and evaluating the rational term from µ we obtain, A (1 + g , + g ; H ) = 2 m [1 2] h i h (4 m − M h ) C ( p , p ; m ) + 2 i . (2.9)where C is the scalar triangle integral, defined in Eq.(A.2). Note that the essential feature leading tothe simple answer was the simplified form of the tree level inputs. In the following section we presentthe tree-level building blocks for one-loop Higgs amplitudes with greater numbers of gluons. Q ¯ Q + n gluon amplitudes Multi-gluon tree amplitudes with a pair of massive fermions have been considered by a number ofauthors [20–22] using BCFW techniques and supersymmetric relations to scalar amplitudes. Howeversince these authors make specific choices of spinors for the massive fermions they are not well suitedfor our purposes. All orders results for tree graphs with n gluons have been given in a convenient formin ref. [23]. In our notation the n +2-point amplitude for a quark-antiquark pair and n positive-helicitygluons is given in four dimensions by, G n ( a, + , + , . . . , n + , b ) = m ¯ u ( a ) γ R u ( b ) [1 | Q n − j =1 (cid:8) p a...j p j +1 + ( s a ...j − m ) (cid:9) | n ]( s a − m )( s a − m ) . . . ( s a ... ( n − − m ) h ih i . . . h n − | n i . (3.1)The important features of the all-positive helicity gluon amplitude are that the amplitude vanishesfor massless quarks and that spin structure of the dependence on the massive quark momenta entersthrough the combination ¯ u ( a ) γ R u ( b ) for all n .For n = 2 the product collapses to unity and we recover the four dimensional version of Eq. (2.5) G ( a, + , + , b ) = m ¯ u ( a ) γ R u ( b )( s a − m ) [1 2] h i . (3.2)The results for larger numbers of gluons are similarly compact. For example, for n = 3 , G ( a, + , + , + , b ) = m ¯ u ( a ) γ R u ( b ) [1 | (cid:0) p a p + ( s a − m ) (cid:1) | s a − m )( s a − m ) h i h i . (3.3) G ( a, + , + , + , + , b ) = m ¯ u ( a ) γ R u ( b ) [1 | (cid:0) p a p + ( s a − m ) (cid:1) (cid:0) p a p + ( s a − m ) (cid:1) | s a − m )( s a − m )( s a − m ) h i h i h i . (3.4) From Eq. (3.1) the all-positive helicity gluon amplitude has the same kinematic structure for all n . Itis therefore useful to contract the Higgs production amplitudes with this structure, M † = m ¯ u ( b ) γ R u ( a ) (3.5)We will interfere this structure with the amplitude for the production of a Higgs + 0,1 or 2 gluons.Summing over the polarizations of the massive quarks X u ( p )¯ u ( p ) = p + m , (3.6)– 4 –e obtain the following results for the interference with zero, one and two gluon amplitudes, H , H and H , M † H ( a ; H, b ) = m (cid:16) m − M h (cid:17) , (3.7) M † H ( a, + g ; H, b ) = m (cid:16) m − M h (cid:17) h q i n [1 | a | q i [1 | a | i − [1 | b | q i [1 | b | i o = m (cid:16) m − M h (cid:17) [1 | ab | | a | i [1 | b | i , (3.8)(where q is an arbitrary light-like vector), M † H ( a, + g , + g ; H, b ) = 2 m (cid:16) m − M h (cid:17) h i n [2 |6 b ( b − 6 a ) | | a | i [2 | b | i− m [2 1] (cid:16) | b | i (( b − p ) − m ) + 1[1 | a | i [2 | b | i − | a | i (( a + p ) − m ) (cid:17)o . (3.9)We take all momenta to be outgoing except for b . We note that in four dimensions the interference ofEq. (3.5) with the Higgs + gluon amplitudes, H n , is always proportional to 4 m − M h . In four dimensions a scalar pentagon integral can be expressed as a sum of five boxes [24, 25]. (Thisreduction formula is described in appendix C). Consequently any attempt to identify the coefficient ofa pentagon integral is inherently a d -dimensional calculation. In d -dimensions we must introduce anextra parameter µ , that describes the magnitude of the loop momentum momentum in the ( d −
