#### The Sudakov form factor at four loops in maximal super Yang-Mills theory

HJE
The Sudakov form factor at four loops in maximal
Rutger H. Boels 0 1 2 5
Tobias Huber 0 1 2 3
Gang Yang 0 1 2 4
0 Chinese Academy of Sciences , Beijing 100190 , China
1 Walter-Flex-Str. 3, 57068 Siegen , Germany
2 Luruper Chaussee 149 , D-22761 Hamburg , Germany
3 Naturwissenschaftlich-Technische Fakultat, Universitat Siegen
4 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics
5 II. Institut fur Theoretische Physik, Universitat Hamburg
The four-loop Sudakov form factor in maximal super Yang-Mills theory is analysed in detail. It is shown explicitly how to construct a basis of integrals that have a uniformly transcendental expansion in the dimensional regularisation parameter, further elucidating the number-theoretic properties of Feynman integrals. The physical form factor is expressed in this basis for arbitrary colour factor. In the nonplanar sector the required integrals are integrated numerically using a mix of sector-decomposition and Mellin-Barnes representation methods. Both the cusp as well as the collinear anomalous dimension are computed. The results show explicitly the violation of quadratic Casimir scaling at the four-loop order. A thorough analysis concerning the reliability of reported numerical uncertainties is carried out.
Extended Supersymmetry; Perturbative QCD
Review and setup
2.1
Infra-red divergent structure of the form factor in N = 4 SYM
2.2 Integrand and integral relations
1 Introduction
2
3
4
5
3.1
3.2
3.3
4.1
4.2
5.1
5.2
5.3
Uniformly transcendental basis
Warm up: a one-loop example
Systematic construction dLog forms
Full form factor in UT basis
UT integrals for the nonplanar form factor
UT integrals for the planar form factor
Numerical integration in the nonplanar sector
Mellin-Barnes representations
Sector decomposition
5.3.1
5.3.2
Rationalisation
Error analysis
Nonplanar cusp and collinear anomalous dimensions
6
Discussion and conclusion
A UT integrals
A.1 UT integrals with 12 lines
A.2 UT integrals with 11 lines
A.3 UT integrals with 10 lines
B Basis of propagators and numerators
50 years after its rst modern formulation by Yang and Mills [1], it remains a hard task to
compute observables even in a perturbative expansion in the coupling constants. Beyond
perturbation theory much less is known in general, with the particular exception of those
the maximally supersymmetric, N = 4, Yang-Mills (SYM) theory based on the SU(Nc)
gauge group, in 't Hooft's planar limit [3]. In this limit, where Nc !
1, remarkable
simpli cations occur. A lighthouse result in this direction is the Beisert-Eden-Staudacher
equation [4]: this equation describes a certain observable known as the planar lightlike
cusp anomalous dimension (CAD) at all values of the coupling in N = 4 and ties into
integrability ideas.
Weak and strong coupling expansions of this anomalous dimension
have been matched to independently obtained results, see e.g. [5{12]. However, beyond the
planar limit much less is known in general despite some very recent progress in [13, 14]. For
the cusp anomalous dimension no nonplanar correction had been computed in any theory
until recently the rst numerical result at four loops in N = 4 was presented by us in [15].
Beyond the AdS/CFT correspondence and especially at weak coupling, the N = 4
super-Yang Mills theory is also a time-tested sandbox to explore computational ideas, such
as those motivated by Witten's twistor string theory [16]. These have ignited a long-running
program to explore the space of on-shell observables, using on-shell methods. This article
is a part of this program, aimed at computing the so-called Sudakov form factor in N = 4
SYM theory. This form factor can be used to isolate several interesting universal functions
that are contained within it. Prime among these is the lightlike cusp anomalous dimension
mentioned above. The cusp anomalous dimension plays a central role in the analysis of
infra-red (IR) divergences, as rst pointed out in [17]. By extrapolating structures found
through three loops a general conjecture was formulated in [18] that the nonplanar part
of the CAD vanishes in any perturbative gauge theory. This became known as quadratic
Casimir scaling of the CAD, see e.g. [18{24]. It was noted that the quadratic Casimir
scaling may be violated to higher orders of perturbative expansion due to the appearance
of higher Casimir operators of the gauge group [25], see also [26]. At strong coupling, this
scaling is known to break down in N = 4 SYM [27]. In addition, instanton e ects break
the scaling [28]. Finally, ref. [15] disproved the conjecture in perturbation theory, see also
the two recent works [
29, 30
] which apply directly to quantum chromodynamics and also
report violation of Casimir scaling.
The Sudakov form factor we consider is an observable which involves two on-shell
massless states and a gauge invariant operator in the stress tensor multiplet in N = 4 SYM,
F =
Z
d4x e iq xhp1; p2jO(x)j0i :
(1.1)
In N = 4 SYM, form factors were rst studied thirty years ago in [31] and revived in
the past few years at weak coupling [32{61] and at strong coupling [62{64]. There have
been interesting recent studies of loop form factors of non-Bogomolnyi-Prasad-Sommer eld
(BPS) operators [65{73]. For reviews, see the theses [74, 75]. The present paper is aimed
at elucidating the evaluation of the integrals that appear in the four-loop Sudakov form
factor, with the expectation that the presented techniques can be applied more widely.
A key idea in this article is to make transparent the transcendentality properties of the
Feynman integrals that make up the Sudakov form factor. It is known quite generally that
at xed orders in the expansion in the dimensional regularisation parameter
of Feynman
integrals only rational linear combinations of certain constants appear. These constants
{ 2 {
are known as multiple zeta values (MZV). In principle, also more general constants such
as Euler sums can appear, but in the known terms of the Sudakov form factor through
to three loops in N = 4 SYM, MZVs are su cient. MZVs have a property known as
transcendental weight which takes integer values. The number of independent MZVs is
small for low weight, and a basis for these constants is formed by (see e.g. [76])
maximal weight terms appear at each order in the -expansion. Assigning to
a
transcendental weight
1, these integrals have a well-de ned overall transcendental weight, and
will be referred to as uniformly transcendental (UT) integrals. The concept of
transcendental weight is important as it is observed in many examples that in N = 4 SYM (and
superstring theory) only terms with maximal weight appear. Although the origin of this is
somewhat ill-understood, it at the very least makes for a useful tool. Moreover, a general
conjecture [8, 77] relates the maximal transcendental terms appearing in QCD directly to
N = 4 SYM for certain quantities. An example of this kind is given by the quark and
gluon form factors in QCD [78{82] and the Sudakov form factor in N = 4 SYM, where the
maximal transcendentality principle was veri ed through to three loops and for all terms
up to transcendental weight eight [37]. Examples for two-loop remainders were also found
in [38, 73].
For the three-loop form factor in N = 4 SYM, an expression in terms of UT integrals
was obtained in [37]. In that case the master integrals were known analytically, facilitating
the analysis. In the four-loop case generically the basis of UT integrals was unknown. In
this article, it will be shown how to identify UT candidates systematically, and how to
write the four-loop Sudakov form factor as a rational linear combination of UT candidate
integrals. The result in the nonplanar sector will then be integrated numerically, yielding
a large list of new integral results. What is surprising is the empirical observation that
obtaining numerical results for UT integrals turns out to be substantially simpler than for
generic non-UT integrals in the class under study, even though the integration techniques
themselves do not make use of the UT property. The result is combined into the nonplanar
cusp and collinear anomalous dimensions at four loops. The result for the cusp anomalous
dimension was rst announced by us in [15], while the result on the collinear anomalous
dimension is new. We comment extensively on the numerics below, making use of the UT
property to inform the error analysis.
