Combined nonrelativistic constituent quark model and heavy quark effective theory study of semileptonic decays of Λ_b and Ξ_b baryons

0.35 MB·English

by  C. Albertus

Combined nonrelativistic constituent quark model and heavy quark effective theory
study of semileptonic decays of Λ and Ξ baryons.
b b
C. Albertus,1 E. Hern´andez,2 and J. Nieves1
1Departamento de F´ısica Moderna, Universidad de Granada, E-18071 Granada, Spain.
2Grupo de F´ısica Nuclear, Facultad de Ciencias, E-37008 Salamanca, Spain.
We present the results of a nonrelativistic constituent quark model study of the semileptonic
decays Λ0 → Λ+l−ν¯ and Ξ0 → Ξ+l−ν¯ (l = e,µ). We work on coordinate space, with baryon
b c l b c l
wavefunctionsrecentlyobtainedfromavariationalapproachbasedonheavyquarksymmetry. We
developanovelexpansionoftheelectroweakcurrentoperator,whichsupplementedwithheavyquark
effectivetheoryconstraints,allowsustopredictthebaryonformfactorsandthedecaydistributions
for all q2 (or equivalently w) values accessible in the physical decays. Our results for the partially
integrated longitudinal and transverse decay widths, in the vicinity of the w = 1 point, are in
5 excellent agreement with lattice calculations. Comparison of our integrated Λb−decay width to
0 experiment allows us to extract the Vcb Cabbibo-Kobayashi-Maskawa matrix element for which we
0 obtainavalueof|Vcb|=0.040±0.005(stat)+−00..000012 (theory)alsoinexcellentagreementwitharecent
2 determination by the DELPHI Collaboration from the exclusive B¯0 →D∗+l−ν¯ decay. Besides for
d l
theΛ (Ξ )−decay,thelongitudinal and transverseasymmetries, and thelongitudinal totransverse
n b b
decay ratio are ha i=−0.954±0.001 (−0.945±0.002) , ha i =−0.665±0.002 (−0.628±0.004)
a L T
J and RL/T =1.63±0.02 (1.53±0.04), respectively.

0
1 PACSnumbers: 14.20.Mr,14.20.Lq,12.39.Hg,12.39.Jh

2
v I. INTRODUCTION
6
0
The understandingofthe non-perturbativestronginteractioneffects inthe exclusiveb csemi-leptonic transition
0 →
is necessary for the determination of the cb (V ) Cabbibo-Kobayashi-Maskawa (CKM) matrix element from the
2 cb
1 experimentally measured rates and distributions. A considerable amount of work has been carried out in the meson
4 sector, where the ideas of heavy quark symmetry (HQS) [1] and heavy quark effective theory (HQET) [2] were first
0 developed. In the theoretical side, there exist lattice calculations [3]–[6], and a large variety of other theoretical
/ analysis (HQET, dispersive bounds, quark model, sum rules, etc.) [7]– [12]. From the experimental point of view
h
∗
t there were also an important activity and CLEO and Belle collaborations have recent measurements of B D
- →
l decays [13]–[15].
c
The discoveryofthe Λ baryonatCERN[16], the discoveryofmostofthe charmedbaryonsofthe SU(3)multiplet
u b
on the second level of the SU(4) lowest 20-plet [17], and the recent measure of the semileptonic decay of the Λ0 [18]
n b
: make the study of the weak interactions of heavy baryons timely. Experimental knowledge of the Λb semileptonic
v decay can lead to an independent estimate of V if the effects of the strong interaction in the decay are understood.
cb
i
X There exists an abundant literature on the subject [19]– [34]. Almost all theoretical approaches applied to the
mesonsectorhavealsobeenexploredforbaryons. Acommondrawbackinmostofthesestudiesistheimpossibilityof
r
a describingthedecaydistributionsforallq2 (qisthefourmomentumtransferredtotheleptonsinthedecay)accessible
values in the physical decay. Thus, lattice calculations and HQET based approaches lead to reliable predictions in
the neighborhood of q2 = (m m )2, conventional sum rule approaches are more reliable near q2 = 0, while
max Λb − Λc
traditionalnonrelativisticconstituentquarkmodels(NRCQM’s)cannotpredictdifferentialdecayratesfarfromq2 .
max
HQS allows theoretical control of the non-perturbative aspects of the calculation around the infinite quark mass
limit. The classification of the weak decay form factors of heavy baryons has been simplified greatly in HQET [35].
In addition, the Λ ,Ξ baryons have a particularly simple structure in that they are composed of a heavy
Q=b,c Q=b,c
quark and light degrees of freedom with zero angular momentum. At leading order in an expansion on the heavy
quark mass only one universalform factor,the Isgur-Wise function, is required to describe the Λ Λ semileptonic
b c
→
decay. In next to leading order,1/m [36], one more universalfunction and one mass parameter are introduced [37].
Q
However, HQS does not determine the universal form factors and the mass parameter, and one still needs to employ
some other non-perturbative methods.
