Improved renormalization constants for B-decay and BB mixing

NUCLEAR

P H V S I CS B

Nuclear Physics B 385 (1992) 502—524 North-Holland

________________

Improved renormalization constants for B-decay and BB mixing A. Borrelli

*

CERN, Geneva, Switzerland and Dipartimento di Fisica, Università di Roma, “La Sapienza”, 1-00185 Rome, Italy

C. Pittori Dipartimento di Fisica, Università di Roma, “La Sapienza “, 1-00185 Rome and INFN Sezione di Roma, Rome, Italy Received 17 February 1992 (Revised 19 May 1992) Accepted for publication 21 May 1992

In this paper we discuss the improvement of lattice matrix elements involving heavy quarks. We then perform the new one-loop matching between continuum and improved lattice operators relevant for the evaluation of B-meson decay constant and BB mixing amplitude.

1. Introduction In recent years much theoretical and experimental effort has been devoted to the search for the t-quark, and to the estimate of its mass, m~.Being members of the same weak isospin doublet as t, b-quarks can be an important channel of information; they also are a possible source for evidence of CP violation, through measurement of the imaginary part of the Cabibbo—Kobayashi—Maskawa mixing matrix. The evaluation of the B-meson decay constant and BB mixing amplitude involves weak matrix elements between hadronic states, which depend non-perturbatively on strong interactions. Although lattice QCD has been successfully used for the evaluation of several non-perturbative quantities, the case of the B-meson requires a different approach. In present lattice calculations the typical range of values for the inverse lattice spacing is a~1 2 3 GeV; thus, owing to —

—

*

Supported by Fondazione Angelo Della Riccia.

0550-3213/92/$05.00 © 1992

—

Elsevier Science Publishers B.V. All rights reserved

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the fact that mb is of the order of 5 GeV, b-quarks are too heavy to propagate on such lattices, and cannot be treated in the same way as lighter ones. A solution to this problem has been suggested in ref. [1] and then developed by Eichten and Hill [21,who proposed an effective theory in which mb is considered infinite, and thus it is analytically removed from the lattice action. The use of the expansion in inverse powers of the heavy-quark mass has drawn much attention in successive works [3]. To compare the results of lattice simulations involving heavy quarks to the experimental values, it is necessary to relate lattice operators in the effective theory to the corresponding renormalized continuum ones. The renormalization constants, calculated with the effective lattice lagrangian for b-quarks and with light quarks treated as Wilson fermions, have been computed at one loop in refs. [4—61for B-decay and in ref. [71for BB mixing. Lattice simulations, however, are affected by systematic errors due to the finiteness of the lattice spacing. A practical method to reduce the cut-off dependence of the lattice hadronic matrix elements involving Wilson fermions has been presented in refs. [8,9]. In this paper we apply the method of improvement to lattice matrix elements involving heavy quarks and we evaluate at one loop the improved lattice renormalization constants of the relevant two- and four-fermion operators. We will consider the renormalization of the generic heavy—light bilinear O~=bFq,

(1)

where F is one the 16 Dirac matrices. We apply our results to the interesting case of the time component of the axial current, which is used to compute on the lattice the pseudoscalar B-meson decay constant fB’ defined as


=

(2)

is the B-meson momentum. We will also consider the renormalization of 2 four-fermion operator O~=~by~qby~q,

(3)

y~=4y,~(1—y5).

(4)

where we have defined

This operator enters in the effective hamiltonian for BB mixing, whose amplitude is usually expressed in terms of the B parameter, defined as OL(l.L)

I B°)

=

~

(5)

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where MB is the B-meson mass; with this definition the BB parameter depends on the scale jx at which the four-fermion operator is renormalized. We recall that the renormalization group invariant BB factor, at one loop and with four active quark flavors, is given by BB

=

(6)

6”25BB(~x).

