Perturbative corrections to B → D form factors in QCD
Received: January
Perturbative corrections to
YuMing Wang 0 1 3 6 7
YanBing Wei 0 1 2 3 5 7
YueLong Shen 0 1 3 4 7
CaiDian Lu 0 1 2 3 5 7
power in 0 1 3 7
Open Access 0 1 3 7
c The Authors. 0 1 3 7
0 Yuquan Road 19 , 100049 Beijing , P.R. China
1 Boltzmanngasse 5 , 1090 Vienna , Austria
2 School of Physics, University of Chinese Academy of Sciences
3 Weijin Road 94 , 300071 Tianjin , P.R. China
4 College of Information Science and Engineering, Ocean University of China
5 Institute of High Energy Physics , CAS
6 School of Physics, Nankai University
7 Songling Road 238 , Qingdao, 266100 Shandong , P.R. China
We compute perturbative QCD corrections to B ! D form factors at leading =mb, at large hadronic recoil, from the lightcone sum rules (LCSR) with Bmeson distribution amplitudes in HQET. QCD factorization for the vacuumtoBmeson correlation function with an interpolating current for the Dmeson is demonstrated explicitly at one loop with the power counting scheme mc encoding information of the hardcollinear dynamics in the abovementioned correlation function are complicated by the appearance of an additional hardcollinear scale mc, compared to the counterparts entering the factorization formula of the vacuumtoBmeson the subleading power correction induced by the threeparticle quarkgluon distribution amplitudes of the Bmeson at tree level employing the background gluon eld approach. The LCSR predictions for the semileptonic B ! D` form factors are then extrapolated to the entire kinematic region with the zseries parametrization. Phenomenological implications of our determinations for the form factors fB+D;0(q2) are explored by investigating the (di erential) branching fractions and the R(D) ratio of B ! D` and by determining the CKM matrix element jVcbj from the total decay rate of B ! D
Heavy Quark Physics; Perturbative QCD; Resummation

correction function for the construction of B !
from factors. Inspecting the
nextto
Contents
1 Introduction The framework 2 3
Twoparticle contributions to the LCSR at O( s)
Perturbative matching functions at NLO
Weak vertex diagram
Dmeson vertex diagram
Wave function renormalization
Box diagram
The NLL hard and jet functions
The NLL LCSR for B ! D form factors
Numerical analysis
Theory input parameters
Predictions for B ! D form factors
Phenomenological implications
Concluding discussion
A Loop integrals
B Spectral representations
C The coe cient functions of
Introduction
QCD and heavyparticle e ective
eld theories. The longstanding tension between the
BR(B !
function performed in [10].
correlation function at leading power in
eld appeared in the
infunction, following closely [15, 19, 20].
the sum rules for B !
in the background eld approach.
power counting mb
, which was further updated in [25] recently. Albeit with
lish the power counting scheme for the external momenta in
=mb with the power
The framework
(n p; n p) = i
b(0)g jB(pB)i ;
where pB
frame of the Bmeson and introduce two lightcone vectors n
and n satisfying n v =
n p =
m2B + m2D
O(mB) ;
p is the transfer momentum and
is a hadronic
scale of order
the diagram in gure 1 with the lightcone OPE and we obtain
; 2P(n p; n p) =
i f~B( ) mB n
i f~B( ) mB n
!c + i 0
!0 + i 0
pion interpolating current in the mc ! 0 limit.
B ! D transition form factors
h0jqn= 5 ujD(p)i = i n p fD ;
(n p; n p) =
i fD mB
2 (m2D=n p
meson correlation function
(n p; n p) de ned in (2.1) at tree level.
h0j (q Ys) (t n) Ysybv
(0)jB(v)i
if~B( ) mB
1 + v= h2 ~B+(t; ) +
~B+(t; ) n=
the soft gauge link is given by
Ys(t n) = P
Exp i gs
dx n As(x n)
f~B( ) =
n p fB+D(q2) + fB0 D(q2)
fB0 D(q2)
at leading power in
the LCSR for the two B ! D form factors at tree level
fB+D; 2P(q2) =
d!0 Exp
fB+D; 2P(q2) + O( s) ;
B(!0) + O( s) ;
leading power in
the sum rules for B !
