2-loop QCD corrections to Bc-meson leptonic constant

0.21 MB·English

by  Andrei I. Onishchenko

B
2-loop QCD corrections to -meson leptonic constant.
c
Andrei I. Onishchenkoa,b and Oleg L. Veretina
a) II. Institute fu¨r Theoretische Teilchenphysik,
Universit¨at Hamburg, 22761 Hamburg, Germany
b) Institute for High Energy Physics, Protvino, Russia
7
Abstract
0
0
We present results for two-loop QCD corrections to the short distance coefficient that gov-
2
erns leptonic decay of B -meson. After brief discussionof the technics of asymptotic expansion
n c
a used to perform this calculation, we give comments on the numerical value of two-loop correc-
J
tions and its impact on NRQCD description of B -meson bound state.
c
2
PACS: 12.39.Jh, 13.25.Gv, 14.40.Lb, 14.40.Nd
2

3
v 1 Introduction
2
3
1 Thedescription of various aspects of Bc-meson physics1 has received recently a lot of attention
2 both from theoretical and experimental sides. B -meson is the only meson within SM, which
c
0
contains two heavy quarks of different flavor. For this reason one might expect, that its
3
0 spectroscopy,productionmechanismsanddecaysdiffersignificantlyfromthoseofcharmonium,
/ bottomonium as well as hadrons with one heavy quark. Indeed, contrary to the case of J/Ψ
h
p and Υ-mesons, Bc-meson is a long living system, decaying trough electroweak interactions.
- To produce¯bc-system in flavor conserving collisions, an additional heavy quark pair should be
p
e created,whatmakestheconsiderationofproductionmechanismcomplexalreadyattheleading
h
order in the coupling constant. The bound state heavy quarks dynamics, being very similar to
:
v that of other quarkonia, at the same time has some differences. With respect to the hierarchy
i
X of dynamical scales, B -meson stands among the families of charmonium c¯c and bottomonium
c
r ¯bb and thus could be used to study both quantitatively and conceptually existing effective low
a
energy frameworks for the description of bound state heavy quark dynamics, like NRQCD [2],
pNRQCD [3] and vNRQCD [4]. So, the study of B -meson gives us an opportunity to test SM
c
in a completely new dynamical regime, explore it at a different angle, what may give us an
additional insight into our understanding of SM in general.
Today, we already have many theoretical predictions, concerning various B -meson prop-
c
erties: spectroscopy [5], production [6] and decay channels [7]. In the present work we will
be interested in radiative corrections to the leptonic decay constant of B -meson, entering
c
expression for its purely leptonic width. This particular decay mode of B -meson is not really
c
interesting for a direct measurement — the necessarily spin-flip makes its branching fraction
very small. However, the same quantity appears in the formulation of two-point QCD sum
rules [8]. And the latter may be used to extract an additional information about parameters
of SM, namely the heavy quark masses. A first step towards systematic formulation of NNLO
B -meson sum rules, treating B -meson as a Coulomb system, was made in [9], where NLO
c c
1For a review see [1]
1
analysis was presented. In order to describe B -meson QCD sum rules at NNLO, one needs
c
both 2-loop matching Wilson coefficient of QCD current with B quantum numbers on cor-
c
responding NRQCD current and two point NRQCD correlator, computed at NNLO. Below
we will present the results of matching procedure. These results are also interesting from the
point of view of further development of computation technics used in dimensionally regulated
NRQCD [10]. Here we are dealing with threshold expansion with two mass scales m and m ,
b c
what is an extension of previously obtained results for matching of vector current with two
equal heavy quark masses [11, 12]. The presence of two mass scales of course makes problem
more complicated, however as we will see the whole calculation could be performed within
the general framework of threshold expansion in dimensionally regulated NRQCD [10]. Here
we would like to stress, that we treat B -meson as a Coulomb system. So, in our matching
c
procedure c-quark is considered to be heavy with respect to factorization scale m > µ .
c fac
As a consequence, it is impossible to make connection of our result with the results on the
matching of heavy-light currents [13]. On the other side the limit m = m is well defined in
b c
ourapproach andinthecase ofvector currentmatchingwith equalquarkmasseswereproduce
already known results of [11, 12].
This paper is organized as follows. In section 2 we introduce necessary notation and
present one-loop results obtained previously [14]. Section 3 contains two-loop results for B -
c
meson leptonic decay constant. Here we also describe shortly various steps of the performed
calculation. And finally, in section 4 we present our conclusion.
