PHYSICAL REVIEW D 66, 034012 ~2002!
Muon transverse polarization in the Kl2g decay in the standard model

V. V. Braguta and A. E. Chalov
Moscow Institute of Physics and Technology, Moscow, 141700 Russia

A. A. Likhoded*
Instituto de Fı´sica Teo´rica–UNESP, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, SP, Brazil

~Received 26 March 2002; published 15 August 2002!

The muon transverse polarization in theK1→m1ng process induced by the electromagnetic final state
interaction is calculated in the framework of the standard model. It is shown that one loop contributions lead
to a nonvanishing muon transverse polarization. The value of the muon transverse polarization averaged over
the kinematical region ofEg>20 MeV is equal to 5.6331024.

DOI: 10.1103/PhysRevD.66.034012 PACS number~s!: 13.88.1e, 13.20.Eb, 13.40.Ks
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I. INTRODUCTION

The study of the radiativeK-meson decays is extreme
interesting in searching for new physics effects beyond
standard model~SM!. One of the most appealing possibilitie
is to probe new interactions, which could lead toCP viola-
tion. Contrary to the SM, whereCP violation is caused by
the presence of the complex phase in the Cabib
Kobayashi-Maskawa~CKM! matrix, theCP violation in ex-
tended models~for instance, in models with three and mo
Higgs doublets! can naturally arise due to the complex co
plings of new Higgs bosons to fermions@1#. Such effects can
be detected by using experimental observables, which
essentially sensitive toT-odd contributions. These observ
ables, for instance, are theT-odd correlation $T

5(1/MK
3 )pW g•@pW p3pW l #% in theK6→p0m6ng decay@2# and

muon transverse polarization (PT) in K6→m6ng. The
search for new physics effects using theT-odd correlation
analysis in theK6→p0m6ng decay will be done in the
proposed OKA experiment@3#, where an event sample o
7.03105 for the K1→p0m1ng decay is expected to be ac
cumulated.

At the moment the E246 experiment at KEK@4# performs
the analysis of the data on theK6→m6ng process to put
bounds on theT-violating muon transverse polarization.
should be noted that the expected value of new physics
tribution to PT can be of the order of.7.031023–6.0
31022 @5,6#, depending on the type of model beyond t
SM. Thus, when one searches for new physics contribut
to PT , it is extremely important to estimate the effects co
ing from so called ‘‘fake’’ polarization, which is caused b
the SM electromagnetic final state interactions and which
natural background for the new interaction contributions.

In this paper we calculate muon transverse polarizatio
the K6→m6ng process, induced by the electromagnetic
nal state interaction in the one-loop approximation of
minimal quantum electrodynamics.

In next section we present the calculations of the mu
transverse polarization taking into account one-loop d

*On leave of absence from Institute for High Energy Physi
Protvino, 142284 Russia. Email address: andre@ift.unesp.br
0556-2821/2002/66~3!/034012~8!/$20.00 66 0340
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grams with final state interactions within the SM. The la
section summarizes the results and conclusions.

II. MUON TRANSVERSE POLARIZATION IN THE
K¿\µ¿ng PROCESS IN SM

The K1→m1ng decay at the tree level of SM is de
scribed by the diagrams shown in Fig. 1. The diagrams
Figs. 1~b! and 1~c! correspond to the muon and kaon brem
strahlung, while the diagram in Fig. 1~a! corresponds to the
structure radiation. This decay amplitude can be written
follows:

M5 ie
GF

A2
Vus* «m* F f Kmmū~pn!~11g5!S pK

m

~pKq!
2

~pm!m

~pmq!

2
q̂gm

2~pmq!
D v~pm!2Gmnl nG , ~1!

where

l m5ū~pn!~11g5!gmv~pm!,

Gmn5 iF v«mnabqa~pK!b2Fa~gmn~pKq!2pK
mqn!, ~2!

