JHEP12(2012)038 and − Springer July 16, 2012 : October 8, 2012 : December 7, 2012 November 9, 2012 : ory involve large : and Rob , c,d Received Revised 10.1007/JHEP12(2012)038 duction methods proved ed spurious terms at one Accepted Published enoNet), by the MICINN Grants doi: d to be actually calculated. ntegrand level for one nd non-planar diagrams. bers of tensor structures and nation with the large number eed and set-up of these calcu- itat de Val`encia, Published for SISSA by 0 Programme CSD2007-00042 (CPAN). loop case. This method is also applicable to Costas G. Papadopoulos − 1 b, [email protected] , [email protected] , Ioannis Malamos, a 1206.4180 QCD Phenomenology, NLO Computations Calculation of amplitudes in perturbative quantum field the R[email protected] a loop amplitudes. We clarify their connection to the so-call − Supported by REA Grant Agreement PITN-GA-2010-264564 (LHCPh 1 Department of Physics, Theory1211 Division, Geneva CERN, 23, Switzerland E-mail: Instituto de Corpuscular, F´ısica Consejo Superior de InvestigacionesParc Cient´ıficas-Univers E-46980 Cient´ıfic, Paterna (Valencia), Spain. NCSR ‘Demokritos’, Agia Paraskevi, 15310, Greece Radboud University, Nijmegen, The Netherlands [email protected] b c d a Verheyen Open Access FPA2007-60323, FPA2011-23778 and Consolider-Ingenio 201 Abstract: Ronald H.P. Kleiss, Counting to one: reducibilityamplitudes of at one- the and integrand two-loop level Keywords: ArXiv ePrint: higher loops, and the results obtained apply to both planar a independent coefficients, how to write such relations at the i loop integrals. Theof complexity of Feynman those diagrams, integrals, maketo in the be combi very calculations helpful, veryEspecially lowering difficult. reduction the at number the Re oflations. integrand integrals In level that this improves nee the article we sp demonstrate, by counting the num loop and discuss their structure in the two two JHEP12(2012)038 3 5 9 1 3 9 11 12 14 14 17 18 such n tors) in 4 ] the authors 2 nman integrals with a n order to understand widely been considered d the Tevatron before it n the dimension. However, ecise theoretical results and .e. with fewer denominators) (2 denominators) in 2 dimen- er (NNLO) calculations with tical point of view, that Next- , ]. A typical integral with 3 n , 2 ...D 1 2 D ]. This implies that (very) large loop integrals 1 1 – 1 – D q d d Z is the denominator of a generic propagator. In [ 2 i m − 2 space-time dimensions, is given by ) i d p ] a pentagon (5 denominators) is reduced to boxes (4 denomina 3 + q = ( i D Reduction techniques form a way out. The idea of reducing Fey 3.3 Comments on3.4 the number of Reduction independent with3.5 coefficients general quadratic terms Reduction with general cubic terms 2.1 Introduction:2.2 reduction with trivial Reduction coefficients with coefficients linear in the loop momentum 3.1 Preliminaries 3.2 Further reduction with linear terms large number of denominatorsat to a one set loop of simpler goes integrals surprisingly (i many years back [ reduce a triangle (integral withsions 3 while denominators) in to [ dimensions. bubbles We see that the result of the reduction depends o denominators, in have to be computed forthe very bottleneck many of Feynman diagrams, such which calculations. has where experimental results is needed.to-Leading-Order It (NLO) is and clear,many Next-to-Next-to-Leading-Ord from external the legs theore have to be considered [ 1 Introduction Modern colliders such aswas the shut Large down) Hadron producethe Collider a output (LHC) of large (an these amount experiments, of comparison experimental between very data. pr I A Pentagons to boxes with the OPP method 4 Conclusions 3 Reduction of two-loop integrands Contents 1 Introduction 2 Reduction at one loop JHEP12(2012)038 - ], nts the 24 , 1 23 ’s vanish D = 4, cutting four ]). d 41 – asis for any integral is 27 is introduced. One can the reduction and mul- ain interest is of course hey try to decompose the at, for s in four-dimensional mo- ). As a consequence, only must also include spurious ple [ y cutting propagators n which proved useful for un- performing decomposition `a egration. Their will rˆole be t the integrand level [ ms of scalar boxes, triangles, out this paper is the use of to scalar integrals with up to of the packages available for d level and we shall see why . The OPP method has been re is, in general, more than a ree and four external legs (in ] comes as a natural combina- ≤ roducts of tree amplitudes. In tegrals the situation is different. 22 r which one or more of the m ]. Instead of working with specific – numerators) is needed in order to 11 18 ting propagators on-shell’. – ] at two loops. multiple cuts 7 48 – 46 – 2 – to decompose a scalar pentagon to boxes. Spu- ], the notion of ] used Lorentz invariance to express tensor one-loop 5 17 point scalar integrals ( – ]). Finding the coefficients is a purely algebraic problem. ], it is well known fact a generic one-loop amplitude is − 4 12 26 m , . It is believed that unitarity methods can also be applied in 25 ] a pentagon-to-boxes decomposition is performed in 4 dimen ’s are quadratic in the loop momentum). 6 = 4), every one-loop amplitude is written in terms of coefficie D spurious terms d a priori ], or using generalized unitarity [ 45 – = 4. 