In this work we determine the non-perturbative corrections to the electroweak Λ Λ matrix element by using
b c
→
different NRCQM’s. We use a spectator model with only one–body current operators, and work in coordinate space,
2
with baryon wave functions recently obtained from a HQS based variational1 approach [38]. We propose a novel
expansion of the electroweak current operator, which allows us to predict the decay distributions for all q2 values
accessibleinthe physicaldecay. Thus,we keepupto firstordertermsinthe internal(small)heavyquarkmomentum
within the baryon, but all orders in the transferred (large)momentum ~q. Some preliminary results were presented in
[40]. Now, we shall further impose (1/m ) accuracy HQET constraints among the form factors to improve on the
Q
O
spectatormodelresults. Thepaperisorganizedasfollows. InSect.IIweintroducetheformfactorsandtheirrelation
to the differential decay width. Those form factors carry all non-perturbative QCD corrections to the semileptonic
Λ and Ξ decays. In Sect. III, we relate baryon wave function with form factors, and introduce the heavy quark
b b
internalmomentumexpansion(Subsect.IIID). AbriefsummaryoftheHQETpredictionsforthesedecaysisoutlined
in Sect. IV, while our results and main conclusions are presented in Sects. V and VI, respectively. Finally, in the
Appendix some detailed formulae can be found.
II. DIFFERENTIAL DECAY WIDTH AND FORM FACTORS
′ ′ ′ ′
We will focus on the Λ (p) Λ (p)l(k )ν¯(k) reaction, where p,p,k and k are the four-momenta of the involved
b c l
→
particles. The generalization to the study of the Ξ baryon semileptonic decay is straightforward. In the Λ rest
b b
frame, the differential decay width reads
d3p′ d3k d3k′
dΓ=8V 2m G2 (2π)4δ4(p p′ k k′)LαβW (1)
| cb| Λc 2E′ (2π)32E (2π)32E′(2π)3 − − − αβ
Λc νl l
where2 m = 2285 MeV, and G = 1.1664 10−11 MeV−2 is the Fermi decay constant. L and W are the leptonic
Λc ×
and hadronic tensors, respectively. The leptonic tensor is given by (in our convention, we take ǫ = +1 and the
0123
metric gµν =(+, , , )):
− − −
L = k′k +k′k g k k′+iǫ k′αkβ (2)
µσ µ σ σ µ− µσ · µσαβ
The hadronic tensor includes all sort of non-leptonic vertices and corresponds to the charged electroweak Λ Λ
b c
→
transition. It is given by
1
Wµσ = Λ ;~p′,sjµ(0)Λ ;~p,r Λ ;p~′,sjσ(0)Λ ;~p,r ∗ (3)
2 h c | cc | b ih c | cc | b i
r,s
X
where r and s are helicity indices and baryon states are normalized so that p~,rp~′,s = (2π)3(E/m)δ3(p~ p~′)δ .
rs
h | i −
Finally the charged current is given by
jµ =Ψ γµ(1 γ )Ψ (4)
cc c − 5 b
with Ψ and Ψ quark fields.
c b
The non-perturbative strong interaction effects are contained in the matrix elements of the weak current, jµ, which
cc
can be written in terms of six invariant form factors F ,G with i=1,2,3, as follows
i i
Λ ;p~′,sjcc(0)Λ ;~p,r =u¯(s)(p~′) γ (F γ G )+v (F γ G )+v′ (F γ G ) u(r)(p~) (5)
h c | µ | b i Λc µ 1− 5 1 µ 2− 5 2 µ 3− 5 3 Λb
n o
′ ′
whereu andu aredimensionlessΛ andΛ Diracspinors,normalizedtou¯u=1,andv =p /m (v =p /m )
Λc Λb c b µ µ Λb′ µ µ Λc
isthefourvelocityoftheΛ (Λ )baryon. Theformfactorsarefunctionsofthevelocitytransferw=v v orequivalently
b c
of q2 = (p p′)2 = m2 +m2 2m m w. In the decay Λ (p) Λ (p′)l(k′)ν¯(k) and for mas·sless leptons, the
− Λb Λc − Λb Λc b → c l
variable q2 ranges from 0 (smallest transfer), which corresponds to w =w =(m2 +m2 )/2m m 1.434, to
max Λb Λc Λb Λc ≈
q2 =(m m )2 (highest transfer, final Λ at rest), which corresponds to w=1.
max Λb − Λc c
1 InRef.[38],wedevelopedarathersimplemethodtosolvethenonrelativisticthree-bodyproblemforbaryonswithaheavyquark,where
wehavemadefulluseoftheconsequencesofHQSforthatsystem. ThankstoHQS,themethodproposedprovidesuswithsimplewave
functions,whiletheresultsobtainedforthespectrumandotherobservablescomparequitewellwiththelengthlyFaddeevcalculations
donein[39].
