Direct CP violation in Λb→n(Λ)π+π- decays via ρ-ω mixing

PUBLISHED VERSION Guo, Xuhong; Thomas, Anthony William

58(9):096013

© 1998 American Physical Society

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15th April 2013

Direct CP violation in L

b

˜ n „ L … p

¹

p

²

decays via r - v mixing

X.-H. Guo*

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia

and Institute of High Energy Physics, Academia Sinica, Beijing 100039, China A. W. Thomas^†

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia

~Received 14 May 1998; published 7 October 1998!

We study direct C P violation in the bottom baryon decaysLb→fr⁰(v)→fp¹p² ( f5n orL). It is found that in these decays via r-v mixing the C P violation could be very large when the invariant mass of the p¹p²pair is in the vicinity of thevresonance. With a typical value N_c52 in the factorization approach, the maximum C P-violating asymmetries are more than 50% and 68% for Lb→np¹p² and Lb→Lp¹p², respectively. With the aid of heavy quark symmetry and phenomenological models for the hadronic wave functions of Lb, L and the neutron, we estimate the branching ratios of Lb→n(L)r⁰.

@S0556-2821~98!06419-4#

PACS number~s!: 11.30.Er, 12.15.Hh, 12.39.Hg, 13.20.He

I. INTRODUCTION

C P violation is still an open problem in the standard model, even though it has been known in the neutral kaon system for more than three decades @1#. The study of C P violation in other systems is important in order to understand whether the standard model provides a correct description of this phenomenon through the Cabibbo-Kobayashi-Maskawa

~CKM!matrix.

Recent studies of direct C P violation in the B meson system@2#have suggested that large C P-violating asymme- tries should be observed in forthcoming experiments. It is also interesting to study C P violation in the bottom baryon system in order to find the physical channels which may have large C P asymmetries, even though the branching ra- tios for such processes are usually smaller than those for the corresponding processes of bottom mesons. The study of C P violation in the bottom system will be helpful for under- standing the origin of C P violation and may provide useful information about the possible baryon asymmetry in our uni- verse. Actually, some data on the bottom baryon Lb have appeared just recently. For instance, OPAL has measured its lifetime and the production branching ratio for the inclusive semileptonic decayLb→Ll^{2¯ X}n @3#. Furthermore, measure- ments of the nonleptonic decay Lb→LJ/c have also been reported @4#. More and more data are certainly expected in the future. It is the purpose of the present paper to study the C P violation problem in the hadronic decaysLb→np¹p² andLb→Lp¹p^2.

The C P-violating asymmetries in the decays we are con- sidering arise from the nonzero phase in the CKM matrix, and hence we have the so-called direct C P violation which

occurs through the interference of two amplitudes with dif- ferent weak and strong phases. The weak phase difference is determined by the CKM matrix elements while the strong phase is usually difficult to control. In Refs. @5,6#, the au- thors studied direct C P violation in B hadronic decays through the interference of tree and penguin diagrams, where r^-v mixing was used to obtain a large strong phase~as re- quired for large C P violation!. The data for e¹e²→p¹p² in the r^-v interference region strongly constrains the r^-v mixing parameters. Gardner et al. established not only that the C P-violating asymmetry in B⁶→r⁶r⁰⁽v⁾→r⁶p¹p² is more than 20% when the invariant mass of thep¹p^2pair is near the v mass, but that the measurement of the C P-violating asymmetry for these decays can remove the mod(p) uncertainty in arg@2V_tdV_tb*/(V_udV_ub*)# @6#. In the present work we generalize these discussions to the bottom baryon case. It will be shown that the C P violation in Lb

hadronic decays could be very large.

In our discussions hadronic matrix elements for both tree and penguin diagrams are involved. These matrix elements are controlled by the effects of nonperturbative QCD which are difficult to handle. In order to extract the strong phases in our discussions we will use the factorization approximation so that one of the currents in the nonleptonic decay Hamil- tonian is factorized out and generates a meson. Thus the decay amplitude of the two body nonleptonic decay becomes the product of two matrix elements, one related to the decay constant of the factorized meson and the other to the weak transition matrix element between two hadrons.