, FIG. 1. Feynman diagrams for theK6→m6ng decay at the tree
level of SM.
©2002 The American Physical Society12-1



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V. V. BRAGUTA, A. E. CHALOV, AND A. A. LIKHODED PHYSICAL REVIEW D 66, 034012 ~2002!
GF is the Fermi constant,Vus is the corresponding CKM
matrix element, f K is the K-meson leptonic constan
pK ,pm ,pn ,q are the kaon, muon, neutrino, and photon fo
momenta, correspondingly, and«m is the photon polarization
vector.Fv andFa are the kaon vector and axial form factor
In Eq. ~2! we use the following definition of Levi-Civita
tensor:e0123511.

The part of the amplitude which corresponds to the str
ture radiation and kaon bremsstrahlung and which will
used further in one-loop calculations, has the form

MK5 ie
GF

A2
Vus* «m* F f Kmmū~pn!~11g5!

3S pK
m

~pKq!
2

gm

m mD v~pm!2Gmnl nG . ~3!

The partial width of theK1→m1ng decay in theK-meson
rest frame can be expressed as

dG5
( uM u2

2mK
~2p!4d~pK2pm2q2pn!

3
d3q

~2p!32Eq

d3pm

~2p!32Em

d3pn

~2p!32En

, ~4!

where summation over muon and photon spin states is
formed.

Introducing the unit vector along the muon spin directi
in muon rest framesW whereeW i ( i 5L,N,T) are the unit vec-
tors along the longitudinal, normal and transverse com
nents of muon polarization, one can write down the ma
element squared for the transition into the particular mu
polarization state in the following form:

uM u25r0@11~PLeWL1PNeWN1PTeWT!•sW#, ~5!

wherer0 is the Dalitz plot probability density averaged ov
polarization states. The unit vectorseW i can be expressed i
terms of the three-momenta of final particles

eWL5
pW m

upW mu
, eWN5

pW m3~qW 3pW m!

upW m3~qW 3pW m!u
, eWT5

qW 3pW m

uqW 3pW mu
. ~6!

With such definition ofeW i vectors,PT , PL , andPN denote
transverse, longitudinal, and normal components of the m
polarization, correspondingly. It is convenient to use the f
lowing variables:

x5
2Eg

mK
, y5

2Em

mK
, l5

x1y212r m

x
, r m5

mm
2

mK
2

,

~7!

whereEg and Em are the photon and muon energies in t
kaon rest frame.

The Dalitz plot probability density, as a function of thex
andy variables, has the form
03401
-

-
e

r-

-
x
n

n
-

r05
1

2
e2GF

2 uVusu2H 4mm
2 u f Ku2

lx2
~12l!Fx212~12r m!S 12x

2
r m

l D G1mK
6 x2~ uFau21uFvu2!~y22ly2lx12l2!

14 Re~ f KFv* !mK
4 r m

x

l
~l21!

14 Re~ f KFa* !mK
4 r mS 22y1x12

r m

l
2

x

l
12l D

12 Re~FaFv* !mK
6 x2~y22l1xl!J . ~8!

Calculating the muon transverse polarizationPT we follow
the original paper@7# and assume that the decay amplitude
CP invariant, and form factorsf K , Fv , andFa are real. In
this case the tree level muon polarizationPT50. When one-
loop contributions are incorporated, the nonvanishing mu
transverse polarization can arise due to the interferenc
tree-level diagrams and imaginary parts of one-loop d
grams, induced by the electromagnetic final state interact

To calculate the imaginary parts of formfactors one c
use theS-matrix unitarity

S1S51 ~9!

and, usingS511 iT, one gets

Tf i2Ti f* 5 i(
n

Tn f* Tni , ~10!

wherei , f ,n indices correspond to the initial, final, and inte
mediate states of the particle system. Further, using thT
invariance of the matrix element one has

Im Tf i5
1

2 (
n

Tn f* Tni , ~11!

Tf i5~2p!4d~Pf2Pi !M f i . ~12!