2 WealsotakemΛb =5624MeV,mΞb =5800MeVandmΞc =2469MeV.
3
The differential decay rates from transversely (Γ ) and longitudinally (Γ ) polarized W’s, are given, neglecting
T L
lepton masses, by (the total width is Γ=Γ +Γ ) [41]
L T
dΓ G2 V 2
T = | cb| m3 w2 1q2 (w 1)F (w)2+(w+1)G (w)2
dw 12π3 Λc − − | 1 | | 1 |
p n o
dΓ G2 V 2
L = | cb| m3 w2 1 (w 1) V(w)2+(w+1) A(w)2
dw 24π3 Λc − − |F | |F |
p n o
V,A(w) = (m m )FV,A+(1 w) m FV,A+m FV,A , FV F (w), FA G (w), i=1,2,3 (6)
F Λb ± Λc 1 ± Λc 2 Λb 3 i ≡ i i ≡ i
h (cid:16) (cid:17)i
where in the last expression the +( ) sign goes together with the V(A) upper index. The polar angle distribution
−
reads [41]:
d2Γ 3 dΓ dΓ
= T +2 L 1+2α′cosθ+α′′cos2θ (7)
dwdcosθ 8 dw dw
(cid:18) (cid:19)n o
whereθ istheanglebetween~k′ andp~′ measuredintheWoff−shell restframe,andα′ andα′′ areasymmetryparameters
which can be expressed as
dΓ da dΓ dΓ da G2 V 2m3
α′ = T T/ T +2 L , T = | cb| Λcq2(w2 1)F (w)G (w) (8)
dw dw dw dw dw − 6π3 dΓT − 1 1
(cid:16) (cid:17) dw
dΓ dΓ dΓ dΓ
′′ T L T L
α = 2 / +2 (9)
dw − dw dw dw
(cid:16) (cid:17) (cid:16) (cid:17)
There are other asymmetry parameters if the successive hadronic cascade decay Λ a+b, where a (J =1/2) and
c a
→
b (J =0) are hadrons, is considered. Two new angles are usually defined, Θ the angle between the Λ momentum
b Λ c
in the Λ rest frame and the a hadron momentum in the Λ rest frame, and χ the relative azimuthal angle between
b c
the decay planes defined by the three-momenta of the l, ν leptons and the three-momenta of the a,b hadrons. The
decay distributions with respect to these two angles read [41]:
d2Γ d2Γ 3π2
1+P α cosθ , 1 γα cosχ (10)
L Λ Λ Λ
dwdcosθΛ ∝ dwdχ ∝ − 32√2
where α is the asymmetry parameter in the Λ hadronic decay (for the non-leptonic decays Λ Λπ and Λ Σπ
Λ c c c
one has: α = 0.94+0.24 [42], 0.96 0.42[43] and α = 0.45 0.32 [42]), a→nd P (longitu→dinal
Λ+c→Λπ+ − −0.08 − ± Λ+c→Σ+π0 − ± L
polarization of the daughter baryon Λ ) and γ are given by
c
dΓ da dΓ da dΓ dΓ da G2 V 2m3
P = T T + L L / T + L , L = | cb| Λc(w2 1) V(w) A(w) (11)
L dw dw dw dw dw dw dw − 12π3 dΓL − F F
(cid:16) (cid:17) (cid:16) (cid:17) dw
G2 V 2 dΓ dΓ
γ = | cb| m3 q2 w2 1 (w+1) A(w)G (w) (w 1) V(w)F (w) / T + L (12)
6√2π3 Λc − F 1 − − F 1 dw dw
(cid:16) p p (cid:8) (cid:9)(cid:17) (cid:16) (cid:17)
The asymmetry parameters introduced in Eqs. (8-9) and Eqs. (11-12) are functions of the velocity transfer w. On
averaging over w, the numerators and denominators are integrated separately and thus we have
G2 V 2m3 wmax
a = | cb| Λc q2(w2 1)F (w)G (w)dw (13)
h Ti − 6π3 Γ − 1 1
T Z0
G2 V 2m3 wmax
a = | cb| Λc (w2 1) V(w) A(w)dw (14)
h Li − 12π3 Γ − F F
L Z0
G2 V 2m3 wmax
γ = | cb| Λc q2(w2 1)21 (w+1) A(w)G1(w) (w 1) V(w)F1(w) dw (15)
h i 6√2π3 Γ − F − − F
Z0
p (cid:8) (cid:9)
a 1 2R a +R a Γ
α′ = h Ti , α′′ = − L/T, P = h Ti L/Th Li R = L (16)
h i 1+2R h i 1+2R h Li 1+R L/T Γ
L/T L/T L/T T
4
q
~r12
~x1
~y1
~r1
0
q
~x2
~y2
~
R
O
CM ~r2
~xh ~yh
~r
Q P
FIG. 1: Definitionofdifferentcoordinatesusedthroughthiswork.
III. BARYON WAVE FUNCTIONS AND FORM FACTORS
Baryonwave functions are taken from our previous work in Ref. [38], where different non-relativistic Hamiltonians
(H) for the three quark (q,q′,Q, with3 q,q′ =l or s and Q=c or b) system of the type
~2
H = mi ∇xi +Vqq′ +VQq+VQq′ (17)
i=q,q′,Q − 2mi!
X
were used. In the above equation mq,mq′ and mQ are constituent quark masses, and the quark-quark interaction
′
terms, V , depend on the quark spin-flavor quantum numbers and the quark coordinates (~x ,~x and ~x for the q,q
ij 1 2 h
and Q quarks respectively, see Fig. 1).