There have been many discussions concerning the plausi- bility of the factorization approach. Since bottom hadrons are very heavy, their hadronic decays are energetic. Hence the quark pair generated by one current in the weak Hamiltonian moves very fast away from the weak interaction point.

Therefore, by the time this quark pair hadronizes into a me- son it is far away from other quarks and is therefore unlikely to interact with the remaining quarks. Hence this quark pair

*Electronic address: xhguo@physics.adelaide.edu.au

†Electronic address: athomas@physics.adelaide.edu.au

0556-2821/98/58~9!/096013~9!/$15.00 58 096013-1 ©1998 The American Physical Society

is factorized out and generates a meson. This argument is based on the idea of ‘‘color transparency’’ proposed by Bjorken @7#. Dugan and Grinstein proposed a formal proof for the factorization approach by constructing a large energy, effective theory @8#. They established that when the energy of the generated meson is very large the meson can be fac- torized out and the deviation from the factorization ampli- tude is suppressed by the energy of the factorized meson.

Furthermore, we will estimate the branching ratios for the decay modesLb→n(L)r⁰. In the factorization approach the decay rates for these processes are determined by the weak matrix elements between Lb and n(L). With the aid of heavy quark effective theory~HQET! @9#it is shown that in the heavy quark limit, m_b→`, there are two independent form factors. We will apply the model of Refs. @10,11# to determine these two form factors and hence predict the branching ratios forLb→n(L)r^0.

The remainder of this paper is organized as follows. In Sec. II we give the formalism for the C P-violating asymme- try inLb→fr⁰⁽v⁾→fp¹p^{2 ( f}5n orL) and calculate the strong phases in the factorization approach. Numerical re- sults will also be shown in this section. In Sec. III we apply the result of HQET and the model of Refs. @10,11#to esti- mate the branching ratios forLb→n(L)r⁰. The results from the nonrelativistic quark model @12# will also be presented for comparison. Finally, Sec. VI is reserved for a brief sum- mary and discussion.

II. CP VIOLATION INLb˜n„L…p¹p²DECAYS A. Formalism for CP violation inLb˜n„L…p¹p² The formalism for C P violation in B meson hadronic decays @5,6# can be generalized to Lb in a straightforward manner. The amplitude, A, for the decay Lb→fp¹p^{2 is}

A5^p¹p^2fuH^TuLb&1^p¹p^2fuH^PuLb&^, ~1!

where H^T and H^P are the Hamiltonians for the tree and penguin diagrams, respectively. Following Refs.@5,6#we de- fine the relative magnitude and phases between these two diagrams as follows:

A5^p¹p^2fuH^TuLb&@11re^ide^if#,

A¯5^p¹p^2f¯uH^TuL¯_b&@11re^ide^2if#, ~2!

whered ^andf are strong and weak phases, respectively.f is caused by the phase in the CKM matrix, and if the top quark dominates penguin diagram contributions it is arg@V_tbV_td*/(V_ubV_ud*)# for b→d and arg@V_tbV_ts*/(V_ubV_us*)# for b→s. The parameter r is the absolute value of the ratio of tree and penguin amplitudes:

r[

U

^{^}^^pp^11pp^22ffuH^uHPTuLuL^bb^&&

U

^{. ~3!}

The C P-violating asymmetry, a, can be written as

a[uAu²2uA¯u²

uAu²1uA¯u²5 22r sind^sinf

112r cosd^cosf1r². ~4! It can be seen explicitly from Eq. ~4! that both weak and strong phases are needed to produce C P violation.r^-v^mix- ing has the dual advantages that the strong phase difference is large ~passing through 90° at thev ^resonance! and well known. In this scenario one has@6#

^p¹p^2fuH^TuLb&⁵_s^gr

rs_vP˜

rvt_v1g_r

s_rt_r, ~5!

