Baryonic rare decays of

PHYSICAL REVIEW D, VOLUME 64, 074001

Baryonic rare decays ofLb\L l¿lÀ

Chuan-Hung ChenDepartment of Physics, National Cheng Kung University, Tainan, Taiwan, Republic of China

C. Q. GengDepartment of Physics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China

~Received 16 April 2001; published 16 August 2001!

We present a systematic analysis for the rare baryonic exclusive decays ofLb→L l 1l 2 ( l 5e,m,t). Westudy the differential decay rates and the dilepton forward-backward, lepton polarization, and variousCPasymmetries with a new simple set of form factors inspired by the heavy quark effective theory. We show thatmost of the observables are insensitive to the nonperturbative QCD effects. To illustrate the effect of newphysics, we discuss our results in an explicit supersymmetric extension of the standard model, which containsnew CP violating phases and therefore induces sizableCP violating asymmetries.

DOI: 10.1103/PhysRevD.64.074001 PACS number~s!: 13.30.Ce, 11.30.Er, 12.60.Jv, 14.20.Mr

erw

caalherao

u

ea

fce

r

ad.-

th

ayrd

ou

ffi-

re

n

t thetheffectnt.the

rm

I. INTRODUCTION

A priority in current particle physics research is to detmine the parameters of the Cabibbo-Kobayashi-Maska~CKM! matrix elements@1# in the standard model~SM!. Be-cause of the CLEO measurement of the radiativeb→sg de-cay @2#, some interest has been focused on the rare derelated tob→sl1l 2 induced by the flavor changing neutrcurrents~FCNCs!. In the SM, these rare decays occur at tloop level and depend on the CKM elements. In the liteture, most studies have been concentrated on the corresping exclusive rareB-meson decays such asB→K (* )l 1l 2

@3#. However, these exclusive modes contain severalknown hadronic form factors, which cannot be measuredthe presentB-meson facilities unlike the kaon cases. Rcently, we have examined the exclusive rare baryonic decof Lb→L l l ( l 5n,e,m,t) @4–6# and found that some othe physical quantities are insensitive to the hadronic untainties.

In this paper, we present a systematic study of the baonic decays ofLb→L l 1l 2. We will explore various pos-sible CP even and odd asymmetries to show how the hronic unknown parameters are factored out in most casesillustrate CP violating effects, we will also discuss an explicit CP violating model with supersymmetry~SUSY!.

The paper is organized as follows. In Sec. II, we giveeffective Hamiltonian for the decays ofLb→L l l and themost general form factors in theLb→L transition. In Sec.III, we derive the general forms of the differential decrates. In Sec. IV, we study the dilepton forward-backwalepton polarization, and variousCP violating asymmetries.We perform our numerical analysis in Sec. V. We presentconclusions in Sec. VI.

II. EFFECTIVE HAMILTONIAN AND FORM FACTORS

In the SM, the effective Hamiltonian forb→sl1l 2 isgiven by

H524GF

A2VtbVts* (

i 51

10

Ci~m!Oi~m!, ~1!

0556-2821/2001/64~7!/074001~16!/$20.00 64 0740

-a

ys

-nd-

n-in-ys

r-

y-

-To

e

,

r

where the expressions of the renormalized Wilson coecientsCi(m) and operatorsOi(m) can be found in Ref.@7#.From Eq.~1!, the free quark decay amplitude is written as

M~b→sl1l 2!5GFaem

A2pVtbVts* F sS C9

eff~m!gmPL

22mb

q2C7

eff~m!ismnqnPRD b lgml

1 sC10gmPLb lgmg5l G ~2!

with PL(R)5(17g5)/2. We note that in Eq.~2!, only theterm associated with the Wilson coefficientC10 is indepen-dent of them scale. In addition to the short-distance~SD!contributions, the long-distance~LD! ones such as that fromthe cc resonant states ofC,C8, etc., are also important fothe decay rate. It is known that for the LD effects in thB-meson decays@8–13#, both the factorization assumptio~FA! and the vector meson dominance~VMD ! approxima-tion have been used. In baryonic decays, we assume thaparametrization of LD contributions is the same as that inB-meson decays. Hence, we may include the resonant e~RE! by absorbing it to the corresponding Wilson coefficieIn this paper as a more complete analysis we also includeLD contributions to the decay ofb→sg, induced by thenonfactorizable effects@14,15#. The effective Wilson coeffi-cients ofC9

eff andC7eff can be expressed as the standard fo

C9eff~m!5C9~m!1Y~z,s8!, ~3!

C7eff~m!5C7~m!1C78~m,q2!, ~4!

where

©2001 The American Physical Society01-1

th

-

s

rsD

th

e

uchvy

d

e-

he

CHUAN-HUNG CHEN AND C. Q. GENG PHYSICAL REVIEW D64 074001

Y~z,s8!5S h~z,s8!13

aem2 (

j 5C,C8kj

pG~ j→ l 1l 2!M j

q22M j21 iM jG j

D2

1

2h~1,s8!~4C314C413C51C6!

21

2h~0,s8!~C313C4!,

C78~m,q2!5Cb→sg8 ~m!1vS h~z,s8!

13

aem2 (

j 5C,C8kj

pG~ j→ l 1l 2!M j

q22M j21 iM jG j

D , ~5!

with

h~z,s8!528

9ln z1

8

271

4

9x2

2

9~21x!u12xu1/2

35 lnUA12x11

A12x21U2 ip for x[4z2/s8,1,

2 arctan1

Ax21for x[4z2/s8.1,

Cb→sg8 5 iasF2

9h14/23~G1~xt!20.1687!20.03C2~m!G ,

G1~x!5x~x225x22!

8~x21!31

3x2 ln x

4~x21!4. ~6!