This article is structured as follows: section 2 contains a review and setup of the
problem. In section 3, uniformly transcendental integrals are discussed both at the general
level as well as for the speci c observable under study. Of special interest is a general
technique for obtaining candidate-UT integrals. The full form factor is expanded in terms
of the UT basis in section 4. In section 5 we discuss the numerical integration of the
appearing integrals in the nonplanar sector, present our results and perform a thorough
analysis of the reported numerical uncertainties. We conclude in section 6. The article
{ 3 {
is supplemented by several appendices. In appendix A we give explicit results of the UT
integrals in the nonplanar sector, while appendix B contains the parametrisation of the
integral topologies in terms of loop and external momenta.
Review and setup
Infra-red divergent structure of the form factor in N = 4 SYM
The perturbative expansion of the Sudakov form factor is xed by supersymmetry and
dimensional analysis as
F = F
tree X1 g2l( q2) l F (l) ;
l=0
(2.1)
HJEP01(28)53
and . The coupling constant is normalised as g2 = g(Y24M)N2c (4 e
where p1; p2 are two on-shell momenta, and q = (p1 + p2) is o -shell. In dimensional
regularisation with D = 4
2 , F (l) is a purely numerical function of gauge group invariants
E ) . We consider explicitly
the SU(Nc) gauge group, although our results apply to any Lie group: up to the order
considered, there is a one-to-one map from Nc to Casimir invariants, see below.
The form factor is free of ultraviolet (UV) divergences, since the operator O in the
stress tensor multiplet is protected. On the other hand, there are IR divergences due to
soft and collinear singularities from the massless states. Setting q2 =
1 and de ning the
normalised form factor as F = 1+Pl1=1 g2lF (l), the IR structure is described in the following
form [83] (exponentiation structure of Sudakov form factor in more general theories was
original studied in [84{87])
log F =
1
X g2l
l=1
" (l)
(l)
#
(2clus)p2 + Gcoll + Fin(l) + O( ) ;
2l
where the leading singularity is determined by the cusp anomalous dimension (CAD)
cusp, and the sub-leading divergence is related to the so-called collinear anomalous
dimension Gcoll.
1
Besides analysing the IR structure of the form factor, one also has to investigate its
colour structure. For a classical Lie-group with Lie-algebra [T a; T b] = if abc T c and structure
constants f abc, the quadratic Casimir operators in the fundamental (F ) and adjoint (A)
representation are de ned via (see e.g. [88])
[T aT a]ij =CF ij ;
f acdf bcd = CA ab ;
respectively. The building block of the quartic Casimir invariant dabcddabcd is the fully
R
R
symmetric tensor
1
6
daRbcd = Tr[TRaTRb TRc TRd + perms.(b; c; d)] ;
1There are di erent conventions of de ning cusp and collinear anomalous dimensions in the literature.
In our convention, the cusp anomalous dimension cusp = P
l c(ul)spg2l is the same as the function f (g) in [4].
(2.2)
(2.3)
(2.4)
{ 4 {
and [TAa]bc =
Here, NA = (Nc2
where R = F; A denotes the fundamental or adjoint representation, with [TFa ]ij = [T a]ij
if abc. The values of the relevant Casimir invariants in the case of gauge
1) is the number of generators of SU(Nc). The colour structure of the
form factor at l loops in N = 4 SYM theory with matter in the adjoint representation is
simply (CA)l up to l = 3. Starting from four loops, the quartic Casimir invariant arises in
addition, and hence in SU(Nc) gauge theory one has, besides the planar (i.e. Ncl
leadingcolour) contribution a nonplanar (i.e. Ncl 2 subleading-colour) correction. Starting from
six loops, additional group invariants appear [42].
CAD, this function needs to be expanded down to
2 at l loops, combined together with
The planar form factor has leading divergence / 1= 2l at l-loop order. To compute the
higher terms in the Laurent expansion in
from lower-loop contributions. As mentioned
above, the rst nonplanar correction starts at four loops, due to the appearance of a quartic
Casimir invariant. The nonplanar part of the four-loop form factor takes the following form
since, upon taking the logarithm in (2.2),
this piece cannot mix with any planar contribution from lower loops. We emphasise that
individual integrals that contribute to F N(4P) will typically have the full 1= 8 divergence. The
cancellation of these higher-order poles in the nal result therefore provides a very strong
constraint on as well as a non-trivial consistency check of the computation.
The form factor exhibits a Laurent expansion in the dimensional regularisation
parameter . In this expansion, each term is expected to be a rational-coe cient polynomial of
Riemann Zeta values n, or their multi-index generalizations, n1;n2;:::, known as multiple
zeta values (MZVs) (see e.g. [76]). In principle, even more general objects such as Euler
sums can appear. However, as mentioned earlier, any analytically known piece of the form
factor does not go beyond MZVs. The MZVs have a transcendentality degree which is the
sum of their indices, Pi ni. Also, the regularisation parameter
is assigned
transcendentality
1. In N = 4 SYM, the nite part of the form factor is expected to have (maximal)
uniform transcendentality, which at l loops is 2l, and which suggests that the CAD at l
loops is of uniform transcendental weight 2l
2. Indeed, the planar CAD at four loops in
N = 4 SYM has transcendentality six and was computed as [9, 10, 12]
(log F )(P4) =
2
4
loop collinear anomalous dimension Gcoll,P was obtained in [89]. Recently, also the analytic
value of this quantity was presented [90].
2.2
Integrand and integral relations
The full four-loop Sudakov form factor including the nonplanar part in N = 4 SYM was
obtained as a linear combination of a number of four-loop integrals in [42] based on
colour{ 5 {
p2
p1
p2
p1
p2
p1
p2
p1
p2
p1
p2
(5)
(10)
ℓ4
p2
p1
p2
p1
p2
p1
p2
p1
p2
p1
p2
ℓ5
ℓ4
(34)
ℓ6
p2
HJEP01(28)53
kinematics duality [91, 92]. Similar ve-loop result was also obtained recently in [57]. For
more details on colour-kinematics duality, see e.g. the lecture [93]. The explicit form of the
integrals for the problem at hand can be found in [42]. There are 34 distinct cubic integral
topologies, each with 12 internal lines, that contribute to the four-loop form factor. They
are labelled (1) { (34) in [42] and we provide them in gures 1{3 for convenience and further
reference throughout the present paper.
The four-loop integrals take the generic form as
I = ( q2)2+4 e4 E
Z idDDl=12 : : :
dDl
4
i D=2 Q12
N (li; pj ) ;
k=1 Dk
(2.7)
where Di are twelve propagators and N (li; pj ) are dimension-four numerators in terms of
{ 6 {
(l4 − p1)2
q
l
Lorentz products of the four independent loop and two independent external on-shell
momenta. For each topology, one needs to pick six additional propagators (i.e. six irreducible
numerators) to form a complete basis, and we label them Dk; k = 13; : : : ; 18. Such a choice
is not unique. Below we use as propagator basis Di(n), where the superscript (n) indicates
the topology, and the subscript i; i = 1; : : : ; 18 refers to the basis given explicitly in
appendix B (see also appendix C of [48]). We de ne D1(n9) = (p1 + p2)2. Any given numerator
can then be represented uniquely in the chosen basis.