42 d The next big step comes from unitarity methods [ In another attempt [ Since a few decades now [ As we noticed before, in the case of one-loop calculations a b The Ossola-Papadopoulos-Pittau (OPP) method [ ‘Cutting’ a propagator means that loop momenta are chosen fo point integrals in terms of 1 = 4). Passarino and Veltman [ − rious terms are termsexplained that, later by when construction, weone vanish consider cannot upon reductions avoid int them. at the integran the evaluation of scalarperform integrals a (integrals one-loop with calculation. trivial that multiply these scalar integrals. The OPP method works a A basis is not known that case and therela are OPP some [ recent papers in that direction, The method is suitablewidely for used a so fully far numerical in implementation many one loop calculationsknown (see in for advance. exam Any one-loopbubbles integral and can tadpoles. be However, written in in the ter case of higher-loop in which means that forterms. these decompositions Then to one be hastiply to possible them find one with a the waythe to appropriate evaluation of calculate scalar them(i.e. the integrals, [ coefficients using of one tion of all the above.four Since denominators every (for integral can be decomposed Feynman diagrams these methods havewhole a big one-loop advantage amplitude in that inrational t coefficients terms of of loop thegeneralized integrals scalar are unitarity given methods integrals. in [ terms B of p derstanding one-loop reduction.what Another we important call fact nowadays ab sions. The importancementum of space this (the so paper called is van that Neerven-Vermaseren basis), it provides a basi decomposable in terms ofd scalar integrals, with one,n two, th so that the integrand becomes singular. One also speaks of ‘put the case the methods we will use can be applied to all dimensions. Our m cut more than one propagator to find these coefficients. Note th propagators essentially determines thesingle loop solution momentum since (the the JHEP12(2012)038 , se (2.1) , 2 j m − 2 j p integration-by-parts external momentum includes. We give an ontribute to a given ≡ 5, -loop integrals does not merator. Let us regard tten as a linear combi- j ese integrands consist of d outgoing momenta and ... P one, but the connection perform the decomposition oefficients times denomina- as existing denominators of , µ s that if one is interested in t the same procedure in the of these coefficients in order or that reason we deal with eduction at one loop. We see nal momenta are simply fixed containing ISP ’s. The layout ous and integrate to zero, but = 1 by writing first the numerator j ls (MI). There might be smaller j µ is called the lso have irreducible scalar products ) + hat ends up with is not necessarily a q µ · j . p j mean a momentum related to a particle p N D not + 2( k , and 2 ··· · µ q – 3 – 2 q q D = 1 2 j D denominators. From this set of integrals, one can in m d − 2 ) j p + as a collection of denominators with arbitrary momenta. We u q ]. with a nontrivial numerator, for instance 51 ) = ( n j – iGraph p 49 + q ( D ≡ j D The loop momentum is denoted by The order of the iGraph is simply the number of propagators it We deal only with scalar iGraphs, that have unity for their nu We define the In this article we prove that any two-loop integral can be wri Two remarks are in order here. The first is that the basis of two example of an iGraph of order 5 (pentagon) below, where the freedom of shifting in every loop momentum. the various diagram topologies. incoming or outgoing in a givenmomenta, amplitude: given what in we some call way exter by the configuration of incoming an nation of integrals with at most 2 integrand-graphs, or iGraphs instead of Feynman diagrams. 2.1 Introduction: reductionWe with start trivial with coefficients integrandsthe of sum any of givenprocess integrands one-loop and coming amplitude. share the from same Th once the topology; instead the Feynman of advantage diagrams is reducing that that every we c single diagram separately. F case of two-loop integrands.of Our the method two is methods not will exactly become the clear. OP 2 Reduction at one loop of this paper is thewhy following. one can We start have unitarity-like withof bases definitions our at and integrals the r (for integrand scalar level tors. cases We the investigate, number using one)for as simple our a algebra, polynomial sum what equation of is to c the have solutions. form Then we repea principle end up with true master integrals with numerators where it must be realized that by this we do an iGraph of order minimal one: the integrals aresets not by of default true Master Integra MI(IBP) and identities at [ two or more loops one can find them by using include only scalar integrals. It(ISP) includes as integrals numerators that (to a somethe power) integral. that cannot In be the rewritten for one-loop higher case loops these this ISP does areconstructing a always no unitarity-like spuri longer basis, hold. the set The of second integrals remark t i JHEP12(2012)038 ) d ≤ 2.4 (2.2) (2.5) (2.4) (2.3) n simply 1, plus µ − ω , n 1 ) such that and eq. ( q − ( 1 (or lower). n and − n ,...,n ) n 2 q , + 1 then ...D , . . . , n ( 1 3 n d T he several denominators. , T ntegrates to zero. Our D ) = 1 2 , ≥ order 1 one-loop iGraphs of order j D µ n 1 s of order + D j (if ) = 1 l. µ + q j ( . The reason is simple: for , − n = 0 for ··· . 4 1 T µ j n p j + D n ) is to take them to be just numbers D p D n q . j − ( ζ + j j j 5 T x D ...D we can find funtions n =2 3 p ( ) ··· j ...D X 4 = 0 for all p q j 4 if ( ζ D 1 j ) = 2 : 3 – 4 – D p ) + q + 2 T 3 n n ( q =2 · D ( j µ j X 2 D 2 T ω ω 1 D T = 0 by a suitable shift of the loop momentum, and 1 1 2 D ) + = p µ D D ω 1 µ + · p k 2 n q p ) + q propagators, so that ( 1 ) ...D ) = ( T all q k 1 4 ( · 1 D D q T 3 ( D be decomposed in the above manner 2 as (‘trivial’ coefficients): D µ µ k q = n cannot ’s are fixed numbers and D ζ 1 . A one-loop pentagon iGraph. The dots serve to distinguish t ··· 1 D The simplest possibility for the functions This immediately leads us to state the following theorem: Consider a one-loop iGraph of order or smaller independent of d where the We say that we can decompose this iGraph and the original iGraph is decomposed into a sum of iGraphs of then would become 0 = 1. there exist a cut through for then we have write the vector vanishes). We can then write so that this nonscalar iGraph decomposes into scalar iGraph Figure 1 possibly a spurious termtreatment of about the which scalar we case do is therefore not sufficiently worry genera since it i Now, we can always arrange for The external momenta are indicated. JHEP12(2012)038 5 2 . (2.9) (2.6) (2.7) (2.8) T = 4, ..., (2.10) d It is easy vectors are = 1 d , j = 4. = 0 d +2 or higher are j = 4 any iGraph of d x d agon in 4-dimensions sition exists in ) is ) exists, then by suitable . 