arXiv:0705.1975v1 [hep-ph] 14 May 2007
DESY 07-064
SFB/CPP-07-19
May 2007
arXiv:0705.1975v1
Heavy-quark production
in massless quark scattering 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 Institute
of Nuclear Physics, NCSR “DEMOKRITOS”
15310 Athens, Greece
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 the quark–
anti-quark–annihilation channel in the limit when all kinematical invariants are large compared
to the mass of the heavy quark. Our result is exact up to terms suppressed by powers of the
heavy-quark mass. The derivation is based on a simple relation between massless and massive
scattering amplitudes in gauge theories proposed recently by two of the authors as well as a direct
calculation of the massive amplitude at two loops. The results presented here form an important
part of the next-to-next-to-leading order QCD contributions to heavy-quark production in hadronhadron collisions.
The hadro-production of heavy quarks, especially of top-quarks, is an important process at
hadron colliders. Thus far, the Tevatron has provided us with a wealth of information on the topquark, most prominently with a precise measurement of its mass. In the future, the LHC is expected
to accumulate very high statistics for the production of t t¯-pairs, approximately 8 · 106 events per
year in the initial low luminosity run [1]. At the LHC the uses of t t¯-pairs are e.g. for energy
scale calibration, for background estimates and last but not least, for precision measurements of
Standard Model parameters.
The present and in particular the anticipated future experimental precision on heavy-quark
hadro-production requires theory predictions to include radiative corrections in Quantum Chromodynamics (QCD) beyond the next-to-leading order (NLO). For instance, at the LHC the total cross
section for t t¯-production at NLO in QCD is accurate to O (15%) only, which has to be contrasted
with expected precision for the top-mass measurement of O (1GeV). Another important process
is the production of b-quarks at moderate to large transverse momentum. NLO QCD corrections
for heavy-quark hadro-production have been known since long, see e.g. Refs. [2–8]. Knowledge
of the radiative QCD corrections at next-to-next-to-leading order (NNLO) will certainly improve
the stability of theory predictions with respect to scale variations and provide a match with precise
parton evolution at NNLO [9, 10].
In this letter, we present results for the virtual QCD corrections at two loops for the pairproduction of heavy quarks in the qq̄-annihilation channel. To be precise, we calculate the interference of the two-loop amplitude with the Born one and we work in the limit of fixed scattering
angle and high energy, where all kinematic invariants are large compared to the heavy-quark mass.
Thus, our result contains all logarithms in the heavy quark mass as well as all constant contributions (i.e. the mass-independent terms). Throughout this letter we neglect power corrections in the
heavy-quark mass.
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 [11, 12] to the case of
massive partons [13]. In essence, it results in an extremely simple universal multiplicative relation
between a massive QCD amplitude in the small-mass limit and its massless version [13]. In this
way, we can largely use for our derivation the results of Ref. [14], where the NNLO QCD corrections to massless quark-quark scattering (i.e. qq̄ → q′ q̄′ ) have been computed. 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. As an added benefit, this approach provides us
with a non-trivial, albeit complicated, check of the massless results. Moreover, it makes it possible
to systematically calculate power corrections in the mass, which can improve the convergence of
the small-mass expansion. This is certainly relevant in the case of the top-quark pair-production at
the LHC (less so, perhaps, for b-quark production).
We would like to emphasize that the agreement between our prediction based on the approach
of Ref. [13] and our direct calculation constitutes the first non-trivial check of this factorization approach at two loops. The formalism of Ref. [13] has recently been also applied to the re-derivation
of the two-loop QED corrections to Bhabha scattering [15] confirming earlier results on the two1
loop radiative photonic corrections [16–19] in that process.
Setting the stage
The pair-production of heavy quarks in the qq̄-annihilation channel corresponds to the scattering
process,
q(p1 ) + q̄(p2 ) → Q(p3 , m) + Q̄(p4 , m) ,
(1)
where pi denote the quark momenta and m the mass of the heavy quark. Energy-momentum
conservation implies
µ
µ
µ
µ
p1 + p2 = p3 + p4 .
(2)
Following the notation of Ref. [14] 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 . It may be
written as a series expansion in the strong coupling αs ,
i
α 2
h
α s
s
|M (2) i + O (α3s ) ,
(3)
|M (1) i +
|M i = 4παs |M (0) i +
2π
2π
where 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 b
+ O (α3s ) ,
(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
[20, 21]
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)
As we work in a general non-Abelian SU(N)-gauge theory we set 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.
The squared amplitude for the process (1) summed over spins and colors 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 .
(6)
Then it is convenient to define the function A (ε, m, s,t, µ) for the spin and color averaged amplitudes as
∑ |M (q + q̄ → Q + Q̄)|2
2
=
1
A (ε, m, s,t, µ) ,
4N 2
(7)
which has a perturbative expansion similar to Eq. (3),
α α 2
s
s
8
3
2 2
4
6
A + O (αs ) .
A (ε, m, s,t, µ) = 16π αs A +
A +
2π
2π
In terms of the amplitudes the expansion coefficients in Eq. (8) may be expressed as
2
t + u2
4
(0)
(0)
2
− ε + O (m) ,
A = hM |M i ≡ 2(N − 1)
s2
6
(0)
(1)
(1)
(0)
A = hM |M i + hM |M i ,
A 8 = hM (1) |M (1) i + hM (0) |M (2) i + hM (2) |M (0) i ,
(8)
(9)
(10)
(11)
where we have neglected powers in the heavy-quark mass m in A 4 . Expressions for A 6 with
the complete heavy-quark mass dependence using dimensional regularization can be obtained e.g.
from Ref. [7, 8]. The loop-by-loop contribution hM (1) |M (1) i in dimensional regularization in A 8
and also with the full heavy-quark mass dependence can be computed with the help of Ref. [22].
In this letter, 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.
Massive amplitudes from QCD factorization
Let us briefly recall the key features of Ref. [13] to calculate loop amplitudes with massive partons
from massless ones. The QCD factorization approach rests on the fact that a massive amplitude
M [p],(m) for any given physical process shares essential properties in the small-mass limit with the
corresponding massless amplitude M [p],(m=0) . The latter one, M [p],(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 factorizing poles in ε in dimensional regularization after the usual ultraviolet renormalization is
performed. In the former case, the soft singularities remain in M [p],(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. [23].
