Two-Loop QCD Corrections to the Heavy-to

arXiv:0809.4687v1 [hep-ph] 26 Sep 2008
ZU-TH 16/08
Two-Loop QCD Corrections to the Heavy-to-Light
Quark Decay
R. Bonciani a,∗ and A. Ferroglia a,†
a
Institut für Theoretische Physik, Universität Zürich, CH-8057 Zurich, Switzerland
Abstract: We present an analytic expression for the two-loop QCD corrections to the
decay process b → u W ∗ , where b and u are a massive and massless quark, respectively,
while W ∗ is an off-shell charged weak boson. Since the W -boson can subsequently decay in
a lepton anti-neutrino pair, the results of this paper are a first step towards a fully analytic
computation of differential distributions for the semileptonic decay of a b-quark. The
latter partonic process plays a crucial role in the study of inclusive semileptonic charmless
decays of B-mesons. The three independent form factors characterizing the bW u vertex are
provided in form of a Laurent series in (d − 4), where d is the space-time dimension. The
coefficients in the series are expressed in terms of Harmonic Polylogarithms of maximal
weight 4, and are functions of the invariant mass of the leptonic decay products of the
W -boson.
Keywords: Heavy Quark Decay, Two-Loop Calculations.
∗
†
Email: [email protected]
Email: [email protected]
Contents
1. Introduction
1
2. Feynman Diagrams and Form Factors
3
3. UV Renormalization
5
4. One-Loop Form Factors
7
5. Two-Loop Form Factors
8
6. Ward Identities
14
7. Conclusions
17
A. The Master Integrals
17
1. Introduction
The measurements of inclusive semileptonic B meson decays, such as B → Xu l ν and
B → Xc l ν, allow a precise determination of the CKM matrix elements |Vub | and |Vcb |.
The latter are relevant for the study of flavor and CP violation in the quark sector (for a
recent review see [1]).
Total decay rates of the B meson are described by a local Operator Product Expansion
(OPE) in inverse powers of the b-quark mass mb . To leading order in 1/mb , the total B
meson decay rate is equivalent to the decay rate of an on-shell b quark, which can be
calculated in perturbation theory [2]. Many authors contributed to the calculation of the
radiative corrections to the total decay rate of b → u l ν and b → c l ν, at O(αS ) [3, 4]
and O(α2S ) [5–11]. However, experimental collaborations need to impose cuts (also severe)
on the kinematic variables. For instance, in charmless semileptonic decays, the need to
suppress the charm background (which is ∼ 50 times larger than the signal) forces one
to restrict the measurements to the “shape-function region”, in which the hadronic final
state has large energy (EX ∼ mb ), but only moderate invariant mass (∼ mb ΛQCD ). It
is therefore of great interest to consider differential decay distributions, from which it is
possible to derive predictions for partial decay rates with arbitrary cuts. In this context,
a first important set of results was obtained in [12], where it is possible to find analytic
expressions for the NLO triple-differential distribution of the semileptonic B → Xu l ν
decay together with several double and single differential distributions for the same process.
The resummation of threshold logarithms to next-to-leading approximation in the b → u
–1–
transition was considered in [13]. Higher order contributions to B → Xu l ν decays were
considered in [14–17] and, very recently, the full NNLO QCD corrections to the partonic
process b → c l ν were obtained in [18]. Since the OPE applies only to sufficiently inclusive
quantities, different frameworks were developed in order to account for effects due to cuts
on the kinematic space [19–22]. In particular, in the shape-function region, Soft Collinear
Effective Theory (SCET) provides an appropriate framework for the evaluation of the
triple-differential distribution of the inclusive semileptonic decay B → Xu l ν. The NLO
analysis of the latter process within the SCET approach is presented in [19,20]. At NNLO,
the situation is more complicated, but the jet and soft functions are known to O(α2S ) in
perturbation theory [23, 24]. The only missing piece is the hard function, which can be
obtained from the two-loop QCD corrections to the decay of a b-quark into a u-quark and
an off-shell W -boson [25]. On the other hand, these virtual corrections can be considered
as a first step towards an exact evaluation of the NNLO QCD corrections to the heavy-tolight quark transition. To complete the latter calculation, it is also necessary to take into
account the real emission.
In this work we focus on the calculation of the two-loop QCD corrections to the decay
process b → u W ∗ . We provide an analytic expression for the three independent vertex form
factors characterizing the coupling of the quark current with the charged weak boson. These
form factors are evaluated by employing a set of techniques which are by now standard in
multiloop calculations (see for instance [26]). We generate the relevant Feynman diagrams
with QGRAF [27]. The form factors are extracted directly from the Feynman diagrams
by means of projector operators. The whole calculation is carried out in Dimensional
Regularization (DR); UV and IR (soft and collinear) divergencies appear as poles in (d−4),
where d is the space-time dimension. Since we work in DR, a prescription for handling
the matrix γ5 in d-dimensions must be chosen. We employed a γ5 which anticommutes
with γµ in d-dimensions. This prescription is appropriate for the case under study, since
it is known that the diagrams that we consider fulfill a canonical (non-anomalous) Ward
identity. After applying the projectors, the contribution of individual Feynman diagrams
to the form factors is given by a combination of dimensionally regularized scalar integrals.
These integrals are related to a small set of master integrals (MIs) by means of the Laporta
algorithm [28]. The MIs are evaluated by employing the Differential Equations method [29]
and they are expressed as Laurent series in (d − 4). The coefficients of the series are
given in terms of Harmonic Polylogarithms (HPLs) [30] of a single dimensionless variable
y = q 2 /m2b = −Ml2 /m2b , where q 2 is the squared momentum carried by the W -boson and
Ml is the lepton pair invariant mass. Since y is negative in the physical region (−1 ≤ y ≤ 0)
we perform an analytic continuation y → −x − i0+ , where now x = Ml2 /m2b , 0 ≤ x ≤ 1.
The analytic continuation is indeed completely trivial, since all the HPLs appearing in the
result are real for −1 ≤ y ≤ 0. The form factors found with the above procedure still
contain UV and IR divergencies. It is possible to get rid of the UV divergencies by means
of the renormalization procedure. We renormalize the form factors in a mixed scheme:
the heavy- and light-quark wave functions and the heavy-quark mass are renormalized
in the on-shell (OS ) scheme, while the strong coupling constant is renormalized in MS
scheme. The results shown in this paper contain IR divergencies. In order to cancel them,
–2–
it is necessary to combine these results with the appropriate jet and soft functions [19].
Analogously, one can add the exact real emission and consider physical observables which
are sufficiently inclusive with respect to the hard and soft radiation.
The paper is structured as follows. In section 2, we introduce the Feynman diagrams
involved in the calculation and we discuss their structure in terms of form factors. In
section 3, it is possible to find the details of the UV renormalization procedure. In sections
4 and 5, we collect the analytic expressions of the UV renormalized one- and two-loop
QCD corrections to the form factors, respectively. The expressions of the bare form factors
as well as the contributions of the individual diagrams to the form factors can be found
in [31]. In section 6, we discuss the Ward identity relevant for the b → u W ∗ decay and we
prove that our form factors satisfy it. We also provide the analytic expression of the oneand two-loop QCD corrections to the scalar vertex in which the pseudo-Goldstone boson
couples to the quarks, since it enters in the Ward identity fulfilled by the bW u vertex. Our
conclusions can be found in section 7. Finally, in appendix A we collect the set of MIs
employed in the calculation.
2. Feynman Diagrams and Form Factors
We consider the decay process b → u W ∗ → u l ν̄l . The bottom quark of mass mb carries a
momentum P and decays in an up-quark (considered as massless) which carries momentum
p and a W -boson of momentum q = P −p. Subsequently, the W -boson decays in the pair lν̄l
of squared invariant mass Ml2 = −q 2 . The mass-shell conditions are such that P 2 = −m2b
and p2 = 0. The Feynman diagrams contributing to the two-loop QCD corrections to the
decay process b → u W ∗ are shown in Fig. 1. The most general vertex correction in the
Standard Model can be described in terms of six form factors Fi and Gi (i = 1, 2, 3):
V µ (P, p) = F1 (q 2 )γ µ +
+
1
i
i
F2 (q 2 )σ µν qν +
F3 (q 2 )q µ + G1 (q 2 )γ µ γ5 +
G2 (q 2 )γ5 q µ
2mb
2mb
2mb
i
G3 (q 2 )γ5 q̃ µ ,
2mb
(2.1)
where q̃µ = Pµ + pµ . The spinors u(p) and u(P ), multiplying Eq. (2.1) from left and right,
respectively, are not written down explicitly. We define σ µν = −i/2[γ µ , γ ν ]. Since the
u-quark is taken as massless, only three of the above form factors are independent. By
replacing
u(p) [γ µ , γ ν ] u(P )qν = 2imb u(p)γ µ u(P ) − 2q̃ µ u(p)u(P ) ,
(2.2)
in Eq. (2.1), we find the following relations among the form factors in Eq. (2.1):
F2 = −G3 ,
F3 = −G2 ,
1
F1 + F2 = G1 .
2
(2.3)
Consequently, using the definitions PL = (1 + γ5 )/2 and PR = (1 − γ5 )/2, we can rewrite
the vertex structure as follows:
V µ (P, p) = 2G1 (q 2 )γ µ PL −
i
i
G2 (q 2 )PR q µ −
G3 (q 2 )PR q̃ µ .
mb
mb
–3–
(2.4)
q
p
P
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Figure 1: Feynman Diagrams for the Two-loop QCD corrections to the b → u W ∗ decay process.
