Muon transverse polarization in the K-l2 gamma decay in the standard model

(1)

Muon transverse polarization in the K

_l2_␥

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

共Received 26 March 2002; published 15 August 2002兲

The muon transverse polarization in the K⫹→␮⫹␯␥ 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 of E_␥⭓20 MeV is equal to 5.63⫻10⫺4.

DOI: 10.1103/PhysRevD.66.034012 PACS number共s兲: 13.88.⫹e, 13.20.Eb, 13.40.Ks

I. INTRODUCTION

The study of the radiative K-meson decays is extremely interesting in searching for new physics effects beyond the standard model共SM兲. One of the most appealing possibilities is to probe new interactions, which could lead to C P viola-tion. Contrary to the SM, where C P violation is caused by the presence of the complex phase in the Cabibbo-Kobayashi-Maskawa共CKM兲 matrix, the CP violation in ex-tended models 共for instance, in models with three and more Higgs doublets兲 can naturally arise due to the complex cou-plings of new Higgs bosons to fermions关1兴. Such effects can be detected by using experimental observables, which are essentially sensitive to T-odd contributions. These observ-ables, for instance, are the T-odd correlation 兵T

⫽(1/MK

3

) pជ_␥•关pជ_␲⫻pជl兴其in the K⫾→␲0␮⫾␯␥ decay关2兴 and

muon transverse polarization ( P_T) in K⫾→␮⫾␯␥. The search for new physics effects using the T-odd correlation analysis in the K⫾→␲0␮⫾␯␥ decay will be done in the proposed OKA experiment 关3兴, where an event sample of 7.0⫻105 for the K⫹→␲0␮⫹␯␥ decay is expected to be ac-cumulated.

At the moment the E246 experiment at KEK关4兴 performs the analysis of the data on the K⫾→␮⫾␯␥ process to put bounds on the T-violating muon transverse polarization. It should be noted that the expected value of new physics con-tribution to PT can be of the order of ⯝7.0⫻10⫺3–6.0

⫻10⫺2 _{关5,6兴, depending on the type of model beyond the}

SM. Thus, when one searches for new physics contributions to PT, it is extremely important to estimate the effects

com-ing from so called ‘‘fake’’ polarization, which is caused by the SM electromagnetic final state interactions and which is a natural background for the new interaction contributions.

In this paper we calculate muon transverse polarization in the K⫾→␮⫾␯␥ process, induced by the electromagnetic fi-nal state interaction in the one-loop approximation of the minimal quantum electrodynamics.

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

dia-grams with final state interactions within the SM. The last section summarizes the results and conclusions.

II. MUON TRANSVERSE POLARIZATION IN THE

K¿\µ¿␯␥ PROCESS IN SM

The K⫹→␮⫹␯␥ decay at the tree level of SM is de-scribed by the diagrams shown in Fig. 1. The diagrams in Figs. 1共b兲 and 1共c兲 correspond to the muon and kaon brems-strahlung, while the diagram in Fig. 1共a兲 corresponds to the structure radiation. This decay amplitude can be written as follows: M⫽ieGF

冑

2Vus*␧␮*

冋

fKm␮u ¯_共p ␯兲共1⫹␥5兲

冉

p_K␮ 共pKq兲 ⫺ 共p␮兲␮ 共p␮q兲 ⫺ qˆ␥␮ 2共p_␮q兲

冊

v共p␮兲⫺G ␮␯_l ␯

册

, 共1兲 where l_␮⫽u¯共p_␯兲共1⫹␥₅兲␥_␮v共p_␮兲, G␮␯⫽iF_v␧␮␯␣␤q_␣共pK兲␤⫺Fa共g␮␯共pKq兲⫺pK␮q␯兲, 共2兲

*On leave of absence from Institute for High Energy Physics, Protvino, 142284 Russia. Email address: andre@ift.unesp.br

FIG. 1. Feynman diagrams for the K⫾→␮⫾␯␥ decay at the tree level of SM.

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GF is the Fermi constant, Vus is the corresponding CKM

matrix element, f_K is the K-meson leptonic constant, pK, p␮, p␯,q are the kaon, muon, neutrino, and photon

four-momenta, correspondingly, and␧_␮is the photon polarization vector. F_v and Faare the kaon vector and axial form factors.

