arXiv:0707.4139v1 [hep-ph] 27 Jul 2007
DESY 07-101
SFB/CPP-07-37
July 2007
arXiv:0707.4139v1
Heavy-quark production
in gluon fusion at two loops in QCD
M. Czakona,b, A. Mitovc and S. Mochc
a Institut
für Theoretische Physik und Astrophysik, Universität Würzburg
Am Hubland, D-97074 Würzburg, Germany
b Department
of Field Theory and Particle Physics, Institute of Physics
University of Silesia, Uniwersytecka 4, PL-40007 Katowice, Poland
c Deutsches
Elektronensynchrotron DESY
Platanenallee 6, D–15738 Zeuthen, Germany
Abstract
We present the two-loop virtual QCD corrections to the production of heavy quarks in gluon fusion.
The results are exact in the limit when all kinematical invariants are large compared to the mass of
the heavy quark up to terms suppressed by powers of the heavy-quark mass. Our derivation uses
a simple relation between massless and massive QCD scattering amplitudes as well as a direct
calculation of the massive amplitude at two loops. The results presented here together with those
obtained previously for quark-quark scattering form important parts of the next-to-next-to-leading
order QCD corrections to heavy-quark production in hadron-hadron collisions.
1
Introduction
The production of heavy quarks at hadron colliders is an important process for a range of reasons.
Experimentally, events with a top-quark, being the heaviest quark known thus far, lead to very
characteristic signatures in a hadronic scattering process. This allows for event reconstruction in
a variety of channels, which e.g. at LHC is supplemented by an anticipated very high statistics
for the production cross section [1, 2]. In this way precision measurements of the top-mass, of
(differential) t t¯-distributions and also tests of the production and the subsequent decay mechanism
(including anomalous couplings and top-spin correlations) can be conducted at Tevatron or LHC.
The precise knowledge of the top-quark parameters has direct impact on precision tests of the
Standard Model, in particular on the Higgs sector due to the large top-Yukawa coupling. Moreover,
events with top-quarks will also make up for a large part of the background in searches for the
Higgs boson or new physics.
In the case of bottom-quark production important measurements are inclusive B-meson and bjet production at moderate to large transverse momentum. At colliders b-flavored jets are produced
for instance in decays of top-quarks, the Higgs boson and numerous particles proposed in extensions of the Standard Model. Differential b-jet distributions are also important for measurements of
parton distributions. Predictions for heavy-quark hadro-production have theoretical uncertainties,
which can even be larger than e.g. the corresponding predictions for the production of light jets.
This is due to radiative corrections in Quantum Chromodynamics (QCD), where typically large
logarithms at higher orders appear. At present, the next-to-leading order (NLO) QCD corrections
for heavy-quark hadro-production are known [3–9] and, once they become available, the complete next-to-next-to-leading order (NNLO) QCD corrections will reduce the uncertainty of theory
predictions by improving the stability with respect to scale variations. For inclusive B-meson production [10, 11], for instance, the formalism of perturbative fragmentation functions of a heavy
quark [12] has already been extended through NNLO by providing the initial conditions [13, 14]
together with the time-like (non-singlet) evolution [15]. Alternative ways devoted to control the
theoretical uncertainty in the case of b-quarks, for example through the definition of dedicated jet
algorithms have been proposed recently [16].
In the hadronic production of heavy quarks large perturbative QCD corrections arise in different kinematical regions. If the heavy quarks are produced with (partonic) center-of-mass energies s close to threshold, then Sudakov-type
logarithms, powers of log(β), appear to all orders
p
2
in perturbation theory, where β = 1 − 4m /s is the velocity of the heavy quark. The necessary
resummation has been carried out to next-to-leading logarithmic (NLL) accuracy [17–19] and has
successfully improved the phenomenology of top-quark production at Tevatron. In the high-energy
regime on the other hand, when the heavy quarks are fast, logarithms in the quark mass m appear.
These log(m)-terms are of collinear origin and dominate cross sections when m becomes negligible in comparison to other kinematical invariants, such as e.g. for the inclusive b-jet spectrum
at large transverse momentum. All-order predictions can be achieved by means of an explicit resummation. The necessary technology for resumming both the incoming and outgoing collinear
logarithms to NLL accuracy is available, see e.g. Refs. [20, 21] and references therein.
1
In this article, we want to further improve the precision of fixed-order perturbative QCD corrections. To that end, we present results for the virtual QCD corrections at two loops for the
pair-production of heavy quarks in gluon fusion. Together with the corresponding results for
quark-quark scattering obtained previously by us [22] these are essential parts of the complete
NNLO QCD corrections. To be precise, we calculate the interference of the two-loop with the
Born amplitude. We work in the limit of fixed scattering angle and high energy, where all kinematic invariants are large compared to the heavy-quark mass m. Thus, our result contains all
logarithms log(m) as well as all constant contributions (i.e. the mass-independent terms) and we
consistently neglect power corrections in m.
In our calculation we employ two different methods. On the one hand, we apply a generalization of the infrared factorization formula for massless QCD amplitudes [23, 24] to the case of
massive partons [25]. In a nut-shell this results in an extremely simple universal multiplicative
relation between a massive QCD amplitude in the small-mass limit and its massless version [25].
In this way, we can largely use for our derivation the results of the NNLO QCD corrections to
massless quark-gluon scattering (i.e. gg → qq̄). These have been computed for the squared matrix element, i.e. the interference of the two-loop virtual corrections with the Born amplitude in
Ref. [26], while results for the individual (independent) helicity configurations of the two-loop
amplitude |M (2) i itself have been given in Refs. [27, 28]. On the other hand, we perform a direct calculation of the relevant Feynman diagrams in the massive case followed by a subsequent
expansion in the small-mass limit by means of Mellin-Barnes representations.
The outline of the article is as follows. In Section 2 we give some basic definitions and present
a short summary of our methods in Section 3. There we briefly explain the essence of the QCD
factorization approach to calculate massive amplitudes and give the relevant formulae. We also
highlight the key steps of the direct evaluation of Feynman diagrams in the small-mass limit and, in
particular, comment on the non-planar topologies. Section 4 contains our results and we conclude
in Section 5. Appendix A gives the explicit result for a massive non-planar scalar integral in the
small mass limit.
2
Setting the stage
The pair-production of heavy quarks in the gluon fusion process corresponds to the scattering
g(p1 ) + g(p2 ) → Q(p3 , m) + Q̄(p4 , m) ,
(1)
where pi denote the gluon and quark momenta and m the mass of the heavy quark. Energymomentum conservation implies
µ
µ
µ
µ
p1 + p2 = p3 + p4 .
(2)
Following the notation of Ref. [26] we consider the scattering amplitude M for the process (1) at
fixed values of the external parton momenta pi , thus p21 = p22 = 0 and p23 = p24 = m2 . The amplitude
2
M may be written as a series expansion in the strong coupling αs ,
i
α 2
h
α s
s
|M (2) i + O (α3s ) ,
|M (1) i +
|M i = 4παs |M (0) i +
2π
2π
(3)
and we define the expansion coefficients in powers of αs (µ2 )/(2π) with µ being the renormalization
scale. We work in conventional dimensional regularization, d = 4 − 2ε, in the MS-scheme for the
coupling constant renormalization. The heavy mass m on the other hand is always taken to be the
pole mass.
We explicitly relate the bare (unrenormalized) coupling αbs to the renormalized coupling αs by
2
i
h
β 0 1 β 1 αs 2
β 0 αs 3
b
+ O (αs ) ,
(4)
+ 2−
αs S ε = αs 1 −
ε 2π
ε
2 ε
2π
where we put the factor Sε = (4π)ε exp(−εγE ) = 1 for simplicity and β is the QCD β-function
[29, 30]
β0 =
11
2
CA − TF n f ,
6
3
β1 =
17 2 5
C − CA TF n f −CF TF n f .
6 A
3
(5)
The color factors are in a non-Abelian SU(N)-gauge theory CA = N, CF = (N 2 − 1)/2N and TF =
1/2. Throughout this letter, N denotes the number of colors and n f the total number of flavors,
which is the sum of nl light and nh heavy quarks.
In the following, we will confine ourselves to the discussion of the squared amplitude for the
process (1), although it should be clear that our approach and the results of the present article can be
easily extended to the (color ordered) partial amplitudes for the individual helicity combinations of
the massive two-loop amplitude |M (2) i itself. There one would rely in particular on Refs. [27, 28].
For convenience, we define the function A (ε, m, s,t, µ) for the squared amplitudes summed over
spins and colors as
∑ |M (g + g → Q + Q̄)|2
= A (ε, m, s,t, µ) .
(6)
A is a function of the Mandelstam variables s, t and u given by
s = (p1 + p2 )2 ,
t = (p1 − p3 )2 − m2 ,
u = (p1 − p4 )2 − m2 ,
and has a perturbative expansion similar to Eq. (3),
α 2
α s
s
8
3
6
4
2 2
A + O (αs ) .
A +
A (ε, m, s,t, µ) = 16π αs A +
2π
2π
(7)
(8)
In terms of the amplitudes the expansion coefficients in Eq. (8) may be expressed as
2
N2
N −1
(N 2 − 1)
4
(0)
(0)
(1 − ε)
−2 2
t 2 + u2 − εs2 + O (m) , (9)
A = hM |M i ≡ 2
N
ut
s
A 6 = hM (0) |M (1) i + hM (1) |M (0) i ,
(10)
A 8 = hM (1) |M (1) i + hM (0) |M (2) i + hM (2) |M (0) i ,
(11)
3
where we have discarded powers in the heavy-quark mass m in A 4 .
The expressions for A 6 have been presented e.g. in Refs. [8, 9] and the loop-by-loop contribution hM (1) |M (1) i at NNLO in A 8 has been published in Ref. [31]. Both results for A 6 and
hM (1) |M (1) i have been obtained in dimensional regularization and with the complete dependence
on the heavy-quark mass. Here we provide for the first time the real part of hM (0) |M (2) i up to
powers O (m) in the heavy-quark mass m.
3
Method
3.1 The massive amplitude from QCD factorization
Let us briefly recall the key findings of Ref. [25] on how to calculate loop amplitudes with massive
partons from purely massless amplitudes. Heuristically, the QCD factorization approach rests on
the fact that a massive amplitude M (m) for any given physical process shares essential properties in
the small-mass limit with the corresponding massless amplitude M (m=0) . The latter one, M (m=0) ,
generally displays two types of singularities, soft and collinear, related to the emission of gluons
with vanishing energy and to collinear parton radiation off massless hard partons, respectively.
These appear explicitly as poles in ε in dimensional regularization after the usual ultraviolet renormalization is performed. In the former case, the soft singularities remain in M (m) as single poles
in ε while some of the collinear singularities are now screened by the mass m of the heavy fields,
which gives rise to a logarithmic dependence on m, see e.g. Ref. [32].
This structure of singularities for massless amplitudes has been clarified to all orders in perturbation theory [24] as all 1/ε terms can be exponentiated, see also Ref. [33]. Similarly, all poles
in ε and log(m) terms for amplitudes with massive partons also obey an all-order exponentiation
with mostly the same anomalous dimensions as in the massless case [25]. Thus, in the small-mass
limit the differences between a massless and a massive amplitude can be thought of as due to the
difference in the infrared regularization schemes. QCD factorization provides a remarkably simple
direct relation between M (m) and M (m=0)
1
(m|0) 2
(m)
× M (m=0) .
(12)
Z[i]
M
=
∏
i∈ {all legs}
The function Z (m|0) is process independent and depends only on the type of external parton, i.e.
quarks and gluons in the case at hand. For external massive quarks Q it is defined as the ratio of
the on-shell heavy-quark form factor and the massless on-shell one, both being known [34–36] to
sufficient orders in αs and powers of ε. An explicit expression for
∞ αs j ( j)
(m|0)
Z[Q] = 1 + ∑
(13)
Z[Q] ,
j=1 2π
up to two loops is given in Ref. [25] (note the different normalization αs /(4π) used there). The
leading n f terms ∼ (n f αs )n for our process Eq. (1), gg → QQ̄, can also be predicted based on the
4
above arguments. Keeping only terms quadratic in nh and/or n f = nh + nl one has up to two loops:
(m|0)
Z[g]
= 1+
where
(2)
Z[g]
(1)
=
αs (1) αs 2 (2)
Z[g] + O (α3s ) ,
Z +
2π [g]
2π
(1) 2
Z[g] +
2
(1)
n f TF Z[g] + O (nh 1 × nl 0 ) ,
3ε
(14)
(15)
(1)
with Z[g] ∼ nh given in Ref. [25] (again note the different normalization αs /(4π) used there). Z[g]
is also equal to the O (αs ) term in the gluon wave function renormalization constant Z3 in Eq. (22).
(m|0)
The relation of Z[g] to Z3 is discussed after Eq. (24) below. To derive Eq. (15) we apply the
definition for Z (m|0) given in Ref. [25], i.e. evaluate the ratio of the gluon form factor with heavyloop insertions and the pure massless gluon form factor [36, 37]. The additional renormalization
constant that enters the effective Hgg vertex (see e.g. Ref. [37] for details) cancels in the ratio and
(m|0)
does not contribute to Z[g] . Exploiting the predictive power of the relation Eq. (12) and applying
it to the process Eq. (1) we get
(1)
(1)
2Re hM (0) |M (2) i(m) = 2Re hM (0) |M (2) i(m=0) + Z[Q] + Z[g] A 6,(m=0)
(2)
(2)
(1) (1)
+ 2 Z[Q] + Z[g] + Z[Q] Z[g] A 4,(m=0) + O (nh 1 × nl 0 ) + O (m) , (16)
which assumes the hierarchy of scales m2 ≪ s,t, u , i.e. we neglect terms O (m). Eq. (16) predicts
the complete real part of the squared amplitude hM (0) |M (2) i(m) except (as indicated) for those
terms, which are linear in nh (the number of heavy quarks) and, at the same time not proportional
to nl (the number of light quarks). These two-loop contributions have been excluded explicitly
also from the definition [25] of Z (m|0) , as one needs additional process dependent terms for their
description.