The form factors are expanded in powers of αs :
3 α 2
α igw
αs
(1l)
s
(2l)
(0l)
s
Gi = √ Vub Gi +
Gi +
Gi + O
.
π
π
π3
2 2
(2.5)
(2l)
The purpose of the present work is to evaluate Gi where i = 1, 2, 3. Vub represents the
CKM matrix element and gw the weak interaction coupling constant. With the normalization chosen in Eq. (2.5), one finds
(0l)
G1
= 1,
and
(0l)
G2
(0l)
= G3
= 0.
(2.6)
The contribution of the virtual two-loop corrections to the hadronic tensor of [12] can be
obtained by meas of the following translation rules:
2
(2l,vir.)
(2l)
(1l) 2
= 2 2G1 + G1
W1
,
(2.7)
mb
1 − x (1l)
(1l) 2
(2l,vir.)
G2 + G3
,
(2.8)
W3
=
4mb
1−x 1
(1l) 2
(2l,vir.)
(1l) 2
(2l)
(2l)
(1l)
(1l)
(1l)
G3
W4
= 2 G3 + G2 + G1 G3 + G2
− G2
,(2.9)
+
4
mb
–4–
(2l,vir.)
W5
=
1−x 2
(2l)
(2l)
(1l)
(1l)
(1l)
(1l)
(1l) 2
G
−
G
+
G
G
−
G
+
G
−
G
. (2.10)
3
2
1
3
2
3
2
8
m2b
Consequently, the hard functions Hij , as defined in [19], can be extracted using the relation
between Eqs. (16) and (17) of the same article. Note that the hadronic form factor W2
does not receive contributions from two-loop virtual corrections.
We write our analytic results in terms of the dimensionless variable
x=−
Ml2
q2
=
,
m2b
m2b
0 ≤ x ≤ 1.
(2.11)
In writing or results, we employ the Harmonic Polylogarithms as defined in [30]; on top
of the canonical weights, we introduce the weights −2 and 2, arising from the integrating
factors 1/(x+2) and 1/(2−x), respectively. Actually, only two HPLs containing the weight
2 appear in the final result; they are
x
= −H(−1, 1 − x) + ln(2) ,
H(2; x) = − ln 1 −
2
Z x
1
H(2, 1, 1; x) =
dt
ln2 (1 − t) =
2(2 − t)
0
3
1
= − ln(1 − x)2 ln(2 − x) − ln(1 − x)Li2 (−1 + x) + Li3 (−1 + x) + ζ (3) ,
2
4
3
= −H(−1, 0, 0, 1 − x) + ζ(3) .
(2.12)
4
For convenience, all the results of the paper, including the renormalized and bare
form factors, as well as the contributions of individuals diagrams, are collected in the file
SemilepFF.txt [31] included in the arXiv submission of the present work.
3. UV Renormalization
The UV renormalization is performed by subtracting the one-loop sub-divergencies and the
two-loop over-all divergencies. We renormalize the heavy- and light-quark wave functions
and heavy-quark mass in the on-shell (OS ) scheme, while the coupling constant αS is
renormalized in the MS scheme.
Neglecting for the time being mass renormalization, the bare and renormalized form
factors satisfy the relation
1
1
2
2
G = Z2,u
Z2,b
Gbare (αbare
s ),
where in the functions G we dropped the subscript i = 1, 2, 3.
The perturbative expansion of the various quantities in the equation above is
igw
G = √ Vub G(0l) + aG(1l) + a2 G(2l) + O a30 ,
2 2
igw
(1l)
(2l)
Gbare = √ Vub G(0l) + a0 Gbare + a20 Gbare + O a30 ,
2 2
(1l)
(2l)
Z2,u = 1 + a0 δZ2,u + a20 δZ2,u + O a30 ,
(2l)
(1l)
Z2,b = 1 + a0 δZ2,b + a20 δZ2,b + O a30 ,
a0 = a 1 + aδZα(1l) + a2 δZα(2l) + O a3 ,
–5–
(3.1)
(3.2)
where we defined
αbare
αs
s
,
a≡
.
π
π
Therefore, the one-loop renormalized amplitude is given by
a0 ≡
(3.3)
1 (1l)
(1l)
G(1l) = Gbare + δZ2,b G(0l) ,
2
(3.4)
(1l)
where we already took into account the fact that δZ2,u = 0 in the on-shell scheme. The
two-loop renormalized amplitude reads instead
1 (2l) 1 (2l) 1 (1l) (1l) 1 (1l) 2
(2l)
(2l)
G
= Gbare + δZ2,b + δZ2,u + δZα δZ2,b −
δZ2,b
G(0l)
2
2
2
8
1 (1l)
(1l)
(1l)
+ δZ2,b + δZα
Gbare .
(3.5)
2
To account for mass renormalization, it is sufficient to add the contribution of the
counter term diagram in Fig. 2 to the r. h. s. of the equation above.
The renormalization constants are the following:
1
11
1
(1l)
(3.6)
− CA + TR (Nl + Nh ) ,
δZα,MS (d) = −C(d)
d−4
6
3
2 (4−d)/2
µ2 CF
(d − 1)
µ
(1l)
δmOS d, m, 2 = m C(d)
,
(3.7)
2
m
m
2 (d − 4) (d − 3)
2 (4−d)/2
µ2 (d − 1)
µ
CF
(1l)
δZ2,b d, 2 = C(d)
,
(3.8)
2
m
m
2 (d − 4) (d − 3)
2 4−d
µ2 µ
CF
1
5
(2l)
δZ2,u d, 2 = C 2 (d)
N
−
−
,
(3.9)
h
m
m2
8
2(d − 4) 24
2 4−d
µ2 µ
CF
1
1
(2l)
2
CF f1 + CA f2 + Nl f3 + Nh f4 ,(3.10)
δZ2,b d, 2 = C (d)
m
m2
2
2
2
where µ is the renormalization scale and the constants f1 , · · · , f4 are [32]
9
433 3
13
51
+
− ζ3 − π 2 ln(2) − π 2 + O(d − 4) ,
−
2
8(d − 4)
32(d − 4) 128 2
16
2
11
101
803 3
π
5
f2 = −
+
−
+ ζ3 −
ln(2) + π 2 + O(d − 4) ,
8(d − 4)2
32(d − 4) 128 4
2
16
59 π 2
9
1
+
+
+ O(d − 4) ,
−
f3 =
2(d − 4)2 8(d − 4) 32 12
1
19
1139 π 2
f4 =
−
+
−
+ O(d − 4) .
(d − 4)2 24(d − 4)
288
3
f1 =
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
The factor C(d) is
C(d) = (4π)
(4−d)/2
d
Γ 3−
2
.
(3.16)
After UV renormalization, the vertex form factors still contain poles in 1/(d−4), which
are associated to soft and collinear singularities.
–6–
(1l)
−
δmOS
m
Figure 2: Mass-renormalization counter-term.
4. One-Loop Form Factors
In this section we collect the analytic expression of the one-loop renormalized form factors
defined in Eq. (2.5). In the formulas below, CF = (Nc2 − 1)/2Nc is the Casimir operator of
the fundamental representation of SU (Nc ), where Nc is the number of colors (in the SM
Nc = 3).
(1l)
The form factor G1 is given by
2 4−d
1
X
2
µ
(1l,i)
(1l)
G1 (d − 4)i + O (d − 4)2 ,
(4.1)
G1 = C(d)
C
F
2
m
i=−2
where the first four coefficients in the expansion in (d − 4) are
(1l,−2)
G1
(1l,−1)
G1
(1l,0)
G1
(1l,1)
G1
= −1 ,
5
= + H(1; x) ,
4
1
3 1 − 3x
H(1; x) − H(0, 1; x) − H(1, 1; x) ,
=− +
2
4x
2
3 1 − 2x
1 − 3x
1 − 3x
1
= −
H(1; x) −
H(0, 1; x) −
H(1, 1; x) + H(0, 0, 1; x)
2
2x
8x
4x
4
1
1
(4.2)
+ H(0, 1, 1; x) + H(1, 0, 1; x) + H(1, 1, 1; x) .
2
2
(1l)
The form factor G2
(1l)
G2
is
2 4−d
1
X
2
µ
(1l,i)
= C(d)
C
G2 (d − 4)i + O (d − 4)2 ,
F
2
m
(4.3)
i=0
with
(1l,0)
G2
(1l,1)
G2
1 2 − 3x
−
H(1; x) ,
x
2x2
1 1 − 3x
2 − 3x
2 − 3x
=− +
H(1; x) +
H(0, 1; x) +
H(1, 1; x) .
2
2
x
2x
4x
2x2
=
(4.4)
(1l)
Finally, the form factor G3 is
2 4−d
1
X
2
µ
(1l,i)
(1l)
C
G2 (d − 4)i + O (d − 4)2 ,
G3 = C(d)
F
2
m
i=0
–7–
(4.5)
where
(1l,0)
G3
(1l,1)
G3
1
H(1; x) ,
2x
1
1
1
= H(1; x) +
H(1, 1; x) +
H(0, 1; x) .
x
2x
4x
=−
(1l)
Note that the IR poles of the form factor G1
the O(α2S ) expansion of the form factor
(4.6)
exponentiate. This means that from
(1l)
F = exp {G1 } ,
(4.7)
we can predict exactly the 1/(d − 4)4 and 1/(d − 4)3 poles of the CF2 part of the two-loop
(2l)
form factor G1 (Eqs. (5.3,5.4) below). Moreover, exponentiating also the finite part of
(1l)
G1 , the double pole of Eq. (5.5) is exactly recovered.
