Stable one-dimensional integral representations of one-loop N-point functions in the general massive case. I -- Three point functions

Abstract

In this article we provide representations for the one-loop three point functions in 4 and 6 dimensions in the general case with complex masses. The latter are part of the GOLEM library used for the computation of one-loop multileg amplitudes. These representations are one-dimensional integrals designed to be free of instabilites induced by inverse powers of Gram determinants, therefore suitable for stable numerical implementations.

Published for SISSA by Springer Received: October 17, 2013 Accepted: November 4, 2013

Published: November 19, 2013

Stable one-dimensional integral representations of one-loop N-point functions in the general massive case. I Three point functions

J.Ph. Guillet,a E. Pilon,a M. Rodgersb and M.S. Zidia,c

aLAPTh, Universit de Savoie, CNRS,

B.P. 110, Annecy-Le-Vieux, F-74941, France

bIPPP, Department of Physics, Durham University,

Durham, DH1 3LE, United Kingdom

cLPTh, Universit de Jijel,

B.P. 98 Ouled-Aissa, 18000 Jijel, AlgrieE-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: In this article we provide representations for the one-loop three point functions in 4 and 6 dimensions in the general case with complex masses. The latter are part of the GOLEM library used for the computation of one-loop multileg amplitudes. These representations are one-dimensional integrals designed to be free of instabilites induced by inverse powers of Gram determinants, therefore suitable for stable numerical implementations.

Keywords: NLO Computations

ArXiv ePrint: 1310.4397

Open Access doi:http://dx.doi.org/10.1007/JHEP11(2013)154

Web End =10.1007/JHEP11(2013)154

JHEP11(2013)154

Contents

1 Introduction 2

2 Outline of the derivation 3

3 det(G) ! 0 whereas det(S) non vanishing 10

4 det(G) ! 0 and det(S) ! 0 simultaneously 124.1 Characterization of the specic kinematics det(G) = 0, det(S) = 0 12

4.2 Behaviour of I43 and I63 when det(G) 0, det(S) 0 14

4.3 Extension to the complex mass case 164.4 A comment on the GOLEM reduction formalism when det(S) = 0 17

5 Summary and outlook 18

A Useful algebraic identities among determinants 18A.1 Preliminaries 18A.2 The identity (3.1) and Jacobi identities for determinants ratios 20

B Kinematics leading to a vanishing det(G) 21B.1 General considerations 22B.2 Focus on N = 3 23

C Spectral features of S for N = 3 23C.1 Eigenvalues 23C.2 Eigenvectors 25

D Analysis of the reduction coe cients (b0)j, bj, B0 and B when det(G) and det(S) ! 0 27

E A relation between the zero eigenmodes of S and G(N) when

det(G) = 0, det(S) = 0 31

F The directional limit det(S) ! 0, det(G) ! 0 of eq. (4.1) is actually isotropic 33

1

JHEP11(2013)154

1 Introduction

The Golem project [1] initially aimed at automatically computing one loop corrections to QCD processes using Feynman diagrams techniques whereby 1) each diagram was written as form factors times Lorentz structures 2) each form factor was decomposed on a particular redundant set of basic integrals. Indeed when the form factors are reduced down to a basis of scalar integrals only, negative powers of Gram determinants, generically noted det(G) below, show up in separate coe cients of the decomposition. These det(G), albeit spurious, are sources of troublesome numerical instabilities whenever they become small. The set of basic integrals used in the Golem approach is such that all coe cients of the decomposition of any form factor on this set are free of negative powers of det(G). Let aside trivial one- and two-point functions, the Golem library of basic functions is instead made of a redundant set involving the functions In3(j1, , j3), In+23(j1), In+24(j1, , j3) and In+44(j1). Here the

lower indices indicate the number of external legs, the upper indices stand for the dimension of space-time, and the arguments j1, , ji labels i Feynman parameters in the numerator

of the corresponding integrand. The strategy is the following. In the phase space regions where det(G) are not troublesome, the extra elements of the Golem set are decomposed on a scalar basis and computed analytically in terms of logarithms and dilogarithms. In the phase space region where det(G) vanishes these extra Golem elements are instead used as irreducible building blocks explicitly free of Gram determinant and provided as one-dimensional integral representations computed numerically.

Much faster and more e cient methods than those relying on Feynman diagrams techniques have been developed, e.g. based on unitarity cuts of transition amplitudes and not individual Feynman diagrams, and/or processing the decompositions at the level of the integrands [28]. Yet these methods still amount to a decomposition onto a set of basic integrals. In this respect the stand-alone relevance of the Golem library of basic functions, initially developed as a part of the Golem approach, remains. Furthermore the decompositions obtained by these new methods project onto a basis of scalar integrals and thus are still submitted to numerical instabilities caused by det(G). The issue of numerical instability is then addressed in various ways ranging from smoothing numerical interpolations over the regions of instabilities [9] to more involved rescue solutions [10, 11]. In [12] the solution adopted is to provide a rescue alternative relying on the Golem decomposition to compute the amplitude in the troublesome kinematic congurations. The Golem library [13], initially designed for QCD, did not include basic functions with internal masses yet provided a convenient way of handling infrared and collinear singularities inherent in the massless case. Its completion with the cases involving internal masses, possibly complex, extends its range of use [14]. This completion shall supply the functions In3(j1, , j3), In+23(j1),

In+24(j1, , j3) and In+44(j1) in the massive cases in a numerically stable with respect

to det(G) issues.

To handle det(G) issues, we advocate the use of one-dimensional integral representations rather than relying on Taylor expansions in powers of det(G). The latter may be thought a priori better both in terms of CPU time and accuracy, however the order up to which the expansion shall be pushed may happen to be rather large. Furthermore, unless

2

JHEP11(2013)154

p1

q1

q3

q2

p3

p2

Figure 1. The triangle picturing the one-loop three point function.

a xed large number of terms, hopefully large enough in all practical cases, be computed, it is not easy to assess a priori the optimal order required to reach a given accuracy. Actually this assessment would demand a quantitative estimate of the remainder as a function of the order of truncation, which, as with the Taylor expansion with Laplace remainder, namely requires the computation of an integral! Originally, we proposed the antipodal option of computing numerically the two- or three-dimensional Feynman integral dening respectively the three- and four point functions, more precisely hypercontour deformations thereof [1] that would be numerically more stable. Yet the computation of these multiple integrals was both slow and not very precise. It is far more e cient both in terms of CPU time and accuracy to evaluate a one-dimensional integral representation, insofar as one is able to nd such a representation. In the case without internal masses, we indeed found such a representation.

The issue which we address here is the extension of this approach of one-dimensional integral representations for our set of basic integrals in the most general case, i.e. with internal complex masses. In this article we treat the case of the three point function. The case of four point functions is more involved therefore it will be elaborated separately in a companion article. We follow the approach developed by tHooft and Veltman in ref. [15]. In a subsequent third article, we will present an alternative approach providing integral representations for both three and four point functions equivalent to the one presented here yet with a number of new features and advantages. The present article is organized as follows. Section 2 sketches the derivation of the three point function leading to our integral representation. Section 3 treats the case when det(G) vanishes whereas the determinant of the kinematic matrix S remains non vanishing. Section 4 elaborates on the more tricky

case when both det(G) and the det(S) vanish. The main body of the text presents the

general arguments whereas the various technical details supporting the latter are gathered in appendices, to make the reading of this article more uent.

2 Outline of the derivation

A generic three point function can be represented by the diagram of gure 1.

Each internal line with momentum qi stands for the propagator of a particle of mass mi. We dene the kinematic matrix S, which encodes all the information on the kinematics

3

JHEP11(2013)154

associated to this diagram by:

Si j = (qi qj)2 m2i m2j (2.1)

The squares of di erences of two internal momenta can be written in terms of the internal masses mi and the external invariants si = p2i so that S reads:

S =

2 m21 s2 m21 m22 s1 m21 m23 s2 m21 m22 2 m22 s3 m22 m23 s1 m21 m23 s3 m22 m23 2 m23

(2.2)

In this section, we will sketch the computation of I43 and I63 using the method of ref. [15]. These two integrals are dened1 by:

I43 = Z

1

JHEP11(2013)154

3

Xi=1zi

!

1

2 z T S z i 1

3

0 Yi=1dzi 1

(2.3)

1

In+23 =

(1 + )

Z

3

Xi=1zi

!

1

2 z T S z i

3

0 Yi=1dzi 1

12 z T S z i

= Idiv3 + I63 (2.4)

where Idiv3 isolates the MS ultra violet pole in , and I63 is the nite part which we will focus on. We may single out any index a in S = {1, 2, 3} and writeza = 1

Xi6=azi (2.5)

The quadratic form z T S z becomes: z T S z =

Xi,j6=aG(a)ij zi zj + 2

Xj6=aV (a)j zj + Saa (2.6)

with

G(a)ij = (Sij Saj Sia + Saa), i, j 6= a (2.7)

V (a)j = Saj Saa j 6= a (2.8)

The matrix G(a) is the 2 2 Gram matrix built from the four-vectors ia = qi qa:

G(a)ij = 2( ia. j a). Its determinant does not depend on the choice of a, and it is also the determinant of the similar Gram matrix built with any subset of two external momenta. We note it simply det(G) without referring to a and unambiguously call it the

1The Feynman contour prescription in the propagators is noted i in order to avoid any confusion with the parameter = (4 n)/2 involved in dimensional regularization.

4

=

(1 + )

Z

3

Xi=1zi

!

1

3

0 Yi=1dzi 1

1 ln

Gram determinant associated with the kinematic matrix S. Specifying for example a = 3,

I43 reads:

I43 = Z

1

0 dz1 Z

1z1

0 dz2

1

2

1

2

Xi,j=1G(3)ij zi zj

2

Xj=1V (3)j zj 12Saa i

(2.9)

The 2 2 Gram matrix G(3) and the column two-vector V (3) are explicitly given by:

G(3) ="

2 s1 s3 s2 + s1 s3 s2 + s1 2 s3

#

(2.10)

JHEP11(2013)154

V (3) =

"

s1 m21 + m23 s3 m22 + m23 #

(2.11)

We then dene

z1 = 1 x z2 = y

and we get:2

I43 = Z

0 dx Z

1

x a x2 + b y2 + c x y + d x + e y + f i

1 (2.12)

I63 =

Z

0 dx Z

x

0 dy ln

a x2 + b y2 + c x y + d x + e y + f i

1

(2.13)

with:

a = s1

b = s3

c = s3 + s2 s1

d = m23 m21 s1

e = s1 s2 + m22 m23

f = m21

(2.14)

Eq. (2.12) is the starting point of the computation of the three point integral in ref. [15], cf. their eq (5.2). We keep the same notations as those of ref. [15] for the di erent quantities, and we closely follow the strategy of ref. [15] for the rst integration. We only sketch these stages. An alternative strategy may be proposed which leads to the sought integral representations in a faster, more straightforward and more transparent way for three point functions, and which can be elaborated for four point functions as well thereby providing a number of interesting features. This alternative will be presented in a subsequent publication.