The two-loop massless amplitudes Re hM (0) |M (2) i(m=0) are computed in Ref. [26]. We have
checked that the finite remainders of the squared two-loop amplitudes obtained after the infrared
subtraction procedure discussed in that reference agree with the corresponding terms constructed
from the two-loop helicity amplitudes calculated in Ref. [27]. We have also found similar agreement between the finite remainders of the qq̄ → q′ q̄′ amplitudes we used in Ref. [22] that we
extracted from Refs. [38, 39].
3.2 Direct computation of the massive amplitude
The direct computation of the massive amplitude proceeds according to the same scheme as in
our previous publication [22], which itself was an evolution of the methodology developed in
Refs. [40–42]. In short, the complete amplitude is reduced to an expression containing only a
small number of integrals with the help of the Laporta algorithm [43]. In a next step, Mellin-Barnes
(MB) representations [44, 45] of all these integrals are constructed, and analytically continued in
the dimension of space-time with the help of the MB package [46] revealing the full singularity
5
structure. A subsequent asymptotic expansion in the mass parameter is done by closing contours
and resumming the integrals, either with the help of XSummer [47], or the PSLQ algorithm [48].
We shall now concentrate on the differences with respect to our previous calculation. First
of all, the number of master integrals is substantially larger, reaching 422, which adds a lot to
the complexity of the computation. This is partly due to the fact, that the symmetry with respect
to the exchange of the gluons generates the same topologies in the t- and u-channels, but more
importantly because of completely new topologies, which come together with the more extended
set of gluon interactions. Whereas in Ref. [22], it was possible to avoid the computation of the
high-energy asymptotics of non-planar graphs, and still have a test of the factorization approach,
we were not able to avoid them here. In fact, this additional complication is due to the single heavyquark loop diagrams of Fig. 1, which are explicitely removed from the factorization approach. The
complete set of non-planar master integrals belonging to this class is depicted in Fig. 3.
×5
×4
Figure 1: Most complicated diagrams of the
single heavy quark loop contribution. The
thick lines are massive.
×2
kµ
1
Figure 3: Non-planar master integrals corresponding to the second diagram of Fig. 1.
The numbers denote the multiplicities of the
integrals of the given topology. All of the
required MB representations can be derived
from the MB representation of the integral
Fig. 2.
5
3
4
2
7
6
Figure 2: Definition of the momentum in the
numerator of the non-planar integral considered in the text, together with the labeling of
the denominators.
The first problem that one has to face when dealing with non-planar integrals is the construction
of MB representations. In the planar case, the iterative loop-by-loop integration has proved to be
the most fruitful. On the other hand, the first non-planar double-box diagram ever computed [45],
6
with massless propagators and on-shell external legs, had its four-dimensional MB representation,
m=0, dim=4
INP
, derived directly from the two-loop Feynman parameter representation. It seems, however, that when masses are involved, the loop-by-loop representations are more compact, as seen
for example in Ref. [49, 50]. However, any asymptotic expansion contains a so-called hard part,
which is obtained by setting all the small parameters to zero, and which in our case would correspond to the massless graph. Following this line of thought, one can derive a representation for
the massless on-shell graph by taking suitable residues in the result presented in Ref. [49]. Unm=0, dim=6
fortunately, one arrives at a six-fold representation, INP
, in clear disadvantage with respect
m=0, dim=4
to INP
. Interestingly, there is an even more severe problem inherent in the loop-by-loop
approach. The leading pole derived in Ref. [45] reads (up to normalization factors irrelevant for
this discussion)
1
2
m=0, dim=4
(17)
INP
= 4 +O 3 ,
ε stu
ε
whereas the six-fold representation gives (with the same normalization)
5
1
m=0, dim=6
INP
= 4 +O 3 .
(18)
2ε stu
ε
This clear discrepancy is only explained when we look at the subleading pole from the six-fold
representation, which contains a logarithmic singularity
1
1
m=0, dim=6
(19)
INP
|u→−s−t ≈ − 3 log(−s − t − u) + O 2 .
ε stu
ε
Obviously, the extension of the integral into the Euclidean domain, performed with the help of
the u parameter, regularized part of the infrared singularity. The only way to obtain the correct
result would be to first take the limit u → −s − t, and only then ε → 0. This, however, is a highly
non-trivial task, as is well known from studies aiming at the derivation of exact expressions of
Feynman integrals in d-dimensions.
In view of all the above arguments, we derived our MB representations directly from the twoloop Feynman parametric representation. In particular, for the scalar integral of Fig. 2, we have
the following six-fold representation
INP = −(−s)
−2ε−3
Z i∞
6
∏
−i∞ i=1
s z1 t z2 u z3
dzi − 2
− 2
− 2
m
m
m
×Γ(−z2 )Γ(−z3 )Γ(−z4 )Γ(−z5 )Γ(−z6 )Γ(−2z1 − z2 − z3 − 2z4 + 1)Γ(−ε − z4 )2
×Γ(z1 + z2 + z3 + z4 )Γ(−ε + z1 + z2 + z3 + z4 − z5 − 1)Γ(−2ε + z1 − z6 − 2)
×Γ(−2ε + z1 + z2 − z5 − z6 − 2)Γ(−2ε + z1 + z3 − z5 − z6 − 2)
×Γ(z2 + z6 + 1)Γ(z3 + z6 + 1)Γ(z5 + z6 + 1)Γ(2ε − z1 + z5 + z6 + 3)
−1
,
× Γ(−2ε − 2z4 )Γ(−3ε − z4 − 1)Γ(−2ε + z1 + z2 + z3 − z5 − 1)2
(20)
where the loop integration is done with the measure eεγE d d k/(iπd/2 ) per loop. We defer the
presentation of the full result of the expansion of this integral to the Appendix A. Here we only
R
7
note, that the leading term of the expansion has a square root singularity, which is a feature of
non-planar graphs, that does not occur in any of the planar integrals considered in this calculation.
Clearly, the disappearance of this square root singularity is a simple test of the correctness of the
calculation.
The second problem that requires care is connected with the choice of the master integrals. In
Fig. 3, we have not only shown the topologies, but also the multiplicities of the masters. The basic
seven-liner needs as much as five different integrals. It is clear that we want to avoid coefficients
containing poles in ε or m2 , since the leading behavior of the integrals is difficult enough to determine, and such poles would be synonymous of higher orders in the respective expansions. After
inspection it turns out that one can take two integrals with second powers of the denominators into
the set, but tensors rank one and two are unavoidable. Although it is possible to generate representations for arbitrary tensors, one ends up with a huge number of four-fold integrals after expansion.
Instead, one can introduce a new ficticious propagator that will have a negative power. For this we
choose the square of the momentum that runs through the crossed box subloop, as shown in Fig. 2.
m2
s
m2
s
k2
s
k4
s2
Figure 4: A suitable choice of integrals that forms together with the underlying scalar integral the
set of five master integrals of the seven line non-planar topology. The momentum kµ has been
defined in Fig. 2, whereas a dot on a line denotes a squared propagator.
The final set of seven-line non-planar master integrals is shown in Fig. 4. Note that the dotted
masters, i.e. those having higher powers of chosen propagators are particularly easy to calculate,
because one only needs the singularities in 1/m2 . All of the representations can be derived from
the following one
a
I NP = (−1) (−s)
−a−2ε+4
Z i∞
8
∏
−i∞ i=1
dzi s z1 t z2 u z3
− 2
− 2
− 2
Γ(ai )
m
m
m
×Γ(−z3 )Γ(−z4 )Γ(−z5 )Γ(−z6 )Γ(−z7 )Γ(−z8 )Γ(−z2 + z4 + z5 )Γ(z1 + z2 + z3 + z6 )
×Γ(a + 2ε − z1 + z7 + z8 − 4)Γ(z2 + z3 − z4 − z5 + z8 + a1 )Γ(z5 + z8 + a4 )Γ(z7 + z8 + a6 )
×Γ(−2z1 − 2z2 − 2z3 + z4 + z5 − 2z6 + a7 )Γ(z2 − z5 + a8 )Γ(−ε − z6 − a13 + 2)
×Γ(−ε − z6 − a24 + 2)Γ(−2z1 − z2 − z3 + z4 − 2z6 + a78 )
×Γ(−ε + z1 + z2 + z3 − z4 + z6 − z7 − a5678 + 2)Γ(−2ε + z1 − z8 − a1234678 + 4)
8
×Γ(−2ε + z1 + z5 − z7 − z8 − a1245678 + 4)
×Γ(−2ε + z1 + z2 + z3 − z4 − z5 − z7 − z8 − a1345678 + 4)
× Γ(−a − 3ε − z6 + 6)Γ(−2z1 − z2 − 2z3 + z4 − 2z6 + a78 )Γ(−2ε − 2z6 − a1234 + 4)
×Γ(−2ε + z1 + z2 + z3 − z4 − z7 − a135678 + 4)
×Γ(−2ε + z1 + z2 + z3 − z4 − z7 − a245678 + 4)
−1
,
(21)
where a = ∑8i=1 ai , aS = ∑i∈S ai with S a subset of 1, ..., 8, and ai with i = 1, ..., 7 are the powers of
the denominators according to the labeling given in Fig. 2, whereas a8 is the power of the additional
denominator 1/k2 . Even though Eq. (21) has two more integrations than Eq. (20), the presence of
the factor 1/Γ(a8 ) makes it necessary to perform first an analytic continuation in a8 to a negative
integer value, which effectively reduces the number of integration variables back to six. In fact,
the above representation can be used to compute any of the master integrals from the full set of
non-planars of Fig. 3.
To complete our exposition of the direct calculation of the amplitude, we have to note that the
renormalization of the bare amplitude requires the on-shell wave function renormalization constant
of the gluon, which is non-vanishing due to the presence of heavy-quark loops. We give it here as
an expression exact in d-dimensions,
2
4
2
b
b
+ as nh TF nh TF
Z3 = 1 + as nh TF −
3ε
9ε2
−4ε5 + 15ε3 + ε2 − 11ε − 3
4ε3 − 7ε − 1
+CA
, (22)
+CF
ε (4ε3 − 8ε2 − ε + 2)
2ε2 (4ε4 − 4ε3 − 13ε2 + 7ε + 6)
where abs is defined by the bare coupling constant αbs and with Sε = (4π)ε exp(−εγE ), and
ε
αbs µ2
b
as =
eεγE Γ(1 + ε) Sε .
2π m2
(23)
In terms of the renormalized coupling αs and expanding in powers of ε through sufficient terms,
the result for Z3 reads
2
2
n 2 2
α 1
π2
1 2 3 µ2
µ
s
2 µ
n T − − log
− ε log
− ε − ε log
Z3 = 1 +
2π h F
3ε 3
m2
3
m2
18
9
m2
n
h4
π2 2
µ2
2 2 o αs 2
µ2
2 2 µ2
π2 i
− ε log
+
ε
ζ
+
log
+
log
+
n
T
n
T
3
h F
h F
18
m2
9
2π
9ε
m2
3
m2
27
h 1
h 4
2 2 µ2
π2 i
15 i
4
µ2
µ2
−
+C
−
−
−
log
−
log
+nl TF − 2 − log
F
9ε
9ε
m2
9
m2
27
2ε
m2
4
h 35
io
2
2
2
2
5 5
1
13 13π
13
µ
µ
µ
+CA
. (24)
− − log
+ log2
+ +
+
log
2
2
2
2
36ε
18ε
m
8ε 4
m
9
m
48 216
The corresponding expression for the wave function renormalization constant Z2 of a light quark
state has been given in Ref. [22].
9
(1)
Our result for Z3 in Eqs. (22) and (24) coincides to first order in αs with Z[g] from Eq. (14).
At second order in αs all known terms (i.e. those quadratic in the number of flavors) in the two
constants are also a complete match (also in d-dimensions). In our direct evaluation of Z3 in
Eq. (22) we observe gauge independence through two loops (within the class of covariant gauges
employed in the calculation), which is consistent with the arguments given in Ref. [25] in favor of
(m|0)
the identification of the constant Z[g] with Z3 evaluated in a physical gauge.
4
Results
We are now ready to present our result for gg → QQ̄ scattering for the interference of the two-loop
and Born amplitude,
1
1
(25)
2Re hM (0) |M (2) i = (N 2 − 1) N 3 A + NB + C + 3 D + N 2 nl El + N 2 nh Eh
N
N
n
n
n2
nn
n2 +nl Fl + nh Fh + l2 Gl + h2 Gh + Nnl 2 Hl + Nnl nh Hlh + Nnh 2 Hh + l Il + l h Ilh + h Ih ,
N
N
N
N
N
which we have ordered according to the power of the number of colors N and the numbers of nl
light and nh = 1 heavy quarks with n f = nl + nh total flavors.
The coefficients A, El , Hl , Hlh , Hh , Il , Ilh, and Ih have been computed with both of our methods.
We have found agreement between the direct computation of the relevant Feynman diagrams in the
small-mass expansion and the QCD factorization approach as given by the universal multiplicative
relation (12). All other terms linear in nh , that includes Eh , Fh and Gh have been obtained by means
of a direct calculations of the massive loop integrals as detailed above. The remaining coefficients
B, C, D, Fl and Gl have been derived by application of the factorization formula.
We choose x = −t/s as the only dimensionless kinematic variable in the problem and we keep
the dependence on the renormalization scale, µ, explicit. We also introduce the following compact
notation
2
s
m
, Ls = log 2 , Lx = log (x) , Ly = log (1 − x) .