5. Two-Loop Form Factors
In this section we collect the analytic expression of the two-loop renormalized form factors
defined in Eq. (2.5). In the expressions below, CA is the Casimir operator of the adjoint
representation of SU (Nc ), CA = Nc , TR is the normalization factor of the color matrices,
TR = 1/2, Nl is the number of massless quarks in the theory, and Nh is the number of
quarks of mass mb . Therefore, for the decay b → u W ∗ in the SM, Nl = 4, and Nh = 1.
In the finite part of the form factors given below, the constant K is a rational number. Its
numerical value is K = 3.32812±0.00002, and its analytical value is likely to be K = 213/64.
We observe that the following formulas involve HPLs of argument x and of maximal weight
4. If desired, the HPLs appearing in the equations below can all be rewritten in terms of
product of Nielsen Polylogarithms of more complicated argument. Because of the chosen
renormalization scheme, our results depend on the renormalization scale µ. In the formulas
below, we employ the following notation:
2
µ
≡ Lµ .
(5.1)
ln
m2b
(2l)
The form factor G1
(2l)
G1
2
can be written as
= C (d)
µ2
m2
4−d
CF
0
X
i=−4
(2l,i)
G1
(d − 4)i + O (d − 4) ,
(5.2)
where the coefficient of the expansion in (d − 4) (up to the finite term) are
(2l,−4)
G1
(2l,−3)
G1
(2l,−2)
G1
= CF
1
,
2
(5.3)
11
1
5
(5.4)
= CF − − H(1; x) − CA + TR Nl ,
4
8
2
49−66Lµ +9ζ(2)
73 1−8x
1
= CF
−
H(1; x) + H(0, 1; x) + 2H(1, 1; x) + CA
32
4x
2
72
–8–
−5 + 6Lµ 1
11
1
H(1; x) + TR Nl
− H(1; x) + TR Nh Lµ ,
(5.5)
12
18
3
3
1 − 8x
3(71 + 8ζ(2) − 16ζ(3)) 13 − 55x
= CF −
+
H(1; x) +
H(0, 1; x)
64
16x
8x
1
3
3−14x
H(1, 1; x)− H(0, 0, 1; x)− H(0, 1, 1; x)−H(1, 0, 1; x)−4H(1, 1, 1; x)
+
4x
4
2
2
1549+1980Lµ −396Lµ +972ζ(2)−1188ζ(3) 67+66Lµ −18ζ(2)
+CA
+
H(1; x)
1728
72
−125 − 180Lµ + 36L2µ − 108ζ(2) 5 + 6Lµ
+TR Nl
−
H(1; x)
432
18
−5Lµ + L2µ − ζ(2) Lµ
+TR Nh
−
H(1; x) ,
12
3
1
(6635(x − 1)3 + (80(x − 1)(−59 − 62x + 3x2 ) − 480 ln (2)(10
= CF
1280(x − 1)3
+
(2l,−1)
G1
(2l,0)
G1
−22x + 5x2 + 4x3 ))ζ(2) − 16(261 − 64K + 41x + 192Kx + 405x2 − 192Kx2
1
−99x3 + 64Kx3 )ζ 2 (2) + 40(52 − 72x + 15x2 + 14x3 )ζ(3)) +
(ζ(2)
4(x − 1)2 x
1
(−49 + 251x − 355x2
−3xζ(2) + x2 ζ(2) + 5x3 ζ(2))H(−1; x) +
32(x − 1)2 x
+153x3 + 12ζ(2) − 160xζ(2) − 148x2 ζ(2) + 24x3 ζ(2) − 16xζ(3) + 32x2 ζ(3)
3(−2ζ(2) − 2xζ(2) + x2 ζ(2))
−16x3 ζ(3))H(1; x) +
H(2; x)
8(x − 1)3
ζ(2) − 5xζ(2) + 3x2 ζ(2) − x3 ζ(2)
1
+
H(0, −1; x) +
(15 − 106x
3
2(x − 1)
32(x − 1)3 x
+248x2 − 138x3 − 19x4 + 72xζ(2) + 152x2 ζ(2) + 48x3 ζ(2))H(0, 1; x)
25 − 134x + 59x2
1 − 3x + x2 + 5x3
+
H(1, 1; x) +
H(−1, 0, 1; x)
16(x − 1)x
2(x − 1)2 x
1+27x−9x2 −25x3 +12x4
5−59x+83x2 −56x3 +30x4
−
H(0, 0,1; x)+
H(0,1,1; x)
3
16(x − 1) x
8(x − 1)3 x
1−x+21x2 −7x3
7−26x
2+2x−x2
−
H(1,
0,
1;
x)
−
H(1,
1,
1;
x)
−
H(2, 1, 1; x)
4(x − 1)2 x
4x
8(x − 1)3
3 + 11x − 5x2 + 3x3
1 − 5x + 3x2 − x3
H(0,
−1,
0,
1;
x)
−
H(0, 0, 0, 1; x)
+
(x − 1)3
8(x − 1)3
9+9x+13x2 −3x3
18x−7x2 +3x3
−
H(0,0,1,1; x)+
H(0,1,0,1; x) +3H(1,0,1,1; x)
3
4(x − 1)
4(x − 1)3
1
7
+ H(0,1,1,1; x)− H(1, 0, 0,1; x)+2H(1,1, 0,1; x)+8H(1,1,1,1; x)
2
2
2
29700Lµ − 447185−142560Lµ −3960L3µ
1
+CA
+
((11880Lµ (x−1)3
103680
103680(x−1)3
−180(x − 1)(517 + 982x + 913x2 ) + 19440 ln (2)(10 − 22x + 5x2 + 4x3 ))ζ(2)
+648(71 − 64K − 631x + 192Kx + 159x2 − 192Kx2 − 125x3 + 64Kx3 )ζ 2 (2)
–9–
+180(−698 + 1338x − 825x2 + 104x3 )ζ(3)) −
(1 − 3x + x2 + 5x3 )ζ(2)
H(−1; x)
8(x − 1)2 x
1
(807+198Lµ −4159x−990Lµ x+198L2µ x+5897x2 +1386Lµ x2
864(x − 1)2 x
−396L2µ x2 −2545x3 −594Lµ x3 +198L2µ x3 −(216 + 1368x + 342x2 + 1206x3 )ζ(2)
+
3(2 + 2x − x2 )ζ(2)
H(2; x)
16(x − 1)3
(1 − 5x + 3x2 − x3 )ζ(2)
1
−
(−33 + 260x + 66Lµ x
H(0, −1; x) +
3
4(x − 1)
144(x − 1)3 x
−384x2 −198Lµ x2 +273x3 +198Lµ x3 −116x4 −66Lµ x4 −18xζ(2)+468x2 ζ(2)
1
(−39+235x+132Lµ x−349x2 −132Lµ x2
+72x4 ζ(2))H(0, 1; x) +
144(x − 1)x
1 − 3x + x2 + 5x3
−72xζ(2) + 72x2 ζ(2))H(1, 1; x) −
H(−1, 0, 1; x)
4(x − 1)2 x
8 + 36x − 33x2 − 20x3
2 − 168x + 219x2 − 62x3
−
H(0,
0,
1;
x)
+
H(0, 1, 1; x)
48(x − 1)3
48(x − 1)3
47 − 49x + 44x2
11
2 + 2x − x2
−
H(1,
0,
1;
x)
−
H(1,
1,
1;
x)
+
H(2, 1, 1; x)
48(x − 1)2
6
16(x − 1)3
1 − 5x + 3x2 − x3
1 + 10x + 4x2
−
H(0,
−1,
0,
1;
x)
−
H(0, 0, 0, 1; x)
2(x − 1)3
8(x − 1)3
1 + 2x + 4x2
1 + 2x + 4x2
1
−
H(0, 0, 1, 1; x) +
H(0, 1, 0, 1; x) + H(1, 0, 0, 1; x)
4(x − 1)3
8(x − 1)3
2
1
(6629+2592Lµ −540L2µ +72L3µ +3420ζ(2)−216Lµ ζ(2) + 720ζ(3))
+TR Nl
5184
57+ 18Lµ −209x−54Lµ x+18L2µ x−90xζ(2)
3−19x−6Lµ x
−
H(1; x)−
H(0, 1; x)
216x
36x
3 − 19x − 6Lµ x
1
1
1
−
H(1, 1; x) + H(0, 0, 1; x) + H(0, 1, 1; x) + H(1, 0, 1; x)
18x
6
3
3
1
2
(−1111 − 1296Lµ + 270L2µ − 36L3µ
+ H(1, 1, 1; x) + TR Nh
3
2592(x − 1)3
+(756x − 1512x2 + 756x3 )ζ(3))H(1; x) +
+7869x + 3888Lµ x − 810L2µ x + 108L3µ x − 14709x2 − 3888Lµ x2 + 810L2µ x2
−108L3µ x2 + 7951x3 + 1296Lµ x3 − 270L2µ x3 + 36L3µ x3 − 414ζ(2) + 108Lµ ζ(2)
−5670xζ(2) − 324Lµ xζ(2) + 9126x2 ζ(2) + 324Lµ x2 ζ(2) − 738x3 ζ(2)
−108Lµ x3 ζ(2) + 504ζ(3) + 1080xζ(3) + 1512x2 ζ(3) − 504x3 ζ(3))
1
+
(−57 − 18Lµ − 89x + 90Lµ x − 18L2µ x + 73x2
216(x − 1)2 x
−126Lµ x2 + 36L2µ x2 + 265x3 + 54Lµ x3 − 18L2µ x3 + 18xζ(2) − 36x2 ζ(2)
3+8x−24x3 −19x4 −(6x−18x2 +18x3 −6x4 )Lµ
H(0, 1; x)
36(x − 1)3 x
1 + 3x + 3x2 − x3
1
H(0, 0, 1; x) .