4) = − ǫ space. We now use unitarity to extract the coefficient of the scalar pentagon integral for the diagramshown in Fig. 2. We express the loop momentum l as, l ν = α p ν + β p ν + γ h | γ ν |
2] + δ h | γ ν |
1] + l νǫ . (3.10)We denote the length of the component of l beyond 4 dimensions, l ǫ , by µ . Placing all five propagatorson their mass shell we obtain the following five equations, l − m = 0 , → − γδ h i [2 1] − m − µ = 0 , determines µ , ( l − p ) − m = 0 , → β = 0 , ( l + p ) − m = 0 , → α = 0 , ( l + p + p ) − m = 0 , → γ h i [3 2]) + δ h i [3 1] + s = 0 , ( l + p + p + p ) − m = 0 , → γ h i [4 2] + δ h i [4 1] + s − s = 0 . (3.11)However, because of the good ultraviolet properties of the pentagon integral, terms of order less than µ will play no part in the limit ǫ → qgggg ¯ q amplitude, Eq. (3.4) and the projectionof the Higgs production vertex, Eq. (3.7). After imposing the mass-shell conditions, all dependenceon the loop momentum drops out and the result for the coefficient of E ( p , p , p , p ; m ) is, m (4 m − M h ) [1 |6 l p ( l + p ) p | h i h i h i = − m (4 m − M h ) tr + { }h i h i h i h i . (3.12)To express this formula (and subsequent formula) we have introduced a notation for the traces ofgamma matrices (defined in detail in Appendix B),tr + { . . . n } = tr { γ R p p . . . p n } . (3.13)The full result for the Higgs + 4 gluon amplitude is given in Eq. (4.3).– 5 – igure 2 . Feynman diagram to illustrate the calculation of coefficient the scalar pentagon integral. Figure 3 . Feynman diagram to illustrate the calculation of the coefficient of the scalar integral E ( p , p , p , p ; m ) We now use a similar method to identify the pentagon coefficient for the hexagon diagram shown inFig. 3. We parameterize the loop momentum as before, Eq. (3.10) and set the same condition on thefive propagators, Eq (3.11). The condition on the propagator l − m serves to fix the length of theloop momentum in the extra dimension. Solving the simultaneous equations for γ and δ we have that, γ = + 1tr (1 , , , h h i [3 1] ( s − s ) − h i [4 1] s i δ = − (1 , , , h h i [3 2] ( s − s ) − h i [4 2] s i (3.14)With this solution for the γ, δ in hand we can evaluate the sixth denominator, d = ( l + p + p + p + p ) − m . The result for d on the cut of the first five denominators is d = − tr (1 , , , , , (1 , , , . (3.15)Not surprisingly, this is inverse of the coefficient which occurs in the reduction of a scalar hexagonintegral to the scalar pentagon integral formed by the first five propagators, see Eq. (D.3). By evaluat-ing any possible numerators factors for the value of l determined by our γ, δ from Eq. (3.14) we obtainthe coefficient of this particular pentagon integral in our real physical amplitude. Thus determining– 6 –entagon coefficients is even easier than box coefficients, because we deal with a linear rather than aquadratic equation. The full result for the Higgs + 5 gluon amplitude is given below in Eq. (4.4). n = 2For the case n = 2 we have the well known result[26, 27] A (1 + g , + g ; H ) = 2 m [1 2] h i h (4 m − M h ) C ( p , p ; m ) + 2 i . (4.1)For n = 2 the same helicity amplitudes are the only non-zero amplitudes. We follow the normalnotation for spinor products,[28] with h ij i [ ji ] = s ij where s ij = p ij = 2 p i · p j for the lightlike momenta p i and p j . The C functions are the scalar triangle integrals, defined along with the box, pentagonand hexagon integrals, D , E and F in Eq. (A.2). n = 3For the case n = 3 the results for all helicities are given in ref. [29]. The result for all positive helicitygluons is given by, A (1 + g , + g , + g ; H ) = m "( m − M h h i h i h i h − s s D ( p , p , p ; m ) − ( s + s ) C ( p , p ; m ) i − s + s h i h i h i ) + ( ) . (4.2)This result of ref. [29] has been