2The argument of the logarithm appearing in eq. (2.13) shall be understood to contain an implicit arbitrary factor 1/M2 with dimension -2 in order to make the argument of this logarithm dimensionless. This arbitrary M2 dependence is cancelled by the corresponding one in the ln(M2/2) involved in the term Idiv3, where 2 is the dimension two parameter introduced by the dimensional regularization of the ultra violet divergence subtracted in Idiv3. In practice the kinematic matrix S is rescaled from the start by its entry of largest absolute value, and so is the Gram matrix G(a), which thereby become both dimensionless. This amounts to specifying M2 to be this normalization parameter.

5

The integration variable y is rst shifted according to y = y + x, the parameter is being chosen such that

b 2 + c + a = 0 (2.15)

in order that the quadratic form of x, y in the integrands of eqs. (2.12), (2.13) become linear in x. Note that the discriminant of eq. (2.15) is minus the Gram determinant det(G).

For all kinematical congurations p1, p2, p3 = p1 p2 involved in one-loop calculations of

elementary processes of interest for collider physics, det(G) is non-positive.3 The roots

of the polynomial (2.15) are thus real in all relevant cases. We split the integral over y and reverse the order of integrations:

Z

1

0 dx Z

1

dy = Z

1

0 dx Z

(1)x

0 dy Z

1

0 dx Z

x

0 dy

JHEP11(2013)154

Z

= 0 dy Z

1

1

Z

0 dy Z

1

dx (2.16)

Since the integrand seen as a function of x and y in eq. (2.16) is now linear in x the integration on x is made straightforward. For I43 eq. (2.16) involves two integrals of the form

Z

1

dx

y/(1)

y/()

1

A

ln

A + B Axmin + B

(2.17)

where A and B are functions of y and xmin(y) = y/(1 ) and xmin(y) = y/()

respectively. As can be traced back to eqs. (2.3), (2.4) the polynomial [a x2 + b y2 + c x y +

d x + e y + f i ] in eqs. (2.12), (2.13) has a negative imaginary part, this holds true also

for complex internal masses. Therefore the numerator and denominator in the argument of the logarithm in eq. (2.17) both have a negative imaginary part, thus the logarithm in eq. (2.17) can be harmlessly split in two terms:

ln

xmin(y)

dx [A x + B]1 =

A + B Axmin + B

= ln (A + B) ln A xmin(y) + B

(2.18)

It is convenient to add and subtract a term ln(C) in the right hand side (r.h.s.) of eq. (2.18),

and split the latter into a sum of two terms

Z

1 dx [A x + B]1

= 1

A

1 [ln (A xmin + B) ln (C)] (2.19)

such that the residue of the fake pole 1/A vanishes in each combination [ln(A+B)ln(C)]/A

and ln(A xmin + B) ln(C)]/A separately. The two terms in the r.h.s. of eq. (2.19) thus

3As seen by exhaustion, the only congurations leading to a positive Gram determinant would require that all three external four-momenta p1, p2, p3 = p1 p2 of the three point function be spacelike. At the one-loop order which is our present concern, each of the three points, through which p1, p2 and p3 respectively ow, shall be connected to an independent tree. In order for p1, p2 and p3 to be all space-like, each of these trees should involve one leg in the initial state: this would correspond neither to a decay nor to a collision of two incoming bodies.

[ln (A + B) ln (C)]

6

lead to integrals over y which are individually well dened and may be safely handled on their own. A similar treatment may be done for I63 adding and subtracting a term C ln(C).

We note that

B|A=0 =

1

2B i

with

B

det(G) det(S)

(2.20)

thus we choose

1

2B i (2.21)

In this way the integration over x yields four terms. By means of an appropriate change of variable, two of them may be further recombined so that each of the integrals I43 and I63 can be written as the sum of three terms. We call these terms sector integrals labelled

I(j), j = 1, 2, 3, they may be put in the following form. For I43 we get:

I43 =

3

JHEP11(2013)154

C =

Xj=1I43(j) (2.22)

with the sector integrals I43(j)of the form

I43(j) = Z

1

0 dz

K(j)()

D(j)z + E(j)

ln F(j)z2 + G(j) z + H(j) i

ln

12B i

(2.23)

The coe cients D(j), , K(j)() being provided by the following table; the dependence of

the K(j)() on is made explicit for further convenience.

sector (1) sector (2) sector (3)

D(1) = (2b + c) D(2) = (2b + c)() D(3) = (2b + c)(1 )

E(1) = (d + e ) + (2a + c ) E(2) = (d + e ) E(3) = (d + e )

F(1) = b F(2) = a F(3) = (a + b + c)

G(1) = (c + e) G(2) = d G(3) = (d + e)

H(1) = f + d + e H(2) = f H(3) = f

K(1)() = 1 K(2)() = K(3)() = (1 ) (2.24)

where a, b, , f have been listed above in eq. (2.14).

Similarly, for I63(j) we have:

I63 =

1

2 +

3

Xj=1I63(j) (2.25)

with I63(j) of the form

I63(j) = Z

1

0 dz

K(j)()

D(j)z + E(j)

F(j)z2 + G(j) z + H(j)

ln F(j)z2 + G(j) z + H(j) i

+ 1

2B ln

12B i

(2.26)

with D(j), , K(j)() given in table (2.24) above.

7

The values of the integrals I43 and I63 do not depend on the particular root =

of eq. (2.15) chosen to perform the rst integration leading to eqs. (2.23), (2.26). As in ref. [15], either of the two roots, say +, may be used to further compute the remaining single integrals in closed form in terms of logarithms and dilogarithms. A symmetrization over would generate an unnecessary doubling of dilogarithms in the closed form that

would be prejudicial regarding CPU time in practice. However the discussion of the behaviours of these integrals when det(G) 0 is made somewhat obscure once one particular

choice is made, and for this purpose it is on the contrary more enlightening to symmetrize expressions (2.23) and (2.26) over , especially in the perspective of providing one di

mensional integral representations free of det(G) instabilities. The dependence comes only from the factors K(j)()/(D(j)z + E(j)), not from the arguments of the logarithms in numerators. Each of the sector integrals in the decomposition of I43, respectively I63, has an explicit dependence of the type:

I =

Z

1

0 dy

K() A + C L

where L stands for the -independent numerators in the integrands of the sector integrals

I4,63(j), and we omit the superscript (j) labelling the sector for simplicity. Symmetrizing

over we get:

I =

1

2

JHEP11(2013)154

Z

0 dy

(K(+) + K() +) A + (K(+) + K()) C + A2 + A C ( + +) + C2

1

L

Let us introduce the following quantities:

Q = + A2 + A C ( + +) + C2

= 1

b (a A2 c A C + b C2)

N = (K(+) + K() +) A + (K(+) + K()) C

Here are the explicit forms corresponding to the di erent sector integrals.

For sector (1), K() = 1, A = 2 b z + e + c, C = c z + d + 2 a, and we get:

Q = 1

b

b z2 (c + e) z + a e2 c e d + b d2 (d + a)

= 1

b

det(G) g(1)(z) + 12 det(S)

(2.27)

N = 1

b [2 b d c e ]

= 1

b b1 det(S) (2.28)

8

For sector (2), similarly, K() = , A = c z + e and C = 2 a z + d, so that:

Q = 1

b a z2 d z + a e2 c e d + b d2

= 1

b

det(G) g(3)(z) + 12 det(S)

1b [2 a e c d]

b b2 det(S) (2.30)

For sector (3), K() = (1 ), A = (2 b + c) z + e and C = (c + 2 a) z + d, so that:

Q = 1

b (a + b + c) z2 (e + d) z + a e2 c e d + b d2

= 1

b

det(G) g(2)(z) + 12 det(S)

1b [2 b d + c d 2 a e c e]

b b3 det(S) (2.32)

where the coe cients bj are dened by

bj =

They are such that

They were introduced in the GOLEM reduction algorithm [1], and the second degree polynomials g(j)(z) are given by

g(1)(z) = b z2 + (c + e) z + (a + d + f) g(2)(z) = a z2 + d z + fg(3)(z) = (a + b + c) z2 + (d + e) z + f

The polynomials g(j)(z) are namely those appearing in the integral representations of the two-point functions corresponding to the three possible pinchings of one propagator in the triangle diagram of gure 1. In what follows we parametrize the g(j)(z) generically as

g(j)(z) = (j) z2 + (j) z + (j) (2.36)

in order to formally handle them all at once when concerned with the zeroes of g(j)(z) + 1/(2B) further below. Let us note that the discriminant j of the second degree polynomial g(j)(z), dened by

j 2(j) 4 (j) (j) (2.37)

9

(2.29)

N =

= 1

JHEP11(2013)154

(2.31)

N =

= 1

3

Xk=1

S1jk (2.33)

Xj=1bj = B = det(G) det(S)

3

(2.34)

(2.35)

turns out to be equal to minus the determinant of the reduced kinematic matrix S{j}. This reduced kinematic matrix corresponds to the pinching of the propagator j in the triangle of gure 1, and is obtained from the matrix S by suppressing line and column j.

Correlatively (j) can be seen as half the reduced Gram determinant associated with the reduced kinematic matrix S{j}.

Equation (2.23) can thus be written:

I43 =

3

Xj=1bj

Z

1 ln g(j)(z)

ln (1/(2 B))

2 B g(j)(z) + 1 (2.38)

Likewise for eq. (2.26):

I63 =

1

2 +

JHEP11(2013)154

3

Xj=1bj

Z

1 g(j)(z) ln g(j)(z)

+ 1/(2 B) ln (1/(2 B))

2 B g(j)(z) + 1 (2.39)

In eqs. (2.38), (2.39), the contour prescription inherited from ( zT Sz i) in eqs. (2.3), (2.4) is implicit: the logarithmic terms ln g(j)(z)

in the numerators stand

. Let us remind that the terms ln(1/(2B)) in the numerators have been introduced in order that the zeroes z(j) of the denominators (2B g(j)(z) + 1) be ctitious poles in each of the sector integrals in any case i.e. the residues vanish: hence ln(1/(2B)) stands for ln(1/(2B) i) as well, furthermore no contour prescription

around the z(j) is needed.

Equations (2.38) and (2.39) are appealing candidates for the integral representations which we seek. Let us examine them more closely when det(G) 0. We shall distinguish

two cases: the generic case when det(G) 0 whereas det(S) remains non vanishing, and

the specic case det(G) 0 and det(S) 0 simultaneously which deserves a dedicated

treatment. Let us subsequently examine these two cases.