A fundamental property of Feynman integrals, as those in equation (2.7), is that they
obey integration-by-parts (IBP) identities [94, 95], which follow from
Z
(integrand) = 0 :
(2.8)
Working out the left-hand side gives a linear relation between di erent integrals. By solving
linear systems of such equations, a generic Feynman integral can be expressed in terms of
a set of basis integrals. This procedure is known as IBP reduction, and the set of basis
integrals is also known as the set of master integrals. The form factor was expressed in
terms of a set of master integrals in [48] using the Reduze code [96].2 The master integrals,
however, have evaded full integration so far due to their overwhelming complexity. In
addition, the full IBP reduction generically leads to coe cients that contain higher-order
poles in . This requires to evaluate the master integrals to higher orders in the expansion,
which further increases the size of the problem. In this paper a di erent strategy will be
used by expanding the form factor in terms of a set of integrals which are each simple
enough to integrate and have -independent prefactors.
A particular subset of the IBP relations turns out to be very useful for our purpose.
These are the IBP relations in which the coe cients in front of integrals are pure rational
numbers and independent of . These `rational IBP' relations have been obtained in [105]
for the form factor presently under study as a subset of the full reduction. An example
is shown in
gure 4. Note that integral relations derived from graph symmetries are a
particular subset of the rational IBP relations.
3
Uniformly transcendental basis
A key idea of the present study is to expand the form factor in a set of integrals that all
have uniform transcendentality (UT), which will be referred to as UT integrals. Such a
2There exist various private and public implementations of IBP reduction, mainly based on Laporta's
algorithm [97], such as AIR [98], FIRE [99{101] and Reduze [96, 102]. See LiteRed [103, 104] for an alternative
approach to IBP reduction.
{ 7 {
representation of the form factor will make manifest the expected maximal
transcendentality property of N
= 4 SYM, and has been achieved at three loops in [37]. As will
be shown in the next section, the UT integrals turn out to be much simpler to integrate
numerically compared to generic non-UT integrals of similar complexity, which is crucial
for the computation at hand.
We will now turn to the question how to nd UT integrals prior to explicitly computing
them. There are basically three ways to show whether an integral is UT.
A UT integral can be written in the so-called dLog form [106, 107].
The leading singularities, or equivalently, the residues at all poles of a UT integral
must always be a constant [107{109]. This is conjectured to be a necessary and
su cient condition.
A set of UT integral basis can lead to certain simple di erential equations [110].
The last point regarding di erential equations is not directly applicable to the Sudakov
form factor at hand since it is a single-scale problem, and thus not `di erentiable'. See
however [
109, 111
] for a work-around by deforming an on-shell leg to be massive, thus
creating a two-scale problem. Below we illustrate the
rst two UT properties using a
simple one-loop example. Then their application to four-loop form factor integrals will
be discussed.
3.1
Warm up: a one-loop example
A one-loop UT example is given by the following scalar triangle integral:
qi2 = qi pj = 0
8i; j and q1 q2 =
p1 p2 ;
{ 8 {
3
I(1) = ( q2)1+ e E
Z
dDl
1
i D=2 l2(l
p1)2(l + p2)2
:
This is a UT integral as evidenced by the explicit result in the
expansion
3
integrand level. In the following, we consider only the integral in four dimensions as
3
I(1) = ( q )
It is convenient to parametrise the loop momenta such that only scalar integration
parameters remain. The four-dimensional loop momentum can be parametrised as
l =
1p1 + 2p2 + 3q1 + 4q2 ;
where pi = i ~i; i = 1; 2 are the external on-shell momenta, and q1, q2 can be chosen as
the two complex solutions to
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
for example, q1 = 1 ~2; q2 = 2 ~1. The integral in the parametric form is
This can be written in the following dLog form
3
which, in terms of momenta, is equivalent to the form
The remaining parameters obviously have only simple poles and the
nal residue is a
constant. One needs to check all di erent orders of taking residues, and in all occurring
poles. In any intermediate step, after taking a residue in a particular parameter, if one
encounters other than a simple pole in a remaining parameter, the integral is not UT.
We would like to emphasise that the simple pole requirement should also apply to poles
at in nity. To be more concrete, consider following simple examples. For the integral
there is a double pole for 2 at 1, thus it is not UT. As for another example,
Z
d 1d 2
1
1
1(1
Z
2
2)2
;
Z
d 1d 2
or
d 1d 2
2
1(1
2)
dLog l2 dLog (l
p1)2 dLog (l + p2)2 dLog
2(l q2) :
HJEP01(28)53
The existence of the dLog representation implies that the integral is UT.
As mentioned above, an alternative way to prove UT property is to consider the leading
singularity. In the parametric form like (3.6), this is equivalent to check the residues
at all poles of the integral: the constant leading singularity property translates to the
simple pole condition for all parameters. Let us explain this in more detail. To check the
simple pole condition, one needs to pick up a certain order of the parameters to take the
residue. Consider the one-loop example, we can rst take residue for 1 at the pole of the
rst propagator
Residue at pole 1 =
d 2d 3d 4
Next, we take the residue for 2 at pole 0
Further residue at pole 2 = 0
!
Z
d 3d 4
1
both have a double pole for 2 at in nity, so they are not UT either.3
3We would also like to point that for amplitudes in N = 4 SYM as studied in [107, 108], an additional
related to the hidden dual conformal symmetry of amplitudes [107, 108]. For the form factor, we must
allow simple pole at in nity.
{ 9 {
(3.6)
(3.7)
(3.8)
(3.10)
(3.11)
(3.12)
The condition that only simple poles are allowed is related to the required existence of
a dLog form where only a logarithmic singularity is allowed. However, it does not require
to
nd the explicit transformation to the dLog form which can be very complicated in
general. This simple-pole condition provides an essential constraint for the construction of
UT integrals below. A related strategy was also used in [107{109].
The aforementioned condition of simple poles is used here both to construct and as well as
to check UT integral candidates. Given a four-loop form factor integral in four dimensions,
there are 16 integration parameters, so in principle there are 16!
2
1013 di erent
orders in which the residues can be taken. Practically therefore, the simple-pole condition
is veri ed by choosing a large number of random orders of taking residues. A non-UT
integral typically fails the UT test well within a few hundred of such random checks.
This UT test strategy can be used to constrain the space of potential UT integrals
when combined with an Ansatz for the numerator. For the four-loop form factor integrals,
one can start with a linear Ansatz of mass dimension four numerators of a given topology.
We then perform the above described residue tests. The requirement of absence of higher
order poles provides linear constraints on the set of coe cients in the Ansatz by computing
the residues of the higher order poles. Solving these linear constraints then yields a smaller
Ansatz, and the process is repeated. For speed, it is better to rst identify a sequence of
residues leading to a higher pole by choosing the Ansatz coe cients to be random integer
numbers. This sequence can then be used to derive the analytic constraint on the full
Ansatz. Below we provide more technical details.