In Eq. 共2兲 we use the following definition of Levi-Civita tensor:⑀0123⫽⫹1.

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

MK⫽ie GF

冑

2Vus*␧␮*

冋

fKm␮u ¯_共p ␯兲共1⫹␥5兲 ⫻

冉

pK ␮ 共pKq兲 ⫺ ␥␮ m␮

冊

v共p␮兲⫺G ␮␯_l ␯

册

. 共3兲

The partial width of the K⫹→␮⫹␯␥ decay in the K-meson rest frame can be expressed as

d⌫⫽

兺

兩M兩2 2mK 共2␲兲 4_␦_共p K⫺p␮⫺q⫺p␯兲 ⫻ d 3_q 共2␲兲3_2E q d3_p ␮ 共2␲兲3_2E ␮ d3_p ␯ 共2␲兲3_2E ␯ , 共4兲

where summation over muon and photon spin states is per-formed.

Introducing the unit vector along the muon spin direction in muon rest frame sជ where eជi (i⫽L,N,T) are the unit

vec-tors along the longitudinal, normal and transverse compo-nents of muon polarization, one can write down the matrix element squared for the transition into the particular muon polarization state in the following form:

兩M兩2_⫽_␳

0关1⫹共PLeជL⫹PNeជN⫹PTeជT兲•sជ兴, 共5兲

where␳0is the Dalitz plot probability density averaged over polarization states. The unit vectors eជ_i can be expressed in terms of the three-momenta of final particles

e ជL⫽ p ជ_␮ 兩pជ␮兩 , eជN⫽ pជ_␮⫻共qជ⫻pជ_␮兲兩pជ␮⫻共qជ⫻pជ␮兲兩 , eជT⫽ qជ⫻pជ_␮ 兩qជ⫻pជ␮兩 . 共6兲 With such definition of eជi vectors, PT, PL, and PN denote

transverse, longitudinal, and normal components of the muon polarization, correspondingly. It is convenient to use the fol-lowing variables: x⫽2E␥ mK , y⫽2E␮ mK , ␭⫽x⫹y⫺1⫺r␮ x , r␮⫽ m_␮2 m_K2, 共7兲

where E_␥ and E_␮ are the photon and muon energies in the kaon rest frame.

The Dalitz plot probability density, as a function of the x and y variables, has the form

␳0⫽ 1 2e 2_G F 2_兩V us兩2

再

4m_␮2兩 fK兩2 ␭x2 共1⫺␭兲

冋

x 2_⫹2共1⫺r ␮兲

冉

1⫺x ⫺r_␭␮

冊册

⫹mK 6 x2共兩Fa兩2⫹兩Fv兩2兲共y⫺2␭y⫺␭x⫹2␭2兲 ⫹4 Re共 fKFv*兲mK 4_r ␮ x ␭ 共␭⫺1兲 ⫹4 Re共 fKFa*兲mK 4 r_␮

冉

⫺2y⫹x⫹2r␮ ␭ ⫺ x ␭ ⫹2␭

冊

⫹2 Re共FaFv*兲mK 6 x2共y⫺2␭⫹x␭兲

冎

. 共8兲

Calculating the muon transverse polarization PT we follow

the original paper关7兴 and assume that the decay amplitude is C P invariant, and form factors fK, Fv, and Fa are real. In

this case the tree level muon polarization PT⫽0. When

one-loop contributions are incorporated, the nonvanishing muon transverse polarization can arise due to the interference of tree-level diagrams and imaginary parts of one-loop dia-grams, induced by the electromagnetic final state interaction. To calculate the imaginary parts of formfactors one can use the S-matrix unitarity

S⫹S⫽1 共9兲

and, using S⫽1⫹iT, one gets Tf i⫺Ti f*⫽i

兺

n

T_{n f}*Tni, 共10兲

where i, f ,n indices correspond to the initial, final, and inter-mediate states of the particle system. Further, using the T invariance of the matrix element one has

Im T_{f i}⫽1 2

兺

n

T_{n f}*T_ni, 共11兲

Tf i⫽共2␲兲4␦共Pf⫺Pi兲Mf i. 共12兲

One-loop diagrams of the SM, which contribute to the muon transverse polarization in the K⫹→␮⫹␯␥ decay, are shown in Fig. 2. Using Eq. 共3兲 one can write down the imaginary parts of these diagrams. For the diagrams in Figs. 2共a兲, 2共c兲 one has Im M₁⫽ie␣ 2␲ GF

