Semileptonic decays of D and B mesons

ELSEVIER

UCLEAR PHYSIC~

Nuclear Physics B (Proc. Suppl.) 47 (1996) 485-488

PROCEEDINGS SUPPLEMENTS

Semileptonic Decays of D and B Mesons* S. Giisken a, K. Schilling a'b , G. Siegert b

a Physics Depar tment , University of Wuppertal , D-42097 Wuppertal , Germany b HRLZ, c /o KFA, D-52425 Jfdich, Germany

We report results of our ongoing investigation concerning semi]eptonlc decays of heavy pseudoscalar mesons into pseudoscalar and vector mesons. Particular attention is paid to uncertainties in the qa and the heavy quark mass dependence of formfactors. Moreover we present a non-perturbative test to the LMK current renormalization scheme for vector current transition matrix elements and find remarkable agreement.

1. I N T R O D U C T I O N

The accurate determination of Kobayashi Maskawa (KM) matr ix dements involving heavy quarks requires control over the low energy QCD parts of the corresponding weak transition ampli- tudes. Although lattice QCD is the per se method to tackle this regime, it suffers from fact that the lattice resolution is still too poor to represent b quarks directly on the lattice.

With our current high statistics project we push forward to a lattice resolution of a -1 _~ 3.2 GeV. This allows for direct calculation of transi- tion amplitudes up to s mass region of (1 - 1.6) x roD, keeping discretization errors small. As direct access to the b mass is still excluded, we work with four heavy quark masses, which provide us with a sufficient lever a rm for the extrapolation to the B meson.

2. L A T T I C E S E T U P

We are working on 24 s × 64 lattice at ~3 : 6.3, with standard Wilson quarks in the quenched ap- proximation. The heavy quark masses are rep- resented by ~ : 0.1200,0.1300,0.1350,0.1400. This covers the physical region 0.8mc _< m~, _< 1.6mc. For the light quark masses we have chosen ~t : 0.1450,0.1490,0.1507,0.1511, which corresponds to 0.8m, < mz _< 3m,. The decay amplitudes have been constructed with the initial meson at rest and the fi-

*Work supported by DFG grant Schi 257/1-4, Schi 257/3- 2, EC contract CHRX-CT92-0051 and Mu 810/3.

hal meson carrying spatial momen ta pa = 2,2_.ix {(0, 0, 0); (1, 0, 0); (1, 1, 0); (1, 1,1); (2, 0, 0)]. + permutations. In order to improve on the groundstate projection we have used Wupperta l "ganssian" smeared wavefunctions for the quark fields. Details can be found in ref.[1].

The simulation has been performed on the 32 node CM5 located at the IAI in Wuppertal . We have created a to ts / of 100 independent gauge configurations, which served for calculation of 2- and 3-point corrdators .

The analysis has been done so far on a sub- set of 60 configurations. Using this da ta set we find ~ c , , : 0.151818(37). The lattice spacing has been determined with three different meth- ods. With 1t. Sommer ' s [2] method we find a -1 = 3.314(29) GeV, extrapolat ion of the vec- Ho tot meson mass to ~¢, , yields a~ 1 = 3.453(217) GeV, and with the stringtension V ~ = 440 MeV we get a~ 1 : 3.210(52) GeV. The results agree nicely within errors. In the following we will use aR. to set the scale.

3. q= D E P E N D E N C E

The KM matr ix elements can be extracted from experimental measurements of the decay widths P and the branching ratios R. In doing so one needs the formfactors 2 F(q 2) , which parametr ise the transition matr ix dements[3]. As the major contribution to the phase space integrals for r and R comes from the region of small q2, one con-

~For the time being F(q ~) denotes generically the form- factors/+,/0, V, At, A2.

0920-5632/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved. PII: S0920-5632(96)00106-5

486 X Giisken et al./Nuclear Physics B (Proc. Suppl.) 47 (1996) 485-488

veniently quotes the values F(qa= 0). Of course knowledge of F(q 2) over the entire q2 region is neccessary in order to calculate R and r accu- rately.

According to our setup of spatial momenta, the calculation provides us with (up to) 5 entries F(q 2) for each mass combination, which we use to extrapolate to F(0) as well as to learn about the functional dependence of F(q2).