Thus, in the small-mass limit the differences between a massless and a massive amplitude can
be thought of as a mere change in the regularization scheme. As an upshot, QCD factorization
provides a direct relation between M [p],(m) and M [p],(m=0) which can be cast in the remarkably
simple and suggestive relation
M [p],(m) =
∏
i∈ {all legs}
1
(m|0) 2
Z[i]
× M [p],(m=0) .
(12)
The function Z (m|0) is process independent and depends only on the external parton, i.e. quarks
in the case at hand. For external massive quarks Q it is defined as the ratio of the on-shell heavyquark form factor and the massless on-shell one, both being known [24–26] to sufficient orders in
3
αs and powers of ε. An explicit expression for
(m|0)
Z[Q]
∞
= 1+ ∑
j=1
α j
s
2π
Z ( j) ,
(13)
up to two loops is given 1 in Ref. [13]. Exploiting the full predictive power of the relation Eq. (12)
and applying it to the process Eq. (1) we get
2Re hM (0) |M (2) i(m) = 2Re hM (0) |M (2) i(m=0) + Z (1) A 6,(m=0) + 2Z (2) A 4,(m=0) ,
(14)
which assumes the hierarchy of scales m2 ≪ s,t, u , i.e. we neglect terms O (m). Eq. (14) predicts
the complete real part of the squared amplitude hM (0) |M (2) i except for those terms, which are
linear in nh , i.e. the number of heavy quarks. These two-loop contributions have been excluded
explicitly from the definition [13] of Z (m|0) , as one needs additional process dependent terms for
their description. Their incorporation for the case of Bhabha scattering was presented in Ref. [15]
in agreement with the direct calculation [27].
Direct calculation of the massive amplitude
An alternative method is the direct calculation of all necessary massive Feynman diagrams together
with an expansion in the small mass. The advantage of this approach is an independent check of
Eq. (14) as well as of the corresponding massless results. Moreover, it also allows for a relatively
easy access to all heavy-quark loop corrections. Presently, these have not been obtained from the
QCD factorization method, since they are related to the process dependent contributions.
We performed our computation using the DiaGen/IdSolver system of one of the authors (M.C.).
After the diagram generation phase, all the integrals have been reduced to a set of 145 masters with
the help of the Laporta algorithm [28] extended by topological symmetry properties, where, however, integrals which are related by a t ↔ u exchange of the Mandelstam variables are considered
as being independent. The evaluation of the masters proceeded similarly to the methodology developed in Ref. [29,30]. The main idea was to construct Mellin-Barnes [31,32] representations for
all the integrals, followed by a subsequent analytic continuation with the MB package [33] and an
expansion in the mass by closing contours. The resulting integrals have then been transformed into
series representations, some two-fold, and resummed with the help of XSummer [34] in the cases
of non-trivial dependence on the kinematic variables. Some constants, though, were not given by
harmonic series and have been computed with the help of the PSLQ algorithm [35]. Needless to
say that all of the above steps were performed fully automatically.
In our calculation, we have only been able to obtain the leading color term and the full dependence on the number of light and heavy fermion species. The reason is that the terms subleading in
color contain contributions from non-planar graphs. Fig. 1 shows the bottleneck cases with 6- and
8-fold Mellin-Barnes representations, for the six and seven liners respectively. The representations
1 Note
the different normalization αs /(4π) of the coupling used in that reference.
4
Figure 1: The most complicated non-planar topologies, together with the multiplicities of the respective
master integrals for qq̄ → QQ̄ scattering. The thick
lines are massive.
Figure 2: A heavy-quark loop
contribution to the wave-function
renormalization of the light quarks.
have been obtained similarly as in Ref. [32], i.e. from the Feynman-parametric representation of
the two-loop graphs and not by integrating loop-by-loop, the strategy adopted for planars. In this
way, the mass expansion generates√
integrals that are at worst as complex as the massless ones, plus
some new terms that behave as 1/ m2 and must cancel in the complete result. The sheer amount
of remaining difficult 4-fold integrals turned out to be presently intractable with the developed
software.
Finally, as mentioned above, we renormalized the mass of the heavy quark and the external
wave functions in the on-shell scheme (besides the MS renormalization of the coupling constant).
The former results are now known up to three-loop level from Ref. [36, 37]. A peculiarity of the
light quark states is that they obtain contributions to the wave-function renormalization constants
from heavy-quark loops, see Fig. 2. In fact, we have
2
α 2
5
m
1 1
s
Z2 = 1 +
.
(15)
−
− log
CF TF nh
2π
4ε 2
µ2
24
Results
We are now in a position to present our result for qq̄ → QQ̄ scattering for the interference of the
two-loop and Born amplitude,
2Re hM (0) |M (2) i =
n
n
1
2(N 2 − 1) N 2 A + B + 2 C + Nnl Dl + Nnh Dh + l El + h Eh + (nl + nh )2 F ,
N
N
N
(16)
which we choose to express by grouping terms 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. As detailed
above, the coefficients A, Dl , El and F have been computed with both methods, i.e. by employing
our universal multiplicative relation (12) between the massive and massless amplitudes as well as
a direct evaluation of the loop integrals in the small-mass expansion. The terms linear in nh , Dh
5
and Eh could also be easily obtained from a direct Feynman diagram calculation, while for B and
C the approach based on factorization proved more powerful.
As the only dimensionless kinematic variable in our problem, we choose x = −t/s and 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) .