B
2 - meson leptonic constant: one loop result
c
The B -meson leptonic decay proceeds through a virtual W-boson. The W-boson couples
c
to the B through the axial-vector part of the charged weak current. All QCD effects, both
c
perturbative and nonperturbative, enter into the decay rate through the decay constant f ,
Bc
defined by the matrix element
h0|¯bγµγ c|B (P)i = if Pµ, (1)
5 c Bc
where |B (P)i is the state of B -meson with four-momentum P. It has the standard covariant
c c
normalization hB (P′)|B (P)i = (2π)32Eδ3(P′−P) and its phase has been chosen so that f
c c Bc
is real and positive.
To separate short-distance (∼ 1/m ) and long-distance (≫ 1/m ) QCD contributions to
Q Q
f we use the formalism of NRQCD [2]. As a result the decay amplitude is factored into
Bc
short-distance coefficients multiplied by NRQCD matrix elements. The Wilson short-distance
coefficients could bethen calculated furtherorder byorder inperturbationtheory by matching
a perturbative calculation in the full QCD with the corresponding perturbative calculation in
NRQCD. It is the subject of this paper to calculate Wilson coefficient in front of NRQCD
current up to 2-loop order in perturbative QCD.
Let us now discuss the matching procedure in some detail. We start with NRQCD la-
grangian describing B -meson bound state dynamics
c
L = L + ψ† iD +D2/(2m ) ψ + χ† iD −D2/(2m ) χ + ..., (2)
NRQCD light c 0 c c b 0 b b
(cid:16) (cid:17) (cid:16) (cid:17)
where L is the usual relativistic lagrangian for gluons and light quarks. The 2-component
light
field ψ annihilates charm quarks, while χ creates bottom antiquarks. The typical relative
c b
velocity v of heavy quarks inside the meson provides a small parameter that can be used as a
nonperturbative expansion parameter.
2
To express the decay constant f in terms of NRQCD matrix elements, we express the
Bc
axial-vector current¯bγµγ c in terms of NRQCD fields. Only the µ = 0 component contributes
5
to the matrix element (1) in the rest frame of the B -meson. This component of the current
c
has the following operator expansion in terms of NRQCD fields:
¯bγ0γ c = C (m ,m )χ†ψ + C (m ,m ) (Dχ )†·Dψ + ..., (3)
5 0 b c b c 2 b c b c
where C and C are Wilson short-distance coefficients that depend on the quark masses m
0 2 b
and m . By dimensionalanalysis, thecoefficient C is proportionalto 1/m2. Thecontribution
c 2 Q
tothematrixelement h0|¯bγ0γ c|B ifromtheoperator(Dχ )†·Dψ issuppressedbyv2 relative
5 c b c
†
to the operator χ ψ , where v is the relative velocity of heavy quarks inside B -meson. The
b c c
dots in (3) represent other operators whose contributions are suppressed by higher powers of
v2.
The short-distance coefficient C and C can be determined by matching perturbative
0 2
calculations of the matrix elements in the full QCD and NRQCD. A convenient choice for
matching is the matrix element between the vacuum and the state |c¯bi consisting of a c and a
¯b on their perturbative mass shells with nonrelativistic four-momenta p and p′ in the center of
momentum frame: p+p′ = 0. The matching condition is
h0|¯bγ0γ c|c¯bi = C h0|χ†ψ |c¯bi + C h0|(Dχ )† ·Dψ |c¯bi + ..., (4)
5 0 b c 2 b c
(cid:12) (cid:12) (cid:12)
(cid:12)QCD (cid:12)NRQCD (cid:12)NRQCD
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
whereQCDand(cid:12)NRQCDrepresentper(cid:12)turbativeQCDandperturbativeN(cid:12) RQCD,respectively.
At leading order in α , the matrix element on the left hand side of (4) is v¯ (−p)γ0γ u (p).
s b 5 c
The Dirac spinors are
E +m ξ
uc(p) = c c p·σ , (5)
s 2Ec Ec+mcξ !
v(−p) = Eb+mb (E−bp+)m·σbη , (6)
s 2Eb η !
whereξ andη are2-componentspinorsandE = m2 +p2. Makinganonrelativisticexpansion
Q Q
of the spinors to second order in p/m , we find
Q
1 m +m 2
v¯ (−p)γ0γ u (p) ≈ η†ξ 1 − b c p2 + ... . (7)
b 5 c b c 8 (cid:18) mbmc (cid:19) !