HereY(z,s8) combines the one-loop matrix elements andLD contributions of operatorsO1–O6 , Cb→sg8 is the absorp-tive part ofb→sg @16# with neglecting the small contribution from VubVus* , z5mc /mb , s85q2/mb

2 , h5as(mW)/as(m), xt5mt

2/mW2 , M j (G j ) are the masse

~widths! of intermediate states,uvu<0.15 describing thenonfactorizable contributions tob→sg decay at q250@14,15#, and the factorskj are phenomenological parametefor compensating the approximations of the FA and VMand reproducing the correct branching ratios ofB(Lb→LJ/c→L l 1l 2)5B(Lb→LJ/c)3B(J/c→ l 1l 2) whenwe study theLb decays. We note that by takingkc.21/(3C11C2) andB(Lb→LJ/c)5(4.762.8)31024, thekj factors in theLb case are almost the same as that inB-meson one@5#. The Wilson coefficients~WCs! at the scaleof m;mb;4.8 GeV are shown in Table I.

Using the form factors given in Appendix A, we write thamplitude ofLb→L l 1l 2 as

M~Lb→L l 1l 2!5GFaem

A2pVtbVts* $H1

mLm1H2mLm

5 %, ~7!

where

07400

e

e

H1m5Lgm~A1PR1B1PL!Lb1L ismnqn~A2PR1B2PL!Lb ,

H2m5Lgm~D1PR1E1PL!Lb1L ismnqn~D2PR1E2PL!Lb

1qmL~D3PR1E3PL!Lb , ~8!

Lm5 l gml ,

Lm5 5 l gmg5l ~9!

with

Ai5C9eff f i2gi

22

2mb

q2C7

efff i

T 1giT

2,

Bi5C9eff f i1gi

22

2mb

q2C7

efff i

T2giT

2,

Di5C10

f i2gi

2,

Ei5C10

f i1gi

2, ~10!

and i 51,2,3.The processes for the heavy to light baryonic decays s

as those withLb→L have been studied based on the heaquark effective theory~HQET! in Ref. @17# and it is foundthat

^L~pL!usGbuLb~pLb!&5uL„F1~q2!1v”F2~q2!…GuLb

,~11!

whereG denotes the Dirac matrix,v5pLb/MLb

is the four-

velocity of Lb , q5pLb2pL is the momentum transfer, an

F1,2 are the form factors. Clearly, there are only two indpendent form factorsF1,2 in the HQET. Comparing with thegeneral forms of the form factors in Appendix A, we get trelations among the form factors as follows:

g15 f 15 f 2T5g2

T5F11ArF 2 ,

g25 f 25g35 f 35gTV5 f T

V5F2

MLb

,

gTS5 f T

S50,

TABLE I. Wilson coefficients for mt5170 GeV, m54.8 GeV.

WC C1 C2 C3 C4 C5

20.226 1.096 0.01 20.024 0.007

WC C6 C7 C8 C9 C10

20.028 20.305 20.15 4.186 24.559

1-2

m

lereonhin-

ngith

s ofe

-B.

BARYONIC RARE DECAYS OFLb→L l 1l 2 PHYSICAL REVIEW D 64 074001

g1T5 f 1

T5F2

MLb

q2,

g3T5

F2

MLb

~MLb1ML!,

f 3T52

F2

MLb

~MLb2ML!, ~12!

where r 5ML2 /MLb

2 . From the CLEO result ofR520.25

60.1460.08 @18#, we know thatuF2u,uF1u. Because of Eq.~12!, only f 1(g1) and f 2

T(g2T) are proportional toF1 and

therefore, they are large, whereas all the others are ssince they are related to the small form factorF2. Further-more, from Eq.~10!, we find that$ f T% and $gT% are associ-ated withC7 which is about one order of magnitude smalthanC9 andC10 so that their effects on the deviation of thresults in the HQET are small. Hence, with the informatiof the HQET, we can make a good approximation for tgeneral form factors of transition matrix elements givenEqs. ~7! and ~10!. Altogether, we have the following relations:

f[f 11g1

2,

f 2T1g2

T

f 11g1.1,

f 12g1

f 11g1.d,

g2

f 2.

g1T

f 1T

.g2

T

f 2T

.1,

f 1T1g1

T

f 11g1

1

q2.

f 21g2

f 11g1. ~13!

07400

all

e

In the HQET, it is easy to show that

d50, r[MLbS f 21g2

f 11g1D5

F2

F11ArF 2

. ~14!

III. DIFFERENTIAL DECAY RATES

In this section we first present the formulas by includithe lepton mass for the double differential decay rates wrespect to the angle of the lepton and the invariant masthe dilepton. In the following we only show the results of thSM with the form factors in Eq.~13!. The general ones including right-handed couplings are presented in Appendix

Introducing dimensionless variables oft5pLbpL /MLb

2 ,

r 5ML2 /MLb

2 , ml5ml /MLb, mb5mb /MLb

, and s

5q2/MLb

2 , the double partial differential decay rates forLb

→L l 1l 2 ( l 5e,m,t) can be written as

d2G

ds dz5

GF2aem

2 l t2

768p5MLb

5 Af~s!A124ml

2

q2f 2RLb

~s,z!,

~15!

where

RLb~s,z!5I 0~s,z!1 zI 1~s,z!1 z2I 2~s,z! ~16!

and

I 0~s,z!526ArsH 22mbrS 112ml

2

q2 D ReC9effC7

eff* 1dF S 112ml

2

q2 D uC9effu21S 126

ml2

q2 D uC10u2G J1

3

4@~12r !22s2#@~2mbr!2uC7

effu21uC9effu21uC10u2#16ml

2t@~2mbr!2uC7effu21uC9

effu22uC10u2#

16Ar ~12t !H 4S 112ml

2

q2 D mb2ruC7

effu21rsF S 112ml

2

q2 D uC9effu21S 122

ml2

q2 D uC10u2G J112S 112

ml2

q2 D mb~ t2r !~11sr2!ReC9effC7

eff* 112S 112ml

2

q2 D mbArsr ReC9effC7

eff*

26H s~12t !~ t2r !21

8@~12r !22s2#J F4mb

2

suC7

effu21sr2~ uC9effu21uC10u2!G

26ml2@2r 2~11r !t#F S 2mb

sD 2

uC7effu21r2~ uC9

effu22uC10u2!G , ~17!

I 1~s,z!53A124ml

2

q2f~s!$s@122Arr2~12r !r2#ReC9

effC10* 12mb~12sr2!ReC7effC10* %, ~18!

