Rare dileptonic decays of Λ_b in a quark model

0.86 MB·English

by  Lonnie Mott

RARE DILEPTONIC DECAYS OF Λ IN A QUARK MODEL
b
L. Mott and W. Roberts
Department of Physics, Florida State University, Tallahassee, FL 32306
Hadronic form factors for the rare weak transitions Λ → Λ(∗) are calculated using a nonrela-
b
tivistic quark model. The form factors are extracted in two ways. An analytic extraction using
single component wavefunctions(SCA) with thequarkcurrent beingreducedto itsnonrelativistic
Pauli form is employed in the first method. In the second method, the form factors are extracted
numerically using the full quark model wave function (MCN) with the full relativistic form of the
quarkcurrent. Althoughtherearedifferencesbetweenthetwosetsofformfactors,bothsetssatisfy
therelationships expected from theheavy quark effective theory (HQET). Differential decay rates,
branchingratiosandforward-backwardasymmetries(FBAs)arecalculatedforthedileptonicdecays
Λ → Λ(∗)ℓ+ℓ−, for transitions to both ground state and excited daughter baryons. Inclusion of
b
2 thelongdistancecontributionsfromcharmoniumresonancessignificantlyenhancesthedecayrates.
1
In the MCN model the Λ(1600) mode is the dominant mode in the µ channel when charmonium
0
resonancesareconsidered; theΛ(1520) modeisalso foundtohaveacomparablebranchingratioto
2
that of the ground state in theµ channel.

n
a
J I. INTRODUCTION

2
1
The weak decays of heavy hadrons have been an important source of information on some of the fundamental

parameters of the Standard Model (SM). In particular, semileptonic decays of heavy hadrons have been important
]
h in the extraction of some Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1]. However, the description of
-t these processes requires a number of a priori unknown form factors that parametrize the uncalculable (to date)
l nonperturbative QCD dynamics. The precision to which these form factors can be calculated or modeled limits the
c
u accuracy with which the CKM matrix elements can be extracted. In this regard, the heavy quark effective theory
n (HQET) [2], quark models [3, 4], QCD sum rules [5], lattice QCD [6], etc. have been employed to improve the
[ modeling of these form factors.

In the same way, decays of heavy hadrons induced by flavor-changing neutral currents (FCNC), the so-called rare
3
v decays,havebeen a subject ofsignificantinterest in recentyears. Processeslike b sγ and b sℓ+ℓ− areforbidden
→ →
9 attreelevel,andatone-looplevelaresuppressedby the Glashow-Iliopoulos-Maiani(GIM)mechanism. Thesedecays
2 thereforereceivetheir maincontributionfromone-loopdiagramswithavirtualtopquarkandaW boson. Therefore,
1 they provide valuable information about the CKM matrix elements V and V .
ts tb
6
Inaddition,becausethesedecaysoccuratlooplevel,theyaresensitivetonewphysicsbeyondtheSM.Intheserare
.
8 decays,newphysicscanappeareitherthroughnewcontributionstotheWilsoncoefficientsthatenterintotheeffective
0 Hamiltonianthatdescribesthesedecays,orthroughnewoperatorsintheeffectiveHamiltonianthatarisefromsources
1
beyondthe SM. For these reasons,these decays arepromising candidates for looking for new physics beyond the SM.
1
Thus, the values of the Wilson coefficients are crucial to the determination of any new physics. As with the case of
:
v semileptonicdecays,theaccuracytowhichtheseparameterscanbeextractedislimitedbyourknowledgeofthe form
i factors that are used to parametrize the hadronic matrix elements. Many experimentally measurable quantities such
X
as branching ratios, forward-backward asymmetries, lepton polarization asymmetries, etc. can be analyzed to more
r
a preciselydetermineSMparametersandtolookfornewphysicsbeyondtheSM.Itisknownthatmostobservableswill
dependsensitively onthe formfactors;thus,the accuracyto whichthese formfactorscanbe calculatedis paramount
to the determination of any new physics inferred from these decays.
There havebeen many theoretical investigationsof rare b s transitions in the the mesonsector [7–52]. Melikhov
→
et al. have used a relativistic constituent quark model to treat the dileptonic decays of the B meson [21]. In [41],
perturbative QCD (pQCD) is used to estimate the form factors for B K( ). Light-cone sum rules (LCSRs) have
∗
→
been employed for these transitions as well [15, 29, 35, 46]. The form factors obtained from the various models
have been used to calculate observables such as branching ratios (BRs), forward-backwardasymmetries (FBAs) and
lepton polarizationasymmetries (LPAs). These observableshave been calculatedboth within the SM andfor various
scenarios that arise beyond the standard model, such as supersymmetric (SUSY) models [29, 34, 42], models with
universal extra dimensions (UEDs) [52], and other new physics (NP) scenarios [44, 46].