(17)
Lm = log
s
µ
The different components now read
x2 x 1
1
21x2 21x
1
1
2
2
A =
− +
−
+ Lx −2x2
+ 3 Lm x − x +
+ Ls −x + x −
+
2 2 4
ε
2
2
4
4
29x2 29x
1
19
1
2
2
2
−
+2x − 1) +
+ 2 Lm Ls −2x + 2x − 1 + Ls x − x +
+ Lm
8
ε
2
6
6
2
23
13
19x
19x
+Lx −2x2 + 2x − 1 +
+
+ Lx 4x2 − 4x + 2 −
+ Ls −
+ 2x2
12
6
6
12
2
x
1
26x2 55x 23
173x2 173x
x
5x 5
2
2
+
−
−
+π − + −
Lx + −
Lx +
− +
2 4
3
6
6
72
72
6 6 12
2
x 1
1
x
205
2x2 2x 1
+ −
+
L3m − + −
+ Lm L2s 2x2 − 2x + 1 + L3s −
−
144
ε
3 3 6
3
3
3
2
1
7x
7x
1
1
2
2
2
2
+ + Lx 4x − 4x + 2 −
+Lm −x + x + Lx x − x +
−
+ Lm Ls −
2
2
3
3
6
2
3
x
3x
x
47x2
1 x
+L2s − + + Lx −4x2 + 4x − 2 −
−
− x2 Lx −
+ Lm
L2x +
2 2
4
4 2
2
12
2
2
11x 1
487x
8x
5
47x 35
487x
−
−
+
−
+
+ Ls −4x2 + 5x −
L2x +
Lx +
12
8
2
3
3
3
36
36
2
2
x
x 1
1
5x
601
3
x 1
x
− +
+x−
+
+
L3x + − + Ly −x2 + −
+
L2x
+π2
3 3 6
72
3
2
2
2 4
4
2
1
5x
5
43x2 151x
2
2 4x
+Li2 (x) −2x + x −
−
+π
− +
Lx +
+ 10 Lx
2
3
12
3
6
12
25
1
23x2
23x2 5x
9907x2 9907x
2
2
+
+π −
+ +
+ Li3 (x) 2x − x +
+ −
−
432
432
72
72 144
2
6
2
2
2x
x 1
2x 1
17x 17
10945
4x2
4 x
3
3
−
− +
− +
+
ζ3 −
+ Lm
+ Lm Ls
+ Lm Ls −
6
12
864
4 4 8
3
3 3
3
2
2
2
4x 2
x 1
11x
x 1
5
x
x
11x
+ −
− +
+
+ Lx − + −
+ L4s
+ L3m −
−
3 3
3 3 6
18
18
3 3 6
36
5
5x2 5x
4x2 4x
2
2
2
+ + Lx −2x + 2x − 1 −
+ + Lx −4x2 + 4x − 2
+Lm Ls −
+ Lm Ls −
3
3
6
3
3
2
2
2
x
14x
8x 4
3x
10
8x
x 1
5
3 14x
2
2
−
+ Lx
− +
−
−
+
+ Lm
Lx +
Lx
− + Ls
3
9
9
3
3 3
9
4 8
2
4
247x2 247x 283
1
23x2 23x 83
2
2
−
+
−
+
+
+ Lm Ls x −
Lx + 2x − 3x Lx +
36
36
72
2
2
2
12
1
ε4
6
2
x 1
14x2 11x 10
37x2 37x
x
5
2
2
−
+
+
+π − + −
Lx +
Lx −
4x − 5x +
2
3
3
3
4
4
3 3 6
2 3 35
x
1
1
x Lx
x 1
2
2
2
−
+ − + Ly −x + −
+ Lm
+
Lx + Li2 (x) −2x + x −
Lx
8
3
2
2 4
4
2
1
x 1
781x2 781x
7x2 x
19x
2
2
2
2
+π x − +
+
+π −
+ −
− 2 Lx −
+ −4x +
4
2 4
72
72
12 3 24
2
7x
10x 5
499
2x2
1
2
−
+
− 2x + 1 L3x
+
ζ3 −
+ Ls −
+Li3 (x) 2x − x +
2
3
3
3
144
3
4x
1
86x2 173x
1
2
2
2
+ Ly 2x − x +
+
+
Lx + Li2 (x) 4x − 2x + 1 Lx + −
+
3
2
3
3
6
2
2
8x
5x 5
2003x
49
2003x
+π2 −
+ −
−
+ Li3 (x) −4x2 + 2x − 1
−
Lx +
3
3 6
3
216
216
2
2
61x 91
17x 17
1
23x
x
919
43x
2
2
+
−
−
+
+π −
+
ζ3 −
+ −x − +
L4x
36
36
72
3
3
6
432
24 48
2
2
7x2 13x
7x
7
5x 11
10x2
x
3
2
+ −
+
+ Ly
− 2x + 1 −
Lx +
− − +
Ly +
18
12
3
12
2
4
8
6
617
101x
3
13x 13
25x2 25x 25
+
+ π2 −
+
−
−
Ly +
+
L2x + S1,2 (x) −2x2
+
12
12
72
6
12
24
1 − x 144
2
260x2 139x
11x 17
25x2 53x 47
x
11
2
2
−
+π −
+
−
−
Lx +
+ Ly π − +
−5x +
2
9
4
9
36
18
3
6
12
2
2
2
52x
31x 31
17845x
x
1
1027
17845x
11x
4
−
+
−
+π −
−
+
+
ζ3 +
Lx +
3
3
6
72
2592
2592
80
240 480
2
3
259x 403
11
2
2
2 1009x
+Li4 (x) 2x + 3x −
−
+
+ S2,2 (x) 2x + 5x −
+π
2
2
432
432
864
11
7x2 13x
13
−
+ Lx −8x2 + 2x − 1 + Ly 2x2 + 5x −
+Li3 (x) −
−
+ Li2 (x) 7x2
3
6
2
6
2
2
7x
13x
11x 17
13
x
11
7x 7
2
2
2
+
+ Ly −2x − 5x +
−
Lx +
+
Lx + π − +
− +
2 4
3
6
2
6
3
6
12
2
11
55x 149
14x
77287
+Ly −2x2 − 5x +
+
+
,
(18)
ζ3 + −
ζ3 +
2
9
18
36
5184
+L2s
B =
2
31x2 31x
1
1
2
+
+ Ly −4x2
−x + x −
+ 3 Lm −2x2 + 2x − 1 + Ls 2x2 − 2x + 1 −
2
ε
4
4
27
1 2
+ 2 Lm Ls 4x2 − 4x + 2 + L2s −2x2 + 2x − 1
+4x − 2) + Lx 6x − 6x + 3 −