(5.6)
+ Lµ H(1, 1; x) −
3
6(x − 1)3
+18x3 ζ(2))H(1; x)+
– 10 –
(2l)
The form factor G2
(2l)
G2
2
is
= C (d)
µ2
m2
4−d
CF
0
X
i=−2
(2l,i)
G2
(d − 4)i + O (d − 4) ,
(5.7)
where
(2l,−2)
G2
(2l,−1)
G2
(2l,0)
G2
1 2 − 3x
= CF − +
H(1; x) ,
(5.8)
x
2x2
9
7(2 − 5x)
(2 − 3x)
3(2 − 3x)
= CF
−
H(1; x) −
H(0, 1; x) −
H(1, 1; x) , (5.9)
2
2
4x
8x
4x
2x2
1
= CF
(−310(x − 1)4 + (60 ln (2)x(−38 + 58x − 40x2 + 11x3 )
80(x − 1)4 x
−20(x − 1)(10 − 120x − 79x2 + 12x3 ))ζ(2) + 16x(125 + 103x)ζ 2 (2)
(2−9x−5x2 +3x3 −3x4 )ζ(2)
H(−1; x)
−5x(−30 + 110x − 80x2 + 27x3 )ζ(3)) +
2(x − 1)3 x2
1
(−32 + 195x − 397x2 + 337x3 − 103x4 + 24ζ(2) − 12xζ(2)
+
16(x − 1)3 x2
3(30−34x+16x2 −3x3 )ζ(2)
+844x2 ζ(2)−76x3 ζ(2)+36x4 ζ(2))H(1; x) +
H(2; x)
4(x − 1)4
1
2(2 + x)ζ(2)
H(0, −1; x) +
(26 − 69x − 68x2 − 58x3 + 166x4
+
4
(x − 1)
16(x − 1)4 x2
8−18x+37x2 +172x3 −49x4
H(1, 1; x)
+3x5 −(448x2 +368x3 )ζ(2))H(0, 1; x) +
8(x − 1)2 x3
2 − 9x − 5x2 + 3x3 − 3x4
30 − 34x + 16x2 − 3x3
+
H(−1,
0,
1;
x)
+
H(2, 1, 1; x)
(x − 1)3 x2
4(x − 1)4
2 − 27x + 4x2 − 48x3 + 66x4 − 15x5
7(2 − 3x)
−
H(0, 0, 1; x) +
H(1, 1, 1; x)
4
2
8(x − 1) x
2x2
10−39x+234x2 −276x3 +86x4 −24x5
4(2 + x)
+
H(0, 1, 1; x) +
H(0, −1, 0, 1; x)
4(x − 1)4 x2
(x − 1)4
2 − 13x − 22x2 − 12x3 + 3x4
(4 + 5x)
−
H(1, 0, 1; x) +
H(0, 0, 0, 1; x)
3
2
2(x − 1) x
(x − 1)4
6(4 + 3x)
3(4 + 3x)
+
H(0, 0, 1, 1; x) −
H(0, 1, 0, 1; x)
(x − 1)4
(x − 1)4
1
+CA
(1320Lµ (x − 1)4 + 20(x − 1)3 (−269 + 242x)
1440(x − 1)4 x
+(540(x − 1)(4 + 50x + 5x2 + 8x3 ) + 540 ln (2)x(38 − 58x + 40x2 − 11x3 ))ζ(2)
+36x(364 + 317x + 108x2 )ζ 2 (2) + 45x(−30 + 110x − 80x2 + 27x3 )ζ(3))
1
(2−9x−5x2 +3x3 −3x4 )ζ(2)
H(−1; x) +
(406+132Lµ −2067x
−
3
2
4(x − 1) x
144(x − 1)3 x2
−594Lµ x+3603x2+990Lµ x2−2629x3−726Lµ x3+687x4 +198Lµ x4 −(144−864x
3(30 − 34x + 16x2 − 3x3 )ζ(2)
−1224x2 −1242x3 + 54x4 )ζ(2))H(1; x) −
H(2; x)
8(x − 1)4
– 11 –
1
(2 + x)ζ(2)
H(0, −1; x) +
(−22 + 145x − 300x2 + 109x3 + 59x4
(x − 1)4
24(x − 1)4 x2
26−151x+14x2 −42x3
H(1, 1; x)
+9x5 −(240x2 +210x3 +72x4 )ζ(2))H(0, 1; x) −
24(x − 1)2 x2
2−9x−5x2 +3x3 −3x4
4+76x−96x2 +16x3 −9x4
−
H(0, 0, 1; x)
H(−1,
0,
1;
x)
+
2(x − 1)3 x2
8(x − 1)4 x
8 + 58x − 24x2 − 36x3 + 3x4
4 + 24x + 11x2 + 3x3
+
H(0,
1,
1;
x)
+
H(1, 0, 1; x)
8(x − 1)4 x
8(x − 1)3 x
30 − 34x + 16x2 − 3x3
24 + 17x + 4x2
−
H(2,
1,
1;
x)
+
H(0, 0, 0, 1; x)
8(x − 1)4
4(x − 1)4
2(2 + x)
8 + 9x + 4x2
−
H(0, −1, 0, 1; x) +
(2H(0, 0, 1, 1; x) − H(0, 1, 0, 1; x))
(x − 1)4
4(x − 1)4
2−3x
19+6Lµ 26−51x+6(2−3x)Lµ
+
H(1; x)+
(H(0,1; x)+2H(1,1; x))
+TR Nl −
18x
36x2
6x2
1
+TR Nh
(−19 − 6Lµ − 164x + 24Lµ x + 393x2 − 36Lµ x2 − 218x3
18(x − 1)4 x
−
+24Lµ x3 +8x4 −6Lµ x4 +(252x − 300x2 +84x3 −36x4 )ζ(2) − (72x + 36x2 )ζ(3))
26 − 223x − 124x2 − 51x3 + (12 − 42x + 48x2 − 18x3 )Lµ
+
H(1; x)
36(x − 1)2 x2
2 − 9x − 21x2 − 13x3 − 3x4
2(2 + x)
−
H(0,
1;
x)
+
H(0,
0,
1;
x)
.
(5.10)
6(x − 1)3 x2
(x − 1)4
(2l)
The form factor G3
(2l)
G3
2
can be written as
= C (d)
µ2
m2
4−d
CF
0
X
i=−2
(2l,i)
G3
(d − 4)i + O (d − 4) ,
(5.11)
with
(2l,−2)
G3
(2l,−1)
G3
(2l,0)
G3
1
= CF
H(1; x) ,
(5.12)
2x
13
1
= CF − H(1; x) −
(H(0, 1; x) + 6H(1, 1; x)) ,
(5.13)
8x
4x
1
((−60(x − 1)x(57+2x) − 60 ln (2)(−22+30x−26x2 +9x3 ))ζ(2)
= CF
80(x − 1)4
1
(51−157x
−48(9+49x+18x2 )ζ 2 (2)−5(22−90x+82x2 −41x3 )ζ(3))−
16(x−1)3 x
+161x2 − 55x3 − (12 − 364x − 436x2 − 28x3 )ζ(2))H(1; x)
(1 + 5x + 5x2 + x3 )ζ(2)
3(−14 + 6x − 2x2 + x3 )ζ(2)
+
H(−1;
x)
+
H(2; x)
2(x − 1)3 x
4(x − 1)4
1
6xζ(2)
H(0, −1; x) +
(11−72x+366x2 −260x3 −45x4 +96xζ(2)
−
(x − 1)4
16(x−1)4 x
1 + 36x + 8x2 − 24x3 − 3x4
+528x2 ζ(2) + 192x3 ζ(2))H(0, 1; x) −
H(0, 0, 1; x)
8(x − 1)4 x
– 12 –
25−174x−x2
1+5x+5x2 +x3
7
H(1,
1;
x)
+
H(−1, 0, 1; x) +
H(1, 1, 1; x)
2
3
8(x − 1) x
(x − 1) x
(2x)
5 − 78x + 4x2 + 70x3 + 8x4
1 + 12x + 32x2 − 3x3
+
H(0, 1, 1; x) −
H(1, 0, 1; x)
4
4(x − 1) x
2(x − 1)3 x
14 − 6x + 2x2 − x3
2(2 + 15x + 4x2 )
−
H(2,
1,
1;
x)
−
H(0, 0, 1, 1; x)
4(x − 1)4
(x − 1)4
12x
2+15x+4x2
2+3x+4x2
H(0,0,0,1; x)−
H(0,−1,0,1; x)+
H(0,1,0,1; x)
−
(x−1)4
(x−1)4
(x−1)4
1
+CA
(60(x−1)3 + (−20(x−1)(56 + 105x + 40x2 ) + 60 ln (2)(−22
160(x−1)4
+
+30x − 26x2 + 9x3 ))ζ(2) − 12(18 + 137x + 108x2 )ζ 2 (2) − 5(−22 + 90x − 82x2
1
(1 + 5x + 5x2 + x3 )ζ(2)
H(−1; x) +
(335 + 66Lµ
+41x3 )ζ(3)) −
3
4(x − 1) x
144(x − 1)3 x
−843x − 198Lµ x + 681x2 + 198Lµ x2 − 173x3 − 66Lµ x3 − (72 + 468x + 2466x2
3(−14ζ(2) + 6xζ(2) − 2x2 ζ(2) + x3 ζ(2))
+126x3 )ζ(2))H(1; x) −
H(2; x)
8(x − 1)4
3xζ(2)
1
+
H(0, −1; x) −
(11 + 22x − 171x2 + 79x3 + 59x4 − (36x
4
(x − 1)
24(x − 1)4 x
20 + 40x − 66x2 − 3x3
+270x2 + 216x3 )ζ(2))H(0, 1; x) −
H(0, 0, 1; x)
8(x − 1)4
1 + 5x + 5x2 + x3
13 + 82x + 58x2
−
H(−1,
0,
1;
x)
−
H(1, 1; x)
2(x − 1)3 x
24(x − 1)2 x
10 + 23x + 9x2
22 + 56x − 62x2 − 7x3
H(0,
1,
1;
x)
−
H(1, 0, 1; x)
−
8(x − 1)4
8(x − 1)3
14 − 6x + 2x2 − x3
2 + 31x + 12x2
+
H(2,
1,
1;
x)
−
H(0, 0, 0, 1; x)
8(x − 1)4
4(x − 1)4
6x
2 + 7x + 12x2
+
H(0, −1, 0, 1; x) −
(2H(0, 0, 1, 1; x) − H(0, 1, 0, 1; x))
(x − 1)4
4(x − 1)4
(25 + 6Lµ )
1
+TR Nl
H(1; x) +
(H(0, 1; x) + 2H(1, 1; x))
36x
6x
(54 − 91x + 20x2 + 17x3 − (36 + 12x − 52x2 + 4x3 )ζ(2) + 36xζ(3))
+TR Nh
6(x − 1)4
25+322x+25x2 +(6−12x+6x2 )Lµ
1+21x+21x2 +x3
H(1;
x)
−
H(0, 1; x)
+
36(x − 1)2 x
6(x − 1)3 x
6x
−
H(0, 0, 1; x) .