Inthe baryonsector,therehavebeenafewtechniquesemployedinthe computationoftheformfactorsforthe rare
Λ Λ transitions [53–62]. Inmuchof whathas been put forth,HQET is usedto reduce the number ofindependent
b
→
form factors to two universal form factors valid for all currents, and a model is then employed to compute these two
formfactors. Aslamet al. [53],Wanget al. [54]andHuanget al. [55]haveusedLCSRstoobtaintheformfactorsfor
Λ Λ. QCD sum rules (QCDSRs) have also been employed in obtaining these form factors [56–59, 62]. A number
b
→
Rare dileptonic decays of Λ in a quark model 2
b
of authors [56, 58, 60] have also used pole model parametrizations(PM) for the form factors. In [61] the formfactors
for these transitions have been estimated using pQCD. The MIT bag model (BM) has been used in [59]. To the best
of our knowledge, no transitions to excited state Λ’s have been explored.
Experimentally, the exclusive radiative transition B K (892)γ was first observedby CLEO [63]. The branching
∗
→
ratio for this mode has been measured recently by both Belle [64] and BaBar [65] with an average branching ratio
(B0 K0 (892)γ) = (4.33 1.9) 10 5. Several other decay modes such as B K γ,K (1430)γ, etc. [66–69]
B → ∗ ± × − → 1 2∗
have been observed as well. The mode B0 φγ has been observed by Belle; its branching ratio was measured to
be (B0 φγ) = (57+22) 10 6 [70]. Brsan→ching ratios for the inclusive process B X γ have been measured by
CLEBO [s71→], BaBar [72]−,1a9nd×Bel−le [73]. From these measurements, the average branch→ing rsatio is [74]
(B X γ)=(3.55 0.24 0.09) 10 4.
s −
B → ± ± ×
Branching ratios for the dileptonic decays have been measured by Belle [75], BaBar [76], and CDF [77]. From the
BaBar and Belle data, (B Kℓ+ℓ ) = (0.45 0.04) 10 6 and (B K (892)ℓ+ℓ ) = (1.08 0.11) 10 6.
− − ∗ − −
B → ± × B → ± ×
Branching ratios for the inclusive process B X ℓ+ℓ have also been measured by both Belle [78] and BaBar [79];
s −
→
the average branching ratio is [80]
B(B →Xsℓ+ℓ−)=(3.66+−00..7767)×10−6.
Note that this means that decays to the K and K account for less than 50% of the rare dileptonic decays of the
∗
B meson. The Belle Collaboration [81] has also measured the forward-backward asymmetry in B K ℓ+ℓ and
∗ −
→
extracted ratios of Wilson coefficients from those data.
Theexperimentalsituationforb stransitionsinthebaryonsectorislessrichthaninthemesonsector. TheCDF
collaborationrecently reportedthe→firstobservationofthe baryonicFCNC processΛ Λµ+µ [82]. The branching
b −
ratio for this mode was measured to be (Λ Λµ+µ ) = (1.73 0.42 0.55) 10→6. The LHCb Collaboration
b − −
[83] estimate that with 2 fb 1 of data takBen in→one year, there shou±ld be 75±0 Λ ×Λγ events and nearly six times as
− b
→
many events in excited hyperons. For the dileptonic decay mode, 800 events are expected for decays to the ground
state Λ; no potential yields were reported for excited states.
In this paper, we examine the rare weak dileptonic decays of Λ baryons to ground state and excited Λ baryons.
b
Some of the motivation for this study has been outlined above. It will also be useful to examine the sensitivity
of some of the lepton asymmetries to the form factors. It has been shown for rare meson decays that there are
observables,suchasleptonpolarizationasymmetries,thatarelargelyindependentoftheformfactorsincertainlimits
[7, 12, 13, 15–18]. Such quantities thus offer largely model-independent ways to examine the physics content of some
of the Wilson coefficients [13].
To this end, we use two approximations to compute the form factors for Λ Λ( ) transitions. First, we use the
b ∗
→
approximation employed in [84, 85]. In that work, analytic form factors for Λb Λ(c∗), Λc Λ(∗), Λb N(∗) and
→ → →
Λ N( ) were calculated using single component wave functions obtained from a variational diagonalization of
c ∗
→
a quark model Hamiltonian. In that calculation, quark operators were reduced to their nonrelativistic Pauli form.
The decay rates obtained using the form factors extracted in that calculation were in reasonable agreement with
experimental results for the semileptonic decays Λ Λ ℓν and Λ Λℓν .
b c ℓ c ℓ
→ →
The second approximation we use for computing the form factors is an extension of this method; we keep the full
relativistic form of the quark spinors and use the full quark model wave function in a numerical extraction of the
formfactors. Becausethis methoduses fewer approximations,the formfactorsobtainedinthis wayshouldgivemore
reliable results than the form factors obtained from the first method. We present the results of both methods to
demonstrate the sensitivity of the observables to the form factors. The goal here is to find observables that may be
less dependent on the form factors.