3 det(G) ! 0 whereas det(S) non vanishing

Let us rst consider the polynomials g(j)(z)+1/(2B) appearing in the denominators of the integrals I4,63(j) in eqs. (2.38), (2.39). Let us rst consider (j) 6= 0, so that g(j)(z) + 1/(2B)

is of degree two. Using the identity

b2j = 2 (j) det(S) det(G) j (3.1)

where j has been dened in eq. (2.37), and the rescaled coe cients

bj bj det(S), j = 1, 2, 3 (3.2)

it is insightful to write the corresponding discriminant of g(j)(z) + 1/(2B) as

e j =

b2j

det(G) (3.3)

for ln g(j)(z) i

10

Identity (3.1) is derived in appendix A. It is an example of the so-called Jacobi identities for determinant ratios, relating the determinant of a matrix and related cofactors i.e. determinants of reduced matrices4 [1618]. Similar identities may be met in the treatment of the four-point function. The zeroes z(j) of g(j)(z) + 1/(2B) are given by

z(j) =

(as commented earlier, det(G) 0). When det(G) 0, both zeroes zj of 2Bg(j)(z)+1 are

dragged away from [0, 1] towards + and respectively. If (j) = 0, g(j)(z) + 1/(2B) is only of degree one, and its unique root z0(j) given by

z0(j) =

is again dragged away from [0, 1] towards when det(G) 0. In either case, as soon as

det(G) becomes small enough each of the integrals

Jj =

Z

3

Xj=1bj

1

2 +

P3j=1 bjJj ln(1/(2B) i) after each term would have been 4In ref. [16], see: entry 107 (III.3) Determinants p. 348-351, in particular paragraph F. Theorems on Determinants, Theorem(3).

11

(j)

2 (j)

bj 2 (j)

p det(G)

(3.4)

JHEP11(2013)154

1 (j)

(j) + 12det(S) det(G)

(3.5)

dz 2Bg(j)(z) + 1

is analytically well dened and numerically safe, and furthermore the following identity holds:

3

Xj=1bjJj = 0 (3.6)

so that the contributions ln(1/(2B) i) sum up to zero in I43 as well as in I63. In this

respect, let us stress that the contributions ln(1/(2B)i) are ctitious from the start.

They were introduced through eq. (2.19) with the custodial concern of separately handling integrals - the sector integrals - with integrands free of poles within the integration domain namely when either of zj is inside [0, 1]. When zj are both outside [0, 1] the introduction of the ln(1/(2B) i) terms is irrelevant and indeed identity (3.6) allows to drop them

explicitly from eqs. (2.38), (2.39). The following integrals

I43 =

1

0

Z

1 ln g(j)(z) i

2 B g(j)(z) + 1 (3.7)

I63 =

3

Xj=1bj

Z

1 g(j)(z) ln g(j)(z) i

2 B g(j)(z) + 1 (3.8)

thus provide suitable integral representations in the case at hand. From a numerical point of view the explicit suppression of the ln(1/(2B) i) terms from integrals (3.7), (3.8)

is preferable since ln(1/(2B) i) when det(G) 0 thus implementing a numeri

cal cancellation of the sum

separately calculated would be submitted to numerical instabilities. Besides, if case some g(j)(z) vanishes at some z(j) inside [0, 1], a possible numerical improvement of the integral representation consists in deforming the integration contour in the complex z plane, to skirt the vicinity of the integrable singularity at z(j), so as to prevent the integrand from becoming large and avoid cancellation of large contributions, according to a one-dimensional version5 of the multidimensional deformation described in section 7 of ref. [1].

4 det(G) ! 0 and det(S) ! 0 simultaneously

This case is more tricky and deserves further discussion. Indeed, when det(S) = 0 and

det(G) = 0, eq. (2.33) dening the parameters bj as

P3k=1S1jk is no longer valid as S1 is not dened, and the parameter B = det(S)/ det(G) is an indeterminate quantity of the

type 0/0, likewise the z(j) are indeterminate quantities not manifestly driven away from the interval [0, 1].

In this subsection we will rst characterize the specic kinematics which leads to such a case. Then we will consider kinematic congurations close to the so-called specic ones above, such that det(G) and det(S) are simultaneously small but non vanishing and we

will study how I43 and I63 behave when det(S) and det(G) both go to zero. Anticipating

on the result, we rewrite both for I43 and I63 the corresponding sums

3

Xj=1bj I3(j) = b3 I3(3) +12(b1 + b2) I3(1) + I3(2)

+ 12(b1 b2) I3(1) I3(2)

JHEP11(2013)154

(4.1)

One of the coe cients bj, say b3 will be shown to have a nite limit whereas b1 and b2 diverge towards innity in a concomitant way such that their sum b1 + b2 has a nite limit. Furthermore, the di erence I3(1) I3(2) will be shown to tend to zero so that the product

(b1 b2)(I3(1) I3(2)) has a nite limit. A well-dened expression is thus achieved in

the double limit det(S) 0, det(G) 0 although some of the ingredients are separately

ill-dened in the limit considered. We will conclude this subsection with a comment in relation with the behaviour of the GOLEM reduction formalism in this case.

4.1 Characterization of the specic kinematics det(G) = 0, det(S) = 0The quantities det(S) and det(G) are polynomials in the kinematical invariants. Hereafter

we propose a presentation which partly linearizes the resolution of the non linear system det(G) = 0, det(S) = 0. This approach, applied here to N = 3, extends to other N, e.g.

N = 4. The determinant det(S) can be written (see appendix A):

det(S) = Saa det(G) + V (a)T

eG(a) V (a) (4.2) 5In broad outline, the contour deformation is contained inside the band 0 Re(z) 1. It departs from the real axis at 0 with an acute angle and likewise ends at 1 in such a way that Im(g(j)(z)) is kept negative

along the deformed contour so that the latter does not cross any cut of ln(g(j)(z) i ). In the case at hand

this type of contour never embraces any of zj as soon as the latter are outside [0, 1], thus no subtraction of illegitimate pole residue contribution at zj has to be cared about.

12

where

eG(a) is the matrix of cofactors6 of G(a); the superscript T refers to matrix transposition. The system det(G) = 0, det(S) = 0 is thus equivalent to the system det(G) = 0,

V (a)T

eG(a) V (a) = 0. Since

G(a)(a) =(a) G(a) = det(G) 1IN (4.3)

the matrices G(a) and(a) are simultaneously diagonalizable. When det(G) = 0 and G(a) has rank7 (N 2) - namely 1 in the N = 3 case at hand - the only eigendirection n(a)

of G(a) associated to the eigenvalue zero is concomitantly the only eigendirection of(a) associated to the only non vanishing eigenvalue of(a). For n(a) properly normalized, (a) = n(a) n(a)T . The condition V (a)T

(a) V (a) = 0 quadratic in V (a) is thus

equivalent to the following linear one:

(n(a)T V (a)) = 0 (4.4)

Let us now consider the condition det(G) = 0. A detailed discussion is provided in appendix B, we only summarize it here for the N = 3 case at hand. A vanishing det(G) happens (i) either when the external momenta p1,2,3 are proportional to each other (ii)

or when there exists a non vanishing linear combination of the external momenta which is lightlike and orthogonal to all of them [19]. Possibility (i) corresponds to degenerate kinematic congurations irrelevant for next-to-leading order (NLO) calculations of collider processes. Let us focus on possibility (ii) further assuming any subset of two of the three external momenta to be linearly independent. To x the ideas, let us consider8 p1 and p3.

If one of them say p1 is lightlike it is namely (proportional to) the lightlike combination sought, whereas p3 shall be spacelike, p2 = p1 p3 is spacelike as well and s2 = s3. If

neither p1 nor p3 are lightlike, both shall be spacelike with s1 = s3, and p2 is (proportional to) the lightlike combination of p1 and p3. Actually, congurations of type (ii) with p3, p1 and p2 linearly independent and all spacelike can also lead to a vanishing det(G), yet such congurations are not relevant for collider processes at NLO,9 we thus discard them.

Let us assume p2 lightlike and orthogonal to p3 and p1 both spacelike: s1 = s3 s+ <

0, s2 = (p2 p3,1) = 0, so that (p1 p3) = s+. We single out line and column 3 of S whose

corresponding G(3) reads:

G(3) = 2 s+ "

6This matrix is sometimes also called adjoint matrix of G(a).

7See comment at the beginning of appendix E regarding (N 1) (N 1) Gram matrices of lower ranks. In the present N = 3 case we discard the degenerate possibility that the 2 2 matrix G(a) has two vanishing eigenvalues which not only makes the cofactor matrix [tildewide]

G(a) vanish identically but also G(a) itself.

This would correspond to a very peculiar kinematics of three lightlike external four-momenta collinear to each other.

8This particular choice corresponds to singling out and erasing line and column 3 in the matrix S and considering the Gram matrix G(3).

9Indeed, at NLO, each of the external legs of the one loop three point function considered has to be connected to separate tree, and all the external legs of at least one of these three trees have to be nal state legs. Therefore the external momentum owing through the corresponding leg of the one loop three point function cannot be spacelike. Such congurations with three spacelike legs could appear only in higher loop diagrams, of which the one loop three point function would be seen as a subdiagram.

13

JHEP11(2013)154

1 1 1 1

#

(4.5)

The normalized eigenvector n(3) associated with the eigenvalue zero is (up to a sign):

n(3) = 12 "

1

1

#

(4.6)

With eqs. (2.11) and (4.6), condition (4.4) imposes the following restriction on the internal masses:

m21 = m22 m2 (4.7)

4.2 Behaviour of I43 and I63 when det(G) 0, det(S) 0Let us assume condition (4.7) and parametrize the departure from the critical kinematics using s2, s (s1 s3)/2 and s+ (s1 + s3)/2. The determinants read:

det(G) = 4 s+ s2 s2 1 4s22

(4.8)

JHEP11(2013)154

det(S) = 2

e

s2 + 4 m2 s2 + m23 s22 s2 s2

(4.9)

where

e is the Kallen symmetric function of s+, m2, m23 given by:

e = s2+ + (m2)2 + (m23)2 2 m2 s+ 2 m23 s+ 2 m2 m23 > 0 (4.10)

The region where det(G) and det(S) are concomitantly small corresponds to |s2|, |s| both small compared with the other kinematical invariants, with |s2/s+| and (s/s+)2 of the

same order, so that det(G) and det(S) can be approximated by:

det(G) = 4 s+ s2 s2

+ (4.11)

det(S) = 2

e

s2 + 4 m2 s2

+ (4.12)

In order to understand in more detail the origin of the diverging contributions to the coe cients bj, the matrix S may be decomposed as follows:

S =

3

Xj=1j v(j) vT(j) (4.13)

Let us address the real mass case rst; we will briey comment at the end of this subsubsection on how the study shall be - only slightly - modied in the complex mass case. Decomposition (4.13) corresponds to the usual diagonalization of S: the v(j) and j, j = 1, 2, 3

are real eigenvectors orthonormalized in the euclidean sense (vT(j) v(k)) = jk and the corresponding real eigenvalues respectively. The labelling of eigenvectors and values is chosen such that i=3 explicitly given by

3 =

s2 + 4 m2

e s2

+ (4.14)

14

is the eigenvalue which vanishes when s2 and s both vanish, whereas the two others

remain nite in this limit. Introducing

e

P3j=1 bj = (eT .b) take the form:

More explicit algebraic expressions of the various ingredients in the relevant regime are gathered in appendices C and D for convenience. They show that (vT(i) e) O(s) so that the components of b = S1 e are individually wild-behaved when det(S) 0 due to

the O(s2/3) contribution along v(3) being O(|3|1/2), although B O(s2/3) remains O(1). A closer look reveals that both b3 O(s2/3) and the combination (b1 + b2) O(s2/3, s2/3) separately remain O(1) whereas (b1 b2) O(s/3) is O(|3|1/2).