A. Intrinsic Hamiltonian
We briefly outline here the procedure followed in [38]. To separate the Center of Mass (CM) free motion, we went
to the heavy quark frame (R~,~r ,~r ), where R~ and ~r (~r ) are the CM position in the LAB frame and the relative
1 2 1 2
′
position of the q (q ) quark with respect to the heavy Q quark. In this frame, the Hamiltonian reads
~2
H = ∇R~ +Hint (18)
−2M
tot
~ ~
Hint = hsip+Vqq′(~r1−~r2,spin)− ∇1m·∇2 + mi (19)
i=q,q′ Q i=q,q′,Q
X X
~2
hsp = ∇i +V (~r ,spin), i=q,q′ (20)
i −2µ Qi i
i
winhtreirnesiMc HtoatmisiltthoneiasunmHoinftqdueasrckribmeasstshees,d(ymnqam+icmsqo′f+thmeQb)a,rµyoq,nq′a=nd(w1/emuqs,eqd′ +a1v/amriaQt)i−on1aalnadpp∇~ro1,a2ch=to∂/s∂o~rlv1,e~r2i.t [T44h]e.
Hint consists ofthe sum oftwo single particle Hamiltonians (hsp), which describe the dynamics of the lightquarks in
i
the mean field created by the heavy quark, plus the light–light interaction term, which includes the Hughes-Eckart
term (~ ~ ). In Ref. [38], several quark-quark interactions, fitted to the meson spectra, were used to predict
1 2
∇ ·∇
charmedand bottom baryonmassesand some static electromagneticproperties. Furthers details can be found there.
3 ldenotes alightquarkofflavoruord
5
B. Λ and Ξ Wave Functions and HQS
b,c b,c
To solve the intrinsic Hamiltonian of Eq. (19), a HQS inspired variational approach was used in Ref. [38]. HQS
is an approximate SU(N ) symmetry of QCD, being N the number of heavy flavors. This symmetry appears in
F F
systems containing heavy quarks with masses much larger than any other energy scale (η = Λ , m , m , m ,...)
QCD u d s
controlling the dynamics of the remaining degrees of freedom. For baryons containing a heavy quark, and up to
corrections of the order ( η ), HQS guarantees that the heavy baryon light degrees of freedom quantum numbers
O mQ
(spin, orbital angular momentum and parity) are always well defined. We took advantage of this fact in Ref. [38] in
choosing the family of variationalwave functions. Assuming that the ground states of the baryonsare in s–waveand
a complete symmetry of the wave function under the exchange of the two light quarks (u,d,s) flavor,spin and space
degreesof freedom(SU(3) quark model), the wavefunctions read(I, andSπ arethe isospin, andthe spin parityof
light
the light degrees of freedom)4
Λ type baryons: I =0, Sπ =0+
• − light
1
Λ ;J = ,M = 00 00 ΨΛQ(r ,r ,r ) Q;M (21)
| Q 2 J i | iI ⊗| iSlight ll 1 2 12 ⊗| Ji
n o
where the spatial wave function, since we are assuming s wave baryons, can only depend on the relative
distances r , r and r = ~r ~r . In addition ΨΛQ(r ,r−,r ) = ΨΛQ(r ,r ,r ) to guarantee a complete
1 2 12 | 1 − 2| ll 1 2 12 ll 2 1 12
symmetry of the wave function under the exchange of the two light quarks (u,d) flavor,spin and space degrees
of freedom. Finally M is the baryon total angular momentum third component5.
J
Ξ type baryons: I = 1,Sπ =0+
• − 2 light
1 1
Ξ ;J = ,M ;M = ls ΨΞQ(r ,r ,r ) sl ΨΞQ(r ,r ,r ) 00 Q;M (23)
| Q 2 J T i √2 | i ls 1 2 12 −| i sl 1 2 12 ⊗| iSlight ⊗| Ji
n o
where the isospin third component of the baryon,M , is that of the light quarkl (1/2 or 1/2for the u or the
T
−
d quark, respectively).
The spatial wave function6, ΨBQ(r ,r ,r ), was determined in [38] by use of the variational principle
qq′ 1 2 12
δ B Hint B =0, and can be easily reconstructed from Tables X and XI of that reference.
Q Q
h | | i
4 An obvious notation has been used for the isospin–flavor (|I,MIiI, |lsi or |sli) and spin (|S,MSiSlight) wave functions of the light
degreesoffreedom.
5 Note,thatSU(3)flavor symmetry(SU(2), inthecaseoftheΛQ baryon)wouldalsoallowforacomponent inthewavefunctionofthe
type
(21112|MQMSMJ) |00iI⊗|1MSiSlight ΘΛllQ(r1,r2,r12)⊗|Q;MQi (22)
MXSMQ n o
with ΘΛllQ(r1,r2,r12) = −ΘΛllQ(r2,r1,r12) (for instance terms of the type r1−r2), and where the real numbers (j1j2j|m1m2m) =
hj1m1j2m2|jmiareClebsh-Gordancoefficients. ThiscomponentisforbiddenbyHQSinthelimitmQ→∞,whereSlight turnsoutto
bewelldefinedandsettozeroforΛQ−typebaryons. ThemostgeneralSU(2)ΛQwavefunctionwillinvolvealinearcombinationofthe
two components, given inEqs. (21) and (22). Neglecting O(η/mQ) corrections, HQSimposes an additional constraint, which justifies
the use of a wave function of the type of that given in Eq. (21) with the obvious simplification of the three body problem. Within a
spectator model for the Λb−decay, in which the light degrees of freedom remain unaltered, and due to the orthogonality in the spin
space, taking into account the Slight =1components ofthe ΛQ wave functions wouldlead to O(η2/m2Q) corrections to the transition
formfactorsofEq.(5).
