Direct extraction of one-loop integral coefficients
arXiv:0704.1835v1  [hep-ph]  15 Apr 2007
SLAC–PUB–12455 UCLA/07/TEP/12
Direct extraction of one-loop integral coefficients ∗
Darren Forde
Stanford Linear Accelerator Center
Stanford University
Stanford, CA 94309, USA,
Department of Physics and Astronomy, UCLA
Los Angeles, CA 90095–1547, USA.
(Dated: 12th April 2007)
Abstract
We present a general procedure for obtaining the coefficients of the scalar bubble and triangle
integral functions of one-loop amplitudes. Coefficients are extracted by considering two-particle and
triple unitarity cuts of the corresponding bubble and triangle integral functions. After choosing
a specific parameterisation of the cut loop momentum we can uniquely identify the coefficients
of the desired integral functions simply by examining the behaviour of the cut integrand as the
unconstrained parameters of the cut loop momentum approach infinity. In this way we can produce
compact forms for scalar integral coefficients. Applications of this method are presented for both
QCD and electroweak processes, including an alternative form for the recently computed three-
mass triangle coefficient in the six-photon amplitude A6(1
−, 2+, 3−, 4+, 5−, 6+). The direct nature
of this extraction procedure allows for a very straightforward automation of the procedure.
PACS numbers: 11.15.Bt, 11.25.Db, 12.15.Lk, 12.38.Bx
∗ Research supported in part by the US Department of Energy under contracts DE–FG03–91ER40662 and
DE–AC02–76SF00515.
http://arXiv.org/abs/0704.1835v1
I. INTRODUCTION
Maximising the discovery potential of future colliders such as CERN’s Large Hadron
Collider (LHC) will rely upon a detailed understanding of Standard Model processes. Dis-
tinguishing signals of new physics from background processes requires precise theoretical
calculations. These background processes need to be known to at least a next-to-leading
order (NLO) level. This in turn entails the need for computation of one-loop amplitudes.
Whilst much progress has been made in calculating such processes, the feasibility of produc-
ing these needed higher multiplicity amplitudes, such as one-loop processes with one or more
vector bosons (W’s, Z’s and photons) along with multiple jets, strains standard Feynman
diagram techniques.
Direct calculations using Feynman diagrams are generally inefficient; the large number
of terms and diagrams involved has by necessity demanded (semi)numerical approaches be
taken when dealing with higher multiplicity amplitudes. Much progress has been made in
this way, numerical evaluations of processes with up to six partons have been performed [1,
2, 3, 4, 5]. On assembling complete amplitudes from Feynman diagrams it is commonly
found that large cancellations take place between the various terms. The remaining result is
then far more compact than would naively be expected from the complexity of the original
Feynman diagrams. The greater simplicity of these final forms has spurred the development
of alternative more direct and efficient techniques for calculating these processes.
The elegant and efficient approach of recursion relations has long been a staple part of the
tree level calculational approach [6, 7]. Recent progress, inspired by developments in twistor
string theory [8, 9], builds upon the idea of recursion relations, but centred around the use of
gauge-independent or on-shell intermediate quantities and hence negating a potential source
of large cancellations between terms. Britto, Cachazo and Feng [10] initially wrote down a set
of tree level recursion relations utilising on-shell amplitudes with complex values of external
momenta. Then, along with Witten [11], they proved these on-shell recursion relations using
just a knowledge of the factorisation properties of the amplitudes and Cauchy’s theorem.
The generality of the proof has led to their application in many diverse areas beyond that
of massless gluons and fermions in gauge theory [10, 13]. There have been extensions to
theories with massive scalars and fermions [14, 15, 16] as well as amplitudes in gravity [12].
Similarly “on-shell” approaches can also be constructed at loop level. The unitarity
of the perturbative S-matrix can be used to produce compact analytical results by “glu-
ing” together on-shell tree amplitudes to form the desired loop amplitude. This unitarity
approach has been developed into a practical technique for the construction of loop ampli-
tudes [17, 18, 19], initially, for computational reasons, for the construction of amplitudes
where the loop momentum was kept in D = 4 dimensions. This limited its applicability to
computations of the “cut-constructible” parts of an amplitude only, i.e. (poly)logarithmic
containing terms and any associated π2 constants. Amplitudes consisting of only such terms,
such as supersymmetric amplitudes, can therefore be completely constructed in this way.
QCD amplitudes contain in addition rational pieces which cannot be derived using such cuts.
The “missing” rational parts are constructible directly from the unitarity approach only by
taking the cut loop momentum to be in D = 4 − 2ǫ dimensions [20]. The greater difficulty
of such calculations has, with only a few exceptions [21, 22], restricted the application of
this approach, although recent developments [23, 24, 25] have provided new promise for this
direction.
The generality of the foundation of on-shell recursion relation techniques does not limit
their applicability to tree level processes only. The “missing” rational pieces at one-loop, in
QCD and other similar theories, can be constructed in an analogous way to (rational) tree
level amplitudes [26, 27]. The “unitarity on-shell bootstrap” technique combines unitarity
with on-shell recursion, and provides, in an efficient manner, the complete one-loop ampli-
tude. This approach has been used to produce various new analytic results for amplitudes
containing both fixed numbers as well as arbitrary numbers of external legs [28, 29, 30].
Other newly developed alternative methods have also proved fruitful for calculating rational
terms [31, 32, 33, 34]. In combination with the required cut-containing terms [35, 36, 37]
these new results for the rational loop contributions combine to give the complete analytic
form for the one-loop QCD six-gluon amplitude.
The development of efficient techniques for calculating, what were previously difficult
to derive rational terms, has emphasised the need to optimise the derivation of the cut-
constructible pieces of the amplitude. One-loop amplitudes can be decomposed entirely in
terms of a basis of scalar bubble, scalar triangle and scalar box integral functions. Deriving
cut-constructible terms therefore reduces to the problem of finding the coefficients of these
basis integrals. For the coefficients of scalar box integrals it was shown in [38] that a
combination of generalised unitarity [19, 39, 40, 41], quadruple cuts in this case, along with
the use of complex momenta could be used, within a purely algebraic approach, to extract
the desired coefficient from the cut integrand of the associated box topology.
Extracting triangle and bubble coefficients presents more of a problem. Unlike for the case
of box coefficients, cutting all the propagators associated with the desired integral topology
does not uniquely isolate a single integral coefficient. Inside a particular two-particle or triple
cut lie multiple scalar integral coefficients corresponding to integrals with topologies sharing
not only the same cuts but also additional propagators. These coefficients must therefore
be disentangled in some way. There are multiple directions within the literature which have
been taken to effect this separation. The pioneering work by Bern, Dixon, Dunbar and
Kosower related unitarity cuts to Feynman diagrams and thence to the scalar integral basis,
this then allowed for the derivation of many important results [17, 18, 19]. More recently the
technique of Britto et. al. [23, 24, 25, 35, 36] has for two-particle cuts and the its extension to
triple cuts by Mastrolia [42], highlighted the benefits of working in a spinor formalism, where
the cut integrals can be integrated directly. Important results obtained in this way include
the most difficult of the cut-constructable pieces for the one-loop amplitude for six gluons
with the helicity configurations A6(+−+−+−) and A6(−+−−++). The cut-constructible
parts of Maximum-Helicity-Violating (MHV) one-loop amplitudes were found by joining
MHV amplitudes together in a similar manner to at tree level [43]. This method has been
applied by Bedford, Brandhuber, Spence and Travaglini to produce new QCD results [37].
In the approach of Ossola, Papadopoulos and Pittau [44, 45] it is possible to avoid the need
to perform any integration or use any integral reduction techniques. Coefficients are instead
extracted by solving sets of equations. The solutions of these equations include the desired
coefficients, along with additional “spurious” terms corresponding to coefficients of terms
which vanish after integrating over the loop momenta.
The many-fold different processes and their differing parton contents that will be needed
at current and future collider experiments suggests that some form of automation, even of
the more efficient “on-shell” techniques, will be required. From an efficiency standpoint,
therefore, we would ideally wish to minimise the degree of calculation required for each step
of any such process. Here we propose a new method for the extraction of scalar integral
coefficients which aims to meet this goal. The technique follows in the spirit of the simplicity
of the derivation of scalar box coefficients given in ref. [38]. Desired coefficients can be
constructed directly using two-particle or triple cuts. The complete one-loop amplitude
can then be obtained by summing over all such cuts and adding any box terms and rational
pieces. Alternatively our technique can be used to extract the bubble and triangle coefficients
from a one-loop amplitude, generated for example from a Feynman diagram. Hence the
technique is acting as an efficient way to perform the integration.
We use unitarity cuts to freeze some of the degrees of freedom of the integral loop mo-
mentum, whilst leaving others unconstrained. This then isolates a specific single bubble
or triangle integral topology and hence its coefficient. Within each cut there remain ad-
ditional coefficients. In the triangle case those of scalar box integrals. In the bubble case
both scalar box and scalar triangle integrals contribute. Disentangling our desired coefficient
from these extra contributions is a straightforward two step procedure. First one rewrites
the loop momentum inside the cut integrand in terms of its unconstrained parameters. In
the triangle case there is a single parameter, and in the bubble case there are a pair of
parameters. Examining the behaviour of the integrand as these unconstrained parameters
approach infinity then allows for a straightforward separation of the desired coefficient from
any extra contributions. The coefficient of each basis integral function can therefore be
extracted individually in an efficient manner with no further computation.