One-loop diagrams of the SM, which contribute to the mu
transverse polarization in theK1→m1ng decay, are shown
in Fig. 2. Using Eq.~3! one can write down the imaginar
parts of these diagrams. For the diagrams in Figs. 2~a!, 2~c!
one has

Im M15
iea

2p

GF

A2
Vus* ū~pn!~11g5!

3E d3kg

2vg

d3km

2vm
d~kg1km2P!Rm~ k̂m2mm!gm

3
q̂1 p̂m2mm

~q1pm!22mm
2

gd«d* v~pm!. ~13!

For the diagrams in Figs. 2~b!, 2~d! one has
2-2



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s

rs

MUON TRANSVERSE POLARIZATION IN . . . PHYSICAL REVIEW D66, 034012 ~2002!
ImM25
iea

2p

GF

A2
Vus* ū~pn!~11g5!E d3kg

2vg

d3km

2vm

3d~kg1km2P!Rm~ k̂m2mm!gd«d*
k̂m2q̂2mm

~km2q!22mm
2

3gmv~pm!, ~14!

where

Rm5 f KmmS ~pK!m

~pKkg!
2

gm

mm
D2 iF v«mnab~kg!a~pK!bgn

1Fa@gm~pKkg!2~pK!mk̂g#. ~15!

To write down the contributions of diagrams shown in Fig
2~e!, 2~f!, one should substituteRm by

Rm5 f KmmS gm

mm
2

~km!m

~kmkg!
2

k̂ggm

2~kmkg!
D ~16!

FIG. 2. Feynman diagrams contributing to the muon transve
polarization at the one-loop level of SM.
03401
.

in expressions~13!,~14!.
Using thex PT Lagrangian@8#, one can derive decay am

plitudes for theK1→p0m1n andp0→gg processes, which
contribute to the imaginary part of the diagram in Fig. 2~g!:

T~K1→p0m1n!52
GF

2
ū~pn!~11g5!~ p̂K1 p̂p!v~pm!,

T~p0→gg!5
aA2

pF
emnlsk1

me1
nk2

ae2
s , ~17!

whereF5132 MeV. It should be noted thatT(p0→gg) is
written at O(p2) level. In addition, the amplitude differs
from the one in Ref.@8# by the sign, since we used th
opposite sign of pseudoscalar octet of mesons. From Eq.~17!
one can write down the imaginary part of the diagram sho
in Fig. 2~g!:

Im M35
GFa

8A2p3F
eE d3kp

2vp

d3km

2vm
d~kp1km2P!

3
ersabqaeb* kr

p

kg
2

ū~pn!~11g5!~ p̂K1 k̂p!

3~ k̂m2mm!gsv~pm!. ~18!

The details of the calculations of integrals entering Eqs.~13!,
~14!, ~18!, and their dependence on kinematical parame
are given in Appendix A.

The expression for the amplitude including the imagina
one-loop contributions can be written as

M5 ie
GF

A2
Vus* «m* F f̃ Kmmū~pn!~11g5!S pK

m

~pKq!
2

~pm!m

~pmq!
D

3v~pm!1F̃nū~pn!~11g5!q̂gmv~pm!2G̃mnl nG , ~19!

where

G̃mn5 i F̃ v«mnabqa~pK!b2F̃a@gmn~pKq!2pK
mqn#.

~20!

The f̃ K , F̃v , F̃a , andF̃n form factors include one-loop con
tributions from diagrams shown in Figs. 2~a!–2~f!. The
choice of the form factors is determined by the matrix e
ment expansion into set of gauge-invariant structures.

As long as we are interested in the contributions of ima
nary parts of one-loop diagrams only~since they lead
to a nonvanishing value of the transverse polarizatio!,
we neglect the real parts of these diagrams a
assume that Ref̃ K ,ReF̃v ,ReF̃a coincide with their
tree-level valuesf K ,Fv ,Fa , correspondingly, and ReF̃n
52 f Kmm/2(pmq). Explicit expressions for imaginary part
of the form factors are given in Appendix B.