1-3


I 2~s,z!523

4f~s!S 124

ml2

q2 D @~2mbr!2uC7effu21uC9

effu21uC10u2#

13

4f~s!S 124

ml2

q2 D F4mb2

suC7

effU21sr2~ uC9effu21uC10u2!G , ~19!

-

e

cn

ab

an

with z5 pB• pl 1 being the angle between the momenta ofLband l 1 in the dilepton invariant mass frame andf(s)5(12r )222s(11r )1s2. Here, for simplicity, we have not displayed the dependence of them scale in effective Wilsoncoefficients. We note that the main nonperturbative QCDfect from f has been factored out in Eq.~15!. The functionRLb

(s,z) is only related to the two parameters ofd and r

which become one in the HQET. Sincer is the ratio of formfactors and insensitive to the QCD models, the QCD effein the baryonic dilepton decays are clearly less significaTherefore, these decay modes are good physical observto test the SM.

After integrating the angular dependence, the invarimass distributions as a function ofs are given by

07400

f-

tst.les

t

dG~Lb→L l 1l 2!

ds

5GF

2aem2 l t

2

384p5MLb

5 Af~s!A124ml

2

q2f 2RLb

~s!, ~20!

where

RLb~s!5G1~s!1G2~s!1G3~s! ~21!

with

G1~s!526ArsH 22mbrS 112ml

2

q2 D ReC9effC7

eff* 1dF S 112ml

2

q2 D uC9effu21S 126

ml2

q2 D uC10u2G J 1F22r S 112ml

2

q2 D24t2S 12

ml2

q2 D 13~11r !tG @~2mbr!2uC7effu21uC9

effu21uC10u2#16ml2t@~2mbr!2uC7

effu21uC9effu22uC10u2#, ~22!

G2~s!56Ar ~12t !H 4S 112ml

2

q2 D mb2ruC7

effu21rsF S 112ml

2

q2 D uC9effu21S 122

ml2

q2 D uC10u2G J112S 112

ml2

q2 D mb~ t2r !~11sr2!ReC9effC7

eff* , ~23!

G3~s!512S 112ml

2

q2 D mbArsr ReC9effC7

eff* 2F2t2S 112ml

2

q2 D 14r S 12ml

2

q2 D 23~11r !tG3F4mb

2

suC7u21sr2~ uC9

effu21uC10u2!G26ml2„2r 2~11r !t…F S 2mb

sD 2

uC7effu21r2~ uC9

effu22uC10u2!G . ~24!

m-

The limits for s are given by

4ml2<s<~12Ar !2. ~25!

From Eqs.~22!–~24!, we see thatr appears either asArr orr2 which is small sincer;0.04 anduru;0.25.

IV. LEPTON AND CP ASYMMETRIES

A. Forward-backward asymmetries

The differential and normalized forward-backward asymetries~FBAs! for the decays ofLb→L l 1l 2 as a functionof s are defined by

1-4

n

ian-cethrv

to

th

rs

aton

tctedicle

i-


dAFB~s!

ds5F E

0

1

dzd2G~s,z!

ds dz2E

21

0

dzd2G~s,z!

ds dzG ~26!

and

AFB~s!51

dG~s!/dsF E0

1

dzd2G~s,z!

ds dz2E

21

0

dzd2G~s,z!

ds dzG ,

~27!

respectively. Explicitly, using Eq.~15!, we obtain

dAFB~s!

ds5

GF2aem

2 l t2

28p5MLb

5 f~s!S 124ml

2

sD f 2RFB~s!

~28!

and

AFB~s!53

2Af~s!A12

4ml2

s

RFB~s!

RLb~s!

~29!

where

RFB~s!5s@122Arr2~12r !r2#ReC9effC10*

12mb~12sr2!ReC7effC10* . ~30!

It is known that the FBA is a parity-odd butCP-even ob-servable, which depends on the chirality of the leptonic ahadronic currents. In order to obtain one power ofz depen-dence, the related differential decay rate should be assocwith Tr LmLn

5 . This explains why the FBAs depend oReC9

effC10* and ReC7effC10* . However, unlike that in the de

cays ofB→Kl 1l 2 where the FBAs are always zero sinthey only involve vector and tensor types of currents,transition matrix elements in the baryonic decays presethe chirality of free quark interaction.

Similar to theB-meson decays@3,19# the FBA in Eq.~29!vanishes ats0 which satisfies the relation

ReC9effC10* 52

2mb

s0

12s0r2

122Arr2~12r !r2ReC7

effC10* .

~31!

We will see later that the vanishing point is only sensitivethe effects of weak interaction.

B. Lepton polarization asymmetries

To display the spin effects of the lepton, we choosefour-spin vector ofl 1 in terms of a unit vector,j, along thespin of l 1 in its rest frame, as

s10 5

pW 1• j

ml, sW15 j1

s10

El 11ml

pW 1 , ~32!

and the unit vectors along the longitudinal and transvecomponents of thel 1 polarization to be

07400

d

ted

ee

e

e

eL5pW 1

upW 1u,

eT5pW L3pW 1

upW L3pW 1u,

eN5eL3eT , ~33!

respectively.Defining the longitudinal and transversel 1 polarization

asymmetries by

Pi~ s!5dG~ ei j51!2dG~ ei j521!

dG~ ei j51!1dG~ ei j521!, ~34!

with i 5L andT, we find that

PL5A124ml

2

q2

RL~s!

RLb~s!

, ~35!

PT53

4pmlA12

4ml2

q2Asf~s!

RT~s!

RLb~s!

, ~36!

where

RL52ReC9effC10* $~12r !21s~11r !22s2

16Arrs~12r 1s!1r2s@2~12r !22s~11r !2s2#%

26mb ReC7effC10* @~12r 2s!~11r2s!14Arrs#,

~37!

RT5@122Arr2r2~12r !#Im C9effC10*

12mb

s~12r2s!Im C7

effC10* . ~38!

Here we do not discuss the normal polarization (PN) becausethe nonperturbative effects from the form factors are largethe smalls region and moreover, the dependence of Wilscoefficients is similar to the invariant mass distribution@6#.We note that the longitudinal lepton polarization ofPL in Eq.~35! is also a parity-odd andCP-even observable just likethe FBA, whereasPT in Eq. ~36! a T-odd one which is re-lated to the triple correlation ofsW1•(pW L3pW 1). In general,PT can be induced withoutCP violation as the cases inB-meson@20# and kaon@21# decays. However, we expecthat they are small. Moreover, such effects can be extraaway while we consider the difference between the partand antiparticle as discussed in the next section.