The rest of this paper is organized as follows: in Section II, we discuss the hadronic matrix elements, decay rates,
and forward-backwardasymmetries. Section III gives a brief overview of HQET and presents HQET predictions for
the relationships among the form factors for the transitions we investigate. In Section IV, we describe the quark
model used to obtain the form factors, including some description of the Hamiltonian. The two methods we employ
to obtain the form factors are also briefly discussed. Numerical results such as form factors, differential decay rates,
branching ratios, and forward-backward asymmetries are presented in Section V. We present our conclusions and
outlook in Section VI. Some details of the calculation are shown in the Appendices.
Rare dileptonic decays of Λ in a quark model 3
b
II. MATRIX ELEMENTS, DECAY RATES, AND FORWARD-BACKWARD ASYMMETRIES
A. Matrix Elements
The amplitude for the dileptonic decay of the Λ baryon can be written
b
G α
i (Λ Λℓ+ℓ )= F emV V (HµL(V)+HµL(A)), (1)
M b → − √2 2π tb t∗s 1 µ 2 µ
where Hµ and Hµ contain the hadronic matrix elements
1 2
2m
Hµ = bC (m )Tµ+C (m )Jµ, (2)
1 − q2 7 b R 9 b L
Hµ = C (m )Jµ. (3)
2 10 b L
The C are the Wilson coefficients,
i
Tµ = Λ(p ,s ) siσµνq (1+γ )b Λ (p ,s ) , (4)
R h Λ Λ | ν 5 | b Λb Λb i
and J is the matrix element of the standard V A current
L
−
Jµ = Λ(p ,s ) sγµ(1 γ )b Λ (p ,s ) . (5)
L h Λ Λ | − 5 | b Λb Λb i
L(V) and L(A) are the vector and axial vector leptonic currents, respectively, written as
µ µ
L(µV) = uℓ(p ,s−)γµvℓ(p+,s+), (6)
−
L(µA) = uℓ(p ,s−)γµγ5vℓ(p+,s+), (7)
−
where s are the spin projections for the lepton and the antilepton.
±
Inouranalysisofthedileptonicdecays,wewillincludethelongdistancecontributionscomingfromthecharmonium
resonances J/ψ, ψ , ... etc. To include these resonant contributions, we replace the Wilson coefficient C in Eq. 2
′ 9
with the effective coefficient
C9eff =C9+YSD(z,s′)+YLD(s′), (8)
wherez =m /m ands =q2/m2. Y containsthe shortdistance(SD) contributionsfromthe four-quarkoperators
c b ′ b SD
far from the charmonium resonance regions and Y are the long distance (LD) contributions from the four-quark
LD
operatorsneartheresonances. TheSDtermcanbecalculatedreliablyintheperturbativetheory,butthesamecannot
be donefor the LDcontributions. The LD contributionsareusually parametrizedusing aBreit-Wignerformalismby
making use of vector meson dominance (VMD) and the factorization approximation (FA). The explicit expressions
for Y and Y are [7, 53, 54, 56–58]
SD LD
Y (z,s) = (3C +C +3C +C +3C +C )h(z,s)
SD ′ 1 2 3 4 5 6 ′
−
1 1
(4C3+4C4+3C5+C6)h(1,s′) (C3+3C4)h(0,s′)+
2 − 2
2
(3C +C +3C +C ), (9)
3 4 5 6
9
3π
Y (s) = (3C +C +3C +C +3C +C )
LD ′ −α2 1 2 3 4 5 6
em
m Γ(j ℓ+ℓ )
j −
k → ,
× jq2 m2+im Γ
j=J/Xψ,ψ′,... − j j j
(10)
Rare dileptonic decays of Λ in a quark model 4
b
where m , Γ , and Γ(j ℓ+ℓ ) are the masses, total widths, and partial widths of the resonances,respectively,
j j −
→
8 8 16z2 2 4z2 4z2 1/2
h(z,s) = lnz+ + 2+ 1
′
−9 27 9s − 9 s − s
′ (cid:18) ′ (cid:19)(cid:12) ′ (cid:12)
(cid:12) (cid:12)
Θ 1 4z2 ln 1+ 1− 4sz′2(cid:12)(cid:12) iπ (cid:12)(cid:12)+2Θ 4z2 1
× (cid:18) − s′ (cid:19) 1−q1− 4sz′2−  (cid:18) s′ − (cid:19)
  q  
 1
arctan ,
×  4z2 1
s′ − 
8 4 q 4 
h(0,s′) = lns′+ iπ,  (11)
27 − 9 9
and k are phenomenological parameters introduced to compensate for VMD and FA; they are chosen so as to
j
reproduce the correct branching ratio of (B K( )V K( )ℓ+ℓ ) = (B K( )V) (V ℓ+ℓ ), for V =
∗ ∗ − ∗ −
B → → B → B →
J/ψ,ψ . SincenoneoftheanalogousbranchingratiosoftheΛ haveyetbeenmeasured,weapplythephenomenological
′ b
factors obtained from decays of the B mesons to the decays of the Λ . For the lowest resonances J/ψ and ψ , we use
b ′
k =1.65 and k =2.36, respectively; for the higher resonances, we use the average of J/ψ and ψ (see [35, 62]).