This paper is organised as follows. In section II we outline the notation used throughout
this paper. In section III we proceed to present the basic structure of a one-loop amplitude
in terms of a basis of scalar integral functions. We describe in section IV our procedure
for extracting the coefficients of scalar triangle coefficients through the use of a particular
loop-momentum parameterisation for the triple cuts along with the properties of the cut
as the single free integral parameter tends to infinity. Section V extends this formalism
to include the extraction of scalar bubble coefficients. The two-particle cut used in this
case contains an additional free parameter and requires an additional step in our procedure.
Finally in section VI we conclude by providing some applications which act as checks of
our method. Initially we examine the extraction of various basis integral coefficients from
some common one-loop integral functions. We then turn our attention to the construction
of the coefficients of some more phenomenologically interesting processes. These include the
three-mass triangle coefficient for the six photon amplitude A6(− + − + −+), as well as
a representative three-mass triangle coefficient of the process e+e− → q+q−g−g+. Finally
we construct the complete cut-containing part of the amplitude A
1−loop
−, 2−, 3+, 4+, 5+)
and discuss further comparisons against coefficients of more complicated gluon amplitudes
contained in the literature.
II. NOTATION
In this section we summarise the notation used in the remainder of the paper. We will
use the spinor helicity formalism [47, 48], in which the amplitudes are expressed in terms of
spinor inner-products,
〈j l〉 = 〈j−|l+〉 = ū−(kj)u+(kl) , [j l] = 〈j+|l−〉 = ū+(kj)u−(kl) , (2.1)
where u±(k) is a massless Weyl spinor with momentum k and positive or negative chirality.
The notation used here follows the QCD literature, with [i j] = sign(k0i k
j )〈j i〉∗ for real
momenta so that,
〈i j〉[j i] = 2ki · kj = sij . (2.2)
Our convention is that all legs are outgoing. We also define,
λi ≡ u+(ki), λ̃i ≡ u−(ki) . (2.3)
We denote the sums of cyclicly-consecutive external momenta by
i...j ≡ k
i + k
i+1 + · · ·+ k
j−1 + k
j , (2.4)
where all indices are mod n for an n-gluon amplitude. The invariant mass of this vector is
si...j ≡ K2i...j . (2.5)
Special cases include the two- and three-particle invariant masses, which are denoted by
sij ≡ K2ij ≡ (ki + kj)2 = 2ki · kj, sijk ≡ (ki + kj + kk)2 . (2.6)
We also define spinor strings,
∣ (/a ± /b)
= 〈i a〉[a j] ± 〈i b〉[b j] ,
∣ (/a + /b)(/c + /d)
= [i a]
∣ (/c + /d)
+ [i b]
∣ (/c + /d)
. (2.7)
III. UNITARITY CUTTING TECHNIQUES AND THE ONE-LOOP INTEGRAL
BASIS
Our starting point will be the general dimensionally-regularised decomposition of a one-
loop amplitude into a basis of scalar integral functions [18, 53]
A1−loopn =Rn+rΓ
(µ2)ǫ
(4π)2−ǫ
biB0(K
cijC0(K
i , K
dijkD0(K
i , K
j , K
.(3.1)
The scalar bubble, triangle and box integral functions are denoted by B0, C0 and D0 respec-
tively, and along with rΓ their explicit forms can be found in Appendix C. The bi, cij and
dijk are their corresponding rational coefficients. Any ǫ dependence within these coefficients
has been removed and placed into the rational, Rn, term. The problem of deriving the
one-loop amplitude is therefore reduced to that of finding the coefficients of these scalar
integral functions and any rational terms when working in D = 4 dimensions.
We are going to consider obtaining these coefficients via the application of various cuts
within the framework of generalised unitarity [19, 39, 40, 41]. In general our cut momenta
will be complex, so for our purposes we define a “cut” as the replacement
(l + Ki)2
→ (2π)δ((l + Ki)2). (3.2)
By systematically constructing all possible unitarity cuts we can reproduce every integral
coefficient of a particular amplitude. Alternatively, application of the same procedure of
“cutting” legs can be used to extract from a one-loop integral the corresponding coefficients
of the standard basis integrals making up that particular integral, in a sense acting as a form
of specialised integral reduction. This approach follows in a similar vein to that adopted by
Ossola, Papadopoulos and Pittau [44].
The most straightforward implementation of the technique we present here is when the cut
loop momentum is massless and kept in D = 4 dimensions. Eq. 3.1 therefore contains, within
the term Rn, any rational terms missed by performing cuts in only D = 4. Approaches for
deriving such terms independently of unitarity cuts exist and so we do not concern ourselves
with these here [23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 44, 45].
As was demonstrated in [38], the application of a quadruple cut, as shown in figure 1, to
A1−loopn uniquely identifies a particular box integral topology D0(K
i , K
j , K
k) and hence its
Kk−Ki − Kj − Kk
FIG. 1: A generic quadruple cut used to isolate the scalar box integral D0(K
coefficient. This coefficient is then given by
dijk =
A1(lijk;a)A2(lijk;a)A3(lijk;a)A4(lijk;a), (3.3)
where lijk;a is the a
th solution of the cut loop momentum l that isolates the scalar box
function D0(K
i , K
j , K
k), there are 2 such solutions. Eq. 3.3 applies as well to the cases
when one or more of the four legs of the box is massless. This is a result of the existence, for
complex momenta, of a well-defined three-point tree amplitude corresponding to any corner
of a box containing a massless leg.
Applying a triple cut to the amplitude A1−loopn does not isolate a single basis integral.
Instead we have a triangle integral plus a sum of box integrals obtained by “opening” a
fourth propagator. This can be represented schematically via
(µ2)ǫ
(4π)2−ǫ
cijC0(K
i , K
j ) +
dijkD0(K
i , K
j , K
k) + . . .
, (3.4)
where the additional terms correspond to “opening” the Ki leg or the Kj leg instead of the
−(Ki + Kj) leg. Similarly in the case of a two-particle cut we again cannot isolate a single
basis integral by itself. Instead we get additional triangle and box integrals corresponding
to “opening” third and forth propagators. Schematically this is given by
(µ2)ǫ
(4π)2−ǫ
biB0(K
i ) +
cijC0(K
i , K
j ) +
dijkD0(K
i , K
j , K
k) + . . .
, (3.5)
where again the additional terms are boxes with the Ki leg or the Kj legs “opened”. Whilst
not isolating a single integral each of the above cuts does single out either one scalar triangle,
in the triple cut case, or one scalar bubble, in the two-particle cut case. Disentangling
these single bubble or triangle integral functions from the contributions of the remaining
basis integrals will allow us to directly read off the corresponding coefficient. Applying all
possible two-particle, triple and quadruple cuts then enables us to derive the coefficients of
every basis integral function.
IV. TRIPLE CUTS AND SCALAR TRIANGLE COEFFICIENTS
A triple cut contains not only contributions for the corresponding scalar triangle integral,
but also contributions from scalar box integrals which share the same three cuts as the
triangle. Of the four propagators of a scalar box integral, three will be given by the three
cut legs of the triple cut loop integral. The forth propagator will be contained inside the cut
integrand in a denominator factor of the form (l − P )2, which corresponds to a propagator
pole. Ideally we want to separate terms containing such poles from the remainder of the
cut integrand. The remaining term will be the scalar triangle integral multiplied by its
coefficient for that particular cut.
The three delta functions of a triple cut constrain the cut loop momentum such that
only a single free parameter of the integral remains, which we label t. We can express the
loop momentum in terms of this parameter using the orthogonal null four-vectors, a
i , with
i = 1, 2, 3, specific forms for these basis vectors are presented in section IVA. The loop
momentum is then given by
lµ = a
0 t +
1 + a
2 . (4.1)
Denominator factors of the cut integrand depending upon the cut loop momentum, can be
written as propagators of the general form, (l − P )2. When these propagators go on-shell
they will correspond to poles in t. These poles will be solutions of the following equation
(l − P )2 = 0 ⇒ 2(a0 · P )t + 2(a1 · P )
+ 2(a2 · P ) − P 2 = 0. (4.2)
If we consider t to be a complex parameter then we can use a partial fraction decomposi-
tion in terms of t to rewrite an arbitrary triple-cut integral. For the extraction of integral
coefficients we need only work with integrals in D = 4 dimensions. We also drop an overall
denominator factor of 1/(2π)4 which multiplies all integrals. The partial fraction decompo-
sition is therefore given, in the case when we have applied a triple cut on the legs l2, (l−K1)2
and (l − K2)2, by
(2π)3
δ(l2i )A1A2A3
=(2π)3
δ(l2i )
[InftA1A2A3] (t) +
poles {j}
Rest=tj A1A2A3
t − tj
 , (4.3)
where li = l −Ki and l0 = l. This is a sum of all possible poles of t, labelled here as the set
{j}, contained in the cut integrand denoted by A1A2A3. Pieces of the integrand without a
pole are contained in the Inf term, originally given in [30], and defined such that
([InftA1A2A3] (t) − A1(t)A2(t)A3(t)) = 0. (4.4)
In general [InfzA1A2A3](t) will be some polynomial in t,
[Inf tA1A2A3] (t) =
i, (4.5)
where m is the leading degree of large t behaviour and depends upon the specific integrand
in question.