3=2=mb1=2 ;
3=2=mb1=2 :
Based upon the canonical behaviour for the Bmeson DA
level sum rules (2.9), di erent from the scaling law fB+D(q2)
O(1) [14] obtained with the
B(!0), one can readily deduce
counting scheme mc
Twoparticle contributions to the LCSR at O( s)
; 2P(n p; n p) =
i f~B( ) mB
!c + i0
Ci;n(n p) Ji;n(n p; !) n + Ci;n(n p) Ji;n(n p; !) n
; k =
s manifestly and
the convenience.
Perturbative matching functions at NLO
gure 2 in the following.
Weak vertex diagram
in gure 2 (a)
; weak= (p k)2
mc2 +i0 (2 )D [(p k +l)2
mc2 +i0][(mbv +l)2
mb2 +i0][l2 +i0]
q(k)n= 5(p=
k= + mc)
k= + =l + mc)
(mbv= + =l + mb)
b(v) ; (3.2)
meson correlation function
(n p; n p) de ned in (2.1) at O( s).
to the abovementioned QCD diagrams at leading power in
=mb will be cancelled out
O(mb)
I1 =
(2 )D [(p
k + l)2
mc2 + i0][(mbv + l)2
mb2 + i0][l2 + i0]
the hard contribution from the weak vertex diagram yields
where the two hard functions are given by
; weak =
Ch(w;neak)(n p) =
Ch(w;neak)(n p) =
2 Li2 1
ln r +
+ 2 ln
d(k)n= 5 b(v)
+ 2 ln
+ 2 r1 ln
2 n (p+l) n
J (w;neak)(n p)=
n p (!
n p (!
1 + r1
2 ln(1 + r1) ln
2 Li2
1 + r1
ln (1 + r1) + 1 + ln2
n p (!
n p (!
+ 1 + ln2(1 + r1)
with r = n p=mb.
=mb
; weak =
n p n k !c (2 )D [n (p+l) n (p k +l)+l2
mc2 +i0][n l+i0][l2 +i0]
with r1 = mc2= [n p n (k
p)] and !
n k. It is apparent that our result of J (w;neak)
of [15], for constructing the sum rules of B !
form factors in the mc ! 0 (i.e., r1 ! 0)
diagram by expanding (3.2) in the soft region
; weak =
q(k)n= 5 b(v) ;
(2 )D [n (p
k + l)
!c + i0][v l + i0][l2 + i0]
Dmeson vertex diagram
in gure 2(b)
; D =
of the partonic DA of the Bmeson at NLO in
s, calculable from the Wilsonline Feynman
terized by the Bmeson DA at leading power in
n p (n p
(2 )D [(p
=ln= 5 (p=
=l + mc)
k= + mc)
mc2 + i0][(l
k)2 + i0][l2 + i0]
=mb,
in [35, 36] yields
(1;)D =
(1); hc =
i f~B( ) mB
B(!) J ( D;n)(n p) + n
B+(!) J +(D;n)(n p) (3.11)
nmcp B+(!) n J +(D;n)(n p)
B(!) + n
where we have introduced the following jet functions
J ( D;n)(n p)= s CF
1 + r2 + r3
1 + r2
2(1 + r2)
1 + r2
1 + r2 + r3
1 + r2
+2 ln (1+r2) + r2 [r2(1+r3) 2]
1 + r3
1 + r2 + r3
1 + r2 + r3
1 + r2
+ ln (1 + r2)
2(2 + r3)(1 + r2 + r3)
(1 + r2)(1 + r3)
(4+r2) ;
1 + r3
r2 ; (3.13)
J +(D;n)(n p)= s CF
r3(1 + r3)
(1 + r2 + r3)2
r2 [2(1 + r3) + r2(2 + r3)]
(1 + r3)2
r3(1+r3)
1 + r3
1+r2 +r3
1 + r2
1 + r2 + r3
1 + r2
1 + r2
1 + r3
r2 =
r3 =
vertex diagram presented in (3.11) are in order.
scheme mc
mb and n p
O( ). This observation can be readily
unO(pmb= )
of mc=
(1); mc =
mc n q(k) n= 5 b(v)
2) (n l)2
(2 )D [(p
mc2 + i0][(l
k)2 + i0][l2 + i0]
energetic charm quark and the light spectator quark yields
hD(p)jc
+ : : : ; (3.18)
contribution to B ! D form factors.
at tree level, the NLO jet function J +(D;n)(n p) must be infrared
nite to validate the
diagram as displayed in (38) of [15].
bution to the Dmeson vertex diagram, at leading power in
=mb, is cancelled out
precisely by the corresponding soft subtraction term.