Concomitantly, since

g(1)(z) = g(z) + s z (1 z) (4.18) g(2)(z) = g(1 z) s z (1 z) (4.19)

where

g(z) = s+ z (1 z) + m2 z + m23 (1 z) (4.20) the quadratic forms g(1)(z) and g(2)(1 z) become both equal to g(z) when s = 0.

The di erence of the two integrals I3(1) and I3(2) in factor of b1 and b2 respectively, in

eqs. (2.38) for I43 and likewise in (2.39) for I63, is (I3(1) I3(2)) O(s). The combination

(b1 b2) (I3(1) I3(2)) is thus (O(s2/3) i.e O(1) as well. In the summary, rewriting

P3j=1 bjI3(j) according to eq. (4.1), each of the three terms b3 I3(j), (b1 + b2) (I3(1) + I3(2)) and (b1 b2) (I3(1) I3(2)) remains bounded and has a nite limit when s 0, s2 0.

Let us however notice that we are taking a double limit. Properly speaking, the limits of each of these three terms in eq. (4.1) which are separately well-dened are directional limits s 0, s2 0 in the {s2, s2} plane keeping the ratio t = s2/s2 xed, i.e. these directional

limits are functions of t. However, the limit of the sum of these three terms in eq. (4.1) is indeed independent of t. This can be easily checked numerically, this can also be proven analytically although this is somewhat cumbersome; a proof is presented in appendix F. The ground reason why this property holds is further understood as follows: were the limit of the sum a directional one, it would imply that the three-point function would be a singulari.e. non analytical function of the kinematical invariants at such congurations. However

15

1 1 1

(4.15)

the column vector b = S1 e and the quantity B =

3

3

b =

Xi=1 1i

vT(i) e v(i) (4.16)

JHEP11(2013)154

B =

Xi=1 1i

vT(i) e 2

(4.17)

the kinematic singularities are characterized by the so-called Landau conditions10 [19, 20]

(see also [17]). For one loop diagrams, these conditions require not only that det(S) = 0,

but also that the eigenvectors associated with the vanishing eigenvalue of S shall have only

non negative components and that their sum be strictly positive. By contrast, in the case at hand, the eigenvector v(3) in the limit where 3 = 0 is, cf. appendix C:

1

1 0

Xj=1i u(j) uT(j) (4.22)

which now reects the so-called Takagi factorization S = U UT in terms of a real non

negative diagonal matrix and a unitary matrix U, instead of a standard diagonalization. The diagonal elements j of are the square roots of the eigenvalues of the hermician matrix S S, whereas the columns u(i) of U are corresponding eigenvectors11 of S S. The

corresponding Takagi factorization of S1 for S invertible reads S1 = U 1 U i.e. in

the tensor product notation:

S1 =

16

v(3)|3=0

(4.21)

such that (eT v(3)|3=0) = 0. The vanishing det(S) in the present case is therefore not

related to a kinematic singularity: the three-point function is regular in the limit considered, in particular this limit shall be uniform i.e. not directional.

4.3 Extension to the complex mass case

The above study was stressed to hold, strictly speaking, for real masses. Actually, it can be extended to the complex mass case with only slight modications. Indeed in the complex mass case, the symmetric matrix S albeit complex admits a decomposition formally

identical eq. (4.13):

S =

3

JHEP11(2013)154

3

Xi=11j u(j) u(j) (4.23)

Identity (4.23) provides the equations for b and B which modify eqs. (4.16), (4.17) in the complex mass case. A study quite similar to the real mass case then follows12 and the same

10For general parametric integrals the Landau conditions provide only necessary conditions to face singularities, either of pinched or end-point type. However for Feynman integrals, Coleman and Norton [21] proved these conditions to be su cient.

11Let us note by passing that, unlike with standard diagonalization, the phases of the vectors u(i) involved in the Takagi factorization shall be adjusted modulo in order to fulll the condition (uT(j) S u(k)) =

j (uT(j) u(k)), because the decomposition involves the transpose of U not its hermician conjugate.

12Technically speaking, the determination of the singular values j and corresponding vectors u(j) may seem somewhat awkward given the algebraically more complicated form of the matrix elements of S S.

Actually we are interested in a practical case where the imaginary parts of the masses - i.e widths of unstable particles in internal lines - are much smaller than the real parts. Therefore, splitting S in real and imaginary parts S = SR i SI, and writing S S = S2R + , with = i[SR, SI]+ S2I, the square roots j of eigenvalues of S S and the corresponding eigenvectors u(j) can be expanded in integer powers of matrix elements of SI , as perturbative deformations of the eigenvalues k and eigenvectors v(k) of SR i.e. the spectral features of the real mass case, by a straightforward application of the formalism of time-independent perturbation theory in Quantum Mechanics.

conclusions hold.

4.4 A comment on the GOLEM reduction formalism when det(S) = 0Let us end this subsection with a comment on the applicability of the GOLEM reduction formalism [1] to congurations such that det(S) = 0. The equation S b = e with S

not invertible can still be solved e.g. introducing the so-called Moore-Penrose Pseudo-Inverse [22] T0 of S(3 = 0) given in the real mass case13 by:

S3=0 =

Xi=1,2i v(i) vT(i) (4.24)

T0 =

Xi=1,21i v(i) vT(i) (4.25)

provided the following compatibility condition to be satised:

[1I S|3=0 T0] e = 0 (4.26)

Still noting v(3) the eigenvector of S with vanishing eigenvalue, the compatibility condi

tion (4.26) reads

eT v(3)

= 0 (4.27)

Condition (4.27) is incompatible with the Landau conditions mentioned earlier which characterizes a kinematic singularity, namely the non negativity of all the components of v(3): thus the formalism breaks down for singular kinematics.

On the other hand, the peculiar congurations such that det(G) = 0, det(S) = 0

examined in the present subsection are non singular and do fulll condition (4.27), and the (non unique)14 solution b reads b = b0 + Ker (S) = b0 + x v(3)|3=0 with x arbitrary

scalar and b0 T0 e, leading to B0 = eT T0 e. The GOLEM formalism thus applies

also, using b = b0 and B = B0, when standing precisely at the peculiar congurations. Yet slightly away from these peculiar congurations the GOLEM ingredients dened by b =

S1 e separately show discontinuities15 w.r.t. those given by b = b0 precisely at the

peculiar congurations; this discontinuity comes from the contribution to b coming from the (divergent) component along v(3), which have no counterpart in b0. Notwithstanding, these individual discontinuities are artefacts in the sense that they cancel out in the reduction formula when put altogether, as discussed above.

13A similar discussion holds in the complex mass case as well with similar expressions cf. the previous paragraph.

14The arbitrary component of b along v(3)|3=0 is irrelevant for any practical purpose. Indeed, the condition (4.27) makes the contribution to b3, hence to B, coming from this component vanish, whereas the nite contribution to b1 b2 from this component is weighted by the vanishing I3 (1) I3 (2) in the decomposition (4.1).

15More precisely b0 is equal to the directional limit s2 ! 0 of b along the direction s = 0 i.e. t = 1. The discontinuities are meant for all other, nite t directions.

17

JHEP11(2013)154

5 Summary and outlook

In this article, we provided a representation of one-loop, 3-point functions in 4 and 6 dimensions in the form of one dimensional representations in the general case with complex masses. These one-dimensional integral representation have the virtue to avoid the appearance of factitious negative powers of Gram determinants, and are therefore numerically stable and remain rather/relatively fast to compute numerically. We addressed the two cases at hand separately: the generic case when det(G) becomes small whereas det(S)

remains nite, and the trickier specic case when both det(G) and det(S) become concomi

tantly small. Here we presented the existence proof for scalar integrals, but the method applies to tensor integrals as well, i.e. loop integrals involving integer powers of Feynman parameters in the denominators of their integrands.

A forthcoming article will continue the present one by the similar treatment of one-loop 4-point functions. The latter proved to be quite more involved than the 3-point case, we thus preferred to split it from the present article. These two will be supplemented by a dedicated treatment of the specic mixed case involving both nite (complex) masses, and some zero masses triggering infrared issues. In the meantime we also found an alternative approach leading to a derivation of integral representation which is perhaps simpler and also makes the algebraic nature of the ingredients involved more transparently related to the GOLEM reduction algorithm both for 3-point and 4-point functions, this approach will be presented in a separate article. Last, this approach will be fully implemented in the next version of the GOLEM95 library in Fortran 95. We will provide various numerical tests of numerical stability at this occasion.

A Useful algebraic identities among determinants

Equation (3.1) used in section 3 relates various ingredients of the reduction formula involving the one loop three point function. Similar identities can be found and used in the case of four point functions. These properties can be traced back to general algebraic identities between the determinant of a square matrix and minors of this matrix, referred to as Jacobi identities for determinant ratios [1618]. This appendix reminds these general identities and species them to the case useful for the present work. Beforehand we remind a few properties useful in this respect.

A.1 Preliminaries

Let us rst recall a few useful properties which we state in general for arbitrary N not just N = 3. Consider the kinematic N N matrix S associated with a given one-loop

N-point diagram generalizing eq. (2.2) for any N. We single out the line and column a, and consider the corresponding (N 1) (N 1) Gram matrix G(a) associated to

S, generalizing (2.7) and the (N 1)-column vector V (a)i generalizing eq. (2.8). Let us

choose a = N to x the ideas and make formulas simpler; the results obtained can be straightforwardly generalized to any a.

18

JHEP11(2013)154

1. Let us subtract the last column of S from every column j, then subtract the last line

from every line i in the intermediate matrix thus obtained, with 1 i, j N 1.

This denes the N N matrix

[hatwide]

S(N) given by:

[hatwide] S(N) =

G(N)ij | V (N)i

+

V

(A.1)

(N) j

| SNN

generalizing eqs. (2.7) and (2.8). The determinants det(S) and det(

[hatwide]

S(N)) are equal.