C. CP asymmetries

In this subsection, we define the following interesting drect CP asymmetries~CPAs! by

1-5

thoner-

s

oue

from. We

:


DG5dG2dG

dG1dG, ~39!

DFB5dGFB2dGFB

dG1dG, ~40!

DPi5

dG~jW•eW i !2dG~jW•eW i !

dG1dG, i 5L,T, ~41!

where we have useddG1dG as the normalization. Theabove four CPAs areCP-odd quantities and they areCPviolating observables. ForDG,FB,PL

in Eqs.~39!–~41!, to dis-play the difference of the physical observable betweenparticle and antiparticle, it is necessary to have the strand weak phases simultaneously in the processes. In thcays ofb→sl1l 2 ( l 5e,m,t), the strong phases are geneated by the absorptive parts of one-loop matrix elementoperatorsO1;O6 and LD contributions. However, sincePTis aT-odd observable and only related to the imaginary cplings, even without strong phases, we still can have nonzvalues ofDT(Lb→L l 1l 2). For b→sl1l 2 and b→ sl 2l 1

decays the Wilson coefficientsC9eff(m) and C7

eff(m) in Eqs.~3! and ~4! can be rewritten as

C9eff~m!5C9

0~m!1 iC9abs~m!,

C9eff~m!5C9

0* ~m!1 iC9abs~m!,

07400

eg

de-

in

-ro

C7eff~m!5C7

0~m!1 iC7abs~m!,,

C7eff~m!5C7

0* ~m!1 iC7abs~m!, ~42!

with

C90~m!5C9~m!1ReY~z,s8!,

C70~m!5C7~m!1ReC78~m,q2!,

C9abs~m!5Im Y~z,s8!,

C7abs~m!5Im C78~m,q2!, ~43!

where we have assumed that the strong phases are allthe SM and there are no weak phases in absorptive partsnote that there is no strong phase inC10.

According to Eqs.~20!, ~29!, ~35!, and ~36!, the CPasymmetries are all related to the following combinations

ReC9effC7

eff* 2ReC9effC7

eff* 52C9absIm C7

012C7absIm C9

0 ,

ReC7,9effC10* 2ReC7,9

eff C10* 52C7,9absIm C10,

Im C7,9effC10* 2Im7,9

eff C10* 52 ImC7,90 C10* 12C7,9

absIm C10,

uC7,9eff u22uC7,9

eff u254C7,9absIm C7,9

0 . ~44!

Explicitly, the CP asymmetries in Eqs.~39!, ~40!, and ~41!are found to be

DG52

RLb~s! H 6mbS 112

ml2

q2 D @2Arrs1~ t2r !~11sr2!#@C9absIm C7

01C7abs Im C9

0#

1F22r S 112ml

2

q2 D 24t2S 12ml

2

q2 D 13~11r !t16ml2tG @4mb

2r2C7absIm C7

01C9abs Im C9

0#

1F22t2S 112ml

2

q2 D 24r S 12ml

2

q2 D 13~11r !t26ml

2

s~2r 2t2tr !G F4mb

2

sC7

absIm C701sr2C9

abs Im C90G

16Arr~12t !S 112ml

2

q2 D @4mb2C7

absIm C701sC9

absIm C90#26ArsdS 112

ml2

q2 D C9absIm C9

0J , ~45!

DFB53

2RLb~s!A12

4ml2

q2f~s!Im C10$s@122Arr2~12r !r2#C9

abs12mb~12sr2!C7abs%, ~46!

DPL52

1

RLb~s!A12

4ml2

q2Im C10$C9

abs@6Arrs~12r 1s!1~112sr2!~12r !21s~12sr2!~11r !2s2~21sr2!#

16mbC7abs@~12r 2s!~11sr2!14Arrs#%, ~47!

1-6


DPT5

3pml

4RLb~s!A12

4ml2

q2Asf~s!H ~ Im C9

0C10* 1C9absIm C10!@122Arr2~s12t22r !r2#

12mb

s~12sr2!~ Im C7

0C10* 1C7absIm C10!J . ~48!

rere

fo

th

re

o

ti

eas

et-

we

nd

ults

tRshe

-

As seen from the above equations,DG is related to ImC7 andIm C9, while DFB , DPL

, andDPTdepend on ImC10. More-

over, for small values ofC9absIm C10 and C7

absIm C10, DPT

would still be sizable because ImC90C10* or ImC7

0C10* wouldbe large.

V. NUMERICAL ANALYSIS

In our numerical calculations, the Wilson coefficients aevaluated at the scalem.mb and the other parameters alisted in Table I of Ref.@5#. From Eq.~13!, we know that themain effects on the deviation of the HQET are fromd. Byusing a proper nonzero value ofd, we will see later that thedeviations of the decay branching ratios ofLb→L l 1l 2 areonly a few percent. Since there is no complete calculationthe form factors of theLb→L transition in the literature, weuse the form factors derived from QCD sum rule underassumption of the HQET, given by

Fi~q2!5Fi~0!

11aq21bq4, ~49!

with the parameters shown in Table I of Ref.@6#. In order toillustrate the contributions of new physics, we adopt thesults of the generic supersymmetric extension of the SM@22#in which

C7SUSY521.75~d23

u !LL20.25~d23u !LR210.3~d23

d !LR ,

C9SUSY50.82~d23

u !LR ,

C10SUSY529.37~d23

u !LR11.4~d23u !LR~d33

u !RL

12.7~d23u !LL , ~50!

and take the following values instead of scanning the whallowed parameter space:

~d23u !LL;0.1,

~d33u !RL;0.65,

~d23d !LR;331022ei (2p/5),

~d23u !LR;20.8ei (p/4), ~51!

where (d i jq )AB ( i , j 51,2,3 andA,B5L,R) denote the pa-

rameters in the mass insertion method, which describeeffects of the flavor violation. The set of the parametersEq. ~51! satisfies with the constraint fromB→Xsg on C7

07400

r

e

-

le

hen

5C7SM1C7

SUSY @22#. Hence, the numerical values of thSUSY contributions to the relevant Wilson coefficients arefollows:

ReC7SUSY.0.06, ImC7

SUSY.20.29,

ReC9SUSY.20.46, ImC9

SUSY.20.46,

ReC10SUSY.4.78, ImC10

SUSY.4.50. ~52!