At leading order in α , the matrix elements on the right hand side of (4) are η†ξ and p2η†ξ .
s b c b c
The short distance coefficients are therefore C = 1 and
0
1
C = − , (8)
2 8 m2
red
where m = m m /(m +m ) is the reduced mass of the system.
red b c b c
In order to determine the short distance coefficients up to order α2, we must calculate the
s
matrix elements on both sides of (4) up to α2. Up to this order it is sufficient to calculate
s
radiative corrections only for the coefficient C , since the contribution proportional to C is
0 2
suppressed by additional power of v2. The coefficient C can be isolated by taking the limit
0
3
p → 0, in which case the matrix element of (Dχ )† ·Dψ vanishes. Such calculation at the
b c
one-loop order was performed in [14], where the following result was obtained for C :
0
α (m ) m −m m
s red b c b
C = 1 + log −2 . (9)
0
π m +m m
(cid:20) b c c (cid:21)
m is the scale of the running coupling constant. To the accuracy of this calculation, any
red
scale of order m or m would be equally correct.
b c
Thus, the final result for the decay constant of the B is
c
if M = C h0|χ†ψ |B i + C h0|(Dχ )†·Dψ |B i. (10)
Bc Bc 0 b c c 2 b c c
with short-distance coefficient C up to next-to-leading order in α as given in (9), while C is
0 s 2
given to leading order in (8). The uncertainties consist of the perturbative errors in the short
distance coefficients and an error of relative order v4 from the neglected matrix elements that
are higher order in v2. The matrix element h0|χ†ψ |B i can be estimated using wavefunctions
b c c
at the origin from nonrelativistic potential models:
3
|h0|χ†ψ |B i|2 ≈ 2M |R(0)|2. (11)
b c c Bc2π
The factor of 2M takes into account the relativistic normalization of the state |B i.
Bc c
3 Two-loop result
In this section we consider the two-loop contribution to B -meson leptonic constant. An
c
analogous calculation, but for the case of vector current and equal quark masses was done
previously in [11, 12]. Here we are following the approach of [11, 12], extending it, where it
is necessary, to the case of two different mass scales involved in our problem. In the case of
B -meson we consider the matching of QCD axial current on the NRQCD one.
c
To compute Wilson short distance coefficient C up to α2 order, we replace B -meson state
0 s c
by afreequark-antiquark pair of on-shellheavy quarkswithsmall relative velocity. Interms of
the on-shell matrix elements for heavy quarks with small relative velocity: p =qm /(m +m )
1 1 2
andp′ = qm /(m +m )(q isthemomentumofthegaugeW-boson)2. Thematchingequation
2 1 2
has the following form
Z Γ =C Z Z−1 Γ +O(v2), (12)
2,QCD QCD 0 2,NRQCD J,NRQCD NRQCD
where Γ and Γ are amputated bare axial-vector vertices in QCD and NRQCD,
QCD NRQCD
correspondingly. Here, we would like to note, that since terms of order v2 are not needed
to determine C (see Eq. (4)), one can set relative momentum to zero and compute Green
0
functions directly at the threshold. Note, that here we deal with partially conserved axial-
vector current and its QCD renormalization constant is given by Z = 1, as in the case of
J,QCD
vector current.
In order to obtain the coefficient C at two loops one has to perform the following steps:
0
1) sum over all contributions to Γ’s, including the one-loop term, multiplied by the two-
loop QCD on-shell wave function renormalization constant [15], Z−1 and one-loop
J,QCD
NRQCD-current renormalization constant (at one-loop it is equal to unity),
2Thisandpreviousdefinitionsofexternalheavyquarkmomentaleadtothesamevalueofthematchingcoefficient,
but latter is more convenient in two-loop calculations.
4
a b c d
e f g h
Figure 1: QCD Feynman diagrams relevant for the calculation of the two-loop corrections to the
B structure constant.
c
2) perform the one-loop renormalization of the coupling and masses.
After these manipulations the remaining poles in ǫ belong to the anomalous dimension of the
corresponding NRQCD current and finite part correspond to short-distance constant C .