′
The matrix elements in Eqs. 4, and 5 contain the currents sγµb, sγµγ b, siσµνb and siσµνγ b. In this work, we
5 5
examine decays to daughter baryons with JP =1/2+, 1/2 , 3/2 ,3/2+, 5/2+. We make this choice because in the
− −
quark model we use to calculate the matrix elements, these states have the most significant overlaps with the initial
ground state within the spectator quark approximation.
A baryon with angular momentum and parity JP may be represented by a generalized Rarita-Schwinger field (or
spinor-tensor) u (p ), with n = J 1/2 indices. These spinor-tensors are symmetric in all of the indices, and
µ1...µn Λ −
satisfy the conditions
p/ u (p ) = m u (p ), γµ1u (p )=0,
Λ µ1...µn Λ Λ µ1...µn Λ µ1...µn Λ
pµ1u (p ) = 0, gµνu (p )=0. (12)
Λ µ1...µn Λ µν...µn Λ
For parity considerations, it is necessary to divide the spinor-tensors into two classes. Those with natural parity are
calledtensor,whilethosewithunnaturalparityarelabeledaspseudo-tensor. Here,astateoftotalangularmomentum
J is said to have natural parity if P =( 1)J 1/2, unnatural parity otherwise.
−
−
For transitions between the ground state and any state with JP =1/2+, the matrix elements of these currents are
Λ sγµb Λ = u(p ,s ) F (q2)γµ+F (q2)vµ+F (q2)vµ u(p ,s ), (13)
h | | bi Λ Λ 1 2 3 ′ Λb Λb
(cid:20) (cid:21)
Λ sγµγ b Λ = u(p ,s ) G (q2)γµ+G (q2)vµ+G (q2)vµ γ u(p ,s ),
h | 5 | bi Λ Λ 1 2 3 ′ 5 Λb Λb
(cid:20) (cid:21)
(14)
Λ siσµνb Λ = u(p ,s ) µνu(p ,s ), (15)
h | | bi Λ Λ T Λb Λb
where we have used v =p /m , v =p /m , and
Λb Λb ′ Λ Λ
µν = H1(q2)iσµν +H2(q2)(vµγν vνγµ)+H3(q2)(v′µγν v′νγµ)+
T − −
H (q2)(vµvν vνvµ). (16)
4 ′ ′
−
The F , G , and H are the form factors of interest, and these are functions of the square of the four-momentum
i i i
transfer q2 =(p p )2 between the initial and final baryons. Since
Λb − Λ
i
σµνγ = εµναβσ , (17)
5 αβ
2
the matrix elements involving the current siσµνγ b can be related to those involving siσµνb.
5
Rare dileptonic decays of Λ in a quark model 5
b
Similarly, the matrix elements for decays to daughter baryons with JP =3/2 are given by
−
Λ sγµb Λ = u (p ,s ) vα F γµ+F vµ+F vµ +
b α Λ Λ 1 2 3 ′
h | | i
(cid:20) (cid:18) (cid:19)
F gαµ u(p ,s ),
4 Λb Λb
(cid:21)
(18)
Λ sγµγ b Λ = u (p ,s ) vα G γµ+G vµ+G vµ +
5 b α Λ Λ 1 2 3 ′
h | | i
(cid:20) (cid:18) (cid:19)
G gαµ γ u(p ,s ),
4 5 Λb Λb
(cid:21)
(19)
Λ siσµνb Λ = u (p ,s ) αµνu(p ,s ), (20)
h | | bi α Λ Λ T Λb Λb
where
αµν = vα H iσµν +H (vµγν vνγµ)+H (vµγν vνγµ)+
1 2 3 ′ ′
T − −
(cid:20)
H4(vµv′ν vνv′µ) +H5(gαµγν gανγµ)+H6(gαµvν gανvµ). (21)
− − −
(cid:21)
For decays to JP =5/2+, the matrix elements are
Λ sγµb Λ = u (p ,s )vα vβ F γµ+F vµ+F vµ +
b αβ Λ Λ 1 2 3 ′
h | | i
(cid:20) (cid:18) (cid:19)
F gβµ u(p ,s ),
4 Λb Λb
(cid:21)
(22)
Λ sγµγ b Λ = u (p ,s )vα vβ G γµ+G vµ+G vµ +
5 b αβ Λ Λ 1 2 3 ′
h | | i
(cid:20) (cid:18) (cid:19)
G gβµ γ u(p ,s ),
4 5 Λb Λb
(cid:21)
(23)
Λ siσµνb Λ = u (p ,s ) αβµνu(p ,s ), (24)
h | | bi αβ Λ Λ T Λb Λb
where
αβµν = vα vβ H iσµν +H (vµγν vνγµ)+H (vµγν vνγµ)+
1 2 3 ′ ′
T − −
(cid:26) (cid:20)
H4(vµv′ν vνv′µ) +H5(gβµγν gβνγµ)+H6(gβµvν gβνvµ) . (25)
− − −
(cid:21) (cid:27)
Thus far, only the matrix elements involving spinors with natural parity, i.e. spinors with parity ( 1)J 1/2, have
−
−
been presented. The equations involving spinors with unnatural parity can be found by inserting γ to the left of the
5
parent baryon spinor in the equations for natural parity.
