A lattice study of $Λ_b$ semileptonic decay

3 Pages·0.1 MB·English

by  Steven Gottlieb

1
A lattice study of Λ semileptonic decay
b
3
Steven Gottlieb a and Sonali Tamhankara∗
0
0
aIndiana University, Bloomington, IN 47405, USA; Theory Group MS106, Fermilab, PO Box 500,
2
Batavia,IL 60510,USA.
n
a
J We present results from a lattice study of the semileptonic decay Λb → Λclνl. We use O(a2,αsa2) improved
2 quenchedlatticesoftheMILCcollaboration, withlatticespacing∼0.13fm. Forthevalencequarks,thetadpole-
2 improved clover action is used, with the Fermilab method employed for the heavy quarks. Form factors are
extracted from the vectoras well as theaxial-vector part of thecurrent.

1
v
2
2
0 1. INTRODUCTION Wise function, ξ(ω). This function is normalised
1 at zero-recoil,ξ(1)=1.
CurrentknowledgeoftheCKMmatrixelement
0
3 Vcb isderivedfromthemesonicdecaysB →D¯∗lν
0 or B → D¯lν. Experimental knowledge of the 2. SIMULATION PARAMETERS
t/ Λb semileptonic decay can lead to an indepen- The simulations are performed on the Asqtad
a
dent estimate of V if the effect of the strong
l cb quenched lattices at β = 8.00 generated by the
- interaction in the decay are understood, e.g., via
p MILC collaboration [2]. These are O(αsa2) im-
e lattice QCD. A first lattice study of the bary- proved 203× 64 lattices, with a−1 = 1.33 GeV,
h onic semileptonic decay was performed by the
as determined from m . We use three light
: UKQCD collaboration [1]. We report our initial ρ
v quark masses near the strange quark mass, κ =
l
i results for the dominant form factors of this de- 0.1343,0.1333,0.1323. Twoheavyquarkκvalues,
X
cay.
0.104 and 0.114 bracket the charm quark, and
ar The semileptonic decay ΛQ → ΛQ′lν can be other two, 0.064 and 0.077 bracket the bottom.
parametrizedin terms ofsix formfactors,F and
i We use the clover action for the valence quarks,
G , for i = 1, 2, 3.
i with a tadpole improved clover coefficient. The
hΛ(s)(v)|J |Λ(r)(v′)i=u¯(s)(v)[γ (F −γ G ) value for the tadpole improvement factor u0 is
Q µ Q′ Q µ 1 5 1 taken from the Landau gauge fixed mean link.
+ v (F −γ G )
µ 2 5 2 Fermilab formalism is used for the heavy quarks.
+ v′(F −γ G )]u(r)(v′). (1) Results are presented for 300 lattices for two-
µ 3 5 3 Q′
point functions, and 237 lattices for three-point
Here J is the weak current and r,s are polari-
µ functions.
sation states of the baryons. Since both Λ and
b
Λ are hadrons containing a single heavy quark,
c 3. TWO-POINT RESULTS
heavy quark effective theory (HQET) is applica-
ble [3]. Hence the matrix element is taken be-
The dispersion relation is shown in Fig. 1.
tween baryons of a given velocity, and the form
The fitted energy values agree very well with the
′
factors are functions of the scalar ω = v · v .
expectation from the lattice dispersion relation.
To leading order in HQET, the combinations
The chiralextrapolationsfor a fixed heavyquark
′
F +v F +v F and G involving the dominant
1 0 2 0 3 1 mass are shown in Fig. 2. The baryon kinetic
form factors F and G can be written in terms
1 1 mass M is estimated as M = M +m −m ,
2 2 1 2 1
of a single function, called the (baryonic) Isgur-
WhereM andm arethebaryonandheavy
2(1) 2(1)
∗PresentedbyS.Tamhankar quark kinetic(rest) masses respectively. We use
2
4.6 5.5 kkkkhhhh====....100107614744
4.4
4.5
4.2
Λ
4 m
3.5
2E 3.8
3.6 2.5
3.4
1.5
3.2
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
3 mq
0 0.5 1 1.5 2
p2 Figure 2. Chiral extrapolations of the measured
heavy baryon masses to the u quark. We have
Figure 1. Dispersion relation for κ =0.114, κ =
h l
used the light quark kinetic mass m for the fit.
0.1323/0.1323. The E here is E , the parameter 2
1
obtained from exponential fits. The line shows
the lattice dispersion relation. 5.5
Qqq
5 Qq
4.5
a linear fit for these extrapolations. In Fig. 3,
4
we have shown the chirally extrapolated baryon
massasafunctionoftheheavyquarkmass,along 3.5
with the coresponding meson masses taken from 3
the MILC collaboration. Our values for m and 2.5
Λb
m are 5.626(36)GeV and 2.300(27)GeV. 2
Λc
1.5
4. THREE-POINT RESULTS 10 0.5 1 1.5 2 2.5 3 3.5 4
mQ
Different form factors contribute to different
Figure 3. The heavy baryon mass, plotted as
matrix elements in the three-point function. For
