$\Lambda_c$ Semileptonic Decays in a Quark Model

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by  Md Mozammel Hussain

Λ Semileptonic Decays in a Quark Model
c
Md Mozammel Hussain and Winston Roberts
∗ †
Department of Physics, Florida State University, Tallahassee, 32306
Hadronic form factors for semileptonic decay of the Λc are calculated in a nonrelativistic quark
model. Thefullquarkmodelwavefunctionsareemployedtonumericallycalculatetheformfactors
to all relevant orders in (1/mc, 1/ms). The form factors obtained satisfy relationships expected
from the heavy quark effective theory (HQET). The differential decay rates and branching frac-
tions are calculated for transitions to the ground state and a number of excited states of Λ. The
branchingfractionofthesemileptonicdecaywidthtothetotalwidthofΛc hasbeencalculatedand
compared with other theoretical estimates and experimental results. The branching fractions for
Λc →Λ∗l+νl →Σπl+νl andΛc →Λ∗l+νl →NK¯l+νl arealsocalculated. Apartfromdecaystothe
7 ground state Λ(1115), it is found that decays through the Λ(1405) provide a significant portion of
1 the branching fraction Λc → Xslνl. A new estimate for f = B(Λ+c → Λl+νl)/B(Λ+c → Xsl+νl) is
0 obtained.
2

b
e
F
I. INTRODUCTION AND MOTIVATION

1
2
Semileptonic decays of hadrons are the main sources for precise knowledge on Cabibo-Kobayashi-Maskawa(CKM)

matrix elements [1]. The form factors that parametrize the non-perturbative QCD effects in these transitions play a
]
h crucial role in the extraction of CKM matrix elements and the precision depends on how well the form factors are
-t calculated.
l A great deal of work has been done on semileptonic decay processes to calculate and improve the modeling of the
c
u form factors. For example, monopole type form factors were used to study semileptonic decay of heavy mesons by
n Wirbel, Stech and Bauer [2]. Isgur, Scora, Grinstein and Wise caculated the semileptonic B and D meson decays in
[ a non-relativistic quarkmodel [3]. Lattice QCD calculationsof semileptonic decay form factorshave been done in ref

[4]. These are a very few out of a huge number of articles. More work has been done on semileptonic meson decays
2
v than baryon decays. Pervin, Roberts and Capstick worked on semileptonic baryon decays of ΛQ [5] and ΩQ [6]) in a
6 constituent quark model. Some baryon decays have also been addressed in QCD sum rules [7], perturbative lattice
7 QCD [8] and a number of other approaches [9].
8 The description of the weak decays of heavy hadrons are somewhat simplified because of the so-called heavy
3
quark symmetry. This was first pointed out by Isgur and Wise [10]. Hadrons containing one heavy quark Q (with
0
m >> Λ ) possess this symmetry, which has been formalized into the heavy quark effective theory (HQET). In
. Q QCD
1 HQET the properties of the hadrons are governed by the light degrees of freedom and are independent of the heavy
0
quarkdegreesoffreedom. For semileptonic decaysofheavyhadrons,HQETreducesthe number ofindependent form
7
factors needed to describe the decays.
1
In this paper, we examine the semileptonic decays of the Λ+ to a number of Λs, including the ground state.
: c
v Because it is the lightest charmed baryon, Λ+ plays an important role in understanding charm and bottom baryons.
c
Xi The lowest-lyingbottom baryonis mostoften detected throughits weakdecay to Λ+c . In addition, the study of allof
theΛ+-typeandΣ -typebaryonsaredirectlylinkedtothe understandingofthegroundstateofΛ+,asthesebaryons
r c c c
a eventually decay into a Λ+.
c
Among the branching fractions of the Λ , (Λ+ pK π+) is used to normalize most of its other branching
c B c → −
fractions. The Particle Data Group (PDG), in their previous version [11] reported that there was no model inde-
pendent measurement of (Λ+ pK π+). Two model-dependent measurements were reported, with two different
results obtained from diffBerenct a→ssump−tions. The model that calculated branching fractions (Λ+ pK π+) from
B c → −
semileptonic decays, estimated that
B(D Xl+ν )
B(Λ+c →pK−π+)=RfF 1+→Vcd 2 l τ(Λ+c ), (1)
|Vcs|
where,
R=B(Λ+ pK π+)/B(Λ+ Λl+ν ),
c → − c → l
∗ [email protected]
† [email protected]
2
f =B(Λ+ Λl+ν )/B(Λ+ X l+ν ),
c → l c → s l
F =B(Λ+ X l+ν )/B(D X l+ν ).
c → s l → s l
They estimated B(Λ+ pK π+) = (7.3 1.4)% with the theoretical estimate of f = F = 1.0 with significant
c → − ±
uncertainties.