2.7 e loop momentum so that 2.7 pendence of the coefficients instead of 4 because = 1 , . In this case, we have a system . d j ons ( , in the loop can be considered µ j terms, which are the terms of the . ) j p k µ x = 0 t µ +2, a solution to the system of the q · q µ 2 n , iGraphs of order d 2 =1 j ing of course in the case q q . j X d ( p j ≥ + , q j,k x n x µ = 1 ν j n =1 q j p j X 4 + 1) variables to be determined in each =1 j µ µ k X ) runs from 1 to d j x – 5 – ), we now have more tensor structures in terms , we must have separately x + 2.9 , q n =1 µ j j X 2.8 n q =1 x µ j X µ ). We see that, at one loop, for must be linearly independent but are otherwise arbi- q = 0 µ j , q ) = 2.8 k x q t + 2 ). We therefore turn to the next simplest possibility for ( 1 j j n =1 T x value of 2.8 j X ) and eq. ( n =1 j ) can always be found. X any 2.7 2 q 2.8 dependence: ) vectors d q is now replaced by 5 (or = 4, therefore 1+4+10+4 = 19 independent tensor structures. j x d ) then we have ) and eq. ( part of the propagators give rise to the dimensional case, the sum in eq. ( 2 2.4 2.7 q d = 0. Then the only solution to the conditions in eq. ( µ ’s, with a linear j The single Here, the four (or T p In the 2 j + 2 eq. ( d decomposable with trivial coefficients since for There are, for to see why onlytimes these the tensor structures appear. The linear de of the loop momentum: we can denote them by the shorthand For a one-loop iGraph of order 5 (or lower) no trivial decompo the 2.2 Reduction with coefficients linear in the loop momentum scaling we can always satisfy eq. ( thus cannot be decomposed that way. For general order 6 or higher can beof decomposed 6 in this equations fairly with trivial way 6 or more unknowns that we can solve. A pent Since this has to hold for and Note that if a nontrivial solution to the homogeneous equati From eq. ( trary. In analogy to eq. ( as linearlyP independent. One way to see this is by shifting th which is unacceptable in eq. ( needed for a basis. We shall work in general dimension, focus assuming that four out of the five external momenta, JHEP12(2012)038 = j ( gator s and linear (2.13) (2.11) (2.12) j ξ precisely , we find m a package = 4 allow k µ n q and µ j p . values for the essential advantage ! now describe how we can ly, any special phase space tion is not so simple since p tures up to constant terms ndent and it is included as ant part of the propagators. 1 + p y, by an approach that may be . random − ). d ! , . . . , ξ . d, k ). We are left with a set of k (

N k + = 1 +1 =0 2.4 p k X d = 5 and the 20 coefficients for +

, j n ! n – 6 – 1) = 19, it would seem that iGraphs of order 5 k , . Values of ) = = 1 (4 j values for the external momenta 1 + k x 0 13 44 2 85 5 13 36 13 26 19 67 19 4 45 26 458 7 26 71 34 90 71 105 161 161 266 322 588 N j − i n, d, k to be solved for is given by d 4 5 6 2 3 k ( M Table 1 : x =1 X

x ξ =1 d random j X . . ) = dimensions, and with the inclusion of tensor up to rank = 1 we have d, k ) of independent tensor structures ( . This can be extended to the inclusion of higher-rank tensor d ν k is purely numerical. We obtain it using the computer-algebr N q unknowns d, k µ term that could be produced from the quadratic part of a propa ( ξ q M 2 N q ) times, and insert all this in eq. ( = 4 and we give the results for various ranks and dimensionalities. matrix d 1 ξ cancellation probing n, d, k which, although not numerically the fastest available, has ). This avoids any possibility of us choosing, coincidental ( × 3 X ξ The number of coefficients We start by generating http://www.maplesoft.com/ 3 = , . . . , n it is not obvious that the 25 coefficients for and 4 are decomposable with linear terms. However, the situa In table equations for the The for the number Note that the highest rank. Thecoming other from terms the constant produce part all of lower the coeffients tensor times struc the const times the constant partthe of trace the part coefficients of is no longer indepe MAPLE Since for other dimensions: in us to actually buildascertain up the the number of 19dubbed independent required structures tensor numericall structures. We 1 ξ point where degeneracy might occur. Then, we choose JHEP12(2012)038 to may 098; . Now, 0 ates at 4 must be ± MAPLE MAPLE q 988 . . ’s is less than T 108 = 5 − q we give the results then a matrix with run down from 150 3 10 2 5 . p , actually give a number to 2 l lines. We conclude that e case of a pentagon decom- ensor structures since one , especially since = 10, possibly since ethod for the calculation of med. We also see that for sition of pentagon to boxes, q 826 p ination, r ( insoluble cases) than the − (one more than the minimum rmed with our MAPLE erations are performed. ). = 4 is not. We now also see the determines the decomposability t. In table tained value is n 6 10 . By letting n . pq MAPLE this − = 15 and p sparse, and anyway we can choose Gaussian ) for it to be decomposable. Note that , and not d, k M ( is N . M – 7 – d = 10 is default in p and a very accurate estimate of n 6 . 