functionoftheheavyquarkmass. Alsoshownare
µ = 0, the dominant contribution to three-point
the heavy-light meson masses, taken from stud-
functions comesfromthe vectorformfactorsand
ies of the MILC collaboration. The bursts corre-
for µ = i, axial-vector form factor G gives the
1 spond to the b and c quark.
dominantcontribution. Wepresentresultsforthe
Isgur-Wise function from vector as well as axial-
vector data. Λ is created at time 0 and Λ is 0.114 → κ , (.1323,.1323)
b c h
annihilated at time t ≡ 16 in lattice units. The 1.05
x κ = 0.64
time at which the current acts is varied, and we 1 κh = 0.77
0.95 κ h= 0.104
study three-point function as a function of this 0.9 κh = 0.114
h
itnimtiaeltyba≡ryot.nFisoratthreesrtesaunltds pthreesefinntaeldbhaerryeo,nthies ξω () 00..078.558
moving with different velocities giving different
0.7
values for ω. On the lattice, one is restricted to 0.65
regionnear ω =1 as data starts getting noisy for 0.6
′ 0.55
high momenta. In this region, v can be approx-
0 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07
imated by 1. Then for an initial baryon of mass ω
′
M decayingto a finalbaryonofmassM moving
Figure 4. Isgur-Wise function from the vector
withamomentum~q,ifweconsiderthesumofthe
current. κ is 0.114 for all these points, and
h1
co-efficients of I and γ , for large t and t −t ,
0 y x y the points corresponding to four different κ are
h
the three-point expression simplifies to
shown with four different symbols.
3
0.114 → κh, (.1323,.1323) Isgur Wise function for light kappa 0.1343/0.1343
1.1 1
κ = 0.64
1.015 κκ hh= = 0 .01.0747 0.9
0.95 κh = 0.114 0.8
0.9 h
ω) 0.85 ω) 0.7
ξ ( 0.8 ξ ( 0.6
0.75
0.7 0.5
0.65
0.4
0.6
0.55 0.3
1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.02 1.04 1.06 1.08 1.1
ω ω
Figure 5. Isgur-Wise function from the axial-
Isgur Wise function for light kappa 0.1323/0.1323
vectorcurrent. Asbefore,κ is0.114forallthese
h1 1
points, and the points corresponding to four dif-
0.9
ferent κ are shown with four different symbols.
h
0.8
′
C(t )= ZlZs(|~q|)e−txM′e(M′−E)ty4M′ ω) 0.7
y 16M′E ξ ( 0.6
(F1(ω)+F2(ω)+F3(ω))2(E+M), (2) 0.5
′ 0.4
where Z and Z are known from the two-point
l s
functions. We fit this to a form Ae−Bt and con- 0.3
1.02 1.04 1.06 1.08 1.1
sider the ratio ω
A[(M′,~0)→(M,~q)] F (ω)+F (ω)+F (ω) Figure 6. Isgur-Wise function for κl =
= 1 2 3 0.1343/0.1343,andκ =0.1323/0.1323,ourhigh-
A[(M′,~0)→(M,~0)] (cid:16) F1(1)+F2(1)+F3(1) (cid:17) est and lowest valueslfor κ .
l
′
Z (|~q|) E+M
· s · . (3) Isgur-Wise function seems to be quite insensitive
(cid:16) Zs′(0) (cid:17) (cid:16) 2E (cid:17) to the heavy quark mass.
We have also studied the light quark mass de-
First factor on the RHS is2 the Isgur-Wise func-
pendence of the Isgur-Wise function. The Isgur-
tion ξˆQQ′. The second and third factors are Wisefunctionisexpectedtofallslowerforsmaller
known from the two-point functions. The third
light quark masses, by a heuristic argument. We
factor may be approximated by 1 to 0.5 per cent
doseesuchatrendinthispreliminarystudy, but
accuracy. The second factor differs from 1 by
it is veryfar fromclearwith the statisticalerrors
upto 10% over our range of ~q. The ratio is inde-
we have. This is shown in Fig. 6.
pendent of the renormalization constant Z be-
V The calculations were done on the IBM SP at
causewehavethesameheavyquarktransitionin
Indiana University. We gratefully acknowledge
both numerator and denominator.
the hospitality of the Fermilab Theory Group.
Our results for the Isgur-Wise function from
thevectorcurrentareshowninFig.4. TheIsgur-
REFERENCES
Wise function obtainted from the axial-vector
current (µ = i case) is shown in Fig. 5. The 1. K. C. Bowler et al., Phys. Rev. D 57, 6948
2We follow Ref. [1] in using this definition. This way of (1998), hep-lat/9709028.
defining the Isgur-Wise function agrees with the conven- 2. C. Bernard et al., Phys.Rev. D 64, 054506
tionaldefinitionuptoO(1/m)corrections,whicharefur- (2001), hep-lat/0104002.
ther multipliedby(1-ω). Forthe smallmomenta accessi-
3. A.V.ManoharandM.B.Wise,HeavyQuark
ble on the lattice, ξˆis a very good approximation to the
baryonicIsgur-Wisefunction. ThesuffixQQ′ emphasizes Physics. Cambridge University Press, 2000,
thattheinfinitemasslimithasnotbeentaken. and references therein.