However,intheirmostrecentrelease,PDG[12]reportsamodelindependentmeasurementof (Λ+ pK π+). A.
Zupancet al. (BelleCollaboration)[13]measuredittobe6.84+0.32%,whileM.Ablikimet al. (BBESIcII→Collab−oration)
0.40
[14] measured it to be 5.84 0.27 0.23%. The PDG fit is 6.3−5 0.33% that leads to a new estimate of
± ± ±
f =B(Λ+ Λl+ν )/B(Λ+ X l+ν )=0.87+0.13,
c → l c → s l −0.17
with the assumption of F =1.0. Pervin,Roberts and Capstick (PRCI) [5] estimated the value of f to be 0.85 0.04.
±
MottandRoberts[15]laterestimatedtheraredecaybranchingfractionsoftheΛ usingtwodifferentmethods. Their
b
results indicated that the results were sensitive to the precision with which the form factors were estimated, and
this further implied that f could be even smaller than 0.85. The semileptonic branching fraction, (Λ+ Λl+ν ) is
reportedtobe 2.8 0.5%withthe assumptionthatthe Λ+ decaysonlytothe groundstate Λ(1115)B. Noc s→emileptolnic
± c
decays to excited Λ have been reported. This provides the motivation for our work.
There have been a number of theoretical articles on the semileptonic decay of Λ+ in recent years. Gutsche et al.
c
used a covariantquark modelto estimate the branching fractionfor Λ Λl+ν [16]. Liu et al. usedQCD light cone
c l
→
sum rules to examine this decay [17], while Ikeno and Oset have examined the semileptonic decay to the Λ(1405),
treating that state as a dynamically generated molecular state [18].
Intheworkpresentedherein,weworkintheframeworkofaconstituentquarkmodel. Suchmodelshavebeenquite
successful in explaining the main features of hadron phenomenology. In computing the form factors for Λ Λ , we
c ∗
→
have deployedtwo approximations. In the first approximation,single component wavefunctions are usedto compute
the analytic form factors for Λ Λ transitions. As in PRCI [5] a variational diagonalization of a quark model
c ∗
→
Hamiltonian was used to extract the single component wave functions and the quark operators were reduced to their
non-relativistic Pauli form. In the second method we keep the full relativistic form of the quark spinors and use the
full quark model wave functions. We believe that this second method provides more reliable numerical values of the
form factors as it uses fewer approximations.
We calculate the decay widths and branching fractions for decays to ground state and a number of excited Λ( ).
∗
We also study the decay widths and branching fractions of two other decay channels, namely Λ+ Σπl+ν and
Λ+ NK¯l+ν , via a set of Λ resonances. c → l
c → l
The rest of this paper is organized as follows: in section II, we discuss the hadronic matrix elements and decay
rates. Section III presents a concise overview of HQET and the relationships predicted by HQET among the form
factors for the transitions we study. In section IV we describe the model we employ to calculate the form factors.
Section V is devoted to discussing the numerical results such as form factors, decay rates and branching fractions.
Section VI presents our conclusions and outlook. A number of details ofthe calculationare shownin the appendices.
II. MATRIX ELEMENTS AND DECAY WIDTHS
A. Semileptonic decay (Λ+ →Λl+ν)
c l
1. Matrix Elements
Fig.1 depicts the semileptonic decay Λ+ Λ( )l+ν . We work in the rest frame of the parent Λ . The transition
c → ∗ l c
matrix element for the decay is
G
= FVcsLµ Λ(p′,s′)Jµ Λc(p,s) , (2)
M √2 h | | i
where V is the CKM matrix element, Lµ = u¯ γµ(1 γ )v is the lepton current and J = s¯γ (1 γ )c is the
cs νl − 5 l µ µ − 5
hadronic current. The momenta of the Λ , Λ, l, ν are labeled as p, p, p and p , respectively. The hadronic matrix