25 determinants range from 1 × will vanish. In an ideal real-number model of computation, ) = 0, but in our actual numerical computation there will be M then gives the rank of Digits is the number of digits specifies in the precision we tell ( M q p = 5 is precisely decomposable, but det − the number of independent combinations of coefficients satur n ξ n , where p − . The number of zero eigenvalues using rounding errors for th We have denoted the limit of decomposability with horizonta The difference The relevant variable is This is almost unavoidable since the matrix Surprisingly, the cancellation probing appears to fail for 4 5 6 zero eigenvalues will have a determinant of order 10 , the determinant of in four dimensions, cannot construct more tensor structures than actually exis the rank can benumber in of general tensor equal structures. (for soluble It cases) can or never smalle be larger than the t of 19), only 17sufficiently independent large combinations can actually be for deeper reason for this: in spite of there being 20 coefficients of the iGraph: the rank must be equal to to 20 in steps of 10, we can obtain of cancellation probing for various q if the number of independent tensor structures that can be fo with linear and quartic terms respectively. integer. We give two examplesthe to zero demonstrate the eigenvalues. use In of both this examples m we consider a decompo Figure 2 position with coefficients linear in the loop momentum. The ob ξ rounding errors. A cancellation of numbers to ‘zero’ will, i that one can easily set the precision with which numerical op we would thus find of order 10 the (absolute values of the 25 have special ways to treat these accuracies ( elimination as an option in any case. use. If the matrix’ determinant is computed by Gaussian elim JHEP12(2012)038 ourse + 1 is and solve d ξ 15, with the ] for exam- . . q 0 18 − ± ξ 98 . ]and [ ). Once a would-be 6 4 = 1 8-4 4-0 6-2 10-6 12-8 14-10 16-12 d 2.13 ecomposition is possible = 188 e case of a pentagon decom- he spurious terms have a q give an example in the ap- therefore the decomposition . is of course not. 8 = 2 gators and spurious terms, we ) for additional random values 6-0 9-1 12-4 15-7 d 18-10 21-13 24-16 d ystem ( . Here, 2.4 3320 − = 3 13 8-0 , given as the difference 20-7 12-1 16-3 triangles (like in [ 24-11 28-15 32-19 d n 3 10 . dimensions, an iGraph of order and d and to 2 = 4 d 19 25-6 10-0 15-1 20-3 30-11 35-16 40-21 d number of linear equations, of which we take – 8 – 26000 − = 5 26 7 10 12-0 18-1 24-3 30-6 infinite . 36-10 42-16 48-22 d for various M = 6 34 ], the linear terms are precisely the spurious terms. We have 14-0 21-1 35-6 28 -3 42-10 49-15 56-22 d 18 ) implies an This ‘global 1=1 test’ then verifies this solution, where of c 7 2.4 1) 2 3 4 5 6 7 8 d, n ( . The rank of N Table 2 . The number of zero eigenvalues using rounding errors for th 350 determinants ranging from 2 1) as it ought to. We conclude that in At this point it must be pointed out that, in all cases where a d In the OPP method [ × In a certain sense, eq. ( d, 7 ( N in principle, we actually have obtained a solution for, the s precisely decomposable with linear terms, but one of order solution is found, it can easilyof be the tested by loop evaluating momentum eq.the ( equality is supposed to hold only up to the precision used pendix. Rewriting our general linearsee terms that in we terms decompose ofple). propa a pentagon It into can boxes beis checked actually that unique. the triangles always cancel, and to note that ourspecific general property linear leading terms to are fewer not tensor exactly structures, those. and T we Figure 3 position with up to quartic in the loop momentum coefficients. 350 them. JHEP12(2012)038 1 n and . If (3.1) (3.2) 1 l , we have 2 l 4 cannot be imply on the ) provided we ↔ of propagators ≤ 2 3 1 3 l , , n n not 2 3 , ) + 1 j , n n p ree different loop lines. 1 mixed propagators n + out. We consider these ropagators for the three ( 2 l is does , with exchange are the external momenta tly several attempts in this ↔ ) s missing since, by shifting, + k d j ) becomes a double copy of one s them further. 1 p 2 propagators containing both. ]). Let us assume that 3 p l we write the total order of the 5 , the number ( p 2 4 + 1 6 46 ) which indicates the number , n ≤ p l p D n + 3 2 2 , satisfying the following equation ) + l 2 + l n 2 2 2 l j l , n , n 43 + , l 2 1 x , 1 1 n 7 l l p , n . and the ( ( 42 1 j + 2 j 2 x n l 8 2 9 p l ,D m 2 p ) or ( , and the +1 + n ) momenta for which all propagators appear- 1 + 2 1 2 l ) = 1 j l 2 + − n l j 2 1 p X + , , n = 2 + p n – 9 – 1 1 2 j l ) 1 l l + j + 1 p l 2 , n l 2 3 3 ) + l ( p j n + ( ( + p i 1 1 D l p l ,D ) 2 + p + 2 ↔ ) 1 1 i l + , l l ) 1 p l ( 1 3 l ) = ( ( D + j , n j ) . 1 p 2 2 x l 3 ( n , l , n + 1 +1 1 D l 2 i + l ( n n ( j 2 n + x n 1 X D n 1 + =1 = n j X j 1 n = n . An example of the two-loop iGraph (4,2,5). One can see the th The propagators depending on both loop momenta are called Such iGraphs can be denoted by the triplet ( The general strategy consists in finding function As in the one-loop case, a generic graph of order are the two loop momenta. We consider three different kind of p 2 Figure 4 relations of the form ( also exchange properly the externaliGraph momenta. as Predictably, Obviously due to the symmetries of the iGraphs, for instance containing only the other loop momentum these are absent the two integralsloop factor integrals. out and The the same problem one happens can in always case arrange anycases other the solved loop loop (by line the momenta i one such loop that techniques) they and factor will not discus ing in the above equation can be simultaneously on-shell. Th decomposed in this way, since there are where for instance associated with the propagators of the diagram. of propagators that contain only the one loop momentum different loop lines of a generic two loop iGraph. l We now turn to thedirection problem of have reducibility appeared at in two the loops. literature Recen (see i.e. [ 3 Reduction of two-loop3.1 integrands Preliminaries JHEP12(2012)038 6 6 in as = 0 ≥ 2 ≥ 1 l l (3.4) (3.3) (3.5) (3.6) 2 3 3 osable n n n d, since + + 1 and 2 n 1 p n l 6, or + ≥ 6 or ··· e massless, it is mposed down to 2 e: instead one has to ≥ + n 1 2 p n 6, . + all five propagators are ≥ 1 , n 1 osable for any phase-space + 3) n diagram of order 5 in two d = 0 + 3 homogeneous equations is ntum of this loop line and take all j t reduce the propagators in this loop d x , 2 can again solve the problem `ala one loop. +1 11 (= 2 n that are independent of 1 + n j Feynman 1 X ≤ x = 0 = n . j 3 ossible. In the case of 5 propagators we already one-loop with trivial terms, taking µ n j = p + j j = 1 `ala x x 2 j n µ +1 +1 j – 10 – as one of the “external” momenta or vice-versa. 