The matrix elements for the tensor currents can be written in a more convenient form. If Eq. 15 is contracted on
both sides with the four-momentum transfer q , use of the equations of motion leads to
ν
Λ siσµνq b Λ =u(p ,s ) FT(q2)γµ+FT(q2)vµ+FT(q2)vµ u(p ,s ), (26)
h | ν | bi Λ Λ 1 2 3 ′ Λb Λb
(cid:20) (cid:21)
where,
FT = (m +m )H (m m v v )H (m v v m )H ,
1 − Λb Λ 1− Λb − Λ · ′ 2− Λb · ′− Λ 3
F2T = mΛbH1+(mΛb −mΛ)H2+(mΛbv·v′−mΛ)H4,
FT = m H +(m m )H (m m v v )H , (27)
3 Λ 1 Λb − Λ 3− Λb − Λ · ′ 4
Rare dileptonic decays of Λ in a quark model 6
b
valid for states with JP =1/2+. For the axial tensor, a similar procedure leads to
Λ siσµνγ q b Λ =u(p ,s ) GT(q2)γµ+GT(q2)vµ+GT(q2)vµ γ u(p ,s ), (28)
h | 5 ν | bi Λ Λ 1 2 3 ′ 5 Λb Λb
(cid:20) (cid:21)
with
GT = (m m )H m (1 v v )H m (1 v v )H ,
1 Λb − Λ 1− Λ − · ′ 2− Λb − · ′ 3
GT = m H m H m H ,
2 Λb 1− Λ 2− Λb 3
GT = m H +m H +m H . (29)
3 Λ 1 Λ 2 Λb 3
Similarly, for transitions to states with JP = 1/2 , the matrix elements for the tensor and axial tensor currents
−
become
Λ siσµνq b Λ = u(p ,s ) FT(q2)γµ+FT(q2)vµ+FT(q2)vµ γ u(p ,s ),
h | ν | bi Λ Λ 1 2 3 ′ 5 Λb Λb
(cid:20) (cid:21)
(30)
Λ siσµνγ q b Λ = u(p ,s ) GT(q2)γµ+GT(q2)vµ+GT(q2)vµ u(p ,s ),
h | 5 ν | bi Λ Λ 1 2 3 ′ Λb Λb
(cid:20) (cid:21)
(31)
respectively, where
FT = (m m )H (m m v v )H (m v v m )H ,
1 Λb − Λ 1− Λb − Λ · ′ 2− Λb · ′− Λ 3
F2T = mΛbH1−(mΛb +mΛ)H2+(mΛbv·v′−mΛ)H4,
FT = m H (m +m )H (m m v v )H , (32)
3 Λ 1− Λb Λ 3− Λb − Λ · ′ 4
and
GT = (m +m )H +m (1+v v )H +m (1+v v )H ,
1 − Λb Λ 1 Λ · ′ 2 Λb · ′ 3
GT = m H m H m H ,
2 Λb 1− Λ 2− Λb 3
GT = m H m H m H . (33)
3 Λ 1− Λ 2− Λb 3
For transitions to states with JP =3/2 , we obtain
−
hΛ(pΛ)|siσµνqνb|Λb(pΛb)i = uα(pΛ,sΛ) vα F1Tγµ+F2Tvµ+F3Tv′µ +
(cid:20) (cid:18) (cid:19)
FTgαµ u(p ,s ), (34)
4 Λb Λb
(cid:21)
hΛ(pΛ)|siσµνγ5qνb|Λb(pΛb)i = uα(pΛ,sΛ) vα GT1γµ+GT2vµ+GT3v′µ +
(cid:20) (cid:18) (cid:19)
GTgαµ γ u(p ,s ), (35)
4 5 Λb Λb
(cid:21)
while for transitions to states with JP =5/2+, we have
hΛ(pΛ)|siσµνqνb|Λb(pΛb)i = uαβ(pΛ,sΛ)vα vβ F1Tγµ+F2Tvµ+F3Tv′µ +
(cid:20) (cid:18) (cid:19)
FTgβµ u(p ,s ), (36)
4 Λb Λb
(cid:21)
hΛ(pΛ)|siσµνγ5qνb|Λb(pΛb)i = uαβ(pΛ,sΛ)vα vβ GT1γµ+GT2vµ+GT3v′µ +
(cid:20) (cid:18) (cid:19)
GTgβµ γ u(p ,s ), (37)
4 5 Λb Λb
(cid:21)
Rare dileptonic decays of Λ in a quark model 7