0
In step 1) calculating two-loop contribution to vector-axial vertex, it is convenient to write
it in the following form
1
A =¯b(p′) γ γ F (q2)− q γ F (q2) c(p). (13)
µ µ 5 1 µ 5 4
m +m
1 2
h i
The formfactors F (q2) and F (q2) can be singled out by imposing appropriate projections on
1 2
them and thus our calculation will reduce to the calculation of diagrams with nonvanishing
numerators.
To simplify matching calculation at 2-loop order we use threshold expansion in relative
velocity v of [10]. The latter is obtained by writing down contributions corresponding to hard
(k ∼ m ), soft (k ∼ m v), potential (k ∼ m v2,k ∼ m v) and ultrasoft (k ∼ m v2)
Q Q 0 Q i Q Q
regions3. The contributions from the last three regions can be easily identified with NRQCD
diagrams, that appear in the calculation of Γ . Thus, they simply drop out from the
NRQCD
matching relation (12) and it will suffice to compute only contribution to threshold expansion
of Γ of the region, where all momenta are hard. Diagrams, whose hard contribution need
QCD
to be computed, are depicted in Fig. 1. Both form factors, introduced above have Coulomb
singularities at the threshold and diverge as 1/v2. However, these singularities appear only in
the soft, potential and ultrasoft contributions, while the hard contribution is well defined in
dimensionalregularizationdirectlyatthethreshold. Oncetherelativevelocityissettozero,the
corresponding loop integrals are considerably simplified and depend only on two scales M and
m, where M is the mass of the heaviest quark and m is the mass of the lighter quark. All loop
integrals with irreducible numerators can then be further reduced to the scalar integrals with
increased space-time dimension [16]. For the scalar integrals we derived recurrence relations
using integration by parts method of [17], which allowed us to reduce all scalar integrals to
sunsets. The sunset integrals obtained were further reduced to master integrals with the use
3
Here k is the integration momentum.
5
of recurrence relations of [18, 19, 20, 21]. We would like to note, that our reduction procedure
is valid for arbitrary values of heavy quark masses, while to calculate some master integrals
an expansion in parameter r = m/M was performed. All master integrals are of the sunset
type discussed in the paper [21]. In particular there we discuss in full details the procedure of
expansion in parameter r. Here, we describe only shortly the calculation procedure and more
detailed exposition will be given elsewhere. The results, obtained for each of diagrams in Fig.
1 could be found in Appendix A.
After performing the steps of renormalization procedure described above, we have the
following MS expression for the anomalous dimension of NRQCR current, which first arises
here at the two-loop order (r = m/M)
dlnZ (1−r)2 π2 α 2
γ = J,NRQCD = − C2 2− +C C s +O(α3). (14)
J,NRQCD dlnµ ( F (1+r)2! F A) 2 (cid:18) π (cid:19) s
This scale dependence is compensated by the scale dependence of the two-loop matching co-
efficient c (M/µ), which is defined as
2
M α (M) α (M) 2
s s
C (α , )= 1+c (M/µ) +c (M/µ) +..., (15)
0 s 1 2
µ π π
(cid:18) (cid:19)
In order to obtain c in analytical form, we perform the expansion of master integrals, con-
2
sidering the parameter r = m/M as small (in the practice r = m /m ∼ 0.3). Below we give
c b
the result of expansion up to O(r2) inclusive. Separating the different colour group factors, in
c (M/µ) in the MS we have:
2
c (M/µ)=C2c +C C c +C T n c +C T c , (16)
2 F 2,A F A 2,NA F F l 2,L F F 2,H
(1−r)2 π2 M2 9 π2 41 91π2 29
c = 2− ln + L2+ L+ L+π2ln(2)− +
2,A (1+r)2! 4 µ2 ! 32 3 32 48 16
17π2 43 23π2 3 21 165 25π2 147
+r L− L−6ζ + − +r2 − L2−3π2L+ L+12ζ − − ,
3 3
" 12 16 24 8# " 8 16 12 16 #
(17)
π2 M2 11 13π2 127 9 1 41π2 23
c = ln + L2+ L+ L− ζ − π2ln(2)+ −
2,NA 4 µ2 ! 16 24 96 4 3 2 96 48
25 π2 65 2π2 49 15 23 19π2 31
+r − L2− L+ L− − +r2 L2− L+ + ,
" 16 24 48 3 32# " 8 16 48 36#
(18)
1 11 5π2 1 L2 L L L2
c =− L2− L+ + +r − +r2 − , (19)
2,L
4 24 144 12 " 2 12# "12 2 #
L2 11L 19π2 1147 L2 L 23π2 103
c =− − − + +r − − −
2,H
4 24 96 288 " 2 12 96 144#
L2 89L 23π2 11527
+r2 − + + + . (20)
" 2 60 96 3600 #
Here ζ = 1.202..., M = m , r = m /m , L ≡ ln(r) and we have taken all fermions with
3 b c b
masses less than m and m as massless. The heavy quark masses, appearing in our formulae
b c
6
are pole quark masses. All calculations were performed with the help of computer algebra
system FORM [22]. As a check for our calculations, we employed a general covariant gauge.