(5.14)
(x − 1)4
(2l)
We checked our results for the form factors Gi (x) i = 1, 2, 3 against the calculation
of Martin Beneke, Tobias Huber, and Xin-Quing Li [33] and we found complete analytical
agreement.
– 13 –
6. Ward Identities
We explicitly checked that the UV renormalized form factors satisfy the on-shell Ward
identity1
u
u
iqµ
W +, µ
− MW
b
φ+
11
= 0,
(6.1)
b
where φ is the charged pseudo-Goldstone boson and the gray circles represents the sum of
all two-loop one-particle-irreducible QCD corrections to the vertices. The Lorentz index
associated to the W -boson is saturated by the boson momentum q µ . In order to satisfy
the relation in Eq. (6.1) it is necessary to renormalize also the factor mb appearing in the
tree-level φ+ ub coupling. The relevant NNLO mass counter term can be found in [32].
The two-loop corrections to the scalar coupling of the pseudo-Goldstone boson to quark
can be absorbed in a single form-factor S, defined as follows
u
φ+
11
=−
mb
S(q 2 ) u(p) (1 − γ5 ) u(P ) .
MW
(6.2)
b
The UV renormalized form factor has the following perturbative expansion in αs :
3 α 2
α αs
igw
s
s
(1l)
(2l)
(0l)
S
+
S
+O
,
+
S = √ Vub S
π
π
π3
2 2
with S (0l) = 1.
The one-loop form factor S (1l) is given by
2 (4−d)/2
1
X
µ
(1l)
S (1l,i) (d − 4)i + O (d − 4)2 .
C
S
= C(d)
F
2
m
(6.3)
(6.4)
i=−2
After UV renormalization (including the renormalization of the Yukawa φ+ ub coupling),
the coefficients of the expansion in (d − 4) are
S (1l,−2) = −1 ,
5
S (1l,−1) = + H(1; x) ,
4
1
1
S (1l,0) = −1 −
H(1; x) − H(1, 1; x) − H(0, 1; x) ,
2x
2
x+1
1
1
(1l,1)
S
= 1+
H(1; x) +
H(1, 1; x) +
H(0, 1; x) + H(1, 1, 1; x)
4x
2x
4x
1
1
1
+ H(1, 0, 1; x) + H(0, 1, 1; x) + H(0, 0, 1; x).
2
2
4
1
(6.5)
(6.6)
(6.7)
(6.8)
It can be proved that the Ward identity is fulfilled already at the level of master integrals, irrespectively
on the analytic expression of the MIs themselves.
– 14 –
The two-loop form factor S (2l) is given by
S
(2l)
2
= C (d)
µ2
m2
4−d
CF
0
X
i=−4
S (2l,i) (d − 4)i + O (d − 4) .
(6.9)
where the coefficient of the expansion in (d − 4) are
S (2l,−4) = CF
S (2l,−3) =
S (2l,−2) =
S (2l,−1) =
S (2l,0) =
1
,
2
(6.10)
11
1
5
(6.11)
CF − − H(1; x) − CA + TR Nl ,
4
8
2
57 2 + 5x
1
49 + 9ζ(2)
CF
+
H(1; x) + H(0, 1; x) + 2H(1, 1; x) + CA
32
(4x)
2
72
11
1
11
5
1
1
− Lµ + H(1; x) + TR Nl − + Lµ − H(1; x) + TR Nh Lµ , (6.12)
12
12
18 3
3
3
2 + 5x
3(47 + 8ζ(2) − 16ζ(3)) 7 + 10x
−
H(1; x) −
H(0, 1; x)
CF −
64
8x
8x
6+5x
1
3
−
H(1, 1; x)− H(0, 0, 1; x)− H(0, 1, 1; x)− H(1, 0, 1; x)− 4H(1, 1, 1; x)
4x
4
2
2
(1549+1980Lµ−396Lµ+972ζ(2)−1188ζ(3)) 67+66Lµ−18ζ(2)
+CA
+
H(1; x)
1728
72
−125 − 180Lµ + 36L2µ − 108ζ(2) 5 + 6Lµ
+TR Nl
−
H(1; x)
432
18
−5Lµ + L2µ − ζ(2) 1
− Lµ H(1; x) ,
+TR Nh
12
3
831
3(13 − 7x) 3 ln (2)(x − 4)(7x − 8)
9(5 − 34x + 11x2 )
CF
+
+
ζ(2)
+
256
16(x − 1)
8(x − 1)2
80(x − 1)2
2
2
4K 2
(74 − 96x + 13x )ζ(3) (1 + 2x + x )ζ(2)
−
+
H(−1; x)
ζ (2) −
5
32(x − 1)2
2(x − 1)x
17 + 8x − 25x2 − (12 − 78x + 30x2 )ζ(2) − (8x − 8x2 )ζ(3)
H(1; x)
−
16(x − 1)x
9(4 − 4x + x2 )ζ(2)
1
ζ(2)
−
H(0, −1; x) +
(11 + 12x − 15x2
H(2; x) −
8(x − 1)2
2
16(x − 1)2 x
4 − x + 5x2
H(1, 1; x)
−8x3 + 12xζ(2) + 24x2 ζ(2))H(0, 1; x) +
8x2
1 + 2x + x2
2 − 13x + 20x2 − 3x3
+
H(−1, 0, 1; x) −
H(0, 0, 1; x)
(x − 1)x
16(x − 1)2 x
10 − 33x + 20x2 + 6x3
14 + 5x
H(1, 1, 1; x) +
H(0, 1, 1; x)
+
4x
8(x − 1)2 x
2 + 5x − 4x2
3(4 − 4x + x2 )
−
H(1, 0, 1; x) −
H(2, 1, 1; x) − H(0, −1, 0, 1; x)
4(x − 1)x
8(x − 1)2
5 − 2x + 3x2
1 − 10x + 3x2
7
−
H(0,
0,
0,
1;
x)
+
H(0, 0, 1, 1; x) + H(0, 1, 1, 1; x)
2
2
8(x − 1)
4(x − 1)
2
– 15 –
4 − 4x + 3x2
1
H(0, 1, 0, 1; x) − H(1, 0, 0, 1; x) + 3H(1, 0, 1, 1; x)
2
4(x − 1)
2
55L2µ 11L3µ
54589 11Lµ
+2H(1, 1, 0, 1; x) + 8H(1, 1, 1, 1; x) + CA −
−
+
−
20736
12
192
288
179 − 250x + 125x2
3 ln (2)(x − 4)(7x − 8) 1067 + 49x
11Lµ
−
−
ζ(2)
−
+
96
16(x − 1)2
576(x − 1)
160(x − 1)2
2K 2
(896 − 1324x + 347x2 )ζ(3) (1 + 2x + x2 )ζ(2)
−
−
H(−1; x)
ζ (2) +
5
576(x − 1)2
4(x − 1)x
1
(708 + 198Lµ − 466x − 198Lµ x − 99L2µ x − 242x2 + 99L2µ x2
+
432(x − 1)x
−216ζ(2) + 738xζ(2) − 684x2 ζ(2) − 378xζ(3) + 378x2 ζ(3))H(1; x)
9(4 − 4x + x2 )ζ(2)
1
1
+
H(2; x) + ζ(2)H(0, −1; x) −
(66 + 56x
2
(16(x − 1) )
4
144(x − 1)2 x
+
+66Lµ x−211x2 −132Lµ x2 +89x3 +66Lµ x3 −(126x−144x2 +72x3 )ζ(2))H(0, 1; x)
78 + 223x + 132Lµ x − 72xζ(2)
1 + 2x + x2
−
H(1, 1; x) −
H(−1, 0, 1; x)
144x
2(x − 1)x
44 − 88x + 53x2
11
40 − 56x + 7x2
H(1,
1,
1;
x)
−
H(0,
0,
1;
x)
−
H(0, 1, 1; x)
−
48(x − 1)2
6
48(x − 1)2
29 − 35x
3(4 − 4x + x2 )
1
+
H(1, 0, 1; x) +
H(2, 1, 1; x) + H(0, −1, 0, 1; x)
2
48(x − 1)
16(x − 1)
2
1
1
1
−
H(0, 0, 0, 1; x) −
H(0, 0, 1, 1; x) +
H(0, 1, 0, 1; x)
2
2
8(x − 1)
4(x − 1)
8(x − 1)2
1
1
(3893 + 1728Lµ − 540L2µ + 72L3µ + 3420ζ(2)
+ H(1, 0, 0, 1; x) + TR Nl
2
5184
48 + 18Lµ + 28x − 9L2µ x + 45xζ(2)
H(1; x)
−216Lµ ζ(2) + 720ζ(3)) +
108x
3 + 5x + 3Lµ x
3 + 5x + 3Lµ x
1
+
H(0, 1; x) +
H(1, 1; x) + H(0, 0, 1; x)
18x
9x
6
Lµ 5L2µ L3µ
1
2
1
−
+
+ H(0, 1, 1; x) + H(1, 0, 1; x) + H(1, 1, 1; x) + TR Nh
3
3
3
3
48
72
2
2
3
Lµ 409 − 747x + 651x − 185x
11407 − 17630x + 8527x
+
−
−
ζ(2)
2592(x − 1)2
24
144(x − 1)3
1
7ζ(3)
+
(48 + 18Lµ + 104x − 36Lµ x − 9L2µ x − 112x2 + 18Lµ x2
−
36
108(x − 1)2 x
+18L2µ x2 + 56x3 − 9L2µ x3 + 9xζ(2) − 18x2 ζ(2) + 9x3 ζ(2))H(1; x)
1
−
(3+14x+3Lµ x−9Lµ x2 − 6x3 + 9Lµ x3 + 5x4 − 3Lµ x4 )H(0, 1; x)
18(x − 1)3 x
Lµ
1
+ H(1, 1; x) + H(0, 0, 1; x) .