(26)
Lm = log
s
µ
The different components now read
A =
2
2
2
1
2
4
2
2
−8+
−8+
−8x + 8x +
+ 3 Lm −8x + 8x +
+ Ls 16x2 − 16x −
1−x
x
ε
1−x
x
1−x
2
4
4
154x
142
154x
65
4
+
+ Ly 8x2 −
+ 4 + Lx 8x2 − 16x + 12 −
−
−
+
+16 −
x
3
3
1−x
x
6(1 − x)
3
65
1
4
1
1
4
+
+2−
+ 16 −
+ 2 L2m 2x2 − 2x −
+ Lm Ls 16x2 − 16x −
6x
ε
2(1 − x)
2x
1−x
x
4
4
2
− 16 +
+2
+L2s −16x2 + 16x +
+ Lm −22x2 + 22x + Ly 4x2 −
1−x
x
1−x
1
ε4
10
2
7
7
8
2
2
+Lx 4x − 8x + 6 −
− 18 +
−8
+
+ Ls 44x − 44x + Ly −16x +
x
2(1 − x)
2x
1−x
7
7
3
8
2
2
+ 36 −
−
+ −4x + 12x − 9 +
L2x
+Lx −16x + 32x − 24 +
x
1−x
x
x
8
104x2 104x
3
+ 18x2 − 41x + 31 −
+
+ L2y −4x2 − 4x +
−1
Lx −
x
9
9
1−x
2
161
161 100
8
2
2
2
2
+8−
+8 −
−
+
+ Ly 18x + 5x −
+π 8x − 8x −
1−x
x
1−x
18(1 − x) 18x
9
2
1
2x
1
2
1
1
2x
1
+
− −
+ −
−4+
L3m
+ L2m Ls −4x2 + 4x +
+ Lm L2s −16x2
ε
3
3
6(1 − x) 3 6x
1−x
x
2
4
32x
8
32
8
32x
22x2 22x
4
− 16 +
−
−
+ −
−
+ L3s
+ L2m
+16x +
1−x
x
3
3
3(1 − x)
3
3x
3
3
1
19
4
4
44x2 44x
1
2
2
+Ly −2x +
− 1 + Lx −2x + 4x − 3 +
+ −
−
−
+ Lm Ls
1−x
x
3(1 − x)
3
3x
3
3
2
20
1
1
4
44x
4
− 4 + Lx −8x2 + 16x − 12 +
+ +
+
+ L2s −
+Ly −8x2 +
1−x
x
3(1 − x)
3
3x
3
44x
8
8
20
1
1
+
+ Ly 16x2 −
+ 8 + Lx 16x2 − 32x + 24 −
− −
−
3
1−x
x
3(1 − x)
3
3x
3
1
3x
140x2
1
3 1
2
2
2
+ Ly −2x +
+
+Lm 2x − +
Lx + −2x + − 1 +
Lx +
2 2x
2
2x
9
2(1 − x) 2
140x
3
3
1
80
80
5x
1
2
2
2
−
+ Ly −2x + +
−
+2−
−
+ π 2x − 2x −
−
9
2
2(1 − x) 2
2(1 − x)
2x
9(1 − x) 9x
2
2
70x
4
496x
20x
496x
6
451
+
− 18 +
+
+ Ls 8x2 − 24x + 18 −
L2x + −
Lx −
+
18
x
3
3
3x
9
9
2
4
4
4
4
20x
− 16 +
− 10x +
−
+π2 −16x2 + 16x +
+ Ly −
1−x
x
3
3(1 − x) 3
205
205 640
1
4x2
6
2
2
+2 +
+
−
− 4x + 3 −
+ −
L3x
+Ly 8x + 8x −
1−x
9(1 − x)
9x
9
3
x
29x
5
2
1
2
2
2
2
+ 4x +
+ Ly 4x − 4x + 3 −
− 10 +
Lx + Li2 (x) 8x − 8x + 6 −
Lx
6
x
6x
x
332x2 514x
11
34x2 44x
116 166
2584x2 2584x
+ −
+
+ π2 −
+
− 11 +
+
−
−
Lx +
9
9
3
3
3x
3
9x
27
27
2
20x
1
7
5
7
2
4x
77x
2
3
2
2
+
−
−
+
−
+Li3 (x) −8x + 8x − 6 +
+ Ly −
+ Ly 4x −
x
3
3
1−x 3
6
6(1 − x) 6
1
1
332x2 50x
2
2
2
2
+ 6 + π 19x − 19x +
+4+
+
+ Ly −
+S1,2 (x) 8x − 8x −
1−x
4(1 − x)
4x
9
3
2
2
11
23
166
184x
40
178
184x
166
34x
+ 8x +
−
−
−
−
+
+π2 −
+
+
3
3(1 − x)
3
9(1 − x)
9
3
3
3(1 − x)
3
46
625 2416
5
5
5
625
5x2 5x
−
−
+
+ +
− +
ζ3 −
+ L4m −
3x
27(1 − x) 27x
27
6
6
24(1 − x) 6 24x
2
4x
1
4
1
1
1
4x
3
2 2
2
+ +
− +
+4−
+Lm Ls −
+ Lm Ls 4x − 4x −
3
3 3(1 − x) 3 3x
1−x
x
2
11
8
32
8
4
16
4
16x2 16x
32x2 32x
4
−
−
+ −
+
+
− +
+ Ls −
3
3
3(1 − x)
3
3x
3
3
3(1 − x)
3
3x
2
2
2x
1
1
4x
1
1
1
1
2x
+L3m Ly
−
+
− +1−
+ −
+ Lx
+
3
3(1 − x) 3
3
3
3x
6(1 − x) 6x 3
1
1
2
2
2
2
2
+ 2 + Lx 4x − 8x + 6 −
− +2
−
+Lm Ls Ly 4x −
1−x
x
1−x x
4
4
4
4
2
2
2
+Lm Ls Ly 8x −
+ 4 + Lx 8x − 16x + 12 −
− +8
−
1−x
x
1−x x
16
16
16
8 16
8
32x2 64x
32x2
3
+
−
+
− 16 +
+ −
+ Lx −
+
+Ls Ly −
3
3(1 − x)
3
3
3
3x
3(1 − x) 3x
3
2
3
1
391x
1
3x 1 3
391x
1
2
2
2
2
+Lm −x + −
+
+ Ly x −
−
Lx + x − + −
Lx −
4 4x
4
2 4x
18
18
4(1 − x) 4
1
1
3
3
505 887
505
5x
2
2
2
+π −x + x +
−1+
+
+
−
+ Ly x − −
+
4(1 − x)
4x
4 4(1 − x) 4
72(1 − x) 72x
36
2
148x
3
148x
1
1
+
+ L2y 4x −
−1
L2x + 4x2 − 3x + 2 −
Lx −
+Lm Ls −4x + 3 −
x
x
9
9
1−x
1
1
61
61 187
3
+π2 −4x2 + 4x +
−4+
+3 +
+ −
+ Ly 4x2 − 5x −
1−x
x
1−x
9(1 − x) 9x
9
2
364x
6
364x
6
−
+L2s −8x2 + 24x − 18 +
L2x + −8x2 + 6x − 4 +
Lx +
x
x
9
9
4
6
4
6
2
2
2
2
2
+Ly −8x − 8x +
− 2 + Ly −8x + 10x +
− 6 + π 16x − 16x −
+ 16 −
1−x
1−x
1−x
x
x
106 376
1
9
3
2 2 3
106
−
+
+ Ly 2x2 − 2x + −
+ Lm − x Lx +
−
L2x
−
9(1 − x)
9x
9
3
4
2 2x
4
61x
3 1
29 21
1165x2
1
2
2
2
2
+ π −2x + 2x − +
Lx + 6x −
+ −
Lx −
+Li2 (x) 4x − 4x + 3 −
x
4
2 2x
2
4x
54
x
1165x
2
1
4x 2
2x
+L2y − − 2 +
+ Li3 (x) −4x2 + 4x − 3 +
+ −
+ L3y −
4
54
x
3
3
3
2
17x
1
23
1
1
13x
2 17x
2
+3 +π
−
−
+ −
+S1,2 (x) 4x − 4x −
+ Ly 6x2 +
1−x
3
3
6(1 − x) 12 6x
4
2
1
3
21
8x
5
11
2
21
8x
2
2
+π −2x + 2x +
−
+
+ +
− +
−
+ −
ζ3
2(1 − x) 2
4(1 − x)
4
3
3 3(1 − x)
3
3x
2
2203 3521
2
8x
2203
+
−
+ 8x − 6 +
+ Ls
L3x + −8x2 + 5x
+
216(1 − x) 216x
108
3
x
400x2 665x
2
4
2
2
2
2
−
+9+
Lx + Li2 (x) −16x + 16x − 12 +
Lx +
+Ly −8x + 8x − 6 +
x
x
x
9
9
2
2
88x
22
1040x
167
1040x
68x
−
+ 22 −
+
+π2
+ 48 −
Lx −
3
3
3x
9x
27
27
4
119
148 119
122x2 122x
+S1,2 (x) −16x2 + 16x +
− 12 + π2 −
+
−
+
−
1−x
9
9
18(1 − x)
9
18x
2
40x
2
14
4
2
3 8x
2
2
2
+ 6 + Ly
−
+
+
+Ly −8x + 11x +
+ Li3 (x) 16x − 16x + 12 −
1−x
3
3
1−x
3
x
+Lm L3s
12
2
22
46
167
167
400x2
2 68x
− 15x + π
− 16x −
+
+
−
+Ly
9
3
3(1 − x)
3
9(1 − x)
9
2
368x
368x
80
356 92
211 55
211
+ −
+
+
−
+
−
+
ζ3 −
3
3
3(1 − x)
3
3x
27(1 − x) 27x 27
2
1
17x
2
3
2x
13x2 x 1
26
22x2
4
+ − +
−
+ Ly −
+ 8x − 6 +
Lx +
+ −
L3x
+
6
6 8 24x
3
12
3
x
3
4x
9
31
15
59x
265x
38x 19 19
+1+
+ π2 12x2 −
+ −
+
2x2 + 2x − +
L2y + −2x2 −
Ly −
2
2x
12
12x
72
3
2
6x
2
2
91
17
3379x
8x
2
5
262x
20x
−
−
+
+ L y π2 −
+ +2−
L2x + −
−
1 − x 72x 12
27
216
3
3
x
2
2
16
272x
68
172x
97 2279
1246x2
17x
2
+ 4x + 10 −
+
− 68 +
+ −
ζ3 − −
Lx −
+π −
9
3x
3
3
3x
18 216x
81
2
18
18
422x
589
3581 589
1246x
422x
2
+S2,2 (x) −32x +
+ 16 −
+π −
+
+
−
+
+
1−x
x
81
27
27
54(1 − x)
108
54x
17x2 7x
173
121
23
3
− 1 + π4
+ −
+
−
+S1,3 (x) −4x2 − 4x +
1−x
30
30 360(1 − x) 180 360x
2
2
x
3
95
9x
1
53
3
3 2x
4 13x
2
+ −
+
− +
+
+Ly
+ Ly
+ Li4 (x) 4x − 12x + 9 −
3
12 4(1 − x) 12
6
2
24(1 − x) 24
x
107x
5
18
31
+S1,2 (x) −4x2 +
+ Ly −16x2 + 12x +
− 11 + Lx 8x2 + 8x − 30 +
+
6
1−x
x
6(1 − x)
71
19
53
367
5
265x
34x
91
−
+ π2 12x2 −
−
+
−
−
+ L2y
−
6
72
3
6(1 − x)
6
72(1 − x)
72
x
31
18
2
59x
2
2
2
+ Ly −8x − 8x + 30 −
+ Lx 12x − 8x + 6 −
−2−
+Li3 (x) 4x +
6
x
x
6x
21
7
59x
31
18
2
2
2
2
+Li2 (x)
−14x + 14x − +
+ Ly 8x + 8x − 30 +
Lx + −4x −
+2+
Lx
2
2x
6
x
6x
2
23
89 9
395x
262x2 271x
2
2
2
+4−
+
+ +
+
+ 52x −
ζ3 + L y −
+π −8x +
1−x
x
6
6(1 − x)
6
x
27
72
2
2
17x
2x
16
109
53
161 18
100x
+π2 −
− −
+
+ 20x +
−
+
+ −
ζ3
9
9
3(1 − x)
9
3
3(1 − x)
3
x
119
16313 39113
2279
16313
+
+
−
,
(27)
−
+
216(1 − x) 216
648(1 − x)
648x
648
B =
2
4
4
2
1
4
4
2
+ 12 −
−8+
− +4−
+ 3 Lm 8x − 8x −
+ Ls
+ 8x2 − 8x
x
1−x
ε
1−x
x
x
1−x
4
4
4
101 77
1
1
77
1
4
+Lx
−8+
−8+
+
−
−1+
+ Ly
−
+ 2 L2m
x
1−x
x
1−x
6(1 − x)
3
6x
ε
2x
2(1 − x)
8
4
4
8
− 24 +
+ L2s − + 8 −
+ Lm 26x2 − 26x
+Lm Ls −16x2 + 16x +
1−x
x
x
1−x
2
4
8
8
4
2
+Ly −4x2 +
−6+
− 10 +
+ 33 −
+ Lx −4x2 + 8x +
−
1−x
x
1−x
x
1−x
x
8
8
11
11
8
8
2
− 38 +
+Ls −16x + 16x + Lx − + 16 −
+ Ly − + 16 −
+
x
1−x
x
1−x
1−x
x
1
ε4
13
2
8
13
5
8
2
2
2
2
+ −8x + x + 1 −
+ 24 −
− 33 +
Lx + −4x + 9x + Ly 16x − 16x −
+
Lx
x
1−x
x
1−x
x
1
16
1
2
28x2 28x
2
2
2
2
+
+
− +
−6
+ Ly −8x + 15x −
+34x − 34x + π −
3
3
3(1 − x)
3
3x
1−x
5
31 313
5
31
1
13
8x2 8x
2
− 28 +
+ +
+ +
−3
+Ly −4x − x +
+
+
L3m −
1−x
x
9(1 − x) 9x
9
ε
3
3
6(1 − x)
5
1
8
16
8
1
8
2
2
3 8
2
+
+ 24 −
− +
+ Lm Ls − + 2 −
+ Lm Ls 16x − 16x −