A full four-loop topology contains 12 lines (i.e. propagators). For many topologies one
can simply ask the following question: which sets of 10- and 11-line integrals can be added
to a given topology in the four-loop form factor such that the sum is UT? Suppose such a
linear combination exists. Then it is obvious this is likely not unique: adding any linear
combination of 10- or 11-line UT integrals will satisfy the same constraint. To nd a basis
for all these UT integrals, we take the form factor numerators N
as they appear in the
N = 4 theory as input and add to this the set of all 10- and 11-line integrals (of which
there are 162). In this case the initial Ansatz looks like
Nansatz = a0N
12
+ X Dj
j=1
19
X aj;kDk
k=13
!
+
X
1 j k 12
bj;kDj Dk ;
(3.13)
where D19 := q2 and the D's are the propagators as given in appendix B. Inserting the
parametrisation (3.4) for each of the four-loop momenta gives a rational expression of 16
-type parameters. Now one needs to identify a sequence of residues yielding a double or
higher pole in the
parameters. Demanding that the pole becomes a simple one yields at
least one constraint equation for the 162 + 1 parameters fa0; aj;k; bj;kg. Explicitly solving
these linear constraint equations gives a smaller Ansatz. Now one repeats by again trying
to
nd a sequence of residues that will yield additional constraints. After a number of
iterations for the integrals in the case at hand, one has obtained a set consisting of one
integral containing the 12-line parts of the form factor contribution and other integrals
which contain at most 11 lines. These are a set of UT candidate (UTC) integrals.
One can also ask the question which UT integrals with unit exponent propagators exist
in a given topology, worrying later about expressing the N = 4 form factor in terms of
these. To answer this question one chooses a more general initial Ansatz such as
Nansatz =
bj;kDj Dk :
X
1 j k 19
(3.14)
Here the simple pole condition will provide a set of linear equations of 190 parameters
fbj;kg. The end-result for this wider initial ansatz will be a set which contains all possible
UT candidate integrals in a given topology (with unit exponents for the propagators).
If, after deriving constraints with a certain number of random checks and no new
further constraints are found in typically a few hundred more random pole checks, the
remaining Ansatz contains a set of good UT candidates.
The choice of initial Ansatz is dictated to a large part by practical ease of subsequent
numerical integration. For many public codes, the numerator of integrals is in general
preferred to be a product of two factors, each quadratic in momenta. If a single such
integral is to contain the full 12-line parts of a particular integral topology k, a necessary
but not su cient condition is to check that the irreducible numerators of a given integral
form a product form separately. Concretely, one sets all propagators of this topology
to zero, and veri es if a product form emerges for the irreducible numerators. In our
chosen set of expressions, the propagators of a topology are always the rst 12 entries (see
appendix B), so to check is:
UTCjtopk Di(k)=0 81 i 12
=? product form ;
where the product form is a quadratic function of Di; i = 13; : : : ; 19. This condition is
satis ed for all topologies in the four-loop form factor under study, except for topologies
(12), (17), (19), and (26). Note this condition is independent of the exact choice of
propagator basis. If this condition is satis ed, then the smaller Ansatz approach of form factor
integral plus 10- and 11-liners has a chance of su cing. This is usually much quicker and
more transparent. If the 12-line parts do not have a product form, the larger Ansatz must
be used. Examples of both possibilities are, for instance, topology (19)
UTCjtop19 Di(19)=0 81 i 12
=
D1(149)D1(169)
D1(139)D1(199) ;
(3.15)
(3.16)
(3.17)
(3.18)
which does not have a product form, and topology (23)
UTCjtop23 Di(23)=0 81 i 12
= (D1(233) + D1(293))2 ;
which does.
the following equation
From a generic set of UT candidates UTCi, the product form can be found by solving
X
i
i UTCi =
j Dj
kDk ;
X
k
X
j
X
i
X
i
for non-trivial parameters ,
and
which are rational numbers. This is a quadratic set
of equations, obtained by matching coe cients of products of D's. Since we are interested
in integrals that can be used to express the form factor in, more constraints can be added
to the problem for speci c purposes. For instance, the constraint can be added that
the twelve-line parts match known form factor numerator contribution in the topology
under study,
i UTCi
FF
Note this constraint only makes sense in a topology where the form factor has a product
form on the left hand side of equation (3.15). Alternatively, one can simply demand one
speci c coe cient to be unity,
i UTCi
This in particular avoids nding trivial solutions to the general problem in equation (3.18)
( i =
j =
k = 0). This constraint is particularly useful when looking for very general
solutions to the quadratic problem, matching only to some terms appearing in the form
factor. Finally, one can add manifest graph symmetry constraints on the UT candidates:
this we did in almost all cases. Which constraint to use in a particular situation depends
on the generality of the solution sought for.
Having set up the quadratic problem (3.18), the rst step is to solve the linear
subproblem for . Then, one can impose graph symmetry patterns on the product form. The
remaining set of quadratic equations can be analysed completely, or a particular solution
can be guessed by computer algebra.4 In several cases, it can be shown that no solution
to a given problem exists. In these cases, after exhausting all options, one can widen
the Ansatz in equation (3.18) by adding a linear combination of ten-line integrals (which
are expected to be simple to integrate). These cases can be clearly seen in the results in
section 4, e.g. (4.7){(4.9). Also, sometimes residual parameter-containing solutions to the
product-form problem are obtained. In these cases educated guesses were employed, aimed
at as parametrically simple as possible integrals.
The result is a list of product-form UT candidates for each topology. The ones listed
in this article have all individually been checked to pass at least 10; 000 simple residue
checks, giving ample evidence for their uniform transcendentality. As will be discussed
later, checking a set of found integrals individually also serves as a useful cross-check on
computational errors.
3.3
dLog forms
Writing a four-loop integral in dLog form will give a direct proof of UT property. However,
the construction of a dLog form for a generic four-loop form factor integral is a di cult
4In Mathematica, these options are represented by the commands Reduce and FindInstance, respectively.
task, and hence this method is more suitable to show the UT property of a given integral
rather than to derive a UT numerator.
A useful strategy to construct a dLog form is loop by loop [107, 108]. With proper
numerators, all one-loop triangle and box integrals can be written explicitly in dLog forms.
For example, the three-mass box is known to have a dLog form (see e.g. [107], k1 is massless,
K2 and K4 are massive)
Z
4
d ` `2(`
k1)2(`
N3m
k1
K2)2(` + K4)2
;
N3m = (k1 + K2)2(k1 + K4)
2
K22K42 ;
(3.21)
(3.22)
with given numerator
which is the Jacobian of the quadruple cut of the box, such that the leading singularity is a
kinematics-independent constant. So when there is a three-mass sub-box in the four-loop
integral, one can write this sub-box in a dLog form, and the remaining integral is a
threeloop integral involving a new propagator 1=N3m. In some topologies, such a procedure
can be done recursively loop by loop, so that the full integral can be written explicitly in
the dLog form. This normally happens when the topology involves at least one box with
at least one massless leg, and has some ladder structure.5 Such cases include topology
(1), (6), (13), (21), (23), (28), as shown in gure 5, whose dLog numerators are given,
respectively, by
(q2)2; (l4
p1)2q2; (`3
p1)2(q2
2`4 p2)
(`4
p1)2(q2
2`3 p2);
(3.23)
[(l3
p1)2]2; [(l3
p1)2]2; (`3
`4
Full form factor in UT basis
Finding an expansion of the full form factor in terms of generic UT candidate integrals can
be obtained by relatively straightforward linear algebra techniques. In addition, we
dis5It is also possible to write a dLog form for four-mass box and three-mass triangle integrals, with
numerators in a square-root form. This makes it di cult to
nd a dLog form for the remaining part, since
cussed above how to nd product-form numerators for candidate UT integrals. Combining
the two involves quite a wealth of choices that can be made in intermediate steps. For the
nonplanar form factor, we rst found a linear combination of 12-line UT candidates which
satis es the requirement that the di erence to the full result contained at most 11-line
integrals. Combining the remaining expression into UT candidates in the nonplanar sector was
then a relatively easy task. In the planar sector, it turned out that more work was required.