(6.13)
3
6
When written in terms of form factors, the Ward identity in Eq. (6.1) reads as follows:
(2l)
(2l)
(2l)
2G1 (x) + xG2 (x) + G3 (x) − 2S (2l) (x) = 0 .
– 16 –
(6.14)
It can be checked that the form factors presented in this paper fulfill Eq. (6.14).
7. Conclusions
In this paper, we presented analytic expressions for the two-loop QCD corrections to the
decay process b → u W ∗ → u l ν. This process is important for the precise determination
of the CKM matrix element Vub and, therefore, for the study of flavor and CP violation
within and beyond the Standard Model of fundamental interactions.
The Lorentz structure of the process is parametrized in terms of three form factors,
whose analytic expression are given in the form of a Laurent series of (d − 4), where d is
the space-time dimension. The coefficients of the series are expressed in the well known
functional basis of harmonic polylogarithms of a single dimensionless variable. The result
can be used in a SCET framework, after combining it with the jet and soft functions
already known in the literature, for a phenomenological determination of |Vub |. The results
presented here are the first step towards a complete determination of the NNLO QCD
corrections to the heavy-to-light quark transition.
Acknowledgments
We are grateful to U. Aglietti, for proposing the subject of the paper, and to M. Beneke
for allowing us to compare our results with the ones obtained by his group. We thank
U. Aglietti and P. Gambino for carefully reading the manuscript and providing us with
valuable feedback. We wish to thank T. Becher, C. Greub, T. Gehrmann, and B. Pecjak
for useful discussions. We are indebted with R. Boughezal and G. Bell for several numerical
and analytical cross-checks of the master integrals. We are grateful to J. Vermaseren for
his kind assistance in the use of FORM [34], and to the authors of the packages AIR [35] and
FIESTA [36], that were employed in partial checks of the calculation.
R.B. wishes to thank the Theoretical Physics Department of the University of Florence
for kind hospitality during a part of this work.
This work was supported by the Swiss National Science Foundation (SNF) under
contract 200020-117602.
A. The Master Integrals
In this Appendix we collect the analytic expressions of the Master Integrals for the Feynman
diagrams of Fig. 3. We provide only eight of them, since the other MIs can be found
in [37, 38]. It must be pointed out that the MIs (a)–(f) in Fig. 3 were already calculated
in [39]. We checked the analytic expressions that we obtained against the results in [39]
and we found complete agreement. Moreover, all the MIs were checked by comparing their
numerical value to the results obtained by direct numerical integration with the sector
decomposition method. The numerical integration was carried out by using the package
FIESTA (see [36]). The checks were done for several values of the variable y.
– 17 –
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 3: Master Integrals needed for the Two-loop QCD corrections. Thick lines represent
massive particles, thin lines represent massless ones.
The explicit expression of the MIs depends on the chosen normalization of the integration measure. The integration on the loop momenta is normalized as follows
Z
1
Dd k =
C(d)
µ2
m2
(d−4)
Z
2
dd k
(4π 2 )
(d−2)
2
,
(A.1)
where C(d) is defined in Eq. (3.16). In Eq. (A.1) µ stands for the ’t Hooft mass of dimensional regularization. The integration measure in Eq. (A.1) is chosen in such a way that
the one-loop massive tadpole becomes
Z
1
m2
Dd k 2
.
(A.2)
=
k + m2
(d − 2)(d − 4)
In the expressions below, K is a rational number (its numerical value is K = 3.32812 ±
0.00002 ∼ 213/64), while a4 = Li4 (1/2) = 0.51747906.... ζ(2) and ζ(3) are the Riemann ζ
function evaluated in 2 and 3 respectively: ζ(2) = 1.6449341..., ζ(3) = 1.2020569....
The expressions of the MIs are the following.
=
0
X
1
Ai (d − 4)i + O(d − 4) ,
m4 (1 + y)2
(A.3)
i=−4
A−4 =
1
,
12
(A.4)
– 18 –
1
H(−1; y) ,
(A.5)
6
7
1
A−2 = − ζ(2) + H(−1, −1; y) ,
(A.6)
48
3
89
7
2
A−1 =
ζ(3) − ζ(2)H(−1; y) + H(−1, −1, −1; y) ,
(A.7)
96
24
3
2
65
7
4
A0 = − ζ 2 (2)K + ζ(3)H(−1; y) − ζ(2)H(−1, −1; y) + H(−1, −1, −1, −1; y)
5
48
12
3
1
− H(−1, 0, 0, −1; y) .
(A.8)
2
A−3 =
=
B0 =
1
B0 + O(d − 4) ,
m2 (1 + y)
(A.9)
27 2
3
1
1
ζ (2) + ζ(2)H(0, −1; y) − H(0, −1, 0, −1; y) + H(0, 0, −1, −1; y)
160
16
16
8
1
− H(0, 0, 0, −1; y) .
(A.10)
16
=
1
X
1
Ci (d − 4)i + O(d − 4)2 ,
m4 (1 + y)
(A.11)
i=−1
1
ζ(2) ,
(A.12)
32
1
1
1
1
C0 =
ζ(3) + ζ(2)H(−1; y) + H(0, −1, −1; y) − H(0, 0, −1; y) ,
(A.13)
64
16
16
16
9
1
3
1
C1 = − ζ 2 (2) + ζ(3)H(−1; y) − ζ(2)H(0, −1; y) + ζ(2)H(−1, −1; y)
80
32
32
8
1
1
3
+ H(−1, 0, −1, −1; y) − H(−1, 0, 0, −1; y) + H(0, −1, −1, −1; y)
8
8
16
9
5
1
(A.14)
− H(0, −1, 0, −1; y) − H(0, 0, −1, −1; y) + H(0, 0, 0, −1; y) .