S(N)) is Laplace-expanded according to its Nth line as:

det([hatwide]

S(N)) = SNN det

JHEP11(2013)154

det([hatwide]

G(N)

+

N1

Xj=1(1)N+j V (N)j det [hatwide]S(N)

{N}

{j}

(A.2)

{N}

{j} is the (N 1) (N 1) matrix obtained from

where [hatwide]

S(N)

[hatwide]

S(N) by suppressing

its line N and column j. Using this notation we have in particular:

G(N) =

{N}

{N} (A.3)

The determinant det

[hatwide] S(N)

[hatwide] S(N)

{N}

{j}

may in turn be Laplace-expanded with respect to

its last column:

det

[hatwide] S(N)

{N}

{j}

=

N1

Xi=1(1)i+(N1)V (N)i (1)N2 (1)i+j [tildewide]G(N)

ij

(A.4)

In the r.h.s. of eq. (A.4), the factor (1)N2 comes from the explicit minus sign in G(N); (1)i+j (

G(N))ij is the minor of the Gram matrix element G(N)ij. Substituting into eq. (A.2) we get:

det(S) = (1)N1 hSNN

det

[tildewide]

G(N) + V (N)T

[tildewide]

G(N) V (N)i(A.5)

hence eq. (4.2).

2. The coe cients bi in the GOLEM N-point reduction algorithm are dened by

N

Xj=1

Sijbj = 1, i = 1, , N (A.6)

Singling out bN in eq. (A.6) corresponding to i = N, and subtracting eq. (A.6) for i = N from eq. (A.6) for every i = 1, , N 1, eq. (A.6) may alternatively be

19

rewritten in terms of G(N) and V (N)i as:

N1

Xj=1bj + bN = B (A.7)

N1

Xj=1G(N)ij bj = B V (N)i, i = 1, , N 1 (A.8)

N1

Xj=1V (N)j bj = 1 B SNN (A.9)

When G(N) is invertible, eq. (A.8) is solved as:

bj = B

N1

Xk=1

JHEP11(2013)154

hG(N)

i1

jk V (N)k = B

hdet

G(N) i1N1

[tildewide] G(N)

jk V (N)k (A.10)

Xk=1

where the matrix [tildewide]

G(N) is the matrix of cofactors. Thus, using eq. (A.5):

N1

Xj=1V (N)jbj = B

hdet

G(N)

i1

V (N)T

[tildewide]

G(N) V (N)

= B (1)N1

det(S)

det(G(N)) B SNN (A.11)

Comparing eqs. (A.11) and (A.9) yields:

B = (1)N1

det(G(N))

det(S)

(A.12)

and bN is obtained by solving eq. (A.7). Introducing

bj bj det(S) (A.13)

and, using eq. (A.4), eq. (A.10) reads:

bj = (1)N1 h

[tildewide]

G(N) V (N)i

j (A.14)

= (1)j+N2 det [hatwide]S(N)

{N}

{j}

, j = 1, , N 1 (A.15)

A.2 The identity (3.1) and Jacobi identities for determinants ratios

As shown below, identity (3.1) is a special case of the following general property [16]. Let A be any nn matrix, and A{i1,,ir}{k1,,kr} the matrix obtained from A by suppressing the lines

i1, , ir and columns k1, , kr. Then, for any i1 < i2 and k1 < k2:

det( A ) det

A{i1,i2}{k1,k2} = det

A{i1}{k1} det

A{i2}{k2} det

A{i1}{k2} det

A{i2}{k1} (A.16)

20

Indeed, let us specify A = [hatwide]

S(N) in the identity (A.16) and give the explicit forms of the other

quantities obtained by suppressing appropriate lines and columns. Let us take any i 6= N.The (N 2) (N 2) matrix

[hatwide] S(N)

{i,N}

{i,N} is nothing but the matrix Gi, thus

det

G{i} det

[hatwide] S(N)

{i,N}

{i,N}

= (1)N2 det

G{i}

(A.17)

Furthermore, , we rst notice that, for A symmetric,

A{i}{k} =

A{k}{i} T

(A.18)

JHEP11(2013)154

thus

det

A{i}{k} = det

A{k}{i}

(A.19)

from eqs. (A.15) and (A.19), we have:

bi

2 =

det

[hatwide] S(N)

{N}

{i}

2

= det

[hatwide] S(N)

{N}

{i}

det

[hatwide] S(N)

{i}

{N}

(A.20)

Besides,

det

[hatwide] S(N)

{N}

{N}

= (1)N1 det(G) (A.21)

In the case at hand, identity (A.16) thus reads:

det

[hatwide] S(N)

| {z }

| {z }

[hatwide] S(N)

{i,N}

{i,N}

det(S)

det

(1)

N2 det(G{i})

= det

[hatwide] S(N)

| {z }

{i}

{i}

[hatwide] S(N)

{N}

{N}

[hatwide] S(N)

{i}

{N}

det

[hatwide] S(N)

{N}

{i}

(A.22)

det(S{i})

det

(1)

N1 det(G)

det

(bi)2

i.e.

b2i = (1)N1 h

det( S ) det

G{i} + det

S{i} det(G)

i

(A.23)

Specifying N = 3 in the present case of interest gives eq. (3.1), with

j = 12 det

G{j}

j = det

S{j}

(A.24)

and where G{j} is the (N 2) (N 2) Gram matrix associated to S{j} and obtained

from it via a procedure similar to the one leading to eq. (A.1). q.e.d.

B Kinematics leading to a vanishing det(G)

This appendix supplements the discussion on the kinematics leading to a vanishing det(G) provided in subsection 4.1.

21

B.1 General considerations

Let us consider a set {pi, i = 1 , N 1} of N 1 four-momenta in Minkowski space,

their Gram matrix16 Gij = 2 (pi pj), and the linear system given byN1

Xj=1Gij xj = 0 , i = 1, , N 1 (B.1)

A vanishing det(G) means the existence of a set of scalars {xj, j = 1, , N 1} not all vanishing and solution of the system (B.1). Multiplying eq. (B.1) for each i = 1, , N 1

by xi and summing over i leads to the condition

l2 = 0 , l

N1

JHEP11(2013)154

Xj=1xj pj (B.2)

which means that (i) l vanishes i.e. the {pi} are linearly dependent momenta, or that (ii)

l is lightlike and eq. (B.1) is the orthogonality condition (l pi) = 0, i = 1, , N 1 [19].

Let us focus on case (ii) assuming furthermore the {pj} to be linearly independent.17 The

orthogonality condition requires that none of the pj be timelike, and pN

PN1j=1 pi is

orthogonal to l too, thus cannot be timelike either.

If one of the pj, say p1, is lightlike, l is proportional to p1, and all the pj6=1 shall

be spacelike. Were pN lightlike it should be p1, which would contradict the extra

assumption of linear independence of the {pi}, i = 1 , N 1, thus pN shall be spacelike

too. These requests impose a steric constraint on N w.r.t. the spacetime dimension d = 4. As seen by decomposing pj as (p0j/p01) p1 + qj for j = 2, , N 1 in a frame chosen such

that p1 = p01(1;~0; , ), with = and qj = (0; ~qj; 0), the maximal number of possibly

independent qj is d 2 = 2 i.e. N shall be 4. Besides, for N 4, NLO calculations

involve no one-loop N-point function with external momenta all spacelike but one lightlike, neither in collision nor in decay processes: alternative (ii) only occurs for N = 3 for any NLO purpose.

If none of the pj j = 1, , N 1 is lightlike, all of them shall be spacelike. The

momentum pN shall be either lightlike - hence proportional to l: one is driven back to the previous case; see the N = 3 case below - or spacelike. The latter case is submitted to a similar steric constraint as above, as seen by trading one of the pj for l; no such conguration matters at NLO whatever N.

In summary, for any practical purpose at NLO, a vanishing det(G) can happen for a linearly independent kinematic conguration only in the case N = 3. Otherwise the congurations with vanishing det(G) correspond to linearly dependent four-momenta.

16The overall factor 2 in the denition of G is actually irrelevant in the present discussion, we keep it only for notation consistency with the bulk of the article.

17If both properties (i) and (ii) are simultaneously fullled, then the rank of the (N 1) (N 1) matrix G is (at most) (N 3), corresponding to quite degenerate congurations. For example for N = 3 this corresponds to all four-momenta lightlike and collinear to each other, for which G identically vanishes. For N = 4 it corresponds to two spacelike and two collinear lightlike external momenta being a linear combination of the two spacelike ones. None of these cases are involved in NLO calculation of processes relevant e.g. for collider physics.

22

B.2 Focus on N = 3

This appendix elaborates on the case N = 3 involved in subsubsection 4.1, with p1 and p3 linearly independent and spacelike. We parametrize the lightlike combination l (dened up to an overall multiplicative constant) as l = p1 x p3. The orthogonality conditions

(implying that l is lightlike) read:

(l p1) = s1 x (p1 p3) (B.3) (l p3) = (p1 p3) x s3 (B.4)

The vanishing det(G) = s1 s3 (p1 p3)2 ensures the compatibility of eqs. (B.3) and (B.4)

in x and

x = (p1 p3)s3 = sign(p1 p3) r

s1s3 (B.5)

The condition det(G) = 0 also implies that s2 = s1 + 2 (p1 p3) + s3 can be written

s2 = s1 sign(p1 p3) s3

2 0 (B.6)

Therefore s2 = 0 i sign(p1 p3) = + and s1 = s3, in which case x = 1 and p2 = l.

Otherwise s2 < 0.

C Spectral features of S for N = 3

This appendix gathers the spectral properties of S for N = 3 which are further used in

appendix D.Accounting for the condition m21 = m22 m2 and the parametrization used in subsec

tion 4.2, the kinematic matrix S reads:

S =

2m2 s2 2 m2 s+ + s (m2 + m23) s2 2 m2 2m2 s+ s (m2 + m23)

s+ + s (m2 + m23) s+ s (m2 + m23) 2 m23

(C.1)

Let us compute the eigenvalues 1,2,3 of S and the corresponding eigenvectors v(1,2,3) in

the regime det(G) 0, det(S) 0 corresponding to s 0, s2 0. Since 3 0

whereas 1,2 remain nonvanishing in this regime, in order to correctly get the coe cients bj and B in eqs. (4.16) and (4.17) respectively in subsection 4.2, we shall keep the leading dependence on s, s2 in 3 and in the components of the corresponding normalized eigen

vector v(3), whereas s and s2 can be safely put to zero to rst approximation in 1,2 and

the corresponding normalized eigenvectors v(1,2). This is the approximation to which we provide the algebraic results below.