confirmed in ref. [30] where it is presented in a notation similar tothe notation of the current paper. This result has been also obtained later by unitarity methods inref. [31]. n = 4Analytical results for the full one-loop amplitude for Higgs + 4 gluons have been calculated for allhelicities by the authors of ref. [32] and are available in MCFM. However simple analytic results havenot been achieved. For the case n = 4 we find the simple expression, A (1 + g , + g , + g , + g ; H ) = m "( m − M h h i h i h i h i h − tr + { } m E ( p , p , p , p ; m )+ 12 (( s + s )( s + s ) − s s ) D ( p , p , p ; m )+ 12 s s D ( p , p , p ; m )+ ( s + s + s ) C ( p , p ; m ) i + 2 s + s + s h i h i h i h i ) + ( ) . (4.3)– 7 – .4 n = 5For the case n = 5 we find, A (1 + g , + g , + g , + g , + g ; H ) = m "( (4 m − M h ) h i h i h i h i h i h X i =1 e ( i ) E ( i ) − s s D ( p , p , p ; m ) −
12 [( s + s )( s + s ) − s s ] D ( p , p , p ; m ) −
12 [( s + s + s )( s + s + s ) − s ( s + s + s )] D ( p , p , p ; m ) − ( s + s + s + s ) C ( p , p ; m ) i − s + s + s + s ) h i h i h i h i h i ) + ( ) , (4.4)where the coefficients of the scalar pentagon integrals are given by, e (1) = m h
12 tr − { } + s s s (tr − { } + s s )tr { } i ,e (2) = − m s s tr − { } tr { } ,e (3) = − m tr + { } tr − { } tr { } ,e (4) = − m tr + { } tr − { } tr { } ,e (5) = − m s s tr − { } tr { } ,e (6) = m h
12 tr − { } + s s s (tr − { } + s s )tr { } i , (4.5)and the pentagon integrals E ( i ) ≡ F ( i )0 correspond to the scalar hexagon integrals with the i th propaga-tor removed, see Eq. (D.2). Note the absence of boxes of the form D ( p , p , p ; m ), D ( p , p , p ; m ), D ( p , p , p ; m ) and D ( p , p , p ; m ) apart from those which would occur if the scalar pentagonsin Eq. (4.4) were expressed as a sum of boxes. The momentum of the Higgs boson is denoted by p such that X i =1 p i = 0 . (4.6)Note that tr { } = − tr { } . This relationship is important to show that the apparentsingularity in e (1) and e (6) in the limit p → E ( p , p , p , p ) = E ( p , p , p , p ). One of the benefits of an analytic formula is that we can investigate the behaviour of the amplitudes invarious limits. In this section we shall present the behaviour of the amplitude in the limit of vanishingHiggs boson momentum and for large top mass. The high energy limits of Higgs + 4 parton amplitudeshave been considered in ref. [33]. – 8 – .1 Soft Higgs limit
The insertion of a soft Higgs boson is performed by the operating on the corresponding multi-gluonamplitude without a Higgs boson with the operator, mv ddm ≡ v m ddm . (5.1)The colour sub-amplitude for scattering of four positive helicity gluons via a loop of quarks has beenpresented by Bern and Morgan[11], A (1 + g , + g , + g , + g ) = − h i h i h m D ( p , p , p ; m ) − i . (5.2)In the limit in which p → A (1 + g , + g , + g , + g ; H ) → − m [1 2] h i [3 4] h ih i h i h i h i× h
12 ( D ( p , p , p ; m ) + D ( p , p , p ; m ) + D ( p , p , p ; m ) + D ( p , p , p ; m ))+ m ( E ( p , p , p , p ; m ) + E ( p , p , p , p ; m ) + E ( p , p , p , p ; m ) + E ( p , p , p , p ; m )) i = − h i [3 4] h ih i h i h i h i (cid:20) m D ( p , p , p ; m ) + 2 m ddm D ( p , p , p ; m ) (cid:21) = − h i [3 4] h ih i h i h i h i m ddm (cid:2) m D ( p , p , p ; m ) (cid:3) , (5.3)since in the limit p → h i [3 4] h i = − s s and ddm D ( p , p , p ; m ) = E ( p , p , p , p ; m ) + E ( p , p , p , p ; m )+ E ( p , p , p , p ; m ) + E ( p , p , p , p ; m ) . (5.4)This demonstrates the expected form in the limit p →
0. Similarly Eq. (4.4) can be studied in thelimit p →
0. In fact, we make use of the existence of this limit to help organise coefficients of scalarpentagons presented in Eq. (4.5).