[cts(/x)]

The main results of this work, which have also been presented in ref. [101,are the following. We choose the values f3 6.0, i.e. ~1tt 0.080 =

~ =a1

=

=

2 GeV,

~,

mb=S GeV,

r= 1,

(7)

where ~ is the continuum renormalization point and r is the Wilson parameter, and we use the two-loop continuum value of the strong coupling constant ~ 0.25, corresponding to fl(4)~ 200 MeV for four active quarks see ref. [121.We find that the relation between the value of fB measured on a lattice with the above parameters, using the improved theory, and the physical continuum value is =

=

—

f~°11t O.89f~. =

(8)

In the unimproved case the corresponding factor would be 0.85. For the combination f~BB(~x),neglecting the mixing with operators with different chirality, the lattice value, computed with the same parameters, is related to the continuum one by (f~BB)COhlt(2GeV)

=

O.8O(f~B~)~tt(2 0eV),

(9)

in the unimproved case the factor would be 0.69. For the correction to the BB(~x) parameter alone, we find B~°~t(2 GeV)

=

1.02B~tt(2GeV)

(10)

in the unimproved case the factor would be 0.99. We have checked that, within the precision of one unit on the last decimal digit, these results are insensitive to variations of ~ in the range 150—300 MeV. Our results are valid in the assumption that log(mba) is not a “large logarithm”, so that the two-loop corrections [13] are small. For a recent two-loop calculation of the anomalous dimension of the axial current, see ref. [14]. The paper is organized as follows. In sect. 2 we discuss the method of improvement in general. In sect. 3 we briefly review the i/mb expansion and the *

Recently the possibility of large failures of perturbation theory resulting from the use of the bare lattice coupling constant as an expansion parameter, has been investigated [11].

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effective field theory. In sect. 4 we show how to apply the improvement to matrix elements involving heavy quarks; we explicitly show in appendix A that on-shell matrix elements of improved two-fermion heavy—light operators at one loop do not have order a corrections. We present in sects. 5 and 6 our results respectively for two- and four-fermion heavy—light operators, comparing them with previous calculations; technical details and analytic formulas will be given in appendix B.

2. Improvement for light quarks In refs. [8,9] it has been shown that it is possible to eliminate all terms which are effectively of order a, from lattice hadronic matrix elements with Wilson fermions. In perturbation theory, since in the scaling limit the squared lattice coupling constant g2 is proportional to 1/log a, these are all the terms of the form (g2IYa log’~a.For Monte Carlo simulations, the practical method to be used can be summarized as follows: one must work with the improved action with nearestneighbour interactions only [15] S~ S~—an ~ ~igar~i(x)P~,,(x)~/i(x), =

(11)

where S~is the “standard” Wilson action: S~=a~ (-

~

+~(x+ ~)(r

[~(x)(r

-

y~)U~(x)~(x + ~)

+

+a~~(x)(m~+4r/a)~(x),

(12)

x

m 0 is the bare light fermion mass and field strength tensor. We define [16]

is the lattice definition of the continuum

t), 1~1 4a ~ 2ig P~—~L——(U,—U,

(13)

where the sum is over the four plaquettes in the ji—r’ plane, stemming from the point x and taken counterclockwise. At the same time, the quark fields in fermion operators must undergo the transformation *

*

There is actually a full family of transformations, that can be described by an arbitrary real parameter z, discussed in detail in rcfs. [17,181,which are equivalent on-shell. In this paper we consider only the transformation defined in eq. (14).

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i~i—*~’=(i_

~ar~)qi, (14)

where the lattice covariant derivative is defined as aD~(x) ~[U~(x)~(x+~) =

-

(15) Thus, for the two-fermion operator of the kind

Or=~IJflhJ,

(16)

the corresponding improved operator is defined as [i7]

O~-=O~.+ ~ar~(x)~Fi/i(x)

—

~ar~(x)F~i(x).

(17)

Analogously, for the four-fermion operators

o~=

±(2

~

~

~)1~

(18)

the improved operator is [181

O~=~((L~xL3~)

+

[(L~2xL34)

+

(L12xL~4)I±(2~-~4)}.

(19)

To shorten the notation, in eq. (19) we have defined L12~~i1(x)y,~fi2(x)

(20)

and the “rotated” operator is defined as L~2~ ~

(21)

On-shell lattice matrix elements of the improved operators (17) and (19) do not have effective 0(a) corrections.

3. The effective field theory for b-quarks Non-perturbative hadronic matrix elements can be estimated by numerical lattice calculations. As already mentioned in the introduction, in the case of fB

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and BB, since (22)

mb>>a,

to treat b-quarks on the lattice it is necessary to use an approach different from the usual one. We will only consider the renormalization of the operators in eqs. (1) and (3) at lowest order in i/mb. At this order in the continuum theory, expanding the heavy-quark propagator in powers of i/mb leads to [ii Sb(x,

t)

=P(~)ô(x)(~(t)exp(_m~t)~Y0 +@(—t) exp(m~t)1 Yo) (23)

where m°bis the bare b-quark mass and =

T exp(igjtdt~A~(0,tt)Ta).