+ Ls
6x
x
1−x
1−x
x
3x
3
3(1 − x)
2
28x
1
2
11
2
1
28x
+
+ Ly 2x2 −
+3−
+5−
− 13
+ Lx 2x2 − 4x −
+
+L2m −
3
3
1−x
x
1−x
x
3(1 − x)
11
8
4
8
68x2 68x
4
2
2
+
+
+ Ly 8x −
+ 12 −
+ 20 −
+ Lm Ls −
+ Lx 8x − 16x −
3x
3
3
1−x
x
1−x
x
4
4
8
8
11
8
8
+
− 22 +
− 16 +
− 16 +
+ L2s 16x2 − 16x + Lx
+ Ly
−
3(1 − x)
3x
x
1−x
x
1−x
3(1 − x)
1
1
2
3
3
3x
2
70 11
+2−
−
+ Lm − −
L2x + 2x2 − x + Ly − + 4 −
−
+ −
3
3x
2
1−x
2x
x
1−x
1−x 2
2
2
3
86x
1
1 1
14x
5
17
5
3x
86x
14x
−
+
+ L2y
−
+ −
+
+
− +
Lx −
+ π2 −
2x
9
9
2 2(1 − x) 2 x
3
3
3(1 − x)
3
3x
1 3
160 88
160
4
3
− −
+
−
+
+ Ls 16x2 − 2x − 2 +
L2x
+Ly 2x2 − 3x −
2(1 − x) 2 x
9(1 − x)
9x
3
x
16
16
110 34
14
116x2 116x
2
2
+ 8x − 18x + Ly −32x + 32x +
− 48 +
+
−
+
+
Lx −
1−x
x
3(1 − x)
3
3x
3
3
80
14
4
34
+ +
+ 12
+ L2y 16x2 − 30x +
+Ly 8x2 + 2x −
3(1 − x)
3
3x
1−x
2
56x
2
32
2
172 10
8
1
2
172
23x
2 56x
−
−
+ −
−
−
+
− −
+π
−
+ −
L3x
3
3
3(1 − x)
3
3x
9(1 − x)
9x
9
3
3(1 − x) 3 3x
4
19x
5
82 11
20
8
2
2
+ 8x −
+ Ly 23x −
− 14 +
+ −
+ 10
−
Lx + Li2 (x) 2x −
6
1−x
x
3(1 − x)
3
2x
1−x
4
11
46 11
7
95x
−
+ 3 L2y + −16x2 + 16x +
− +
Lx +
−22x +
Ly − 4x2 +
x
1−x
3(1 − x)
3
3x
6
2
2
14x
11
31
179 10
238
184x
20x
+
−
+
−
−
+π2
−
Lx +
3
3
3(1 − x)
3
9(1 − x) 18x
9
9
4
8
4
8
184x
+S1,2 (x) −2x −
+ 12 −
− 10 +
+ Li3 (x) −2x +
−
1−x
x
1−x
x
9
2
2
8
226x
259
121 259
226x
3 23x
2
−
−8+
+
+
−
+
+Ly
+π −
3
3(1 − x)
3x
9
9
36(1 − x)
9
36x
2
77x
11
193 20
65 11
47x
2
2
2 20x
2
+Ly 8x −
−
+
−
+π
− 18x + −
+ Ly −4x −
6
2(1 − x)
6
3x
6
3
3
3x
179
193 238
158 67
820 2569
55
820
−
+
−
−
+
+
−
+ −20x2 + 22x +
ζ3 +
18(1 − x)
18
9x
3(1 − x)
3
3x
27(1 − x) 27x
54
2
29
17
16x
5
5
16x
17
+ −
−
−
+6−
+L4m 2x2 − 2x −
+ L3m Ls
24(1 − x) 12 24x
3
3
3(1 − x)
3x
14
1
16
16
8
4
32x2 32x
4
1
3
4
−2+
+
+
− 16 +
+ Lm Ls −
+ Ls − + −
x
1−x
3
3
3(1 − x)
3x
3x 3 3(1 − x)
2
2
2
2x
2
1
4x
1
5
2
1
2x
2x
3 2x
+Lm
− + Ly −
+
−1+
+ +
− +
+ Lx −
−
3
3
3
3(1 − x)
3x
3
3 3(1 − x) 3 3x
1−x
2
4
2
4
1
2
2
2
2
−6+
− 10 +
+ Lm Ls 4x − 4x + Ly −4x +
+ Lx −4x + 8x +
+4
+2 −
x
1−x
x
1−x
x
8
4
8
6
4
6
2
2
2
2
+Lm Ls 8x − 8x + Ly −8x +
− 12 +
− 20 +
+
+ Lx −8x + 16x +
+
1−x
x
1−x
x
1−x x
16
16
32
16 32
16 32
32x2 32x
3
+
+ Lx − + −
+ Ly − + −
−
+Ls −
3
3
3x
3
3(1 − x)
3x
3
3(1 − x)
3
3x
1
1
1
3 3
1
x
3
+L2m
+
−1+
−2+
+ +
L2x + −x2 + + Ly
+
Lx
4
2(1 − x)
4x
2
x
1−x
2(1 − x) 4 4x
227x2
1
1 1
3
1 3
3x
227x
3x
2
2
+
+ Ly − +
− +
+ Ly −x + +
+ +
−
9
4
4(1 − x) 4 2x
9
2
4(1 − x) 4 2x
2
8x
11
19
11
1127 161
1127
2 8x
− −
+ −
−
+
−
+π
3
3
12(1 − x)
6
12x
72(1 − x)
72x
4
1
4
3
6
40x2
4
2
2
2
−4+
−8+
+3+
Lx + −4x + 2x + Ly
+
Lx +
+Lm Ls 3x +
1−x
x
x
1−x
1−x
x
9
1
2
6
40x
3
+L2y −3x +
−1+
+ Ly −4x2 + 6x +
+1+
−
1−x
x
9
1−x
x
2
28x
10
34 10
122
122
4
2
2
2 28x
−
−
+ −
−
+ 22 + Ls −16x + 2x + 2 −
−
L2x
+π
3
3
3(1 − x)
3
3x
9(1 − x)
9x
x
16
4
12
16
2
2
+ −8x + 18x + Ly 32x − 32x −
+ 48 −
− 22 +
−
Lx + 24x2 − 24x
1−x
x
1−x
x
2
32
2
4
56x2 56x
2
2
2
+
+
− +
− 12
+ Ly −16x + 30x −
+π −
3
3
3(1 − x)
3
3x
1−x
2
12
106 56
x
1
1
106
2x
4
2
− 12 −
+
−
− +
+
+
+ Lm
+Ly −8x − 2x +
1−x
x
9(1 − x)
9x
9
3
3 3(1 − x) 3
1
1
1
31
3
5x
1
5x
3
2
−
−1+
+ −
Lx + − + Ly −2x + −
+
L2x
3x
4
2 1−x
2x
2(1 − x)
4
2x
4
1
1
3
1
3
2
2
+Li2 (x) −4x + 5x −
+2−
−2+
−8+
Lx +
Ly +
Ly − 6x2
1−x
x
x
1−x
x
1−x
3
47 109
4
1028x2 1028x
5x
35x
2
2
+ π 2x − + 3 −
+ −
−
+
Lx +
+
2
2
2x
1 − x 4x
4
27
27
2
3
13
1
17x
13
23
13
17x
2 5x
2
+Ly
−
+ +
+
−
+ −
+π −
4
2(1 − x)
2
2x
3
3
12(1 − x)
6
12x
2
4
1
2 1
2x
1
+3−
−x−
+ +
+ L3y
+S1,2 (x) −4x2 + 3x −
1−x
x
3
3(1 − x) 3 3x
4
1
3
5
3x
11x
2
2
2
2
+Li3 (x) 4x − 5x +
−2+
+ π 2x − −
+
+ Ly −6x −
1−x
x
2
2 2(1 − x) 2
63 4
14
23
4703 1997
88x2 91x
4703
47
− +
+
+
− 28 +
−
+
+ −
ζ3 −
+
4(1 − x)
4
x
3
3
3(1 − x)
3x
216(1 − x) 216x
36
+L2m L2s
15
46x
16
2 4
10
6
8
2
3
+Ls
−
+ +
+ 28 −
Lx + −16x + 10x + Ly −46x +
+
3
3(1 − x) 3 3x
1−x
x
1−x
8
16
14
11
− 20 +
− 6 L2y + 32x2 − 32x
L2x + Li2 (x) −4x +
Lx +
44x −
−51 +
x
1−x
x
1−x
22
22
22
62
113 13
212
40x2 28x
−
+ 60 −
−
+
−
+
−
Ly + 8x2 − 28x + π2 −
+
Lx
1−x
x
3
3
3(1 − x)
3
9(1 − x)
9x
9
4
16
16
8
46x
104x
104x2
3
+ Ly −
+
+ 16 −
+ Li3 (x) 4x −
+ 20 −
+
−
9
3
3(1 − x)
3x
9
1−x
x
16
6
92x2 92x
11
8
− 24 +
− 57 +
−
+ L2y −16x2 + 22x +
+ π2
+S1,2 (x) 4x +
1−x
x
1−x
x
3
3
2
61
44
61
130 22
193
113
40x
−
− −
+ 36x −
+
−
+ Ly 8x2 + 12x + π2 −
+
18(1 − x)
9
18x
3
3
3x
9(1 − x)
9
2
25x
316 134
80
802
57x
110
80
212
+
−
−
+
+
+ 40x2 − 44x −
ζ3 −
+
+
9x
3(1 − x)
3
3x
27(1 − x) 27x
27
6
8
2
2
50x
5
9
2x
37x
5x
2
1
8
19
+ +
+
+ Ly −
+ +
−4+
−
L4x +
−
3(1 − x) 12 8x
3
2
3
3 1−x
3x
2(1 − x)
121 23
5
22
385 11
9
155x
3
2
2
2
−
+
+5+
+
+
−
Lx +
8x − 27x +
Ly + −2x −
Ly
12
4x
2(1 − x)
x
3
3(1 − x)
12
4x
23
5
4
83
1
44x2 4x
695
1093
x2 539x
+ π2
+ −
+ −
− −
+
+
L2x
+ −
2
36
3
3
6(1 − x) 6 3x
36(1 − x) 8x (x − 1)2
72
8x2 61x
395x
11
8
2
17
307 89
3
+
−
+
−
− −
−
−
+
Ly +
L2y + −x2 + x
3
3
3(1 − x) 3 3x
6
2(1 − x)
6
6x
7
100 41
142 194
194
749x
88x2 151x
2
+
+
−
+
−
+
+
Ly − 8x2 +
+π −
3
3
3(1 − x)
3
3x
9(1 − x)
9
9x
18
2
221x
73
19 101
214 92
80
16x
+
+
− −
+
−
+ −8x2 − 16x −
ζ3
+π2
3
18
18(1 − x)
6
18x
3(1 − x)
3
3x
839
3457
78
78
308x2 308x
430
+
−
−
+ S2,2 (x) 40x +
− 20 −
Lx +
+
27(1 − x) 216x
216
27
27
1−x
x
2
74
9
119x
23
109 19
2
3 2x
+Li4 (x) −36x + 37x −
+ 96 −
−
+
+
−
+ Ly
1−x
x
3
6
4(1 − x)
12
2x
2
2
268x
557
347
7
371x
9
281
8
4 43x
4 25x
−
+
−
+
−
+
+
−
+π
+ Ly
18
45
180(1 − x)
90
10x
6
24
8(1 − x)
24
3x
9
74
1
+S1,3 (x) 36x2 − 35x +
− 97 +
+ π2 − (432 log(2) + 551)x2
1−x
x
54
1
1
2(27 log(2) − 332) 2(27 log(2) − 332)
+ (432 log(2) + 551)x + (1505 − 288 log(2)) +
+
54
36
27(1 − x)
27x
2
2
503x
92x
4
101 23
347 1093 1
83
2 44x
2 x
+
+π
−
−
+
−
−
+
− 2
+Ly
−
2
36
3
3
3(1 − x)
6
6x
8(1 − x)
72
36x
x
61x
30
82
11
34
+S1,2 (x) −4x2 −
+ Ly −16x2 + 11x +
− 65 +
− 38 +
+ Lx 16x2 − 8x −
3
1−x
x
1−x
x
23
355 15
44
85x
4
+
+
−
+ Ly −16x2 + 64x −
+ 10 −
+ Li3 (x) 4x2 −
2(1 − x)
6
x
3
1−x
x
16
44
2
209 23
7
15
79x
+Lx 52x − 58x +
− 26 −
−
−
−
+
+ Li2 (x)
−34x2 +
1−x
x
1−x
6
2x
2
1−x
4
44
209 23
85x
15
13
2
2
2
+ Ly 16x − 64x +
− 10 +
+
+
Lx + −4x +
−
Lx
−22 +
2x
3
1−x
x
1−x
6
2x
34
34
69
51 8
64x2 137x
+π2 42x −
− 21 +
+
−
+ −
+ −
ζ3
3(1 − x)
3x
3
3
2(1 − x)
2
x
2
413x
101
130
73
113
461x
2 16x
2
+π
−
−
+
+
+ 24x2 − 43x −
+Ly −8x −
18
3
18
18(1 − x)
9
18x
3(1 − x)
307 38
3803 430
15317 3353
839
15317
+
−
+
+
−
+
,
ζ3 +
−
3
3x
216(1 − x)
216
27x
648(1 − x)
648x
324
C =
2
2
1
2
1
2
1
2
2
−4+
−2+
Lm
+ −4+
+ 2 Lm −2x + 2x +
x
1−x
x
1−x
ε
2(1 − x)
2x
4
4
2
2
2
2
2
+Lm Ls − + 8 −
+ Lm −4x + 4x + Lx − + 4 −
+ Ly − + 4 −
x
1−x
x
1−x
x
1−x
11
11
4
2
2
1
5
4
2
+
− 17 +
+ Ls − + 8 −
+ − +1−
Lx + − + 5 −
Lx
2(1 − x)
2x
x
1−x
x
1−x
x
1−x
5
1
8
5
2
5
2
2
2
2
2
+ −
+ Ly − + 1 −
+ 2x + π 2x − 2x −