Wave function renormalization
we obtain
; wfc =
{ 10 {
J (w;nfc)(n p)=
J (w;nfc; 1)(n p)=
J (w;nfc; 2)(n p)=
1 + r1
2 r1 J (w;nfc; 2)(n p) ;
+r12 ln
r1(r1 +4) ln
ln (1+r1)+1 r1 ;
3 ln (1+r1)+r1 +5 ; (3.22)
T (0) =
T (0) = 0 ;
renormalization of the external quarks
Box diagram
computed as
; box= igs2CF
given by
C(b;wnf)(n p) =
+ 3 ln
(2 )D [(pb +l)2
=l)n= 5 (p=
mb2 +i0][(p k +l)2
k= + =l + mc)
mc2 +i0][(k l)2 +i0][l2 +i0]
(p=b + =l + mb)
hardcollinear region to the leading power in
=mb yields
(1;)b;ohxc =
2) n l n=
2 mb n=] 5 b(v)
n (p + l) n
[n (p + l) n (p
k + l) + l2
mc2 + i0][n l n(l
k) + l2 + i0][l2 + i0]
projector of the Bmeson leads to
(1;)b;ohxc =
J (b;onx) =
i f~B( ) mB
s CF (1 + r1)(1 + r3)
1 + r1
1 + r1
J +(b;onx) =
s CF (1 + r1)(1 + r3) r
r12) ln (1 + r1)
1 + r1
1 + r1
as presented in (52) of [15], in the mc ! 0 limit.
=mb can be
further extracted from (3.26) as follows
B(!) J (b;onx)(n p) +
B+(!) J +(b;onx)(n p) ;
ln (1 + r1) + 1 +
1 ln (1
r4 + r1)
n p (!
1 + r1
(1;)b;osx =
(2 )D [v l + i0][n (p
k + l)
!c + i0][(k
l)2 + i0][l2 + i0]
=l) n= 5 n=
which is precisely the same as the soft subtraction term
T (0) computed with the
of the Bmeson in HQET.
The NLL hard and jet functions
renormalized hard coe cients including the O( s) corrections
C+;n = C+;n = 1 ;
C ;n = 1 + Ch(w;neak) +
= 1
C ;n = Ch(w;neak) =
+ 5 ln
2 Li2 1
1+r2 +r3
r2r3 ln
; (3.33)
r2(1+r3)
J+;n =
s CF (1+r2 +r3)2
r3(1+r3)
mc (1+r2 +r3)2
1+r2 +r3
r2 [2(1+r3)+r2(2+r3)]
(1+r3)2
r (1+r2 +r3)
(1+r3)2
1+r2 +r3
+r22(r3 +2) ln
J ;n = 1 ;
n p (! n p)
(1+r2 +r3)2
(1+r2)(1+r3)
1+r2 +r3
+2 ln2 (1+r2 +r3) 4 ln (1+r2 +r3) ln (1+r3)
= 1+ s CF
n p (! n p)
+ ln2 (1+r3)+
+2 ln (1+r2)
ln2 (1+r2)+r2
4 Li2 1+r3 1+r2 +r3
2(1+r2) + r2 (r2 +2(1+r3))
1 ln (1+r2 +r3)
1+r2 +r3
+2 Li2
(1+r3)2
ln (1+r3)+
1+r2 +r3 (1+r3)2 1+r3
(1+r2)(r3(1+r2) 2)
r2 1+r2 +r3 + 1+r3
ln (1+r2)
matching coe cients of the weak current q
5) b from QCD onto SCET, as
disjet functions for the massless hardcollinear quark.
it is then straightforward to write down
; 2P(n p; n p) =
i f~B( ) mB
1 + r2 + r3
+ 4 ln
n p (!