(24)

A~is the time component of an external gauge field, T denotes time ordering and are the color matrices normalized by tr(T’~T”) ~ab Except for the factor exp(—m~t)(exp(m~t)),the heavy quark (antiquark) has been replaced by a static color source. Eichten and Hill [21have proposed an effective action from which the propagator (23) can be obtained. A fixed momentum (mg, 0) is subtracted from the momentum of the heavy quark in order to eliminate the remaining trivial dependence on m~.In this theory, heavy quarks and antiquarks are considered as independent fields, because at infinite mb, bb pair creation cannot happen. In the four-component formalism the heavy-quark spinors have the form Ta

=

b=(~)

and are projected out by ~(i

+

(25)

y~)•The antiquark spinors are the row vectors (oo~=(~1~2))

(26)

and are projected out by ~(i ye). The continuum effective lagrangian for b-quarks in the euclidean space—time is —

L~=bt(ia 0+gA0)b

(27)

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and the one for antiquarks is L1~,=b(ia0+gA0)bt.

(28)

Using this formalism, we can treat heavy quarks as two-component spinors, as long as we remember that y~applied to a b-field is + 1 and to a b-field is 1. On the lattice we use the same discretized actions as in ref. [41, —

Sb

=ia3Ebt(n)[b(n)

—

S

UJ(n

—

O)b(n

—

_U0(n_O)bt(n_O)],

13=ia3Eb(n)[bt(n)

(29)

where O is the unit versor of the time component. With this choice the free heavy-quark propagator in momentum space is

(-i/a)[exp(~0a)

-1] +ie~

(30)

One uses the definition (29) to avoid the doubling problem, that would occur with a symmetric derivative.

4. Improvement for heavy quarks It is possible to show that for order-a improvement of on-shell matrix elements, no modification of the static quark propagator is needed [191. Let us consider the usual lattice heavy-quark propagator [ii for t > 0 in the static approximation S~,att(X t)

1 =

+

2

y0

H(x,

t),

(31)

where we have defined H(x,

t) =Piatt(~)ö(X)O(t)

(32)

and Piatt(~) U0(t)U0(t —1)... U0(1). =

The quantity H(x,

t)

(33)

is the solution of the discretized differential equation

~[H(x,

t)-UJ(x, t-i)H(x,

t-1)] =~(x).

(34)

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and BB mixing

509

This equation differs from the continuum one by terms of order a, as can be seen from the free case; in momentum space we find H~1(p0)

=

~[exp(ip0a)

—

iJ

=p0

+

~ip~a +

(35)

O(a2).

In order to eliminate the order-a terms, we can add next-to-nearest neighbour interactions and modify eq. (34) as follows t(x, t)

t- i)H(x,

t-

1)

4U0

-

+

~U t(x, t-2)H(x, t-2)] 0

=

(36) In this case we get H~1(p 0)

=

~[2

exp(+i~0a) ~exp(+2ip0a) —

—

~] ~p0+O(a~).

(37)

In this way eq. (36) and the corresponding improved free propagator coincide with the continuum up to terms of order a. The key observation is that the solution of eq. (36) is of the form H(x,

t) =Piatt(~(X)O(t)f(t)~

(38)

where f(t) tends exponentially to 1 for t We thus conclude that, for on-shell matrix elements, i.e. t the improved heavy-quark propagator coincides with the unimproved one. On the other end, in the effective field theory formalism, we show explicitly in appendix A that, following the prescriptions of sect. 3 for light quarks only, without modifying the heavy sector of the theory, the on-shell matrix elements of any heavy—light bilinear do not have effective order a contributions at one loop. Summing up, in order to eliminate order a terms from on-shell matrix elements of the heavy—light operators of eqs. (1) and (3), the recipe is the following: (i) use the nearest-neighbour improved action, eq. (ii), for light quarks; (ii) use the usual effective action, eq. (29), for heavy quarks; (iii) “rotate” only the light-quark fields appearing in the operators according to eq. (14), i.e. use the improved operators —~ ~.

—~ ~,

Oj.

