ELSEVIER

UCLEAR PHYSIC~

Nuclear Physics B (Proc. Suppl.) 53 (1997) 408-410

PROCEEDINGS SUPPLEMENTS

A first s t u d y of t he s emi - l ep ton i c decay of t he Ab ba ryon .

Nicoletta Stella ~ for the U K Q C D Collaboration

Physics Depar tment , University of Southampton, SO17 1BJ Southampton, UK

We present the preliminary results of the first Lattice study of the baryonic Isgur and Wise function obtained from the matrix element of the weak current between A-baryon external states. Its dependence on the heavy and light quark masses is studied. Some result on the semi-leptonic decay Ab --+ A~ + Iv are given.

We present the results of the first non- per turbat ive s tudy of the semi-leptonic decay Ab -+ A~+lu, carried out using Lattice QCD. This study will provide an independent measurement of the CKM matr ix element Vcb, as experimental da ta become available.

We evaluate the hadronic matr ix element of the weak current Jt~ = ~%~(1 - ~/~)b, computing the correlators

C(t= ) =Ee-i#'~ (O Q (x)(9 Q (0)} ( i) £

c . (t~, t~) =}--]e -~(~~+ ¢~) <0 Q (x)( J. (V) )O c~ (0)>

on 60 243 x 48 lattices at /3 = 6.2, and using the O ( a ) - i m p r o v e d fermion action [1]. In (1) O Q is the interpolating operator of the AQ. In this s tudy we consider only correlators with initial and final momenta either zero or Pmin -- 27r/La and transitions with equal initial and final heavy quark masses mQ. The matr ix element can be decomposed into six form factors (FF), which are invariant fimctions of w = v • v'

(AQ (v')l J,, IA~ ) (v)> = ~(v')r~(F~ v - ~F~)~(v).

' . In this basis, one can with Fi = ~ , , v , and v , use the Heavy Quark Effective Theory (HQET) analysis to relate the six FF to the Isgur-Wise (IW) function [2], through the correction coeffi-

cients N~ 5)

N~(co)~Qe, (co)= r~" (co)N~ (co)~QQ, (co) = r~ (co)

~QQ, (co) is explicitly fiavour-dependent, but still normalized to 1 and protected by Luke's theorem a t w = 1.

We will s tudy the quantities

^ FiA(co) N5(1) E~F~v(co) E~N,(1) ~QQ'(co) = F A ( 1 ) N 5 ( w ) - E i F v ( 1 ) y ] ~ N ~ ( c o )

(2)

which are independent of the current renormaliza- tion constants, and exhibit a stable signal. The equality (2) is valid up to O ( 1 / m Q ) , and we ne- glect higher order corrections.

^

The explicit flavour-dependence of ~QQ, is stud- ied by linearizing it about co = 1

~QQ,(co) = 1 +p2( l -co) (3)

and measuring the slope p2 as a function of mQ. By keeping the light quark masses fixed to ~ = 0.14144, around that of the strange quark, and varying mQ around the charm mass, we obtain

/

p2 = 2.4 +4_4 at ~Q = ~Q = 0.121

! p2 = 2 . 4 +4_ at ~Q = ~ Q =0 .125

p2 = 2 . 4 +43_ at ~Q = ~ =0 .129

p2 2.4 +~ at nQ = t~Q = 0.133, (4)

^

suggesting that the flavour dependence of ~QQ, can be neglected at our masses or above. A global fit, shown in Figure 1, to the four sets of deter- minations, yielding p2 = 2.4 + 4, is our best esti- mate.

According to HQET, the IW function only de- pends on the quantum numbers of light quarks. Previous studies on the lattice [3] demonstra ted that such a dependence is not negligible in the case of mesons, where the "brown muck" contains only one light quark. Thus we expect to measure an even stronger dependence of the baryonic IW function.