1 1 + n n x n n 1 2 X X l = = j j n =1 j X = = , n j µ ) = 4). Furthermore, with linear terms we see, from the j x d p ) is nonlinear with respect to the external momenta, due to = 11, since by shifting we can arrange 2 j d n n x + =1 3.2 2 j 1 X 4 (= = 0, the only solution to the 2 n n + =1 n j 1 ≤ X p n 3 , + 2 , that we only have to consider two-loop iGraphs with + 4 conditions, so that the minimum size of a trivially decomp 1 , and this fails the inhomogeneous equation. On the other han 9 d n ··· , in fact, be decomposed, but not by the method described abov + can if all internal lines in this self-energy Feynman diagram ar +1 , . . . , n +4. In four dimensions, scalar iGraphs can therefore be deco 1 8 n d p = 1 j )) now reads 3.2 The requirement for trivial decomposition (for A word of caution is in order here. We may have a case where In case there at least 6 propagators in one loop line we can firs This diagram = 0, 9 8 = 11. Again in analogy, for j line with constant coefficients andknow then that continue further adding if p coefficients linear that terms depend on that the depend other loop only momentum to on zero, the we loop mome use IBP techniques. is trivially decomposable (for iGraph is 2 and possible to choose thesimultaneously two on-shell. loop momenta components such that In total there are 2 dimensions: any subset of an iGraph is itself an iGraph, any iGraph with one-loop discussion and mass configuration. A counterexample is the other hand that iGraphs of higher order must always be decomp x eq. ( n as well as and then decide to perform a decomposition Since the solution of the eq. ( the loop momentum, for instance, and JHEP12(2012)038 in we 2 ) (3.7) n i p ins, for appears + j 1 l p may be an dimensions, j d p · 1 l (possibly, but not c number of ISP’s raph of order ). We use general i he loop momentum t m. The integrals of refore not very large: 3.2 menta of the integrals th an iGraph of order as a linear combination grals were always scalar. j . p ny vector can either recon- wer. It is not obvious that 2 j ) (3.8) ient to express any two-loop re difficult to calculate than j t a basis t m gators of the type ( · y different classes of functions. g, in the case a denominator is s commonly accepted that this ability holds for the non scalar + 2 ith fewer denominators. In case e, in general, the simple form of ops this is not the case anymore. l 2 j dying scalar integrals is sufficient ( p ij c − 2 . Then, the product 1 j l 2 X ) j − p ] 2 j ) + j + t m 2 · l ( − 1 – 11 – l 2 ( D ) j ij b p + ]. But not always; the ISP’s of an integral are more j X 1 = 4. l 42 + d i a = [( ’s can form a basis, one can rewrite j i = p ) may, however, not be present in the integral in case p (ISP) [ i · j x p 1 l constants with respect to the loop momenta. Since in + 2 1 ij l = 3, 19 for coefficients. As in the one-loop case, we give a table that conta c ( d n D , and + 1) ij d , b vectors to construct such a basis, it is obvious that for an iG i a d ’s and manage to reconstruct denominators. There is a specifi i = 2, 10 for p d Contrary to the one-loop case, scalar integrals are not suffic A note is in order here. In the one loop case the resulting inte Again, we want to write, if possible, the number 1 as in eq. ( The number of two-loop iGraphs that we have to consider is the + 3. Like in the one loop case, we now add coefficients linear in t d we need start with (2 a scalar integral with aa specific non-scalar number integral of with denominators feweris is denominators; the mo however case. it i in any diagram andthe one resulting can basis have can some have numerators freedom with in ISP’s how in to some write po the with the necessarily, the external momenta in the iGraph) and we have of the The denominator irreducible scalar product struct denominators or be areconstructed spurious the term. remaining After integral integratin the is term a is scalar spurious integral it w One vanishes after can integration. always In use two dotto lo products write relations of like the loop momenta with the mo The reason is that any contraction of the loop momentum with a in a propagator of another type such as linear terms in the sense that in every dimension we construc complicated to write. For example, if there are enough propa and hope for further reductions. With trivial decomposition we2 see that we can always end up wi 3.2 Further reduction with linear terms one with the same denominators. integral. However, as far assince reducibility if is the concerned, scalar stu integral is decomposable, then decompos the diagram such that the iGraphs again, and the emerging integrals will belong to ver 4 for matrix inversion, the resultant decomposition will not hav JHEP12(2012)038 ) j p · 3.8 i l )) and d ( T + 2 case is 4) iGraph in d , 1 , = 2 ases. Reducibility ), we see that the n 3.6 = 1 10 . To see this, we note 9-0 onstrate here a way to ne to have a complete 12-2 15-5 2 d ndependent coefficients. r of independent tensor σ more the reason why we he coefficients of eq. ( res (denoted by rs, there are three external y calculated by cancellation and ssume the (4 e some examples. s point, we are one step away = 2 27 ns every 15-0 20-0 25-1 30-3 35-8 1 d σ with the momentum we borrowed. i l = 3 46 + 1 21-0 28-0 35-1 42-3 49-6 56-10 63-17 d d for linear terms. 