c l ′ l νl
element is defined as
H = Λ( ) J Λ . (3)
µ ∗ µ c
h | | i
Thehadronicmatrixelementsareparametrizedintermsofanumberofformfactors. Fortransitionsfromtheground
state Λ (JP = 1+) to the ground state Λ (JP = 1+), the matrix elements for the vector (V ) and axial-vector (A )
c 2 2 µ µ
3
FIG. 1: Semileptonic decay Λ+ Λ( )l+ν
c → ∗ l
✰
❧
✗
(cid:0)
✰
✄
❝
✄
currents are, respectively,
pµ p′µ
Λ(p′,s′)Vµ Λc(p,s) =u¯(p′,s′) γµF1+ F2+ F3 u(p,s), (4)
h | | i m m
(cid:18) Λc Λ (cid:19)
Λ(p,s)A Λ (p,s) =u¯(p,s) γ G + pµ G + p′µ G γ u(p,s), (5)
′ ′ µ c ′ ′ µ 1 2 3 5
h | | i m m
(cid:18) Λc Λ (cid:19)
where the F ’s and G ’s are the form factors and s(s) is the spin of Λ (Λ). The matrix elements for transitions to a
i i ′ c
daughter baryon with JP = 3− are
2
Λ(p,s)V Λ (p,s) =u¯α(p,s) pα γ F + pµ F + p′µ F +g F u(p,s), (6)
′ ′ µ c ′ ′ µ 1 2 3 αµ 4
h | | i m m m
(cid:20) Λc(cid:18) Λc Λ (cid:19) (cid:21)
Λ(p′,s′)Aµ Λc(p,s) =u¯α(p′,s′) pα γµG1+ pµ G2+ p′µ G3 +gαµG4 γ5u(p,s). (7)
h | | i m m m
(cid:20) Λc(cid:18) Λc Λ (cid:19) (cid:21)
The Rarita-Schwinger spinor u¯α satisfies the conditions
p′αu¯α(p′,s′)=0, u¯α(p′,s′)γα =0, u¯α(p′,s′)/p′ =mΛ3/2u¯α(p′,s′). (8)
The corresponding matrix elements for transitions to a daughter baryon with Jp = 5+ are
2
Λ(p,s)V Λ (p,s) =u¯αβ(p,s) pα pβ γ F + pµ F + p′µ F +g F u(p,s).
′ ′ µ c ′ ′ µ 1 2 3 βµ 4
h | | i m m m m
Λc(cid:20) Λc(cid:18) Λc Λ (cid:19) (cid:21)
Λc(p′,s′)Aµ Λc(p,s) =u¯αβ(p′,s′) pα pβ γµG1+ pµ G2+ p′µ G3 +gβµG4 γ5u(p,s).
h | | i m m m m
Λc(cid:20) Λc(cid:18) Λc Λ (cid:19) (cid:21)
The spinor u¯αβ satisfies the conditions
p u¯αβ(p,s)=p u¯αβ(p,s)=0, u¯αβ(p,s)γ =u¯αβ(p,s)γ =0,
′α ′ ′ ′β ′ ′ ′ ′ α ′ ′ β
u¯αβ(p′,s′)/p′ =mΛ5/2u¯αβ(p′,s′), u¯αβ(p′,s′)gαβ =0.
Here we have shown the hadronic transition matrix elements for the decays to daughter baryons with natural parity.
For decays to states with unnatural parity, the matrix elements are constructed by switching γ from the equations
5
defining the G to the equations defining the F .
i i
2. Decay Width
The differential decay rate for the transition Λ Λ( )l+ν is
c ∗ l
→
1 d3p d3p d3p (2π)4δ4(p p p p )
dΓ= 2 l νl ′ − ′− l− νl , (9)
2m |M| 2E2E 2E (2π)3(2π)3(2π)3
Λc l νl ′
4
where
G2 1
|M|2 = 2F|Vcs|22 Hµ†HνLµ†Lν,
spins
X
G2
= F V 2H Lµν. (10)
cs µν
4 | |
2 is the squared amplitude averagedover the initial spins (the factor of 1) and summed over the final spins.
|M| 2
The most general Lorentz form of the hadronic tensor can be written as
H =αg +β P P +β P L +β L P +β L L +iγǫ P L , (11)
µν µν PP µ ν PL µ ν LP µ ν LL µ ν µνρσ ρ σ
where we have defined P =p and L=p p. The lepton tensor is
′ ′
−
Lµν =8[pµpν +pµpν gµν(p .p )+iǫµναβp p ]. (12)
l νl νl l − l νl lα νlβ
Integrating over the lepton momenta allows us to write the lepton tensor as
d3p d3p
l νl Lµν = dΩ (Agµν +ALµLν), (13)
(2π)3(2π)32E2E l ′
Z l νl Z
where q2 =(p p)2 and
′
−
(q2 m2)2(2q2+m2) (q2 m2)2(q2+2m2)
A= − l l , A′ = − l l . (14)
− 384π6q4 192π6q6
The complete expression for the differential decay rate becomes
dΓ V 2 G2 (q2 m2)2
= | cs| F λ1/2(m2 ,m2,q2) − l 6αq2+β 2q2 (P L) m2 +m2 4(P L) m2 (15)
dq2 192 π3m3 Λc Λ 4q4 − PP · − Λ l · − Λ
Λc (cid:18) h (cid:0) (cid:1) (cid:0) (cid:1)i
+ β (P L)+β (P L)+β q2 3m2 ,
LP · PL · LL l
h i (cid:19)
where P L = 1(m2 m2 q2) and λ1/2(x,y,z) = (x2 +y2 +z2 2xy 2yz 2zx)1/2. When contracted with
· 2 Λc − Λ − − − −
the lepton tensor, all of the βs (except β ) are proportional to powers of the lepton mass m and thus give small
PP l
contributions to the decay rate. The complete form of β is given in appendix E1.