We note that the contributions of the minimal supersymmric standard model~MSSM! to b→sl1l 2 can be found inRefs.@23# and @24#.

To show the typical values of various asymmetries,define the integrated quantities as

Q5Esmin

smaxQ~s!ds, ~53!

whereQ denotes the physical observables withsmin54ml andsmax5(12Ar )2.

A. Decay rates and invariant mass distributions

We now discuss the influences ofd, r, and v on thebranching ratios~BRs! of Lb→L l 1l 2 decays in detail. Theeffects ofkj for compensating the assumption of the FA aVMD have been analyzed in Ref.@5#. In Table II, we showthe BRs by choosing different sets of parameters. Our resare given as follows.

~1! By taking udu50.05 which means 10% away from thain to the HQET, we clearly see that the deviations of the Bare only 4–6 %. It is a good approximation to neglect texplicit d term in Eqs.~17!, ~22!, and ~45!. Hence, f 5( f 11g1)/2, which also owns thed effect, is the main nonperturbative part.

TABLE II. BRs ~in the unit of 1026) for various parameterswith v50 and neglecting LD effects.

Parameter Lb→Le1e2 Lb→Lm1m2 Lb→Lt1t2

HQET 2.23 2.08 1.7931021

d50.05 2.36 2.21 1.8631021

d520.05 2.09 1.96 1.7131021

r50,d50 2.52 2.38 2.6631021

C750,d50 2.36 2.34 2.2331021

C752C7SM,d50 3.34 3.19 2.7631021

1-7

,d

th

is-atve

of

ot

the

n

a-

sen-ive

ma-esd-

epe


~2! If r50, the effects are about 10% fore andm modesbut 48% for thet one.

~3! If one neglects the contribution fromC7, the influ-ences onB(Lb→L l 1l 2) for e, m, andt modes are about 512, and 24 %, respectively. However, taking the magnituof C7 is the same as the SM but with an opposite sign;deviations are all over 50%.

The contributions of the parameterv to the BRs ofLb→L l 1l 2 are listed in Table III and the invariant mass dtributions are shown in Fig. 1. From Table III, it is clear ththe nonfactorizable effects are small on the BRs. Howethose directly related tov effects such asPT andCP asym-

TABLE III. BRs ~in the unit of 1026) without LD effects withdifferent values ofv.

Mode Lb→Le1e2 Lb→Lm1m2 Lb→Lt1t2

v50.15 2.24 2.12 1.8931021

v50.0 2.23 2.08 1.7931021

v520.15 2.25 2.06 1.7131021

FIG. 1. BRs as a function ofq2/MLb

2 for ~a! Lb→Lm1m2 and~b! Lb→Lt1t2. The curves with and without resonant shapes rresent including and no LD contributions, respectively. The dashsolid, and dash-dotted curves stand forv50.15, 0, and20.15,respectively.

07400

ee

r,

metries will have large influences.As for the new physics contributions, using the values

the SUSY model in Eq.~52!, we show the results in TableIV. Although the deviations of the BRs to the SM are nsignificant, they have a large effect on the lepton andCPasymmetries which will be shown next.

B. Forward-backward and lepton polarization asymmetries

From Eq.~14! andR5F2 /F1.20.25 in the HQET, wehave thatr.20.26. We note thatr is defined by the ratio ofthe form factors and it is expected to be insensitive toQCD models. Withsmax.0.64, we obtainsmaxr

2.0.04, (12r )r2.0.06, and 2Arr.0.2. Using these values, one casimplify Eqs.~30! and ~31! to

RFB~s!.s~122Arr!ReC9effC10* 12mb ReC7

effC10* ~54!

and

ReC9effC10* .2

2mb

s0~122Arr!ReC7

effC10* , ~55!

respectively. It is easy to see thats0 is only sensitive to theWilson coefficients. The result is similar to the case inB→K* l 1l 2 decay @3,19# where the approximation of thelarge energy effective theory~LEET! @25# is used. As for thelepton polarization asymmetries, with the same approximtion, Eqs.~37! and ~38! can also be reduced to

RL.2ReC9effC10* @11s22s216Arrs~11s!#

26mb ReC7effC10* @12s14Arrs#, ~56!

RT.~122Arr!Im C9effC10* 1

2mb

sIm C7

effC10* ,

~57!

respectively. Hence, the lepton asymmetries are all moresitive to the Wilson coefficients than the nonperturbatQCD effects.

It is worth mentioning that the effects ofv, introduced forthe LD contributions tob→sg and absorbed toC7

eff , willchange ReC7

eff in the SM such thats0 is also shifted. There-fore, in terms ofs0, we can also theoretically determinev bycomparing the result with that ofv50. Another interestingquantity is aT-odd observable ofPT which is proportional toC10 Im C7

eff in the SM. Because of the enhancement ofC10, anonzero value ofv will modify PT enormously. As for theother asymmetries, the effects are insignificant. The estitions of integrated lepton asymmetries with different valuof v in the SM are displayed in Table V and the corresponing distributions are shown in Figs. 2–4.

-d,

TABLE IV. BRs ~in unit of 1026) in the generic SUSY model.

Model Lb→Le1e2 Lb→Lm1m2 Lb→Lt1t2

SUSY 2.47 2.24 1.7931021

1-8

to

Sres.

n

t

est

i-ofced

.ence

e

-

D

on


To illustrate the new physics effects, the integrated lepasymmetries in the generic SUSY model withv are listed inTable VI and their distributions as a function ofq2/MLb

areshown in Figs. 5–7. From the figures, we see that SUeffects make the shapes of lepton asymmetries quite diffefrom that in the SM. We summarize the results as follow

~1! Since the SD contributions toC9C10* and ReC7C10* are21.40 and21.35, respectively, which violate the conditioin Eq. ~55!, the vanishing point is removed.