In the large top mass limit we obtain the following results for the scalar integrals C ( p , p ; m ) = − m − ( p + p + p )24 m + O (cid:18) m (cid:19) , (5.5) D ( p , p , p ; m ) = 16 m + ( s + s + p + p + p + p )60 m + O (cid:18) m (cid:19) , (5.6) E ( p , p , p , p ; m ) = − m + O (cid:18) m (cid:19) . (5.7)– 9 –sing these expansions we obtain the expected form [34, 35] for the tree graphs in the effective theory. A (1 + g , + g ; H ) = + 23 M h h i h i , (5.8) A (1 + g , + g , + g ; H ) = − M h h i h i h i , (5.9) A (1 + g , + g , + g , + g ; H ) = + 23 M h h i h i h i h i , (5.10) A (1 + g , + g , + g , + g , + g ; H ) = − M h h i h i h i h i h i . (5.11) The results of the paper have shown that, having simple expressions for the component tree graphamplitudes in hand, it is feasible to extract compact expressions for the Higgs boson + n -partonamplitudes for n ≤
5. The results with all gluon helicities taken to be the same, display simplepatterns. One is tempted to try and extend these results to even higher n , but in view of the limitedphenomenological importance of higher n we have not succumbed to this temptation. The resultsfor n = 4 and n = 5, after extension to all helicities, offer the prospect of fast and stable numericalevaluation. Acknowledgements
We would like to acknowledge useful discussions with Simon Badger and Nigel Glover. RKE gratefullyacknowledges the hospitality and partial support of the Mainz Institute for Theoretical Physics (MITP)during the completion of this work.
A Integrals
We define the denominators of the integrals as follows d = l − m + iε ,d = ( l + p ) − m + iε = ( l + q ) − m + iε ,d = ( l + p + p ) − m + iε = ( l + q ) − m + iε ,d = ( l + p + p + p ) − m + iε = ( l + q ) − m + iε ,d = ( l + p + p + p + p ) − m + iε = ( l + q ) − m + iε ,d = ( l + p + p + p + p + p ) − m + iε = ( l + q ) − m + iε . (A.1)– 10 –he p i are the external momenta, whereas the q i are the off-set momenta in the propagators. In termsof these denominators the integrals are, C ( p , p ; m ) = 1 iπ Z d l d d d ,D ( p , p , p ; m ) = 1 iπ Z d l d d d d ,E ( p , p , p , p ; m ) = 1 iπ Z d l d d d d d ,F ( p , p , p , p , p ; m ) = 1 iπ Z d l d d d d d d . (A.2) B Definitions of γ -matrix traces In order to obtain compact expressions for the coefficients of the scalar integrals, we define the followingtraces of γ -matrices. tr { . . . n } = tr { γ p p . . . p n } , tr + { . . . n } = tr { γ R p p . . . p n } , tr − { . . . n } = tr { γ L p p . . . p n } , tr { . . . n } ≡ tr + { . . . n } − tr − { . . . n } , (B.1)with γ R/L = (1 ± γ ) /
2. For the special case of lightlike vectors we have thattr + { . . . n } = [1 2] h i [3 4] . . . h n i , tr − { . . . n } = h i [2 3] h i . . . [ n . (B.2)In the case of lightlike vectors, the traces with γ can be written as differences of spinor strings,tr( γ p p p p ) = (cid:0) [1 2] h i [3 4] h i − h i [2 3] h i [4 1] (cid:1) , (B.3)tr( γ p p p p p p ) = (cid:0) [1 2] h i [3 4] h i [5 6] h i − h i [2 3] h i [4 5] h i [6 1] (cid:1) . (B.4)In the case where external vectors are not light-like, (e.g. in our case the Higgs momentum p ), thespinor expressions must be modified using momentum conservation, e.g. Eq. (4.6) for the five gluoncase. C Reduction of scalar pentagon integrals to boxes