An obscuring factor is the existence of many relations between di erent integrals from the
rational IBP relations. A choice that works is given below. This choice was driven by the
attempt to nd as simple expressions as possible and to express the end-result in as small a
number of integrals as possible. This includes both aiming at graph-symmetric expressions
as well as trying to nd an expansion involving only small integer or half-integer expansion
coe cients. This necessarily involves some heuristics. It would be very interesting to nd
concise target integral expressions more easily, ideally driven by integration convenience or
accuracy, but this would lead us beyond the scope of this work.
One important result that follows is that both the planar as well as the nonplanar
sector of the form factor can be expressed in terms of rational (i.e. -independent) linear
combinations of UT integral candidates. We regard this as strong evidence for the maximal
transcendentality of the form factor. By extension, this implies maximal transcendentality
for the cusp and collinear anomalous dimensions at the four-loop order in maximal SYM
theory, both in the planar and nonplanar sectors. Moreover, the smallness of the
expansion coe cients clearly suggests this expansion is natural. In the nonplanar sector we have
checked explicitly that the form factor integrals found originally in [42] when taken as
complete topologies can only be expressed in terms of UT integrals in one unique combination
of the 14 topologies: the one in which they appear. This provides a cross-check on the
symmetry and colour factors.
4.1
UT integrals for the nonplanar form factor
Below we list 23 UT integrals I1(n)
23 that combine into the nonplanar form factor. The
superscript (n) denotes the twelve propagators from topology (n) in
gure 2. In this
notation, we only have to list the numerator of each integral. Moreover, each integral Ii(ni)
gets multiplied by a rational pre-factor ci according to
The nonplanar form factor is then obtained as
where the prefactor 48=Nc2 = 2
24=Nc2 is the normalisation stemming from the
permutational sum of external legs and the colour factor [42], and the UT integrals are
I(21) = [(`3
1
I(22) = (`3
2
p1)2]2
(4.2)
(4.3)
(4.4)
We note that integrals I1 11, I12 18, and I19 23, are 12-, 11-, and 10-line integrals,
respectively.
The integral I5(25) in topology (25) is the only one which does not carry the symmetry
of the topology explicitly. This was done to arrive at a simpler form to integrate. In
general topologies (25) and (26) are the hardest topologies to
nd UT integrals which are
reasonably compact. Note that topologies (31) through (34) do not appear: there are no
UT candidate integrals at all in these topologies.
UT integrals for the planar form factor
(n)
Similar to the nonplanar part, we also provide an expansion of the planar form factor in
terms of 32 UT integrals Ip;1 32. To distinguish from the nonplanar integrals, we add `p' in
subscription to denote it is for the planar form factor. Each integral Ip;i
by a rational pre-factor cp;i according to
(ni) gets supplemented
The planar form factor is then obtained as
32
X
i=1
F
(4)
P
= 2
cp;i Ip(n;ii) ;
where the prefactor 2 is the normalisation stemming from the permutational sum,6 and
the UT integrals are (as in nonplanar case, we only indicate the numerator)
(1)
6Note that unlike the nonplanar case, there is no color factor contribution.
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
The full four-loop form factor can be obtained as:
F (4) = F (4) + F N(4P) :
P
(4.60)
5
Numerical integration in the nonplanar sector
Although preferably one would want analytic results for the integrals that appear in the
four-loop form factor, they appear to be somewhat beyond the current state-of-the-art for
computing integrals analytically. Two promising analytic approaches are: (1) the detour
via introducing an additional scale and subsequent use of di erential equations [109, 112,
113], and (2) the nite integral approach of [114]. In both approaches, the IBP reduction
seems to be the main bottleneck. For instance, the latter would also enable the use of
dimensional recurrences [79, 115], but requires the solution to the IBP problem of so-called
four-dotted integrals.
In the present work, we choose a numerical approach. While numerical integration of
the four-loop form factor integrals remains quite hard for generic numerators, we make the
surprising empirical observation that UT integrals are numerically much easier to integrate
than generic numerators of the class under study. We may o er an intuitive explanation
for this. The constant leading transcendentality criterion used to nd candidate UT
integrals guarantees that these integrals have very mild singularity properties. An algorithm
like sector decomposition is bound to be more e cient in cases where internal singularities
are simpler. Note however that sector decomposition algorithm works in Feynman
parameter space, whereas constant leading singularity criteria are applied in parametric form
like (3.6). Whatever the precise origin, the relative simplicity of UT integrals is a boon for
explicit computation, leading to a remarkable reduction in intermediate expression sizes
and integration times. Moreover, the obtained coe cients in the expansion appear to be
numerically much smaller than for generic integrals; this is bene cial for reducing potential
cancellation errors.
Because of the physical motivation, we will only focus on integration of the integrals in
the nonplanar sector of the form factor. We leave integration of the integrals in the
colourplanar sector to future work, mostly because all terms of the latter through to O(
are already known: the
f8;6;5;4;3g poles are dictated by contributions from lower loops
1
)
according to eq. (2.2), and the cusp [4, 9, 12] and collinear [89, 90] anomalous dimensions
are already known analytically.
5.1
Mellin-Barnes representations
Mellin-Barnes (MB) representations constitute a powerful tool for evaluating Feynman
integrals [116{118]. They rely on the fact that one can factorise sums of terms at the cost
of introducing line integrals in the complex plane. The basic formula reads
1
(A1 + A2 + : : : + Am)
=
c1+i 1
Z dw1
2 i
c1 i 1
cm 1+i 1
Z
2 i
dwm 1 Aw1
1
cm 1 i 1
Awmm 11 Am
w1 ::: wm 1
The curves are usually straight lines parallel to the imaginary axis whose constant real
parts are chosen such as to separate all left from all right poles of -functions. This is
achieved by choosing the real parts of all MB variables wi together with that of
such
that the arguments of all -functions have positive real parts. The poles in
are then
extracted by analytical continuation to
! 0, for which several algorithms exist [118{
122]. Subsequently, the terms can be Laurent-expanded about
= 0 and integrated, which
proceeds mostly numerically with MB.m [121], but also examples of analytical evaluation of
MB integrals exist [123, 124]. parameters, e.g. like To get from a loop integral to an MB representation, one rst introduces Feynman
After integration over k1 the remaining terms can be factorised using eq. (5.1), and
subsequently integrated over the xi via
Z 1
0
dx1dx2
dxn x1a1 1 xa2 1
2
xan 1
n
1
n
X xi
i=1
=
(a1) (a2)
(a1 + a2 + : : : + an)
(an) : (5.3)
The procedure is then repeated until all loop momenta are integrated out. In our case
where no kinematic thresholds are present one has to obtain positive de nite terms at all
stages of the calculation if q2 is space-like (we put q2 =
1 for de niteness). Moreover, all
terms in the
expansion are real. For planar topologies this so-called loop-by-loop approach
is always applicable, and we will refer to MB representations coming exclusively from
positive de nite terms as valid MB representations. However, valid MB representations
for a given loop integral are not unique, even their dimensionality can di er depending on
the order the loop momenta are integrated over.