To test our FORM programs we also have reproduced already known results for the sort
distance corrections to the vector current with equal masses.
Now let us discuss the expression for 2-loop anomalous dimension of B -meson NRQCD
c
current. It is an exact formula in a sense, that we resumed a whole series in r. In the limit of
equal quark masses it differs from the result obtained in the case of vector current. This fact
is explained by the contribution, coming from Breit–Fermi potential, namely spin-spin and
spin-orbital heavy quark interaction, which in the case of B -meson has a different expression
c
compared to that computed for J/Ψ and Υ-mesons [23]. This result for anomalous expression
could be also obtained considering NNLO corrections to the Coulomb Green function, given
by Breit–Fermi and non-abelian potentials.
Numerically, the scale dependence of the two-loop matching coefficient (16) is large. We
expect it to be compensated by the scale dependence of B -meson wave function computed
c
within the lattice NRQCD framework with the scale dependence of the results introduced by
imposing ultraviolet cutoff/factorization scale. The results of nonrelativistic potential models
should be considered as being obtained at a scale typical for B -meson bound state dynamics
c
µ ∼ 1 GeV. For numerical estimates we consider µ = 1 GeV as an adequate bound state scale
and use its value for factorization scale µ . In the case of B -meson and pole quark masses
fac c
(m = 4.8 GeV, m = 1.65 GeV) we have
b c
α (m ) α (m ) 2
f = 1−1.48 s b −24.24 s b fNR (21)
Bc " (cid:18) π (cid:19) (cid:18) π (cid:19) # Bc
Here we see that 2-loop corrections are large and constitute ≈ 100% of 1-loop correction. Even
less favorable situation was detected earlier for J/Ψ states [11]. The variation of factorization
scale does not help to solve this problem. So, at this point one may conclude that application
of NRQCD to B -meson as well as for J/Ψ is not well justified. However, while in the case of
c
J/Ψ-meson 2-loop corrections far exceed 1-loop, one may speculate that situation in B -meson
c
is marginal, in a sense that one may still obtain sensible results using NRQCD framework
as well as treating B -meson as a heavy-light system. This way one may study the change
c
in heavy quarks dynamics between bottomonium and charmonium families. Another possible
reason for such large correction is our use of particular renormalization scheme MS, which
may turn out not to work well for heavy quark bound state systems. To prove or disprove this
statement one has to calculate 2-loop corrections to any relation between physical observables,
from which quarkonium wave function at the origin is eliminated [11].
4 Conclusion
Inthispaperwepresentedtheresultsforthetwo-loopradiativecorrectionstotheshortdistance
coefficient in matching between QCD and NRQCD for the B -meson currents. It turns out
c
that the numerical value of these corrections are large. There are several conclusions, which
could be made here. First NRQCD description is inadequate not only for charmonium family
but also for B -meson. To see whether it is true, we suppose to perform a systematical study
c
of NRQCD B -meson sum rules at NNLO in one of our next publications. Second, is it the
c
use of MS renormalization scheme, which makes things look nasty?. To answer this question
one needs to compute some other decay mode of either J/Ψ or B -mesons at 2-loop level and
c
analyze perturbative corrections to the ratio of these decay modes.
7
We thank K. Chetyrkin, A. Davydychev, A. Grozin, M. Kalmykov, V. Kiselev, O. Tarasov
and A. Pivovarov for fruitful discussion of the topics discussed in this paper. We especially
grateful to V. Smirnov for encouragement, constant interest to this work and help with nu-
merous checks of our calculation. This work was supported by DFG Forschergruppe ”Quan-
tenfeldtheorie, Computeralgebra and Monte-Carlo-Simulation” (contract FOR 264/2-1) and
by BMBF under grant No 05HT9VKB0.