The muon transverse polarization can be written as

PT5
rT

r0
, ~21!

e

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V. V. BRAGUTA, A. E. CHALOV, AND A. A. LIKHODED PHYSICAL REVIEW D 66, 034012 ~2002!
where

rT52mK
3 e2GF

2 uVusu2xAly2l22r mFmmIm~ f̃ KF̃a* !

3S 12
2

x
1

y

lxD1mmIm~ f̃ KF̃v* !S y

lx
2122

r m

lxD
12

r m

lx
Im~ f̃ KF̃n* !~12l!1mK

2 x Im~ F̃nF̃a* !~l21!

1mK
2 xIm~ F̃nF̃v* !~l21!G . ~22!

It should be noted that Eq.~20! disagrees with the expressio
for rT in Ref. @9#. In particular, the terms containing ImFn
are missing in therT expression given in Ref.@9#. Moreover,
calculating the muon transverse polarization we took i
account the diagrams shown in Figs. 2~e!–2~g!, which have
been neglected in Ref.@9#, and which give the contribution
comparable with the contribution from other diagrams in F
2.

III. RESULTS AND DISCUSSION

For the numerical calculations we use the following fo
factor values:

f K50.16 GeV, Fv5
0.095

mK
, Fa52

0.043

mK
.

The f K form factor is determined from experimental data
kaon decays@10#, and theFv ,Fa ones are calculated at th
one loop-level in the chiral perturbation theory@11#. It
should be noted that our definition forFv differs by a sign
from that in Ref.@11#. With this choice of form factor values
the decay branching ratio Br(K6→m6ng), with the cut on
photon energyEg>20 MeV, is equal to53.331023, which
is in good agreement with the PDG data.

The three-dimensional distribution of muon transverse
larization, calculated in the one-loop approximation of SM
shown in Figs. 3 and 4.PT , as function of thex and y
parameters, is characterized by the sum of individual con
butions of diagrams in Figs. 2~a!–2~f!, while the contribu-
tions from diagrams 2~a!–2~d! @12# and 2~e!–2~f! are com-
parable in absolute value, but they are opposite in sign
that the totalPT(x,y) distribution is the difference of thes
group contributions and in absolute value it is about o
order of magnitude less than each individual one of thos

It should be noted that the value of muon transverse
larization is positive in the whole Dalitz plot region. Ave
aged value of transverse polarization can be obtained by
tegrating the function 2rT /G(K1→m1ng) over the
physical region, and with the cut on photon energyEg
.20 MeV it is equal to

^PT
SM&55.6331024. ~23!

Let us note that the obtained numerical value of the av
aged transverse polarization andPT(x,y) kinematical depen-
dence in Dalitz plot differ from those given in Refs.@9,13#.
03401
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-

n-

r-

Note that in Ref.@13# only the diagram shown in Fig. 2~g!
was calculated and the result for that diagram does not c
cide with ours.

As it was calculated in Ref.@9#, thePT value varies in the
range of (20.1–4.0)31023 for cuts on the muon and pho
ton energies 200,Em,254.5 MeV, 20,Eg,200 MeV.
We have already mentioned above that~1! the authors of
Ref. @9# did not take into account terms containing the ima
nary part of theFn form factor ~contributing torT), which,
in general, are not small being compared with others, and~2!
the authors of Ref.@9# omitted the diagrams, shown in Fig
2~e!–2~f!, though, as was mentioned above, their contrib
tion to PT is comparable with the one of diagrams in Fig
2~a!–2~d!, and ~3! the authors of Ref.@9# did not take into
account the diagram shown in Fig. 2~g!.

All these points lead to serious disagreement between
results and results obtained in Ref.@9#. In particular, our
calculations show that the value of the muon transverse
larization has positive sign in the whole Dalitz plot regio
and its absolute value varies in the range of~0.0–1.5)
31023, and thePT dependence on thex,y parameters is
different from that in Ref.@9#.