After applying the three delta functions constraints we see that taking the residue of
A1A2A3 at a particular pole, t = t0, removes any remaining dependence upon the loop
momentum. Hence we can write
δ(l2i )
Rest=t0 A1A2A3
t − t0
∼ lim
[(t − t0)A1A2A3]
δ(l2i )
t − t0
. (4.6)
Where on the right hand side of this we understand the integral,
d4l, as over the parame-
terised form of l in terms of t and the three other degrees of freedom. In the cut integrand
the only source of poles in t is from propagator terms of the type 1/(l − P )2. Generally
each such propagator, when on-shell, contains two poles due to the quadratic nature, in t,
of eq. (4.2). If we label these solutions t± then we can write a triple-cut scalar box in terms
of these poles as
δ(l2i )
(l − P )2 ∼
t+ − t−
δ(l2i )
t − t+
δ(l2i )
t − t−
. (4.7)
From comparing this to eq. (4.6) we see that all residue terms of eq. (4.3) simply correspond
to pieces of triple-cut scalar box functions multiplied by various coefficients.
Therefore we can associate all residue terms with scalar boxes, meaning that our triple
cut amplitude can be written simply as
(2π)3
δ(l2i )A1A2A3 = (2π)
boxes {l}
0 . (4.8)
This is a sum over the set {l} of possible cut scalar boxes, Dcut0 , and their associated
coefficients, dl, along with a power series in positive powers of t. In eq. (4.8) we have
integrated over the three delta functions after performing the integral transformation from
lµ to t, the Jacobian of which, and any additional factors picked up from the integration
is contained in the factor Jt. The limit m of the summation is the maximum power of
t appearing in the integrand, which in turn is the maximum power of l appearing in the
numerator of the integrand. In general for renormalisable theories, such as QCD amplitudes,
m ≤ 3.
We must now turn our attention to answering the question of what do the remaining
terms correspond to? To do this we need to understand the behaviour of the integrals
over positive powers of t. There is a freedom in our choice of the parameterisation of the
cut-loop momentum. This freedom extends, as we will prove in section IVA, to choosing
a parametrisation where the integrals over all positive powers of t vanish. Doing this then
reduces the cut integrand to
(2π)3
δ(l2i )A1A2A3 = (2π)
dtJt +
boxes {l}
0 . (4.9)
The remaining integral is now simply that of a triple-cut scalar triangle, multiplied by the co-
efficient f0. For the triple-cut scalar triangle integral, C
i , K
j ), given by −(2π)3
dtJt,
the triple cut form of eq. (C4), we find that its corresponding coefficient is given simply by
cij = − [InftA1A2A3] (t)
, (4.10)
which is just the first term in the series expansion in t of the cut-integrand at infinity.
The simplicity of this result relies crucially upon two facts. The first is that on the
triple cut the integral is sufficiently simple that it can be decomposed into either a triangle
contribution or a box contribution. This is important as it allows us to easily distinguish
between the two types of term. As an example consider a linear box which contains a
numerator factor constructed such that it vanishes at the pole contained in the denominator,
but without being proportional to the denominator itself. To which basis integral does this
term contribute to? In the simplest case such a term would look like
δ(l2i )
〈lW 〉
〈lP 〉 =
δ(l2i )
〈aW 〉(t − t0)
〈aP 〉(t− t0)
〈aW 〉
〈aP 〉
δ(l2i ),
and hence must contribute entirely to the triangle integral, it contains no box terms. Here we
have chosen a simplified loop momentum parameterisation in terms of two basis spinors |a+〉
and |a+〉 such that 〈lP 〉 = t〈aP 〉+ 〈aP 〉. This then contains a pole in t at t0 = −〈aP 〉/〈aP 〉
and we have chosen the spinor |W+〉 such that 〈aW 〉 = −t0〈aW 〉.
The second crucial fact is the vanishing of the other integrals over t so that the complete
scalar triangle integral is given by only the remaining integral over t0. Hence the coefficient
is given by a single term. Furthermore, the use of a complex loop momentum also means
that we can apply this formalism to the extraction of scalar coefficients corresponding to
one- and two-mass triangles as well as three-mass triangles. As discussed above for the case
of box coefficients, this is a result of the possibility of a well-defined three-point vertex when
using complex momentum, enabling in these cases the construction of non-vanishing cut
integrands.
A. The momentum parameterisation
We wish to compute the coefficient of the scalar triangle singled out by the triple cut
given in figure 2. The cut integral when written in terms of tree amplitudes is
l1 = l − K1
l2 = l − K2
c3 − 1
c2 − 1
c1 − 1
FIG. 2: The triple cut used to compute the scalar triangle coefficient of C0(K
(2π)3
δ(l2i )A
c3−c1+2
(−l, c1, . . . , (c3 − 1), l1)Atreec2−c3+2(−l1, c3, . . . , (c2 − 1), l2)
× Atreen−c2+c1+2(−l2, c2, . . . , (c1 − 1), l), (4.11)
with l1 = l − K1 = l − Kc1...c3−1 and l2 = l − K2 = l + Kc2...c1−1, so that K1 = Kc1...c3−1 and
K2 = −Kc2...c1−1.
Our first step will be to find a parameterisation of l in terms of the single free integral
parameter remaining after satisfying all three of the cut delta functions constraints,
l2 = 0, l21 = (l − K1)2 = 0, and l22 = (l − K2)2 = 0. (4.12)
Each of the three legs can be massive or massless. We will deal with the general case of three
massive legs explicitly here. The cases with massless legs are then easily found by setting
the relevant mass in the parameterisation to zero. We will find it very convenient to express
lµ in terms of a basis of momentum identical to the momenta l1 and l2 used by Ossola,
Papadopoulos and Pittau [44]. We will write these momenta in the suggestive notation K♭1
and K♭2 and define them via
1 = K
2 = K
1 , (4.13)
with γ = 〈K♭,−1 | /K
1 〉 ≡ 〈K
2 | /K
2 〉 and Si = K2i . Each momentum K♭1, K♭2 is
the massless projection of one of the massive legs in the direction of the other masslessly
projected leg. A more practical definition of K♭1 and K
2, in terms of the external momenta
alone, can be found by solving the above equations for K♭1 and K
2, so that in terms of S1,
S2, K
1 and K
2 we have
1 − (S1/γ)K
1 − (S1S2/γ2)
2 − (S2/γ)K
1 − (S1S2/γ2)
. (4.14)
In addition γ can be expressed in terms of the external momenta,
γ± = (K1 · K2) ±
∆, ∆ = (K1 · K2)2 − K21K22 . (4.15)
When using eq. (4.10) we must average over the number of solutions of γ. In the three-mass
case there are a pair of solutions. For the one- and two-mass cases, when either K21 = 0 or
K22 = 0, then there is only a single solution.
After satisfying the three constraints given by eq. (4.12) we write the spinor components
of lµ in terms of our basis K♭1 and K
〈l−| = t〈K♭,−1 | + α01〈K
〈l+| = α02
〈K♭,+1 | + 〈K
2 |, (4.16)
where
α01 =
S1 (γ − S2)
(γ2 − S1S2)
, α02 =
S2 (γ − S1)
(γ2 − S1S2)
. (4.17)
Written as a four-vector, lµ is given by
lµ = α02K
1 + α01K
〈K♭,−1 |γµ|K
2 〉 +
α01α02
〈K♭,−2 |γµ|K
1 〉. (4.18)
We can also use momentum conservation to write component forms for the other two cut
momenta li with i = 1, 2,
〈l−i | = t〈K
1 | + αi1〈K
〈l+i | =
〈K♭,+1 | + 〈K
2 |, (4.19)
where the αij are given in Appendix A.
A final point is that after having integrated over the three delta function constraints and
performed the change of variables to the momentum parameterisation of eq. (4.16) we have
the factor Jt = 1/(tγ) contained in eq. (4.8). We always associate this factor with the scalar
triangle integral and so its explicit form does not play a role in our formalism.
B. Vanishing integrals
As we have remarked previously, the simplicity of the method outlined here rests crucially
upon the properties of the momentum parameterisation we have used. The key feature is the
vanishing of the integrals over t. It can easily be shown that within our chosen momentum
parameterisation, of section IVA, any integral of a positive or negative power of t vanishes.
Following an argument very similar to that used by Ossola, Papadopoulos and Pittau [44]
we use 〈K♭,±1 | /K1|K
2 〉 = 0, 〈K
1 | /K2|K
2 〉 = 0 and 〈K
1 |γµ|K
2 〉〈K
1 |γµ|K
2 〉 = 0, to
show that
〈K♭,−1 |/l|K
l2l21l
= 0 ⇒
= 0 for n ≥ 1,
〈K♭,−2 |/l|K
l2l21l
= 0 ⇒
dtJtt
n = 0 for n ≥ 1. (4.20)
The vanishing of these terms then leads directly to our general procedure, encapsulated in
eq. (4.10), which is to simply express the triple cut of the desired scalar triangle in the
momentum parameterisation given by eq. (4.16) and then take the t0 component of a series
expansion in t around infinity.