For crossed topologies, the situation is more complicated as one encounters cases in
which the loop-by-loop approach yields polynomials in the Feynman parameters xi which
are not positive de nite, even in the absence of kinematic thresholds. Consequently, the
MB integrals will be highly oscillating and hence their numerical evaluation will be di cult
to handle, although steps in this direction have been undertaken [125{128].
One way of circumventing this problem in the case of crossed topologies is to not
integrate over the loop momenta one by one, but to simultaneously integrate over all loop
momenta. This is done by means of the Symanzik graph polynomials U and F [129{132],
which at L loops are homogeneous of order L and L + 1, respectively. In the absence of
kinematic thresholds they are positive de nite and hence automatically lead to valid MB
representations. The price to pay is the fact that the number of terms in U and F scales
as L!. As L increases this therefore quickly leads to MB representations that are too
highdimensional to be integrated in practice. A partial remedy to this problem is to group the
f 8; 6; 5; 4g it is expected that the numerical coe cients can be written as a rational
number times f1; 2; 3; 4g. Hence, by dividing the numerical result by the appropriate
MZV constant, a numerical result is obtained which should be expressible as a rational
number. For the case at hand, we typically have at least ve to six digits available and
the found integers have on the order of three digits in numerator and denominator. This
indicates that the obtained rational numbers are reasonable, which gets supported by the
fact that their contribution in the nal result of the nonplanar form factor cancels exactly.
In appendix A the results of the rationalisation are listed.
For f 3; 2g the UT property still holds, but at these orders there are two MZVs of
transcendentality 5 and 6 respectively. For weight 5 these could for instance be taken to be
2 3 and 5, and one can attempt a solution with the PSLQ algorithm [143], for instance
through Mathematica's command FindIntegerNullVector. The appropriate integer
relation then contains three unknowns: one for the numerical result, and two for the MZVs.
For integer coe cients to be reliably isolated one needs much more digits in these cases,
certainly more than 10. Since we have typically only four to ve digits available at these
orders, the PSLQ algorithm is currently not feasible. Moreover, many of our numerical
results were obtained using sector decomposition where the price of integration roughly scales
quadratically with increasing precision. This makes PSLQ unfeasible for the coe cients at
orders f 3; 2; 1g within the numerical setup employed here. It would be highly interesting
to obtain high precision numerics at these orders, or even better of course analytic results
that do not rely on PSLQ.
5.3.2
Error analysis
Since numerical integration methods are used, a thorough discussion of the errors in these
integrals is called for. Both for MB as well as for sector decomposition methods an error
is reported. As is well-known, if an e cient MB representation can be found, the error in
its integration is in general small, especially compared to sector decomposition. For the
integrals at hand typically a di erence in precision of three to ve digits arises. Hence, the
discussion here will focus on sector decomposition.
FIESTA employs the CUBA [142] integration library. Although we have cross-checked
some simple integrals as well as leading expansion coe cients of more complicated ones,
most of the coe cients needed for the cusp anomalous dimension at order
2 were obtained
using exclusively the VEGAS [141] algorithm. VEGAS employs an adaptive sampling
algorithm. It should be noted that the integrals under study do not have any physical
singularities, and do not have to be analytically continued, two common sources of error.
For su ciently many evaluation points, the VEGAS error is of Gaussian type. To check
that this regime is reached, one evaluates the integrals for several evaluation points settings.
In the Gaussian regime, the error scales as 1=peval points. For all integrals in the set
integrated here, this was reached very quickly. In rare cases involving much more complicated
integrals, it has been reported in [
144
] that the error in FIESTA can be underestimated. In
those cases the central value of certain coe cients changed outside the reported error with
increasing evaluation points. We have checked for this as well, and have never observed
variations outside of reported error upon increasing the number of evaluation points for the
f 6; 5; 4g orders. (a) Plot of cases IPSLQ
FIESTA error
IFIESTA > 0. (b) Plot of cases IFIESTA
IPSLQ > 0. A
FIESTA error
logarithmic scale is used for the vertical axis, and all ratios larger than 200 are not shown in the
gures. We can see that all ratios are larger than unity, which suggests that the FIESTA errors are
conservative estimates. Besides, we
nd that the deviation of FIESTA results from PSLQ results
are both positive and negative, which indicates that there is no source of systematic errors.
integrals under study. Several simpler integrals have been computed using SecDec with the
DIVONNE and CUHRE algorithms as a further crosscheck. More cross-checks for integral
I(21) and I1(360) follow from available MB results, as well as an exact result for integral I1(21).
1
For the leading coe cients of the individual integrals an additional cross-check is
enabled by their UT properties: having obtained a product of a rational number times a zeta
value for the leading coe cients from the expansion (see section 5.3.1), one can use this to
obtain an estimate of the true precision. For this, we compute the ratio between FIESTA
errors and the assumed `true' errors obtained by comparing to the PSLQ result at order
f 6; 5; 4g, namely,
FIESTA errork
Ik;PSLQ
Ik;FIESTA
;
(5.9)
where k labels the 23 integrals in section 4.1. The results are plotted in gure 6. Two
panels are provided for positive and negative deviations separately. Note that for all 23
integrals, all absolute ratios are larger than one, corresponding to reported FIESTA errors
larger than the discrepancy between PSLQ result and numerical integration. Moreover, by
comparing gure 6(a) and
gure 6(b), it is clear there is no de nite sign of the deviation:
positive and negative deviations are about as likely. If this had been di erent, this might
have indicated a systematic error.
Finally, physics provides a strong cross-check of the numerics. The leading coe cient
8
of the nonplanar form factor should be of order
2, while individual integrals generically
contribute from order
. Hence, in the sum there should be numerical cancellations
between the integrals to give zero within error bars for the rst six orders of expansion,
down to
3
.
With the errors added in quadrature and the result for the sum of the
central value, one can compare to the exact answer, 0, for these coe cients. These results
are contained in table 1 and clearly indicate that reported errors are not underestimated,
giving further support for our error analysis.
In total, the above analysis shows that the errors reported by FIESTA are stable and
in general conservatively estimate the errors for the form factor integrals in the present
study. This strongly indicates that the
nal error for CAD is not underestimated either,
and hence there is no need to manually in ate the reported uncertainty. Conservatively,
we will interpret the FIESTA reported error as the standard deviation of a Gaussian error.
For a true single standard deviation in a Gaussian error, one would expect deviations from
the true result to exceed the standard deviation of the Gaussian distribution roughly 32%
of the time, while here this never occurs. As a consequence of the error interpretation, the
obtained errors are added in quadrature. For reference, also the result of adding errors
linearly is provided, which is recommended in cases which involve a small systematic error.