PP
B. Λc →Λ∗l+νl →Σπl+νl/NK¯l+νl
We include six Λ( ) in this calculation. We denote these as Λ , i = 1...6. In this notation, Λ = Λ(1115) 1/2+;
∗ i 1
Λ = Λ(1600) 1/2+; Λ = Λ(1405) 1/2 ; Λ = Λ(1520)3/2 ; Λ = Λ(1890) 3/2+; Λ = Λ(1820) 5/2+. With the
2 3 − 4 − 5 6
exception of Λ , these excited Λ are not stable particles and will decay strongly to Σπ or NK¯. Thus we study the
1 i
four-body decays, Λ Λ l+ν Σπl+ν and Λ Λ l+ν NK¯l+ν as shown in Fig. 2(a, b). There are other
c i l l c i l l
→ → → →
contributions to each of these four-body final states, two of which are shown in Fig. 2 (c, d). However, in each case,
the intermediate resonanceis very heavy andvery far from the mass shell. Thus, we expect these contributions to be
small.
1. Kinematics
Fig 3 shows the kinematic diagram for the four-particle decay Λ Σπl+ν . We define
c l
→
P p +p , Q p p , L p +p , (16)
Σ π Σ π l ν
≡ ≡ − ≡
so that p =P +L. In the restframe of the Λ , the back-to backmomenta P~ andL~ define a commonz-axis. In the
Λc c
rest frame of the daughter hadrons, θ is the polar angle between the pion momentum and P~. Similarly, in the rest
h∗
frame of the lepton pair, θ is the polar angle between the lepton momentum and L~. φ is then the angle between
l∗ ∗
the lepton and hadron planes.
5
FIG. 2: (a) shows the semileptonic decay Λ+ Λ l+ν followed by the strong decay Λ Σπ; (b) shows the
semileptonic decay Λ+ Λ l+ν followed bcy th→e stron∗g delcay Λ NK¯; (c) shows the strong→decay Λ+ Σ π fol-
lowed by the semileptcon→ic de∗caylΣ Σl+ν ; (d) shows the stron→g decay Λ+ D N followed by thecse→mile∗cptonic
decay D K¯l+ν . ∗c → l c → ∗
∗ l
→
❧✰ ❧✰
✗(cid:0) ✗(cid:0)
✄✰❝ ✙ ✄✰❝ ❑✖
✁
✁ ✄
✄
✭❛✮ ✭❜✮ ◆
✝
✙ ❧✰ ❧✰
✗(cid:0) ◆ ✗(cid:0)
✰ ✰
✄❝ ✁ ✄❝ ✁
✝❝ ❉
✝ ❑✖
✭✂✮ ✭❞✮
FIG. 3: Kinematics for the process Λ Σπlν. The lepton momenta define the lepton plane, while the momenta of
c
→
the hadrons define the hadron plane.
✙
✣
(cid:0)
✒❤
✒❧
✗❧
✝
In the overallrest frame of Λ , the momenta P and L are
c
1 1
P = (m2 +S q2), 0, 0, λ1/2(m2 ,S ,q2) ,
2m Λc Σπ − 2m Λc Σπ
(cid:16) Λc Λc (cid:17)
1 1
L= (m2 S +q2), 0, 0, λ1/2(m2 ,S ,q2) .
2m Λc − Σπ −2m Λc Σπ
(cid:16) Λc Λc (cid:17)
In the rest frame of the daughter hadrons, the momenta p and p are
Σ π
1 1 1
p = (S m2 +m2), λ1/2(S ,m2,m2)sinθ , 0, λ1/2(S ,m2,m2)cosθ ,
π 2√S Σπ − Σ π 2√S Σπ Σ π h∗ 2√S Σπ Σ π h∗
Σπ Σπ Σπ
(cid:16) (cid:17)
1 1 1
p = (S +m2 m2), λ1/2(S ,m2,m2)sinθ , 0, λ1/2(S ,m2,m2)cosθ .