V. TWO-PARTICLE CUTS AND SCALAR BUBBLE COEFFICIENTS
In the same spirit as the triangle case we now wish to extract the coefficients of scalar
bubble terms using, in this case, a two-particle cut. Now a two-particle cut will contain in
addition to our desired scalar bubble both scalar boxes and triangles, all of which need to
be disentangled. What we will find, though, is that naively applying the technique as given
for the scalar triangle coefficients will not give us the complete scalar bubble contribution.
The reason for this is straightforward to see. A two-particle cut places only two constraints
on the loop momentum and so we can parameterise it in terms of two free variables, which we
will label t and y. Consider rewriting the cut integrand in a partial fraction decomposition
in terms of y. Schematically, therefore, the two-particle cut of the legs l2 and (l −K1)2 can
be written as
(2π)2
δ(l2i )A1A2 =(2π)
dtdyJt,y
[InfyA1A2] (y) +
poles {j}
Resy=yj A1A2
y − yj
 ,(5.1)
where again {j} is the sum over all possible poles, this time in y, and Jt,y contains any
terms from the change into the parameterisation of y and t as well as any pieces picked up
by integrating over the two delta functions. So far this seems to be similar to the triangle
case, but with the residue terms now corresponding to triangles as well as boxes. As we
have two parameters though we can consider a further partial fraction decomposition, this
time with t, giving
(2π)2
δ(l2i )A1A2 =
(2π)2
dtdyJt,y
[Inft [InfyA1A2] (y)] (t) +
Inft
poles {j}
Resy=yj A1A2
y − yj
 (t)
poles {l}
Rest=tl [InfyA1A2] (y)
t − tl
poles {j},{l}
Rest=tl
Resy=yj A1A2
t − tl
 , (5.2)
where here {l} is the sum over all possible poles in t. The general dependence of the cut
integral momentum, lµ, on the free integral parameters t and y can be written in terms of
null four-vectors a
i with i = 0, 1, 2, 3, 4 such that l
2 = 0. An explicit form for these will be
presented in section VA. We then define lµ by
1 + ya
2 + ta
3 + a
4 . (5.3)
Again residues of pole terms will correspond to the solutions of (l − P )2 = 0 and hence it
is straightforward to see that the final term of eq. (5.2), containing the sum of residues in
both y and t, has both of these free parameters fixed. Any such terms must contain at least
one propagator pole. Also the numerator will be independent of any integration variables,
as both y and t are fixed. Thus all such terms will correspond to purely scalar triangle and
scalar box terms. Looking at the second and third terms of eq. (5.2) we might also, at least
initially, want to associate these terms with contributions to scalar triangle terms only and
hence naively conclude that only the first term of eq. (5.2) contributes to the scalar bubble
coefficient. This assumption though would be wrong.
The crucial difference between the single residue terms of eq. (5.2) and those of eq. (4.3)
is the parameterisation of the loop momentum which is being used. Taking the residue of
a pole term at a particular point y freezes y such that we force a particular momentum
parameterisation upon these triple-cut terms. Importantly, in general this particular forced
momentum parameterisation is such that the integrals over t in the second and third terms
of eq. (5.2) now no longer vanish.
If only scalar triangle contributions came from the integrals over t then this would not
be an issue; we could just discard these terms as not relevant for the extraction of our
bubble coefficient. What we find though, through a simple application of Passarino-Veltman
reduction techniques, is that these integrals contain scalar bubble contributions, B0, with
coefficients b,
dtJ ′tt
n = bB0 + c C0, (5.4)
where J ′t is the relevant Jacobian for this parameterisation of the loop momentum and c is
the coefficient corresponding to the scalar triangle contribution, C0. We cannot therefore
simply discard the residue pieces of eq. (5.2), as we could in the triangle case, if we want to
derive the full scalar bubble coefficient. Furthermore, there is an additional complication.
We will see that the integrals over powers of y contained in the first term of eq. (5.2) also
do not vanish in general and hence must also be taken into account.
There is a limit to the maximum positive powers of y and t that appear in the rewritten
partial-fractioned decomposition of the integral. For renormalisable theories, such as QCD,
up to three powers of t appear for triangle coefficients and up to four powers of y for bubble
coefficients. Therefore the power series in y and t of the Inf operators will always terminate
at these fixed points. It is then straightforward, as we will discuss in section VD and
section VB, to derive the general form for all possible non-vanishing contributing integrals,
over powers of y and t, in terms of their scalar bubble contributions.
Calculation of the scalar bubble coefficient therefore requires a two stage process. First
take the Infy and Inft pieces of the cut integrand and replace any integrals over y with
their known general forms, as we shall see integrals proportional to t will vanish. Secondly
compute all possible triple cuts that could be generated by applying a third cut to the two-
particle cut we are considering. To these terms then apply, not the parameterisation we used
in section IV, but the parameterisation forced upon us by taking the residues of the poles in
y, which we will derive in section VC. This is equivalent to calculating all the contributions
from the residues of the partial fraction decomposed cut integrand of eq. (5.2). Within these
terms we then replace any integrals of powers of t with their known general forms. Finally
we sum all the contributing pieces together to get the full scalar bubble contribution and
hence its coefficient. Our final result for assembling the bubble coefficient is then given by
eq. (5.28).
A. The momentum parameterisation for the two-particle cut
We want to extract the scalar bubble coefficient obtainable from the application of the
two-particle cut given in figure 3. This two-particle cut can be expressed in terms of tree
amplitudes as
(2π)2
δ(l2i )A
c2−c1+2
(−l, (c1 + 1), . . . , c2, l1)Atreen−c2+c1+2(−l1, (c2 + 1), . . . , c1, l),(5.5)
with l1 = l − Kc1+1...c2 = l − K1.
A bubble can be classified entirely in terms of the momentum of one of its two legs, which
we label K1, and so we will find it useful to express the cut loop momentum l in terms of
c2 + 1
c1 + 1
l1 = l − K1
FIG. 3: The two-particle cut for computing the scalar bubble coefficient of B0(K
the pair of massless momenta K♭1 and χ defined via
1 = K
χµ, (5.6)
here γ = 〈χ±| /K1|χ±〉 ≡ 〈χ±| /K♭1|χ±〉. The arbitrary vector χ can be chosen independently
for each bubble coefficient as a result of the independence of the choice of basis representation
for the cut momentum. In the two-particle cut case we have only two momentum constraints
l2 = 0, and l21 = (l − K1)2 = 0, (5.7)
and so we have two free parameters which we will label y and t. The loop momentum can
then be expressed in terms of spinor components as
〈l−| = t〈K♭,−1 | +
(1 − y) 〈χ−|,
〈l+| = y
〈K♭,+1 | + 〈χ+|. (5.8)
Written as a four-vector lµ is
lµ = yK
(1 − y)χµ + t
〈K♭,−1 |γµ|χ−〉 +
(1 − y)〈χ−|γµ|K♭,−1 〉. (5.9)
We can also use momentum conservation to write a component form for the other cut
momentum. We have
〈l−1 | = 〈K
1 | −
〈χ−|,
〈l+1 | = (y − 1) 〈K
1 | + t〈χ+|. (5.10)
Furthermore after rewriting the integral in this cut-momentum parameterisation and
integrating over the two delta function constraints we find the following simple result for
the constant Jt,y contained in eq. (5.1), namely Jt,y = 1.
B. Non-vanishing integrals
In the case of the scalar triangles of section IVB crucial simplifications occurred as a result
of our chosen cut momentum parameterisation. Any integral over a power of t vanished,
leaving only a single contribution corresponding to the desired coefficient. For the scalar
bubble coefficient things are not quite as simple.
We can use 〈K♭,±1 | /K1|χ±〉 = 0 as well as 〈K
1 |γµ|χ±〉〈K
1 |γµ|χ±〉 = 0 to show that
〈χ−|/l|K♭,−1 〉n
l2l21
= 0 ⇒
dtdy tn = 0,
〈K♭,−1 |/l|χ−〉n
l2l21
= 0 ⇒
(1 − y)n = 0. (5.11)
Hence the integrals over all positive and negative powers of t vanish,
dtdy tn = 0 for n 6= 0. (5.12)
Integrals over positive powers of y, contained within the double Inf piece of the first term
of eq. (5.2), will not vanish. These integrals are straightforwardly derivable with the aid
of identities involving the four vector nµ = K
1 − (S1/γ)χµ which satisfies the constraints
(K1 · n) = 0 and n2 = −S1. It is then possible to show the following relations in D = 4
dimensions, and remembering that Jt,y = 1,
(l · n)2m−1
l2l21
= 0 ⇒
)2m−1
(l · n)2m
l2l21
= S2m1 B
= S2m1 B̃
(l · K1)2m
l2l21
= (2m + 1)S2m1 B
dtdy = (2m + 1)S2m1 B̃
PV , (5.13)
where BmPV and B̃
PV are Passarino-Veltman reduction coefficients, the explicit forms of which
are not needed. Solving these equations for the integral of ym leads to the result
dtdy ym =
m + 1
dtdy for m ≥ 0. (5.14)
Contributions to our desired scalar bubble coefficient from the double Inf piece of eq. (5.2)
therefore come not only from the single constant t0y0 term but also from terms proportional
to integrals of t0ym. This is not the end of the story. As described above, there can be further
contributions from the second and third residue terms generated in the decomposition of
eq. (5.2). We could proceed from the cut integrand to explicitly calculate these residue
terms. However as we will shall see, a more straightforward approach is to derive these
terms by relating them to triple cuts.