^p¹p^2fuH^PuLb&5 g_r

s_rs_vP˜

rvp_v1g_r

s_rp_r, ~6! where t_V (V5r ^or v) is the tree and p_V is the penguin amplitude for producing a vector meson, V, byLb→f V, g_r is the coupling for r⁰→p¹p^2, P˜

rv is the effective r^-v mixing amplitude, and s_V²¹ is the propagator of V,

s_V5s2m_V²1im_VGV, ~7! with

A

s being the invariant mass of thep¹p^{2 pair.}

P˜

rv is extracted @13,14# from the data for e¹e²

→p¹p²@15#when

A

s is near thev mass. Detailed discus- sions can be found in Refs. @6,13,14#. The numerical values are

ReP˜

rv~m_v²!5235006300 MeV², ImP˜

rv~m_v²!523006300 MeV².

We stress that the direct coupling v→p¹p² is effectively absorbed intoP˜

rv, where it contributes some s-dependence.

The limits on this s-dependence, P˜

rv(s)5P˜

rv(m_v²)1(s 2m_v²)P˜

rv8 ^(m_v²), were determined in the fit of Gardner and O’Connell, P˜

rv8 ^(m_v²⁾50.0360.04 @13#. In practice, the ef- fect of the derivative term is negligible.

From Eqs.~1!,~2!,~5!,~6!one has

re^ide^if5P˜

rvp_v1s_vp_r P˜

rvt_v1s_vt_r . ~8! Defining

p_v

t_r [r8^{ei~dq1f!, t}_t^v

r[a^eida, _p^pr

v[b^eidb, ~9! whereda, db anddq are strong phases, one has the follow- ing expression from Eq.~8!

re^id5r8^eidqP˜rv1be

id_bs_v s_v1P˜

rva^{eida. ~10!} It will be shown that in the factorization approach, for both Lb→np¹p^{2 and} Lb→Lp¹p^{2, we have} ~see Sec.

II C for details!

X.-H. GUO AND A. W. THOMAS PHYSICAL REVIEW D 58 096013

096013-2

a^eida51. ~11! Letting

b^eidb5b1ci, r8^eidq5d1ei, ~12! and using Eq.~10!, we obtain the following result when

A

^s

;m_v:

re^id5 C1Di

~s2m_v²1ReP˜

rv!²1~ImP˜

rv1m_vG_v!², ~13! where

C5~s2m_v²1ReP˜

rv!$^d@ReP˜

rv1b~s2m_v²!2cm_vG_v# 2e@ImP˜

rv1bm_vG_v1c~s2m_v²!#% 1~ImP˜

rv1m_vG_v!$^e@ReP˜rv1b~s2m_v²!2cm_vG_v# 1d@ImP˜

rv1bm_vG_v1c~s2m_v²!#%^, D5~s2m_v²1ReP˜

rv!$^e@ReP˜

rv1b~s2m_v²!2cm_vG_v# 1d@ImP˜

rv1bm_vG_v1c~s2m_v²!#% 2~ImP˜

rv1m_vG_v!$^d@ReP˜

rv1b~s2m_v²!2cm_vG_v# 2e@ImP˜

rv1bm_vG_v1c~s2m_v²!#%^{. ~14!} b^{eidb and r}8^eidq will be calculated later. Then from Eqs.

~13!and~14!we obtain rsind^{, rcos}d and r. In order to get the C P-violating asymmetry a in Eq.~4!sinf ^{and cos}f ^are needed.f is determined by the CKM matrix elements. In the Wolfenstein parametrization @16#, and in the approximation that the top quark dominates the penguin diagrams, we have

~sinf!ⁿ5 h

A

@r~12r!2h²#²1h^2,

~cosf!ⁿ5 r~12r!2h²

A

@r~12r!2h²#²1h^{2, ~15!} for Lb→np¹p^{2, and}

~sinf!^L52 h

A

@r~11l²r!1l²h²#²1h^2,

~cosf!^L52 r~11l²r!1l²h²

A

@r~11l²r!1l²h²#²1h^2,

~16! for Lb→Lp¹p². Note that here, and in what follows, all the quantities with the superscript n ~or L) represent those for Lb→nr⁰⁽v⁾ @orLb→Lr⁰⁽v^)].