i0 d ln
(1 + r2 + r3)2
(1 + r2)(1 + r3)
1 + r2 + r3
evolution equation of the Bmeson DA
B(!; ) in the absence of the lightquark mass e ect
hf~B( ) B(!; )i =
(1)(!; !0; ) can be found in [39, 40], we can readily
where the renormalization kernel H
compute the last term of (3.38) as
= i f~B( ) mB
i0 d ln
Substituting (3.40) into (3.38) immediately leads to
{ 14 {
; 2P(n p; n p) = O( s2) ;
the RG evolution of the Bmeson DA
as a hardcollinear scale hc
0 being a hadonic scale of O( ), from
B(!; ), since we will take the factorization scale
mb which is quite close to the hadronic scale 0
static Bmeson decay constant f~B( )
dC ;n(n p; )
df~B( )
cusp( s) ln
+ ( s) C ;n(n p; ) ;
= ~( s) f~B( ) ;
(1 + r2 + r3)2
(1 + r2)r3
accuracy, the cusp anomalous dimension
cusp( s) needs to be expanded at the threeloop
computed as
torization formula for the correlation function
; 2P(n p; n p) =
i hU2(n p; h2; ) f~B( h2)i mB
!c +i0
C+;n(n p; ) J+;n(n p; !; ) n +C+;n(n p; ) J+;n(n p; !; ) n
taken at a hardcollinear scale O(pmb ).
where h1 and
h2 are the hard scales of O(mb) and the factorization scale
needs to be
The NLL LCSR for B ! D form factors
collected in appendix B yields
spectral representation of the factorization formula for
obtained in the above, which
; 2P(n p; n p) =
i hU2(n p; h2; ) f~B( h2)i mB
C+;n(n p; ) e+;n(!0; ) n + C+;n(n p; ) e+;n(!0; ) n
where we have introduced the \e ective" DA
hardcollinear QCD corrections to the correlation function
ie;k(!0; ) (i =
e+;n(!0; ) =
! + !c
!c) (!0
(! + !c)2
(! + !c
+ r (!0
!c) (!0
!c) (!0
!c) +
e+;n(!0; ) =
e ;n(!0; ) =
e ;n(!0; ) =
B+(!0 !c) (!0 !c)
!c(2! + !c) 1
!(! + !c)2 !0
!c) (!0
!c) (2;)n(!0) + B(!0) (3;)n(!0) + B(0) (4;)n(!0)
d! B(!) (5;)n(!; !0) +
fD Exp
n p fB+D; 2P(q2) ; fB0 D; 2P(q2)
U2(n p; h2; ) f~B( h2)
d!0 Exp
{ 16 {
meson correlation function
(n p; n p) de ned in (2.1) at tree level.
C+;n(n p; ) e+;n(!0; ) + C ;n(n p; ) e ;n(!0; )
gluon eld method [46{48]
(k= + mc)
(u x) ;
with G
; 3P(n p; n p) =
~2;n(u; !; )
~2;n(u; !; )
~3;n(u; !; )
where the coe cient functions ~i;k(u; !; ) are given by
~2;n(u; !; ) = 2 (u
1) [ V (!; ) +
A(!; )] ;
~2;n(u; !; ) =
V (!; ) + (2 u
~3;n(u; !; ) = 2 (2 u
1) XA(!; )
2 YA(!; ) :
element on the light cone [49, 50]
h0jq (x) G
(u x) bv (0)jB(v)i x2=0
f~B( ) mBZ 1
operator on the lefthand side. The following conventions
XA(!; ) =
YA(!; ) =
twist3 DA
V (!; ) which was shown to be completely integrable
tion function
the Borel transformation lead to the following sum rules
fD Exp
F3P;n(!s; !M )
fB+D; 3P(q2) ; fBD; 3P(q2)
0
F3P;n(!s; !M ) ;
F3P;n(!s; !M ) =
F3P;n(!s; !M ) =
~2;n(u; !; )
~2;n(u; !; )
~3;n(u; !; )
~3;n(u; !; )
u= !0 !c !
~2;n(u; !; )
u= !s !c !
u= !s !c !
~2;n(u; !; )
u= !0 !c !
threeparticle Bmeson DA, already at tree level.
threeparticle Bmeson DA in the endpoint region [7]
XA(!; )
YA(!; )
by a factor of
fB0 D(q2) = fBD; 2P(q2) + fB0 D; 3P(q2) ;
0
and (4.6), respectively.