A 2 loop bare hard contribution
Here we collected results for hard contributions of diagrams, shown in Fig. 1. Below, to
save space, we present only coefficient of (α /π)2(eγEM2/(4πµ2))−2ε for each of the digrams,
s
presented above. Though our calculations were performed in general covariant gauge, we
present here for brevity only the results in the Feynman gauge.
• diagram 1 (Fig. 1a)
C2{ 9 + 1 − 3 − π2 − 9L +r[9L − π2]+r2[π2− 9L]
F 32ε2 ε 64 4 8 8 2 8 !
9L2 11 L 299π2 41 3 9L π2 15
+ + π2L+ −3ζ − − +r −3L2+ π2L− −6ζ + −
3 3
4 12 2 192 128 " 2 8 3 32#
+r2 7L2 − 19π2L+ 491L +12ζ − 49π2 − 401 }, (22)
3
" 8 6 48 24 36 #
• diagram 2 (Fig. 1d, with selfenergy for quark with mass M)
C2{ 3 (1− 2)+ 1 1[3L + 3]− 3L − 3 −r[3L + 3]+r2[9L + 3]
F 32ε2 r ε r 8 8 8 64 8 4 8 4
(cid:18) (cid:19)
1 3 3 π2 13 9 13 45π2 2041
− L2+ L+ + + L2+ L+ +
r "8 4 32 12# 8 8 64 384
+r 3L2− 17L+ π2 − 63 +r2 −5L2+ 31L+ π2 + 1001 }, (23)
"8 4 4 32# " 8 6 4 288 #
• diagram 3 (Fig. 1d, with selfenergy for quark with mass m)
C2{ 3 (1−2r)+ 1 −3L− 51 +r[3L+ 9]−r2[15L+ 3]
F 32ε2 ε 8 64 2 8 8 4
(cid:18) (cid:19)
3 247π2 1057 29 11π2 37
+ L2+3L+ + −r 3L2+ L+ +
4 192 384 " 16 96 48#
+r2 15L2− 11L− 5π2 − 21 }, (24)
" 4 8 8 32#
• diagram 4 (Fig. 1e, with two gluons coupled to quark with mass M)
C C { 15 + 1 19 − 9 L+r[ 9 L− π2]+r2[π2 − 9 L]
F A 64ε2 ε 128 16 16 16 16 16 !
9 3 7π2 329 15 π2 11 3 25π2 21
+ L2− L− + +r − L2+ L+ L− ζ − −
3
16 16 128 256 " 16 6 16 4 48 64#
8
+r2 13L2− π2L− 7 L+ 3ζ + 7π2 − 115 }, (25)
3
"16 6 96 4 16 144#
• diagram 5 (Fig. 1e, with two gluons coupled to quark with mass m)
C C { 15 + 1 −15L− π2 + 19 +r[ 9 L+ π2]−r2[ 9 + π2]
F A 64ε2 ε 16 16 128 16 16 16 16 !
15 π2 5 3 221π2 473 3 π2 3 3 π2 21
+ L2+ L− L− ζ − + +r − L2− L+ L+ ζ + −
3 3
8 4 16 4 384 256 " 2 4 16 4 12 64#
+r2 2L2+ π2L− 83L− 3ζ + π2 + 353 }, (26)
3
" 4 96 4 8 288#
• diagram 6 (Fig. 1c, with two gluons coupled to quark with mass M)
C (C −2C ){ − 9 + 1 3 L− 27 − 3 rL+ 3 r2L + 3 L2+ 5L− 3 ζ
F A F 64ε2 ε 16 128 16 16 16 8 16 3
(cid:18) (cid:19)
π2 33π2 287 3 3 π2 π2 9
+ ln(2)+ + +r L2−L− ζ + ln(2)+ −
3
8 128 256 "16 8 4 12 64#
+r2 1 L2− 11L+ 3ζ − π2 ln(2)+ 5π2 + 71 }, (27)
3
"16 48 8 4 48 36#
• diagram 7 (Fig. 1c, with two gluons coupled to quark with mass m)
C (C −2C ){ − 9 + 1 9 L− 27 − 3 rL+ 3 r2L − 9L2+ 3L− 3 ζ
F A F 64ε2 ε 16 128 16 16 8 4 16 3
(cid:18) (cid:19)
5 77π2 215 3 25 3 3 9π2 53
+ π2ln(2)+ + +r L2+ L− ζ − π2ln(2)− −