−2x + Ly − + 5 −
x
1−x
x
1−x
6(1 − x) 3 6x
6
6
1
10
7
1
1
7
2
3
2
2
+
+ − 17 +
+ −
+4−
Lm 2x − 2x −
+ Lm Ls 4x − 4x −
1−x x
ε
6(1 − x)
3
6x
1−x
x
4
4
1
1
10
1
1
+Lm L2s
−8+
−2+
−2+
+ L2m 2x2 − 2x + Lx
+ Ly
−
x
1−x
x
1−x
x
1−x
3(1 − x)
4
4
58 11
11
4
4
23 10
2
−8+
−8+
+ −
+ Lm Ls 8x − 8x + Lx
+ Ly
−
+ −
3
3x
x
1−x
x
1−x
3(1 − x)
3
3x
4
4
1
2
3
3
x
2
x
+L2s
−8+
−4+
− + 3 Lx
+ Lm − −
L2x + Ly
− +
x
1−x
2 2x
x
1−x
2 1 − x 2x
1
1
3
5 3
x
2
2 x
−6x + Ly
−
−
−
+ +
+ Ly
+ 6x
2 2(1 − x) 2
2 2(1 − x) 2 x
2
8x
11
11
80 41
4
80
2
2 8x
− −
+5−
− +
−2+
−
+ Ls
L2x
+π
3
3 6(1 − x)
6x
9(1 − x) 9x 18
x
1−x
10
4
4
2
10
2
2 4
+
− 10 +
−2+
− 10 +
Lx + 4x + Ly
+ Ly
− 4x
x
1−x
x
1−x
x
1−x
5
5 5
16
5
14 58
10
14
2
2
+π −4x + 4x +
− +
− +
− +
−
+
L3x
3(1 − x)
3
3x
3(1 − x) 3x
3
3x 3 3(1 − x)
x
5
5 25
4
1
2
2
+ − −
− +
−1+
L2x + Li2 (x) − + 2 −
Lx +
L2y
2 3(1 − x) 3 6x
x
1−x
x
1−x
5
5
4
10 77
x
2
1
5
2
−6+
− +
− +
Ly − + π x −
+
Lx − 6x2
+
x
1−x
2
3(1 − x) 3 3x
1−x
x
6
4
2
2
4
5
10 5
+S1,2 (x) − + 2 −
−2+
− +
+ Li3 (x)
+ L3y
x
1−x
x
1−x
3x 3 3(1 − x)
25
13
5
2 2
10
1 37
x
4
2 x
2
+
− −
− −
+ +
+Ly
+ Ly π −x +
+ −
2 6(1 − x)
6
3x
3(1 − x) 3 3x
2 1−x x
3
1
ε3
2
17
(28)
28
139 28
9
121 73
7
121
20x2 20x
−
−
+
−
−
−
+ − + 12 −
ζ3 −
+6x + π
3
3
9(1 − x)
18
9x
x
1−x
18(1 − x) 18x 18
7x2 7x
19
13
19
20
7
7
4
3
2
+Lm −
+ +
− +
− +
+ Lm Ls −4x + 4x +
6
6 24(1 − x)
6
24x
3(1 − x)
3
3x
2
1
16
8
2x
8
2x
1
−4+
+
+ Lm L3s − + −
+ L3m −
+L2m L2s −4x2 + 4x +
1−x
x
3x
3
3(1 − x)
3
3
2
1
1
3
1
3
1 2
+Lx − + −
−2+
+ Ly − + −
+
3x 3 3(1 − x)
3x 3 3(1 − x)
2(1 − x)
2x
2
2
3
3
2
2
2
2
−8+
+ Ly − + 4 −
+
+Lm Ls −4x + 4x + Lx − + 4 −
x
1−x
x
1−x
1−x
x
4
4
4
16
8
4
+Lm L2s −8x2 + 8x + Lx − + 8 −
+ Ly − + 8 −
− 12 + L3s − +
x
1−x
x
1−x
3x
3
8
1
1
3
3
3
x
x
7x2
1
−
+
+ −
+ L2m
L2x + Ly − + 2 −
+ −
Lx −
3(1 − x)
4 4x
x
1−x
4 2(1 − x) 4x 2
2
2
1
1
3
5
3
5x
13
7x
5x
x
x
+
− −
+ +
+ Ly − +
+ + π2 −
+L2y − +
4 4(1 − x) 4
4 4(1 − x) 4 2x
2
3
3 12(1 − x)
13
739 643
4
1
4
739
−3 +
+
−
+
+ Lm Ls x +
L2x + Ly − + 8 −
+x
12x
72(1 − x) 72x
36
x
x
1−x
3
6
1
3
6
2
2
+ − 6 Lx + 12x + Ly −x +
+ 1 + Ly −x +
−5−
− 12x
−
1−x x
1−x
1−x
x
16x2 16x
11
11
61 25
4
61
2
2
2
+π −
+
+
− 10 +
+ +
+
+ Ls − + 2 −
L2x
3
3
3(1 − x)
3x
9(1 − x) 9x
9
x
1−x
4
10
2
4
4
10
2
2
Lx − 4x + Ly − + 10 −
+ Ly − + 2 −
+ 4x
+ − + 10 −
x
1−x
x
1−x
x
1−x
16
5
1
1
2
2
5
1
x
2
2
+π 4x − 4x −
+ −
+ − 12 + Lm
+
− +
+
L3x
3(1 − x)
3
3x
1−x x
3 3(1 − x) 3 3x
1
1
3
x
3 13
1
3
+ Ly − −
+x+ −
L2x +
− +2−
L2y + − + 8 −
Ly
2 2x
x
2
x
1−x
x
1−x
1
3
4
31
2
1
9x
33x2
2 x
+
−2+
− + 15 Lx + Li2 (x) −x +
−4+
− −
Lx −
+π
2 1−x
2x
4
1 − x 4x
1−x
x
2
8
25
8
11
2
1
1
33x
3
x
+π2
− +
−
− +
+ L2y −x +
+ L3y − +
+
3x
4
3(1 − x)
1−x
2
3 3(1 − x) 3 3x
2
2
1
2
3
3
1
1
x
2
+Li3 (x) x −
+4−
−5+
− +
+ S1,2 (x) x +
+ Ly π − +
1−x
x
1−x
x
2 2(1 − x) 2 x
31
4 51
146 52
2797
49
2797
9x
2
− +
+
−
+
+ 32x − 33x −
ζ3 +
+ −
4
4(1 − x) x
4
3(1 − x)
3
3x
216(1 − x) 216x
2983
4
20
12
8
4
10 10
3
2
−
+7−
−4+
+ Ls − + −
Lx + x −
Lx + Li2 (x)
Lx
108
3x
3
3(1 − x)
1−x
x
x
1−x
4
2
10
10
8
10
4
2
2
+
− +2−
+ −
Ly + − + 12 −
Ly + π −2x +
+x
x
1−x
x
1−x
3(1 − x)
3
3x
2
9
10
8
20 10
4
−
+ − 22 Lx + 12x2 + L3y − + −
+ Li3 (x) − + 4 −
1−x x
3x
3
3(1 − x)
x
1−x
2
18
4
4
4
8
12
40x2 40x
2
2
−4+
+8−
+
+
+ Ly −x −
− 12x + π −
+S1,2 (x)
x
1−x
1−x
x
3
3
3(1 − x)
17
14
4
4 4
2
18
9
8
2
− +
+ +
− 21 −
− 24 +
+ Ly −x + π 2x −
+
+
ζ3
3
3x
3(1 − x) 3 3x
1−x
x
x
1−x
88
5x
88 7
17
3 25
2
3x
1 1
+
+ +
−
+ −
− +
+
L4x + Ly
−
9(1 − x) 9x 9
24 6(1 − x) 2 24x
3x 3 3(1 − x)
4
47
17
31
8
37
23x
5
3x 5 9
11x2
−
+
−
− −
−
L3x +
− + −
L2y +
Ly +
6(1 − x) 12x
6
2
2 2x
4
6(1 − x) 3 12x
2
x
5
1
91
991
1
128
49x
+π2 − −
+1−
+
+
+
−
−
L2x
2
3 6(1 − x)
6x
8
9(1 − x) 72x (x − 1)
9
x
5
1 5
15
6
7x
3
+
−
− −
+8−
Ly + − −
L2y + −11x2 + 11x
3 3(1 − x) 3 3x
2 2(1 − x)
x
26
23 4
53
35
29x
11
2
2 4x
−
+ −
− +
− 12 Ly + π −2x +
−
+π
3
3(1 − x)
3
x
9(1 − x) 18 18x
8
10
16
103 473
12
8
12
2
+
− 28 +
+
+
+ 12 −
ζ3 +
Lx − 38x + S1,2 (x) Lx −6x −
x
1−x
1 − x 24x
18
1−x
x
10
53
5 59
9
7
9x
16
+Ly −x +
+7+
+ −
− 20 −
− +
+ Li4 (x) −x −
1−x
x
2
6(1 − x) 3x
6
1−x
x
9
16
25
41 17
47
3x
5x
+S1,3 (x) −x +
+ 21 +
+ −
−
+ L4y − −
+ L3y
1−x
x
24 24(1 − x) 24 6x
4 12(1 − x)
5
6
217
199
6
53x2 19x
25
4
−6+
+
+
+
+ 38x + S2,2 (x) 12x −
+π −
+ −
12 6x
1−x
x
45
20
360(1 − x) 180
2
39x
1
2 5
1069 91 1
97
991
2 x
2 11x
−
+π
−
+ −
−
+ + 2
−
+ Ly
+
360x
2
8
3 6(1 − x) 3 6x
72(1 − x)
72
9x x
1
1
1
432 log(2) − 1379
+π2
(48 log(2) + 107)x2 − (48 log(2) + 107)x + (648 log(2) − 1175) −
6
6
54
108(1 − x)
1379 − 432 log(2)
x
2
5
18
6
2
+
+
−4+
+ 12 −
+ Li2 (x)
Lx + Ly −6x −
108x
2 1−x
2x
1−x
x
5
53 43
14
4 14
6
8x
2
9x
+ −
−
− +
+ 14 +
L x + π2
+ Li3 (x) Lx
+ +
2
3(1 − x) 6x
3
3 3(1 − x) 3 3x
x
1−x
9x
6
18
53 43
139
383 16
5
79x
2
− + Ly 6x +
− 12 +
− +
−
+
−
−
+ 44x −
ζ3
2
1−x
x
3(1 − x) 6x
3
2
6(1 − x)
6
x
29x
89 11
12
1631
35
3
103
2
+Ly
+ π 2x +
− +
− 23 −
+
+ −5x −
ζ3 +
8
18(1 − x) 18 9x
1−x
x
24(1 − x)
72
8
937
5369
937
+
+
−
,
(29)
+
x
216(1 − x) 216x
216
D =
1
1
1
1
1
1
1
2
2
−1+
Lm − + 1 −
+ Lm − + 2 −
+π
−
2x
2(1 − x)
x
1−x
2x
2(1 − x)
2(1 − x)
1
1
1
1
1
1
1
1
− +1 +
−1+
−2+
−1+
L3m
+ L2m Ls
+ L2m
2x
ε
2x
2(1 − x)
x
1−x
x
1−x
3
2
1
1
1
2
1 1
3
2
−4+
− +
−
−
+Lm Ls
+ Lm
Lx +
Lx + Ly
x
1−x
2x 2 1 − x
2x 2
2(1 − x) 2
1
ε2
19
2 4
2
1
1
1
1
1
2 1
2
+π
− +
− +
+ Ly
+ 2 + Ls π − + 2 −
+
3x 3 3(1 − x)
x 2 2(1 − x)
x
1−x
1−x
1
1
1
2
1
3
2
1
1
− +
L2x + π2 − + 1 −
+ −
L x + L y π2 − + 1
+ −2 +
x
2x 2 1 − x
x
1−x
2x 2
x
1
1
7
1
1
1 1
3
1
1
−
−
− +
+
+ π2 − − −
+ L2y
−
1−x
2(1 − x) 2
3x 3 3(1 − x)
x 2 2(1 − x)
2(1 − x)
1
7
7
1
1
7
1
1
4
3
2 2
− + 2 + Lm −
+ −
+ Lm Ls − + 2 −
+ Lm Ls − + 2 −
2x
24x 12 24(1 − x)
x
1−x
x
1−x
1
2
2
2
2
2
2
+L3m − + −
+ L2m Ls − + 2 −
+ Lm L2s − + 4 −
3x 3 3(1 − x)
x
1−x
x
1−x
1 1
1 1
1
1
3
3
1
1
2
2
2
+Lm − + −
−
−
Lx +
Lx + Ly
+ Ly − + −
4x 4 2(1 − x)
4 4x
4 4(1 − x)
2x 4 4(1 − x)
5
5
13
9
2
13
3
1
5
2
2
+ −
− +
−
+ Lm Ls − + 1 −
Lx + 1 −
Lx
+π −
12x 6 12(1 − x)
8(1 − x) 8x 4
x
1−x
x
3
1
4
2
4 8
1
2
2
2
+Ly 1 −
−2
+ Ly − + 1 −
+π − + −
− 4 + L s π2
1−x
x
1−x
3x 3 3(1 − x)
x
1
1 1
1 1
1
2
1
1
1
+
− + 2 + Lm − + −
− +
−
L3x + Ly
−
1−x
1−x x
3x 3 3(1 − x)
2x 2 1 − x
2(1 − x)
3
1
2
1
5 9
1 1
2
2
− + 1 Lx + Li2 (x)
−1+
Lx + π − + −
+ −
Lx
2x
x
1−x
2x 2 1 − x
4x 4
1
9
1
17
5
17
1 1
−
+ −
+
+ π2 −
+Ly π2 − + −
x 2 2(1 − x)
4(1 − x) 4
12x 2 12(1 − x)
1
2
1
1
3
2
1
3
2
+Li3 (x) − + 1 −
+ Ly − + −
+ Ly − + 1 −
x
1−x
3x 3 3(1 − x)
2x
2(1 − x)
2
9
1
10
11 19
11
1
+S1,2 (x)
−1+
− 17 +
− +
+
ζ3 −
+ Ls − + 1
x
1−x
x
1−x
8(1 − x) 8x
4
x
4
2
3
3
2
2
2 2
2 4
−2+
−2+
+1
Lx + π
− + 1 Lx + Ly π
−
−
1−x
x
1−x
x
x
1−x
1−x
2
1
2
1
1
5
2
1
2 14
2
2
+Ly − + 1 −
+ +
+ −4 + − + −
+π
+
L4x
x
1−x
3x