6 Itsnormalizationisgivenby
2 +∞ +∞ +1 2
1= d3r1 d3r2 ΨBqqQ′(r1,r2,r12) =8π2 dr1 r12 dr2 r22 dµ ΨBqqQ′(r1,r2,r12) (24)
Z Z (cid:12) (cid:12) Z0 Z0 Z−1 (cid:12) (cid:12)
whereµisthecosineoftheanglefo(cid:12)(cid:12)rmedby~r1 and(cid:12)(cid:12)~r2. (cid:12)(cid:12) (cid:12)(cid:12)
6
C. The hΛ ;~p′,s|jcc(0)|Λ ;p~,ri and hΞ ;~p′,s|jcc(0)|Ξ ;p~,ri Matrix Elements
c µ b c µ b
We will first focus on the Λ Λ matrix element. Within a NRCQM and considering only one–body current
b c
→
operators (spectator approximation) we have in the Λ rest frame
b
E′ m m
Λ ;~p′,sjα(0)Λ ;~0,r = Λc d3q d3q d3q d3q′ b c u¯(s)(~q′)γα(1 γ )u(r)(~q )
D c | cc | b E smΛc Z 1 2 h h rEb(~qh)rEc(~qh′)h c h − 5 b h i
[φΛc(~q ,~q ,~q′)]∗φΛb(~q ,~q ,~q ) (25)
× p~′ 1 2 h ~0 1 2 h
′
with~p =~p ~q = ~q,andu andu charmandbottomquarkDiracspinors. The wavefunctionsinmomentumspace
c b
− −
appearing in the above equation are the Fourier transformed of those in coordinate space
d3x d3x d3x
φΛQ(~q ,~q ,~q )= 1 2 h e−i(q~1·~x1+q~2·~x2+q~h·~xh) ψΛQ(~x ,~x ,~x ) (26)
P~ 1 2 h Z (2π)32 (2π)32 (2π)32 P~ 1 2 h
where the spatial wave function of the Λ baryon with total momentum P~ (see Eq. (18)) is given by
Q
eiP~·R~
ψΛQ(~x ,~x ,~x )= ΨΛQ(r ,r ,r ) (27)
P~ 1 2 h (2π)32 ll 1 2 12
with ΨΛQ(r ,r ,r ) defined in the previous subsection. The actual calculations are done in coordinate space, and
ll 1 2 12
we find
E′
DΛc;−~q,s|jcαc(0)|Λb;~0,rE = smΛΛcc Z d3r1d3r2eiq~·(mq~r1+mq′~r2)/Mtcot[ΨΛllc(r1,r2,r12)]∗
m m
b c u¯(s)(~l′)γα(1 γ )u(r)(~l ) ΨΛb(r ,r ,r ) (28)
× sEb(~l )sEc(~l′) c − 5 b ll 1 2 12
n h io
with the operators ~l = i~ +i~ and ~l′ = ~l ~q acting on the Λ intrinsic wave function. Finally, the flavor of
the light quarks (q,q′) ar∇e~ru1p an∇d~r2down and M−c = m +m +mb, with m = m as dictated by SU(2)–isospin
tot u d c u d
symmetry.
The Ξ Ξ matrix element is easily obtained from the results above, by using ΨΞQ and m instead of ΨΛQ and
b → c ls s ll
mq′ =mu =md, respectively.
D. Heavy Quark Internal Momentum Expansion and Form Factor Equations
Taking~qinthepositivez directionandbycomparingbothsidesofEq(28)forthespinflipα=1or2andspinnon-
′ ′
flipα=0andα=3components,allformfactorsF sandGscanbefound. Themainproblemliesontheoperatorial
nature of the righthand side of Eq.(28), whichrequires of some approximationsto make its evaluationfeasible. Non
relativistic expansions of the involved momenta in Eq. (28) are usually performed [28], but this is only justified near
q2 . With the Λ baryon at rest, ~l in Eq. (28) is an internal momentum which is much smaller than any of the
max b
heavy quark masses. On the other hand, the transferred momentum ~q, which coincides, up to a sign, with the total
momentum carriedout by the Λ baryon,canbe large(note that ~q =m √w2 1 and at q2 =0, ~q(w =w )
c | | Λc − | max |≈
m /2). We haveexpanded the righthand side ofEq. (28), neglecting secondorder terms in~l, but keeping all orders
Λb
in ~q. For instance, this expansion for the charm quark energy gives: E (~l′) E (~q)(1 ~l ~q/E2(~q))+ (~l2/m2),
c ≈ c − · c O Q
with E (~q) E = (m2 +~q2)1/2. Thanks to this novel expansion of the electroweak current operator, in which ~q
is exactcly tr≡eatedc, we acre able to predict the decay distributions for all q2 values accessible in the physical decays,
improvinginthis mannerontheexistingNRCQMcalculations. Finally, wegetthe formfactorsfromtwo(vectorand
′ ′
axial)subsetsofthree equationswiththreeunknowns(F sandGs). Forthe Λ Λ transition,these equationsare
b c
→
compiled in Table I. The hat form factors and the dimensionless baryon integrals ( and ) appearing in the table
I K
are given by
′ 1 1 ′ 1 1
Fˆ(w) = EΛc +mΛc 2 2Ec 2 F (w), Gˆ (w)= EΛc +mΛc 2 2Ec 2 G (w), i=1,2,3 (29)
i 2E′ E +m i i 2E′ E +m i