8
ε
2
35
47x
47x
49x2 49x
2
2
+Lm −
+
+ Ly −4x + 4x − 2 + Lx 6x − 6x + 3 −
−
+ Ls
6
6
12
6
6
67x2
15x 15
37
−
+ −6x2 +
L2x +
+Lx −12x2 + 12x − 6 + Ly 8x2 − 8x + 4 +
12
2
4
3
29
143x
415x2 415x
52x2 49x 20
+ Ly 4x2 − 4x + 2 +
+
+ Ly −
+
−
−
Lx −
6
3
36
36
3
3
3
1
ε4
7
2
2x 1
17
1
1
17x2 17x 17
2
3 2x
2
−
+
− +
+ Ly 2x − x +
−
+
Lm
+π
12
12
24
2
9
ε
3
3
3
2
4x 2
3
2
2
3 4x
2
2
2
+Lm Ls −4x + 4x − 2 + Ls
− +
+ Lm 2x − 2x + Lx −3x + 3x −
3
3 3
2
2
25x
25x
−
+ Lx −12x2 + 12x − 6 + Ly 8x2 − 8x + 4
+Ly 2x2 − 2x + 1 + 1 + Lm Ls
3
3
2
5
9x
9x
3x
13
2
2
2
+ + Ly −8x + 8x − 4 + Lx 12x − 12x + 6 −
+ Ls −
+ Lm
+
6
2
2
4
2
2
2
7x
3
9x
16x
1
16x
+ L2y x −
+ Ly −2x2 + x + 1 + π2
L2x + 3x2 −
Lx −
+
−
4
2
3
2
3
6
2
7x
7
55x
5
46x
15
− +
+
+ Ly −8x2 + 8x − 4
+
+ Ls 12x2 − 15x +
L2x + −
6 12
4
2
3
3
2
2
85x
17x 17
14
85x
17x
16x2
2
2
2
−
+ Ly −4x + 2x − 1 + π −
+
−
−
Lx +
+ Ly
3
18
18
6
6
12
3
10x 4
15x
31
3
9
x 1
−
−
+ Ly 3x2 − +
−
+ −x2 − 3x +
L3x +
−
L2x
3
3
18
2
2
2 4
4
50x2 137x
1
31x2 41x 41
3
2
2
2
− x Ly − Ly −
+
+π −
+
−
Lx +
+Li2 (x) 6x − 3x +
2
2
3
12
3
6
12
2
419x
419x
5
3
−15) Lx −
+
+ L2y 4x −
+ Li3 (x) −6x2 + 3x −
+ S1,2 (x) −4x2 + 6x − 3
108
108
2
2
2
2
2
2
7x 7
101x
91
193x
11x
86x
2 8x
2 263x
3 2x
− +
−
−
−
+π
−
+π
+ Ly
+Ly
3
3
6
72
72
144
3
6
3
3
2
2
2
91x
73x 73
x 1
4x
47
413
x
4x
11
−
+
+
+
+
ζ3 +
+ L4m − + −
+ L3m Ls −
+
6
2
6
6
12
108
2 2 4
3
3
2
2
2
2
8x 4
2x 1
11x
2x
2
8x
11x
2x
2x
− +
+ −
−
+ Ly −
+
− + LmL3s
+ L4s −
+ L3m
3
3
3 3
3
3 3
18
18
3
3
2
1
x
1
1
x
−
+ Lx x2 − x +
−
+ L2m Ls − + + Ly −4x2 + 4x − 2 + Lx 6x2 − 6x + 3
3
2
36
3 3
2
2
1
1
14x
14x
2
2
3 16x
2
− + Lm Ls −
+
+ Ly −8x + 8x − 4 + Lx 12x − 12x + 6 −
+ Ls
6
3
3
3
9
2
9x
16x
16x 8
3 3x
2
16x
+ Lx −8x2 + 8x − 4 + Ly
−
+
−
+
+ L2m
L2x +
−
9
3
3
3
9
8
4
4
2
2
2
3x
x
2x 1
157x
157x
2x
x 1
2 1
2
2
−
+ Ly
−
+π −
+ −
Lx −
+
+ Ly x − −
2
36
4 2
36
3
3
3
2 2
7x2 7x
3
229
− 3x L2x + 9x − 6x2 Lx + 7x2 + L2y (1 − 2x) − 7x + π2 −
+
+ Lm Ls
−
72
2
3
3
2
11x
2x
7
2
15
−
+ Ly 8x2 − 8x
−
+ Ly 4x2 − 2x − 2 −
+ L2s −12x2 + 15x −
L2x +
6
3
2
3
3
8
17x2 17x 17
40x2 40x
28x2 34x
+4) −
−
+ π2
−
+
−
Lx +
+ L2y 4x2 − 2x + 1 + Ly
3
9
9
6
6
12
3
3
161
3
3x
3x 3
26
+ Ly 3x2 − +
+
+ Lm −x2 L3x +
−
L2x + Li2 (x) 6x2 − 3x
+
3
36
2
2
4
4
2
8
57x
1
3x 3
115x2
115x
3
2
2
2
2
+ π −3x + −
+ Ly x −
Lx + 12x −
+ 6 Lx −
+
+
2
4
2 4
9
2
9
2
2
3
4x 2
2x
9x
+Li3 (x) −6x2 + 3x −
− +
−x
+ S1,2 (x) −4x2 + 6x − 3 + L3y
+ π2
2
3
3 3
4
37x2 28x 14
3
5
67
5
13x
2
2
2
+ π 2x − 3x +
−
+
+ Ly −8x +
−
+
ζ3 −
−
24
2
2
2
3
3
3
18
23x
5
1
+Ls 2x2 + 6x − 3 L3x + −
+ Ly −6x2 + x −
+
L2x + Li2 (x) −12x2 + 6x − 3 Lx
3
2
6
2
2
181x
41x 41
68
1957x2
100x
2 62x
2 4
2
−
+π
−
+
+ Ly
+
Lx −
+ (2x − 1)Ly + 2Ly +
3
6
3
3
6
3
54
3
2
2
14x 7
229x 421
67x
1957x
4x
2x
+ L3y −
+
−
−
+
+
+ π2
+ S1,2 (x) 8x2 − 12x
−
3
54
3
3
3
36
36
72
172x2
16x2 22x 11
97
2
2
+6) + Li3 (x) 12x − 6x + 3 + Ly −
+ 57x + π −
+
−
−
3
3
3
3
3
1
553
x
x2 35x
32x2
91x2 73x 73
+
−
+ Ly −
ζ3 −
+ 3x2 + −
L4x + − −
+ −
3
3
6
27
8 16
18
12
3
2
2
5x 17
29x 13
7x
x
95x2
16x 8
7
365x
−
+ −
−
−
+ π2
+
+
L3x +
L2y +
Ly +
3
3
4
2
4
8
6