3
3(1 − x)
1−x x
8x 24 3(1 − x)
5
13
3
2
1
13
9
4 1
3
2
+ −
− −
− −
Lx +
− + −
Ly +
Ly + 3x2
12x 4 3(1 − x)
x 2 2(1 − x)
4x
4
2(1 − x)
2 1
5
1
17
1
43
1
5
2
2
+π
− −
− −
+
−
− 3x −
Lx +
L3y
3x 3 3(1 − x)
4(1 − x) 8x (x − 1)2
8
3x 3
3
4
5
11
4
5
7
2
2
2
2 1
−
− +
Ly + −6x + 6x + π − + −
− 10 Ly + π
+
2 2x
3x 3 3(1 − x)
x
3
3(1 − x)
6
2
19 27
1
1
1
1
−6x +
−
+ +
ζ3 +
Lx + Li2 (x)
− + −
L2x
1−x x
1 − x 8x
8
2x 2 1 − x
2
5
27
3
6
16
14
+ Ly − + 2 −
+ −
−
Lx + S1,3 (x) − + 1 +
x
1−x
1 − x 2x
2
x
1−x
16
2
1
6 27
5
8
+S1,2 (x) Lx − + 2 −
− −
+ Ly − + 1 +
+
x
1−x
x
1−x
2(1 − x) x
2
2
20
3
14
6
2
5
27
6
2
16
+Li4 (x) − − 1 +
−
−2+
− +
+ Li3 (x) Lx
+ Ly
+
x
1−x
x 1−x
x
1−x
1 − x 2x
2
3
13
5
1
29
11
2
1
2
+ −
+ L4y − + −
+ π4 −
+L3y − − −
3x 4 12(1 − x)
3x 24 8(1 − x)
30x 60 60(1 − x)
14
14
2
43
5
1
17
1 1
2
2
2
+S2,2 (x)
−
+ − − 2
+ Ly 3x − 3x + π − − +
−
x
1−x
3x 3 3(1 − x)
8(1 − x)
8
4x x
1
13 − 24 log(2) 24 log(2) − 13
13
15 9
2
2
+π 3x − 3x + (13 − 8 log(2)) −
+
− +
+
ζ3
2
12(1 − x)
12x
x
2 2(1 − x)
11
1
5
1 21
4
2
19
47
+Ly π2
− +
+ −
+ 6x + − + 1 −
ζ3 +
−
3x
3
1−x
x
1−x
8(1 − x) x
8
8(1 − x)
47 57
(30)
− + ,
8x
4
El
28x2 28x
7
28
7
1
1
1
2
=
−
−
+ −
+4−
+ 2 Lm 4x − 4x −
+ Ls −8x2 + 8x
3
3
3(1 − x)
3
3x
ε
1−x
x
2
2
2
2
2
+
−8+
− 2 + Lx −4x2 + 8x − 6 +
+ 12x2 − 12x + Ly −4x2 +
−
1−x
x
1−x
x
3(1 − x)
1
4 1
2
8 2
22 2
1
4x2 4x
8x2 8x
2
+ +
− +
+ +
− +
+ −
+
Lm −
+ Lm Ls −
3
3x
ε
3
3 3(1 − x) 3 3x
3
3
3(1 − x) 3 3x
2
8x
2
8
2
20
62 20
88x2
44x2 44x
2 8x
− −
+ −
+
+
− +
+ Lm −
+ Ls
+Ls
3
3 3(1 − x) 3 3x
9
9
9(1 − x)
9
9x
9
2
2
4
4
16x
4
124 40
8x
88x
40
8x
+ Ly
−
+
−
+4−
+
−
−
+ Lx
−
9
3
3(1 − x) 3
3
3
3x
9(1 − x)
9
9x
2
2
4x
56x
4x
1
97x 26 37
802x
1
1
802x
+ − +1−
−
+ −
+
+ L2y
−
−
L2x +
Lx −
3
3x
9
9
3
9x
27
27
3 3(1 − x) 3
2
1
5x
37
37
581
581
56x
1
−2+
− −
+
+
+ Ly
+
+π2 −2x2 + 2x +
2(1 − x)
2x
9
3 9(1 − x)
9
54(1 − x) 54x
2
2
32x
11
38 11
64x
22
76 22
64x
991
2 32x
−
−
+ −
−
−
+ −
+ Lm
+ Lm Ls
−
27
9
9
9(1 − x)
9
9x
9
9
9(1 − x)
9
9x
2
2
2
64x
64x
22
76 22
50x
2x
1
2
2x
50x
+L2s −
+
+
− +
−
+ π2 −
+ +
−
+ Lm
9
9
9(1 − x)
9
9x
27
27
3
3
6(1 − x) 3
2
2
89
73
80x
44
44
73
80x
64x
64x2
1
+ −
−
+ Ly −
+
−
−
+ Ls
+ Lx −
+
6x
54(1 − x) 27 54x
27
27
9
9(1 − x)
9
9
2
44
4x
1
4
1
56
10
128x
10
4x
− 12 +
+ +
− +
+ +
+
+ π2 −
+
9
9x
9
9 9(1 − x) 9 9x
27(1 − x) 27 27x
2
8x
16x2 133x
1
2
11 10
x
2
2
+
−x + 3 −
−3−
+ −
−
Ly +
Ly − +
Lx +
x
3
3x
9 1 − x 9x
3
27
54
2
2
4
4x 4
35
47
2170x
2x
2x
−2+
− +
+Lyπ2
+ π2
+ +
Lx +
3
3x
9
3 3
9
54x
81
8
2
2
11
11
16x
11x
4
+S1,2 (x) Lx −4x + 12 −
−
−
+
−
−
+π −
x
3
3(1 − x) 3
90
45(1 − x) 45
2
11
31 2
8
8
146x
2170x
2 146x
2 x
+
− +
+ S2,2 (x) 8x −
−4+
−
−
+π
+Ly
9 9(1 − x)
9
x
81
1−x
x
27
27
1
ε3
21
109
307 109
16x
2
8
−
+
−
+ Ly 4x − 12 +
+ Li3 (x) −
+6+
54(1 − x)
54
54x
3
x
3x
8
4
2 4
2
16x
2 4x
−
− +
− −6
+ Lx Ly −4x + 12 −
+
+Li2 (x) π
3
3(1 − x) 3 3x
x
3
3x
40x
2
4
2x2 8x 2
8
16x2 23x
+ −8x2 +
+
− 14 +
+
+ π2
+ +
ζ3 + L y
+ −4x + 12 −
ζ3
3
1−x
3x
27
18
9
9
9
x
181
1487 4289
1487
35
+
−
+
,
(31)
−
+
54(1 − x)
54
162(1 − x) 162x
162
Eh
1
8
32
8
8
32
8
32x2 32x
32x2 32x
= 2 Lm
−
−
+ −
−
−
+ −
+ Ls
ε
3
3
3(1 − x)
3
3x
3
3
3(1 − x)
3
3x
4
4
4
4
1
− 16 +
− 16 +
Lm Ls −16x2 + 16x +
+ L2s −16x2 + 16x +
+
ε
1−x
x
1−x
x
2
2
2
128x
128x
8
8
32x
8
8
16x
16x
+Lm
−
+ Ly −
+
−
+
−8+
+ Lx −
−
9
9
3
3(1 − x) 3
3
3
3x
9(1 − x)
2
2
2
8
128x
8
8
32x
8
128x
16x
16x
80
−
+ Ly −
+
−
+
−8+
+ Ls
+ Lx −
+ −
9
9x
9
9
3
3(1 − x) 3
3
3
3x
2
2
80
8
8x
8x
2
8 2
8
8x
8x
2
+ −
− + π2 −
+ +
− +
−
+
+
9(1 − x)
9
9x
3
3
9
9
9(1 − x) 9 9x
3
44x2 44x
11
44 11
2
8 2
8x2 8x
3
2
+Lm −
+
+
− +
+ +
− +
+ Lm Ls −
9
9
9(1 − x)
9
9x
3
3
3(1 − x) 3 3x
2
2
40x
10
40 10
112x
28
112 28
2 40x
3 112x
−
−
+ −
−
−
+
−
+Lm Ls
+ Ls
3
3
3(1 − x)
3
3x
9
9
9(1 − x)
9
9x
2
166x
166
83
83
2
2
166x
2
2
2
+
+ Ly 4x −
+ 2 + Lx 4x − 8x + 6 −
−
+
+
+Lm −
9
9
1−x
x
18(1 − x)
9
18x
2
2
2
104x
16
16
64x
16
32x
10
32x
104x
+
+ Ly
−
+
−
+ 16 −
+Lm Ls −
+ Lx
−
9
9
3
3(1 − x)
3
3
3
3x
9(1 − x)
32 10
32
16x2 16x
4
4
2
2
2
− −
+
+ Ly 8x −
+ 4 + Lx 8x − 16x + 12 −
+ Ls −
−
9
9x
9
9
1−x
x
9(1 − x)
2
2
1
161x 44 59
598x
88x
56 32
598x
4x
−
+ −
+
+ −
+ Lm − + 1 −
L2x +
Lx −
9
9x
3
3x
9
9
3
9x
9
9
2
2
1
1
20x
5
20
5
5x
20x
88x
4x
−
−
+
+
− +
−
+ π2 −
+ Ly
+L2y
3 3(1 − x) 3
9
9
9(1 − x)
9
9x
9
3
2
59
59
121 662
2
22x 16 10
16x
121
8x
−
+
+
−
−
+ −
+
+ Ls − + 2 −
L2x +
Lx
9(1 − x)
9
6(1 − x)
6x
9
3
3x
3
3
3
3x
2
2
2
8
2
8x2 8x
380x2 380x
2 8x
2
+
+ Ly
−
−
+ +
− +
+π −
−
9
9
3 3(1 − x) 3
3
3 3(1 − x) 3 3x
10
10
109 130
2
109
4 2 3
16x2 10x
−
−
+
+
−
−x + 3 −
+Ly
+
− x Lx +
L2y
3
3
3(1 − x)
3
9(1 − x)
9x
3
9
x
2
4x
188x2 893x
4x
2
2
19
13
4
11x
2 2x
2
+
+ −2−
+
+
−
−
+ Ly π
−2+
Ly −
Lx +
3
3
3x
18
1 − x 18x
3
27
54
3
3x
22
11
11
176 323
3485x2
11x
8x2 4x 2 2
4
− + +
−
+π −
+
−
+
Lx −
+π −
9
9
3 9x
9
54x
81
90
45(1 − x) 45
19
89 2
8
8
3485x
4x2 8x 4
2 11x
3
+Ly
+
− +
+ S2,2 (x) 8x −
−4+
+ −
+
+ Ly −
18
18(1 − x) 18 x
81
1−x
x
9
9
9
2
2
4
140x
32
8x
4
8
140x
− 8x + Lx −4x + 12 −
+
−
−
−
+ π2
+S1,2 (x)
3
x
3(1 − x) 3
27
27
27(1 − x)
2
32
8x
4x
8x
4
8
83
− + Ly 4x − 12 +
+ Li3 (x) −
+4+
+ Li2 (x) π2
+ −
27 27x
3
3
x
3x
3
2
2
4
8x
152x2 104x
4
4
8
8x
− +
+ + Ly −4x + 12 −
−
+ Lx
−4−
+
−
3(1 − x) 3 3x
3
3
x
3x
9
9
2
2
32
92 44
47x
20x
2
2
8x
188x
−
+ −
+
+ π2 −
+
+
−
ζ3 + L y
9(1 − x)
9
9x
27
18
9
9
9(1 − x) 3
8
539
5447 9913
323
5447
+ −4x + 12 −
+
+
−
,
(32)
ζ3 −
+
x
54(1 − x)
54
324(1 − x) 324x
162
2
Fl
7
14
7
2
2
2
2
1
2
− +
−6+
+ 2 Lm −4x + 4x +
+ Ls − + 4 −
− 4x2
=
3x
3
3(1 − x)
ε
1−x
x
x
1−x
2
2
5
5
2
2
+4x + Lx − + 4 −
− 10 +
+ Ly − + 4 −
+
x
1−x
x
1−x
3(1 − x)
3x
2
2
4x
2
2
8x
4
4
4
1
8x
2 4x
2 2
− −
+2−
− −
+4−
−
Lm
+ Lm Ls
+ Ls
+
ε
3
3
3(1 − x)
3x
3
3
3(1 − x)
3x
3x 3
2
40
28 40
8x
8
4
4
44x2 44x
8x
2
−
−
+ −
− + Lx
− +
+ Lm
+ Ls
+
3(1 − x)
9
9
9(1 − x)
3
9x
3
3
3x 3 3(1 − x)
4 8
4
20 34
2
1
34
x
+Ly
− +
− +
−
+
+ − +
L2x
3x 3 3(1 − x)
9(1 − x)
9
9x
3 3(1 − x) 3
4
46
28 29
2
2
x
26x2
4 8
2 x
− +
+ −
+ Ly
− +
− +
Lx +
+ Ly
3x 3 3(1 − x)
3 9(1 − x) 9x
9
9
3 3 3x
28
32 46
17
25
17
16x2 16x
26x
346
x
+
− +
+ π2
−
−
+ −
+Ly
−
−
3 9(1 − x)
9
9x
9
9
9
18(1 − x)
9
18x
27(1 − x)
346 578
22
22
44
64x2 64x
32x2 32x
2
−
+
+
+
−6+
+
+
− 12
+ Lm −
+ Lm Ls −
27x
27
9
9
9(1 − x)
9x
9
9
9(1 − x)
2
22
2x
1
1
22 32
44
50x2 50x
2 2x
2
+
+π
− −
+1−
+ Ls − + −
+ Lm −
+
9x
9x
9
9(1 − x)
27
27
3
3 3(1 − x)
3x
2
44 64
32
73
64x
44
64x
73
− +
+
+ Lx − + −
+ Ls −
+
27(1 − x)
9
27x
9
9
9x
9
9(1 − x)
44
2
1
340
56
44 64
56
1
2
+Ly − + −
−
+
+π − + −
+
9x
9
9(1 − x)
9x 9 9(1 − x)
27(1 − x)
27
27x
4
x
2
23
47
3
49
4x
2
2
+
− 2 Ly + − +
+
−
−
Ly + 2x − +
L2x
x
3 3(1 − x)
3
9 9(1 − x) (x − 1)2
9
4x 8
4
8
70 44
44
7x
2
2
2 4
− +
−
+ −
+
Ly + −4x + 4x + π
−
Ly +
3
3 3x