b
where FT and GT are given by
i i
F1T = −(mΛb +mΛ)H1−(mΛb −mΛv·v′)H2−(mΛbv·v′−mΛ)H3−mΛbH5,
FT = m H +(m m )H +(m v v m )H m H ,
2 Λb 1 Λb − Λ 2 Λb · ′− Λ 4− Λb 6
FT = m H +(m m )H (m m v v )H ,
3 Λ 1 Λb − Λ 3− Λb − Λ · ′ 4
FT = (m m )H +(m m v v )H ,
4 Λb − Λ 5 Λb − Λ · ′ 6
GT = (m m )H m (1 v v )H m (1 v v )H +m H +m H ,
1 Λb − Λ 1− Λ − · ′ 2− Λb − · ′ 3 Λb 5 Λ 6
GT = m H m H m H ,
2 Λb 1− Λ 2− Λb 3
GT = m H +m H +m H m H ,
3 Λ 1 Λ 2 Λb 3− Λ 6
GT = (m +m )H +m (1+v v )H . (38)
4 Λb Λ 5 Λ · ′ 6
For JP =3/2+, a state with unnatural parity, we have
hΛ(pΛ)|siσµνqνb|Λb(pΛb)i = uα(pΛ,sΛ) vα F1Tγµ+F2Tvµ+F3Tv′µ +
(cid:20) (cid:18) (cid:19)
FTgαµ γ u(p ,s ), (39)
4 5 Λb Λb
(cid:21)
hΛ(pΛ)|siσµνγ5qνb|Λb(pΛb)i = uα(pΛ,sΛ) vα GT1γµ+GT2vµ+GT3v′µ +
(cid:20) (cid:18) (cid:19)
GTgαµ u(p ,s ), (40)
4 Λb Λb
(cid:21)
with
FT = (m m )H (m m v v )H (m v v m )H m H ,
1 Λb − Λ 1− Λb − Λ · ′ 2− Λb · ′− Λ 3− Λb 5
FT = m H (m +m )H +(m v v m )H m H ,
2 Λb 1− Λb Λ 2 Λb · ′− Λ 4− Λb 6
F3T = mΛH1−(mΛb +mΛ)H3−(mΛb −mΛv·v′)H4,
FT = (m +m )H +(m m v v )H ,
4 − Λb Λ 5 Λb − Λ · ′ 6
GT = (m +m )H +m (1+v v )H +m (1+v v )H +m H +m H ,
1 − Λb Λ 1 Λ · ′ 2 Λb · ′ 3 Λb 5 Λ 6
GT = m H m H m H ,
2 Λb 1− Λ 2− Λb 3
GT = m H m H m H m H ,
3 Λ 1− Λ 2− Λb 3− Λ 6
GT4 = −(mΛb −mΛ)H5−mΛ(1−v·v′)H6. (41)
We can now use these redefined tensor form factors to write Eqs. 2 and 3 in a simplified form.
For transitions to states with J =1/2, these two equations become
H1µ = u(pΛ,sΛ) γµ A1+B1γ5 +vµ A2+B2γ5 +v′µ A3+B3γ5 u(pΛb,sΛb),
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(42)
Hµ = u(p ,s ) γµ D +E γ +vµ D +E γ +vµ D +E γ u(p ,s ).
2 Λ Λ 1 1 5 2 2 5 ′ 3 3 5 Λb Λb
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(43)
For transitions to states with J =3/2,
Hµ = u (p ,s ) vα γµ A +B γ +vµ A +B γ +vµ A +B γ
1 α Λ Λ 1 1 5 2 2 5 ′ 3 3 5
(cid:20) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
+gαµ A +B γ u(p ,s ), (44)
4 4 5 Λb Λb
(cid:18) (cid:19)(cid:21)
Hµ = u (p ,s ) vα γµ D +E γ +vµ D +E γ +vµ D +E γ
2 α Λ Λ 1 1 5 2 2 5 ′ 3 3 5
(cid:20) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
+gαµ D +E γ u(p ,s ), (45)
4 4 5 Λb Λb
(cid:18) (cid:19)(cid:21)
Rare dileptonic decays of Λ in a quark model 8
b
p−=(Eℓ,~pℓ)
pΛb =(EΛb,~ph) θ
pΛ=(EΛ,p~h)
p+=(Eℓ,−p~ℓ)
FIG. 1: Kinematics of thedilepton rest frame.