C.1 Eigenvalues

The characteristic polynomial PS(s) of S is:

PS(s) det (S s 1I3)=

s3 (tr(S)) s2 +1 2

23

JHEP11(2013)154

h(tr(S))2 tr S2

i

s det(S)

(C.2)

The small eigenvalue 3. The eigenvalue 3 vanishing as det(S) may be extracted

from eq. (C.2) rewritten

3 = 2 det(S)

(tr(S))2 tr (S2)

+ 2 tr(S)

(tr(S))2 tr (S2)

23

2 (tr(S))2 tr (S2)

33 (C.3)

by an iteration generating an expansion in integer powers of det(S). The leading term

of this expansion, itself truncated to keep only the leading dependencies in s2 and s, is

given by:

3 = 2 det(S)|

trunc

h(tr(S))2 tr (S2)is=s2=0+ (C.4)

JHEP11(2013)154

Using

(tr(S))2 = 22 2 m2

2 + 22

2 m23

2 (C.5)

tr S2

2 m2 2m23 +

+ (2m23)2

+ 4 (s)2 (8 m2) s2 + 2 (s2)2 (C.6)

we have:

= 22

n

s+ m2 + m23

2

+ (2m2)2

o

tr (S)2 tr S2

= 4

e (1 + ) (C.7) = 1

e

s2 (2 m2) s2 +1 2s22

(C.8)

whereas

det(S) = 2

e

s2 + 4m2

e s2

m23

e s2

e s2

1

+ s2

(C.9)

We further truncate

htr (S)2 tr S2

i

s=s2=0 = 4

e (C.10)

det(S)|

trunc = 2

e

s2 + 4m2

e s2

+ (C.11)

The eigenvalue 3 thus has the following approximate expression:

3 =

s2 + 4m2

e s2

+ (C.12)

in which the terms dropped are of order s22, s2 s2, s4 and higher.

The two non vanishing eigenvalues 1,2. The two other eigenvalues 1,2 are obtained from the factorization of PS(s) as:

PS(s) = (s 3) s2 A s + B

(C.13)

24

which requires

A + 3 = tr(S) (C.14)

B + 3 A =

1

2

h(tr(S))2 tr S2

i

(C.15)

3 B = det(S) (C.16)

The approximation corresponding to s = s2 = 0 in eqs. (C.14)(C.16) replaces

A A = tr(S) = 2 2 m2 + m23

(C.17)

B B =

1

2

h(tr(S))2 tr S2

i

e (C.18) and the zeroth order approximations of 1,2 are given by:

12 = 2 m2 + m23

s=s2=0 = 2

q

2 m2 + m23

2 + 2

e (C.19)

C.2 Eigenvectors

The eigenvector v3 associated with 3. The components x, y, z of v3 are solutions of the degenerate system:

3 + 2 m2

x + s2 2 m2

y + s+ + s (m2 + m23)

z = 0 (C.20)

s2 2 m2

x 3 + 2 m2

y + s+ s (m2 + m23)

z = 0 (C.21)

JHEP11(2013)154

s+ + s (m2 + m23)

x + s+ s (m2 + m23)

y 3 + 2 m23

z = 0 (C.22)

Subtracting (C.20) from eq. (C.21) yields:

(s2 + 3) (x y) 2 s z = 0 (C.23)

Since

(4m2)

e s2eq. (C.23) tells that z = O(s(x y)): x and y being bounded, z thus vanishes at least

as O(s) in the limit s 0. We shall keep the leading dependence on s and s2 in the

components of v(3).Up to an overall normalization, x, y and z are given by:

x =

e + 2 (m2 + m23) 3 + 2 s+ (m2 + m23)

s s2 + (C.24)

y =

s2 + 3

e 2 (m2 + m3) 3 1 + 8 m2 m23

e

s2 + (C.25)

e s2 + (C.26)

In eqs. (C.24)(C.26), the dependence on s2 has been traded for s and 3 up to terms

neglected at the approximation retained. Introducing18

v(3) =

1 2

z = 4 m2

4 m2

s s+ (m2 + m23)

1

s+ (m2 + m23)

1 , l(3) =

s+ (m2 + m23)

4 m2

(C.27)

18In what follows it is not necessary to normalize the vector l(3) to 1.

25

the unnormalized eigenvector vunnorm(3) given by eqs. (C.24)(C.26) can be written:

vunnorm(3) =

e2 1

2 (m2 + m23)

e 3 s

+ (m2 + m23)

e s

4 m2 m23

e2 s2!

v(3)

l(3) + (C.28)

The vector v(3) is the normalized eigenvector of S associated with the eigenvalue 3 = 0

when det(S) = 0. Let us notice that (lT(3) v(3)) = 0 and (eT v(3)) = 0 where the

vector e was dened in eq. (4.15) in subsection 4.2. Once normalized by N3 (vunnormT(3)

vunnorm

+ s 1

s+ (m2 + m23)

e s!

e2 (1 + O(s)) l(3) + (C.29)

The O(s2) terms are no more explicited in eq. (C.29) as they would contribute in ap

pendix D beyond the level of approximation retained only. The departure of v(3) from v(3) in eq. (C.29) does not depend explicitly on 3, it only depends on s.

The eigenvectors v(1)(2) associated with

JHEP11(2013)154

(3) )1/2, the eigenvector v(3) is given by:

v(3) = 1 + O(s2)

v(3)

s

12 . The components x12 , y12 , z12 of the eigenvectors associated with 12 , are solutions of the degenerate system

(12 + 2 m2) x12 + (s2 2 m2) y12 + s+ + s (m2 + m23)

z12 = 0 (C.30)

(s2 2 m2) x

12 (

12 + 2 m2) y12 + s+ s (m2 + m23)

z12 = 0 (C.31)

s+ + s (m2 + m23)

x12 + s+ s (m2 + m23)

y12 (12 + 2 m23) z12 = 0 (C.32)

Subtracting eq. (C.30) from eq. (C.31) yields:

(s2 + 12 ) (x12 y

12 ) 2 s z

12 = 0 (C.33)

12 ) thus vanishes at least as O(s)

in the limit s 0. In the zeroth order approximation corresponding to s = s2 = 0,

(x12 y

12 ) vanishes. Substituting this into eq. (C.32) the latter becomes:

2 s+ (m2 + m23)

Since |s2| |

12 | 6

= 0 and z12 remains bounded, (x12 y

x

12 + 2 m23 z = 0 (C.34)

which involves

(12 + 2 m23) =

(2 m2 m23) q

(2 m2 + m23)2 + 2

e

Up to an overall normalization factor to be xed below, x12 , y12 and z12 are given by:

x12 = y12 = (12 + 2 m23) (C.35)

z12 = 2 s+ (m2 + m23)

(C.36)

26

The condition (vT(1) v(2)) = 0 is fullled by eqs. (C.35), (C.36) since: x1 x2 + y1 y2 + z1 z2

= 2 (1 + 2 m23)(2 + 2 m23) 4 s+ (m2 + m23)

2= 0 (C.37)

Identity (C.37) will be used in appendix D. The normalization factor N

12 required to

12 || to 1 is given by:

N12 =

h2 (12 + 2 m23)2 + 4 s+ (m2 + m23)

normalize ||v

2 i1/2

JHEP11(2013)154

=

h 2

12 + 2 m23 (1 2)

i1/2 (C.38)

Let us dene the angle by

cos = 2 (1 + 2 m23) N1 (C.39) sin = 2 s+ (m2 + m23)

N1 (C.40)

The normalized eigenvectors v(1)

(2)

read:

v(1) =

12 cos

12 cos

sin

, v(2) =

12 sin

12 sin

cos

(C.41)

Together with v(3) given by eq. (C.27) above these eigenvectors dene an orthonormal

basis namely the eigenbasis of S when det(S) = 0. The overall signs have been chosen

such that the orientation is direct i.e. det [v(1), v(2), v(3)] = +1.

D Analysis of the reduction coe cients (b0)j, bj, B0 and B when det(G) and det(S) ! 0

This appendix provides a detailed analysis of the reduction coe cients (b0)j, bj, B0 and

B when det(G), det(S) 0 providing the technical back-up to the discussion in subsec

tions 4.2 to 4.4. Introducing the vectors

n1 =

1 0 0

, n2 =

0 1 0

, n3

0 0 1

(D.1)

the components of b dened by eq. (4.16) can be expressed in the limit 3 0 in terms of

those of b0 = T e introduced in subsection 4.4 as:

bj(3 0) = (b0)j + 13 eT v(3) nT

j

v(3)

+ (D.2)

(b0)j = 11 eT v(1)

nT

j

v(1)

+ 12 eT v(2)

nT

j

v(2)

(D.3)

27

where the column vector e has been dened by eq. (4.15), and where in eq. (D.2)

stand for evanescent terms in the limit considered. As the saying goes, a tedious but straightforward algebraic juggling, sketched below, leads to the following expressions for the sought coe cients.

(i) (b0)3

Using 1 2 = 2

e, we get:

(b0)3 =

2

2

e

eT v(1) nT3 v(1) 1 2

eT v(2) nT3 v(2) (D.4)

e

JHEP11(2013)154

This involves

eT v(1) nT3 v(1) =

2 cos sin sin (D.5)

eT v(2) nT3 v(2) =

2 sin + cos cos (D.6)

(b0)3 takes the form:

(b0)3 =

1

2

e

12 (1 + 2)

+

12 cos2 sin2

+ 2 sin cos

(1 2)

(D.7)

With cos , sin from eqs. (C.38)(C.40) and using identity (C.37), we have:1

2 cos2 sin2

+ 2 sin cos

=

1(1 2)

2 s+ (m2 + m23) + (2 m2 m23)

(D.8)

whereas 1 + 2 = 2(2 m2 + m23). Finally (b0)3 reads:

(b0)3 = 1

e s

+ (m2 + m23)

+ (2 m2)

= 1

2

e eT l(3)

(D.9)

(ii) b3(3 0)

The extra bit to be added to (b0)3 to get b3(3 0) is (eT v(3)) (nT3 v(3)). At the

order of approximation retained, (eT v(3)) O(s), (nT3 v(3)) O(s) and in both

terms the relevant contribution19 comes from the component s/(

e2) l(3) of v(3)

in eq. (C.29). Since (nT3 l(3)) = 4 m2 we have:

13 eT v(3)

nT

3 v(3)

= 4 m2

e

s2

3

1

2

e eT l(3)

+ (D.10)

19Notice that (nT3 v(3) ) = (eT v(3) ) = 0.

28

The combination of eqs. (D.9) and (D.10) involves:

1 + 4 m2

e

s2

3 =

s23 + (D.11)

so that

b3(3 0) =

s2 3

1

2

e eT l(3)

+ (D.12)

(iii) (b0)1 + (b0)2

We have:

JHEP11(2013)154

(b0)1 + (b0)2 =

2

2

e

eT v(1) (n1 + n2)T v(1)

1

2

e

eT v(2) (n1 + n2)T v(2) (D.13)

It involves

eT v(1)

(n1 + n2)T v(1)

= 2 cos sin 2 cos (D.14)

eT v(2) (n1 + n2)T v(2) =

2 sin + cos

2 sin (D.15)

(b0)1 + (b0)2 takes the form:

(b0)1 + (b0)2 =

1

2

e

n

(1 + 2)

h

cos2 sin2

2 sin cos i

(1 2)

o

(D.16)

With cos , sin from eqs. (C.38)(C.40) and using identity (C.37), we have: cos2 sin2

2 sin cos = 1

(1 2)

2 s+ (m2 + m23) + 2 (2 m23) + (1 + 2) (D.17)

Finally (b0)1 + (b0)2 reads:

(b0)1 + (b0)2 = 1

e

s

+ (m2 + m23)

+ (2 m23)

(D.18)

(iv) (b1 + b2)(3 0)

The extra bit added to (b0)1 +(b0)2 to obtain (b1 +b2)(3 0) is proportional to (eT

v(3)) ((n1+n2)T v(3)). Here again,20 (eT v(3)) O(s), ((n1+n2)T v(3)) O(s) and

in both terms the relevant contribution comes from the component s/(

e2) l(3) of v(3) in eq. (C.29). The product of these two contributions provides the term sought.