=

—

~ar(~F~q)

(39)

and ~

(40)

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5. Renormalization of improved two-fermion heavy—light operators In this section we present the one-loop perturbative calculation of the lattice renormalization constants of the improved heavy—light bilinear defined in eq. (39). Following refs. [2,4,5],we perform the calculation in two steps, first relating the generic heavy—light bilinear in the continuum full theory, eq. (i), to the corresponding operator in the effective theory O~=bt(1,0)Fq,

(41)

which has the same tree-level matrix element between an incoming light quark and an outgoing heavy quark. The second step is the matching of the continuum effective theory to the lattice effective one. All the continuum quantities have already been calculated and the result of the first-step matching, in the MS renormalization scheme and with the definition of always anticommuting y~,is [21 °F (i

+

2/m~)

16~2C~4H

—

~)log(~

~HG+ ~H2_HHP_4})O~, (42)

where N2—i

4

2N We use the same notations of ref. [2], defining CF=

(43)

~

HF=y~Fy,~~, GF=y 0Fy0.

(44)

H’ is the derivative of H with respect to d in d dimensions. As for the second step, for the unimproved case we obtain, in agreement with ref. [41,

o~=(i +

16

2c~[~log(~a)+ ~ _A~I)O~tt,

where the constant A1, expressed in terms of the quantities defined in ref. (see also appendix B) A1=d1+d2G+~(e+f).

(45)

[41,is (46)

In the improved case, we must first consider the diagrams in fig. 1 with improved Feynman rules for the light quark—gluon vertex. Besides these diagrams we have to compute those of fig. 2, corresponding to the “rotated” term of the operator ~.

*

In figs. 2, 4, 5, with the symbol ® we denote the

—

~arFØ~ insertion.

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(b)

(a)

(c) Fig. 1. One-loop vertex and self-energy diagrams of local two-fermion operator.

Summing all the contributions, the result of the second step in the improved case is

o~t~(i +

2cF[~log(~a)+ ~ _Ar_4])(onhatt.

16

(47)

The analytical expression of the new constant A~ is given in appendix B, eq. (B.12). In the interesting case of the time component of the axial current, used to determined f~,at ~t a’, we obtain 1) (i + a ~cont (a~~ [— ~log(~2/m~) 2] =

=

—

Iattf

+ as

~a

—1\

) (5 ‘.4

—A YsY~ —A’Y0Y5)\ f~ k YOY5)\latt

48

Eq. (48) contains the contributions of the two steps and gives the final relation between lattice effective and continuum full theories.

A (b)

(c)

Fig. 2. One-loop diagrams for the vertex correction due to the rotated part of the improved two-fermion operator.

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B-decay andBB mixing

1 and A~~

TABLE

Numerical values of the finite constants ~ r

1.00 31.60 —5~23

~

5 for several values of the Wilson parameter r

0.75 31.21 —5.30

0.50 30.65 —4.77

0.25 29.36 —3.07

0 24.30 0

We have computed numerically the constants A~0~5 and ~ with a error of less than 5% and their numerical value, as a function of the Wilson parameter r, is given in table 1. Usually [20], fB is extracted from numerical simulations by fitting (tr( 1

7OSq(O,

=C exp(—~EBt)

t)U0(t)U0(t— 1)...U0(1)))

(49)

tatt

where C is a constant, while zIEB contains linear divergencies lim a~E~

X.

—

(50)

a—*O

In perturbation theory the constant X in eq. (50) coincides with the coefficient of the linearly divergent term in the self-energy, .~(p0),of the heavy quark (including the tad pole) X=ai~(p0=0)

=

(51)

1~2CFi~O,

where i.~0=19.95.

(52)

For the analytical expression of i.~ see appendix B. When using a fit of the form (49), the heavy-quark self-energy contributes to A~~5 with a reduced value, see refs. [4,6],

e~ e =

—

i.~0 4.53.

(53)

=

In this case, as anticipated in the introduction, we find that at f3 a~~tt 0.080,

=

6.0, i.e.

=

~x=a’=2GeV,

mb=5GeV,

r=1

and using the two-loop continuum values of the strong coupling constant

(54) a~0~~t =

/

A. Borrelli, C. Pittori

0.25, corresponding to ~

=

B-decay and BB mixing

513

200 MeV for four active quarks, the lattice value of

fB is related to the physical one by fcont JB

— —

~Iatt AJB

where ZA=O.89.

(56)

6. Renormalization of improved four-fermion operator i~B 2 =

We now consider the four-fermion operator of eq. (3) °L

~(by~q)(Jy~q).