We would like to remark that the muon transverse pol
ization for the same process was calculated in Ref.@14#,
where the contributions from diagrams 2~e!, 2~f!, and 2~g!
were taken into account. However, our result differs from
one obtained in Ref.@14#: PT value has opposite sign in
comparison to ours and in numerical calculation the aut
of Ref. @14# used constantf p instead off K in Eq. ~1!. Since

FIG. 3. The 3D Dalitz plot for the muon transverse polarizati
as a function ofx52Eg /mK andy52Em /mK in the one-loop ap-
proximation of SM.

FIG. 4. Level lines for the Dalitz plot of the muon transver
polarizationPT5 f (x,y).
2-4



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y

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th

ng

hich

MUON TRANSVERSE POLARIZATION IN . . . PHYSICAL REVIEW D66, 034012 ~2002!
the calculation is produced atO(p4) level, one needs to us
f K , as have been done in our paper. The kinematical st
tures for diagrams Figs. 2~a!–2~g! in Ref. @14# coincide with
ours.

ACKNOWLEDGMENTS

The authors thank Dr. V. V. Kiselev and Dr. A. K
Likhoded for fruitful discussion and valuable remarks. T
authors are also grateful to F. Bezrukov and D. Gorbunov
their remarks on the sign of the form factorFv in our previ-
ous results and R. Rogalyov for fruitful discussion. Th
work was in part supported by the Russian Foundation
Basic Research, Grants Nos. 99-02-16558 and 00-15-96
Russian Education Ministry, Grants No. RF E00-33-062 a
CRDF MO-011-0. The work of A. A. Likhoded was partiall
funded by a FapesP Grant No. 2001/06391-4.

APPENDIX A

For the integrals, which contribute to Eqs.~14! and ~15!,
we use the following notations:

P5pm1q, ~A1!

dr5
d3kg

2vg

d3km

2vm
d~kg1km2P!.

We present below either the explicit expressions for in
grals, or the set of equations, which being solved, give
parameters, entering the integrals:

J115E dr5
p

2

P22mm
2

P2
,

J125E dr
1

~pKkg!
5

p

2I
lnS ~PpK!1I

~PpK!2I D ,

where

I 25~PpK!22mK
2 P2,

E dr
kg

a

~pKkg!
5a11pK

a1b11P
a.

The a11 andb11 parameters are determined by the followi
equations:

a1152
1

~PpK!22mK
2 P2 S P2J112

J12

2
~PpK!~P22mm

2 ! D ,

b115
1

~PpK!22mK
2 P2 S ~PpK!J112

J12

2
mK

2 ~P22mm
2 ! D ,

E drkg
a5a12P

a,
03401
c-

r

r
5,

d

-
e

E drkg
akg

b5a13g
ab1b13P

aPb,

where

a125
~P22mm

2 !

2P2
J11,

a1352
1

12

~P22mm
2 !2

P2
J11,

b135
1

3 S P22mm
2

P2 D 2

J11.

J15E dr
1

~pKkg!@~pm2kg!22mm
2 #

52
p

2I 1~P22mm
2 !

lnS ~pKpm!1I 1

~pKpm!2I 1
D ,

J25E dr
1

~pm2kg!22mm
2

52
p

4I 2
lnS ~Ppm!1I 2

~Ppm!2I 2
D ,

where

I 1
25~pKpm!22mm

2 mK
2 ,

I 2
25~Ppm!22mm

2 P2.

E dr
kg

a

~pm2kg!22mm
2

5a1Pa1b1pm
a ,

a152
mm

2 ~P22mm
2 !J21~Ppm!J11

2~~Ppm!22mm
2 P2!