(cid:18) Λc (cid:19) (cid:18) c c(cid:19) (cid:18) Λc (cid:19) (cid:18) c c(cid:19)
7
Vector
′
α=0, spin non–flip Fˆ +Fˆ + EΛcFˆ = I+ ~q2K mc − 1
1 2 mΛc 3 2(Ec+mc) Ec2 mb
(cid:16) (cid:17)
α=3, spin non–flip EΛ′c|+~qm| ΛcFˆ1+ m|~qΛ|cFˆ3 = Ec|~q+|mIc − |~q2|K mEc2c + m1b
(cid:16) (cid:17)
α=2, spin flip EΛ′c|+~qm| ΛcFˆ1 = Ec|~q+|mIc − |~q2|K mEc2c − m1b
(cid:16) (cid:17)
Axial
′
α=0, spin non–flip EΛ′c|+~qm| Λc −Gˆ1+Gˆ2+ mEΛΛccGˆ3 = −E|c~q+|mIc + |~q2|K mEc2c + m1b
(cid:16) (cid:17) (cid:16) (cid:17)
α=3, spin non–flip Gˆ − ~q2 Gˆ = I+ ~q2K mc − 1
1 mΛc EΛ′c+mΛc 3 2(Ec+mc) Ec2 mb
(cid:16) (cid:17)
α=1, spin flip (cid:0) (cid:1)Gˆ = I+ ~q2K mc + 1
1 2(Ec+mc) Ec2 mb
(cid:16) (cid:17)
TABLE I: Equations used todeterminethe Λb→Λc transitionformfactors. Thehat formfactorsand baryonintegrals (I andK )are
giveninEqs.(29)–(31).
I(w) = d3r1d3r2eiq~·(mq~r1+mq′~r2)/Mtcot[ΨΛllc(r1,r2,r12)]∗ΨΛllb(r1,r2,r12) (30)
Z
K(w) = ~q12 d3r1d3r2eiq~·(mq~r1+mq′~r2)/Mtcot[ΨΛllc(r1,r2,r12)]∗[~l·~q]ΨΛllb(r1,r2,r12) (31)
Z
For degenerate transitions (m = m = m ), the baryon factors (w) and (w) are related, ie 2 (w)/ (w) =
b c Q
I K K I
(mq+mq′)/(mq+mq′+mQ), as can be deduced froma integrationby parts in Eq.(31). Bymeans of a partialwave
expansion and after a little of Racah algebra, the integrals get substantially simplified. Explicit expressions can be
found in the Appendix.
Baryon number conservation implies that F(1) = F (1) = 1 in the limit of equal baryon states. The first
i i
equation of Table I leads to F (1) = (1), since w = 1 implies ~q = 0. Besides, (1) accounts for the overlap
i i I P | | I
between the charmed and bottom baryon wave functions and therefore it takes the value 1 for equal baryon states,
P
accomplishing exact baryon number conservation. In general, vector current conservation for degenerate transitions
imposes the restriction F (w) = F (w), which is violated within the spectator approximation assumed in this work.
2 3
Thus for instance at zero recoil, we find F (1) F (1) = 1 m /M , and thus we do not get vector current
2 − 3 − ΛQ tot
conservationbecause of baryonbinding terms. Two body currents induced by inter–quarkinteractions are needed to
conserve the vector current.
The corresponding Ξ decay quantities are obtained from the above expressions by means of the substitutions
b
mentionedattheendofSubsect.IIIC. Notethat, and dependonboththeheavyandlightflavors,hence,andfor
the sake of clarity, from now on we will use the noItationKcb or cb for the Λ and Ξ decays , and a similar notation
IΛ IΞ b b
for the factors.
K
IV. HQET AND FORM FACTORS
When allenergyscalesrelevantinthe problemaremuchsmallerthanthe heavyquarkmasses,HQS is anexcellent
tool to understand charm and bottom physics. Close to zero recoil (w =1) and at leading order in the heavy quark
mass expansion, only one universal (independent of the heavy flavors)form factor, the Isgur-Wise function7 (ξren) is
required to describe the Λ Λ semileptonic decay. To next order, 1/m , one more universal (χren) function and
b c Q
one mass parameter (Λ¯) are→introduced. These functions, and also the form–factors, depend on the heavy baryon
7 Notethat,thoughcalledinthesamemanner,becauseofthedifferentlightcloud,thisfunctionisdifferenttothatenteringinthestudy
ofB→DandB→D∗ semileptonictransitions.
8
w N N N N N5 N5 N5
1 2 3 i i 1 2 3
1.00 1.49 −0.36 −0.10 1.03 0.99 −0.42 0.15
P
1.11 1.40 −0.32 −0.09 0.99 0.94 −0.37 0.13
1.22 1.32 −0.30 −0.09 0.93 0.91 −0.34 0.12
1.33 1.26 −0.27 −0.08 0.91 0.88 −0.31 0.11
1.44 1.20 −0.25 −0.07 0.88 0.85 −0.28 0.10
TABLE II:Correctionfactors(takenfromRef.[7])fortheΛb→Λc decayformfactors
light cloud flavor, and thus in general they will be different for Ξ transitions, though one expect small deviations
−
thanks to the SU(3)-flavor symmetry.