3 3x
9(1 − x)
9
9x
18
2
7
5
1
41
203
22
220
130x
4 11
2 7x
−
− +
+
−
+π
−
−
Lx +
+π
3 3(1 − x) 9 9x
27(1 − x) 27x
54
27
45 45(1 − x)
1
ε3
23
16
16
16
4
16
+S2,2 (x)
−
− 18 + S1,2 (x) Lx
−8
+ Li3 (x) Ly 8 −
− 2x +
1−x
x
x
3(1 − x)
x
130x
197
9 197
56x2 56x
4
32x 35
2
2
+π −
+
+
− +
−
+ Ly 2x2 −
−2x − + 20 −
3x
27
27
27
54(1 − x) 2 54x
9
9
47 3
8
4
8
16
+ − 2 + Li2 (x) π2
−
− 8 + 2x −
+ 18
+ Lx Ly
9x x
3(1 − x) 3x
x
3(1 − x)
16
2
1
16
7
7x
7x
10
2
+ 22 −
+ −
− 8 ζ3
ζ3 + L y π − +
− +
+ 2x −
3(1 − x)
x
3 9(1 − x)
9
3x
18
x
41
220 91
709 113
709
+
−
−
+
+
,
(33)
+
27(1 − x) 27x 27
81(1 − x) 81x 162
Fh
8
8
16
8
16
8
1
16x2 16x
=
− +
− +
+
Lm
+ Ls
+
L2m −
3x
3
3(1 − x)
3x
3
3(1 − x)
ε
3
3
2
4
16
4
16x
8
8
8
4
4
16x
2
+
− +
+
−
+ −
+ Lm Ls −
+ Ls − + 8 −
3(1 − x)
3
3x
3
3
3(1 − x) 3 3x
x
1−x
2
8
16x
16
8
16
8
8
16x
20
+
+ Lx − + −
+ Ly − + −
+
+Lm −
3
3
3x
3
3(1 − x)
3x
3
3(1 − x)
9(1 − x)
2
16x
8
16
8
8 16
16x
8
112 20
+
+
+ Lx − + −
+ Ls −
+ Ly − + −
−
9
9x
3
3
3x
3
3(1 − x)
3x
3
3(1 − x)
2
112 20
4
2
52x
8
74
8
20
2
2
3 52x
−
+
−
−
+ −
+
+π − + −
+ Lm
9(1 − x)
9
9x
9x 9 9(1 − x)
9
9
3(1 − x)
9
3x
2
40x
40x
4
44 4
4
4
4
+L2m Ls
−
−
+ −
+ +
+ Lm L2s 8x2 − 8x +
3
3
1−x
3
x
3(1 − x) 3 3x
2
2
2
56
28
86x
2
2
3 28
2 86x
− +
−
+ Lx
−4+
−4+
+Ls
+ Lm
+ Ly
9x
9
9(1 − x)
9
9
x
1−x
x
1−x
2
193 25
40x
16
16 32
40x
25
+
−
−
+ Lx
− +
+ Lm Ls
−
3(1 − x)
9
3x
3
3
3x
3
3(1 − x)
16 32
16
196 32
4
32
4
2
2
+Ly
− +
+
−
−8+
−
+ Ls 8x − 8x + Lx
3x
3
3(1 − x)
9(1 − x)
9
9x
x
1−x
4
4
4
80 14
2
1
x
14
+Ly
−8+
+ +
−
+
+ Lm − +
L2x + Ly
x
1−x
9(1 − x)
9
9x
3 3(1 − x) 3
3x
2
4
68
50 61
x 2 2
8
x
458x
+ −
+ L2y
− +
− +
− +
Lx +
3 3(1 − x)
3 9(1 − x) 9x
9
27
3 3 3x
2
50
64 68
28x
4
38
4
458x
713
x
2 28x
+
− +
+π
−
−
+ −
−
−
+Ly
3 9(1 − x)
9
9x
27
9
9
3(1 − x)
9
3x
27(1 − x)
713 1481
4
2
16
8
16
2x
8
2x
−
+
−
− +
+ Ls − +
L2x + Ly
− +
27x
27
3 3(1 − x) 3
3x
3
3(1 − x)
3 3(1 − x)
2
4
4
16
2x 4
32x2 32x
4 2
4x
4x
2x
+ L2y
− +
+
+
−
+ +
Lx −
+ Ly
+ + π2
3x 3
3
3
3 3x
3 3(1 − x) 3x
3
9
9
14
44 14
124 184
2
2 2
124
2x
−
+ −
−
+
+ −
−
+ − +
L3x
9(1 − x)
9
9x
9(1 − x)
9x
9
9
9(1 − x) 9 9x
23x
53
1
3
35
4
2
2
− 2 Ly + 8Ly + 2x −
+
− −
−
L2x
+
x
18
9(1 − x) x (x − 1)2 18
1
ε2
24
4x
2
2
8
10 20
20
2
2 4
2
+
+
−4+
−
+ −
Ly + −4x + 4x + π
−
Ly
3 3(1 − x)
x
3 3x
9(1 − x)
9
9x
2 4
229 709
20
50
1091x2
31x
2
+ π 2x −
+ −
+
−
−
Lx +
+
18
9(1 − x) 9 9x
27(1 − x) 27x
54
27
11
16
22
16
8x
4
4
16
+π4
−
−
−8 − −
− + 24
+ S2,2 (x)
+ S1,2 (x) Lx
45 45(1 − x)
1−x
x
x
3
3(1 − x) x
4
4
64
2
2
8x
1091x
2x
16
+ −
−
+
− +
+ L3y
−
+Li3 (x) Ly 8 −
x
3
1 − x 3x
3
9
9(1 − x) 9x
27
2
22x
22x
53
26
53
1
11 53 3
49x
2
2
2
+π −
+
+
+ +
−
− + − 2
+ Ly 2x −
9
9
27(1 − x) 27 27x
18
1−x
9
9x x
8
8
8x
4
4
64
20
8x
16
+Li2 (x) π2
−
−8 + −
− +
+
+ Lx Ly
+
3(1 − x) 3x
x
3 1 − x 3x
3
3 9(1 − x)
16
20 20
229
50
31x
4
80 44
+ −
+
− 8 ζ3 +
−
ζ3 + Ly π2 −2x −
−
+ +
9
9x
9(1 − x)
9
9x
18
x
27(1 − x) 27x
308
4243 5821
4243
−
−
+
,
(34)
−
27
162(1 − x) 162x
81
Gl
1
1
2
1
1
1
1
1
2
=
− +
Lm − + 2 −
− +2−
+
Lm
x
1−x
x
1−x
ε
3x 3 3(1 − x)
2 4
2
20
4
2
20 22
2
+Lm Ls
− +
− +
− +
+ Lm
+ Ls
3x 3 3(1 − x)
9x
9
9(1 − x)
3x 3 3(1 − x)
1 1
2
1
4
4 8
1
1 1
2
2
− +
−
−
− +
Lx +
Lx + Ly
+π
+
3x 3 3(1 − x)
x 3
1−x 3
9x 9 9(1 − x)
2
1
1
37
19
11
37
11 16
2
2
+Ly
− +
+
−
+
+ Lm − + −
3x 3 3(1 − x)
18(1 − x) 18x
9
9x
9
9(1 − x)
22 32
22
1
73
7
1 1
73
2
+Lm Ls − + −
− +
−
+
+ Lm π
−
9x
9
9(1 − x)
6x 3 6(1 − x)
54(1 − x) 54x 27
1 73
22
2
11
22 32
12
241
+Ls − + −
− +
−
+
6−
L2y +
Ly −
9x
9
9(1 − x)
x
3x
3
3(1 − x)
9(1 − x) 9x
58
203
4
20x
8
9
− 4 − 2 L y + π2 −
+
+
L2x +
−4x + 8 −
L2y + π2
+
2
(x − 1)
9
x
x
3
9(1 − x)
13 1
48
13 17
8
18
8
2
+ −
− −
−
− 16x +
Lx + Li2 (x) π
+ Lx Ly 24 −
− 8x
9
9x
1 − x 6x 18
x 1−x
x
28
2
170
11
11
22
241 9 203
4
2
+
+ −
−
+ 2+
−
+π
+ Ly −
3(1 − x) 3x
3
15(1 − x) 15
9x
x
9
9(1 − x)
44
275
44
48
48
48
+π2 −
−
−
−
− 24 + 8x
+ S2,2 (x)
+ Li3 (x) Ly
27x
27
27(1 − x)
x
1−x
x
2
170
28 194
2
48
28
− +
+ −
+ S1,2 (x) Lx 24 −
+ 8x +
+ (−8x
−
3(1 − x) 3x
3
x
3(1 − x) 3x
3
28
170
2
1
47 58
48
20x
+
−
+
−
− +
ζ3 + Ly 16x + π2
+ 24 −
ζ3
3(1 − x)
3
3x
3
9(1 − x)
9
9x
x
305 18
23
80
13
23
−
+
+
− ,
(35)
−
+
6(1 − x)
18
x
54(1 − x) 54x 27
1
ε2
25
Gh
Hl
Hlh
1
8
4
8
4
8
4
4
4
4
2
=
Lm − + −
+ Lm Ls − + −
+ Lm − + −
ε
3x 3 3(1 − x)
3x 3 3(1 − x)
3x 3 3(1 − x)
8
4
26
13
10
4
10 20
3 13
2
+Ls − + −
− +
− +
+ Lm
+ Lm Ls
3x 3 3(1 − x)
9x
9
9(1 − x)
3x
3
3(1 − x)
2
43
67
14
67
14 20
2
2 2
−4+
− +
− +
+ Lm
+ Lm Ls
+Lm Ls
x
1−x
18x
9
18(1 − x)
3x
3
3(1 − x)
2
2
1
1 1
1 1
1
2 2
2
+Ls
−4+
− +
−
−
+ Lm
Lx +
Lx + Ly
x
1−x
3x 3 3(1 − x)
x 3
1−x 3
2 1
1
14
7
337 265
337
7
2
2
+Ly
− +
− +
+
−
+π
+
3x 3 3(1 − x)
9x
9
9(1 − x)
54(1 − x) 54x
27
2
2 2
2
4
2
8
8 16
2
2
2
+Ls
− +
−
−
− +
Lx +
Lx + Ly
+π
3x 3 3(1 − x)
x 3
1−x 3
9x
9
9(1 − x)
2
2
5
2
2
4
2
5
12
4
3
2
− +
+ −
− +
+
+
Lx +
6−
L2y
+Ly
3x 3 3(1 − x)
3(1 − x) 3x 3
9x 9 9(1 − x)
x
235
1
9
391
4
2
2
2 8
−24Ly −
−
+
+
− 4 − 2 Ly
Lx +
−4x + 8 −
Ly + π
9(1 − x) 18x (x − 1)2
18
x
x
20x
64
10
2
5
7
8
8
18
+π2 −
+
+ +
− +
−
− 16x +
Lx + Li2 (x) π2
3
9(1 − x)
9
9x
1 − x 2x 18
x 1−x
11
22
48
8
235 9 391
+Lx Ly 24 −
− 56
+ π4
−
+ 2+
− 8x +
+ L2y −
x
1−x
15(1 − x) 15
9x
x
18
31
7
2
2
48
48
4
7
1
− +
−
+ π2 − − −
+ L3y
+ S2,2 (x)
−
18(1 − x)
9x
3
9(1 − x)
9x 9 9(1 − x)
x
1−x
48
8
8
48
+Li3 (x) Ly
− 24 + 8x −
+ 56 + S1,2 (x) Lx 24 −
+ 8x + − 64
x
1−x
x
x
8
2
50 64
48
2 20x
+ −8x +
− 56 ζ3 + Ly 16x + π
+
− +
+ 24 −
ζ3
1−x
3
9(1 − x)
9
9x
x
5
281 18
1013 1013
1013
−
−
+
+
−
,
(36)
+
2(1 − x)
18
x
108(1 − x) 108x
54
4
16
4
8
8
8
16x2 16x
1 16x2 16x
8x2 8x
=
+
+
− +
−
−
+ −
+
−
+
−
9
9
9(1 − x)
9
9x
ε
9
9
9(1 − x) 3 9x
9
9
10
4
4
− + ,
(37)
+
9(1 − x)
9
9x
1
ε2
1
8
32
8
8
32
8
32x2 32x
32x2 32x
=
+
+
− +
+
+
− +
Lm −
+ Ls −
ε
9
9
9(1 − x)
9
9x
9
9
9(1 − x)
9
9x
2
2
16x
16x
4
16
4
32x
8
32
8
32x
+L2m
−
−
+ −
−
−
+ −
+ Lm Ls
9
9
9(1 − x)
9
9x
9
9
9(1 − x)
9
9x
2
16x
4
16
4
16
16 16
32x2 32x
2 16x
−
−
+ −
−
−
+ −
+ Lm
+Ls
9
9
9(1 − x)
9
9x
9
9
9(1 − x)
3
9x
26
+Ls
2
− ,
9
Hh
2
16
16 16
8x
2
8
2
8x2 8x
32x2 32x
2 8x
−
−
+ −
+ +π
− −
+ −
−
9
9
9(1 − x)
3
9x
9
9
27 27 27(1 − x) 27 27x
(38)
4
16
4
8
32
8
16x2 16x
32x2 32x
=
+
+
− +
+
+
− +
−
+ Lm Ls −
9
9
9(1 − x)
9
9x
9
9
9(1 − x)
9
9x
2
16x
4
16
4
16x
+
+
− +
,
(39)
+L2s −
9
9
9(1 − x)
9
9x
L2m
Il =
Ilh
1
ε2
8
4
8
8
4
4
1 8
4
− +
,
− + −
+
− −
9x 9 9(1 − x)
ε 9x 9 9(1 − x)
9x 9(1 − x)
(40)
8
16
8
8
4
8 16
8
4
1
2
− +
Lm − + −
+ Ls − + −
+ Lm
=
ε
9x
9
9(1 − x)
9x
9
9(1 − x)
9x 9 9(1 − x)
8
16
8
8
4
16
16 16
2 4
+Lm Ls
− +
− +
− +
+ Ls
+ Lm
9x
9
9(1 − x)
9x 9 9(1 − x)
9x
9
9(1 − x)
16 16
16
4
2
2
+Ls
− +
− +
+ π2
,
(41)
9x
9
9(1 − x)
27x 27 27(1 − x)
Ih =
L2m
8
4
16
8
4
4 8
8
4
2
+ Lm Ls − + −
+ Ls − + −
.