In our analysis we have tested three different methods to fit our data: Two of them are guided by the pole dominance hypothesis

F ( q ' ) - F(0)., (1) 1 1 -

where we a) took a fixed pole mass m,, as ex- tracted from the corresponding 2-point correla- tors, and b) treated , q as a free parameter. The third Ansatz assumes linear behaviour

F(q 2) : a q- bq :~ . (2)

Fig. 1 shows a typical example of our formfactor data for weak transitions of a heavy-light pseu- doscalax meson into light-light pseudoscalar and vector mesons (HI -~ ll), together with the dif- ferent fits. It turns out that all Ansi tze describe the data reasonably well and lead to compatible results at q2 = 0. Therefore the good news is that we are able to determine the value of F(q2= 0) to come out independent of the formfactor model.

On the other hand, however, our data does not clearly distinguish between the different func- tional forms. This is due to the statistical noise of the data and gives rise to systematic uncertain- ties.

A similar situation is found for the decays of a heavy-light pseudoscalar meson into heavy-light pseudoscalar and vector mesons (Hl --+ H'l). Again, the results at q2 : 0 are almost unal- tered by the fit method, but sizeable uncertainties concerning the functional form of F(q 2) over the entire q2 range remain.

4. R E N O R M A L I Z A T I O N

In order to convert the above lattice data into continuum results, we need the renormalisation

1

O.i

~ 2

' 2~ " 4', " ~ ; ; " ,~,

2 2

1.S

O.S

4 4 J " "Q~' " '0 .1 " ; " 0:1 " 'N' q~

V

Figure 1. q2 dependence of form/actors for HI -+ II transitions. The mass of the heavy initial quark corresponds to r~ = 0.1350 and the masses of light final and the light spectator quark is given by ~l - ~,j,e¢ = 0.1490. The solid line represents the fit to eq.1 with fized m~, the dashed line keeps rn~ free and the dotted line eorreponds to a fit to eq.2.

constants Zv and ZA of the vector and axialvec- tor currents. It has been pointed out by C.W. Bernard [4] tha t the renormalization constant of the local vector current Zv depends strongly on the mass of the quarks involved. Followin 8 his ar- guments this undesired mass dependence can be removed if the (standard) ~ normalization of the quark fields is replaced by the LMK[5] pre- scription.

As a first step to check this issue s with our data we study the ratio

m,)lV IP(mH, (3) RR,H(p'~ = (P(m~, m~)lVol°~lP(mH, mi)>p

V0 l~c(¢°') is the O'th component of the lo- cal(conserved) vector current, and P denotes a pseudoscalar eigenstate.

R~rs(0~ yields directly the renormali~ation fac- tor of V0/°¢, as V0 c ~ ' is unaltered in that instance. For the s tudy of decays we are rather interested

3.ee al,o [s].

X Gfisken et al./Nuclear Physics B (Proc. Suppl.) 47 (1996) 485-488 487

in the ratios RH/,(O) and RHH(p') however. The LMK predictions for these ratios read

1 _ _8 ~__~L R . . ( ~ ) : ~ ,o , , , . . , . (4)

2~H ~o~ 1 _ _, ~_~_

4 tt©~,it R~, , (~) : ~ , 2,~ (5)

n~.(~ = n ~ , ( ~ ' ) . (6)

Taking the standard normalization one gets

2 r

2 .a M 1,a w • • • w &

1.4 I_

I J

L21"4, f '~

0.8 ............................... 0A 0J 0A J-

..i.... [,,.,I,,..h,.*h,,.J.,.,h,.,J .... 0,, 0.2 0.3 0.4 0J$ 0.| 0.7 O.II 0-# 0 0.10J G,30,40AO.4G,7UO,I

.~- 1/20/,~- 1 /~ m,- 1/20/,r- 1/,~

Figure 2. The ratios RHH(P-~ (left) and Rh , (O) as a function of the initial (x~) and final (~1) quark- mass. The solid line represents the L M K predic- tion, the dashed line corresponds to the standard scheme.

meson mass. As the exact mass dependence of the

Y

@.S / 0.2 0.4 G.dl 0.8 0.2 0.4 0.6 0.8

I

°I

/ ¥ %.