,

b15
~Ppm!~P22mm

2 !J21P2J11

2@~Ppm!22mm
2 P2#

,

The integrals below are determined by the parameters, w
can be obtained by solving the sets of equations:

E dr
kg

a

~pKkg!@~pm2kg!22mm
2 #

5a2Pa1b2pK
a1c2pm

a ,
2-5



ppendix

V. V. BRAGUTA, A. E. CHALOV, AND A. A. LIKHODED PHYSICAL REVIEW D 66, 034012 ~2002!
a2~PpK!1b2mK
2 1c2~pKpm!5J2 , a2~Ppm!1b2~pKpm!1c2mm

2 52
1

2
J12, a2P21b2~PpK!1c2~Ppm!5~pmq!J1 ,

E dr
kg

akg
b

~pKkg!@~pm2kg!22mm
2 #

5a3gab1b3~PapK
b1PbpK

a !1c3~Papm
b1Pbpm

a !1d3~pK
apm

b1pK
bpm

a !

1e3pm
apm

b1 f 3PaPb1g3pK
apK

b ,

4a312b3~PpK!12c3~Ppm!12d3~pKpm!1g3mK
2 1e3mm

2 1 f 3P250,

c3~pKpm!1b3mK
2 1 f 3~PpK!2a150, c3~PpK!1d3mK

2 1e3~pKpm!2b150, a31b3~PpK!1d3~pKpm!1g3mK
2 50,

b3~pKpm!1c3mm
2 1 f 3~Ppm!52

1

2
b11, b3~Ppm!1d3mm

2 1g3~pKpm!52
1

2
a11,

a3P212b3P2~PpK!12c3P2~Ppm!12d3~Ppm!~PpK!1e3~Ppm!21 f 3~P2!21g3~PpK!25~pmq!2J1 ,

E dr
kg

akg
b

~pm2kg!22mm
2

5a4gab1b4~Papm
b1Pbpm

a !1c4PaPb1d4pm
apm

b ,

a41d4mm
2 1b4~Ppm!50, b4mm

2 1c4~Ppm!52
1

2
a12, 4a412b4~Ppm!1c4P21d4mm

2 50,

a4P212b4P2~Ppm!1c4~P2!21d4~Ppm!25
~P22mm

2 !2

4
J2 ,

E dr
kg

akg
bkg

d

~pm2kg!22mm
2

5a5~gabpm
d 1gdapm

b1gbdpm
a !1b5~gabPd1gdaPb1gbdPa!1c5pm

apm
bpm

d 1d5PaPbPd

1e5~Papm
bpm

d 1Pdpm
apm

b1Pbpm
d pm

a !1 f 5~PaPbpm
d 1PdPapm

b1PbPdpm
a !,

2a51c5mm
2 1e5~Ppm!50, a5mm

2 1b5~Ppm!52
1

2
a13, b51e5mm

2 1 f 5~Ppm!50,

d5~Ppm!1 f 5mm
2 52

1

2
b13, 6a51c5mm

2 12e5~Ppm!1 f 5P250,

3a5P2~Ppm!13b5~P2!21c5~Ppm!31d5~P2!313e5P2~Ppm!213 f 5~P2!2~Ppm!5
~P22mm

2 !3

8
J2 .

For the rest of the integrals the following notations are used:

Pkp5
1

2
~P21mp

2 2mm
2 !, dr5

d3kp

2vp

d3km

2vm
d~kp1km2P!.

In terms of this notation the integrals can be rewritten as follows:

J35E dr

kg
2

52
p

4Pq
LnU2~Pq!Pkp12~Pq!APkp

2 2mp
2 P22mp

2 P2

2~Pq!Pkp22~Pq!APkp
2 2mp

2 P22mp
2 P2U , J45E dr5

p

P2
A~Pkp!22mp

2 P2.

APPENDIX B

Here we present the expressions for imaginary parts of form factors as the functions of parameters, calculated in A
A.
034012-6



MUON TRANSVERSE POLARIZATION IN . . . PHYSICAL REVIEW D66, 034012 ~2002!
Im f̃ K5
a

2p
f K@24a3~pKq!14a2mm

2 ~pKq!22b3mm
2 ~pKq!14c2mm

2 ~pKq!24c3mm
2 ~pKq!22d3mm

2 ~pKq!22e3mm
2 ~pKq!

