Perturbative matching of heavy-light currents with NRQCD heavy quarks

UCLEAR PHYSIC~

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 73 (1999) 360-362

ELSEVIER

Perturbative Matching of Heavy-Light Currents with NRQCD Heavy Quarks Colin J. Morningstar a and Junko Shigemitsu b ~Physics Department, University of California at San Diego, La Jolla, CA 92093, USA. bphysics Department, The Ohio State University, Columbus, OH 43210, USA. We present further results for one-loop matching of heavy-light axial and vector currents between continuum QCD and a lattice theory with NRQCD heavy quarks and massless clover quarks.

1.

Introduction

Lattice studies of hadronic matrix elements require matching between operators in full continuum QCD and those in the lattice theory being simulated. In the present article we will focus on an effective lattice theory which combines nonrelativistic (NRQCD) heavy quarks and clover light quarks. We wish to match the theories correct through O(M~, a~M , a ap). For heavy meson (e.g. B and B*) leptonic and semileptonic decays, the relevant operators are heavy-light vector and axial vector currents, denoted in the continuum theory as V, = ~% h and A~, = ~TsT~,h. The first question that arises concerns the number and type of operators required in the effective theory. For L A0 and V0 ] one finds 3 operators in the effective theory through O ( ~ ) j/o) = q(x)

r~ Q(x),

(r~ = -y~o or "~o)

For [ Ak and Vk J one has 5 operators in the effective theory through O(M~)

j(o) = q(x) rk Q(x),

(Fk = 757k

or

7k)

j(1) = ~Mq(X) Fk ~ . V Q(x),

j(2)= ~ ( ~ ) ~ . ~

~0r~ Q(~),

5(3) = ~-~(x) r, vk Q(~),

(r, = ~

or

i)

J(k4) : ~-~q(x) ~kr, Q(x). Again there is an O(a ap) discretization correction to ~k r(°)" jdisc = 2 aM j(2) and j(o)

/-Ak/Vk ~rdiac l(O)'irnp --- J(k°) + (~ kCZ~diac )Jk

> ~k

The desired relation between QCD hadronic matrix elements and nonperturbatively determined lattice current matrix elements is hence,

j[U __ ~M(l(X) r~ ~ . v Q(~), 2

J~) = ~4(~) 7. ~ ~0r~ Q(x),

j=o plus a dimension 4 discretization correction at O(aap) :

4

jdlsc = --a q(x) "7. ~ 70Ft Q(x) = 2 aM J(t 2)

j=0

j o)

-~.

4°)+ ~

~.Ao/Vo~ --dlsc

(the spinors h and Q are related via a FoldyWouthuysen transformation).

and similarly for V0 and Ak. The matching coefficients, Cj, have the following perturbative expansion,

0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0920-5632(98)00598-2

CJ Morningstar, J Shigemitsu/Nuclear Physics B (Proc. SuppL) 73 (1999) 360--362 Cj = l + c~ pj + O(a 2) Cj = ol pj -b O(ot 2)

j = O, 1 j > 1

I

0

I

,0,

Results for One-Loop Coefficients

In figures 1. and 2. we plot the one-loop coefficients P0 and p j / ( 2 a M ) , j > 0, for all four current types, Vk, A~, A0, and V0 versus the inverse dimensionless bare heavy quark mass [1]. The "bursts" show P0 with log(aM) fixed to log(2), where the choice of a M -- 2 is somewhat arbitary, corresponding to the value of the bare dimensionless heavy quark mass appropriate for the physical b-quark on a -1 ~ 2GeV lattices. With the logarithmic dependence taken out one can compare with P0 for the static theory, as indicated for V~ and A012]. One also sees that p2/(2aM) (the "diamonds") goes to a non-vanishing value as a M ~ oo. This is due to the mixing with the discretization correction, jd~sc. 4.

Applications

Matching coefficients presented here have already been applied to several studies of B and B* meson decay constants [3-6]. In Fig.3 we show results for the vector meson decay constant a3/~fyv/-M-y on dynamical H E M C G C configurations[5]. The squares represent tree-level and the crosses and diamonds one-loop results using (~w(q*) with q* -- 7r/a and q* = 1/a respectively. The physical B* meson corresponds

I I

I

l ~ t J l l l l

Vk

o

Some Calculational Details

In the continuum theory we employ on-shell renormalization with naive dimensional regularization and the M S scheme. A gluon mass is introduced to handle IR divergences that eventually cancel between continuum and the lattice. The light quark mass is set equal to zero. On the lattice we worked mainly with an NRQCD action correct through O ( p / M ) . Results with higher order relativistic corrections also exist. The lattice light quarks are massless clover fermions. The light and heavy quark actions and the lattice current operators are all tadpole improved. 3.

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The goal is to calculate the pj's. 2.

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361

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××

- 1 -~:~ I

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0.0

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0.5 Fig. 1

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i.O

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I

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1.5

1/aM o

Figure 1. One-loop Coefficients pj. crosses : P0 squares : p l / 2 a M ; diamonds : p2/2aM (includes ~disc) ; octagons : p3/2aM ; fancy squares : p4/2aM ; bursts : P0 with log(aM) fixed at log(2).

to 1/aMo ~ 0.5. The one-loop correction is a 13 -~ 21% effect depending on q*. This is a slightly larger decrease than is found for the pseudoscalar decay constant based on A0. Acknowledgements : This work was supported by DOE Grants DE-FG03-90ER40546 and DEFG02-91ER40690 and by a NATO grant CRG 941259. REFERENCES

1. For A0 see also C. Morningstar and J. Shigemitsu, Phys. Rev. D57, 6741 (1998). 2. A. Borrelli and C. Pittori; Nucl. Phys. B385, 502 (1992). 3. A. Ali Khan et hi., Phys. Rev. D 56, 7012 (1997). 4. A. Ali Khan et al., Phys. Lett. B427, 132 (1998). 5. S. Collins et hi., in preparation. 6. J. Hein et al. in preparation.

C.J. Morningstar, ,I.. Shigemitsu/Nuclear Physics B (Proc. Suppl.) 73 (1999) 360-362

362

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Figure 2. Same as Fig.1 for Ak, Ao and Vo.

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Figure 3. squares : tree-level ; crosses : av(~r/a) diamonds: ay(1/a).