~2! Due to the factor ofmb /s, from Fig. 7, we see thaIm C7

effC10* has a large effect onPT in the smalls region.

TABLE V. Integrated lepton asymmetries in the SM without Leffects.

Parameter Mode 102AFB 102PL 102PT

v50.15 Lb→Lm1m2 214.37 59.50 0.11Lb→Lt1t2 23.98 10.70 0.53

v50.0 Lb→Lm1m2 213.38 58.30 0.07Lb→Lt1t2 23.99 10.84 0.39

v520.15 Lb→Lm1m2 212.24 56.70 0.04Lb→Lt1t2 24.00 10.94 0.23

FIG. 2. Same as Fig. 1 but for the FBAs.

07400

n

Ynt

~3! In the SUSY model,PT could reach 1 and 10 % for thlight lepton andt modes, which are only 0.2 and 3 % at moin the SM, respectively.

C. CP asymmetries

In the SM, forb→sl1l 2, the relevant CKM matrix ele-ment isVtbVts* which is real under Wolfenstein’s parametrzation. Nonzero CPAs will indicate clearly the existencenew physics. We remark that the CPAs can be in fact induby the complex CKM matrix elementVubVus* which is alsothe source of the direct CPA inB→Xsg in the SM. However,we expect that such effects to the CPAs inb→sl1l 2 aresmaller than that inB→Xsg where the CPA is less than 1%The main reason for the smallness is because of the presof C9 andC10 contributions to the rates ofb→sl1l 2, whichare absent inB→Xsg.

With the values in Eq.~52!, the averaged CPAs in thgeneric SUSY model forLb→L l 1l 2 are listed in Table VIIand their distributions as a function ofs5q2/MLb

2 are shown

in Figs. 8–11. The results are given as follows.~1! From Eqs.~45! and~48!, we see that the terms corre

sponding toC7absIm C7

0 and ImC70C10* 1C7

absIm C10 are as-

FIG. 3. Same as Fig. 1 but for the longitudinal polarizatiasymmetries.

1-9

r

o

wtr

o

e

the

t of

s

nic

therd,

ym

Y

of

.


sociated with a factor ofmb /s. If sizable imaginary partsexist, in the smalls region the distributions will be signifi-cant. For this reason, in Fig. 8~a! one finds thatDG(s) forLb→L l 1l 2 ( l 5e,m) increase ass decreases. On the othehand, if the term withmb /s in Eq. ~48! is dropped, the dis-tributions ofDPT

(s) for e andm modes do not contain zervalue. We note that with the values in Eq.~52!, the maineffect onDG(s) in the smalls region is fromCb→sg8 .

~2! DPL(s) for all lepton channels andDPT

(s) for the t

one could be over 10%, while the remainingCP asymme-tries are at the level of a few percent. We remark that ifcan scan all the allowed SUSY parameters, the asymme

FIG. 4. Same as Fig. 1 but for the transverse polarization asmetries.

TABLE VI. Integrated lepton asymmetries in the generic SUSmodel withv50.

Mode 102AFB 102PL 102PT

Lb→Lm1m2 210.53 24.46 20.57Lb→Lt1t2 21.84 4.40 22.51

07400

eies

exceptDPT(s) for lighter lepton modes would reach up t

10%.~3! It is known thatDPT

(s) is aT-odd observable and th

other CPAs belong to the directCP violation which needsabsorptive parts in the processes. This is the reason whydistributions ofDG(s), DFBA(s), andDPL

(s) around the REregion have the similar shapes but are different from thaDPT

(s). Moreover, all the direct CPAs are sensitive tov

unlike the cases of theCP conserving lepton asymmetriediscussed in Sec. V B.

VI. CONCLUSIONS

We have done a systematic study on the rare baryodecays ofLb→L l 1l 2 ( l 5e,m,t). For theLb→L transi-tion, we have related all the form factors withF1 andF2, andwe have found thatd50 andr.R[F2 /F1, in the limit ofthe HQET. Inspired by the HQEF, we have presenteddifferential decay rates and the dilepton forward-backwalepton polarization and four possibleCP violating asymme-

- FIG. 5. FBAs in the generic SUSY model as a functionq2/MLb

2 for ~a! Lb→Lm1m2 and~b! Lb→Lt1t2. The solid anddashed curves stand for the SM and SUSY model, respectively

1-10

e

av

ymseh

ce-

cC

the

fol-

on ym-

or


tries in terms of the parametersf , d, andr. We have shownthat the nonfactorizable effects for the BRs andCP-evenlepton asymmetries are small but large forPT and the directCPAs. We have also demonstrated that most of the obsables such asAFB , PL,T , andDa (a5G,FB,PL , andPT)are insensitive to the non-perturbative QCD effects. We hillustrated our results in the specificCP violating SUSYmodel. We have found that all the directCP violating asym-metries are in the level of 1–10 %. To measure these asmetries at thens level, for example, in the tau mode, at lea0.5n23(10921010) Lb decays are required. It could bdone in the second generationB-physics experiments, sucas the CERN Large Hadron Collider~LHCb!, ATLAS, andCMS at the LHC, and BTeV at the Tevatron, which produ;1012 bb pairs per year@26#. Finally we remark that measuring these CPAs at a level of 1022 is a clear indication ofa newCP violation mechanism beyond the SM.

ACKNOWLEDGMENTS

This work was supported in part by the National ScienCouncil of the Republic of China under Contract Nos. NS

FIG. 6. Same as Fig. 5 but for the longitudinal polarizatiasymmetries.

07400

rv-

e

-t

e-

89-2112-M-007-054 and NSC-89-2112-M-006-033 andNational Center for Theoretical Science.

APPENDIX A: FORM FACTORS ANDDECAY AMPLITUDES

For the exclusive decays involvingLb(pLb)→L(pL), the

transition form factors can be parametrized generally aslows:

FIG. 7. Same as Fig. 5 but for the transverse polarization asmetries.

TABLE VII. CP asymmetries in the generic SUSY model fdifferent values ofv.