B. The effective Hamiltonian

With the operator product expansion, the effective Hamil- tonian relevant to the processesLb→fr⁰⁽v^{) is}

H_DB515G_F

A

²

F

^{VubVuq*~c1O1u1c2O2u!}

2V_tbV_tq*

(

_i5¹⁰₃ ^ciOi

G

^{1H.c., ~17!}

where the Wilson coefficients, c_i(i51, . . . ,10), are calcu- lable in perturbation theory and are scale dependent. They are defined at the scale m'm_b in our case. The quark q could be d or s for our purpose. The operators O_i have the following expression

O₁^u5q¯

ag_m~12g5!u_b¯u

bg^m~12g5!b_a, O₂^u5q¯gm~12g5!uu¯g^m~12g5!b,

O₃5q¯gm~12g5!b

(

_q8 ^{¯q8gm~12g5!q8,}

O₄5q¯_agm~12g5!b_b

(

_q8 ^{¯qb8gm~12g5!qa8,}

O₅5q¯gm~12g5!b

(

_q8 ^{¯q8gm~11g5!q8,}

O₆5q¯_ag_m~12g5!b_b

(

_q8 ^{¯qb8gm~11g5!qa8,}

O₇53

2¯qgm~12g5!b

(

_q8 ^eq8¯q8g^m~11g5!q8^, O₈53

2¯q

ag_m~12g5!b_b

(

q8 e_q8¯q

b8g^m~11g5!q_a8^,

O₉53

2¯qgm~12g5!b

(

_q8 ^eq8¯q8g^m~12g5!q8^, O₁₀53

2¯q

ag_m~12g5!b_b

(

_q

8 e_q8¯q

b8g^m~12g5!q_a8^,

~18! wherea ^andb are color indices, and q8⁵u, d, s quarks.

In Eq. ~18!O₁^u and O₂^u are the tree level operators. O₃–O₆ are QCD penguin operators, which are isosinglet. O₇–O₁₀ arise from electroweak penguin diagrams, and they have both isospin 0 and 1 components.

The Wilson coefficients, c_i, are known to the next-to- leading logarithmic order @17,18#. At the scale m5m_b 55 GeV their values are

c₁520.3125, c₂51.1502, c₃50.0174, c₄520.0373, c₅50.0104, c₆520.0459, c₇521.050310²⁵, c₈53.839310²⁴,

c₉520.0101, c₁₀51.959310²³. ~19! To be consistent, the matrix elements of the operators O_i should also be renormalized to the one-loop order. This re- sults in the effective Wilson coefficients, c_i8, which satisfy the constraint

c_i~m!^^Oi~m!&5c_i8^^Oi

tree&^, ~20!

where^^Oi(m⁾&are the matrix elements, renormalized to one- loop order. The relations between c_i8 ^{and c}i read

c₁85c₁, c₂85c₂, c₃85c₃2P_s/3, c₄85c₄1P_s, c₅85c₅2P_s/3, c₆85c₆1P_s, c₇85c₇1P_e,

c₈85c₈, c₉85c₉1P_e, c₁₀8 5c₁₀, ~21! where

P_s5~as/8p!c₂@10/91G~m_c,m^,q2!#,

P_e5~aem/9p!~3c₁1c₂!@10/91G~m_c,m^,q2!#, with

G~m_c,m^,q2!54

E

0 1

dxx~12x!lnm_c²2x~12x!q² m^{2 ,} where q²is the momentum transfer of the gluon or photon in the penguin diagrams. G(m_c,m^,q2) has the following ex- plicit expression@19#:

Re G52

3

S

^{lnmm2c225324mq2c21}

S

^112mq2c2

D

3

A

^124mq^2c2 ln

11

A

¹24~m_c²/q²! 12

A

¹24~m_c²/q²!

D

^,

Im G522

3p

S

^112mq2c2

D A

^{124mq2c2. ~22!}

Based on simple arguments for q² at the quark level, the value of q² is chosen in the range 0.3,q²/m_b²,0.5 @5,6#. From Eqs.~19!,~21!and~22!we can obtain numerical val- ues of c_i8^{. When q2/m}b

