$ F.J.SEVIER

IUCL~AR PHYSICS

Nuclear Physics B (Pine. Suppl.) 73 (1999) 378-380

PROCEEDINGS SUPPLEMENTS"

Current Renormalisation in NRQCD for Semi-leptonic B --, D Decays Peter Boyle, Christine Davies a

~Dept. of Physics and Astronomy, University of Glasgow, Glasgow, UK ; UKQCD Collaboration

We present a calculation of the renormMisation constants for the tempora l vector current, ZVo and spatial axial current, ZA~,, to O ( ~ - ) for B --* D transi t ions using the O(--~) N R Q C D action for both b and c quarks evaluated for a large range of mass parameters . Considerations for the renormalisat ion of the spatial vector current and the tempora l axial current are discussed and initial results for a mixed lat t ice current are presented for the spatial vector current.

1. I n t r o d u c t i o n

The semi-leptonic B --* D, D* decay is phe- nomenologically interesting since its Feynman amplitude involves the poorly know CKM matrix element Vcb

i M(B --+ XI-P) = --~V¢bfitTu(1 -- 75)vvH. (1)

Hu = (x(p')l~7,(1 - 75)biB(p)), X = D, D* (2)

where the hadronic tensor Hu, and its Lorentz decompostion into form factors is calculable using lattice QCD.

We perform the matching of the Lattice to con- t inuum MS currents to 1 loop using the on-shell scheme in Feynman gauge with vanishing exter- nal spatial momenta. In the on-shell scheme the wavefunction renormalisation and vertex correc- tions individually contain (cancelling) infra-red divergences which we control using a fictitious gluon mass, A. We use dimensional regularisa- tion to control the ultra-violet behaviour in the continuum calculation.

[ ,+1 ] 1 6Ak/ =7k75 3e_--Z-i-log'-8 + O ( ~ - 2) (5)

6Ao/~-#~ : 70% [3~_--~loge-8]

+ (6)

Here the currents V0 and A~ can be renor- malised to O ( ~ ) in the usual manner, however the currents Vk and A0 will involve operator mix- ing at this order.

3. l - l o o p L a t t i c e C o r r e c t i o n

We use the O ( ~ ) lattice NRQCD action [1] for both b and c quarks,

a~NRQCD "= c t ( x ) ¢ ( x )

_ ¢ t ( ~ + O (1 - ~'"~ (1 - . - o ~ - ~ _ ~ (7) 2 ] 2n ] Uo '

× (1 ~Ho~" (1 - a~H~ - 2 . , 2 , ¢ ( = )

A H0 - 2M0 6H =-CBg°' 'B ' 2M0 (8)

2. C o n t i n u u m C a l c u l a t i o n

Expanding the l-loop correction to the currents to O(M ~ ) , and writing e = M~b, we obtain

L [3e+le- 1 ] lvl-1 6Vo/3~ r~ = 70 loge - 4 - 2 + O(TT~-,~ ) (3)

r 3

2_a_[ " ~ 9. 2 d_.d~_~ / + Mb L~--:-i - (,-1)~1 (4)

"]- Me Le---:'i " (e -1)2j + O ( ~ - ~ )

for which the Feynman rules may be found in reference [2].

The Foldy-Wouthysen transformation of the continuum current corrections and dimensional arguments for which currents can contribute at O ( ~ ) suggest we take the bases of operators in Table 1 for the lattice currents.

For those currents that are unmixed, the calcu- lation is similar to [4], and we write the renormal- isation constant Zr = 1 + ~sZ[~ ] = 1 + 6Z M--ff - 8zlr at, where the lattice integrals for 6Z lat were

0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0920-5632(98)00604-5

B Boyle, C. Davies~Nuclear Physics B (Proc. Suppl.) 73 (1999) 378-380 379

Table 1 Lattice Current Bases

Vk Ao Ak o.a

:

1 J2Vk = a_:_V_~r, j ~ o : _~"V 2M_2 x 2M~, O. k

j = _ v ~ j O _ 2 M ~ Mb ~.

computed numerically using VEGAS [3], per- forming the temporal loop momentum integra- tion analytically for those contributions that were infra-red divergent.

In the case of the spatial vector and tempo- ral axial currents it is necessary to evaluate the derivatives of the diagrams with respect to the ex- ternal momentum to match with the continuum. This is done by numerically evaluating the inte- gral of the analytically taken derivative, resolving the pieces of the Pauli structure by taking differ- ent derivative directions.

In the continuum, both the vertex correction to the current ~Fb, and the wavefunction renormali- sation contained a logarithmic divergence propor- tional to ~ log--~F where the sign is positive in the vertex correction. The same infra-red diver- gences were found in the lattice vertex correction and wavefunction renormalisation so that as one would expect the infra-red behaviour of the the- ory is identical to that in the continuum.

3.1. V0 a n d Ak C u r r e n t s The l-loop contribution to the renormalisation

constant ZVo is plotted in Figure 1 for various values of e = MM~ and Mb. Here the stabilisation parameter n has been chosen in a mass dependent manner such that M0 x n > 3. The lattice current is the conserved current of the lattice action, and the continuum current is conserved, so that there is no correction on the line e = 1.