22 f 3mm
2 ~pKq!14a2~pKq!~pmq!24b3~pKq!~pmq!24c3~pKq!~pmq!24 f 3~pKq!~pmq!#1

a

2p
Fa@8a4~pKq!

28a5~pKq!28b5~pKq!18b4mm
2 ~pKq!14c4mm

2 ~pKq!22c5mm
2 ~pKq!14d4mm

2 ~pKq!22d5mm
2 ~pKq!

26e5mm
2 ~pKq!26 f 5mm

2 ~pKq!112b4~pKq!~pmq!18c4~pKq!~pmq!14d4~pKq!~pmq!24d5~pKq!~pmq!

24e5~pKq!~pmq!28 f 5~pKq!~pmq!#1
a

2p
Fv@8a4~pKq!28a5~pKq!28b5~pKq!18b4mm

2 ~pKq!14c4mm
2 ~pKq!

22c5mm
2 ~pKq!14d4mm

2 ~pKq!22d5mm
2 ~pKq!26e5mm

2 ~pKq!26 f 5mm
2 ~pKq!112b4~pKq!~pmq!18c4~pKq!~pmq!

14d4~pKq!~pmq!24d5~pKq!~pmq!24e5~pKq!~pmq!28 f 5~pKq!~pmq!#,

Im F̃a5
a

2p
f KS a2mm

2 12c2mm
2 2c3mm

2 22d3mm
2 2e3mm

2 2
a1mm

2

~pmq!
2

b1mm
2

~pmq!
1

2b4mm
2

~pmq!
1

c4mm
2

~pmq!
1

d4mm
2

~pmq!
D 1

a

2p
Fv@8a424a5

212b522a1mm
2 14b4mm

2 15c4mm
2 2c5mm

2 2d4mm
2 23d5mm

2 25e5mm
2 27 f 5mm

2 12a1~pKpm!24b4~pKpm!

24c4~pKpm!12d5~pKpm!12e5~pKpm!14 f 5~pKpm!12a1~pKq!22b4~pKq!24c4~pKq!12d5~pKq!12 f 5~pKq!

24a1~pmq!16b4~pmq!110c4~pmq!26d5~pmq!22e5~pmq!28 f 5~pmq!#1
a

2p
Fa@26a412a51c4mm

2 2d4mm
2

2d5mm
2 2e5mm

2 22 f 5mm
2 12a1~pKpm!24b4~pKpm!24c4~pKpm!12d5~pKpm!12e5~pKpm!14 f 5~pKpm!

12a1~pKq!22b4~pKq!24c4~pKq!12d5~pKq!12 f 5~pKq!12c4~pmq!22d5~pmq!22 f 5 ~pmq!#.

Im F̃n5
a

2p
f KF4a1mm12a3mm12b1mm1b11mm22b4mm22c4mm2J12mm22J2mm2b2mK

2 mm1g3mK
2 mm22a2mm

3

2c2mm
3 1c3mm

3 1 f 3mm
3 22a2mm~pKpm!22b2mm~pKpm!12b3mm~pKpm!22c2mm~pKpm!12d3mm~pKpm!

12J1mm~pKpm!12b3mm~pKq!2
a12mm

3

~pmq!2
2

J11mm
3

~pmq!2
2

a12mm

~pmq!
2

2a4mm

~pmq!
1

J11mm

~pmq!
2

a11mK
2 mm

2~pmq!
1

3a1mm
3

~pmq!
1

3b1mm
3

~pmq!

1
b11mm

3

2~pmq!
2

2J2mm
3

~pmq!
2

b11mm~pKpm!

~pmq!
1

J12mm~pKpm!

~pmq!
2

b11mm~pKq!

~pmq!
1

J12mm~pKq!

~pmq!
22a2mm~pmq!12c3mm~pmq!