Numerical analysis
{ 19 {
1j according
to the conformal transformation [64]
z(q2; t0) =
mD)2 will be
tion leads to the suggested parametrization [32, 61]
fB+D(q2) =
fB+D(0)
q2=m2Bc
z(0; t0)k
z(q2; t0)N
z(0; t0)N
0:009) GeV [65], and the zseries
expanwe will employ the following parametrization
fB0 D(q2) =
fB0 D(0)
q2=m2Bc(0)
1 + X ~bk z(q2; t0)k
z(0; t0)k
with the aid of the LCSR predictions at qm2in
2
qmax and the zseries
parametrizakinematic range 0
mD)2 with theory uncertainties in
gure 7 where recent
0:009) GeV [5], and we will only keep
indicated by the shaded regions.
fB+D(z) = fB+D(0) 1
fB0 D(z)
fB+D(z)
1:0036 1 + 0:0068 ! + 0:0017 !2 + 0:0013 !3 ; (5.12)
{ 26 {
Central value
fB+D(0)
from the variations of theory inputs.
where we have introduced the following conventions
z = p
1 + !
! =
m2B + m2D
2 mB mD
the form factors of B ! D` at qm2in
leads to
2
qmax onto the CLN parametrization (5.12)
fB+D(z = 0) = 1:22
= 1:07+00::0181 ;
large theory uncertainties for the slop parameter .
Phenomenological implications
! = 1
r = mD=mB : (5.13)
m2B jp~Dj
where jp~Dj =
the magnitude of the threemomentum of the Dmeson and
2bc is
EW = 1 +
' 1:0066
for the B ! D` decays [69, 70].
(pink band) and B ! D
(blue band) as a
[t1; t2]
GeV2
[0:00; 0:98]
[0:98; 2:16]
[2:16; 3:34]
[3:34; 4:53]
[4:53; 5:71]
(t1; t2)
(10 12 GeV)
(t1; t2)
(10 12 GeV)
[t1; t2]
GeV2
this work
Belle [71]
this work
Belle [71]
1:00+00::2281
1:09+00::2271
0:97+00::2116
0:85+00::1163
0:72+00::1019
[5:71; 6:90]
[6:90; 8:08]
[8:08; 9:26]
[9:26; 10:45]
[10:45; 11:63]
0:57+00::0086
0:43+00::0054
0:28+00::0022
0:14+00::0011
0:03+00::0000
compared with the Belle
measurements from [71].
{ 28 {
measurements [71, 72].
can be obtained straightforwardly
In particular, our predictions for the total decay width of B ! D
in units of 1=jVcbj2
(0; 11:63 GeV2) = 6:06+10::1982
10 12 GeV ;
from which the exclusive determinations of jVcbj are achieved
jVcbj = <
39:2+33::43 th
[BaBar 2010]
[Belle 2016]
q2 shape of B ! D
displayed in table 1 are plotted in
gure 8, including also the recent experimental
measurements for the combination of B+
`(t1; t2) =
R(t1; t2) =
Rtt12 dq2 d (B ! D
Rtt12 dq2 d (B ! D
)=dq2
)=dq2
accuracy of
form factors at di erent q2 for future work.
{ 29 {
GeV2
[4:00; 4:53]
[4:53; 5:07]
[5:07; 5:60]
[5:60; 6:13]
[6:13; 6:67]
[6:67; 7:20]
[7:20; 7:73]
[7:73; 8:27]
[8:27; 8:80]
[8:80; 9:33]
[9:33; 9:86]
[9:86; 10:40]
[10:40; 11:63]
this work
{ 30 {
this work
and for the binned
distributions of
calculations presented in [74].
branching fractions of two semileptonic decay channels
BR(B ! D
BR(B ! D
= 0:305+00::002225 ;
(SM) at the 1
space region 0
in the
phase
Concluding discussion
form factors with the power counting scheme mc
mb , at leading power in
tion between the vector and scalar B ! D form factors.
0 numerically
decay rates and of the ratio
R(t1; t2) by applying our determinations of the form factors
{ 31 {
with the HFAG average value in [2].
ties of the Bmeson DA
B(!; ) at two loops (e.g., the eigenfunctions and the analytical
with X = ( ; )
the general properties of e ective eld theories.
Acknowledgments
Loop integrals
vacuumtoBmeson correlation function (2.1) at O( s).