C. The momentum parameterisation for triple cut contributions
We wish to relate the contributions to the bubble coefficient of the residue pieces, sep-
arated in the decomposition of eq. (5.2), to triple cuts in a specific basis of the cut-loop
momentum. To find this basis we will apply the additional constraint
(l + K2)
2 = 0, (5.15)
to the two-particle cut momentum of section VA. Note that here we label the “K2” leg as K2
in contrast to (−K2) as we did in the triangle coefficient case of section IVA. This constraint
corresponds to the application of an additional cut which would appear as δ((l+K2)
2) inside
the integral. This additional constraint, applied to the starting point of the two-particle cut
loop momentum, forces us to use K♭1 and χ as the momentum basis vectors of l. Importantly,
this differs from the basis choice for the triple cut momenta developed in section IVA, which
leads to the differing behaviour of these triple-cut contributions.
The presence of y in both 〈l−| and 〈l+| directs us for reasons of efficiency to choose to
use eq. (5.15) to first constrain y, leaving t free. Looking at eq. (5.9) we see that as lµ is
quadratic in y then there are two solutions to this constraint, y±, which are given by
2S1〈χ−| /K2|K♭,−1 〉
γ〈K♭,−1 | /K2|K
1 〉 − S1〈χ−| /K2|χ−〉
t + S1〈χ−| /K2|K♭,−1 〉
S1〈χ−| /K2|K♭,−1 〉 + 2tγ (K1 · K2)
− 4S1S2γt
tγ − 〈χ−| /K2|K♭,−1 〉
. (5.16)
On substituting these two solutions into the two-particle cut momentum of eq. (5.8) we
obtain our desired triple-cut momentum parameterisation.
Our final step is then to relate the triple-cut integrals defined in this basis to the residue
terms of eq. (5.2). Rewriting the triple cut integral after the change of momentum parame-
terisation and integrating over all but the third delta function gives the general form
(2π)3
dtdy J ′t (δ(y − y+) + δ(y − y−))M(y, t), (5.17)
where M(y, t) is a general cut integrand and
J ′t =
S1〈χ−| /K2|K♭,−1 〉 + 2tγ (K1 · K2)
− 4S1S2γt
tγ − 〈χ−| /K2|K♭,−1 〉
. (5.18)
Upon examination of a general residue term we find that it corresponds to an integral of the
(2π)2i
dtdyJt,y Res
M(y, t)
(l + K2)2
≡ −(2π)
dtdy J ′t (δ(y − y+) + δ(y − y−))M(y, t),(5.19)
and hence that residue contributions are given, up to a factor of (−1/2), by the triple cut.
This result applies equally when S2 = K
2 = 0, corresponding to a one or two-mass
triangle, when the appropriate scale is set to zero in eq. (5.16) and eq. (5.18). The momentum
parameterisation in this simplified case is contained in Appendix B.
D. More non-vanishing integrals and bubble coefficients
There is a direct correspondence between a triple cut contribution and a residue contri-
bution. The sum of all possible triple cuts, which contain the original two-particle cut, will
therefore correspond to the sum of all residue terms. We must now examine how such terms
contribute to the bubble coefficient itself.
Unlike for the case of triple cut integrands as parameterised in section IVA we will find
that there are contributions, specifically in this case bubble coefficient contributions, coming
from the integrals over t. To see this let us investigate the integrals over t in more detail.
As an example consider extracting the scalar bubble term coming from a two-mass linear
triangle (with the massless leg K2 so that S2 = 0). We would start from a two-particle cut
which, after decomposing as eq. (5.1), would give
(2π)2
δ(l2i )
〈K−2 |/l|a−〉
(l + K2)2
(5.20)
= (2π)2
δ(l2i )
[lK2]
= (2π)2
dtdyJ ′t
[K♭1a]
[K♭1K2]
[χK2]
y[K♭1a]+t[χa]
[χK2]
[K♭1K2]
[K♭1K2] + [χK2]
The first term of this is clearly not the complete coefficient, and so we need to obtain the
bubble contribution contained within the second term. Consider reconstructing this term
using a triple cut with the cut loop momentum parameterised in a form given by setting y
equal to its value at the residue of the pole of this second term. This triple cut term is given
−(2π)
δ(l2i )〈K−2 |/l|a−〉
= −(2π)
dtJ ′t
[χK♭1][K2a]
[K♭1K2]
〈K−2 | /K1|K−2 〉t+
〈χ−| /K2|K♭,−1 〉
, (5.21)
where we have used the parameterisation of lµ given by eq. (B9) and added an extra overall
factor of i which would come from the additional tree amplitude in a triple cut.
Of this triple cut integrand only the first, t dependent, term can give anything other than
a scalar triangle contribution. To derive the result of this integral over t we will, as we have
done previously, use our parameterisation of the cut momentum, eq. (5.9), to pick out the
integral as follows
〈χ−|/l|K♭,−1 〉
l2l21(l + K2)
≡ (2π)3γ
dtJ ′tt. (5.22)
Using Passarino-Veltman reduction on the single tensor integral on the left hand side of
this as well as dropping anything but the contributing bubble integrals of our particular cut
leaves us with the result
dtJ ′tt =
(2π)3
S1〈χK2〉[K2K♭1]
γ〈K−2 | /K1|K−2 〉2
Bcut0 (K
1 ), (5.23)
where Bcut0 (K
1 ) is the cut form of the scalar bubble integral of eq. (C1). This non-vanishing
result for the integral over t, in contrast to that of section IVA, is a direct consequence of
the cut momentum parameterisation forced upon us when taking the residues contained in
the two-particle cut integrand with which we started.
On substituting the result of eq. (5.23) into eq. (5.21) we find that we can write eq. (5.20),
using the bubble integral given in eq. (C1), as
(2π)2
δ(l2i )
〈K−2 |/l|a−〉
(l + K2)2
=−i [χK
1][K2a]
[K♭1K2]
〈χ−| /K2|K♭,−1 〉
〈K−2 | /K1|K−2 〉
Bcut0 (K
1 )+i
[K♭1a]
[K♭1K2]
Bcut0 (K
〈K−2 | /K1|a−〉
〈K−2 | /K1|K−2 〉
Bcut0 (K
1), (5.24)
which is the known coefficient of the scalar bubble contained inside the linear triangle.
Of course, if we had chosen χ = K2 from the beginning, then the first term on the left
hand side of eq. (5.20) would have been the complete bubble coefficient. In general, if we
are able to rewrite a two-particle cut integrand such that each term contains only a single
propagator then we can always choose a different χ = K♭2, defined via
2 = K
〈K♭,−1 | /K2|K
1 , (5.25)
for each term individually such that there are no contributions from the residue terms.
Whether this is both feasible and a more computationally effective approach than calculating
the residue contributions through the use of triple cuts would depend upon the cut integrand
in question.
In general we will be considering processes which contain terms with powers of up to t3,
so we will need to know these integrals. Again these can be found using a straightforward
application of tensor reduction techniques. When all three legs in the cut are massive these
integrals over t are given, after dropping an overall factor of 1/(2π)3 witch always cancels
out of the final coefficient, by
T (j) =
dtJ ′tt
)j〈χ−| /K2|K♭,−1 〉j(K1 · K2)j−1
Sl−12
(K1 · K2)l−1
Bcut0 (K
1). (5.26)
Simply taking the relevant mass to zero gives the forms in the one and two mass cases. ∆
was previously defined in eq. (4.15) and we have
C11 =
C21 = −
, C22 = −
C31 = −
(K1 · K2)2
, C32 =
, C33 =
. (5.27)
Also for later use we define T (0) = 0.
E. The bubble coefficient
We have now assembled all the pieces necessary to compute our desired scalar bubble
coefficient, bj , corresponding to the cut scalar bubble integral B
j ). It is given in general
not as the coefficient of a single term but by summing together the t0ym terms from both the
double Inf in y followed by t as well as residue contributions which we derive by considering
all possible triple cuts contained in the two-particle cut. The coefficient is given by
bj = −i [Inft [InfyA1A2] (y)] (t)
t→0, ym→ 1
{Ctri}
[InftA1A2A3] (t)
tj→T (j)
, (5.28)
where T (j) is defined in eq. (5.26) and the sum over the set {Ctri} is a sum over all triple
cuts obtainable by cutting one more leg of the two-particle cut integrand A1A2.
When computing with eq. (5.28) there is a freedom in the choice of χ. A suitable choice of
which can simplify the degree of computation involved in extracting a particular coefficient.