The reduction of the scalar pentagon integrals E to a sum of the five boxes obtained by removingone propagator has been presented in ref. [25]. We present the result here for completeness. E ( w − m ) = E (1) [2∆ − w · ( v + v + v + v )]+ E (2)0 v · w + E (3)0 v · w + E (4)0 v · w + E (5)0 v · w , (C.1)where the vectors v i are expressed in terms of the totally antisymeetric tensor ε , v µ = ε µ,q ,q ,q , v µ = ε q ,µ,q ,q , v µ = ε q ,q ,µ,q , v µ = ε q ,q ,q ,µ ,w µ = r v µ + r v µ + r v µ + r v µ , (C.2)– 11 –nd the box integrals are, E (1)0 = D ( p , p , p ; m ) ,E (2)0 = D ( p , p , p ; m ) ,E (3)0 = D ( p , p , p ; m ) ,E (4)0 = D ( p , p , p ; m ) ,E (5)0 = D ( p , p , p , ; m ) , (C.3)where p ij = p i + p j . The r i are the residues when the dot products of the offset momenta and theloop momenta, q i · l , are expressed in terms of differences of propagators, q .l = 12 [ d − d − r ] , q .l = 12 [ d − d − r ] ,q .l = 12 [ d − d − r ] , q .l = 12 [ d − d − r ] . (C.4) D Reduction of scalar hexagons integrals to pentagons
The reduction of the scalar hexagon integrals F to a sum of the six pentagons obtained by removingone propagator can be derived following the techniques of ref. [24, 25]. We denote by F ( i )0 the sixpentagon integrals obtained by removing the i th propagator from the hexagon integral, F ( p , p , p , p , p ; m ) = X i =1 c ( i ) F ( i )0 . (D.1)Explicitly we have that, F (1)0 ≡ E (1) = E ( p , p , p , p ; m ) ,F (2)0 ≡ E (2) = E ( p , p , p , p ; m ) ,F (3)0 ≡ E (3) = E ( p , p , p , p ; m ) ,F (4)0 ≡ E (4) = E ( p , p , p , p ; m ) ,F (5)0 ≡ E (5) = E ( p , p , p , p ; m ) ,F (6)0 ≡ E (6) = E ( p , p , p , p ; m ) , (D.2)where p ij = p i + p j . Translating the results of ref. [25] to the notation of Eq. (B.1) we find (see alsoref. [36]), c (1)12345 = +tr { } / tr { } ,c (2)12345 = − tr { (1 + 2) 3 4 5 } / tr { } ,c (3)12345 = +tr { } / tr { } ,c (4)12345 = − tr { } / tr { } ,c (5)12345 = +tr { } / tr { } ,c (6)12345 = − tr { } / tr { } . (D.3)In this equation we have used an obvious extension of the notation of Eq.(B.1),tr { (1 + 2) 3 4 5 } ≡ tr( γ ( p + p ) p p p ) . (D.4)– 12 –xpressed in this form it is manifest that X i =1 c ( i )12345 = 0 . (D.5) References [1] J. M. Lindert, K. Kudashkin, K. Melnikov and C. Wever,
Higgs bosons with large transverse momentumat the LHC , Phys. Lett.
B782 (2018) 210 [ ].[2] S. P. Jones, M. Kerner and G. Luisoni,
Next-to-Leading-Order QCD Corrections to Higgs Boson Plus JetProduction with Full Top-Quark Mass Dependence , Phys. Rev. Lett. (2018) 162001 [ ].[3] V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt and D. Zeppenfeld,
Higgs + 2 jets via gluon fusion , Phys. Rev. Lett. (2001) 122001 [ hep-ph/0105129 ].[4] V. Del Duca, W. Kilgore, C. Oleari, C. Schmidt and D. Zeppenfeld, Gluon fusion contributions to H +2 jet production , Nucl. Phys.
B616 (2001) 367 [ hep-ph/0108030 ].[5] F. Campanario and M. Kubocz,
Higgs boson production in association with three jets via gluon fusion atthe LHC: Gluonic contributions , Phys. Rev.
D88 (2013) 054021 [ ].[6] N. Greiner, S. H¨oche, G. Luisoni, M. Sch¨onherr, J.-C. Winter and V. Yundin,
Phenomenologicalanalysis of Higgs boson production through gluon fusion in association with jets , JHEP (2016) 169[ ].[7] R. V. Harlander, T. Neumann, K. J. Ozeren and M. Wiesemann, Top-mass effects in differential Higgsproduction through gluon fusion at order α s , JHEP (2012) 139 [ ].[8] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, One loop n point gauge theory amplitudes,unitarity and collinear limits , Nucl. Phys.
B425 (1994) 217 [ hep-ph/9403226 ].[9] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,
Fusing gauge theory tree amplitudes into loopamplitudes , Nucl. Phys.
B435 (1995) 59 [ hep-ph/9409265 ].[10] Z. Bern, L. J. Dixon and D. A. Kosower,
Progress in one loop QCD computations , Ann. Rev. Nucl. Part. Sci. (1996) 109 [ hep-ph/9602280 ].[11] Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity , Nucl. Phys.