12
6
72
6
2
103x 103
x 1
1027
4x 2
7
2x
43x2
−
+
+
+ −
−
−
L2x +
−
L3y +
L2y + π2 −
12
24
1−x
144
3
3 3
2 4
3
2
2
153x
149x 155
15x 11
304x
125x
−
+
+ π2 −
+
+
+
+ 2 Ly −
+ −54x2 + 45x
2
4
9
4
18
36
36
1
1181
38959x2 38959x
45
−
+ S2,2 (x) −10x2 + 25x −
ζ3 −
Lx +
+ Li4 (x) −6x2
−
2
72
648
648
2
2
29x 101
1
25x 25
7x2 2x 13
4
3
4 77x
2
−
+ −
−
+
+ Ly −x +
+ Ly −
+π
−x +
2
12
24
9
9 36
45
24
240
281x
19x 19
163 5
+S1,3 (x) 4x2 + 2x − 1 + L2y −
+ π2 −4x2 +
−
+
−
36
3
6
72
x
2
2
14x
47x
3
46
x
+S1,2 (x)
−
+ Ly 8x2 − 8x + 4 + Lx 10x2 − 9x −
+
+ Li3 (x) −
3
3
2
3
3
29
21x
15
23x
+ Ly −10x2 − 3x +
+ Lx 24x2 − 10x + 5 +
+ Li2 (x)
−21x2 +
+
6
2
6
2
2
21
x
23x
29
15
x 25
−
−
+ Ly 10x2 + 3x −
L2x +
−
Lx + π2 −5x2 − +
4
3
6
2
6
6 12
2
2
2
28x 169
829x
28x 25
43x
98x2
50x
520x
+
−
−
+ π2 −
+
−
+
ζ3 + L y
+
18
9
18
9
18
9
3
3
3
2
2353
1585x
605
3797x
107x 71
2
2
+
+
+ 2x − 2x + 1 log(2) −
ζ3 +
+π −
−
3
6
36
216
108
216
41473
+
,
(19)
1296
9
x2 x 1
1
1
1
5x2 5x
2
2
− +
− + Lx −4x2 + 4x
+ 3 Lm x − x +
+ Ls −x + x −
+
C =
2 2 4
ε
2
2
2
2
1
1
2
2
2
2
−2) + Ly 4x − 4x + 2 + 1 + 2 Lm Ls −2x + 2x − 1 + Ls x − x +
+ Lm 3x2 − 3x
ε
2
2
2
2
2
+Lx −4x + 4x − 2 + Ly 4x − 4x + 2 + 1 + Ls −5x + 5x + Ly −8x + 8x − 4 + Lx 8x2
7
73x2
2
−8x + 4) − 2] + 6x − 7x +
L2x + −10x2 + 11x + Ly −12x2 + 12x − 6 − 4 Lx +
2
8
2
53
13x 13
13x
7
73x
+ π2 −
+
−
+ L2y 6x2 − 7x +
+ Ly 10x2 − 9x + 3 +
−
8
4
4
8
2
16
2
2
1
x 1
2x 1
x
2x
+
+ −
L3m − + −
+ Lm L2s 2x2 − 2x + 1 + L3s −
+ L2m −x2 + x
ε
3 3 6
3
3
3
1
2
2
+Ly −2x + 2x − 1 + Lx 2x − 2x + 1 −
+ Lm Ls −6x2 + 6x + Ly −8x2 + 8x − 4
2
2
2
2
+Lx 8x − 8x + 4 − 2 + Ls 5x − 5x + Lx −8x2 + 8x − 4 + Ly 8x2 − 8x + 4 + 2
37x
7
37x2
7x2 7x
1
2
2
2 1
2
− x Lx + 3x − 2x Lx +
+ Ly
−x −
+π −
+ −
+Lm
2
4
2
4
6
6
12
25
+Ly 2x2 − x − 1 +
+ Ls −12x2 + 14x − 7 L2x + 20x2 − 22x + Ly 24x2 − 24x + 12
8
2
73x
73x
13x2 13x
+8) Lx −
+
+ Ly −20x2 + 18x − 6 + L2y −12x2 + 14x − 7 + π2
−
4
4
2
2
2
3
3
2x
13
53
+ 3x −
+
−
+
L3x + −6x + Ly −2x2 − 2x + 1 +
L2x + Li2 (x) −4x2
4
8
3
2
2
2
31x 31
55x
3
2 46x
2
2
+π
−
+
Ly + 3Ly − 24x +
− 10 Lx
+2x − 1) Lx +
3x −
2
2
3
3
6
3L2y 429x2 429x
32x2 17x 29
2x2 5x 5
2
3
+
−
+π −
+
−
− +
+ Ly −
+ S1,2 (x) 4x2
−
2
16
16
3
3
24
3
3 6
41x
46x2 61x 61
13
2
2
2
−6x + 3) + Li3 (x) 4x − 2x + 1 + Ly 24x −
+π −
+
−
+
2
3
3
6
2
2
2
2
28x 14
x 1
2x 1
34x
283
x
2x
+
−
− +
− +
+ −
ζ3 +
+ L4m
+ L3mLs
3
3
3
32
4 4 8
3
3 3
2
2
2
2
4x 2
x 1
2x 1
2x
2x
2x
4x
4 x
3
3
+ −
− +
+ −
−
+ Ls
+ Lm Lx −
+ Ly
+Lm Ls −
3
3 3
3 3 6
3
3 3
3
3
1
1
+
+
+ L2m Ls 2x2 − 2x + Lx −4x2 + 4x − 2 + Ly 4x2 − 4x + 2 + 1 + Lm L2s 6x2
3
6
16x2 16x
10x2 10x
2
3
2
+
+ Ly −
+
−6x + Lx −8x + 8x − 4 + Ly 8x − 8x + 4 + 2 + Ls −
3
3
3
3
8
x
4
3x
5x2
x 1
16x2 16x 8
−
−
+
−
+ L2y
+ Lx
−
+ L2m
L2x + x2 −
Lx −
3
3
3
3
3
2 4
2
2
2
2
2x 1
5x
3
x 1
1
2 2x
2
− +
+ + Ly −x + +
+π
−
+ LmLs (2x − 1)L2x + 4x2
−
4
2
2 2
3
3 3
4
1
ε4
10
2
7x 7
25
37x2 37x
2
2
2 7x
+
+ Ly (2x − 1) + Ly −4x + 2x + 2 + π
− +
−
−6x) Lx −
2
2
3
3
6
4
2
73x
73x
+L2s 12x2 − 14x + 7 L2x + −20x2 + 22x + Ly −24x2 + 24x − 12 − 8 Lx +
−
4
4
2 3
2
53
13x 13
2x Lx
13x
+
−