2 + 9 2 = 4 69 d 27-0 36-0 45-1 54-3 63-6 72-10 81-15 90-21 99-30 d ). As we mentioned, terms of the form – 12 – 3.8 +1 iGraph. In four dimensions, we can decompose ) = 2 d d ( = 5 96 55-1 66-3 77-6 33-0 44-0 T as a linear combination of the four propagators and an 88-10 99-15 d . As in table = 2 110-21 121-28 132-36 143-47 j ). For all iGraphs defined by eq. ( n p · i 3.2 l = 6 Table 3 127 65-1 78-3 91-6 39-0 52-0 d 195-55 104-10 111-15 130-21 143-28 156-36 169-45 182-55 ) d 5 6 7 8 9 3 4 ( n 15 10 11 12 13 14 T a line distinguishes between reducible and non-reducible c 3 In table We can rewrite the terms in eq. ( is explicitly checked bynumber eq. of ( independent coefficientsstructures, becomes when equal reducibility to is the attained. numbe In all dimensio every dimension we worked with, the number of tensor structu ISP. The ISP in that case would be the product of either reconstruct denominators either become ISP. Let us a four dimensions. It has in principle two ISP that we call reducible with linear terms to a the number of independentprobing coefficients as that described we above. have explicitl that in any ofmomenta. the loop We can lines always thatbasis “borrow” and consists a write of fourth any four product one denominato from the other li As shown, we do ithave as numerically many but independent we coefficients wouldestimate as like this we to number; do understand for find. the We case try to of dem linear terms we will giv every integral down to integralsfrom with the 9 limiting denominators. value At of thi 8. 3.3 Comments on theThe number most of difficult independent part coefficients in our counting is always the number of i Repeating for the other loop line we get the two ISP’s. Using t JHEP12(2012)038 and (3.9) . In 1 2 (3.10) ) 1 = 72 σ i σ × products of two 3 + 45 × of the OPP method 10 f the type ( ould avoid terms like hen, rewriting ce ts of two denominators n iGraph of order 10 we s of the terms with the reducible. Adding more the tensor structures for 2 ic power in loop momenta e same way one can count . ) ich means that we have to } ible that special properties s propagators and ISP’s in 2 raints to cancel, namely the 1 clear that it is safer to find l { · . (1) j e a lot of equations by putting j 2 1 l D . σ D ( i i 2 ) ys a role in the counting. b , 2 D D 2 2 i l i + , σ ) ≥ 9 ≥ 1 2 9 1 l X j,j X σ j,j · , σ 1 9 1 b 9 =1 l 9 =1 i X ( i X , + 1 + 2 1 l · } 6 = 66 independent coefficients, which is the ) + ) 2 2 ,...,D a – 13 – 2 there is a constant coefficient in front of every of l 1 − , σ i · , σ = D 1 1 1 ( D l } } ( , σ 2 , σ 2 ∝ 1 , { (1 , σ 4 2 i i , σ 1 10 1 D D , l D , σ 2 2 , σ 1 l 9 1 =1 =1 10 2 1 { i i X X { , l 4 1 l 1 = 1 = ) becomes 3.9 general numbers. In our particular example, we have 9 2 b , 1 ) we can either get a denominator times a constant or an ISP, or b 3.2 This is actually the point where the number of dimensions pla in the form of true ISP’s of every subdiagram’s contribution . We expect something similar to the one-loop case to happen t 2 i 10 2 highest number of denominatorsexist, to as cancel. again in For theautomatically that one-loop tensor it case structures where is to spurious poss zero. term solv σ their number numericallyhigher since cases. there are Thethe way a of example lot rewriting above of the isto overlaps general two still linear in loops, not terms oneD the a would OPP find method. the ISP’s In of an every extension subdiagram and w that case we havepropagators 69 we independent do coefficients notthe get and more independent this independent coefficients graph coefficients. in is In all th dimensions although it is these different terms: denominators. We write this equation schematically as This means that by writing We still need them, though, in the first term to produce terms o where a, in eq. ( We don’t need to go up to 10 in the product of 2 denominators sin terms. As a result we end up with 72 number we get numerically ascan well. prove that If eq. we ( now try to decompose a and ended with uptimes to some quartic constants. powers Theseput since higher extra we powers constraints have on have these our to coefficients. produc cancel, We wh have 6 such const coefficients. However, we started with a problem with up to cub JHEP12(2012)038 . n r 5 (3.11) (3.13) . With d allowing . ν 2 ) )) l l l t t µ 1 l · · ) ible cases, ac- 2 ) 2 2 l l l k t · )( )( · k k 1 t 1 l t l ) · · k adratic terms in the 1 )( 2 t l l j . 2 (3.12) · t on in 2 dimensions. We )( appens if we add cubic, )( · 1 and ( j )) j l − t 1 t k l ν 1 t · )( · l ( ! j · 1 µ 2 here is no such solution in 3 1 t e able to find the number of l cally, for this particular case l l 2 onal here because of the extra · ( 1 d ijk l ( 2 ) 2 mensions. We put our findings l d 1 k )( l t k j ( ijkl

ijkl · t ≤ = 2. In that case, there is more q h j · X 2 l ijk l k 2 d 1 ≤ d l ≤ n . )( k ( j X + 18 k − j ) + X coefficients in total, not all of them of ≤ t ≤ j j 3 j · X t l ! ijk n · f + 1) X 1 d/ l 2 d 2 d ( ) + × l ) + l ( j,k t j ) and the original number of coefficients with X ) + t

ijk l · ij d structure from + 20 + 3 t · f c ( 1 · – 14 – l 2 2 2 + 1) ν 2 ) + T l d d j l 1 ( k )( j,k l d X µ 2 X + 28 t k l ij t · )( 2 1 c k · l ! 2 + 3 t l ) + ) + ) is not valid for 2 j · l j k 2 ) = (2 3 d d 3 + 10 )( X t t 1 ( d d / j )( l · · ( t 3 j T 1