However, we emphasise that there is no sign of systematic errors in the case at hand.
6
Discussion and conclusion
In this article a set of tools and techniques have been discussed for the integration of
four-loop form factor integrals, especially focussed on the nonplanar sector of the Sudakov
form factor in maximally supersymmetric Yang-Mills theory. This sector contains among
others information on the nonplanar correction to the cusp anomalous dimension. Four
loops is the rst time a nonplanar correction enters into the form factor as well as into
the cusp and collinear anomalous dimensions. Although conjectures existed that the CAD
vanished generically in gauge theories, our results, rst announced in [15], show this is not
the case. In this article we also present the rst numerical result for the nonplanar collinear
anomalous dimension. The numerics of especially the latter result leave quite some room
for improvement. Even more interesting would be to obtain an analytic result. Besides
settling conjectures, of much wider interest is how the results reported in this article were
obtained: the tools and techniques are certainly applicable to a wider context than just
this particular computation in this particular theory.
Inspired by similar computations in the literature [106{109], an algorithm was
presented to
nd complete sets of uniformly transcendental integrals in a given set of
topologies. The algorithm is based on the conjecture that these integrals always have constant
leading singularities. Importantly, the algorithm stabilises to a result in
nite time in our
current Mathematica implementation. A surprising amount of uniformly transcendental
integrals were found for each integral topology for the problem at hand. With some
combination techniques, a set of integrals was obtained to express the maximally supersymmetric
form factor in. However, the number of UT integrals involved in this physical problem is
much smaller than the total number of UT integrals in each topology. This points towards
applications of these integrals beyond maximal supersymmetric Yang-Mills. Intriguingly,
the numbers obtained are comparable to the total number of IBP master integrals. It would
be very interesting to explore this further, but this will have to involve IBP-reducing the
pure, non-supersymmetric Yang-Mills form factor, which is beyond currently (publicly)
available technology.
Having obtained a suitable basis of UT master integrals to express the form factor in,
the next step is the integration of these integrals. A pleasant surprise is the observation that
even though many integration techniques such as sector decomposition spoil UT properties
in intermediate steps, the UT integrals appear to be much easier to integrate than generic
integrals in the form factor class.
Within sector decomposition, this manifests itself in
term counts which are an order of magnitude better. This in turn leads to much more
compact expressions in the integration steps which lead to much improved performance in
both speed and accuracy. Intuitively, this corresponds well to the notion that UT integrals
are inherently simple. More mathematically, the absence of higher order singularities in
the integrand in parametric form (as discussed in section 3.1) translates very likely to
less singular integrands in Feynman parameter form. This in turn should then explain
the observed much improved behaviour of sector decomposition methods. It would be
interesting to explore this further, especially a criterion which would allow one to decide
if an integral is UT in Feynman parameter form would be highly desired. Since there are
considerably fewer integrations in Feynman parameter form than in parametric form, this
is potentially even much more powerful.
Special attention is paid to the numerical integration of the form factor integrals in the
nonplanar sector. Apart from the central value, the error analysis in numerical applications
is important.
Here the UT property of the integrals informs the error analysis. The
integration of leading coe cients allows one to check the error analysis by using the PSLQ
algorithm to
nd the exact value of the integrals. This combination of number theory
and numerical integration shows that the errors reported by FIESTA are in general very
conservative estimates. Added to knowledge of a single exact integrals and several results
obtained using Mellin-Barnes integrals, this gives comprehensive evidence for our error
analysis for the computation of the nonplanar cusp and collinear anomalous dimensions at
four loops.
Acknowledgments
It is a pleasure to thank Sven-Olaf Moch, Andreas von Manteu el and Robert Schabinger
for discussions. This work was supported by the German Science Foundation (DFG) within
the Collaborative Research Center 676 \Particles, Strings and the Early Universe". GY is
supported in part by the Chinese Academy of Sciences (CAS) Hundred-Talent Program, by
the Key Research Program of Frontier Sciences of CAS, and by Project 11647601 supported
by National Natural Science Foundation of China.
A
UT integrals
A.1
UT integrals with 12 lines
For the UT integrals we use the parametrizaton in terms of loop momenta from [48] and
the normalisation used by FIESTA, i.e. we work in D = 4
2 -dimensional Minkowskian
space-time and our integration measure is e E dD`=(i D=2) per loop. Moreover, we set
(p1 + p2)2 =
1 and suppress the fact that the -expansion continues in all equations.
Below we give our numerical results as well as the PSLQ up to
4 order.
I(21) =
1
ℓ3
p1
[(`3
The integral I1(21) is known analytically from [109]. Our numerical results obtained by MB
and FIESTA agree with the analytical one well within error bars.
0:001736111111111111 0:04569261296800628(1) 0:2100817041401606(1)
1:6253638839586(7)
8:5855125581(10)
44:566338023(40)
0:00173611
0:0000000004(837)
0:0456926(14)
0:210082(17)
4
+
+
1:62537(18)
8:5853(19)
44:564(20)
3
+
6
2
2
6
+
6
:
+
+
2
5
5
1:46
;
(`3
+
+
q
8
ℓ3
=
8
(22)
I2;PSLQ =
1
=
(23)
I3;PSLQ =
8
4
ℓ4 p1
ℓ6
p2
4
q
ℓ3
7
3
p1
p2
+
+
+
7
3
0:00520833
0:000000003(130)
0:4340801(26)
2:291419(35)
9:56243(42)
51:4505(51)
333:021(67)
1705:78
+
+
1
144 8
8
4
;
5
;
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
I(24) =
4
(A.8)
(A.9)
(`3
`3
;
(24)
I4;PSLQ =
5
I6;PSLQ =
I(26) =
7
ℓ3
8
4
q ℓ3 ℓ4 ℓ5 p1
8
+
1
ℓ6
p2
8
4
q ℓ3 ℓ4 ℓ5 p1
ℓ6
p2
p1
p2
ℓ5
+
p2
n
7
3
n[(`3
3
7
+
+
`
2
4
+
2
6
2
6
(`3
(p1
+
(`3
`4
p1)2
(`6
p1)2
(`3
p1
`4)2
p2)2o
1:16310(3)
2:90880(35)
5
4
12:2720(43)
29:708(57)
3185:60
2:63
(26)
I7;PSLQ =
1
`
2
4
(`4
`3
p1)2
(`3
+
1
31719532 44 + O( 3) :
(29)
I10;PSLQ =
=
+
8
4
1
+
2
1173283 5 +
126596 44 + O( 3) :
6
+
0:015625
0:00000001(14)
0:3426942(17)
1:377357(20)
0:41430(24)
18:1972(33)
155:896(52)
1304:61(93)
1
(`3
`4
`4)2
8
ℓ3
8
q
4
3
ℓ4
q
ℓ3
`
3
2
`
2
4
(`4
`3
I8;PSLQ =
I(28) =
9
=
(28)
I9;PSLQ =
Topology 29:
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
+
7
2
`
3
2
7
3
ℓ5
ℓ6
p1
p2
ℓ3
8
4
p1
p2
ℓ4
p1
ℓ6 p2
+
+
2
+
7
7
3
6
2
6
;
+
`4
5
;
(`3
+
p1)2
5
5
;
;
I1(310) =
ℓ3
8
(`3
`4
p2)2 [(p1
`4)2
(`3
p1)2]
0:00347222
0:05140419
0:2601674
1:5145009
6
5
17:34721164(4)
133:31287(3)
671:48(24)
4
:
This result was obtained with MB. FIESTA performs poorly in this topology.