Since ((n1 + n2)T l(3)) = 2 (s+ (m2 + m23)), we have:

13 eT v(3) (n

1 + n2)T v(3)

= 1

3

s2

e

s+(m2+m23)

e eT l(3)

+ (D.19)

20Notice that ((n1 + n2)T v(3)) = 0.

29

Rewriting

s+ (m2 + m23)

eT

l(3)

= 2

e + (4 m2)

s

+ (m2 + m23)

+ (2 m23)

(D.20)

we get:

13 eT v(3)

(n

1 + n2)T v(3)

= 2 s2 3

e +

s2

3

4 m2

(

e)2

s

+ (m2 + m23)

+ (2 m23)

+ (D.21)

The combination of eqs. (D.18) and (D.21) using eq. (D.11) leads to:

(b1 + b2)(3 0)

= 2 s2 3

JHEP11(2013)154

e

s2 3

1

e

s

+ (m2 + m23)

+ (2 m23)

+ (D.22)

(v) B(3 0)

As a check, let us combine eqs. (D.12) and (D.22). We get:

(b1 + b2 + b3)(3 0) =

2 s2

e 3

s2

e 3

n s

+ (m2 + m23)

+ (2 m2)

+ + (m2 + m23)

s

+ (2 m23)

o

+

= 22 s+s2 s2

2

e 3

+ (D.23)

where we recognize the identity

B = det(G)

det(S)

for 3 0 since the numerator and the denominator of the r.h.s. of eq. (D.23) are

the expressions of det(G) and det(S) respectively, at the approximation retained cf.

eqs. (4.8), (4.9) and (C.12).

(vi) B0

Combining eqs. (D.9) and (D.18) we also get B0 = (b0)1 + (b0)2 + (b0)3:

B0 = 1

e s

+ (m2 + m23)

+ (2 m2)

s

+ + (m2 + m23)

+ (2 m23)

e (D.24) Notice that B0 happens to coincide with the limit t of B seen as a function of

t = s2/s2, as given by eq. (F.2) in appendix F.

30

= 2 s+

(vii) (b0)1 (b0)2Since n1 n2 = 2 v(3), (n1 n2)T v(j)

= 0, j = 1, 2 thus we have:

(b0)1 (b0)2 =

2

2

e eT v(1)

(n

1 n2)T v(1)

1

2

e eT v(2)

(n

1 n2)T v(2)

= 0 (D.25)

(viii) (b1 b2)(3 0)

Given eq. (D.25), (b1 b2)(3 0) is given by:

(b1 b2)(3 0) = 13 eT v(3) (n

1 n2)T v(3)

(D.26)

Whereas (eT v(3)) = 0, ((n1 n2)T v(3)) = 2 6= 0. This makes (b1 b2)(3 0)

diverge. More precisely, since ((n1 n2)T l(3)) = 0, the O(s) terms in the r.h.s. of

eq. (D.27) cancel and, from eq. (C.29) and we get:

(n1 n2)T v(3)

= 1 + O(s2)

2 + (D.27)

As (eT v(3)) = O(s), the O(s2) correction in eq. (D.27) leads to a contribution to

(b1 b2)(3 0) which is O(s3/3) i.e. beyond the approximation retained. We

thus keep:

(b1 b2)(3 0) = 13 eT v(3) (n

JHEP11(2013)154

1 n2)T v(3)

+

=

s

e 3

eT l(3)

+ (D.28)

with (eT l(3)) given by eq. (D.9). This makes (b1 b2)(3 0) diverge as s/3

which is O(s1) O(1/23) when both det(G) and det(S) tend to zero.

E A relation between the zero eigenmodes of S and G(N) when

det(G) = 0, det(S) = 0

We specify a = N to x the ideas. When det(G) = 0, condition (4.4) is equivalent to the condition det(S) = 0 according to eq. (4.2) only if G(N) has rank (N 2). On the

other hand when G(N) has a lower rank, its cofactor matrix

eG(N) vanishes identically thus det(S) = 0 again. However, as already mentioned, Gram matrices G(N) with ranks (N 3) for N = 3, 4 correspond to quite peculiar and degenerate kinematics irrelevant

for NLO processes, thus we do not provide any more detail about this case here. We focus on the generic case for which the Gram matrix has rank (N 2) i.e. exactly one

vanishing eigenvalue.

When det(G) and det(S) vanish simultaneously the eigenvectors v(N) and n(N) corre

sponding to the eigenvalues zero of S and G(N) respectively, happen to be simply related.

31

To see this, using eqs. (2.7), (2.8) let us write the components i = 1, , N of S v for any

N-column vector v as:

(S v)i =

PN1

j=1 G(N)ij vj

+

PN1

j=1 V (N)j vj

+ V (N)i + SNN

PN

j=1 vj

+ SNN

if i 6= N

if i = N (E.1)

As argued in subsection 4.1, the eigenvector n(N) fullls condition (4.4). Therefore the ansatz

v(N)j n(N)j, j = 1, , N 1 , v(N)N

N1

Xj=1n(N)j (E.2)

PN1

j=1 V (N)j vj

PN

j=1 vj

JHEP11(2013)154

PNj=1 v(N)j = 0 by construction. If S has rank (N 1), the eigendirection of S associated to the eigenvalue

zero is thus identied.

Conversely, consider v such that

N

Xj=1

Sij vj = 0 (E.3)

is solution of system (E.1). Furthermore it satises the property

and dene

N

Xj=1vj (E.4)

Using eq. (E.1), eqs. (E.3), (E.4) may be written:

N1

Xj=1G(N)ij vj = V (N)i, i = 1, , N 1 (E.5)

N1

Xj=1V (N)j vj = SNN (E.6)

If = 0, the (N 1)-column vector ni vi, i = 1, , N 1 is an eigenvector of G(N)

associated to the eigenvalue zero and fullling condition (4.4).

Let us conclude these considerations with the following remarks.

1. We recall that, at the one loop order which we are concerned with, the Landau conditions [19, 20] characterizing the appearance of kinematic singularities require vi 0 for all i = 1, , N and > 0. One may hastily infer that, an eigendirection

zero of S associated with a vanishing det(G) is not associated with a kinematic

singularity as characterized by the Landau conditions. This does hold true if S has

rank (N 1).

32

2. Let us however note that if S has two vanishing eigenvalues with corresponding

linearly independent eigenvectors v(1) and v(2) both such that

PNj=1 v(i)j 6= 0, their

PNj=1 v(2)j = . The (N 1)-column vector n dened by ni = v(1)i v(2)i, i = 1, , N 1 then fullls21 G(N)n =

0 and condition (4.4).

In particular, v(1) and v(2) may both fulll the Landau conditions corresponding to piled-up singularities. The so-called double parton singularity [23] is one interesting case of this kind. It occurs for the four-point function with opposite light-like and opposite timelike legs and with internal masses all vanishing, for which det (S) det(G)2, when the two lightlike momenta are incoming head-on and the

two timelike external momenta are outgoing back-to-back in the transverse plane w.r.t. the incoming direction.22

3. In practice we shall however stress that such a degeneracy of the zero eigenvalue of S

is very peculiar. Beside the double parton singularity, this situation happens to occur for N = 4 with three of the four internal masses equal, for very peculiar degenerate kinematics involving two spacelike momenta, and two lightlike momenta orthogonal to the spacelike ones, collinear to each other and being linear combination of the spacelike ones. . . This quite degenerate kinematics namely corresponds to a Gram matrix with rank 1 only. As already said, such an odd case is actually irrelevant regarding NLO processes at colliders, thus do not deserve any further detail here.

PNj=1 v(1)j =

F The directional limit det(S) ! 0, det(G) ! 0 of eq. (4.1) is actually isotropic

This appendix presents an analytical proof that, whereas each of the three terms involved in eq. (4.1) are separately functions of t in the directional limit s 0, s2 0 with

s2/s2 = t xed, the limit of their sum is actually independent of t. For this purpose we

compute the t-derivative of this sum in this limit and prove it to vanish identically in t. We provide an explicit proof for I43; the I63 case, albeit more cumbersome, can be handled in a completely similar way.

21If S has rank N 2, such eigenvectors v(1) and v(2) can always be found even if S is complex. In the latter case, consider linearly independent eigenvectors u1 and u2 of S S associated with the eigenvalue zero: their respective complex conjugates v(1) = u1 and v(2) = u2 are linearly independent eigenvectors of

S associated with the eigenvalue zero. The matrix G(N) being symmetric real, the eigenvector n of G(N) built as described shall be made real by an overall phase shift.

22The fact that such conguration leads to a vanishing det(G) does not contradict the classication of the eligible kinematics provided in appendix B. This appendix focused on the kinematical congurations corresponding to linearly independent sets of four-momenta. On the contrary the double parton scattering singularity appears for coplanar congurations of linearly dependent four-momenta.