+ L2y 12x2 − 14x + 7 + Ly 20x2 − 18x + 6 +
+ Lm
+π2 −
2
2
4
8
3
1
1
19x
+ −x + Ly −2x2 + x −
+ π2 2x2
+
L2x + Li2 (x) −4x2 + 2x − 1 Lx + −8x2 +
2
2
2
189x
1
189x2
5x2 2x 1
2x2 4x
2 1
2
3
+ Ly
−x −
+π −
+ +
+
− 4 Lx +
+ Ly −
−x +
2
8
2
8
3
3
4
3
3
2
13x
−
+ π2 −2x2 + 3x
+ S1,2 (x) 4x2 − 6x + 3 + Li3 (x) 4x2 − 2x + 1 + Ly 8x2 −
3
2
5
44x2 38x 19
115
4x2
3
+
−
− 6x + 3 L3x + 12x + Ly 4x2
+
+ −
ζ3 +
+ Ls −
−
2
2
3
3
3
16
3
92x2
2
2
2
2
2
+4x − 2) − 3) Lx + Li2 (x) 8x − 4x + 2 Lx + (3 − 6x)Ly − 6Ly + 48x − 55x + π −
3
2
62x 31
429x
429x
+
−
+
+ Li3 (x) −8x2 + 4x − 2 + S1,2 (x) −8x2
+ 20 Lx + 3L2y −
3
3
8
8
2
2
2
10x 5
34x 29
2
2 92x
2 64x
3 4x
+
−
−
+
+π
+ Ly −48x + 41x + π
+12x − 6) + Ly
3
3
3
3
3
12
3
2
2
68x
5x
56x 28
1
x
283
122x 61
+
−
+
− 13 +
ζ3 −
+ −3x2 − +
L4x +
−
3
3
3
3
3
16
12 24
3
7x
17
7x 7
1
13x 13
+ + Ly 12x2 −
+
−
L3x +
−7x2 + −
L2y + −5x2 −
Ly
3
3
6
12
2
4
4
3
3 3x
35
58x2 34x 17
3
83x
2
2
2
+π −
+
−
+
−
+
Lx + 2x − 4x + 2 Ly +
L2y
−
4
3
3
3
1−x
8
4 2
128x2
125x
7
116x2 89x 89
−
+
+ π2 25x2 − 13x +
− 6 Ly − 48x2 +
+
+ π2
3
3
6
2
6
3
2
164x 82
2479x
97
2479x
−
+
−
+ S1,3 (x) −36x2 + 62x − 31 + L4y −3x2
ζ3 −
Lx +
3
3
4
32
32
2
2x 11
191
71x 71
5x
829x2 191x
3
4
−
+ +
−
+
+ Ly −
+π −
+ S2,2 (x) 8x2
+
12
24
3
3 12
720
720
1440
20x2 29x 29
37 3
2
2 37x
2
−56x + 28) + Li4 (x) 12x − 14x + 7 + Ly
+π −
+
−
− +
4
3
3
6
8
x
2
2
2
2
+S1,2 (x) 10x − 11x + Lx −16x + 56x − 28 + Ly 8x − 8x + 4 + Li3 (x) 10x + 3x
11
2
2
2
L2x
22x − 11x +
+Lx −28x + 18x − 9 + Ly 16x − 20x + 10 − 1 + Li2 (x)
2
265x2
76x2 52x 26
−
+
+ −
+ −10x2 − 3x + Ly −16x2 + 20x − 10 + 1 Lx + π2
3
3
3
6
2
140x
259x 293
61
128x
67x
−
+ π2 −25x2 + 30x −
+
+
ζ3 + Ly 48x2 −
+ −
6
12
2
6
3
3
11
247
39
1621
1631x2 285x
70
2
2
+
+ −2x + 2x − 1 log(2) −
,
ζ3 +
+π −
+
−
3
4
48
16
96
64
Dl =
Dh
(20)
2
2
2
2
2x
x
x 1
x 1
x 1
x
x
1
5x2 5x
− + Lx
− + −
+ 2 Lm − + −
+ Ls − + −
+
2 2 4
ε
3 3 6
3 3 6
9
9
3
2
2
2x 1
5x
2x 1
19
1
5x
1
2x
+ −
− +1
− +
+
+
Lm Ls −
+ L2s x2 − x +
+ Lm
3
3
36
ε
3
3
3
2
3
3
2
2
2
2
40x
4x
10x
40x
4x 2
649x
10x
37
649x
+Ls −
+
+ Lx
− +
−
+ Lx −
+
−2
−
+
9
9
3
3
3
18
108
108
3
3
2
2
x
1
2x 1
589
11x2 11x 11
2x
2
3 x
2
+
−
− +
− +
+π −
+
+ Lm
+ Lm Ls
36
36
72
216
9 9 18
3
3
3
2
2
2
4x
4x 2
8x 4
23x 23
8x
23x
+Lm L2s
− +
+ −
+
−
+ L3s −
+ L2m −
+ Lm Ls −4x2
3
3
3
9
9
9
18
18
36
2
2
2
2
11x
8x 4
73x
5
1
x
8x
73x
2
2 11x
−
+ Lx −
+ −
−
+π −
+ Ls
+
+ Lm
+4x −
3
9
9
3
3 3
9
18
18
6
2
2
2x 1
1
22x 10
91x
20x
17x2
43
91x
x
−
−
+
−
+ π2
+
+ Ls
L2x +
Lx +
+ −
6 12
36
3
3
3
3
3
18
18
18
2
3
2
13x
2x 1
17x 17
19
2x Lx
5
4x2
2x
+
+ −
+ Ly −
+ −
−
+
+
+
L2x + Li2 (x) −
18
36
4
9
9
3
3
3
9
3
2
2
2
4x 2
17x 17
2639x
25x
56x
29
2639x
+ −
+ 8x + π2
−
+
+
Lx + −
−
Lx −
3
3
9
9
9
18
9
216
216
2
2
2
14x 137
4x 2
73x 73
4x
73x
3937
2x
−
+
− +
+
−
,
+ Li3 (x)
+ −
ζ3 −
+π2
27
27
216
3
3
3
9
9
18
432
(21)
1
ε3
2
2x2 2x 1
x
4x2 4x 2
10x2 10x 5
1
x
=
+ −
+ −
−
+
Lm −
+ Ls −
+
+
L2m − +
3
3
3
3
3
3
9
9
9
ε
3 3
2
2
5x
4x 2
1
4x
50x2
1
5x
+ + Lx
− +
+
+ Ls −
− + L2s 2x2 − 2x + 1 + Lm −
6
9
9
3
3
3
18
9
2
2
2
8x
20x
8x 4
131x
20x 10
50x
19
131x
+ Lx
− +
−
+ Lx −
+
−
+
−
+
9