Σ 2√S Σπ Σ− π −2√S Σπ Σ π h∗ −2√S Σπ Σ π h∗
Σπ Σπ Σπ
(cid:16) (cid:17)
In the rest frame of the lepton pair, the lepton momenta are
1 1 1
p = (q2+m2), (q2 m2)sinθ , 0, (q2 m2)cosθ ,
l 2 q2 l 2 q2 − l l∗ 2 q2 − l l∗
(cid:16) (cid:17)
p p p
6
1 1 1
pν = 2 q2(q2−m2l), −2 q2(q2−m2l)sinθl∗, 0, −2 q2(q2−m2l)cosθl∗ .
(cid:16) (cid:17)
p p p
2. Matrix Elements
The hadron matrix elements for the decays Λ Λ lν BMlν , where B is a baryon with JP =1/2+ and M is
c i l l
→ →
a pseudoscalar meson, can be written as
(B(p )M(p )) J Λ (p ) =u¯(p )ΥsR(P) iu(P +L). (17)
h B M i| µ| c Λc i B Jµ
In this expression,Υs representsthe strongdecay vertex, p andp are the momentaof the daughterbaryonB and
B M
mesonM, respectively,R(P)is the propagatorwith momentumP. J is the weakcurrentleadingto the weakdecay,
µ
while i is the matrix element for the semileptonic decay Λ Λ , written in terms of the form factors of section
Jµ c → i
IIA1. In this notation, the momenta of eqn. 16 are more generally written as
P p +p , Q p p , L p +p . (18)
B M B M l ν
≡ ≡ − ≡
When the intermediate baryon has JP =1/2+, the hadron matrix elements are
(P +L) P
B(p )M(p )Vi Λ (P +L) =g u¯(p )γ (P/ +Mi) i γ Fi+ µFi+ µ Fi u(P +L),
h B M | µ| c i ΛiBM B 5 Γ ∇ µ 1 m 2 m 3
(cid:20) Λc Λi (cid:21)
(P +L) P
B(p )M(p )Ai Λ (P +L) =g u¯(p )γ (P/ +Mi) i γ Gi + µGi + µ Gi γ u(P +L), (19)
h B M | µ| c i ΛiBM B 5 Γ ∇ µ 1 m 2 m 3 5
(cid:20) Λc Λi (cid:21)
where i = 1/(P2 Mi2) and Mi = m iΓ /2, with m and Γ the mass and total decay width of the Λ ,
∇ − Γ Γ Λi − i Λi i i
respectively. g is the strong coupling constant for the decay Λ BM.
For an intermΛieBdMiate state with JP =3/2 , the hadron matrix eleim→ents are
−
p (P +L) (P +L)
B(p )M(p )Vi Λ (P +L) =g u¯(p )γ MαRαβ(P) i β γ Fi+ µFi
h B M | µ| c i ΛiBM B 5m ∇ m µ 1 m 2
M (cid:20) Λc (cid:18) Λc
P
+ µ Fi +g Fi u(P +L),
m 3 βµ 4
Λi (cid:19) (cid:21)
p (P +L) (P +L)
B(p )M(p )Ai Λ (P +L) =g u¯(p )γ MαRαβ(P) i β γ Gi + µGi
h B M | µ| c i ΛiBM B 5m ∇ m µ 1 m 2
M (cid:20) Λc (cid:18) Λc
p
+ µ Gi +g Gi γ u(P +L), (20)
m 3 βµ 4 5
Λi (cid:19) (cid:21)
where Rαβ(P) is the Rarita-Schwinger tensor for a massive spin 3/2 propagator,which takes the form
1 2PαPβ γαPβ γβPα
Rαβ(P)= (P/ +Mi) gαβ γαγβ − .
− Γ − 3 − 3m2 − 3m
(cid:20) Λi Λi (cid:21)
For an intermediate state with JP =5/2+, the hadronic matrix elements are
pα′pβ′ (P +L) (P +L) (P +L)
B(p )M(p )Vi Λ (p ) =g u¯(p )γ M MRαβ (P) i α β γ Fi+ µFi
h B M | µ| c Λc i ΛiBM B 5 m2 α′β′ ∇ m m µ 1 m 2
M Λc (cid:20) Λc (cid:18) Λc
P
+ µ Fi +g Fi u(p),
m 3 βµ 4
Λi (cid:19) (cid:21)
pα′pβ′ (P +L) (P +L) (P +L)
B(p )M(p )Ai Λ (P +L) =g u¯(p )γ M MRαβ (P) i α β γ Gi + µGi
h B M | µ| c i ΛiBM B 5 m2 α′β′ ∇ m m µ 1 m 2
M Λc (cid:20) Λc (cid:18) Λc
P
+ µ Gi +g Gi γ u(P +L), (21)
m 3 βµ 4 5
Λi (cid:19) (cid:21)
where Rαβ (P) is the Rarita-Schwinger propagator tensor for a massive particle with total angular momentum 5/2
α′β′
[19].