WecompileheresomeusefulresultsfromRef.[7,45],wheremoredetailscanbefound. Including1/m corrections
Q
the Λ Λ form factors factorize in the form
b c
→
F (w) = N (w)ξˆ (w)+ (1/m2), G (w)=N5(w)ξˆ (w)+ (1/m2), i=1,2,3 (32)
i i cb O Q i i cb O Q
Λ¯ Λ¯ w 1
ξˆ (w) = ξren(w)+ + 2χren(w)+ − ξren(w) (33)
cb
2m 2m w+1
(cid:16) b c(cid:17)(cid:20) (cid:21)
where the coefficients N ,N5 contain both radiative (Cˆ ,Cˆ5)8 and 1/m corrections. Λ¯ is the binding energy of the
i i i i Q
heavy quark in the corresponding Λ baryon (Λ¯ = m m ) and because of the dependence on the heavy quark
ΛQ − Q
masses, ξˆ is no longer a universal form factor. The function χren(w) arises from higher–dimension operators in the
cb
HQET Lagrangian, and vanishes at zero recoil. Both functions ξˆ and ξren are normalized to one at zero recoil.
cb
The numerical values of the correction factors N ,N5 depend on the value of Λ¯, which is not precisely known. We
i i
reproducehere(TableII)Table4.1ofRef.[7],wherethese correctionfactorsaregivenforallbaryonvelocitytransfer
w accessibleinthe Λ Λ lν¯ decay9. The parametersΛ¯/2m andΛ¯/2m weresetto 0.07and0.24,respectively. At
b c l b c
→
zero recoil, Luke’s theorem [36] protects the quantities F(w)= F (w) and G (w) from (1/m ) corrections
i i 1 O Q
F(1)= F (1)=η + (1/m2), PG (1)=η + (1/m2) (34)
i V O Q 1 A O Q
i
X
where η andη are entirely determined by shortdistance corrections(ie, N5(1)=Cˆ5(1) and N (1)= Cˆ (1))
V A 1 1 i i i i
which are in principle well known, since they are computed using perturbative QCD techniques. The second relation
might be used to extract a model independent (up to 1/m2 corrections) value of V fromPthe measurPement of
Q | cb|
semileptonic Λ decays near zero recoil, where the rate is governedby the form factor G . From Eq. (6), one finds
b 1
1 dΓ G2 V 2
lim = | cb| m3 (m m )2η2 + (1/m2) (35)
w→1√w2 1dw 4π3 Λc Λb − Λc A O c
−
V. RESULTS
To obtain the wave functions for the Λ and Ξ baryons,we will use different NRCQM interactions whose details
Q Q
can be found in Refs. [38]. Following the notation of this reference, we will refer to them as AL1, AL1χ, AL2, AP1,
AP2 and BD. Their free parameters had been adjusted in the meson sector [46, 47, 48]. The potentials considered
differ in the form factors used for the hyperfine terms, the power of the confining term10 (p = 1, as suggested by
lattice QCD calculations [50], or p = 2/3 which for mesons gives the correct asymptotic Regge trajectories [51]), or
8 Theyareknownuptoorderα2s(zlnz)n,wherez=mc/mb istheratiooftheheavy–quark massesandn=0,1,2
9 NotethevaluesforthosecorrectionfactorsaresomewhatdifferentfromtheonesquotedinRef.[45]
10 The force which confines the quarks is still not well understood, although it is assumed to come from long-range non-perturbative
featuresofQCD[49].
9
the use of a form factor in the One Gluon Exchange (OGE) Coulomb potential [52]. All of them provide reasonable
∗ ′ ′ ∗ ∗
and similar masses and static properties for Λ ,Σ ,Σ ,Ξ,Ξ ,Ξ ,Ω and Ω baryons [38].