250.3,

c₁8520.3125, c₂851.1502 c₃852.433310²²11.543310²³i, c₄8525.808310²²24.628310²³i, c₅851.733310²²11.543310²³i, c₆8526.668310²²24.628310²³i, c₇8521.435310²⁴22.963310²⁵i, c₈853.839310²⁴,

c₉8521.023310²²22.963310²⁵i,

c₁₀851.959310²³, ~23! and when q²/m_b²50.5,

c₁8520.3125, c₂851.1502 c₃852.120310²²15.174310²³i, c₄8524.869310²²21.552310²²i, c₅851.420310²²15.174310²³i, c₆8525.729310²²21.552310²²i, c₇8528.340310²⁵29.938310²⁵i, c₈853.839310²⁴,

c₉8521.017310²²29.938310²⁵i,

c₁₀851.959310²³, ~24! where we have taken as(m_Z)50.112, aem(m_b)51/132.2, m_b55 GeV and m_c51.35 GeV.

C. CP violation inLb˜n„L…p¹p²

In the following we will calculate the C P-violating asym- metries in Lb→n(L)p¹p². With the Hamiltonian in Eq.

~17! we are ready to evaluate the matrix elements. In the factorization approximation r⁰⁽v) is generated by one cur- rent which has the proper quantum numbers in the Hamil- tonian.

X.-H. GUO AND A. W. THOMAS PHYSICAL REVIEW D 58 096013

096013-4

First we consider Lb→nr⁰⁽v). After factorization, the contribution to t_rⁿ from the tree level operator O₁^u is

^r⁰ⁿuO₁^uuLb&5^r⁰u~¯ uu !u0&^ⁿu~¯ b!uLd b&[T, ~25!

where (u¯ u) and (d¯b) denote the V-A currents. Using the Fierz transformation the contribution of O₂^u is (1/N_c)T.

Hence we have

t_rⁿ5

S

^c11N1cc2

D

^{T. ~26!}

It should be noted that in Eq. ~26! we have neglected the color-octet contribution, which is nonfactorizable and diffi- cult to calculate. Therefore, N_cshould be treated as an effec- tive parameter and may deviate from the naive value 3. In the same way we find that t_vⁿ5t_rⁿ, hence, from Eq.~9!,

~a^eida!ⁿ51. ~27! In a similar way, we can evaluate the penguin operator contributions p_rⁿ and p_vⁿ with the aid of the Fierz identities.

From Eq.~9!we have

~b^eidb!ⁿ5

22

S

^c481N1cc38

D

¹³

S

^c781N1cc88

D

¹

S

^31N1c

D

^c981

S

^11N3c

D

^c108

2

S

^21N1c

D

^c3812

S

^11N2c

D

^c4814

S

^c581N1cc68

D

^{1c781N1cc881~c982c108 !}

S

^12N1c

D

^{, ~28!}

~r8^eidq!ⁿ52

2

S

^21N1c

D

^c3812

S

^11N2c

D

^c4814

S

^c581N1cc68

D

^{1c781N1cc881~c982c108 !}

S

^12N1c

D

2

S

^c11N1cc2

D U

^{VVubtbVVtd*ud*}

U

^{, ~29!}

where

U

^{VVubtbVVtd*ud*}

U

⁵

A

^{@~r1~2l122r/2!2!~rh221#2}h¹²!^h2. ~30! ForLb→Lr⁰⁽v), the evaluation is the same. Defining

^r⁰LuO₁^uuLb&⁵^r⁰u~¯ uu !u0&^^{Lu~¯bs !uL}b&^{[˜ ,T ~31!}

we have

t_r^L5

S

^c11N1cc2

D

^{T˜ . ~32!}

After evaluating the penguin diagram contributions we obtain the following results:

~a^eida!^L51, ~33!

~b^eidb!^L5

3

S

^{c781N1cc881c981N1cc108}

D

4

S

^{c381N1cc481c581N1cc68}

D

^{1c781N1cc881c981N1cc108 , ~34!}

~r8^eidq!^L52

4

S

^{c381N1cc481c581N1cc68}

D

^{1c781N1cc881c981N1cc108}

2

S

^c11N1cc2

D U

^{VVubtbVVts*us*}

U

^{, ~35!}

where

U

^{VVubtbVVts*us*}

U

⁵

A

^{@r~11ll22~rr!121lh22h!2#21h2. ~36!}

It can be seen from Eqs.~28!and~34!that b ^anddb are determined solely by the Wilson coefficients. On the other hand, r8 ^anddq depend on both the Wilson coefficients and the CKM matrix elements, as shown in Eqs. ~29! and~35!. Substituting Eqs. ~27!, ~28!,~29!, ~33!,~34!, ~35!into Eqs.