冑

2Vus*u ¯_共p ␯兲共1⫹␥5兲 ⫻

冕

d 3_k ␥ 2␻_␥ d3k_␮ 2␻_␮␦共k␥⫹k␮⫺P兲R␮共kˆ␮⫺m␮兲␥ ␮ ⫻ qˆ⫹pˆ␮⫺m␮ 共q⫹p␮兲2⫺m␮2 ␥␦_␧ ␦ *v共p_␮兲. 共13兲

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ImM2⫽ ie␣ 2␲ G_F

冑

2Vus*u ¯_共p ␯兲共1⫹␥5兲

冕

d3k_␥ 2␻_␥ d3k_␮ 2␻_␮ ⫻␦共k␥⫹k␮⫺P兲R␮共kˆ␮⫺m␮兲␥␦␧␦* kˆ_␮⫺qˆ⫺m_␮ 共k␮⫺q兲2⫺m␮2 ⫻␥␮_v_共p ␮兲, 共14兲 where R_␮⫽ f_Km_␮

冉

共pK兲␮ 共pKk␥兲 ⫺ ␥␮ m_␮

冊

⫺iFv␧␮␯␣␤共k␥兲 ␣_共p K兲␤␥␯ ⫹Fa关␥␮共pKk␥兲⫺共pK兲␮kˆ␥兴. 共15兲

To write down the contributions of diagrams shown in Figs. 2共e兲, 2共f兲, one should substitute R_␮by

R_␮⫽ fKm␮

冉

␥␮ m_␮⫺ 共k␮兲␮ 共k␮k␥兲 ⫺ kˆ_␥␥_␮ 2共k_␮k_␥兲

冊

共16兲 in expressions 共13兲,共14兲.

Using the␹PT Lagrangian关8兴, one can derive decay am-plitudes for the K⫹→␲0␮⫹␯ and␲0→␥␥ processes, which contribute to the imaginary part of the diagram in Fig. 2共g兲:

T共K⫹→␲0␮⫹␯兲⫽⫺GF

2 ¯u共p␯兲共1⫹␥5兲共pˆK⫹pˆ␲兲v共p␮兲, T共␲0→␥␥兲⫽␣

冑

2

␲F ⑀␮␯␭␴k1␮e␯1k2␣e2␴, 共17兲 where F⫽132 MeV. It should be noted that T(␲0→␥␥) is written at O( p2) level. In addition, the amplitude differs from the one in Ref. 关8兴 by the sign, since we used the opposite sign of pseudoscalar octet of mesons. From Eq.共17兲 one can write down the imaginary part of the diagram shown in Fig. 2共g兲: Im M3⫽ GF␣ 8

冑

2␲3Fe

冕

d3k_␲ 2␻_␲ d3k_␮ 2␻_␮␦共k␲⫹k␮⫺P兲 ⫻⑀ ␳␴␣␤_q ␣e␤*k␳␲ k_␥2 u ¯_共p_␯_兲共1⫹_␥₅_兲共pˆ_K_⫹kˆ_␲_兲 ⫻共kˆ␮⫺m␮兲␥␴v共p␮兲. 共18兲

The details of the calculations of integrals entering Eqs.共13兲,

共14兲, 共18兲, and their dependence on kinematical parameters

are given in Appendix A.

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

M⫽ieGF

冑

2Vus *_␧_␮*

冋

f˜Km␮u¯共p␯兲共1⫹␥5兲

冉

p_K␮ 共pKq兲 ⫺ 共p␮兲␮ 共p␮q兲

冊

⫻v共p␮兲⫹F˜n¯u共p␯兲共1⫹␥5兲qˆ␥␮v共p␮兲⫺G˜␮␯l␯

册

, 共19兲 where G ˜␮␯_⫽iF˜ v␧␮␯␣␤q␣共pK兲␤⫺F˜a关g␮␯共pKq兲⫺pK␮q␯兴. 共20兲

The f˜K, F˜v, F˜a, and F˜nform factors include one-loop

con-tributions from diagrams shown in Figs. 2共a兲–2共f兲. The choice of the form factors is determined by the matrix ele-ment expansion into set of gauge-invariant structures.