0.2 0.4 0.41 0.8 0.2 0.4 O.S 4kip 1,~lpt 1AI h

Figure 3. Mass dependence of form~actors at q2 : O. Crosses denote the data. The solid line refers to fit method a), the dashed line to b) and the dotted line to c). The eztrapolated results are depicted at l i m b as filled circles (method a), upper triangles (method b), and lower triangles (method c).

Rhu = RHH : Z~, °c. In fig. 2 we display the measured ratios RuR(p-~ and RhR(O) together with the LMK and standard predictions. We have inserted ,~o¢ = 1 - o.s2_2 --4~y , Zz°~ = 1 - 0.17408192, and ~ , = 1, using the boosted coupling g - 2 ( ~ ) : go2(1/3TrPgv) + 0.02461. The LMK prediction describes the data remark- ably well within the entire mass range, whereas the standard prediction fails drastically. In the following we will therefore apply the LMK scheme to our formt'actor results, although a similar ratio test for the spacellke components is still missing.

5. M A S S D E P E N D E N C E

The determination of the formfactors for B me- son decays requires extrapolation in the heavy

formfactors is still unknown we have used several AnsKtse for the extrapolation, in order to check on systematic uncertainties:

a) a + b ,~Ps

F ( m p s , q2 = O) = b) t a b ~ r ; ( + ~-~. ~ ) (7) b c) ~vr~77(a + ~ - ~ ) .

In fig.3 we display the formfactors F ( q 2 = 0) for I l l --~ II transitions, as a function of l i m p s . Ob- viously the Ans~tse (eq.'s 7) lead to consistent results at raps -- roB.

We mention that the transitions HI -+ H ' l ex- hibit a much weaker mass dependence, and again the results at the B mass are unaltered by the fit method.

488 S. Giisken et al./Nuclear Physics B (Proc. Suppl.) 47 (1996) 485-488

Table 1 Form factors at q 2 : 0 for weak decays into pseu- dosc~ far mesons.

mode

D - - * g

D - + K

B - + D

Bo -+ Do

/+(o)

0.68(13)~ °

o.n(12)+ °

0.50(14)+57

1.o1(6o)

0.99(42)

/o(O)

0.67(6)+~

0.20(3)+]

0.58(29)

0.85(24)

Table 2 Formfactors at q= : 0 for weak decays into vector ne so~ .

mode V(0) At(0) A2(0)

D ---+ p 1.31(25)_+~ 0.59(7)_+~ 0.83(20)_+~ 2

D- K* 1.34(24)+ 0.61(6)+I 0.83(20)+ '

B p 0.e1(23)+ 0.16(4)_+I 0.72(35)+_ °

B --~/3* 1.84(1.10) 0.97(32) 2.76(1.58)

B, -+ D* 1.84(1.10) 0.91(24) 2.25(1.12)

6. R E S U L T S

In tables 1 and 2 we quote our results for the formfactors at q2 = 0, which have been obtained after extra- and interpolation of the heavy quark mass to the b and c quark and extra- and in- terpolation of the light quark mass to mq : 0 and mq : m,t,anas respectively. The errors in- clude both statistical (in brackets) and system- atic uncertainties 4. The results for D decays are in good agreement with experiment and other lattice calculations (see e.g. [7]), giving confi- dence to our analysis method. For B --* ~v, p de- cays however, the situation is much less settled. The data obey the condition ]+(0) = Yo(0) on s 2~ level only, indicating systematic uncertainties arising from the extrapolation in the heavy mass. A careful study of this issue will be included in the analysis with full statistics.

REFERENCES

1. S. G~sken et al., Nucl. Phys. B (Proc. Suppl.) 42 (1995)412 and references therein.

2. R. Sommex, Nucl. Phys. B 411 (1994) 839. 3. V. Lubics et al.; Nucl. Phys. B356 (1991)301,

and references therein. 4. C.W. Bernard; Nucl. Phys. B(Proc. Suppl.)

34(1994)47. 5. A.S. Kronfeld, Nucl. Phys. B(Proc. Suppl.)

30(1993)445; P.B. Mackenzie, Nucl. Phys. B(Proc. Suppl.) 30(1993)35.

6. A. Vladikas; these proceedings. 7. K. Schilling et al., HEP-LAT 9507002, HLRZ

38-95.

4 This being sn intermediate sns/ysis, we have omitted in some cases to quote systematic uncertainties.