3
8 128 256 "8 32 8 4 32 32#
+r2 − 3 L2− 5L+ 3ζ + 3π2ln(2)+ 11π2 + 137 }, (28)
3
" 16 4 8 4 24 64 #
• diagram 8 (Fig. 1b)
C (C −2C ){ 1 3 − π2 + 7 π2L− 5L− 15ζ − 3π2ln(2)− 3π2 − 35
F A F 3
ε 16 16! 24 8 8 4 16 12
3 π2 5 3 π2 π2 59
+r L2+ L− L+ ζ + ln(2)− +
3
"16 24 32 4 2 32 64#
+r2 −11L2− π2L+ 11L− 3ζ − π2 ln(2)− 35π2 − 2119 }, (29)
3
" 16 12 6 4 2 48 576 #
• diagram 9 (Fig. 1f)
C C { 19 + 1 33 − 19L+ 19rL− 19r2L + 19L2− 7 L+ 95π2 + 1315
F A 128ε2 ε 256 32 32 32 16 32 768 512
(cid:18) (cid:19)
−r 19L2+ 5 L +r2 19L2+ 5 L }, (30)
16 64 16 64
(cid:20) (cid:21) (cid:20) (cid:21)
9
• diagram 10 (Fig. 1g)
C C { 1 + 1 3 − 1 L+ 1 rL− 1 r2L + 1 L2− 1 L+ 5π2 + 73
F A 128ε2 ε 256 32 32 32 16 32 768 512
(cid:18) (cid:19)
+r 1 L− 1 L2 +r2 1 L2− 1 L }, (31)
64 16 16 64
(cid:20) (cid:21) (cid:20) (cid:21)
• diagram 11 (Fig. 1h, with light quark loop)
C T n { − 1 + 1 1L− 1 − 1r(1−r)L −L2− 5π2 − 67 +r(1−r) L2+ 1L },
F F l 8ε2 ε 2 16 2 48 32 4
(cid:18) (cid:19) (cid:20) (cid:21)
(32)
• diagram 12 (Fig. 1h, with quark of mass M in the loop)
C T { − 1 + 1 1L+ 5 − 1r(1−r)L − 1L2− 1L− π2 − 3
F F 4ε2 ε 2 48 2 2 3 8 32
(cid:18) (cid:19)
+r 1L2+ 1L+ π2 − 211 +r2 −1L2− 13L− π2 + 6007 }, (33)
"2 3 6 144# " 2 30 6 3600#
• diagram 13 (Fig. 1h, with quark of mass m in the loop)
C T { − 1 + 1 L+ 5 − 1r(1−r) −2L2− 2L+ 13π2 − 563
F F 4ε2 ε 48 2 3 96 288
(cid:18) (cid:19)
+r L2+ 1L− 7π2 + 3 +r2 −L2+ 5L+ 13π2 }, (34)
" 4 32 4# " 4 32 #
where L ≡ lnr and n is the number of light fermions.
l
References
[1] I. P. Gouz, V. V. Kiselev, A. K. Likhoded, V. I. Romanovsky and O. P. Yushchenko,
Phys. Atom. Nucl. 67 (2004) 1559 [Yad. Fiz. 67 (2004) 1581];
V. V. Kiselev, [hep-ph/0211021]; S. S. Gershtein et al., [hep-ph/9803433];
S.S.Gershtein, V.V.Kiselev, A.K.LikhodedandA.V.Tkabladze, Phys.Usp.38(1995)
1 ; [Usp. Fiz. Nauk 165 (1995) 3].
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T. Mannel, G.A. Schuler, Z. Phys. C 67 (1995) 159.
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N. Brambilla, A. Pineda, J. Soto and A. Vairo, Phys. Rev. D 60 (1999) 091502.
[4] A. V. Manohar and I. W. Stewart, Phys. Rev. D 62 (2000) 074015;
A. V. Manohar and I. W. Stewart, Phys. Rev. D 63 (2001) 054004.
[5] E. J. Eichten and C. Quigg, Phys. Rev. D 49 (1994) 5845;
V. V. Kiselev, A. K. Likhoded and A. V. Tkabladze, Phys. Rev. D 51 (1995) 3613.
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