33

components can be rescaled in order that

JHEP11(2013)154

In the limit s 0, s2 0, s2/s2 = t xed, eq. (4.1) reads:

3

Xj=1bj I3(j) =

"t (s+ + (m2 m23)) (t

e + 4 m2)

Z

1 ln(m2) ln(S0)

2 B m2 + 1

+ t (s+ + (m23 m2)) 2(t

e + 4 m2)

Z

1 ln(g(z)) ln(S0)2 B g(z) + 1

+ s+ + (m23 m2)(t

e + 4 m2)

Z

1

0 dz

(4 B z (1 z)ln(g(z)) ln(S0)(2 B g(z) + 1)2 z (1 z)g(z)22 B g(z) + 1)#(F.1)

where

JHEP11(2013)154

B =

2 (1 t s+) t

e + 4 m2 , S0 =

1

2 B i (F.2)

e = s+ (m2 + m23)

2 4 m2 m23 (F.3) g(z) = s+ z2 + s+ + m2 m23

z + m23 (F.4)

The kinematic parameter

e was already dened by eq. (4.10), the equivalent form (F.3) is given here for convenience, as eq. (F.4) does for the function g(z) previously dened by eq. (4.20) and fullling eqs. (4.18) and (4.19). To keep formulas compact, let us introduce the following notations: H(z, t) = ln(g(z))ln(S0), D(z, t) = 4 (1t s+) g(z)+t

e+4 m2, m = (m23 m2), T1 = (1 t s+), T2 = m s+, T3 = m + s+. Di erentiating eq. (F.1)

w.r.t. t leads to:

d dt

3

Xj=1bj I3(j) = [P1 + P2 + P3 + P4 + P5] (F.5)

with

P1 = 1 4

T2 T 21 S0

P2 = (1 + T1 t m)

Z

1 (g(z))2 H(z, t)

D(z, t)2

P3 = 1

Z

1 T 22 (1 + T1 t m) + 4 H(z, t) T3S0 T 21

D(z, t)

4 S0 T 21

P4 = 2 T2

Z

1 z (1 z) (g(z) T 22 + 4 s+ g(z) S0 T1 H(z, t) + (g(z))2 S0T1) g(z) D(z, t)2

T1 S0

Z

1 z (1 z) (g(z))2 H(z, t)

D(z, t)3

where g(z) = dg(z)/dz. To derive eq. (F.5), we have used that:

D(z, t)

t = (g(z))2

We will not compute any of these integrals over z explicitly: we will instead integrate by parts to iteratively decrease the powers of D(z, t) in denominators, starting with P5

34

P5 = 16 T1 T2

which involves the highest power, and proceed to a step by step cancellation of terms on the way. For this purpose we note that the partial z-derivative of D(z, t) is g(z) times a z-independent factor:

D(z, t)z = 4 T1 g(z) (F.6)

Integrating P5 by parts and noticing that the boundary term vanishes due to the z (1 z)

factor, we get:

P5 = 2 T2

Z

1

0 dz

H(z, t) g(z) (2 z 1) 2 s+ z (1 z) H(z, t)

z (1 z) (g(z))2 g(z)

JHEP11(2013)154

1D(z, t)2 (F.7)

Accounting for eq. (F.7), let us now collect all the terms with denominator D(z, t)2 in

eq. (F.5). We get:

P2 + P4 + P5 =

Z

1 2 z (1 z) T 32 S0 T1 T3 D(z, t) H(z, t)T1 S0 D(z, t)2 (F.8)

Comparing eq. (F.8) and the equation which gives P3 above, we see that the contribution proportional to H(z, t) cancels out in the sum

P5i=2 Pi which reads:

5

1 8 T1 T2 z (1 z) + D(z, t) (1 + T1 t m)

D(z, t)2 (F.9)

To further decrease the power of D(z, t)2 in eq. (F.9), we notice that

z (1 z) =

14 s2+ T2

Xi=2Pi = T 224 S0 T 21

Z

2 m m t s+ t s2+

(g(z))2

+2 m T2 g(z) + T3 s+ D(z, t)

(F.10)

Inserting eq. (F.10) in eq. (F.9), we get:5

Xi=2Pi = Q1 + Q2 + Q3 (F.11)

with

Q1 = T 22 s2+ t + m t s+ 2 m 2 T1 S0 s2+

Z

1 (g(z))2

D(z, t)2

Q2 =

T 32 m T1 S0 s2+

Z

1

0 dz

g(z)

D(z, t)2

Q3 =

T 224 T 21 S0 s+

Z

1 2 T1 T3 s+ (1 + T1 m t)

D(z, t)

Again, an integration by parts of Q1 and Q2 using eq. (F.6) gives:

Q1 = T 22 (s2+ t + m s+ t 2 m)

8 T 21 S0 s2+

g(1)

D(1, t)

g(0)

D(0, t)

Z

1

0 dz

2 s+ D(z, t)

Q2 =

T 32 m4 T 21 S0 s2+

1 D(1, t)

1 D(0, t)

35

The integrals of terms proportional to 1/D(z, t) in Q1 and Q3 cancel against each other. Besides, the denitions of D(z, t) and g(z) lead to

g(1) = T2 , g(0) = T3 (F.12)

D(1, t) = t T 22 , D(0, t) = t T 23 4 m (F.13) Substituting in eq. (F.11), we nd:

5

Xi=2

Pi =

T24 T 21 S0= P1 (F.14)

Hence

d dt

q.e.d.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] T. Binoth, J.P. Guillet, G. Heinrich, E. Pilon and C. Schubert, An algebraic/numerical

formalism for one-loop multi-leg amplitudes, http://dx.doi.org/10.1088/1126-6708/2005/10/015

Web End =JHEP 10 (2005) 015 [http://arxiv.org/abs/hep-ph/0504267

Web End =hep-ph/0504267 ]

[http://inspirehep.net/search?p=find+EPRINT+hep-ph/0504267

Web End =INSPIRE ].

[2] Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory

amplitudes, unitarity and collinear limits, http://dx.doi.org/10.1016/0550-3213(94)90179-1

Web End =Nucl. Phys. B 425 (1994) 217 [http://arxiv.org/abs/hep-ph/9403226

Web End =hep-ph/9403226 ]

[http://inspirehep.net/search?p=find+EPRINT+hep-ph/9403226

Web End =INSPIRE ].

[3] Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes

into loop amplitudes, http://dx.doi.org/10.1016/0550-3213(94)00488-Z

Web End =Nucl. Phys. B 435 (1995) 59 [http://arxiv.org/abs/hep-ph/9409265

Web End =hep-ph/9409265 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9409265

Web End =INSPIRE ].[4] R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4

super-Yang-Mills, http://dx.doi.org/10.1016/j.nuclphysb.2005.07.014

Web End =Nucl. Phys. B 725 (2005) 275 [http://arxiv.org/abs/hep-th/0412103

Web End =hep-th/0412103 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0412103

Web End =INSPIRE ].[5] R. Britto, E. Buchbinder, F. Cachazo and B. Feng, One-loop amplitudes of gluons in SQCD,

http://dx.doi.org/10.1103/PhysRevD.72.065012

Web End =Phys. Rev. D 72 (2005) 065012 [http://arxiv.org/abs/hep-ph/0503132

Web End =hep-ph/0503132 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0503132

Web End =INSPIRE ].[6] C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, Unitarity cuts and Reduction

to master integrals in d dimensions for one-loop amplitudes, http://dx.doi.org/10.1088/1126-6708/2007/03/111

Web End =JHEP 03 (2007) 111

[http://arxiv.org/abs/hep-ph/0612277

Web End =hep-ph/0612277 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0612277

Web End =INSPIRE ].[7] G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar

integrals at the integrand level, http://dx.doi.org/10.1016/j.nuclphysb.2006.11.012

Web End =Nucl. Phys. B 763 (2007) 147 [http://arxiv.org/abs/hep-ph/0609007

Web End =hep-ph/0609007 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0609007

Web End =INSPIRE ].[8] P. Mastrolia, G. Ossola, C. Papadopoulos and R. Pittau, Optimizing the reduction of

one-loop amplitudes, http://dx.doi.org/10.1088/1126-6708/2008/06/030

Web End =JHEP 06 (2008) 030 [arXiv:0803.3964] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0803.3964

Web End =INSPIRE ].[9] G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: a program implementing the OPP

reduction method to compute one-loop amplitudes, http://dx.doi.org/10.1088/1126-6708/2008/03/042

Web End =JHEP 03 (2008) 042 [arXiv:0711.3596]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:0711.3596

Web End =INSPIRE ].

36

JHEP11(2013)154

3

Xj=1bj I3(j) = 0 (F.15)

[10] R.K. Ellis, W. Giele and G. Zanderighi, Semi-numerical evaluation of one-loop corrections,

http://dx.doi.org/10.1103/PhysRevD.73.014027

Web End =Phys. Rev. D 73 (2006) 014027 [http://arxiv.org/abs/hep-ph/0508308

Web End =hep-ph/0508308 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0508308

Web End =INSPIRE ].[11] A. Denner and S. Dittmaier, Reduction schemes for one-loop tensor integrals,

http://dx.doi.org/10.1016/j.nuclphysb.2005.11.007

Web End =Nucl. Phys. B 734 (2006) 62 [http://arxiv.org/abs/hep-ph/0509141

Web End =hep-ph/0509141 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0509141

Web End =INSPIRE ].[12] G. Cullen et al., Automated One-Loop Calculations with GoSam,

http://dx.doi.org/10.1140/epjc/s10052-012-1889-1

Web End =Eur. Phys. J. C 72 (2012) 1889 [arXiv:1111.2034] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.2034

Web End =INSPIRE ], http://gosam.hepforge.org/

Web End =http://gosam.hepforge.org/ .[13] T. Binoth, J.-P. Guillet, G. Heinrich, E. Pilon and T. Reiter, Golem95: a numerical program

to calculate one-loop tensor integrals with up to six external legs,

http://dx.doi.org/10.1016/j.cpc.2009.06.024

Web End =Comput. Phys. Commun. 180 (2009) 2317 [arXiv:0810.0992] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0810.0992

Web End =INSPIRE ].[14] G. Cullen et al., Golem95C: a library for one-loop integrals with complex masses,

http://dx.doi.org/10.1016/j.cpc.2011.05.015

Web End =Comput. Phys. Commun. 182 (2011) 2276 [arXiv:1101.5595] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1101.5595

Web End =INSPIRE ].[15] G. t Hooft and M. Veltman, Scalar one loop integrals, http://dx.doi.org/10.1016/0550-3213(79)90605-9

Web End =Nucl. Phys. B 153 (1979) 365

[http://inspirehep.net/search?p=find+J+Nucl.Phys.,B153,365

Web End =INSPIRE ].

[16] Encyclopedic Dictionary of Mathematics, vol. I, MIT Press, Cambridge U.S.A. (1993).[17] F. Boudjema and L.D. Ninh, bbH production at the LHC: Yukawa corrections and the

leading Landau singularity, http://dx.doi.org/10.1103/PhysRevD.78.093005

Web End =Phys. Rev. D 78 (2008) 093005 [arXiv:0806.1498] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.1498

Web End =INSPIRE ].[18] H.W. Turnbull, The theory of determinants, matrices ans invariants, third edition, Dover

Publications Inc., New York U.S.A. (1960).[19] R.J. Eden, P.V. Landsho , D.I. Olive and J.C. Polkinghorne, The analytic S-matrix,

Cambridge University Press, Cambridge U.K. (1966).[20] C. Itzykson and J.B. Zuber, Quantum Field Theory, Mcgraw-hill, New York U.S.A. (1980).[21] S. Coleman and R. Norton, Singularities in the physical region, http://dx.doi.org/10.1007/BF02750472

Web End =Nuovo Cim. 38 (1965) 438

[http://inspirehep.net/search?p=find+J+NuovoCim.,38,438

Web End =INSPIRE ].

[22] S. Barnett, Matrices: Methods and Applications, Oxford Applied Mathematics and

Computing Sciences Series, Clarendon Press, Oxford U.K. (1990).[23] Z. Nagy and D.E. Soper, Numerical integration of one-loop Feynman diagrams for N-photon

amplitudes, http://dx.doi.org/10.1103/PhysRevD.74.093006

Web End =Phys. Rev. D 74 (2006) 093006 [http://arxiv.org/abs/hep-ph/0610028

Web End =hep-ph/0610028 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0610028

Web End =INSPIRE ].

37

JHEP11(2013)154