This was studied fixing the heavy quark mass to ~Q = n O, = 0.129, i.e. very close to the charm quark, and considering the three light hopping-parameter combinations (~11 = ect2 = 0.14144),(~ll = 0.14226,~c~2 = 0.14144), and

0920-5632(97)/$17.00 © 1997 Elsevier Science B.'~ All rights reserved. PII: S0920-5632(96)00673-1

N. Stella/Nuclear Physics B (Proc. Suppl.) 53 0997) 408-410 409

Table 1

& co Qq, co &Q, 0.14144/ A 1.037 0.94 +3 1.08 0.88 +

0.14144 V 1.037 o.96 233 1.08 o.90 +_88 0.14144/ A 1.040 0.93 +a_ 1.08 0.86 +~')_~

+ 1 0 0.14226 V 1.040 0.97 +3_ 1.08 0.90 - m + 1 8 0.14226/ A 1.043 0.93 +5_6 1.09 0.82 _,8

0.14226 V 1.043 0.95 +~ 1.09 0.80 +~3 - - , - - 1 4

Chira l / A 1.048 0.92 + r r 1.10 0.90 +22 - - 2 0

Chiral V 1.048 0.96 +9 s _ 1.10 0.95 +19 -19

cd

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

^ ^ ~oQ' co ~QQ'

1.02 +11 1.15 0.68 +7 - 1 1 - 6

0.98 + t l -12 1.1.5 0.62 +8 - 7

0.97 +i5 1.16 0.71 +9 - 1 6 -- 9 + 9 0.86 +]6_17 1.16 0.65 - 8

0.85 +')4 1.18 0.75 +13 - - 2 5 - - 1 2

0.63 +27 1.18 0.68 +ta --27 --ii

1.02 +23 1.19 0.78 +17 - - 2 4 - - 1 7

0.79 +26 1.19 0.74 +]6 - - 3 2 -15

^ Est imates of ~QQ, (co) form both axial and vector form factors, at nQ = n~ = 0.129 and at three values of the light quark masses and at the chiral limit. Errors on co are on the digit beyond the last shown.

~" 1 e l - I . . . . I . . . . I 0 r degenerate transition

~ I O xQ=0.1gl X ~Q=0.1~9

i 1.0 ~7 o gq=O,125 ~:~ ~Q=olaa _

g '

0 . t3

0.6

g I . . . . I , , 0,4 '

1 0 1.1 1.2 6)

Figure 1. ~QQ,(W) from degenerate transit ions, vector and axial current . Different graphical sym- bols denote different heavy quarks.

(eq, = ~12 = 0.14226). By fitting together the quanti t ies (2) to Eqn. (3), it was found:

p 9 = 2.4 + 4 - 4 at nn,2 --- 0.14144, 0.14144

p2 = 2.0 +5 - 5 at ~n,2 = 0.14144, 0.14226

p g = l 7 +6 • - s at ~11,2 --- 0.14226,0.14226 (5)

The slope p e at the chiral limit,

p2 1.2 + 8 : - - l I ' ( 6 )

is ob ta ined by ex t rapola t ing the three est imates of bo th co and ~QQ, (co) linearly in the sum of the

two light quark masses, Our results are presented in Table 1.

Our est imate of the IW flmction can, in turn, be used to quantify the 1 /mQ corrections, affect- ing / '1 and F2,3, in terms of the baryonic binding energy 5, as detailed in [4]. There, it was found tha t A = 0.37 + 0.11 GeV. Its value is necessary to reconstruct the FF at the physical limit, i.e. for the decay A~, --+ Ac + eu.

Wi th the result tha t {QQ,(co) is flavour- independent the FF depend on the quark masses

only th rough the factors N / ' ) . Given our limited

knowledge of the functional form of {QQ, (w), we can only model the FF as linear functions of co

F A , V ( w , m b , m c ) = ,ha,V _ fiA,V(CO _ 1) (7)

where the normalizat ions ~]),',a and the new ~A V slopes P i ' are related to the coefficients

N} 8) (co, rob, m,~) and to the slope of the IW flmc- tion by

,IV, A = N}5) (1), p,,-VA =P2'IiV'A dN}5) ( ~=,

~VA Our results for f l i ' and ~ly 'A are shown in Ta- ble 2.