while for those with J =5/2,
Hµ = u (p ,s )vα vβ γµ A +B γ +vµ A +B γ +vµ A +B γ
1 αβ Λ Λ 1 1 5 2 2 5 ′ 3 3 5
(cid:20) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
+gβµ A +B γ u(p ,s ), (46)
4 4 5 Λb Λb
(cid:18) (cid:19)(cid:21)
Hµ = u (p ,s )vα vβ γµ D +E γ +vµ D +E γ +vµ D +E γ
2 αβ Λ Λ 1 1 5 2 2 5 ′ 3 3 5
(cid:20) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
+gβµ D +E γ u(p ,s ). (47)
4 4 5 Λb Λb
(cid:18) (cid:19)(cid:21)
For states with natural parity spinors,
2m
A = bC FT +C F ,
i − q2 7 i 9 i
2m
B = bC GT C G ,
i − q2 7 i − 9 i
D = C F , E = C G , (48)
i 10 i i 10 i
−
and for states with unnatural parity,
2m
A = bC GT C G ,
i − q2 7 i − 9 i
2m
B = bC FT +C F ,
i − q2 7 i 9 i
D = C G , E =C F . (49)
i 10 i i 10 i
−
B. Decays Rates and Forward-Backward Asymmetries
The decay rate is given by
1 d3p 1
dΓ= f (2π)4δ(4) p p 2. (50)
2m  (2π)32E   Λb − f|M|
Λb f f f
Y X
   
Forthecaseofunpolarizedbaryons, 2 isthesquaredamplitudeaveragedovertheinitialpolarizationandsummed
|M|
over the final polarizations.
For dileptonic decays, the squared averageamplitude is
G2α2
2 = F em V V 2(HµνL +HµνL +HµνL +HµνL ). (51)
|M| 24π2 | tb t∗s| a aµν b bµν c cµν d dµν
Rare dileptonic decays of Λ in a quark model 9
b
For unpolarized leptons, the leptonic tensors are
Laµν = L(µV)†L(νV) =4 p+µp−ν +p+νp−µ−(p−·p++m2ℓ)gµν , (52)
spin
X (cid:2) (cid:3)
Lbµν = L(µA)†L(νA) =4 p+µp−ν +p+νp−µ−(p−·p+−m2ℓ)gµν , (53)
spin
X (cid:2) (cid:3)
Lcµν = L(µV)†L(νA) =4iεµναβpαpβ+, (54)
−
spin
X
L = L(A) L(V) =4iε pαpβ. (55)
dµν µ † ν µναβ +
−
spin
X
The hadronic tensors Hµν are
f
Haµν = H1µ†H1ν, Hbµν = H2µ†H2ν, (56)
pol pol
X X
Hcµν = H1µ†H2ν, Hdµν = H2µ†H1ν. (57)
pol pol
X X
The most general Lorentz structure for each of these hadronic tensors is
Hµν = αfgµν +βf QµQν +βf Qµqν +βf qµQν +βf qµqν +iγfεµναβQ q , (58)
f − ++ +− −+ −− α β
where Q p +p and q =p p is the 4-momentum transfer.
≡ Λb Λ Λb − Λ
We carry out our calculations in the dilepton rest frame (see Fig. 1). In this frame
m m
E = Λb(1 r+sˆ), E = Λb(1 r sˆ),
Λb 2√sˆ − Λ 2√sˆ − −
m φ(sˆ) m m
p = Λb , E = Λb√sˆ, p = Λb sˆψ(sˆ).
h ℓ ℓ
2 sˆ 2 2
r
p
p is the 3-momentum of either baryon in this frame. In addition, we have defined
h
sˆ q2/m2 , r m2/m2 , mˆ m /m ,
≡ Λb ≡ Λ Λb ℓ ≡ ℓ Λb
φ(sˆ) = (1 r)2 2(1+r)sˆ+sˆ2, ψ(sˆ)=1 4mˆ2/sˆ.