As long as we are interested in the contributions of imagi-nary parts of one-loop diagrams only 共since they lead to a nonvanishing value of the transverse polarization兲,

we neglect the real parts of these diagrams and

assume that Ref˜K,ReF˜v,ReF˜a coincide with their

tree-level values fK,Fv,Fa, correspondingly, and ReF˜n

⫽⫺ fKm␮/2( p␮q). Explicit expressions for imaginary parts

of the form factors are given in Appendix B.

The muon transverse polarization can be written as P_T⫽␳T

␳0

, 共21兲

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

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where ␳T⫽2mK 3_e2_G F 2_兩V us兩 2_x

冑

_␭y⫺␭2_⫺r ␮

冋

m␮Im共 f˜KF˜a*兲 ⫻

冉

1⫺2 x⫹ y ␭x

冊

⫹m␮Im共 f˜KF˜v*兲

冉

y ␭x ⫺1⫺2 r_␮ ␭x

冊

⫹2_␭xr␮Im共 f˜KFñ*兲共1⫺␭兲⫹mK 2 x Im共FñFã*兲共␭⫺1兲 ⫹mK 2_xIm_共F˜ nF˜v*兲共␭⫺1兲

册

. 共22兲

It should be noted that Eq.共20兲 disagrees with the expression for ␳T in Ref. 关9兴. In particular, the terms containing Im Fn

are missing in the␳Texpression given in Ref.关9兴. Moreover,

calculating the muon transverse polarization we took into account the diagrams shown in Figs. 2共e兲–2共g兲, which have been neglected in Ref. 关9兴, and which give the contribution comparable with the contribution from other diagrams in Fig. 2.

III. RESULTS AND DISCUSSION

For the numerical calculations we use the following form factor values: f_K⫽0.16 GeV, F_v⫽0.095 mK , F_a⫽⫺0.043 mK . The f_K form factor is determined from experimental data on kaon decays 关10兴, and the F_v,Fa ones are calculated at the

one loop-level in the chiral perturbation theory 关11兴. It should be noted that our definition for F_v differs by a sign from that in Ref.关11兴. With this choice of form factor values the decay branching ratio Br(K⫾→␮⫾␯␥), with the cut on photon energy E_␥⭓20 MeV, is equal to ⫽3.3⫻10⫺3, which is in good agreement with the PDG data.

The three-dimensional distribution of muon transverse po-larization, calculated in the one-loop approximation of SM is shown in Figs. 3 and 4. PT, as function of the x and y

parameters, is characterized by the sum of individual contri-butions of diagrams in Figs. 2共a兲–2共f兲, while the contribu-tions from diagrams 2共a兲–2共d兲关12兴 and 2共e兲–2共f兲 are com-parable in absolute value, but they are opposite in sign, so that the total PT(x,y ) distribution is the difference of these

group contributions and in absolute value it is about one order of magnitude less than each individual one of those.

It should be noted that the value of muon transverse po-larization is positive in the whole Dalitz plot region. Aver-aged value of transverse polarization can be obtained by in-tegrating the function 2␳T/⌫(K⫹→␮⫹␯␥) over the

physical region, and with the cut on photon energy E_␥

⬎20 MeV it is equal to

具

P_TS M

典

⫽5.63⫻10⫺4. 共23兲

Let us note that the obtained numerical value of the aver-aged transverse polarization and PT(x,y ) kinematical

depen-dence in Dalitz plot differ from those given in Refs. 关9,13兴.

Note that in Ref. 关13兴 only the diagram shown in Fig. 2共g兲 was calculated and the result for that diagram does not coin-cide with ours.

As it was calculated in Ref.关9兴, the PTvalue varies in the

range of (⫺0.1–4.0)⫻10⫺3 for cuts on the muon and pho-ton energies 200⬍E_␮⬍254.5 MeV, 20⬍E_␥⬍200 MeV. We have already mentioned above that 共1兲 the authors of Ref.关9兴 did not take into account terms containing the imagi-nary part of the Fn form factor 共contributing to␳T), which,

in general, are not small being compared with others, and共2兲 the authors of Ref. 关9兴 omitted the diagrams, shown in Figs. 2共e兲–2共f兲, though, as was mentioned above, their contribu-tion to PT is comparable with the one of diagrams in Figs.