The decay rates can be wri t ten in terms of the FF, in the velocity basis, t h rough the helicity am- plitudes H [5]:

dF G.)F 9 ,z .),~ q'2 M'~,. X / ~ - 1) d w - ( ~ ) a ~ V~b z ~ X (9)

410 N. Stella /Nuclear Physics B (Proc. Suppl.) 53 (1997) 408-410

fi 1 .8 + 9 - 0 . 4 + 2 -0 .10 +47 1.3 + 8 + ~ + 6 - - 1 5 - - 1 -- - - 1 2 -0 .4 _. 0.16 - l o ~/ 1.28 . . . . . +6 -0 .19 +4 -0 .06 +~ 0.99 -0 .24 +~ 0.09 +2.)

Table 2 Normalization and slope of the physical Form Factors , relevant for the Ab decay.

co ..... I 1.1 1.15 1.20 1.25 rP'~rt i +9 1.03 +22 1.4 +5 1.8 + r 9 * t o t 0 . 5 7 _ 7 --20 -- --

Table 3 Partial decay rates for the At.

(IH1/2,112 + IH_1/2,_112 + IH1/2,ol 2 + 1H-1/2,ol2),

where the subscripts in the helicity amplitudes refer to the polarization of the W boson and of the daugther baryon A¢, respectively. The upper integration limit on the decay rates extends to co _ 1.43, which is beyond the range of velocity transfer accessible to us (w E [1.0, 1.2]). We thus define the partially-integrated decay rate,

Fpart("' / ....... ~_ , . . . . . . ) = 8co (8)

as a function of the upper limit of integration. In Table 3, we present our results for the quantities

F pa~t (co .. . . . ) [10 *a s - 1] (9) iv ,,l"

for several values of co~ ..... The masses are taken from the experiments.

At present, a direct comparison of our results with experiments is not possible. In fact, even if the semi-leptonic decay of At was observed by various experiments [6], a measurement of the de- cay rate is not yet available. The problem of de- termining the rate of the At semi-leptonic decays has been addressed making use of different models (Infinite Momentum Frame, Quark Model, Dipole form factors ) [5],[7]. Their predictions, for the total rate, integrated up to the end-point, are re- ported in Figure 2, and compared with the func- tion Pp~rt(co ..... ). To evaluate this function we have assumed [Vcbl = 0.044.

Finally, we note that many other interesting quantities, such as asymmetry parameters (see for example [8]) and the ratio of the longitudinal

' I

10 toLal r a t e

/

5

1 .0 1.2

COma x

i i

I

I

I

~ Dipole fI IMF

~ Qua rk Mod

1 I

, I! . . . . 1.4 1 .6

Figure 2. Total decay rate with error" bands, for the process At --+ A~+lu, as a function of the limit of integration COmax. A comparison with several model estimates is shown at the end-point.

to transverse rates, could be computed and con- fronted with the upcoming experiments. How- ever, all these quantities are non-trivial only at O(1 /m~) or O(co2): in both cases beyond the pre- cision reached in the present study.

R E F E R E N C E S

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2 3

4 5

6

7

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B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259, 572 (85). M. Neubert; Phys. Rept. 245 (94) 245. K.C. Bowler et al., The UKQCD Collabora- tion, Phys. Rev. D 52, 5067 (95). UKQCD Collaboration, in preparation. J.G. K6rner, D. Pirjol and M. Kr/imer; Prog. Part. Nucl. Phys. 33 (94) 787. ALEPH Collaboration et al.,Phys. Lett. B 278, 367 (1992); M.A. Ivanov and V.E. Lyubovitskij; hep- ph/9502202. A.F. Falk and M. Neubert, Phys. Rev. D 47, 2982 (93).