[n (p + l) n (p
k + l) + l2
mc2 + i0][n l + i0][l2 + i0]
n (p + l)
{ 32 {
I2;a = 2
r3(1 + r3)
I2;b = 21r33 (2 + 2r2 r3)r3 2(1 + r2)2 ln 1 + r2 + r3
1 + r2
p2 ln(1 + r2 + r3) + 3
2r32 (1 + r3)2
2(1r+2 r3) ; (A.5)
(1 + r2)2 ln2 1 + r2 + r3
1 + r2
+ 12+r32r2 ln2 1 + r2 + r3
1 + r2
r2 [22r+3(r13+(2r+3)r22)] ln 1 + r2
r2
2r32(1 + r3)2
r32 + r2 + 2r2r3) ln 1 + r2 + r3
1 + r2
1 + r2 + r3
1 + r2
1 + r3
1 + r2 + r3
[d l] [l2 + i0][(p l)2
mc2 + i0][(l k)2 + i0]
p2 [k I4;a + p I4;b] ;
1 + r2
r3 (1 + r3)
= I5;a (p=
k=) + I5;b mc ;
ln [(1 + r2)(1 + r2 + r3)] + 2
ln (1 + r2)
1 + r2 + r3
; (A.10)
I4; =
I4;a =
1 + r3
1 + r2
I4;b =
r3(1 + r3)
1 + r2 + r3 ln (1 + r2 + r3)
1 + r2 ln (1 + r2) +
1 + r3
I5 =
k + l)2
k= + =l) + D mc
mc2 + i0][l2 + i0]
(n p)2 (1+r2 +r3)2
1 + r2
(1 + r3)2
I5;a =
I5;b = 4
I6;a =
I6;b =
+Li2 1
ln (1+r1)+1 r1 ;
ln (1+r1)+
1+r1 r4
n (p+l)
n p (! n p)
n l n (p+l)
Here, D = 4
2 , the integration measure is de ned as follows
1 ln (1 r4 +r1)+ (1 r12) ln (1+r1)
ln (1+r1)
: (A.15)
d! (!0 !c)
!+!c !02
!0 ! !c +i0 r3(1+r3)
!0 ! !c +i0 1+r3
(1+r2 +r3)2
1+r2 +r3
(!+!c)2 P !0 ! !c !0
and we also introduce the conventions
r1 =
r2 =
r3 =
r4 =
Spectral representations
+ !!c0 B+(!0)+ 0(!0)
!0 ! !c +i0
r2 (r3 r2) ln r2 B+(!)
d! (!0)
!+!c P !0 ! !c
!(!+!c)
!0 ! !c +i0 r2 B+(!)
!c (!0 !c) B+(!0 !c)
{ 35 {
1 (1+r2 +r3)2
d! (!0 !c) P !0 !
(!0 !c) :
d! (!0 !c)
!(!+!c) !02
+ 0(!0) ln
= (!0) (!c !0)
(!0) (!c !0)
B+(!0) (!0) + (!0)
r3(1+r3)
r2 (1+r2 +r3) ln(1+r2 +r3) B+(!)
d! (!c +! !0) (!0) !c
r2 (1+r2) ln(1+r2) B+(!)
B+(!0 !c) (!0 !c)
!+!c !0 ! !c
!0 ! !c +i0 !0
(1+r3)2
! (!+!c)2 !0
!0 ! !c +i0 1+r3
(!0) (!c !0)
!+!c P !0 ! !c
B+(!0 !c) (!0 !c)
1+r2 +r3 r22 (1+r3)2
{ 36 {
= !c2
n p (! !0) d!
d! (!0) (!c !0)
(! !0)
d! (!0 !) P !0 ! !c
!0 ! !c +i0
(1+r2 +r3)2
(1+r2)(1+r3)
n p (! !0)
!0 ! !c +i0 1+r2 +r3
n p (! !0)
!c (!c !0) ln
!0 ! !c +i0 r22 (r3 +2)
0(!0) ln
r2(1+r2 +r3) B+(!)
!0 ! !c +i0
!!c0 B+(!0) !c (!0)
B(!0 !c) (!0 !c) :
2 (!0 !c) (!+!c !0)
n p (! !0)
B(!0 !c) (!0 !c)
!0 ! !c +i0
ln2(1+r3) B(!)
B(!0 !c) (!0 !c) :
(! !0) (!0) B(!)
!0 ! !c +i0
d! ln2 !+!c !0
d! (!+!c !0) (!0) ln
ln(1+r2 +r3) ln(1+r3) B(!)