Particular choices of χ can eliminate the need to calculate the second term of eq. (5.28)
completely, as discussed in section VD. We also note that there are choices of χ which
eliminate the need to evaluate the first term of eq. (5.28), so that the coefficient comes
entirely from the second term of eq. (5.28) instead.
VI. APPLICATIONS
To demonstrate our method we now present the recalculation of some representative
triangle and bubble integral coefficients. We also discuss checks we have made against other
various state-of-the-art cut-constructable coefficients contained in the literature.
A. Extracting coefficients
To highlight the application of our procedure to the extraction of basis integral coefficients
we consider deriving the coefficients of some simple integral functions which commonly
appear, for example, in one-loop Feynman diagrams.
1. The triangle coefficient of a linear two-mass triangle
First we consider deriving the scalar triangle coefficient of a linear two-mass triangle with
massive leg K1, massless leg K2, and a and b arbitrary massless four-vectors not equal to
K2. This is given by the integral
〈a−|/l|b−〉
l2(l − K1)2(l + K2)2
. (6.1)
Extracting the triangle coefficient requires cutting all three propagators of the integrand.
We do this here by simply removing the “cut” propagator as we are interested only in the
integrand. This leaves only
〈a−|/l|b−〉. (6.2)
Rewriting this integrand in terms of the parameterisation of eq. (4.16) gives
α01〈a−| /K2|b−〉 + t〈aK♭1〉[χb]
. (6.3)
As S2 = 0 we see that α01 = S1/γ and that γ = 2(K1 · K2). Then taking the t0 component
of the [Inft] of this in accordance with eq. (4.10) leaves us with our desired coefficient
〈K−2 | /K1|K−2 〉
〈a−| /K2|b−〉, (6.4)
which matches the expected result.
2. The bubble contributions of a three-mass linear triangle
Consider a linear triangle with in this case three massive legs, so now K2 is massive but
again a and b are arbitrary massless four-vectors,
〈a−|/l|b−〉
l2(l − K1)2(l + K2)2
. (6.5)
Extracting the bubble coefficient of the integral B0(K
1) is done by cutting the two propa-
gators l2 and (l−K1)2. Again cutting the legs is done by removing the relevant propagators
from the integrand so that it is given by
〈a−|/l|b−〉
(l + K2)2
. (6.6)
As this contains a single propagator, and therefore a single pole, we could choose to set
χ = K♭2 (as defined in eq. (5.25)), before performing the series expansions in y and t. For
this choice of χ the bubble coefficient comes entirely from the two-particle cut. Using the
first term of eq. (5.28) gives directly
γ〈a−| /K♭1|b−〉
γ2 − S1S2
− S1〈a
−| /K♭2|b−〉
γ2 − S1S2
, (6.7)
where γ = 〈K♭,−2 | /K♭1|K
2 〉, a result which is equivalent to the expected answer.
In order to demonstrate the procedure of using triple cut contributions in extracting a
bubble coefficient we will now reproduce this by assuming χ 6= K♭2. For this case the first
term of eq. (5.28) then gives
− i 〈aχ〉[K
〈χ−| /K2|K♭,−1 〉
, (6.8)
which upon choosing χ = a vanishes and so the complete contribution will come from the
triple cut pieces of eq. (6.5). Cutting the remaining propagator in eq. (6.6) gives us the
single triple cut term which will contribute. The integrand of this is given, after multiplying
by an additional factor of i which would come from the third tree amplitude if this was a
triple cut, by
〈al〉[lb]
+ 〈al〉[lb]
(y+ + y−)〈aK♭1〉[K♭1b] + 2t〈aK♭1〉[ab]
, (6.9)
where we have set χ = a. From eq. (5.16) we have
y+ + y− =
〈K♭,−1 | /K2|K
1 〉 − S1〈a−| /K2|a−〉
t + S1〈a−| /K2|K♭,−1 〉
S1〈a−| /K2|K♭,−1 〉
, (6.10)
Hence taking the [Inft] of the cut integrand, eq. (6.9), and dropping any terms not propor-
tional to t leaves
it〈a−| /K1|b−〉
γ〈K♭,−1 | /K2|K
1 〉 − S1〈a−| /K2|a−〉
S1〈a−| /K2|K♭,−1 〉
[K♭1b]
, (6.11)
which after inserting the result for the t integral given by eq. (5.26) and substituting this
into the second term of eq. (5.28) gives for our desired coefficient
−| /K1|b−〉
〈K♭,−1 | /K2|K
〈a−| /K2|a−〉 +
〈a−| /K2|K♭,−1 〉
[K♭1b]
= −i 1
(K1 · K2)〈a−| /K1|b−〉 − S1〈a−| /K2|b−〉
, (6.12)
where ∆ was given in eq. (4.15). This matches both the expected result and eq. (6.7).
B. Constructing the one-loop six-photon amplitude A6(1
−, 2+, 3−, 4+, 5−, 6+)
Recently an analytic form for the last unknown six-photon one-loop amplitude was ob-
tained by Binoth, Heinrich, Gehrmann and Mastrolia in ref. [46]. This result was used
to confirm a previous numerical result [50]. More recently still further corroboration has
been provided by [45]. Here we reproduce, as an example, the calculation of the three-mass
triangle and bubble coefficients, again confirming part of these results.
Firstly it is a very simple exercise to demonstrate by explicit computation that all bubble
coefficients vanish. If we were to use the basis of finite box integrals, as defined in [35], then
there is only a single unique three-mass triangle coefficient, a complete explicit derivation
of which we now present. Starting from the cut in the 12 : 34 : 56 channel shown in figure 4
we can write the cut integrand as
FIG. 4: Triple cut six-photon amplitude in the 12 : 34 : 56 channel.
16A4(−l−hq , 1−, 2+, lh22,q)A4(−l−h22,q , 3−, 4+, lh11,q)A4(−l−h11,q , 5−, 6+, lhq ), (6.13)
with all unlabelled legs photons and l1 = l − K56 and l2 = l + K12. The overall factor of
16 comes from the differing normalisation conventions between QCD colour-ordered ampli-
tudes and QED photon amplitudes. Both helicity choices h = h1 = h2 = ± give identical
contributions. Written explicitly, eq. (6.13) is
〈l1〉2〈l23〉2〈l15〉2
〈l2〉〈l22〉〈l14〉〈l24〉〈l6〉〈l16〉
. (6.14)
After inserting the momentum parameterisation of eq. (4.16) this becomes
t〈K♭12〉 + α01〈K♭22〉
t〈K♭12〉 + α21〈K♭22〉
t〈K♭14〉 + α11〈K♭24〉
t〈K♭11〉 + α01〈K♭21〉
t〈K♭13〉 + α21〈K♭23〉
t〈K♭15〉 + α11〈K♭25〉
t〈K♭14〉 + α21〈K♭24〉
t〈K♭16〉 + α01〈K♭26〉
t〈K♭16〉 + α11〈K♭26〉
) . (6.15)
Applying eq. (4.10) implies taking only the t0 piece of the [Inft] of this expression. Averaging
over both solutions leaves us with our form for the three mass triangle coefficient
− 16i
〈K♭11〉2〈K♭13〉2〈K♭15〉2
〈K♭12〉2〈K♭14〉2〈K♭16〉2
, (6.16)
where K♭1 depends upon the form of γ± as given in eq. (4.14). Numerical comparison with
the analytic result of [46] shows complete agreement.
C. Contributions to the one-loop A6(1
q , 2
q , 3
−, 4+; 5−e , 6
e ) amplitude
This particular amplitude was originally obtained by Bern, Dixon and Kosower in [19].
Making up this amplitude are many box, triangle and bubble integrals along with rational
terms. Here we will recompute one particular representative three-mass triangle coefficient
in order to highlight the application of our technique to a phenomenologically interesting
process.
Following the notation of [19], we wish to calculate the three-mass triangle coefficient of
I3m3 (s14, s23, s56) ≡ C0(s14, s56) of the F cc term. The only contributing cut is shown in figure
5. We begin by writing down the triple cut integrand for this case
FIG. 5: Triple cut in the 14 : 23 : 56 channel.
A4(−l−h11,q̄ , 5−e , 6+e , lh22,q)A4(−l−h22,q , 4+, 1+q , lhg )A4(−l−hg , 2−q , 3−, lh11,q), (6.17)
where l1 = l − K23 and l2 = l + K14. Only when h = −, h1 = + and h2 = + do we get a
contribution. It can be written explicitly as
〈l25〉2〈ll2〉2〈23〉2
〈14〉〈56〉〈4l2〉〈2l〉〈ll1〉〈l1l2〉
. (6.18)
Rewriting this in terms of the loop momentum parametrisation of eq. (4.16) gives
t〈K♭15〉 + α21〈K♭25〉
)2 〈23〉2
1 − s23
〈14〉〈56〉
t〈4K♭1〉 + α21〈4K♭2〉
t〈2K♭1〉 + α01〈2K♭2〉
. (6.19)
The two solutions of γ are given by γ± = −(K23 · K14) ±
(K23 · K14)2 − s23s14, the αij ’s
are given in Appendix A.