− + −
9x 9 9(1 − x)
9x
9
9(1 − x)
9x 9 9(1 − x)
(42)
Notice that apart from the classic polylogarithms up to weight four, our results are also expressed in terms of Nielsen polylogarithms
Sn,p (x) =
5
(−1)n+p−1
(n − 1)!p!
Z 1
dy
0
logn−1 (y) log p (1 − xy)
.
y
(43)
Conclusions
In the present article we have computed the two-loop virtual QCD corrections to the production of
heavy-quarks for the gluon fusion process in the ultra-relativistic limit. Previously we had already
obtained the corresponding results for quark-quark scattering [22]. Taken together, these results
complete the two-loop radiative QCD corrections to heavy-quark production in hadron-hadron
collisions in the limit when all kinematical invariants are large compared to the heavy-quark mass.
The remaining channel with a gq (or gq̄) initial state involves at most one-loop corrections in a
27
consistent NNLO treatment. At one-loop the complete structure of the singularities as well as all
large logarithms in the heavy-quark mass m can therefore be entirely treated with the methods of
Ref. [32].
Our derivation relies on the combination of two completely different methods and we have
ensured substantial overlap between them. In this way we have had mutual and highly non-trivial
cross checks on our direct calculation of massive Feynman diagrams and on the QCD factorization
approach [25]. In particular through the latter method we have had the possibility to relate to
various massless [26–28, 35, 36] and massive results [34] available in the literature and we have
found consistency.
The results for gluon fusion in the present article (and Ref. [22] for quark-quark scattering) are
of direct relevance whenever power corrections in the heavy-quark mass are negligible. This is
certainly the case for hadro-production of bottom pairs over a large kinematical range at colliders
and, to a lesser extent perhaps, for t t¯-production at LHC energies. Here, possible improvements
would come from the systematic computation of power corrections in the heavy-quark mass, i.e.
terms proportional to (m2 /s)k with k ≥ 1 to improve the convergence of the small-mass expansion.
This could be achieved, for instance, by extending the methods of Sec. 3 for the direct calculation
of massive Feynman diagrams to higher powers in m.
Finally, it is clear, that our result for hM (0) |M (2) i still has to be combined with the tree-level
2 → 4, the one-loop 2 → 3 as well as the square of the one-loop 2 → 2 processes hM (1) |M (1) i in
order to yield physical cross sections. Some of the matrix elements including the full mass dependence can be easily generated, others have become available in the literature only rather recently,
see e.g. Refs. [31, 51]. The combination of all these contributions enables the analytic cancellation of the remaining infrared divergences as well as the isolation of the initial state singularities.
The latter will have to be absorbed into parton distribution functions of, say, the proton in order
to match with a precise parton evolution at NNLO [52, 53]. All these remaining steps are necessary prerequisites e.g. to the construction of numerical programs which then provide NNLO QCD
estimates of observable scattering cross sections for heavy-quark hadro-production.
A M ATHEMATICA file with our results can be obtained by downloading the source from the
preprint server http://arXiv.org. The results are also available from the authors upon request.
Acknowledgments: We are grateful to C. Anastasiou, Z. Bern, L. Dixon and N. Glover for
communication on the results of Refs. [26–28, 39] and for useful discussions. We would also
like to thank V.A.Smirnov for an interesting exchange of opinions on MB representations of nonplanar graphs. M.C. and A.M. thank the Alexander von Humboldt Foundation for support through
a Sofja Kovalevskaja Award and a research grant, respectively. S.M. acknowledges contract VHNG-105 by the Helmholtz Gemeinschaft. This work was also partially supported by the Deutsche
Forschungsgemeinschaft in Sonderforschungsbereich/Transregio 9.
28
A Asymptotics of the massive non-planar scalar integral
Here, we give the leading high energy behavior of the non-planar scalar integral with a massive
loop corresponding to the MB representation Eq. (20)
s3+2ε INP = q
π2
m2
s
−π2 + iπ [−Lm + Lx + Ly + 4 log(2)]
x(1 − x)
7
5
7
4
5
4
7
7
3
4
+
−
+ Lm Ly − −
+ Lx − −
−
+Lm
12x 12(1 − x)
3x 3(1 − x)
3x 3(1 − x)
3(1 − x) 3x
3
3
4
4
6
6
4
4
+Lm 2
+
+
+
−
Lx 2 + Ly
−
Lx + Ly
2x 2(1 − x)
x 1−x
1−x x
1−x x
3
3
2
8
1
8
1
2
2 2
+Ly
+
+
+
−
+π
+
+ Lm
Lx 3
2x 2(1 − x)
x 1−x
1−x x
3(1 − x) 3x
3
1
12
10
1
12
4
3
2
+ Ly − −
+
−
−
Lx 2 +
− −
Ly 2 +
L y + π2
x 1−x
x
x 1−x
x
1−x
1 − x 3x
16
16
4
1
10
1
4
2
3
2 2
+
−
−
−
Lx + π − −
+ Ly
+ Ly π
1−x
x
3x 3(1 − x)
3x 3(1 − x)
x 3(1 − x)
4Ly 2
16
20
32
32
5
20
1
16
+
+
−
−
−
+
ζ3 −
+ −
Lx 4
−
1−x
x
x
1−x
1−x 1−x
x
12x 12(1 − x)
2Ly
4
2
4
1
3
1
2
2
+ Ly
−
+
+
+ π2
−
+
Lx 3 +
Ly 2 +
3(1 − x) 3x
1 − x 3x
2x 2(1 − x)
x
3x 1 − x
8
2
2
6
11
16
6
16
1
2
3
2
2
+
−
−
−
Lx +
Ly + − −
Ly + π
−
Ly
x
3x 3(1 − x)
x 1−x
1 − x 3x
1−x
x
6
18
32
4
10
2
16
4
+
−
+
−
+
ζ3 −
Lx + Li2 (x)
Lx 2
+π2
3x 1 − x
1−x
x
1−x
x
1−x x
6
6
16
14
1
12
14
4
2
2
2
+ Ly
−
+
−
−
+
Lx + π
+ Ly π
1−x x
1−x
x
3(1 − x) 3x
3(1 − x) x
8
12
24
6
8
16
12
6
16
+
−
−
−
−
+ Li4 (x)
+ Li3 (x) Ly
+ Lx
−
1−x
1−x
x
x 1−x
x
1−x
1−x
x
24
12
1
31
47
2
5
+S1,3 (x)
−
−
−
+ Ly 4 −
+ π4
+ Ly 3
1−x
x
12x 12(1 − x)
180x 36(1 − x)
x
14
8
4
14
4
12
16
4
2 4
+
−
−
+
+π
+ S1,2 (x) Lx
+ Ly
+
+
3(1 − x)
x 1−x
1−x
x
1−x x
1−x
x
20
8
20
8
4
20
4
32
6
+S2,2 (x)
−
+
−
+ − −
ζ3 + L y π2
+
ζ3 +
x
1−x
x 1−x
x 3(1 − x)
x 1−x
1−x
16
64
64
−
+
+
x
1−x
x
2Ly
5
6
6
1
5
12Lx
1
3
2 2Lx
+iπ Lm − −
+
−
−
+
+ Lm
+ Lm
Lx 2 +
3x 3(1 − x)
x
1−x 1−x x
x 1−x
x
12Ly
1
10
16
16
2
10
4
1
+
+
+
+
+ π2
+
+ − −
Lx 3
+Ly 2
x 1−x
3x 3(1 − x)
1−x 1−x
x
3x 3(1 − x)
4Ly 10π2 16
1
1
3
6
16
1
2
2
+ Ly
−
−
−
−
−
Lx +
Lx + Ly π
−
x 1−x
x
x
3x
x
3x 1 − x
1−x
29
6
2
4
6
2
4Lx
2
3
+S1,2 (x) − −
+
+ Ly − −
+ Li3 (x)
+ Li2 (x) −
x 1−x
3x 3(1 − x)
x 1−x
1−x
2
2
6Ly
18
2
4
14
2π
16
16
4
2
+
+
+
−
−
−
+
ζ3 −
+Ly − −
x 1−x
1−x x
x
1−x
1 − x 3(1 − x) 1 − x
x
r !
m2
+O
.
s
References
[1] ATLAS, CERN-LHCC-1999-015, ATLAS-TDR-015 (1999), (available at http://cdsweb.cern.ch/).
[2] CMS, CERN-LHCC-2006-021, CMS-TDR-008-2 (2006), (available at http://cdsweb.cern.ch/).
[3] P. Nason, S. Dawson and R.K. Ellis, Nucl. Phys. B303 (1988) 607
[4] P. Nason, S. Dawson and R.K. Ellis, Nucl. Phys. B327 (1989) 49
[5] W. Beenakker et al., Phys. Rev. D40 (1989) 54
[6] W. Beenakker et al., Nucl. Phys. B351 (1991) 507
[7] M.L. Mangano, P. Nason and G. Ridolfi, Nucl. Phys. B373 (1992) 295
[8] J.G. Körner and Z. Merebashvili, Phys. Rev. D66 (2002) 054023, hep-ph/0207054
[9] W. Bernreuther et al., Nucl. Phys. B690 (2004) 81, hep-ph/0403035
[10] M. Cacciari and P. Nason, Phys. Rev. Lett. 89 (2002) 122003, hep-ph/0204025
[11] M. Cacciari et al., JHEP 07 (2004) 033, hep-ph/0312132
[12] B. Mele and P. Nason, Nucl. Phys. B361 (1991) 626
[13] K. Melnikov and A. Mitov, Phys. Rev. D70 (2004) 034027, hep-ph/0404143
[14] A. Mitov, Phys. Rev. D71 (2005) 054021, hep-ph/0410205
[15] A. Mitov, S. Moch and A. Vogt, Phys. Lett. B638 (2006) 61, hep-ph/0604053
[16] A. Banfi, G.P. Salam and G. Zanderighi, (0400), arXiv:0704.2999 [hep-ph]
[17] N. Kidonakis and G. Sterman, Nucl. Phys. B505 (1997) 321, hep-ph/9705234
[18] R. Bonciani et al., Nucl. Phys. B529 (1998) 424, hep-ph/9801375
[19] N. Kidonakis et al., Phys. Rev. D64 (2001) 114001, hep-ph/0105041
[20] R.D. Ball and R.K. Ellis, JHEP 05 (2001) 053, hep-ph/0101199
[21] G. Marchesini and E. Onofri, JHEP 07 (2004) 031, hep-ph/0404242
[22] M. Czakon, A. Mitov and S. Moch, Phys. Lett. B651 (2007) 147, arXiv:0705.1975 [hep-ph]
[23] S. Catani, Phys. Lett. B427 (1998) 161, hep-ph/9802439
[24] G. Sterman and M.E. Tejeda-Yeomans, Phys. Lett. B552 (2003) 48, hep-ph/0210130
[25] A. Mitov and S. Moch, JHEP 05 (2007) 001, hep-ph/0612149
[26] C. Anastasiou et al., Nucl. Phys. B605 (2001) 486, hep-ph/0101304
[27] Z. Bern, A. De Freitas and L.J. Dixon, JHEP 06 (2003) 028, hep-ph/0304168
[28] E.W.N. Glover and M.E. Tejeda-Yeomans, JHEP 06 (2003) 033, hep-ph/0304169
30
(A.1)
[29] T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, Phys. Lett. B400 (1997) 379, hep-ph/9701390
[30] M. Czakon, Nucl. Phys. B710 (2005) 485, hep-ph/0411261
[31] J.G. Körner, Z. Merebashvili and M. Rogal, Phys. Rev. D73 (2006) 034030, hep-ph/0511264
[32] S. Catani, S. Dittmaier and Z. Trocsanyi, Phys. Lett. B500 (2001) 149, hep-ph/0011222
[33] S. Mert Aybat, L.J. Dixon and G. Sterman, Phys. Rev. D74 (2006) 074004, hep-ph/0607309
[34] W. Bernreuther et al., Nucl. Phys. B706 (2005) 245, hep-ph/0406046
[35] S. Moch, J.A.M. Vermaseren and A. Vogt, JHEP 08 (2005) 049, hep-ph/0507039
[36] T. Gehrmann, T. Huber and D. Maitre, Phys. Lett. B622 (2005) 295, hep-ph/0507061
[37] S. Moch, J.A.M. Vermaseren and A. Vogt, Phys. Lett. B625 (2005) 245, hep-ph/0508055
[38] C. Anastasiou et al., Nucl. Phys. B601 (2001) 318, hep-ph/0010212
[39] A. De Freitas and Z. Bern, JHEP 09 (2004) 039, hep-ph/0409007
[40] M. Czakon, J. Gluza and T. Riemann, Phys. Rev. D71 (2005) 073009, hep-ph/0412164
[41] M. Czakon, J. Gluza and T. Riemann, Nucl. Phys. B751 (2006) 1, hep-ph/0604101
[42] S. Actis et al., (2007), arXiv:0704.2400 [hep-ph]
[43] S. Laporta, Int. J. Mod. Phys. A15 (2000) 5087, hep-ph/0102033
[44] V.A. Smirnov, Phys. Lett. B460 (1999) 397, hep-ph/9905323
[45] J.B. Tausk, Phys. Lett. B469 (1999) 225, hep-ph/9909506
[46] M. Czakon, Comput. Phys. Commun. 175 (2006) 559, hep-ph/0511200
[47] S. Moch and P. Uwer, Comput. Phys. Commun. 174 (2006) 759, math-ph/0508008
[48] H.R.P. Ferguson and D.H. Bailey, (1992), (see e.g. http://mathworld.wolfram.com/PSLQAlgorithm.html).
[49] V.A. Smirnov, Nucl. Phys. Proc. Suppl. 135 (2004) 252, hep-ph/0406052
[50] G. Heinrich and V.A. Smirnov, Phys. Lett. B598 (2004) 55, hep-ph/0406053
[51] S. Dittmaier, P. Uwer and S. Weinzierl, Phys. Rev. Lett. 98 (2007) 262002, hep-ph/0703120
[52] S. Moch, J.A.M. Vermaseren and A. Vogt, Nucl. Phys. B688 (2004) 101, hep-ph/0403192
[53] A. Vogt, S. Moch and J.A.M. Vermaseren, Nucl. Phys. B691 (2004) 129, hep-ph/0404111
31