(30)
I11;PSLQ =
1
ℓ5
ℓ6
p1
p2
ℓ3
8
4
p1
p2
1
2
7
2
1
2
2
2
ℓ3
ℓ4
8
q
Topology 28:
+
p1)2
0:0303819
0:00000002(87)
0:625418(2)
2:824274(22)
7:64568(40)
(`4
6
`6)2
`24 + `52 + 2 (p1 + p2)2
5
4
(A.22)
(A.23)
HJEP01(28)53
(A.24)
(A.25)
;
(A.26)
(A.27)
3
22:7148(82)
0:160(47)
1354:58(99)
:
This result is obtained by combining FIESTA and MB results.
(28)
I13;PSLQ =
13
192 6
1015 3
(`3 `4)2 2 (`3 `4 p2)2 +(`6 p1)2 (`4 `6)2 +`42
0:0112847
0:000000001(95)
0:299858(2)
0:848669(24)
0:86617(24)
10:3884(22)
107:036(19)
184:841
1:038
7
3
+
+
2
6
+
+
5
(`4
2
6
+
+
+
1:68939(17)
81:404(26)
+
5
;
`6)2
(`4
p2)2
p1)2
+
2
1547263 44 + O( 3) :
(`3
p1
0:036458333
0:5997155452(1)
2:2622043108(1)
0:828653725(2)
5
4
19:82059(28)
94:8794(349)
232:242(541)
;
271 3
144 5
+
4694 44 + O( 3) :
I1(249) =
Topology 30:
I14;PSLQ =
q
7
q
q
192 8
I16;PSLQ =
6
(A.28)
(A.29)
(A.30)
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
ℓ4
p1
ℓ6 p2
ℓ4
p1
ℓ6 p2
ℓ3
ℓ3
2
5
8
4
ℓ4
ℓ5
ℓ6
ℓ5
ℓ6
3
35 2
96 6
ℓ4
2
5
ℓ4
ℓ5
ℓ6
p1
p2
p1
p2
p1
p2
q
1
ℓ3
8
ℓ3
ℓ3
0:03756430(4)
0:1042870(7)
1:64150(1)
8:56434(14)
35:4679(216)
;
+
1
2
1
2
1
2
(`3
3
7
6
2
4
;
p1
(`6
(`4
(`3
0:001391(11)
4:07561(14)
35:6750(18)
211:233(25)
1162:74(39)
7
3
2
3
(`3
+
+
`26 (p1
`4)2 =
0:00173611111
0:165635722(1)
0:74850303(1)
4:1564218626(4)
+
4
ℓ4 p1
ℓ6
p2
ℓ4 p1
ℓ6
p2
ℓ5
ℓ5
p1
p2
p1
p2
1:34678628(2)
6:89677(9)
p2)2
+
8
I18;PSLQ =
1
UT integrals with 10 lines Topology 22:
I19;PSLQ =
I2(202) =
1
q
5
ℓ3
8
3
ℓ3
8
8
ℓ3
ℓ3
I2(214) =
I2(224) =
0:00868056
0:0000000009(316)
0:211328(1)
0:637202(14)
4:06623(11)
48:3099(8)
242:796(6)
819:895(471)
(24)
I21;PSLQ =
+
2
2
6
5
;
2
p1
6
6
`5)2 (`3
`25 (`3
p1
p2)2
7
3
7
3
5
5
;
5
;
(A.36)
(A.37)
HJEP01(28)53
(A.38)
(A.39)
:
(A.40)
(A.41)
(A.42)
(A.43)
(A.44)
I2(238) =
+
2
I23;PSLQ =
4
58 44 + O( 3) :
ℓ3
ℓ5 ℓ6
ℓ4
p1
p2
B
Basis of propagators and numerators
This appendix contains the basis of 12 propagators and 6 irreducible numerators, which
are used in section 4.2. The numbering of the equations corresponds to the topologies
in
gure 1{2. In each case, the rst twelve entries parametrise the twelve propagators of
the respective integral and the last six entries the chosen numerators. We have de ned
q = p1 + p2.
fl6; l5; l4; l3; l6
l3; p1
l4;
fl6; l5; l4; l3; l5
l4; l6
l5; p1
l4;
l5; l3
l3 + q; l6 + q; l5 + q; l3 + l4
l5 + q;
l5; l3
l4 + q; l6 + q; l3 + q; l4 + l5
p1; l3
fl6; l5; l4; l3; l3
l4; l5
p2; l3
l5; l3
(A.45)
(A.46)
HJEP01(28)53
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
fl6; l5; l4; l3; p1 l5; l4 l5; l4 l3; p2 l6;
fl6; l5; l4; l3; p2 l6; l4 l5; l4 l3; p1 l5;
fl6; l5; l4; l3; l4 l5; p1 l5; l4 l3; l3+q;
l3 l5; l3 p2; l4 p1; l4 p2; l5 l6; l6
l3+l4+l5; l3+q; l4 l5+q; l4 l6+q;
l4 l5; l3 l6; l4 l6; l3 p2; l4 p2; l5 p2g;
fl6; l5; l4; l3; p2 l6; p1 l4; l6 l5; l3 l4;
4 l5+l6+p1; l3 l5+q; l4 l5+q;
l3 l6; l3 p2; l4 p2; l5 p1; l5 p2; l6 p1g;
l3+q; l3 l4+l5 l6; l
3 l5+q; l4 l6+q;
l3 l6; l3 p2; l4 l5; l4
p2; l5 p1; l6 p1g;
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
4 l6; l3+q; l3+l4 l5+q; l3+l4 l5+l6+p1;
l3 p2; l3 l5; l4 l6; l4
p2; l5 p2; l6 p1g;
fl6; l5; l4; l3; p1 l5; l5 l4; p2 l6; l3 l4;
4 l5; l3+l6+p1; l3+l5+p1;
l3 p2; l4 p2; l5 p1; l6 p1; l5 p2; l4 l6g;
l3 p2; l4 l6; l4 p2; l5 p1; l5 p2; l6 p1g;
fl6; l5; l4; l3; l5 l4; l3 l4; p1 l5; p2 l6;
l3 l5; l3 l6; l5 l6; l4 p1; l4 p2; l5 p2g;
l3 l5; l3 l6; l5 l6; l4
p1; l4 p2; l5 p2g;
(B.17)
(B.18)
(B.19)
(B.20)
(B.21)
(B.22)
(B.23)
(B.24)
(B.25)
(B.26)
l
4
l
l3 + q; l3
l4; l4
l
fl6; l5; l4; l3; l3
l4; p2
l5 + p1; l3 + q; l3 + l4 + l5 + p2; l3 + l4 + l5 + l6;
l5 + q; l3 + l4
l6 + p2;
l4; l5 + l6; p2
l6; p1
l4;
l5 + p1; l3 + q; l3 + l4
l6 + p2; l
l6 + q;
l5 + p1; l3 + l4 + l5 + p2; l3 + l4
l6 + p2;
(B.27)
(B.28)
(B.29)
(B.30)
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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