2共a兲–2共d兲, and 共3兲 the authors of Ref. 关9兴 did not take into account the diagram shown in Fig. 2共g兲.

All these points lead to serious disagreement between our results and results obtained in Ref. 关9兴. In particular, our calculations show that the value of the muon transverse po-larization has positive sign in the whole Dalitz plot region and its absolute value varies in the range of 共0.0–1.5)

⫻10⫺3_{, and the P}

T dependence on the x, y parameters is

different from that in Ref.关9兴.

We would like to remark that the muon transverse polar-ization for the same process was calculated in Ref. 关14兴, where the contributions from diagrams 2共e兲, 2共f兲, and 2共g兲 were taken into account. However, our result differs from the one obtained in Ref. 关14兴: PT value has opposite sign in

comparison to ours and in numerical calculation the author of Ref.关14兴 used constant f_␲ instead of fK in Eq.共1兲. Since

FIG. 3. The 3D Dalitz plot for the muon transverse polarization as a function of x⫽2E_␥/mK and y⫽2E␮/mKin the one-loop

ap-proximation of SM.

FIG. 4. Level lines for the Dalitz plot of the muon transverse polarization PT⫽ f (x,y).

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the calculation is produced at O( p4) level, one needs to use fK, as have been done in our paper. The kinematical

struc-tures for diagrams Figs. 2共a兲–2共g兲 in Ref. 关14兴 coincide with ours.

ACKNOWLEDGMENTS

The authors thank Dr. V. V. Kiselev and Dr. A. K. Likhoded for fruitful discussion and valuable remarks. The authors are also grateful to F. Bezrukov and D. Gorbunov for their remarks on the sign of the form factor F_v in our previ-ous results and R. Rogalyov for fruitful discussion. This work was in part supported by the Russian Foundation for Basic Research, Grants Nos. 99-02-16558 and 00-15-96645, Russian Education Ministry, Grants No. RF E00-33-062 and CRDF MO-011-0. The work of A. A. Likhoded was partially funded by a FapesP Grant No. 2001/06391-4.

APPENDIX A

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

P⫽p_␮⫹q, 共A1兲 d␳⫽d 3_k ␥ 2␻_␥ d3k_␮ 2␻_␮␦共k␥⫹k␮⫺P兲.

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

冕

d␳ k␥ ␣ 共p␮⫺k␥兲2⫺m␮2 ⫽a1P␣⫹b1p␮␣, a1⫽⫺ m_␮2共P2⫺m_␮2兲J2⫹共Pp␮兲J11 2共共Pp_␮兲2⫺m_␮2P2兲 , b1⫽ 共Pp␮兲共P2⫺m␮2兲J2⫹P2J11 2关共Pp_␮兲2⫺m_␮2P2兴 ,

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

冕

d␳ k␥ ␣ 共pKk␥兲关共p␮⫺k␥兲2⫺m␮ 2_兴⫽a2P ␣_⫹b 2pK␣⫹c2p_␮␣,

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a2共PpK兲⫹b2mK 2_⫹c 2共pKp␮兲⫽J2, a2共Pp␮兲⫹b2共pKp␮兲⫹c2m_␮2⫽⫺ 1 2J12, a2P 2_⫹b 2共PpK兲⫹c2共Pp␮兲⫽共p␮q兲J1,