Parameter Mode 102DG 102DFBA 102DPL102DPT

v50.15 Lb→Lm1m2 2.05 22.62 6.48 20.53Lb→Lt1t2 1.83 20.79 1.94 22.01

v50.0 Lb→Lm1m2 1.59 21.89 5.00 20.47Lb→Lt1t2 1.38 20.59 1.53 22.21

v520.15 Lb→Lm1m2 1.05 21.06 3.34 20.40Lb→Lt1t2 0.89 20.37 1.06 22.41

1-11

-


^LusgmbuLb&5 f 1uLgmuLb1 f 2uLismnqnuLb

1 f 3qmuLuLb, ~A1!

^Lusgmg5buLb&5g1uLgmg5uLb1g2uLismnqng5uLb

1g3qmuLg5uLb, ~A2!

^LusismnbuLb&5 f TuLismnuLb1 f T

VuL~gmqn

2gnqm!uLb1 f T

S~Pmqn

2Pnqm!uLuLb, ~A3!

^Lusismng5buLb&5gTuLismng5uLb

1gTVuL~gmqn2gnqm!g5uLb

1gTS~Pmqn2Pnqm!uLg5uLb

,

~A4!

FIG. 8. Same as Fig. 5 but forDG .

07400

whereP5pLb1pL , q5pLb

2pL and form factors,$ f i% and

$gi%, are all functions ofq2. Using the equations of the motion, we have

~ML1MLb!uLgmuLb

5~pLb1pL!muLuLb

1 i uLsmnqnuLb, ~A5!

~ML2MLb!uLgmg5uLb

5~pLb1pL!muLg5uLb

1 i uLsmnqng5uLb. ~A6!

The form factors for dipole operators are derived as

^LusismnqnbuLb&5 f 1TuLgmuLb

1 f 2TuLismnqnuLb

1 f 3TqmuLuLb

, ~A7!

FIG. 9. Same as Fig. 5 but forDFBA .

1-12

hea

pli-

nly.


^Lusismnqng5buLb&5g1TuLgmg5uLb

1g2TuLismnqng5uLb

1g3TqmuLg5uLb

, ~A8!

with

f 2T5 f T2 f T

Sq2,

f 1T5@ f T

V1 f TS~ML1MLb

!#q2,

f 1T52

q2

~MLb2ML!

f 3T ,

g2T5gT2gT

Sq2,

g1T5@gT

V1gTS~ML2MLb

!#q2,

g1T5

q2

~MLb1ML!

g3T . ~A9!

FIG. 10. Same as Fig. 5 but forDPL.

07400

We now give the most general formulas by including tright-handed coupling in the effective Hamiltonian withcomplete set of form factors. The free quark decay amtudes forb→sl1l 2 are given by

H~b→sl1l 2!5GFaem

A2pVtbVts* F sgm~C9

LPL1C9RPR!b lgml

1 sgm~C10L PL1C10

R PR!b lgmg5l

22mb

q2sismnqn~C7

LPR1C7RPL!b lgml G ,

~A10!

where CiL and Ci

R ( i 57,9,10) denote the effective Wilsocoefficients of left- and right-handed couplings, respectiveWith the most general form factors in Eqs.~A1!, ~A2!, ~A7!,and ~A8!, and the effective Hamiltonian in Eq.~A10!, thetransition matrix elements for the decays ofLb→L l 1l 2 areexpressed as

FIG. 11. Same as Fig. 5 but forDPT.

1-13


M~Lb→L l 1l 2!5GFaem

A2VtbVts* $@Lgm~A1PR1B1PL!Lb

1L ismnqn~A2PR1B2PL!Lb] l gml

1@Lgm~D1PR1E1PL!Lb

1L ismnqn~D2PR1E2PL!Lb

1qmL~D3PR1E3PL!Lb# l gmg5l %,

~A11!where

Ai5C9R f i1gi

22

2mb

q2C7

Rf i

T2giT

21C9

L f i2gi

2

22mb

q2C7

Lf i

T 1giT

2,

Bi5C9L f i1gi

22

2mb

q2C7

Lf i

T2giT

21C9

R f i2gi

2

22mb

q2C7

Rf i

T 1giT

2,

07400

Di5C10R f i1gi

21C10

L f i2gi

2,

Ei5C10L f i1gi

21C10

R f i2gi

2. ~A12!

APPENDIX B: DIFFERENTIAL DECAY RATES

Using the transition matrix elements in Eq.~A11!, thedouble differential decay rates can be derived as

dG

dsdz5

GF2aem

2 l t2

768p5MLb

5 Af~s!A124ml

2

q2f 2RLb

~s,z!,

~B1!

where

RLb~s,z!5I 0~s,z!1 zI 1~s,z!1 z2I 2~s,z! ~B2!

with

I 0~s,z!526ArsF S 112ml

2

q2 D ReA1B1* 1S 126m1

2

q2 D ReD1E1* G13

4@~12r !22s2#~ uA1u21uB1u21uD1u21uE1u2!

16ml2t~ uA1u21uB1u22uD1u22uE1u2!112ml

2MLbAr ~12t !~ReD1D38* 1ReE1E38* !

112ml2MLb

~ t2r !~ReD1E38* 1ReD3E18* !

16MLbArs~12t !F S 112

ml2

q2 D ~ReA1A2* 1ReB1B2* !1S 122ml

2

q2 D ~ReD1D2* 1ReE1E2* !G26MLb

s~ t2r !F S 112ml

2

q2 D ~ReA1B2* 1ReA2B1* !1S 126ml

2

q2 D ~ReD1E2* 1ReD2E1* !G26MLb

2 Ars2S 112ml

2

q2 D ReA2B2* 26ML b

2 Ars2 S 126ml

2

q2 D ReD2E2*

26MLb

2 H s~12t !~ t2r !21

8@~12r !22s2#J ~ uA2u21uB2u21uD2u21uE2u2!26MLb

2 ml2@2r 2~11r !t#~ uA2u21uB2u2

2uD2u22uE2u2!112ml2MLb

2 st~ReD2D38* 1ReE2E38* !112ml2MLb

2 Ars~ReD2E38* 1ReD38* E2!,

I 1~s,z!53sf~s!$2~ReA1D1* 2ReB1E1* !1MLb@Ar ~ReA1D2* 2ReB1E2* !1~ReA1E2* 2ReB1D2* !