16
32
32
C−1 =
=
1
X
i=−2
Di (d − 4)i + O(d − 4)2 ,
1
,
8
5
(1 + y)
=− +
H(−1; y) ,
16
8y
(A.15)
D−2 =
(A.16)
D−1
(A.17)
– 19 –
D0 =
1
5
3
1
19
+ ζ(2) − H(−1; y) + H(−1, −1; y) − H(0, −1; y)
32 " 16
16
16
16 #
"
5
5
3
1
1
1
ζ(3)
+ − H(−1; y) + H(−1, −1; y) − H(0, −1; y) +
y
16
16
8
(1 + y) 64
3
3
1
1
ζ(2) ln(2) − ζ(2)H(−2; y) − H(−2, −1, −1; y) − H(0, −1, −1; y)
16
16
16
16
#
1
(A.18)
+ H(0, 0, −1; y) ,
16
−
65
7
5
3
19
5
−
ζ(3) − ζ(2) − ζ(2) ln(2) + H(−1; y) + ζ(2)H(−1; y)
64 128
32
32
32
32
5
15
1
3
− ζ(2)H(−2; y) + H(0, −1; y) − H(−1, −1; y) − H(−2, −1, −1; y)
32
32
32
32
5
1
3
1
+ H(−1, −1, −1; y) − H(−1, 0, −1; y) − H(0, −1, −1; y) + H(0, 0, −1; y)
16"
8
32
16
3
15
5
1 19
H(−1; y) + ζ(2)H(−1; y) − H(−1, −1; y) + H(0, −1; y)
+
y 32
32
32
16
#
5
3
3
1
+ H(−1, −1, −1; y)− H(−1, 0, −1; y)− H(0, −1, −1; y)+ H(0, 0, −1; y)
16
16
16
8
"
1
1
5
3
1
33 2
1
+
− ln4 (2) −
ζ(3) + ζ(2) ln(2) + ζ(2) ln2 (2) +
ζ (2) − a4
(1 + y)
96
128
32
16
640
4
5
3
3
7
−
ζ(3) − ζ(2) ln(2) H(−1; y) +
ζ(3) + ζ(2) H(−2; y)
64
16
32
32
3
5
3
− ζ(2)H(0, −1; y) + ζ(2)H(−1, −2; y) − ζ(2)H(−2, −1; y)
32
16
32
1
1
1
+ H(−2, −1, −1; y) + H(0, −1, −1; y) − H(0, 0, −1; y)
32
32
32
3
1
1
− H(−2, −1, −1, −1; y) + H(−2, −1, 0, −1; y) + H(−1, −2, −1, −1; y)
16
16
16
1
1
3
+ H(−1, 0, −1, −1; y) − H(−1, 0, 0, −1; y) − H(0, −1, −1, −1; y)
16
16
16
#
1
3
1
(A.19)
+ H(0, −1, 0, −1; y) + H(0, 0, −1, −1; y) − H(0, 0, 0, −1; y) .
8
8
32
D1 = −
=
2
1 X
Ei (d − 4)i + O(d − 4)3 ,
m2
(A.20)
i=−1
E−1 = −
1
H(−1; y) ,
8y
"
(A.21)
#
1
3
E0 =
H(−1; y) − H(−1, −1; y) + H(0, −1; y)
8y
2
– 20 –
"
#
1
3ζ(2) + H(−1, −1; y) ,
−
16(2 + y)
"
1
1
3
3
E1 =
− H(−1; y) − ζ(2)H(−1; y) + H(−1, −1; y)
y
8
32
16
(A.22)
5
3
1
3
H(−1, −1, −1; y) + H(−1, 0, −1; y) − H(0, −1; y) + H(0, −1, −1; y)
16
16
8
16
"
#
5
1
3
3
1
− ζ(3) + ζ(2) ln(2) + ζ(2)H(−2; y)
− H(0, 0, −1; y) +
8
(1 + y)
64
16
16
"
#
1
7
1
1
1
+ H(−2, −1, −1; y) + H(0, −1, −1; y) − H(0, 0, −1; y) +
ζ(3)
16
16
16
(2 + y) 32
−
5
1
3
3
ζ(2) − ζ(2)H(−1; y) + H(−1, −1; y) − H(−1, −1, −1; y)
16
32
16
16
#
1
+ H(−1, 0, −1; y) ,
(A.23)
16
"
1
1 1
9
3
3
3
E2 =
+
ζ(3) + ζ(2) + ζ(2) ln(2) H(−1; y) +
+ ζ(2) H(0, −1; y)
y 8 128
32
32
8 32
3
7
3
3
−
+ ζ(2) H(−1, −1; y) + ζ(2)H(−1, −2; y) − H(−1, 0, −1; y)
16 32
32
16
1
3
5
+ H(0, 0, −1; y) − H(0, −1, −1; y) + H(−1, −1, −1; y)
8
16
16
1
9
5
+ H(−1, −2, −1, −1; y) − H(−1, −1, −1, −1; y) + H(−1, −1, 0, −1; y)
32
16
16
7
5
5
+ H(−1, 0, −1, −1; y) − H(−1, 0, 0, −1; y) + H(0, −1, −1, −1; y)
16
32
16
#
3
3
1
− H(0, −1, 0, −1; y) − H(0, 0, −1, −1; y) + H(0, 0, 0, −1; y)
16
16
8
"
1 4
5
3
1
33 2
1
1
ln (2) + ζ(3) − ζ(2) ln(2) − ζ(2) ln2 (2) −
ζ (2) + a4
+
(1 + y) 96
64
16
16
640
4
5
3
3
7
+
ζ(3) − ζ(2) ln(2) H(−1; y) −
ζ(3) + ζ(2) H(−2; y)
64
16
32
16
3
5
3
+ ζ(2)H(0, −1; y) + ζ(2)H(−2, −1; y) − ζ(2)H(−1, −2; y)
32
32
16
1
1
1
− H(0, −1, −1; y) + H(0, 0, −1; y) − H(−2, −1, −1; y)
16
16
16
3
1
1
+ H(−2, −1, −1, −1; y) − H(−2, −1, 0, −1; y) − H(−1, −2, −1, −1; y)
16
16
16
1
1
3
− H(−1, 0, −1, −1; y) + H(−1, 0, 0, −1; y) + H(0, −1, −1, −1; y)
16
16
16
#
1
3
1
− H(0, −1, 0, −1; y) − H(0, 0, −1, −1; y) + H(0, 0, 0, −1; y)
8
8
32
+
– 21 –
"
1
7
3
45 2
5
1
− ζ(3) − ζ(2) −
ζ (2) +
ζ(3) + ζ(2)
+
(2 + y)
32
16
128
8
32
3
9
3
1
+ ζ(2) ln(2) H(−1; y) −
+ ζ(2) H(−1, −1; y) + ζ(2)H(−1, −2; y)
16
16 32
16
3
1
1
+ H(−1, −1, −1; y) − H(−1, 0, −1; y) + H(−1, −2, −1, −1; y)
16
16
16
7
3
5
− H(−1, −1, −1, −1; y) + H(−1, −1, 0, −1; y) + H(−1, 0, −1, −1; y)
16
16
32
#
1
− H(−1, 0, 0, −1; y) .
(A.24)
8
1
X
1
= 2
Fi (d − 4)i + O(d − 4)2 ,
m (1 + y)
(A.25)
i=−1
1
1
ζ(2) + H(0, −1; y) ,
(A.26)
8
8
7
3
1
3
1
F0 = − ζ(3) − ζ(2) ln(2) + ζ(2)H(−1; y) − ζ(2)H(−2; y) − H(−2, −1, −1; y)
64
16
8
16
16
1
3
1
+ H(−1, 0, −1; y) + H(0, −1, −1; y) − H(0, 0, −1; y) ,
(A.27)
8
16
8
1
1
227 2
1
3
7
F1 = − ln4 (2) + ζ(2) ln2 (2) +
ζ (2) − a4 − ζ(3)H(−1; y) + ζ(3)H(−2; y)
96
16
640
4
16
32
ζ(2)
3
+
[3H(0, −1; y) + 4H(−1, −1; y) − 5H(−2, −1; y)] − H(−2, −1, −1, −1; y)
32
16
1
1
1
+ H(−2, −1, 0, −1; y) + H(−1, −1, 0, −1; y) + H(−1, 0, −1, −1; y)
16
8
4
3
5
3
− H(−1, 0, 0, −1; y) + H(0, −1, −1, −1; y) − H(0, −1, 0, −1; y)
16
16
16
1
3
(A.28)
− H(0, 0, −1, −1; y) + H(0, 0, 0, −1; y) .
16
8
F−1 =
=
1
X
i=−2
Gi (d − 4)i + O(d − 4)2 ,
(A.29)
1
,
(A.30)
8
5
(A.31)
G−1 = − ,
16
19
1
1
1
G0 =
− ζ(2) − H(0, −1; y) −
[z3 + H(0, 0, −1; y)] ,
(A.32)
32 16
16
8(y + 1)
3
5
1
5
1
65
G1 = − + ζ(3) + ζ(2) − ζ(2)H(−1; y) + H(0, −1; y) − H(−1, 0, −1; y)
64 32
32
16
32
16
G−2 =
– 22 –
"
1
3
1
7 2
1
ζ(3) +
ζ (2)
− H(0, −1, −1; y) + H(0, 0, −1; y) +
8
32
(1 + y) 16
160
1
1
1
1
+ ζ(3)H(−1; y) − ζ(2)H(0, −1; y) + H(0, 0, −1; y) + H(−1, 0, 0, −1; y)
8
16
16 #
8
1
1
(A.33)
− H(0, −1, 0, −1; y) − H(0, 0, −1, −1; y) .
16
4
1
X
1
= 2
Ji (d − 4)i + O(d − 4)2 ,
m (1 + y)
(A.34)
i=−1
1
1
ζ(2) + H(0, −1; y) ,
(A.35)
8
8
3
1
1
1
J0 = − ζ(3) + ζ(2)H(−1; y) + H(−1, 0, −1; y) + H(0, −1, −1; y)
16
8
8
4
3
− H(0, 0, −1; y) ,
(A.36)
16
23 2
1
1
1
J1 =
ζ (2) − ζ(3)H(−1; y) + ζ(2)H(−1, −1; y) + H(−1, −1, 0, −1; y)
160
16
8
8
1
1
1
+ H(−1, 0, −1, −1; y) − H(−1, 0, 0, −1; y) + H(0, −1, −1, −1; y)
4
16
2
1
3
1
− H(0, −1, 0, −1; y) − H(0, 0, −1, −1; y) + H(0, 0, 0, −1; y) .
(A.37)
4
8
32
J−1 =
References
[1] E. Gardi, arXiv:0806.4524 [hep-ph].