3
3
3
9
27
27
9
9
9
2
2
2
2
x
x 1
7x
7
4x 2
101
4x
2
3 7x
2
2 2x
+π − + −
− +
− +
+
+ Lm
+ Lm Ls
+ Lm Ls
3 3 6
54
9
9
18
3
3
3
3
2
2
2
4x
14x 7
37x
4x 2
2x 1
43
37x
14x
+
−
−
+ Lx −
+ −
− +
+ L3s −
+ L2m
+
3 3
9
9
9
18
18
3
3
3
36
2
2
2
1
4x
8x 4
7x
2
8x
4x
2
2 7x
− + Lx −
+ −
− + Lx −4x + 4x − 2 +
+
+ Ls
+Lm Ls
9
9
3
3
3
9
3
3
6
2
2
2
76x
20x
2x 1
76x
20x 16
46x
40x
9
+Lm
−
+ Lx −
+
−
−
−
+
+ Ls
L2x +
9
9
9
9
9
2
3 3
9
9
2
2
3
2
199x
13x
2x
199x
349
2x Lx
1
2x
14
−
+ π2 x2 − x +
+ −
+ Ly −
+
Lx +
+
+
+
9
27
27
2
54
9
9
3
3
1
ε2
12
2
16x 8
4x2 4x 2
5
224x2 272x
1
2 8x
2
+ −
+
+π
−
+
+
Lx + Li2 (x) −
Lx + −
−
3
9
3
3
3
27
27
3
9
9
2
2
2
130
17x
4x
113x
41x
83
4x 2
113x
−
−
+ π2
−
+
− +
Lx +
+ Li3 (x)
27
54
54
54
54
108
3
3
3
2
76x 38
149
76x
+
−
,
(22)
ζ3 −
+ −
9
9
9
108
El
2
2
x2 x 1
x 1
x 1
1
5x2 5x
x
x
4x2 4x
=
− +
− +
− +
+ + Lx −
+
+ 2 Lm
+ Ls
−
2 2 4
ε
3 3 6
3 3 6
9
9
3
3
2
2
4x
2x
4x 2
2x 1
19
1
2
1
− +
− +
+ Ly
−
+
Lm Ls
+ L2s −x2 + x −
−
3
3
3
3
36
ε
3
3 3
2
2
8x 4
40x2 40x
8x
8x2 8x 4
5x2 5x
+ − 1 + Ls
−
+ Lx −
+ −
− +
+ Ly
+Lm −
3
3
9
9
3
3
3
3
3
3
2
2
2
37
20x
20x2
649x
20x
59x 59
59x
649x
+
+
+ Ly −
+
− 4 + π2
−
+
−
+ Lx
18
108
108
3
3
36
36
72
3
2
2
2
589
x
1
2x 1
4x
x
4x
2x
20x
+4 −
+ −
+
+ L3m − + −
+ L2m Ls −
+ Lm L2s −
−
3
216
9 9 18
3
3 3
3
3
2
2
2
8x 4
23x 23
5
2
8x
23x
55x
− +
−
+
− + L3s
+ L2m
+ Lm Ls 4x2 − 4x +
+ L2s −
3
9
9
9
18
18
36
3
9
2
2
2
55x
16x 8
16x 8
73x
23
16x
16x
73x
+
+ Ly −
+
−
−
+
+
+ Lx
−
+ Lm −
9
3
3
3
3
3
3
9
18
18
2
2
2 4x
x
1
44x 20
43
40x
1027x2
x
− +
−
+
−
+π2
−
+ Ls
L2x + −
Lx +
6 6 12
36
3 3
3
3
3
54
2
2
4x
89x 89
16
1027x
89x
787
40x
2
2 2
−
+π −
+
−
− 12x +
−
+ Ly
+
+Ly
3
3
54
18
18
36
3
3
108
2
2
4x
26x
4x 2
8x 4
8x
112x2
4
10
+ Ly
− +
− +
− x2 L3x +
−
L2x + Li2 (x)
Lx +
9
9
3
3 3
9
3
3
3
9
2
2
50x
26x 16
34x 17
13963x
58
13963x
−16x + π2 −
+
−
+
+ L2y
−
+
Lx −
9
9
9
9
648
648
9
9
2
2
2
8x 4
8x 4
8x 4
8x
8x
4x
+ −
+ −
− +
+Li3 (x) −
+ S1,2 (x) −
+ L3y
3
3
3
3
3 3
9
9
9
2
2
2
202x 631
80x
22x 11
26
112x
2 50x
2 250x
−
+
+
+π
−
+
+ Ly −
−
+π
27
27
216
9
9
9
3
3
9
2
13x
13x 13
10069
+
−
+
,
(23)
ζ3 −
9
9
18
1296
Eh =
1
ε3
2
2
2
2x 1
4x 2
x 1
10x2 10x 5
1
2x
4x
2 x
− +
− +
+
−
− +
Lm
+ Ls
+−
+
Lm
3
3
3
3
3
3
9
9
9
ε
3 3 6
2
5x
8x2 8x 4
8x2 8x 4
5x
1
2
2
+Ls −2x + 2x − 1 + Lm
− + Lx −
+ −
− +
+ Ly
−
9
9
3
3
3
3
3 3
18
2
2
2
2
50x
16x 8
16x 8
19
131x
16x
16x
50x
−
+ Lx −
+
−
−
+
+ Ly
+
−
+Ls
9
9
3
3
3
3
3
3
9
27
1
ε2
13
2
5x 5
40x2 40x 20
40x2 40x 20
131x
2 5x
+ Ly −
+
−
− +
−
+
+π
+ Lx
+
27
9
9
9
3
3
6
9
9
9
2
2
2
101
7x
7
4x 2
2x 1
7x
2x
4x
3
2
2
−
+ −
+ −
+ −
+ Lm −
+ Lm Ls −
+ Lm Ls −
54
9
9
18
3
3 3
3
3
3
2
2
2
2
14x 7
37x
8x 4
8x 4
8x
8x
37x
2
3 14x
−
+
+
+ Ly −
+ −
− +
+ Lm −
+ Lx
+Ls
9
9
9
18
18
3
3 3
3
3 3
2
2
2
4x
16x 8
16x 8
2
16x
16x
4x
43
+ + Ly −
+
−
−
+
+ Lm Ls −
+ Lx
−
−
36
9
9
3
3
3
3
3
3
9
2
2
47
65x
76x
65x
76x
+
+ Ly −8x2 + 8x − 4 + Lx 8x2 − 8x + 4 −
+
+ Lm −
+L2s −
9
9
18
9
9
2
2
40x
40x 32
40x 32
9
80x2
40x
2 4x
+Ly −
+
−
−
+
−
+ Lx
−
+ Ls
L2x + −
9
9
9
9
9
9
2
3
3
9
2
2
2 4x
92x 28
451x
451x
5
80x
−
+ L2y
−
+ π2 −5x2 + 5x −
+
Lx +
−
+ Ly