t · 1 2 )( d l l ) + j · ). The coefficients depend on the number of propagators C 2 ( t j l d ) shows that it always evalutes to integers for integer )( 2 t · ( ij ( l j d · = 8 b ( 1 t ( 1 l · 1 ijk ) = 4 C l ( T j e d 2 ( ijkl l X ( k p ( ij ijkl T ≤ b k + j g X ijk ≤ l i j j X e + a ). As we see, we can decompose in 2 dimensions an iGraph of orde ≤ X k k l X ≤ ≤ = 3.2 + j j X X a line distinguishes again between reducible and non-reduc i i 4 + + + a x = . i 4 x The second form for In table fewer independent structures to be constructed. quadratic terms we start with (2 course being independent.independent Using coefficients cancellation for probing different we iGraphsin ar in table different di We give the number of tensor structures Notice that the expression for general quadratic terms, one can completely reconstruct the overlap between the highest tensor structures. More specifi We focus now on 3include and general 4 cubic dimensions terms since and we our finished coefficients the become reducti and 4 dimensions andquartic in terms and this so case on.properties we The that have two-dimensional lower to case the is investigate excepti number what of h tensor3.5 structures. Reduction with general cubic terms cording to eq. ( to lower order iGraphs as indicated by unitarity. However, t We try tocoefficients. reduce In our this iGraphs case the further coefficients with become the use of general qu 3.4 Reduction with general quadratic terms JHEP12(2012)038 . 5 . The (3.15) n . + 1 ! 1 d ensions and we find

9 = 2 60 45-3 60-6 75-15 90-30 d ! = 2, where the same can bic terms we are able to + 1 (3.14) ensions, and managed to iGraph as expected from wer iGraphs with general d we can actually solve the 2 d to dimension 6, in table is is the number of tensor ! d ll the tensor structures can 11

1 d = 3 144 84-3 rse for 128-6 d

140-16 168-32 196-53 224-80 252-108 + 16 ! + 20 3 d ! = 4, and we get a valid decomposition: = 4 270

d 2 d 135-4 180-6 d 225-18 270-38 315-65 360-98 for quadratic terms. 405-136 450-180 495-225 540-270 6 + 1 = + 1 or higher, using cubic terms. We believe 2 ) and the original number of coefficients with

= 8 d/ d d ( n – 15 – T ) coefficients, not all of them being independent. 6 + + 77 = 5 d / 448 ( 2 198-3 264-6 d ! 1 330-20 396-44 462-77 d 3 + 1) 528-118 594-166 660-220 726-279 792-344 858-410 C 3. 3 d d/ ≥ . As in table

d . 3 + 71 + 11 / = 6 3 2 686 d + 68 d 273-3 364-6 d 546-50 637-89 455-22 728-138 819-196 910-262 ). The coefficients depend on the number of propagators 1001-335 1092-414 1183-498 1274-588 1365-679 Table 4 ! d ( 4 d 1 3 + 4 ) 3 + 22 / d C / 6 7 8 9 3 4 5 3 ( n 10 11 12 13 14 15

4 d ) is valid for T d d 16 ( T ) = (4 ) = 2 d d ( ( dimensions we start with 1 T C d In http://codepad.org/zT4wUxCJ 11 expression for system decomposing any iGraph ofcubic order terms, 7 and in perform 3decompose the dimensions any 1=1 to two-loop test. lo iGraph This in means 3 that dimensions with to cu up to a 2 We give the number of tensor structures general cubic terms unitarity. In the same way, we can investigate We run the MAPLE codethat in out the case of of 588structures the original iGraph as of coefficients, well. order 360 7 Using are in another 3 independent. PYTHON-based dim program, Th be reconstructed. Actually wedecompose did every the integral same of in generic 5, order 6, 2 7, and 8 dim from our original 1485 coefficients, 831 are independent and a be achieved with only quadratic terms. We put our findings, up that this is a general result for any dimension, except of cou JHEP12(2012)038 d L , Q , ) cannot d CU are always ≤ d 3 , 2 , 1 n = 2 e and non-reducible 114 d umerical method, we 105-15 140-32 175-61 210-96 onsider such configura- ctively. e-loop iGraph of order one- and two-loop ampli- uch that for specific values ns, two notes are in order. uch a decomposition is not r higher than 2 e for that values of the loop = 3 6 + 1 n to deal with iGraphs in the 360 mposition was achieved. That on can be achieved. d e points, the integrals are still 420-88 d/ or the one loop case and quartic 504-152 588-228 672-312 educible from the non-reducible or smaller with at two loops propagators can be d 6 + d / 2 are not. We specify, in the cases the d = 4 831 d d for cubic terms. 1155-352 1320-497 1485-654 1650-819 2 that describes the reducibility of the iGraphs 3 + 71 / 6 3 d – 16 – = 5 1631 d in one loop and 2 3 + 22 2574-971 / . As in table 2860-1237 3146-1515 3432-1801 d 4 d ) = 2 Table 5 = 6 d ( 2876 d T 5005-2157 5460-2592 5915-3039 6370-3494 ) d 5 6 7 8 9 3 4 ( n 10 11 12 13 14 T , the horizontal line distinguishes again between reducibl 5 stand for cubic, quadratic, linear and constant terms respe co Summarising our findings, we give table Secondly, in section 2.1 we have a theorem stating that any on With this table we finish our analysis for the reducibility of In table be decomposed in theincluded way coefficients we of proposed. even higherfor rank As the (quadratic, an two-loop cubic case) extra f and check asis of it an was our extra expected, n no proof further that deco our numerical method is robust. or smaller (and similarly any two-loop iGraph of order 2 tions. We are interested infirst general place). cases (that In wasreducible our the via experience, reaso IBP in identities. these special phase-spac set to zero simultaneously.momenta That the would equation lead 1 to = problems 1 is sinc not satisfied. However, we do not c and Firstly, there may beallowed. specific An example phase-space of points suchof a for the phase-space which point loop could s momenta, be more one s than tudes at the integrand level. Before we move to our conclusio iGraph is reducible, the smallest rank for which the reducti reducible, while the ones with less or equal to 2 we investigated. The horizontal linecases is in to all distinguish dimensions. the As r we described, the iGraphs of orde cases. JHEP12(2012)038 12 r terms ity-based proof that reductions ecomposed using IBP rated (in the two-loop hat one has to take our dratic, linear and constant ed in this way, is clearly wo-loop amplitudes at the · · · L L = 2 Q co co ach subdiagram seperately d NOT NOT ms are needed (quadratic for se linear terms and the OPP’s e interplay between OPP and or every subdiagram. he decomposability of Feynman might or might not be spurious erms of propagators (this would · · · L L = 3 co co CU d NOT NOT · · · L L = 4 co co CU d NOT NOT – 17 – · · L L = 5 co CU d NOT NOT ] are actually valid and survive the global 