[2] M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. 41 (1985) 120 [Yad. Fiz. 41 (1985)
187].
J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B 247 (1990) 399.
I. I. Y. Bigi, N. G. Uraltsev and A. I. Vainshtein, Phys. Lett. B 293 (1992) 430
[Erratum-ibid. B 297 (1993) 477] [arXiv:hep-ph/9207214].
I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein, Phys. Rev. Lett. 71 (1993)
496 [arXiv:hep-ph/9304225].
A. F. Falk and M. Neubert, Phys. Rev. D 47 (1993) 2965 [arXiv:hep-ph/9209268]. Phys. Rev.
D 47 (1993) 2982 [arXiv:hep-ph/9209269].
A. V. Manohar and M. B. Wise, Phys. Rev. D 49 (1994) 1310 [arXiv:hep-ph/9308246].
B. Blok, L. Koyrakh, M. A. Shifman and A. I. Vainshtein, Phys. Rev. D 49 (1994) 3356
[Erratum-ibid. D 50 (1994) 3572] [arXiv:hep-ph/9307247].
T. Mannel, Nucl. Phys. B 413 (1994) 396 [arXiv:hep-ph/9308262].
M. Neubert, Int. J. Mod. Phys. A 11 (1996) 4173 [arXiv:hep-ph/9604412].
[3] T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652.
S. M. Berman, Phys. Rev. 112 (1958) 267.
– 23 –
[4] Y. Nir, Phys. Lett. B 221 (1989) 184.
M. Jezabek and J. H. Kuhn, Nucl. Phys. B 314 (1989) 1.
V. Aquila, P. Gambino, G. Ridolfi and N. Uraltsev, Nucl. Phys. B 719 (2005) 77
[arXiv:hep-ph/0503083].
[5] M. E. Luke, M. J. Savage and M. B. Wise, Phys. Lett. B 343 (1995) 329
[arXiv:hep-ph/9409287]. Phys. Lett. B 345 (1995) 301 [arXiv:hep-ph/9410387].
[6] P. Ball, M. Beneke and V. M. Braun, Phys. Rev. D 52 (1995) 3929 [arXiv:hep-ph/9503492].
[7] A. Czarnecki and K. Melnikov, Nucl. Phys. B 505 (1997) 65 [arXiv:hep-ph/9703277]. Phys.
Rev. Lett. 78 (1997) 3630 [arXiv:hep-ph/9703291]. Phys. Rev. D 59 (1999) 014036
[arXiv:hep-ph/9804215].
[8] J. Franzkowski and J. B. Tausk, Eur. Phys. J. C 5 (1998) 517 [arXiv:hep-ph/9712205].
[9] I. R. Blokland, A. Czarnecki, M. Slusarczyk and F. Tkachov, Phys. Rev. Lett. 93 (2004)
062001 [arXiv:hep-ph/0403221]. Phys. Rev. D 71 (2005) 054004 [arXiv:hep-ph/0503039].
[10] A. Pak, I. R. Blokland and A. Czarnecki, Phys. Rev. D 73 (2006) 114009
[arXiv:hep-ph/0604233].
[11] T. van Ritbergen, Phys. Lett. B 454 (1999) 353 [arXiv:hep-ph/9903226].
[12] F. De Fazio and M. Neubert, JHEP 9906 (1999) 017 [arXiv:hep-ph/9905351].
[13] U. Aglietti, Nucl. Phys. B 610 (2001) 293 [arXiv:hep-ph/0104020].
[14] E. Gardi, JHEP 0502 (2005) 053 [arXiv:hep-ph/0501257].
[15] J. R. Andersen and E. Gardi, JHEP 0601 (2006) 097 [arXiv:hep-ph/0509360].
[16] U. Aglietti, G. Ricciardi and G. Ferrera, Phys. Rev. D 74 (2006) 034004
[arXiv:hep-ph/0507285]. Phys. Rev. D 74 (2006) 034005 [arXiv:hep-ph/0509095]. Phys. Rev.
D 74 (2006) 034006 [arXiv:hep-ph/0509271].
[17] P. Gambino, E. Gardi and G. Ridolfi, JHEP 0612 (2006) 036 [arXiv:hep-ph/0610140].
[18] K. Melnikov, arXiv:0803.0951 [hep-ph].
[19] S. W. Bosch, B. O. Lange, M. Neubert and G. Paz, Nucl. Phys. B 699 (2004) 335
[arXiv:hep-ph/0402094].
[20] B. O. Lange, M. Neubert and G. Paz, Phys. Rev. D 72 (2005) 073006
[arXiv:hep-ph/0504071].
[21] U. Aglietti, et al., Phys. Lett. B 653 (2007) 38 [arXiv:0707.2010 [hep-ph]].
[22] P. Gambino, P. Giordano, G. Ossola and N. Uraltsev, JHEP 0710 (2007) 058
[arXiv:0707.2493 [hep-ph]].
[23] T. Becher and M. Neubert, Phys. Lett. B 633 (2006) 739 [arXiv:hep-ph/0512208].
[24] T. Becher and M. Neubert, Phys. Lett. B 637 (2006) 251 [arXiv:hep-ph/0603140].
[25] B. Pecjak, contribution to ICHEP ’08, 34th International Conference on High Energy
Physics, Philadelphia, PA, USA.
– 24 –
[26] R. Bonciani, P. Mastrolia and E. Remiddi, Nucl. Phys. B 676 (2004) 399
[arXiv:hep-ph/0307295].
U. Aglietti, R. Bonciani, G. Degrassi and A. Vicini, Phys. Lett. B 595 (2004) 432
[arXiv:hep-ph/0404071].
W. Bernreuther, et al., Nucl. Phys. B 706 (2005) 245 [hep-ph/0406046]; Nucl. Phys. B 712
(2005) 229 [hep-ph/0412259]; Nucl. Phys. B 723 (2005) 91 [hep-ph/0504190]. Phys. Rev. D
72 (2005) 096002 [hep-ph/0508254]. Phys. Rev. Lett. 95 (2005) 261802 [hep-ph/0509341].
S. Moch, J. A. M. Vermaseren and A. Vogt, JHEP 0508 (2005) 049 [arXiv:hep-ph/0507039].
Phys. Lett. B 625 (2005) 245 [arXiv:hep-ph/0508055].
T. Gehrmann, T. Huber and D. Maitre, Phys. Lett. B 622 (2005) 295
[arXiv:hep-ph/0507061].
R. J. Gonsalves, Int. J. Mod. Phys. E 17 (2008) 870 [arXiv:0807.0828 [hep-ph]].
[27] P. Nogueira, J. Comput. Phys. 105 (1993) 279.
[28] S. Laporta and E. Remiddi, Phys. Lett. B 379 (1996) 283 [hep-ph/9602417].
S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033].
F.V. Tkachov, Phys. Lett. B 100 (1981) 65.
K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B 192 (1981) 159.
[29] A.V. Kotikov, Phys. Lett. B 254 (1991) 158; Phys. Lett. B 259 (1991) 314; Phys. Lett. B
267 (1991) 123;
E. Remiddi, Nuovo Cim. A 110 (1997) 1435. [hep-th/9711188];
M. Caffo, H. Czyz, S. Laporta and E. Remiddi, Acta Phys. Polon. B 29 (1998) 2627;
[hep-th/9807119]; Nuovo Cim. A 111 (1998) 365. [hep-th/9805118];
T. Gehrmann and E. Remiddi, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329];
M. Argeri and P. Mastrolia, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037].
[30] A B. Goncharov, Math. Res. Lett. 5 (1998), 497-516.
D.J. Broadhurst, Eur. Phys. J. C 8 (1999) 311 [hep-th/9803091].
E. Remiddi and J.A.M. Vermaseren, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237].
T. Gehrmann and E. Remiddi, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173].
J. Vollinga and S. Weinzierl, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259].
D. Maı̂tre, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152]. hep-ph/0703052.
[31] SemilepFF.txt, file included in the arXiv submission.
[32] K. Melnikov and T. van Ritbergen, Nucl. Phys. B 591,(2000) 515 [arXiv:hep-ph/0005131].
[33] M. Beneke, T. Huber, and X. Li, in preparation.
[34] J.A.M. Vermaseren, Symbolic Manipulation with FORM, Version 2, CAN, Amsterdam, 1991;
“New features of FORM” [math-ph/0010025].
[35] C. Anastasiou and A. Lazopoulos, JHEP 0407, (2004) 046 [arXiv:hep-ph/0404258].
[36] A. V. Smirnov and M. N. Tentyukov, arXiv:0807.4129 [hep-ph].
[37] R. Bonciani, A. Ferroglia, T. Gehrmann, D. Maitre and C. Studerus, JHEP 0807 (2008) 129
[arXiv:0806.2301 [hep-ph]].
[38] J. Fleischer, M. Y. Kalmykov and A. V. Kotikov, Phys. Lett. B 462 (1999) 169
[arXiv:hep-ph/9905249].
M. Argeri, P. Mastrolia and E. Remiddi, Nucl. Phys. B 631 (2002) 388
[arXiv:hep-ph/0202123].
– 25 –
J. Fleischer, A. V. Kotikov and O. L. Veretin, Nucl. Phys. B 547 (1999) 343.
U. Aglietti and R. Bonciani, Nucl. Phys. B 668 (2003) 3.
[39] G. Bell, arXiv:0705.3133 [hep-ph]. Nucl. Phys. B 795 (2008) 1 [arXiv:0705.3127 [hep-ph]].
– 26 –

Download Report