9
9
27
3
3
27
2
9
2
2
68x 16
26x
8x
4x 2
4x
301
4 2 3
10
2
−
+
+ Ly
− +
+
+ − x Lx +
−
Lx + Li2 (x)
9
9
54
9
9
3
3 3
9
3
448x2 544x
260
5809x2
8x 4
16x2 32x 16
−
+ π2 −
+
−
Lx +
+
Lx −
− +
3
3
27
27
3
9
9
27
162
2
2
16
8x 4
8x 4
8x
8x
5809x
2 26x
+ Ly
−
+ −
+ −
+ Li3 (x) −
+ S1,2 (x) −
+
162
9
9
3
3 3
3
3
3
2
2
2
8x 4
391x 301
352x
16x2
448x
4x
487x
− +
−
+
+
+ π2
+L3y
+ π2
+ Ly −
9
9
9
54
54
108
27
27
3
2
64x 32
16x
16x 8
164
5023
−
+
−
+
,
(24)
−
+
ζ3 −
9
9
27
9
9
9
324
F =
L2s
4x2 4x 2
100x2 100x
50
4x2 4x 2
40x2 40x 20
2
− +
+
−
−
+π −
+ −
+ Ls −
+
+ .
9
9 9
27
27
27
81
81
9
9
9
81
(25)
Notice that our results are expressed not only in terms of the classic polylogarithms up to weight
four, but also in terms of Nielsen polylogarithms
(−1)n+p−1
Sn,p (x) =
(n − 1)!p!
Z 1
dy
0
logn−1 (y) log p (1 − xy)
.
y
(26)
Conclusions
In this letter, we have presented the two-loop virtual QCD corrections to the production of heavyquarks in the light quark annihilation channel in the ultra-relativistic limit. Our results form a
crucial part of the NNLO predictions for heavy-quark production in hadron-hadron collisions.
However, depending on the mass of the heavy quark (bottom or top) and the kinematics of the
process under consideration power corrections in the heavy-quark mass may have to be considered
14
as well. Our results have been derived by combining two completely different methods which have
substantial overlap. This provides direct and highly non-trivial checks of the QCD factorization
approach of Ref. [13], of the direct calculation of massive Feynman diagrams and, last but not
least, also on the available masseless results of Ref. [14].
In order to yield physical cross sections, 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. While some of the matrix elements (including the full mass dependence) can be easily
generated, others became available in the literature only rather recently, see e.g. Refs. [22,38]. 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 which need to be absorbed into
parton distribution functions. This is a necessary prerequisite e.g. to the construction of numerical
programs which provide NNLO QCD estimates of observable scattering cross sections.
Finally, while we have chosen to work in this letter with squared matrix elements, it should, of
course, be clear that with the help of Refs. [39,40] analogous results can be derived for the massive
two-loop amplitude |M (2) i itself.
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 acknowledge useful discussions with N. Glover, J. Gluza, Z. Merebashvili and T. Riemann. The work of M.C. was supported by the Sofja Kovalevskaja Award
of the Alexander von Humboldt Foundation and by the ToK Program “ALGOTOOLS” (MTKDCD-2004-014319). A.M. thanks the Alexander von Humboldt Foundation for support and S.M.
acknowledges contract VH-NG-105 of the Helmholtz Gemeinschaft. This work was supported in
part by the Deutsche Forschungsgemeinschaft in Sonderforschungsbereich/Transregio 9.
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16