12 f 3mm~pmq!G1
a

2p
FvS 2a4mm24a5mm12b13mm24b5mm22a1mm

3 1c4mm
3 2c5mm

3 2d4mm
3 2d5mm

3 23e5mm
3

23 f 5mm
3 12a1mm~pKpm!22c4mm~pKpm!12d4mm~pKpm!12d5mm~pKpm!12e5mm~pKpm!14 f 5mm~pKpm!

22c4mm~pKq!12d5mm~pKq!12 f 5mm~pKq!1
3a13mm

~pmq!
1

b13mm
3

~pmq!
2

b13mm~pKpm!

~pmq!
2

b13mm~pKq!

~pmq!
22a1mm~pmq!

12c4mm~pmq!22d4mm~pmq!22d5mm~pmq!22e5mm~pmq!24 f 5mm~pmq! D 1
a

2p
FaS 26a4mm18a5mm2b13mm

18b5mm24b4mm
3 22c4mm

3 1c5mm
3 22d4mm

3 1d5mm
3 13e5mm

3 13 f 5mm
3 12a1mm~pKpm!22c4mm~pKpm!

12d4mm~pKpm!12d5mm~pKpm!12e5mm~pKpm!14 f 5mm~pKpm!22c4mm~pKq!12d5mm~pKq!12 f 5mm~pKq!

2
3a13mm

~pmq!
2

b13mm
3

2~pmq!
2

b13mm~pKpm!

~pmq!
2

b13mm~pKq!

~pmq!
26b4mm~pmq!24c4mm~pmq!22d4mm~pmq!12d5mm~pmq!

12e5mm~pmq!14 f 5mm~pmq! D ,
034012-7



V. V. BRAGUTA, A. E. CHALOV, AND A. A. LIKHODED PHYSICAL REVIEW D 66, 034012 ~2002!
Im F̃v5
a

2p
f KS a2mm

2 1c3mm
2 1e3mm

2 1
a1mm

2

~pmq!
1

b1mm
2

~pmq!
2

2b4mm
2

~pmq!
2

c4mm
2

~pmq!
2

d4mm
2

~pmq!
D 1

a

2p
Fa@6a422a528b51c4mm

2

2d4mm
2 2d5mm

2 2e5mm
2 22 f 5mm

2 22a1~pKpm!14b4~pKpm!14c4~pKpm!22d5~pKpm!22e5~pKpm!24 f 5~pKpm!

22a1~pKq!12b4~pKq!14c4~pKq!22d5~pKq!22 f 5~pKq!12c4~pmq!22d5~pmq!22 f 5~pmq!#1
a

2p
Fv@28a4

14a514b512a1mm
2 24b4mm

2 23c4mm
2 1c5mm

2 2d4mm
2 1d5mm

2 13e5mm
2 13 f 5mm

2 22a1~pKpm!14b4~pKpm!

14c4~pKpm!22d5~pKpm!22e5~pKpm!24 f 5~pKpm!22a1~pKq!12b4~pKq!14c4~pKq!22d5~pKq!22 f 5~pKq!

14a1~pmq!26b4~pmq!26c4~pmq!12d5~pmq!12e5~pmq!14 f 5~pmq!#.

The contribution to imaginary parts coming from the diagram shown in Fig. 2~g! may be written as follows:

Im f̃ K50,

Im F̃v52
a

8p3F
S 3J4mm

2

4P2
2

J4mm
4 mp

2

8~pmq!2P2
1

J3mm
4 mp

4

8~pmq!2P2
2

3J4mm
2 mp

2

8~pmq!P2
1

J3mm
2 mp

4

4~pmq!P2
1

2J4~pmq!

P2 D u@P22~mm1mp!2#,

ImF̃a52ImF̃v ,

Im f̃ n52
a

8p3F
S 23J4mmmp

2

2P2
1

J3mmmp
4

P2
2

J4mm
5 mp

2

4~pmq!2P2
1

J3mm
5 mp

4

4~pmq!2P2
2

5J4mm
3 mp

2

4~pmq!P2
1

J3mm
3 mp

4

~pmq!P2D u@P22~mm1mp!2#.
,

P

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03401
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