− − − ℓ
Performing the contractions,the differential decay rate becomes
d2Γ m G2α2
= Λb F em V V 2 φ(sˆ)ψ(sˆ) (sˆ,zˆ), (59)
dsˆdzˆ 213π5 | tb t∗s| F0
p
where zˆ=cosθ. In these decays, 4mˆ2 sˆ (1 √r)2 and 1 zˆ 1. The normalized rate (sˆ,zˆ) has the form
ℓ ≤ ≤ − − ≤ ≤ F0
(sˆ,zˆ)= (sˆ)+zˆ (sˆ)+zˆ2 (sˆ), (60)
0 0 1 2
F I I I
where
(sˆ)=αaA +βa A +αbB +βb B +βb B +βb B +βb B , (61)
I0 α ++ ++ α ++ ++ +− +− −+ −+ −− −−
and
A = 4m2 (2mˆ2+sˆ), A =2m4 φ(sˆ),
α Λb ℓ ++ Λb
B = 4m2 (sˆ 6mˆ ), B =2m4 φ(sˆ)+4mˆ2(2(1+r) sˆ) ,
α Λb − ℓ ++ Λb ℓ −
B+− = B−+ =8m4Λbmˆ2ℓ(1−r), B−− =(cid:0) 8m4Λbmˆ2ℓsˆ. (cid:1) (62)
The terms proportional to zˆand zˆ2 are
(sˆ) = 4m4 sˆ φ(sˆ)ψ(sˆ)(γc+γd), (63)
I1 Λb
I2(sˆ) = −2m4Λbpφ(sˆ)ψ(sˆ)(β+a++β+b+), (64)
Rare dileptonic decays of Λ in a quark model 10
b
respectively. Integrating out the zˆdependence from Eq. 59, the decay rate becomes
dΓ m G2α2
= Λb F em V V 2 φ(sˆ)ψ(sˆ) (s), (65)
dsˆ 212π5 | tb t∗s| R0
p
where
1
(sˆ)= (sˆ)+ (sˆ). (66)
0 0 2
R I 3I
The functions and are given in Eqs. 61 and 64, respectively. The explicit forms of the coefficients α, β , and
0 2
γ are given in CI. I ±±
The forward-backwardasymmetry (FBA) is defined as
1 1 d2Γ 0 d2Γ
(sˆ)= dzˆ dzˆ . (67)
FB
A dΓ/dsˆ dsˆdzˆ− dsˆdzˆ
(cid:20)Z0 Z−1 (cid:21)
Using Eqs. 59 and 60, we obtain
(sˆ)
1
(sˆ)= I , (68)
AFB 2 (sˆ)+ 1 (sˆ)
I0 3I2
where is given in Eq. 63. Since this observable is wri(cid:2)tten as a ratio,(cid:3)it might be expected to be less dependent on
1
I
the form factors than the differential decay rate. HQET considerations may make it even less model-dependent [57],
so that it may provide a (somewhat) model-independent way of extracting the Wilson coefficients. Since also
FB
A
depends on the chirality of the hadronic and leptonic currents, this observable is also sensitive to any new physics
effects beyond the Standard Model.
III. HQET
Heavy quark effective theory (HQET) is an effective tool for examining the phenomenology of hadrons containing
a single heavy quark. It has been applied to hadronic matrix elements in many processes at higher and higher order
in the 1/m expansion, with m being the mass of the heavy quark. In this section we present the leading-order
Q Q
relationships among the HQET form factors and those presented in Section IIA for the heavy to light transitions
Λ Λ( ). The properties of the spinor-tensors that are used to represent the daughter baryons have been discussed
b ∗
→
near the start of section IIA.
To leading order in HQET, transitions between a heavy baryon and a light one are described in terms of only two
form factors. For any current operatorΓ, the hadronic matrix elements involving tensor states can be written as [86]
hΛ(∗)(pΛ(∗))|sΓb|Λb(v)i=uµ1...µn(pΛ(∗))Mµ1...µnΓu(v) (69)
where v is the velocity of the Λb baryon and Mµ1...µn is the most general tensor consistent with HQET that can be
constructedfromthe availablekinematic variables. We maynot use any factorsofγµi, pµi , orgµiµj in constructing
Λ(∗)
Mµ1...µn; therefore, it must take the form
Mµ1...µn =vµ1...vµn ξ1(n)(v·pΛ(∗))+v/ξ2(n)(v·pΛ(∗)) , (70)
(cid:20) (cid:21)
Forthecaseofapseudo-tensordaughterbaryon,thematrixelementhasthesameformasEq. 69exceptthatMµ1...µn
is now a pseudo-tensor. A pseudo-tensor when sandwiched between spinors can be constructed by multiplying an
ordinary tensor by γ ; thus, for transitions involving a pseudo-tensor spinor,
5
Mµ1...µn =vµ1...vµn ζ1(n)(v·pΛ(∗))+v/ζ2(n)(v·pΛ(∗)) γ5. (71)
(cid:20) (cid:21)
Using these forms of the matrix elements, the form factors defined in Section IIA are found to satisfy the following
relations in the limit when the b quark is infinitely heavy.
For any state with JP =1/2+,
F = G =H =H =0, F =G = H =2ξ(0),
3 3 3 4 2 2 − 2 2
F = ξ(0) ξ(0), G =H =ξ(0)+ξ(0), (72)
1 1 − 2 1 1 1 2