7
We need to cast the matrix elements from the previous three equations into a more generalform that makes it easier
to organize the calculation. The most general form of the contribution of the ith state to the matrix element for the
four-body decay Λ+ Λ l+ν BMl+ν can be written
c → i l → l
16
Mi =u¯(p ) ci u(P +L),
ν B  jOj
j=1
X
 
where the Lorentz-Dirac operators are
i
O
=γ , =P/γ , =P , =P/P , =L , =P/L , =Q , =P/Q ,
1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 7 ν 8 ν
O O O O O O O O
=γ γ , =P/γ γ , =P γ , =P/P γ , =L γ , =P/L γ , =Q γ , =P/Q γ .
9 ν 5 10 ν 5 11 ν 5 12 ν 5 13 ν 5 14 ν 5 15 ν 5 16 ν 5
O O O O O O O O
Because of the forms of the propagators in eqns. 19 - 21, there are no terms containing L/ or Q/ among the . For
i
O
the cases of eqns. 20 and 21, the meson momentum pM can be replaced by P −pB, and any factors of /pB can be
commuted leftwarduntil they are adjacentto the spinor u¯(p ). The Dirac equationcanthen be used to write this as
B
the scalar m .
B
The ci can be written
j
g
ci = ΛiBM (Ci,FF +Ci,GG ). (22)
j P2 Mi2 jk k jk k
− Γ Xk
where k runs from 1 to 3 for spin 1 states and from 1 to 4 for states with higher spin.
2
In the above, we have shown the forms for the Λ states with natural parity. For the states with unnatural parity,
i
the weak and strong vertices each acquire an extra multiplicative factor of γ .
5
3. Decay Width
The differential decay rate for the decay Λ+ BMl+ν is,
c → l
1 G2 d3p d3p d3p d3p
dΓ= F V 2 B M l νlδ4(p p p p p )H Lµν, (23)
2m 2 | cs| 2E 2E 2E2E Λc − B − M − l− νl µν
Λc B M l νl
The hadron tensor that arises from each intermediate state i can be written as,
Hi = Mi Mi =αig +βi P P +βi P Q +βi Q P +βi Q Q
µν µ† ν µν PP µ ν PQ µ ν QP µ ν QQ µ ν
spins
X
+βi Q L +βi L Q +βi L L +βi P L +βi P L
QL µ ν LQ µ ν LL µ ν PL µ ν PL µ ν
+iγiǫµνρδP Q +iγiǫµνρδL P +iγiǫµνρδL Q +iγiǫσµρδL P Q P +iγiǫσµρδL P Q Q
a ρ δ b ρ δ c ρ δ d σ ρ δ ν e σ ρ δ ν
+iγiǫσµρδL P Q L +iγiǫσνρδL P Q P +iγiǫσνρδL P Q Q +iγiǫσνρδL P Q L .
f σ ρ δ ν g σ ρ δ µ h σ ρ δ µ k σ ρ δ µ
In this expression,
16
αi = aijkcij†cik, (24)
j,k=1
X
with similar forms for all of the other coefficients. The terms in γ do not contribute to the decay rates that we
i
consider, due to the symmetry of the lepton tensor.
For the process Λ BMlν we examine the contribution from each Λ individually, as well as the coherent
c l i
→
contribution of all the Λ . For the coherent sum, we write
i
16 6 16
M =u¯(p ) u(p )=u¯(p ) ci u(p ), (25)
ν B  CjOj Λc B  jOj Λc
j=1 i=1 j=1
X X X
   
8
which ultimately leads to
6
= ci. (26)
Cj j
i=1
X
Integrating over the lepton momenta, and making use of eqn. (13) leads to
dΓ V 2 4πG2
dS dq2dθ dθ dφ =| c2s| 128m3 SF sinθhλ1/2(m2Λc,SB2M,q2)λ1/2(SB2M,m2B,m2M) α 4A+A′q2
BM h l Λc BM (cid:18) h i
+β AS +A(P L)2 + β +β A(P Q)+A(P L)(Q L)
PP BM ′ PQ QP ′
· · · ·
h i h ih i
+β A(Q Q)+A(Q L)2
QQ ′
· ·
h i
+ (β +β )(P L)+(β +β )(Q L)+β q2 A+Aq2 , (27)
LP PL LQ QL LL ′
· ·
h ih i(cid:19)
where
P P =p2 S ,
· Λ∗ ≡ BM
P Q=m2 m2 ,
· B− M
P L=(m2 S q2)/2,
· Λc − BM −
L L=q2,
·
Q Q=2m2 +2m2 S ,
· B M − BM
1
Q L= (m2 m2 )(m2 S q2)+cosθ λ1/2(m2 ,S ,q2)λ1/2(S ,m2,m2 ) .