~12!, ~13!, ~14!we can obtain (rsind^{)n(L) and (rcos}d^)n(L). Then in combination with Eqs. ~15! and ~16! the C P-violating asymmetries a can be obtained.

In the numerical calculations, we have several parameters:

q², N_c, and the CKM matrix elements in the Wolfenstein parametrization. As mentioned in Sec. II B, the value of q²is chosen in the range 0.3,q²/m_b²,0.5@5,6#.

The CKM matrix elements should be determined from experiment. l is well measured @20# and we will use l 50.221 in our numerical calculations. However, due to the large experimental errors at present, r ^and h are not yet fixed. From b→u transitions

A

r²1l²50.36360.073

@21,22#. In combination with the results from B⁰-B¯⁰ mixing

@23#we have 0.18,h,0.42@22#. In our calculations we use h50.34 as in Refs. @5,6#. Recently, it has been pointed out

@24# that from the branching ratio of B⁶→hp^{6 a negative} value forr is favored. Hence we will user520.12, corre- sponding to h50.34. These values lead to fⁿ5126° and f^L5272° from Eqs. ~15!and~16!.

The value of the effective N_c should also be determined by experiments. The analysis of the data for B→Dp^{, B6}

→vp^{6and B6}→v^K6indicates that N_cis about 2@25,26#. For theLbdecays, we do not have enough data to extract N_c at present. Finally, we use m_b55 GeV, m_c51.35 GeV, as(m_Z)50.112 andaem(m_b)51/132.2 to calculate the Wil- son coefficients, c_i8, as discussed in Sec. II B@see Eqs.~23! and ~24!#. The numerical values of b^{, r}8^, db and dq for Lb→nr^0andLb→Lr⁰are listed in Tables I and II, respec- tively.

In Figs. 1 and 2 we plot the numerical values of the C P-violating asymmetries, a, for Lb→np¹p^{2 and} Lb

→Lp¹p², respectively, for N_c52. It can be seen that there is a very large C P violation when the invariant mass of the p¹p² is near thev ^{mass. For} Lb→np¹p² the maximum C P-violating asymmetry is a_maxⁿ 5266% (q²/m_b²50.3) and a_maxⁿ 5250% (q²/m_b²50.5), while for Lb→Lp¹p^2, a_max^L 568% (q²/m_b²50.3) and a_max^L 576% (q²/m_b²50.5). It would be very interesting to actually measure such large C P-violating asymmetries.

Although N_cis around 2 for B decays, it might be differ- ent in the Lb case. We also calculated the numerical values when N_c53. It is found that, in this case, we still have large C P violation forLb→np¹p^{2, with a}max

n 5252% (q²/m_b² 50.3) and a_maxⁿ 5240% (q²/m_b²50.5). However, for Lb

→Lp¹p^{2, a}maxL is much smaller, only about 6%.

III. BRANCHING RATIOS FORLb˜n„L…r⁰ In this section we estimate the branching ratios for Lb

→fr⁰. In the factorization approach,r⁰is factorized out and hence the decay amplitude is determined by the weak tran- FIG. 1. The C P-violating asymmetry for Lb→np¹p² with N_c52. The solid~dotted!line is for q²/m_b²50.3~0.5!.

FIG. 2. The C P-violating asymmetry for Lb→Lp¹p² with Nc52. The solid~dotted!line is for q²/m_b²50.3~0.5!.

TABLE I. Values ofb, r8^, d_b^anddqforLb→nr⁰.

Nc q²/m_b² b r8 d_b dq

2 0.3 0.339 1.149 23.096 0.0769

2 0.5 0.328 1.011 22.935 0.297

3 0.3 0.649 2.537 23.103 0.0766

3 0.5 0.629 2.233 22.970 0.296

TABLE II. Values ofb, r8^, d_b^anddqforLb→Lr⁰.