1Ar ~ReA2D1* 2ReB2E1* !2~ReA2E1* 2ReB2D1* !#1MLb~12r !~ReA2D2* 2ReB2E2* !%,

I 2~s,z!523

4f~s!S 124

ml2

q2 D ~ uA1u21uB1u21uD1u21uE1u2!13

4MLb

2 f~s!S 124ml

2

q2 D ~ uA2u21uB2u21uD2u21uE2u2!,

~B3!

1-14

ns-he


where D385D32D2 and E385E32E2, and z5 pB• pl 1 de-notes the angle between the momentum ofLb and that ofl 1

in the dilepton invariant mass frame.

APPENDIX C: FORWARD-BACKWARD AND LEPTONASYMMETRIES

From Eq. ~A11!, the functions ofRLband RFB in the

differential and normalized FBAs forLb→L l 1l 2 in Eqs.~28! and ~29! are given by

RLb~s!5

1

2E21

1

dzRLb~s,z! ~C1!

G

od

v.

07400

and

RFB~s!52Re~A1D1* 2B1E1* !1MLb@Ar Re~A1D2* 2B1E2* !

1Re~A1E2* 2B1D2* !1Ar Re~A2D1* 2B2E1* !

2Re~A2E1* 2B2D1* !#

1MLb

2 ~12r !Re~A2D2* 2B2E2* !, ~C2!

respectively. We can also define the longitudinal and traverse lepton polarization asymmetries. Explicitly, by tdefinition of Eq.~34!, we get

PL521

RLb~s!A12

4ml2

s$„s~11r 2s!1~12r !22s2

…Re~A1D1* 1B1E1* !

1MLb

2 s@2s~11r 2s!12~12r !222s2#Re~A2D2* 1B2E2* !26sAr @Re~A1E1* 1B1D1* !

1MLb

2 s Re~A2E2* 1B2D2* !#13sML~12r 1s!@Re~A1D2* 1B1E2* !1Re~A2D1* 1B2E1* !#

23sMLb~12r 2s!@Re~A1E2* 1B1D2* !21e~A2E1* 1B2D1* !#%, ~C3!

PT53

4pmlA12

4ml2

q2Asf~s!

1

RLb~s!

$2Im~A1D1* 2B1E1* !1ML@ Im~A1D2* 2B1E2* !1Im~A2D1* 2B2E1* !#

1MLb@ Im~A1E2* 2B1D2* !2Im~A2E1* 2B2D1* !#1MLb

2 ~12r !Im~A2D2* 2B2E2* !%. ~C4!

v.

l.

l.

@1# N. Cabibbo, Phys. Rev. Lett.10, 531 ~1963!; M. Kobayashiand T. Maskawa, Prog. Theor. Phys.49, 652 ~1973!.

@2# CLEO Collaboration, M.S. Alamet al., Phys. Rev. Lett.74,2885 ~1995!.

@3# For a recent review, see A. Ali, P. Ball, L.T. Handoko, andHiller, Phys. Rev. D61, 074024~2000!.

@4# Chuan-Hung Chen and C.Q. Geng, Phys. Rev. D63, 054005~2001!.

@5# Chuan-Hung Chen and C.Q. Geng, Phys. Rev. D63, 114024~2001!.

@6# Chuan-Hung Chen and C.Q. Geng, hep-ph/0101201.@7# G. Buchalla, A.J. Buras, and M.E. Lautenbacher, Rev. M

Phys.68, 1230~1996!.@8# N.G. Deshpande, J. Trampetic, and K. Panose, Phys. Re

39, 1462~1989!.@9# C.S. Lim, T. Morozumi, and A.T. Sanda, Phys. Lett. B218,

343 ~1989!.@10# A. Ali, T. Mannel, and T. Morozumi, Phys. Lett. B273, 505

~1991!.@11# P.J. O’Donnell and K.K.K. Tung, Phys. Rev. D43, R2067

~1991!.@12# F. Kruger and L.M. Sehgal, Phys. Lett. B380, 199 ~1996!.

.

.

D

@13# C.Q. Geng and C.P. Kao, Phys. Rev. D54, 5636~1996!.@14# J.M. Soares, Phys. Rev. D53, 241 ~1996!.@15# D. Melikhov, N. Nikitin, and S. Simula, Phys. Lett. B430, 332

~1998!.@16# J.M. Soares, Nucl. Phys.B367, 575~1991!; H.A. Asatrian and

A.N. Ioannissian, Phys. Rev. D54, 5642~1996!.@17# T. Mannel, W. Roberts, and Z. Ryzak, Nucl. Phys.B355, 38

~1991!; T. Mannel and S. Recksiegel, J. Phys. G24, 979~1998!.

@18# CLEO Collaboration, G. Crawfordet al., Phys. Rev. Lett.75,624 ~1995!.

@19# G. Burdman, Phys. Rev. D57, 4254~1998!.@20# C.Q. Geng and C.P. Kao, Phys. Rev. D57, 4479~1998!.@21# P. Agrawal, J.N. Ng, G. Be´langer, and C.Q. Geng, Phys. Re

Lett. 67, 537 ~1991!; Phys. Rev. D45, 2383~1992!.@22# F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini, Nuc

Phys.B477, 321 ~1996!; E. Lunghi, A. Masiero, I. Scimemi,and L. Silvestrini,ibid. B568, 120 ~2000!.

@23# S. Bertolini, F. Borzumati, A. Masiero, and G. Ridolfi, Nuc

Phys. B353, 591 ~1991!; P. Cho, M. Misiak, and D. Wyler,Phys. Rev. D54, 3329~1996!.

1-15

,

n,


@24# A. Ali, G.F. Giudice, and T. Mannel, Z. Phys. C67, 417~1995!; J. Hewett and J.D. Wells, Phys. Rev. D55, 5549~1997!; T. Goto, Y. Okada, and Y. Shimizu,ibid. 58, 094006~1998!.

07400

@25# J. Charles, A.L. Yaouanc, L. Oliver, O. Pe`ne, and J.C. RaynalPhys. Rev. D60, 014001~1999!.

@26# N. Harnew, ‘‘Proceedings of Heavy Flavors 8,’’ Southampto1999.

1-16

Documents

Baryonic rare decays of