· 2S B − M Λc − BM − h Λc BM BM B M
BM
h i
III. HEAVY QUARK EFFECTIVE THEORY
The heavy quark effective theory (HQET) has been a very useful tool in the study of the electroweak decays of
hadrons containing one heavy quark. In this effective theory, the matrix elements are expanded in increasing orders
of 1/m , where m is the mass of the heavy quark. This expansion has facilitated the extraction of CKM matrix
Q Q
elements with decreasing model dependence.
Hadronscontainingasingle charmorbeauty quarkareconsideredtobe heavyhadronsas the massm >>Λ .
Q QCD
Forsuchhadrons,HQETreducesthenumberofindependentformfactorsrequiredtodescribethetransitionsmediated
by electroweak transitions that change a heavy quark of one flavor into a heavy quark of different flavor. At leading
order in the 1/m expansion, such heavy to heavy transitions require a single form factor, the so-called Isgur-Wise
Q
function. This is the case independent of the total angular momentum of the daughter hadron (we assume that
the parent hadron is a ground-state hadron), integer (meson) or half-integer (baryon). For transitions between a
ground-state heavy hadron and a light one, HQET is not as powerful. However, for transitions between a heavy
baryon (ground state) and a light one, HQET indicates that a pair of form factors is all that is needed to describe
the transition, independent of the angular momentum of the daughter baryon.
The semileptonic decays Λ Λ fall into this second category, and are therefore described by two independent
c ∗
→
formfactors. We mayrepresentoneoftheselightbaryonsofangularmomentumJ byageneralizedRarita-Schwinger
field uµ1...µn(p) where n = J 1/2. This field is symmetric under exchange of any pair of its Lorentz indices, and
−
satisfies the conditions
p/uµ1...µn(p)=mΛuµ1...µn(p),γµ1uµ1...µn(p)=0,
p uµ1...µn(p)=0,uµ...µn(p)=0.
µ1 µ
The matrix element we are interested in is
hΛ∗(p′)|s¯Γc|Λ+c (p)i=u¯µ1...µnMµ1...µnΓu(p), (28)
where Γ = γµ or γµγ defines vector or axial vector current and M is a tensor. The most general tensor can
5 µ1µ2...µn
be constructed as
M =v ...v A , (29)
µ1µ2...µn µ1 µn n
9
where An is the most general Lorentz scalar that can be constructed. This takes the form
A =ξ(n)+v/ξ(n), (30)
n 1 2
where v =p/m is the velocityof the parentbaryon. For the transitions to daughterbaryonswith unnaturalparity,
Λc
M must be a pseudo-tensor. This is easily constructed by including a factor of γ , so that
µ1µ2...µn 5
M =v ...v ζ(n)+v/ζ(n) γ . (31)
µ1µ2...µn µ1 µn 1 2 5
(cid:18) (cid:19)
A. Form Factors
The matrix elements can be written in terms of six general form factors for spin 1±, or eight general form factors
2
for spin 3± and 5+, as shown in Section IIA1. Comparing the predictions of HQET with the most general form of
2 2
the matrix elementsleadsto a numberofrelationsamongthe generalformfactorsF /G andthe HQETformfactors
i i
ξ /ζ .