1 = 1 test. L L = 6 co 43 CU d NOT NOT , 42 , respectively. For one-loop graphs, the inclusion of linea d 6 7 8 9 3 4 5 n 16 10 11 12 13 14 15 = 2 n and = 8) that generic iGraphs can be decomposed down to the unitar d d ]). = 43 n . Reducibility of iGraphs. CU, Q, L and co stand for cubic, qua Our work is basically the starting point of an OPP method andWe a note once again that the resulting integral basis, obtain In the same way that one-loop OPP has different spurious terms f 12 = 2). If one wants to design a two-loop OPP method, it is clear t general linear, quadratic, cubic termslead and rewrite to them contributions in t with less denominators) and ISP’s of e identities. Achieving aIBP proper will level certainly of open understandingintegrand the of level. road th for an efficient reduction of t not a minimal one. We are aware of cases that could be further d spurious terms. Atd the two-loop level, ultimately cubic ter is sufficient, and we have elucidated the relation between the leading to scalar and non(see scalar i.e. integrals, [ where the latter Table 6 limit of By introducing the notion of iGraphs,amplitudes we have at investigated t one andcase, two up loops. to In both cases we have demonst terms respectively. 4 Conclusions that were conjectured in [ JHEP12(2012)038 (A.4) (A.5) (A.1) (A.2) (A.3) ) + , , , , . 5 ) ) ) ) ) of the denomi- 4 5 1 2 3 , p i 4 , p , p , p , p , p p 3 4 5 1 2 terms must vanish + , p , 2 µ , p , p , p , p , p ) ) q p 2 3 4 5 1 2 n ( 2 ... p p p p p p q µ ( ( ( ( ( , p ǫ menta µ µ µ µ µ 1 + ǫ ǫ ǫ ǫ ǫ ) + − q 2 , p ( − − − − − ) 5 p 5 . ) ) ) ) ) , p 5 1 2 3 4 ) + ). The ) = , p 4 4 5 q 4 p , p , p , p , p , p ...D ( ( 3 4 5 1 2 , p A.4 , p ) 4 , p 3 D 4 2 3 ] , p , p , p , p , p p ) p 2 2 3 4 5 1 , p 1 ,D , p ( 5 2 p p p p p S ) 3 µ + ( ( ( ( ( 2 1 . ǫ , p ˜ µ µ µ µ µ , p d q ( , p . 4 ǫ ǫ ǫ ǫ ǫ 1 ( , p 2 µ p 5 + p q , p ( D ( 3 2 4 ) , p ǫ ) + ) + ) + ) + ) + = 0 ) = 0 1 d 4 5 1 2 3 4 µ i D i , p p ) = ) 2 p A 5 , p , p , p , p , p , p 3 i . – 18 – + 3 4 5 1 2 3 p ˜ )) + , p d ) ) + [ 5 p , p =1 4 q 1 ( 5 1 i ( X , p , p , p , p , p 2 ( + p 4 p p 2 3 4 5 1 , p i ( D p p p p p , p 3 5 X , q ( ( ( ( ( + + 1 ,D 4 µ µ µ µ µ D , p ) ] we get q q p ǫ ǫ ǫ ǫ ǫ , p 2 5 ( ( ) = 5 p 18 + n − − − − − ( D D p , p ] ] ( µ 4 ) ) ) ) ) 5 1 4 ǫ 5 1 2 3 4 , q S S 3 , p D 5 1 3 − p , p , p , p , p , p ˜ ˜ d d 4 5 1 2 3 ) , . . . , p . , p + 5 2 + + 2 1 , p , p , p , p , p 3 4 5 1 2 p , p 5 1 , p B , q ( p p p p p 3 d d 1 2 4 ( ( ( ( ( [ p p + µ µ µ µ µ , p ( D ǫ ǫ ǫ ǫ ǫ 2 µ 1 n which means + p A ( = = = = = D q 1 = [ q µ ( µ µ 1 µ 2 µ 3 µ 4 µ 5 ǫ ǫ q A A A A A = 1 S We compare the polynomials of the two sides of eq. ( for any value of we try to decompose this pentagon We acknowledge useful discussions with Prof. M. Czakon. A Pentagons to boxesBy with defining the OPP method Acknowledgments to the 5 following boxes We write the spurious terms in the following form nators sum up to zero: We then have: By shifting the loop momenta we can always arrange that the mo Using the OPP master formula [ JHEP12(2012)038 . ˜ d (A.9) (A.6) (A.7) (A.8) , and the ). They )) 2 4 q p A.4 · q )( 5 . A ) ) ) 4 3 4 − w on we call them , p , p , p 4 ) 3 2 3 4 o terms, the A nt of the vectors are zero , p , p , p sum , p 1 1 2 2 p p p ( ( ( ) + ( , p 3 µ µ µ 1 , , , . ǫ ǫ ǫ p p ) ) ) system: · ( 5 5 5 µ ˜ ˜ ˜ q d d d 4 ǫ . = 0 = 0 = 0 = 0 ) p )( 5 − − + . 5 5 5 5 5 5 ˜ ˜ . d d ˜ ˜ ˜ ˜ 4 4 4 − µ 5 d d d d A ˜ ˜ ˜ µ i d d d 3 A = + + − − + , p 5 A − − + 4 ˜ 4 ) d 4 4 4 4 ˜ 3 ˜ i + 4 d d ˜ ˜ ˜ ˜ 3 3 − d d d d p ˜ ˜ A 3 d d + ) + 2 = 0 · ˜ + = d 1 + − + p i ) is satisfied. q − + 3 p + 4 3 3 3 3 ( ˜ ··· parts of the right-hand side of eq. ( A − ˜ · d ˜ ˜ ˜ − 2 2 d 3 d d d – 19 – ) + ( ˜ ˜ 4 ˜ 2 d d q 5 d 4 q 1 2 =1 A.5 + ˜ 5 d i =1 = − + X p p i )( − X µ 1 + − − · 2 2 5 µ 2 − + 4 − 2 ˜ ˜ ˜ 1 A 2 2 d d d ˜ q ˜ A d ˜ ˜ ˜ 1 1 1 d dq d d = ˜ ˜ ˜ 4 d d d )( + = − + 5 5 + + 4 − 1 − − − 1 p 1 1 ˜ A ˜ 1 1 1 d ˜ d d ˜ ˜ ˜ A d d d 4 +( +( +( − ) (( − − − 2 µ A ˜ ) = ( are equal eq. ( A.3 dq ( µ i i ˜ d A = i ˜ d µ i ( A ) i X p · q ( 5 =1 i X µ ˜ dq term. We look at the latter term We have: This is actually the only solution. To see that, consider the Notice that from eq. ( In 4 dimensions, for this sum to be zero, all coefficients in fro One can solve the system and find The only solution is when all coefficients are equal and from no ν q µ q cancel as well but one has to be careful since they come from tw which means that if all Substituting we get (the vectors are linearly independent). This results in the Then we take a look at the quadratic in JHEP12(2012)038 2 q ] , in the (A.12) (A.10) (A.11) , which ˜ ommons d − µ ]. e-time (2009) 007. + µ (1984) 241 , (1979) 365 SPIRE ) + ) and one hep-ph/9403226 IN 2 4 [ i p p d · · q q B 137 B 153 t makes the solution to )( )( point gauge theory 4 3 namely constant part, . N , p , p EPS-HEP2009 ) 3 2 4 annihilation into (1994) 217 (1965) 181 [ , p , p , p − 1 1 d reproduction in any medium, 3 e and the system has a solution. PoS e 40 , , p p Phys. Lett. , ( ( + ]. ) ) , p , . , e Nucl. Phys. µ µ 4 4 2 ) ) ǫ ǫ , B 425 One loop 4 3 , p , p , p − − 3 2 1 , p , p ) ) p SPIRE 3 2 ) we are left with 5 , p , p 1 3 ( 2 1 IN ǫ p p [ , p , p p p 2 · · 1 1 ( ( A.8 p p Nuovo Cim. q q ˜ µ µ dq ( ( ǫ ǫ , 5 )( )( µ µ 5 5 4 4 ǫ ǫ Nucl. Phys. − − − , – 20 – , p , p = 3 2 = = 5 = = 5 ]. µ i (1979) 151 Large loop integrals , p , p 5 5 5 5 2 1 A A A A A ) p p i ( ( p µ µ − − − − SPIRE One loop corrections for · ǫ ǫ solving 9 out of 10 equations in one go. The total number 1 2 3 4 Scalar one loop integrals IN B 160 q is again used. With this substitution we have: terms all vanish owing to the Schouten identity except for [ 2 A A A A ( = q 4 ν 5 µ p q =1 i q X µ ) − q µ 4 Special class of Feynman integrals in two-dimensional spac 3 ˜ dq p , p 3 − (1965) 299 Nucl. 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