(! !0) (!0)
!+!c !0
1 Z 1
!0 ! !c +i0
!+!c !0
!0 ! !c +i0
!+!c
1+r2 +r3
(!+!c !0) (!0 !c) (!c !0)
(1+r2)2 ln (1+r2) B(!)
B(!0 !c) (!0 !c)
B(!0 !c) (!0 !c) !c ln
!0 ! !c +i0
ln2(1+r2 +r3) B(!)
(!c !0) (!0) B(0)
d! ln2 !+!c !0
(!+!c !0) (!0)
(!0) (!c !0)
(!0) (!c !0)
(!c !0) (!0) ln
+P ! !0 +P ! !0 d!
= (!c !0) (!0) ln
!0 ! !c +i0
!0 ! !c +i0
(1+r3)2
(!0) ln
B+(!0) (!0)+ B+(!0 !c) (!0 !c)
d! (!+!c !0) (!0) ln
!+!c !0
!0 ! !c +i0
d! (!c !0) (!0)
d! (! !0) (!0)
d! (!+!c !0) (!0) ln
!+!c !0
!0 ! !c +i0 1+r2 +r3
ln (1+r3) B(!)
B(!0 !c) (!0 !c)
B(!0) (!0)
(!0 !c) !c ln
d! (! !0) (!0)
!0 ! !c +i0
!0 ! !c d! B(!) :
ln2 (1+r2) B(!)
B(!0 !c) (!0 !c)
d! (!c !0) (!0) ln
r2 ln r2
!0 ! !c +i0 1+r2 +r3
(!0 !c) ln
!0 d!0 B(!0 !c)
ln (1+r2 +r3) B(!)
ln (1+r2) ln (1+r3) B(!)
B(!0 !c) (!0 !c)
!0 d!0 B(!0 !c)
+r2 +2 ln r2 B(!)
(!c !0) (!0)
(1+r3)2 1+r3
(!0 !c) + 0(!0) ln
(!c !0) (!0)
(!c !0) (!0)
!02 +2 !0!c !2
c
d! (!0) P !0 ! !c
B(!0 !c) (!0 !c)
!2 +4 ! !c +2 !c2
(!+!c)2
!+!c !0
(!+!c !0) (!0 !c)
(!+!c !0)
!0 ! !c +i0
!+!c !0
1+r2 +r3
+ (!0 ! !c) ln
(!+!c !0) (!0 !c) Li2
!0 ! !c +i0
!+!c !0
(!0 ! !c) ln
= Li2
d! (!0 !c) ln
!0 !c P !0 ! !c B(!) :
(!0 !c) B(0)+
d!0 B(!0 !c) (!0 !c) :
{ 40 {
B(!0 !c) (!0 !c)
!0 +!c
B(!0) (!0)+
B(!0 !c) (!0 !c)
e ;n(!0; ) de ned in (3.49).
(1;)n(!0) = ln
4 (!+!c !0) (!0) ln
+ (!0) (!0 ! !c)
(2;)n(!0) = 2 !c 3 ln
(3;)n(!0) = 2 ln2
ln2 !c + 2 (!c !0) ;
n p (!0 !c)
2 (!0 !) ln
+ (!c !0) (!0)
d!0 f (!0) + g(!0) =
d!0 f (!0) ++ g(!0) =
The coe cient functions of
2 ln2 !c +ln2 !0 !c
(!+!c)2
+4 (!+!c !0) (!0 !c) ln
2 (!0)
+2 (!0)
+ (!0 !c)
(! !0)
(!c !0)+ (! !0)
!c +2 ! 1
!+!c !0
(!0) !c ln
!+!c !0
!+!c
!+!c
{ 41 {
(!c !0) (!0) ;
!+!c !0
n p (! !0)
+4 (!0 !c) ln
!+!c
n p (!
+2 (!0) (! +!c !0) (!0 !) ln2
! +!c !0
4 (! +!c !0) (!0) ln
2 (!c +!
!0) (!0) ln
4 (! +!c !0) (!0 !c) Li2
+ 4 (!0 !
!c) ln
! +!c !0
! +!c !0
!c) (!0)
! +!c !0
(!0) (! +!c !0) ;
(!0 !) (!0 !c) (! +!c !0)+ (!
!0) (!0)
! +!c !0
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