N_c q²/m_b² b r8 d_b dq

2 0.3 0.299 9.925 20.0611 0.0675

2 0.5 0.332 8.833 20.235 0.257

3 0.3 3.086 3.715 21.766310²⁴ 6.353310²³ 3 0.5 3.087 3.668 26.071310²⁴ 0.0216

X.-H. GUO AND A. W. THOMAS PHYSICAL REVIEW D 58 096013

096013-6

sition matrix elementsLb→f . In the heavy quark limit, m_b

→`, it is shown in the HQET that there are two form factors for Lb→f @27#:

^^f~p_f!uq¯gm~12g5!buLb~v!&

5u¯_f~p_f!@F₁~v•p_f!1v”F₂~v•p_f!#

3g_m~12g5!u_L

b~v!, ~37!

where q5d or s; u_f and u_L

b are the Dirac spinors of f and Lb, respectively; p_f is the momentum of the final baryon, f, andvis the velocity ofLb. In order to calculate F₁ and F₂ we need two constraints.

In Ref.@28# the author proposed two dynamical assump- tions with respect to the meson structure and decays: ~i! in the rest frame of a hadron the distribution of the off-shell momentum components of the constituents is strongly peaked at zero with a width of the order of the confinement scale;~ii!during the weak transition the spectator retains its momentum and spin. These two assumptions led to the result that the matrix element of the heavy to light meson transition is dominated by the configuration where the active quarks’

momenta are almost equal to those of their corresponding mesons. This argument is corrected by terms of order 1/m_b andLQCD/E_f, and hence is a good approximation in heavy hadron decays. Some relations among the form factors in the heavy to light meson transitions are found in this approxima- tion. In Ref. @10# the above approach is generalized to the baryon case and a relation between F₁ and F₂ is found.

Another relation between F₁and F₂comes from the over- lap integral of the hadronic wave functions of Lb and f . In the heavy quark limit Lb is regarded as a bound state of a heavy quark b and a light scalar diquark@ud# @10,11,29#. On the other hand, the light baryon f has various quark-diquark configurations @30# and only the q@ud# component contrib- utes to the transition Lb→f . This leads to a suppression factor, C_s, which is the Clebsch-Gordan coefficient of q@ud#. C_s51/

A

2 for n and C_s51/

A

^{3 for} L, respectively

@30#. In the quark-diquark picture, the hadronic wave func- tion has the following form:

ci~x₁,kW_'!5N_ix₁x₂³exp@2b²„kW_'²₁m_i²~x₁2x_0i!²…#,

~38! where i5Lb, n or L; x₁, x₂ (x₂512x₁) are the longitu- dinal momentum fractions of the active quark and the di- quark, respectively; kW_' is the transverse momentum; N_i is the normalization constant; the parameter b is related to the average transverse momentum, b51.77 GeV and b 51.18 GeV, corresponding to ^^k_'²&^1/25400 MeV and

^^k_'²&^1/25600 MeV respectively; and x_0i (x_0i512e^/mi, e

is the mass of the diquark!is the peak position of the wave function. By working in the appropriate infinite momentum frame and evaluating the good current components @10,11#, another relation between F₁ and F₂ is given in terms of the overlap integral of the hadronic wave functions ofLb and f . Therefore, F₁ and F₂ are obtained as the following,

F₁52E_f1m_f1m_q

2~E_f1m_q! C_sI~v!,

F₂5 m_q2m_f

2~E_f1m_q!C_sI~v!, ~39! where I(v) is the overlap integral of the hadronic wave functions of Lb and f ,

I~v!5

S

^{v12 1}

D

^{7/4y29/2@AfK6~}

A

^2be!#21/2

3exp

S

^22b2e2vv2111

D E

^{2y22b2be/Ae/v1Av111dz}

3exp~2z²!

S

^y2

A

^2bv1e12z

DS

^z1

A

^2bv1e1

D

^6,

~40! and y5bm_f

A

v11, with v being the velocity transfer v 5v•p_f/m_f and A_f and K₆ defined as

A_f5

E

0 1

dxx⁶~12x!²exp@22b²m²_f~x2e^/mf!²#,

K₆~

A

^2be!5

E

2A2be

`

dxexp~2x²!~x1

A

^2be!⁶. ~41! It should be noted that in Eqs.~40!and~41!we have taken the limit m_b→`.