The l-loop correction to the spatial axial cur- rent is in Figure 2. The mass dependence is stronger since the line for c = 1 is not protected.

3.2. Vk C u r r e n t We define the l-loop lattice mixing matr ix Zi j ,

(c(p')lS, Ib(p)) = ~ ] Z# f~f (9) 1

0.2

0.1

0.0

0.0

. . . . I . . . . ] ~/~patial Vector Current

, ' . ' : ," I , ' , , ' , I , ," 0.2 0.4 1/(a M d

Figure 1. Z~2 as a function of ~ and M~" The

lines correspond to M~ = 0 .1 , . . . , 1.0 in incre-

ments of 0.1, with the M~ = 0.1 curve topmost.

where the f ~ are the continuum analogs of the lattice operators given in Table 1. We define the contribution to the mixing matr ix arising from the lattice vertex correction, ~ij, via

I ~lat ~lat ~.. ]

Zij = 6ii + ors --~lat 6 6 - - ;]lat g. g. - l-~m b i l j l "1- ~raeVt2uj2

We then invert the mixing matr ix and match to the continuum. Here the coefficients Bi may be inferred from equation 4, and the Z, , factors arise from matching the bare mass in lattice currents to the pole mass in the tree level continuum current.

( B1 - ~t%¢o ~ _¢~ J dl v W - - --~11 -- ~21 -- Zrnb

-- 2t"2bc " ~ ¢~ J J2 + 1 q- o~, -~12 - ~22 -- Zme

-1- a , ( B 3 - - ~ 1 3 - - ~ 2 3 ) J3

+ 0~, ( B 4 - ~ 1 4 - ~ 2 4 ) J 4 (11)

The diagonal elements ~jj contain infra-red di- vergences which cancel with those in the lattice wavefunction renormalisations. Those currents appearing at tree level also carry tadpole im- provement counter terms. We therefore rewrite

380 P. Boyle, C Davies~Nuclear Physics B (Proc. Suppl.) 73 (1999) 378-380

0.2

,91

0.0

- 0 . 2

0 . 0

. . . . I . . . . I ' ' Spatial Axial Current

.

i I i i i I i

0.2 0.4

1/(a M 0

Figure 2. Z [11 Ak as a function of ~bb and MM-~. The

lines correspond to M~ = 0 .1 , . . . , 1.0 in incre-

ments of 0.1, with the M~ = 0.1 curve topmost.

Equation 11 in terms of IR divergence and tad- pole counter term free quantities, as follows:

- {z¢o +

Vk M--g --- 1+4 , --ZmbTT~llTI~21 } ] g l - - E l i - - grab ] J

+ l + a , - Z r n ~ ~I~,TI ~2 2 - -%22 - - ~ r n c ! J

+ ~, ( B 3 - ~13 - ~23) J3

+ 0~, (B4 -- ~14 -- ~24) ,]4 (12)

Preliminary calculations of the mixed lattice have been performed on a few mass values for degenerate quarks. The degenerate case may in fact be of use in certain lattice simulations to obtain the Isgur-Wise function, however the re- sults should really be considered illustrative of the method, and the calculation over a similar param- eter regime to our previous results for the other currents will be performed. In this case the coeffi- cients of J1 and J2, and of J3 and J4 are identical, and we only present the coefficients for J1 and J3 in Table 2. We evaluate the tadpole improvement counter terms using u~q(plaq) = ~ and take val- ues for Zm from [2]. Also, for the degenerate case B1 ---- B2 ---- -3-2; and B3 -- B4 ----- - 1 , so that

Table 2 One Loop Correction Coefficients

2 -0.6581(5) -0.2516(1) -0.4156(6) 4 -0.9219(5) -0.2041(1) -0.0030(6) 10 -1.2662(5) -0.0963(1) -0.2711(8)

Table 3 One Loop Correction Coefficients

MQ,n C1 62 2,2 -0.776(4) 0.145(1) 4,2 -0.571(4) 0.098(1) 10,1 -0.247(4) -0.010(1)

when the mixed current is written as

= (1 + + :2)

+ a,C2 (J3 + J4), (13)

we obtain the coefficients in Table 3.

4. F u t u r e D i r e c t i o n

We are nearing completion of the calculation of the mixed current for Vk, while there is a smaller basis of operators for A0, so that the cor- responding calculation should be somewhat eas- ier. Thereafter we shall perform a similar calcu- lation for the b --, c transition using the NRQCD action for the b quark, and the O(a) improved Wilson action for the c quark. The extension of the calculation to O(~-~) would be of interest.

5. A c k n o w l e d g e m e n t s

We would like to thank the Physics Depart- ment and the ITP, UCSB for their hospitality while this work was carried out. PB is funded by PPARC grant PP/CBA/62, CD thanks the Fulbright commission and the Leverhulme trust.

R E F E R E N C E S

1. G.P. Lepage et. al. Phys. Rev. D, 46, 4052, (1992).

2. C. Morningstar, J. Shigemitsu Phys. Rev. D, 57, 6741, (1998).

3. G.P. Lepage J. Comput. Phys. 27 192 (1978).

4. C.T.H. Davies, B.A. Thacker Phys. Rev. D, 48, 1329 (1993).