The application of eq. (4.10) involves taking [Inft] of eq. (6.19), dropping all but the t
component of the result and then averaging over both solutions of γ giving the coefficient
γ〈K♭15〉2〈23〉2
1 − S1
〈14〉〈56〉〈4K♭1〉〈2K♭1〉
, (6.20)
where again K♭1 depends upon γ±. Numerical comparison against the solution for this
coefficient presented in [19],
〈2−| /K14 /K23|5+〉2 − 〈25〉2s14s23
〈14〉[23]〈56〉〈2−| /K14|3−〉〈2−| /K34|1−〉
+ flip, (6.21)
shows complete agreement, where the operation flip is defined as the exchanges 1 ↔ 2,
3 ↔ 4, 5 ↔ 6, 〈ab〉 ↔ [ab].
The remaining triangle and bubble coefficients can be derived in an analogous way. We
have computed a selection of these coefficients for A6(1
q , 2
q , 3
−, 4+; 5−e , 6
e ), along with
coefficients of other amplitudes given in [19], and find complete agreement.
D. Bubble coefficients of the one-loop 5-gluon QCD amplitude A5(1
−, 2−, 3+, 4+, 5+)
This result for the 1-loop 5 gluon QCD amplitude A5(1
−, 2−, 3+, 4+, 5+) was originally
calculated by Bern, Dixon, Dunbar and Kosower in [18]. It contains neither box nor triangle
integrals, only bubbles. We need therefore only compute bubble coefficients. There are only
a pair of such coefficients, with masses s23 and s234 = s51.
For the first cut in the channel K1 = K23 we have, for the sum of the two possible helicity
configurations, the two-particle cut integrand
〈23〉〈45〉〈51〉
〈1l1〉2〈1l〉〈2l〉〈2l1〉2
〈4l1〉〈3l1〉〈ll1〉2
, (6.22)
and for the second, in the channel K1 = K234,
〈23〉〈34〉〈51〉
〈1l1〉2〈1l〉〈2l〉〈2l1〉2
〈4l1〉〈5l1〉〈ll1〉2
. (6.23)
Focus upon the K1 = K23 cut initially. There are two pole-containing terms in the
denominator of this cut. We could choose to partial fraction these terms and then pick χ =
K2 in each case to extract the coefficient. Instead though we will derive the coefficient using
triple cut contributions. Choosing χ = k1 so that after inserting the cut loop momentum
parameterisation of eq. (5.8) the cut integrand becomes
2γ2〈1K♭1〉
S21〈23〉〈45〉〈51〉
〈2K♭1〉 − S1γ
t〈2K♭1〉 + S1γ (1 − y) 〈21〉
〈3K♭1〉 − S1γ
〈4K♭1〉 − S1γ
) , (6.24)
and hence produces no [Infy[Inft]] term. Consequentially the two-particle cut contribution
to the bubble coefficient vanishes. The same choice of χ similarly removes all two-particle
cut contributions in the channel K1 = K234 from the corresponding scalar bubble coefficient.
Examining the triple cuts of the bubble in the K23 channel shows only two possible
contributions, again after summing over both contributing helicities, given by
〈45〉〈51〉
[3l][3l2]〈1l1〉〈1l〉2〈2l1〉〈2l〉
〈ll1〉〈l1l2〉[ll2]〈l4〉
, (6.25)
when K2 = k3 and
− 2i〈23〉〈51〉
[4l][4l2]〈1l1〉〈1l2〉2〈2l1〉〈2l〉2
〈ll1〉〈l1l2〉[ll2]〈5l2〉〈3l〉
, (6.26)
when K2 = k4. In both cases K2 is massless and is of positive helicity so we use the
parameterisation of the triple cut momenta for y+ given in eq. (B2). Then along with
setting χ = k1 gives for the first triple cut integrand
2i〈1K♭1〉2〈23〉
〈13〉〈34〉〈45〉〈51〉
〈1−|/2|3−〉
〈1K♭1〉
〈3−| /K23|3−〉
〈1K♭1〉
〈13〉 〈23〉+
, (6.27)
and for the second
− 2i〈1K
1〉2〈24〉2
〈23〉〈34〉〈45〉〈51〉〈14〉
〈4−| /K23|4−〉
〈14〉 −
〈1−| /K23|4−〉
〈1K♭1〉
〈1K♭1〉
〈14〉 〈24〉+
S1〈21〉
.(6.28)
Applying these integrands to the second term of eq. (5.28) by taking [Inft], dropping any
terms not proportional to t and then performing the substitution ti → T (i) gives for the
coefficient of the first triple cut simply 1
Atree5 , and for the second triple cut
〈1+|/2/4 /K23|1+〉2
〈4−| /K23|4−〉2
s12 −
〈1+|/2/4 /K23|1+〉
〈4−| /K23|4−〉
. (6.29)
After following the same series of steps as above for the second bubble coefficient with
K1 = K234 we find only a single triple cut contributing term corresponding to K2 = k4. This
is related to the second triple cut coefficient derived above via the replacement K23 → K234
and swapping the overall sign.
After combining the three triple cut pieces above we arrive at the following form for the
cut constructable pieces of this amplitude
(4π)2−ǫ
Atree5 B0(s23)
〈1+|/2/4 /K23|1+〉2
〈4−| /K23|4−〉2
〈1+|/2/4 /K23|1+〉
〈4−| /K23|4−〉
(B0(s23)−B0(s234))
, (6.30)
which can easily be shown to match the result given in [18].
While this example is particularly simple we have also performed additional compar-
isons against other results in the literature. Such tests include the cut constructible pieces
of all two-minus gluon amplitudes with up to seven external legs, originally obtained in
[18, 37]. Additionally we find agreement for the case when, with six gluon legs, three
are of negative helicity and adjacent to each other and the remainder are positive helic-
ity, which was originally obtained in [49]. We have also successfully reproduced the known
three mass triangle coefficients in N = 1 supersymmetry for A6(1−, 2+, 3−, 4+, 5−, 6+) and
−, 2−, 3+, 4−, 5+, 6+), originally obtained in [35].
VII. CONCLUSIONS
The calculation of Standard Model background processes at the LHC requires efficient
techniques for the production of amplitudes. The large numbers of processes involved along
with their differing partonic makeups suggests that as much automation as possible is de-
sired. In this paper we have presented a new formalism which directs us towards this goal.
Coefficients of the basis scalar integrals making up a one-loop amplitude are constructed in
a straightforward manner involving only a simple change of variables and a series expansion,
thus avoiding the need to perform any integration or calculate any extraneous intermediate
quantities. The main results of this paper can be encapsulated simply by eq. (4.10) and
eq. (5.28) along with the cut loop momentum given by eq. (4.16), eq. (5.8) and eq. (5.16).
Although this technique has been presented mainly in the context of using generalised
unitarity [19, 39, 40, 41] to construct coefficients, and hence the cut-constructible part of
the amplitude, it can also be used as an efficient method of performing one-loop integration.
Using the idea of “cutting” two, three or four of the propagators inside an integral, we
isolate and then extract scalar basis coefficients. This procedure then allows us to rewrite
the integral in terms of the scalar one-loop basis integrals, hence giving us a result for the
integral.
Different unitarity cuts isolate particular basis integrals. For the extraction of triangle
integral coefficients this means triple cuts and for bubble coefficients we use a combination
of two-particle and triple cuts. Extracting the desired coefficients from these cut integrands
is then a two step process. The first step is to rewrite the cut loop momentum in terms of a
parameterisation which depends upon the remaining free parameters of the integral after all
the cut delta functions have been applied. Triangle coefficients are then found by taking the
terms independent of the sole free integral parameter as this parameter is taken to infinity.
Bubble coefficients are calculated in a similar if slightly more complicated way. The pres-
ence of a second free parameter in the bubble case means that we must take into account, not
only the constant term in the expansion of the cut integrand as the free integral parameters
are taken to infinity, but also powers of one of these parameters. The limit on the maximum
power of lµ appearing in the cut integral restricts the appearance of such terms and hence
we need consider only finite numbers of powers of these free parameters. Additionally it can
also be necessary to take into account contributions from terms generated by applying an
additional cut to the bubble integral. The flexibility in our choice of the cut-loop momentum
parameterisation allows us to directly control whether we need compute any of these triple
cut terms. Furthermore we can control which of these triple cut terms appears, in cases
when their computation is necessary.
As we consider the application of this procedure to more diverse processes than those
detailed here, we should also investigate the “complexity” of the generated coefficients.
In the applications we have presented we can see that we produce “compact” forms with
minimal amounts of simplification required. This is important if we are to consider further
automation. The straightforward nature of this technique combined with the minimal need
for simplification means that efficient computer implementations can easily be produced. As
a test of this assertion we have implemented the formalism within a Mathematica program
which has been used to perform checks against state-of-the-art results contained in the
literature. Such checks have included various helicity configurations of up to seven external
gluons as well as the bubble and three-mass triangle coefficients of the six photon A6(−+−+
−+) amplitude. In addition representative coefficients of processes of the type e+e− → qqgg
have been successfully obtained.
Our procedure as presented has mainly been in the context of massless theories. Funda-
mentally there is no restriction to the application of this to theories also involving massive
fields circulating in the loop. Extensions to include masses should require only a suitable
momentum parameterisation for the cut loop momentum; the procedure is then expected to
apply as before.