Q Q Q Q Q Q Q
FortheΛ decaywewillpayanspecialattentiontotheAL1andAL1χinter–quarkpotentials. TheAL1potential
b
−
is based on a phenomenological inter–quark interaction which includes a term with a shape and a color structure
determined from the OGE contribution, and a confinement potential. The second model (AL1χ) includes the same
heavyquark–lightquarkpotentialastheAL1model,whilethelightquark–lightquarkisbuiltfromtheSU(2)chirally
inspired quark-quark interaction of Ref. [53] which includes a pattern of spontaneous chiral symmetry breaking, and
that was applied with great success to the meson sector in Ref. [48],
From the experimental side, the Λ semileptonic branching fraction into the exclusive semileptonic mode was
b
measured in DELPHI to be [18]
Br(Λ0 Λ+l−ν¯)= 5.0+1.1(stat)+1.6(syst) (36)
b → c l −0.8 −1.2
A remark is in order here, the perturbative QCD corre(cid:0)ctions have been neglec(cid:1)ted in Ref. [18], i.e. the correction
factors N ,N5 are computed with Cˆ =Cˆ5 =1 and Cˆ =Cˆ5 =0, and a functional form of the type
i i 1 1 2,3 2,3
ξˆ (w)=e−ρˆ2(w−1) (37)
cb
is also assumed in that reference, where it is also found that11
ρˆ2 =2.0+0.8 (38)
−1.1
wherealluncertaintiesquotedinRef.[18]havebeenaddedinquadratures. Ontheotherhand,thebranchingfraction
given by the Particle Data Group is [17]
Br(Λ0 Λ+l−ν¯ + anything)=(9.2 2.1)% (39)
b → c l ±
which is hardly consistent to that quoted in Eq. (36). Nevertheless, none of the values quoted in Eqs. (36) and (39)
correspondtodirectmeasurements. Wewillassumehere,anerrorweightedaveragedvalue12ofthosegiveninEqs.(36)
and (39)
Br(Λ0 Λ+l−ν¯) =(6.8 1.3)% (40)
h b → c l iavg ±
The total Λ0 width is given by its lifetime τ =1.229 0.080 ps [17] and thus one finds
b Λ0b ±
Γ(Λ0 Λ+l−ν¯)=(5.5 1.4)1010s−1 (41)
b → c l ±
Besides, data from Z decays in DELPHI have been searched for B¯0 D∗+l−ν¯ decays. These events are used to
d → l
measure the CKM matrix element V [15]
cb
| |
V =0.0414 0.0012(stat) 0.0021(syst) 0.0018(theory) (42)
cb
| | ± ± ±
Let us first examine the bare NRCQM predictions without including HQET constraints.
A. NRCQM Form Factors
In Fig. 2 we present the Λ Λ form factors obtained from the AL1 inter–quark interaction (left) and also the
b c
→
predictions for the ξˆ function (right) as extracted from any of the form factors (ξˆ = F /N ,G /N5,F/ N , i =
cb cb i i i i i i
1,2,3)showninthe left panel. The correctionfactorsN ,N5 aretakenfromTableII. Severalcommentsareinorder:
i i
P
As expectedfromHQS,the formfactorsF , F , G andG aresignificantlysmaller thanthe dominantonesF
2 3 2 3 1
•
and G .
1
11 Notethat−ρˆ2 isnottheslopeattheoriginoftheuniversalIsgur-Wisefunctionξren(w)introducedinEq.(33).
12 Weaddinquadraturesthestatisticalandsystematicuncertainties quotedinEq.(36).
10
1.2 1.4
F
1 NRCQM: AL1 1.2 NRCQM:AL1 FF21
F3
0.8 F 1 G1
F1 G3
0.6 F2 w) 0.8 G2
F3 (b
0.4 G1 ˆξc 0.6
G3
0.2
G2 0.4
0
0.2
0.2
− 1 1.15 1.3 1.45 1 1.15 1.3 1.45
w w
FIG. 2: NRCQMΛb→Λc formfactors(left)andξˆcb function(right)fromtheAL1inter–quarkinteraction.
Recalling the discussion of Subsect. IIID on vector current conservation for degenerate transitions, one must
•
conclude that the NRCQM predictions for the F and F form factors are not reliable at all, since their sizes
2 3
are comparable to the expected theoretical uncertainties, 1 m /M , affecting them. Presumably, one
O − ΛQ tot
should draw similar conclusions for the axial G and G form factors. As clearly seen in Fig. 2 the ξˆ (w)
2 3 (cid:0) (cid:1) cb
functions obtained from the F , F , G and G form factors substantially differ among themselves and are in
2 3 2 3
complete disagreement to those obtained from the F and G form factors.
1
NRCQM predictions for the vector F and axial G form factors are much more reliable, and lead to similar ξˆ
1 cb
•
functions, with discrepancies smaller than around4%. Such discrepancies can be attributed either to (1/m2)
O Q
corrections,not included in ξˆ , or to deficiencies of the NRCQM. Lattice results of Ref. [19] for these two form
cb
factors, though have large errors, are in good agreement with the results shown in Fig. 2.
B. HQET and NRCQM Combined Analysis.
Toimprovethe NRCQMresults,weproceedasfollows. We assumethe NRCQMestimate ofthe vectorformfactor
F (F = F +F +F ) to be correct for the whole range of velocity transfers accessible in the physical decay13, and
1 2 3
use it to obtainthe flavordepending ξˆ function. Nowby using Eq.(32)andthe HQETcoefficientsN ,N5 compiled
cb i i
in Table II, we reconstruct the rest of form factors, in terms of which we can predict the longitudinal and transverse
differentialdecaywidths andthe asymmetryparametersdefined inSubsect.II. We willestimate the theoreticalerror
of the present analysis by accounting for the spread of the results obtained when all calculations are repeated by
determining ξˆ from the NRCQM G form factor and/or by using different inter–quark interactions.
cb 1
1. Λ Decay
b
Results ofourHQETimprovedNRCQManalysisforthe Λ decayarecompiledinFig.3andTables III andIV. In
b
thefirstofthetables,wegivethetotalandpartiallyintegratedsemileptonicdecaywidths,splitintothecontributions
totheratefromtransversely(Γ )andlongitudinally(Γ )polarizedW’s,andthevalueoftheflavordependingξˆ (w)
T L cb
13 Letusremindhere,thattheNRCQMgivescorrectly F(1)inthecaseofdegenerate transitions.