It can be shown thatv53.03 forLb→nr^{0 and}v52.58 forLb→Lr^{0. Taking}e5600 MeV, from Eq.~40!, we find that Iⁿ50.0258(0.0509) for b51.77 GeV²¹ (b 51.18 GeV²¹), and I^L50.0389(0.0781) for b 51.77 GeV²¹ (b51.18 GeV²¹). Substituting these num- bers into Eq.~39!we obtain the following results,

F₁ⁿ520.0199~20.0393!, for b51.77~1.18! GeV²¹, F₂ⁿ50.00168~0.00332!, for b51.77~1.18! GeV²¹,

~42! and

F₁^L50.0245~0.0492!, for b51.77~1.18! GeV²¹, F₂^L520.00205~20.00411!, for b51.77~1.18! GeV²¹,

~43! where we have taken m_d50.35 GeV and m_s50.50 GeV.

To estimate the branching ratios for Lb→n(L)r^{0 we} only take the O₁^uand O₂^uterms in the Hamiltonian~17!, since they give dominant contributions. In the factorization ap- proach, the amplitude for Lb→n(L)r⁰ has the following form:

A~Lb→f1r⁰!5G_F

A

^{2VubVuq*a1^r0u¯ugm~12g5!uu0&}

3^^fuq¯g^m~12g5!buLb&^{, ~44!}

where

a₁5c₁1 1

N_cc₂. ~45!

In Eq.~44!r⁰has been factorized out and the matrix element

^r⁰u¯ug_m⁽¹2g5)uu0& is related to the decay constant f_r.

From Eq.~44!the branching ratios forLb→n(L)r^{0 can be} obtained directly @10,31#. Taking f_r5216 MeV, V_ub 50.004, V_us50.22, V_ud50.975 and a₁50.28 ~correspond- ing to N_c;2) we obtain

B~Lb→nr⁰!5

H

^{1.614.143310102289 for bfor b551.18 GeV1.77 GeV2211,,}

~46!

and

B~Lb→Lr⁰!5

H

^{1.233.0633101022910 for bfor b551.18 GeV1.77 GeV2211,.}

~47! In Ref.@12#theLb→n(L) transition matrix elements are calculated in the nonrelativistic quark model. The form fac- tors f_i and g_i, which are defined by (q5p_L

b2p_f)

^^f~p_f!u¯qg_m~12g5!buLb~p_L

b!&

5¯u_f$^f1~q²!gm1i f₂~q²!smnqⁿ1f₃~q²!q_m

2@g₁~q²!g_m1ig₂~q²!s_mn^qn1g₃~q²!q_m#g5%^uL_b,

~48! are found to be f₁(0)50.045, f₂(0)520.024/m_L

b, f₃(0) 520.011/m_L

b, g₁(0)50.095, g₂(0)520.022/m_L

b, g₃(0) 520.051/m_L

b for Lb→n, and f₁(0)50.062, f₂(0) 520.025/m_L

b, f₃(0)520.008/m_L

b, g₁(0)50.108, g₂(0) 520.014/m_L

b, g₃(0)520.043/m_L

b for Lb→L. Pole

dominance behavior for the q² dependence of the form fac- tors is assumed:

f_i~q²!5 f_i~0!

S

^12mq2V2

D

^{2, gi~q2!5}

S

^1g2i~m0q!2A2

D

^{2, ~49!}

where for b→d, m_V55.32 GeV, m_A55.71 GeV, and for b→s, m_V55.42 GeV, m_A55.86 GeV. Substituting Eqs.

~48!and~49!into Eq.~44!we find that

B~Lb→nr⁰!56.33310²⁸, B~Lb→Lr⁰!54.44310²⁹.

~50!

These results are bigger than those in Eqs. ~46! and ~47!. Combining the predictions in these two models we expect that B(Lb→nr⁰) is around 10²⁸and B(Lb→Lr^{0) is about} 10²⁹. For comparison, in B decays, the branching ratio for B²→p²r⁰is of the order 10²⁶@32#, and for B²→r²r^0the branching ratio is about 10²⁵ @3