In conclusion therefore we believe that the technique presented here shows great potential
for easing the calculation of needed one-loop integrals for current and future colliders.
Acknowledgements
I would like to thank David Kosower for collaboration in the early stages of this work
and also Zvi Bern and Lance Dixon for many interesting and productive discussions as well
as for useful comments on this manuscript. I would also like to thank the hospitality of
Saclay where early portions of this work were carried out. The figures were generated using
Jaxodraw [51], based on Axodraw [52].
APPENDIX A: THE TRIPLE CUT PARAMETERISATION
In this appendix we give the complete detail of the triple cut parameterisation along with
some other useful results. The three cut momenta are given by
〈l−i | = t〈K
1 | + αi1〈K
2 |, 〈l+i | =
〈K♭,+1 | + 〈K
2 |, (A1)
α01 =
S1 (γ − S2)
(γ2 − S1S2)
, α02 =
S2 (γ − S1)
(γ2 − S1S2)
α11 = α01 −
= −S1S2 (1 − (S1/γ))
γ2 − S1S2
, α12 = α02 − 1 =
γ(S2 − γ)
γ2 − S1S2
α21 = α01 − 1 =
γ(S1 − γ)
γ2 − S1S2
, α22 = α02 −
= −S1S2 (1 − (S2/γ))
γ2 − S1S2
,(A2)
along with the identities α01α02 = α11α12 and α01α02 = α21α22. When written as four-vectors
the cut momentum are given by
i = αi2K
1 + αi1K
〈K♭,−1 |γµ|K
2 〉 +
αi1αi2
〈K♭,−2 |γµ|K
1 〉. (A3)
From these parameterised forms we have the following spinor product identities
[ll1] =
α12 − α02
[K♭1K
2] = −
[K♭2K
〈ll1〉 = t(α11 − α01)〈K♭1K♭2〉 = −
〈K♭1K♭2〉,
[ll2] =
α22 − α02
[K♭1K
2] = −
[K♭2K
〈ll2〉 = t(α21 − α01)〈K♭1K♭2〉 = −t〈K♭1K♭2〉,
[l1l2] =
α22 − α12
[K♭1K
1 − S2
[K♭2K
〈l1l2〉 = t(α11 − α21)〈K♭1K♭2〉 = −t
1 − S1
〈K♭1K♭2〉. (A4)
and we note that
1 − S2
1 − S1
γ = −γ − S1S2
+ S1 + S2 = (K1 − K2)2 = S3, (A5)
and so with l ≡ l0 we have 〈lilj〉[ljli] = Si+j, as expected.
APPENDIX B: THE TRIPLE CUT BUBBLE CONTRIBUTION MOMENTUM
PARAMETERISATION WHEN K22 = 0
In this appendix we give the forms for the triple cut momentum of section VC in the
case when S2 = 0, i.e. we have a one or two mass triangle. Firstly in these cases the K2 leg
is attached to a three-point vertex and so the amplitude for this will contain either [K2l] or
〈K2l〉 depending upon the helicity of K2. This means that in the positive helicity case only
the delta function solution δ(y − y+) survives and for a negative helicity K2 the δ(y − y−)
survives. We have for both solutions
J ′t =
S1〈χ−| /K2|K♭,−1 〉 + tγ〈K−2 | /K1|K−2 〉
) . (B1)
The momentum parameterisation for the y+ solution is given in spinor components by
〈l−| = 〈χK
〈χK2〉
〈K−2 |, 〈l+| = 〈K
1 | −
〈χK2〉
〈K−2 | /K1, (B2)
and as a 4-vector by
〈χK♭1〉
2〈χK2〉
S1〈χK2〉
〈K−2 |γµ /K1|K+2 〉 + 〈K−2 |γµ|K
. (B3)
The other momenta are given by
〈l−1 | = t
〈χK♭1〉
〈χK2〉
〈K−2 | −
〈χ−|, 〈l+1 | = −
S1〈χK2〉
〈K−2 | /K1,
〈l−2 | =
〈χK♭1〉
〈χK2〉
〈K−2 |, 〈l+2 | = −
〈χ−| /K3
〈χK♭1〉
〈χK2〉
〈K−2 | /K1. (B4)
where we have moved the overall factor of t from 〈l−1 | to 〈l+1 | to avoid the presence of a 1/t
term for aesthetical reasons. The spinor products formed from these are given by
〈ll1〉 =
〈χK♭1〉, [ll1] = [K♭1χ],
〈ll2〉 = 0, [ll2] = −
〈χK2〉
〈χK♭1〉
[lK2],
〈l1l2〉 = −
〈χK♭1〉, [l1l2] = −
[K♭1χ]. (B5)
and we see that again, as expected, with l = l0, we have 〈lilj〉[ljli] = Si+j . As we have
massless legs some spinor products will consequentially vanish. In the two-mass case these
〈ll2〉 = 0, 〈lK2〉 = 0, [l2K2] = 0, (B6)
and for the one-mass case
[l1l2] = 0, 〈ll2〉 = 0, 〈lK2〉 = 0, 〈l2K2〉 = 0, [l1K3] = 0, [l2K3] = 0, (B7)
where K3 is the momentum of the third leg.
The momentum parameterisation for the y− solution is given in spinor components by
〈l−| = t
〈K+2 | /K1 +
〈χ−|, 〈l+| = [χK
〈K+2 |, (B8)
and as a 4-vector by
[χK♭1]
2[K2K
〈K+2 | /K1γµ|K−2 〉 +
〈χ−|γµ|K−2 〉
. (B9)
The other momenta are given by
〈l−1 | =
〈K+2 | /K1, 〈l+1 | = −t
[K♭1χ]
〈K+2 | − 〈K
〈l−2 | =
[χK♭1]
〈K♭,+1 | /K3 +
〈K+2 | /K1, 〈l+2 | =
[χK♭1]
〈K+2 |. (B10)
The spinor products formed from these are given by
〈ll1〉 =
〈χK♭1〉, [ll1] = [K♭1χ],
〈ll2〉 =
[χK♭1]
〈K2l〉, [ll2] = 0,
〈l1l2〉 =
〈K♭1χ〉, [l1l2] = [χK♭1]. (B11)
and again 〈lilj〉[ljli] = Si+j as expected. The vanishing spinor products in the two mass case
[ll2] = 0, [lK2] = 0, [l2K2] = 0, (B12)
and in the one mass case
〈l1l2〉 = 0, [ll2] = 0, [lK2] = 0, [l2K2] = 0, 〈l1K3〉 = 0, 〈l2K3〉 = 0. (B13)
APPENDIX C: THE SCALAR INTEGRAL FUNCTIONS
The scalar bubble integral with massive leg K1 given in figure 6 is defined as
FIG. 6: The scalar bubble integral with a leg of mass K21 .
1 ) = (−i)(4π)2−ǫ
d4−2ǫl
(2π)4−2ǫ
l2(l − K1)2
, (C1)
and is given by
ǫ(1 − 2ǫ)(−K
−ǫ = rΓ
− ln(−K21 ) + 2
+ O(ǫ), (C2)
Γ(1 + ǫ)Γ2(1 − ǫ)
Γ(1 − 2ǫ) . (C3)
The general form of the scalar triangle integral with the masses of its legs labelled K21 ,
K22 and K
3 given in figure 7 is defined as
FIG. 7: The scalar triangle with its three legs of mass K21 , K
2 and K
1 , K
2) = i(4π)
d4−2ǫl
(2π)4−2ǫ
l2(1 − K1)2(l − K2)2
, (C4)
and separates into three cases depending upon the masses of these external legs. In the one
mass case we have K22 = 0 and K
3 = 0 and the corresponding integral is given by
1 , K
2 ) =
(−K21 )−1−ǫ =
(−K21 )
− ln(−K
ln2(−K21 )
+ O(ǫ), (C5)
If two legs are massive the integral, assuming K23 = 0, is given by
1 , K
(−K21 )−ǫ − (−K22 )−ǫ
(−K21 ) − (−K22 )
(−K21 ) − (−K22 )
− ln (−K
1 ) − ln (−K22 )
ln2 (−K21 ) − ln2 (−K22 )
.(C6)
Finally if all three legs are massive then the integral is as given in [53, 54]
1 , K
1 + iδj
1 − iδj
− Li2
1 − iδj
1 + iδj
+ O(ǫ), (C7)
where
K21 − K22 − (K1 + K2)2√
−K21 + K22 − (K1 + K2)2√
−K21 − K22 + (K1 + K2)2√
, (C8)
∆3 = −(K22 )2 − (K22)2 − (K23)2 + 2(K21K22 + K23K21 + K22K23) = −4∆, (C9)
with ∆ given by eq. (4.15).
The general form for a scalar box function is given by
1 , K
2 , K
3) = (−i)(4π)2−ǫ
d4−2ǫl
(2π)4−2ǫ
l2(l − K1)2(l − K2)2(l − K3)2
. (C10)
The solution of this integral is split up into classes depending upon the masses of the external
legs. These solutions are labelled as zero mass I0m4 , one mass I
4 , two mass hard, I
4 , two
mass easy I2me4 , three mass I
4 and four mass I
4 integrals. The results for which can be
found in the literature, for example in [17].
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