B467 (1996) 479[ hep-ph/9511336 ].[12] R. Britto, F. Cachazo and B. Feng,
Generalized unitarity and one-loop amplitudes in N=4super-Yang-Mills , Nucl. Phys.
B725 (2005) 275 [ hep-th/0412103 ].[13] D. Forde,
Direct extraction of one-loop integral coefficients , Phys. Rev.
D75 (2007) 125019 [ ].[14] S. D. Badger,
Direct Extraction Of One Loop Rational Terms , JHEP (2009) 049 [ ].[15] G. Ossola, C. G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals atthe integrand level , Nucl. Phys.
B763 (2007) 147 [ hep-ph/0609007 ].[16] F. Cascioli, P. Maierhofer and S. Pozzorini,
Scattering Amplitudes with Open Loops , Phys. Rev. Lett. (2012) 111601 [ ].[17] R. K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi,
One-loop calculations in quantum field theory:from Feynman diagrams to unitarity cuts , Phys. Rept. (2012) 141 [ ].[18] G. ’t Hooft and M. J. G. Veltman,
Scalar One Loop Integrals , Nucl. Phys.
B153 (1979) 365.[19] R. K. Ellis and G. Zanderighi,
Scalar one-loop integrals for QCD , JHEP (2008) 002 [ ]. – 13 –
20] P. Ferrario, G. Rodrigo and P. Talavera,
Compact multigluonic scattering amplitudes with heavy scalarsand fermions , Phys. Rev. Lett. (2006) 182001 [ hep-th/0602043 ].[21] K. J. Ozeren and W. J. Stirling, Scattering amplitudes with massive fermions using BCFW recursion , Eur. Phys. J.
C48 (2006) 159 [ hep-ph/0603071 ].[22] J.-H. Huang and W. Wang,
Multigluon tree amplitudes with a pair of massive fermions , Eur. Phys. J.
C72 (2012) 2050 [ ].[23] A. Ochirov,
Helicity amplitudes for QCD with massive quarks , JHEP (2018) 089 [ ].[24] D. B. Melrose, Reduction of Feynman diagrams , Nuovo Cim. (1965) 181.[25] W. L. van Neerven and J. A. M. Vermaseren, Large loop integrals , Phys. Lett. (1984) 241.[26] H. M. Georgi, S. L. Glashow, M. E. Machacek and D. V. Nanopoulos,
Higgs Bosons from Two GluonAnnihilation in Proton Proton Collisions , Phys. Rev. Lett. (1978) 692.[27] F. Wilczek, Decays of Heavy Vector Mesons Into Higgs Particles , Phys. Rev. Lett. (1977) 1304.[28] L. J. Dixon, A brief introduction to modern amplitude methods , in
Proceedings, 2012 European School ofHigh-Energy Physics (ESHEP 2012): La Pommeraye, Anjou, France, June 06-19, 2012 , pp. 31–67,2014, , DOI.[29] R. K. Ellis, I. Hinchliffe, M. Soldate and J. J. van der Bij,
Higgs Decay to τ + τ − : A Possible Signatureof Intermediate Mass Higgs Bosons at the SSC , Nucl. Phys.
B297 (1988) 221.[30] U. Baur and E. W. N. Glover,
Higgs Boson Production at Large Transverse Momentum in HadronicCollisions , Nucl. Phys.
B339 (1990) 38.[31] J. S. Rozowsky,
Feynman diagrams and cutting rules , hep-ph/9709423 .[32] T. Neumann and C. Williams, The Higgs boson at high p T , Phys. Rev.
D95 (2017) 014004 [ ].[33] V. Del Duca, W. Kilgore, C. Oleari, C. R. Schmidt and D. Zeppenfeld,
Kinematical limits on Higgsboson production via gluon fusion in association with jets , Phys. Rev.
D67 (2003) 073003[ hep-ph/0301013 ].[34] S. Dawson and R. P. Kauffman,
Higgs boson plus multi - jet rates at the SSC , Phys. Rev. Lett. (1992) 2273.[35] L. J. Dixon, E. W. N. Glover and V. V. Khoze, MHV rules for Higgs plus multi-gluon amplitudes , JHEP (2004) 015 [ hep-th/0411092 ].[36] T. Binoth, J. P. Guillet, G. Heinrich, E. Pilon and C. Schubert, An Algebraic/numerical formalism forone-loop multi-leg amplitudes , JHEP (2005) 015 [ hep-ph/0504267 ].].