冕

d␳ k␥ ␣_k ␥ ␤ 共pKk␥兲关共p␮⫺k␥兲2⫺m_␮ 2 兴⫽a3g ␣␤_⫹b 3共P␣pK␤⫹P␤pK␣兲⫹c3共P␣p_␮␤⫹P␤p_␮␣兲⫹d3共pK␣p␮␤⫹pK␤p␮␣兲 ⫹e3p_␮␣p_␮␤⫹ f3P␣P␤⫹g3pK␣pK␤, 4a3⫹2b3共PpK兲⫹2c3共Pp␮兲⫹2d3共pKp␮兲⫹g3mK 2_⫹e 3m␮ 2_{⫹ f} 3P2⫽0, c3共pKp␮兲⫹b3mK 2_{⫹ f} 3共PpK兲⫺a1⫽0, c3共PpK兲⫹d3mK 2_⫹e 3共pKp␮兲⫺b1⫽0, a3⫹b3共PpK兲⫹d3共pKp␮兲⫹g3mK 2_⫽0, b3共pKp␮兲⫹c3m_␮2⫹ f3共Pp␮兲⫽⫺ 1 2b11, b3共Pp␮兲⫹d3m␮ 2_⫹g 3共pKp␮兲⫽⫺ 1 2a11, a3P2⫹2b3P2共PpK兲⫹2c3P2共Pp␮兲⫹2d3共Pp␮兲共PpK兲⫹e3共Pp␮兲2⫹ f3共P2兲2⫹g3共PpK兲2⫽共p␮q兲2J1,

冕

d␳ k␥ ␣_k ␥ ␤ 共p␮⫺k␥兲2⫺m␮2 ⫽a4g␣␤⫹b4共P␣p_␮␤⫹P␤p␣_␮兲⫹c4P␣P␤⫹d4p_␮␣p_␮␤, a₄⫹d₄m_␮2⫹b₄共Pp_␮兲⫽0, b₄m_␮2⫹c₄共Pp_␮兲⫽⫺1 2a12, 4a4⫹2b4共Pp␮兲⫹c4P 2_⫹d 4m␮ 2_⫽0, a4P2⫹2b4P2共Pp␮兲⫹c4共P2兲2⫹d4共Pp␮兲2⫽ 共P2_⫺m ␮ 2_兲2 4 J2,

冕

d␳ k␥ ␣_k ␥ ␤_k ␥ ␦ 共p␮⫺k␥兲2⫺m␮2 ⫽a5共g␣␤p␮␦⫹g␦␣p␮␤⫹g␤␦p␮␣兲⫹b5共g␣␤P␦⫹g␦␣P␤⫹g␤␦P␣兲⫹c5p␮␣p␮␤p␮␦⫹d5P␣P␤P␦ ⫹e5共P␣p_␮␤p_␮␦⫹P␦p_␮␣p_␮␤⫹P␤p_␮␦p␣_␮兲⫹ f5共P␣P␤p_␮␦⫹P␦P␣p_␮␤⫹P␤P␦p_␮␣兲, 2a5⫹c5m_␮2⫹e5共Pp␮兲⫽0, a5m_␮2⫹b5共Pp␮兲⫽⫺ 1 2a13, b5⫹e5m␮ 2_{⫹ f} 5共Pp␮兲⫽0, d5共Pp␮兲⫹ f5m_␮ 2_⫽⫺1 2b13, 6a5⫹c5m␮ 2_⫹2e 5共Pp␮兲⫹ f5P2⫽0, 3a5P2共Pp␮兲⫹3b5共P2兲2⫹c5共Pp␮兲3⫹d5共P2兲3⫹3e5P2共Pp␮兲2⫹3 f5共P2兲2共Pp␮兲⫽ 共P2_⫺m ␮ 2_兲3 8 J2.

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

Pk_␲⫽1 2共P 2_⫹m ␲ 2_⫺m ␮ 2_{兲, d}_␳_⫽d 3_k ␲ 2␻_␲ d3k_␮ 2␻_␮␦共k␲⫹k␮⫺P兲. In terms of this notation the integrals can be rewritten as follows:

J3⫽

冕

d␳ k_␥2⫽⫺ ␲ 4 Pq Ln

冏

2共Pq兲Pk_␲⫹2共Pq兲

冑

Pk_␲2⫺m_␲2P2⫺m_␲2P2 2共Pq兲Pk_␲⫺2共Pq兲

冑

Pk_␲2⫺m_␲2P2⫺m_␲2P2

冏

, J4⫽

冕

d␳⫽ ␲ P2

冑

共Pk␲兲 2_⫺m ␲ 2_P2_. APPENDIX B

⫺6a4m␮⫹8a5m␮⫺b13m␮ ⫹8b5m␮⫺4b4m␮ 3_⫺2c 4m␮ 3_⫹c 5m␮ 3_⫺2d 4m␮ 3_⫹d 5m␮ 3_⫹3e 5m␮ 3_{⫹3 f} 5m␮ 3_⫹2a 1m␮共pKp␮兲⫺2c4m␮共pKp␮兲 ⫹2d4m␮共pKp␮兲⫹2d5m␮共pKp␮兲⫹2e5m␮共pKp␮兲⫹4 f5m␮共pKp␮兲⫺2c4m␮共pKq兲⫹2d5m␮共pKq兲⫹2 f5m␮共pKq兲 ⫺3a13m␮ 共p␮q兲 ⫺ b13m_␮ 3 2共p_␮q兲 ⫺ b13m␮共pKp␮兲共p␮q兲 ⫺ b13m␮共pKq兲共p␮q兲 ⫺6b4m␮共p␮q兲⫺4c4m␮共p␮q兲⫺2d4m␮共p␮q兲⫹2d5m␮共p␮q兲 ⫹2e5m␮共p␮q兲⫹4 f5m␮共p␮q兲

冊

,

(8)

Im F˜_v⫽ ␣ 2␲fK

冉

a2m␮ 2_⫹c 3m␮ 2_⫹e 3m␮ 2_⫹a1m␮ 2 共p␮q兲 ⫹ b1m␮ 2 共p␮q兲 ⫺ 2b4m␮ 2 共p␮q兲 ⫺ c4m␮ 2 共p␮q兲 ⫺ d4m␮ 2 共p␮q兲

冊

⫹ ␣ 2␲Fa关6a4⫺2a5⫺8b5⫹c4m␮ 2

⫺d4m_␮2⫺d5m_␮2⫺e5m2_␮⫺2 f5m_␮2⫺2a1共pKp␮兲⫹4b4共pKp␮兲⫹4c4共pKp␮兲⫺2d5共pKp␮兲⫺2e5共pKp␮兲⫺4 f5共pKp␮兲

⫺2a1共pKq兲⫹2b4共pKq兲⫹4c4共pKq兲⫺2d5共pKq兲⫺2 f5共pKq兲⫹2c4共p␮q兲⫺2d5共p␮q兲⫺2 f5共p␮q兲兴⫹ ␣ 2␲Fv关⫺8a4 ⫹4a5⫹4b5⫹2a1m␮ 2_⫺4b 4m␮ 2_⫺3c 4m␮ 2_⫹c 5m␮ 2_⫺d 4m␮ 2_⫹d 5m␮ 2_⫹3e 5m␮ 2_{⫹3 f} 5m␮ 2_⫺2a 1共pKp␮兲⫹4b4共pKp␮兲 ⫹4c4共pKp␮兲⫺2d5共pKp␮兲⫺2e5共pKp␮兲⫺4 f5共pKp␮兲⫺2a1共pKq兲⫹2b4共pKq兲⫹4c4共pKq兲⫺2d5共pKq兲⫺2 f5共pKq兲 ⫹4a1共p␮q兲⫺6b4共p␮q兲⫺6c4共p␮q兲⫹2d5共p␮q兲⫹2e5共p␮q兲⫹4 f5共p␮q兲兴.

The contribution to imaginary parts coming from the diagram shown in Fig. 2共g兲 may be written as follows: Im f˜_K⫽0, Im F˜_v⫽⫺ ␣ 8␲3_F

冉

3J4m␮ 2 4 P2 ⫺ J4m␮ 4 m_␲2 8共p_␮q兲2_P2⫹ J3m␮ 4 m_␲4 8共p_␮q兲2_P2⫺ 3J4m␮ 2 m_␲2 8共p_␮q兲P2⫹ J3m␮ 2 m_␲4 4共p_␮q兲P2⫹ 2J₄共p_␮q兲 P2

冊

␪关P 2_⫺共m ␮⫹m␲兲2兴, ImF˜a⫽⫺ImF˜v, Im f˜n⫽⫺ ␣ 8␲3F

冉

⫺3J4m␮m_␲2 2 P2 ⫹ J3m␮m_␲4 P2 ⫺ J4m_␮5m_␲2 4共p_␮q兲2P2⫹ J3m_␮5m_␲4 4共p_␮q兲2P2⫺ 5J4m_␮3m_␲2 4共p_␮q兲P2⫹ J3m_␮3m_␲4 共p␮q兲P2

冊

␪关P 2_⫺共m ␮⫹m␲兲2兴.

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