i i
For spin 1+, these relationships are
2
F =ξ(0) ξ(0),G =ξ(0)+ξ(0),F =G =2ξ(0),F =G =0. (32)
1 1 − 2 1 1 2 2 2 2 3 3
For spin 1−, they are
2
F = (ζ(0)+ζ(0)),G = (ζ(0) ζ(0)),F =G = 2ζ(0),F =G =0. (33)
1 − 1 2 1 − 1 − 2 2 2 − 2 3 3
For spin 3−, they are
2
F =ξ(1) ξ(1),G =ξ(1)+ξ(1),F =G =2ξ(1),F =G =0,F =G =0. (34)
1 1 − 2 1 1 2 2 2 2 3 3 4 4
For spin 3+, the relationships are
2
F = (ζ(1)+ζ(1)),G = (ζ(1) ζ(1)),F =G = 2ζ(1),F =G =0,F =G =0. (35)
1 − 1 2 1 − 1 − 2 2 2 − 2 3 3 4 4
For spin 5+, they are
2
(2) (2) (2) (2) (2)
F =ξ ξ ,G =ξ +ξ ),F =G =2ξ ,F =G =0,F =G =0. (36)
1 1 − 2 1 1 2 2 2 2 3 3 4 4
B. Decay Width
At leading order in HQET, the differential decay rates take simple forms for all the excited states we discuss. This
general form is
dΓ m2 m m2 1 m2
=ΦJX A +A l ξ2+ Λ B +B l ξ ξ + C +C l ξ2 , (37)
dq2 1 2 q2 1 m 1 2 q2 1 2 m2 1 2 q2 2
(cid:20)(cid:18) (cid:19) Λc (cid:18) (cid:19) Λc (cid:18) (cid:19) (cid:21)
where ΦJ is a dimensionless quantity that depends on the angular momentum of the daughter baryon. X,A ,B ,C
i i i
are, respectively,
4π3 G2
X = F v 2λ1/2(m2 ,m2,q2),
m3 2 | cs| Λc Λ
Λc
A = m4 +m2 q2 2m2 +m4 +m2q2 2q4 ,
1 Λc Λc − Λ Λ Λ −
(cid:20) (cid:18) (cid:19) (cid:21)
A = 2m4 m2 4m2 +q2 +2m4 m2q2 q4 ,
2 Λc − Λc Λ Λ− Λ −
(cid:20) (cid:18) (cid:19) (cid:21)
10
2
B =2 m4 2m2 m2 2q2 + m2 q2 ,
1 Λc − Λc Λ− Λ−
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
2
B =4 m4 +m2 q2 2m2 + m2 q2 ,
2 Λc Λc − Λ Λ−
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
3
C = m4 m2 +2q2 m2 2m4 3m2q2+q4 + m2 q2 ,
1 Λc Λ − Λc Λ− Λ Λ−
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
3
C = m4 2m2 +q2 +m2 4m4 +3m2q2+q4 +2 m2 q2 .
2 Λc Λ Λc − Λ Λ Λ−
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
ThedecaywidthforstateswithtotalspinJ doesnotdependonparity. TheΦJ forstateswithangularmomentum
J are
Φ1/2 =4,
2
Φ3/2 = λ m2 ,m2,q2 ,
3m2 m2 Λc Λ
Λc Λ
1 (cid:0) (cid:1)
Φ5/2 = λ2 m2 ,m2,q2 .
10m4 m4 Λc Λ
Λc Λ
(cid:0) (cid:1)
IV. THE MODEL
A. Wave Function Components
In our model, a baryon state has the form
Λ (p~,s) =3 3/2 d3p d3p CAΨS q (p~ ,s )q (p~ ,s )q (p~ ,s ) , (38)
| Q i − ρ λ ΛQ| 1 1 1 2 2 2 3 3 3 i
Z
where Λ is a flavored baryon (Λ+ or Λ) having a flavored quark (c or s) Q, which may or may not be considered
Q c
heavy. q (p~ ,s ) is the creationoperator for quark q with momentum ~p and spin s . q (p~ ,s )q (p~ ,s )q (p~ ,s ) is
i i i i i i 1 1 1 2 2 2 3 3 3
the three quarkstatewith quarksq havingmomentaandspins (p~ ,s ). p~ = 1 (p~ |p~ )andp~ = 1 (p~ +p~ 2ip~ )
i i i ρ √2 1− 2 λ √6 1 2− 3
are the Jacobi momenta. CA is the antisymmetric color wave function and ΨS = φ ψ χ is a symmetric
ΛQ ΛQ ΛQ ΛQ
combination of flavor, momentum and spin wave functions. For Λ the flavor wave function is
Q
1
φ = (ud du)Q. (39)
ΛQ √2 −
This is antisymmetric under the exchange of the first two quarks, so the spin-space wave function must also be
antisymmetric under such exchange.
The total spin of a system of three spin-1 particles can be either 3 or 1. The maximally stretched spin states are
2 2 2
χS (+3/2)= ,
3/2 |↑↑↑i
1
χρ (+1/2)= ( ),
1/2 √2 |↑↓↑i−↓↑↑i
1
χλ (+1/2)= ( + 2 ),
1/2 −√6 |↑↓↑i |↓↑↑i− |↑↑↓i
where the superscript S indicates that the state is totally symmetric under the exchange of any pair of quarks, while
ρ, λ denote the mixed-symmetric states that are antisymmetric and symmetric under the exchange of first two spins,
respectively.
The momentum-spacewavefunctionψ canbe constructedfromthe Clebsch-Gordansumofthe productofwave
ΛQ
functions of the two jacobi momenta p , p with total angular momentum L~ =~l +~l ,
ρ λ ρ λ
ψ (p ,p )= CLML ψ (p )ψ (p ). (40)
LMLnρlρnλlλ λ ρ lρm,lλML−m nρlρm ρ nλlλML−m λ
m
X