(57)

To be definite we focus on the matrix element K~°IOLIB°),

(58)

with a B°(bq)in the initial state and a ~°(~b) in the final one. As in sect. 5, we proceed in two steps: the first one is the matching between continuum full and continuum effective theory; again, the result of this first step is the same as in previous calculations [7](we use the same notation),

=

+

~

[—6 log(~t2/m2)+

CLI )o~+ ~CSOSeff,

(59)

where Os=+(bPL q)(6PL q) is a new operator that mixes with

°L

(60)

at one-loop; we define 1 ±~5 2

(61)

The constants in eq. (59) are CL= —14,

C 5= —8.

The continuum effective operators are defined as t(1, 0) y~q)(~(0,1) y~q) O~= (b

(62)

(63)

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B-decay and BB mixing

(a)

(b) Fig. 3. One-loop vertex and self-energy diagrams of local four-fermion operator.

and O~= (b~(1,0)

~L

q)(~(0,1)

‘~‘Lq).

(64)

In reproducing this result, special care must be taken of the renormalization convention used; we follow the choice of ref. [7]and work in the MS naive dimensional renormalization scheme, with fully anticommuting y~.Note, however, that Fierz transformations are not well defined for d ~ 4: for the correct projection of divergent diagrams on physical operators, see ref. [21]. As we have seen, new operators appear in the continuum when we renormalize °L~The

same happens on the lattice, where we find the mixing with the operators OR=~(by~q)(~y~q)

(65)

and ON~[(b y~q)(b y~q)+(b y~q)(b y~q)

+2(bPR q)(bPL q) -l-2(T’PL q)[bPR q)].

(66)

In the unimproved case the graphs considered are those in fig. 3; the result, in agreement with ref. [7], is 2a2)+DL])O~tt+ 0~ît

(i

+ 162

[4 log(~

1

tt+ 2DNO~

12DRO~tt, (67)

DL, DR, DR, are the same constants defined in ref. [7](see also appendix B).

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IN (b)

(a)

xx..

(c) Fig. 4. One-loop diagrams for the vertex correction due to the rotated part of the improved four-fermion operator. Fermion—fermion vertex contribution.

In the improved theory, we compute the same diagrams calculated with the improved light-quark—gluon vertex. Along with these, we have new graphs (figs. 4 and 5) corresponding to the rotated part of the operator. The result for the

M (a)

/\. (b) Fig. 5. One-loop diagrams for the vertex correction due to the rotated part of the improved four-fermion operator. Fermion—fermion—gluon vertex contribution.

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B-decay and BB mixing

improved lattice operator corresponding to the continuum effective O~ is

o~=

+ i6~2

[4 log(~2a2)+DL+D~I)(OL)tatt

+ 162 (DN +DN)ON

+ i6~2 (DR

+D~)O~tt.

(68)

Now, adding the results of the two steps, we obtain the lattice effective operator corresponding to °L in the continuum full theory; by choosing ~ a~ we get =

a~0~~t(al)

0L(1+

4

a~att(a_l) +

+

[_6log(~2/m2)+C~]

(DL+D~))(OL)1att+

atatt(a_l) 4 C~O~att

a~att(a~l) atatt(a (DN+D~)o~tt+ S

t)

(DR+D~)O~tt. (69)

Numerical values of DL, DN, DR and of D~,D~,D~,as a function the Wilson parameter r, can be found in table 2. For their analytical form, and for the discussion of the various diagrams, see appendix B. We use the reduced value e~ 4.53 see eq. (53) and discussion thereof for the heavy-quark self-energy and we choose /3 6.0, i.e. a~tt 0.080, =

jx=a’=2GeV,

—

—

=

=

mb=5GeV,

r=1,

(70)

using the two-loop continuum value of the strong coupling constant a~,0flt 0.25, corresponding to A~4~ 5= 200 MeV for four active quarks. In such a way, as anticipated in the introduction, we find that the coefficient of (O~)~att is 0.80. =

TABLE 2 Numerical values of the finite constants DLRN and parameter

r DN DR DL D~ D~

1.00 —65.50 —14.40 —1.61 16.25 0.44 —2.58

0.75 —63.80 —14.50 —1.77 13.73 2.22 —2.27

D~RN

0.50 —62.00 —14.00 —1.92 10.18 3.48 —1.72

for several values of the Wilson

0.25 —59.80 —11.40 —1.59 4.91 3.36 —0.82

0 —58.10 0 0 0 0 0

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Neglecting the contribution of operators with different chirality, this same factor relates the lattice value of f~BB to the continuum one at a scale ~t 2 GeV. To evaluate the correction factor for the BB parameter alone at the same scale we subtract the correction to f~,and obtain 1.02. =

We warmly thank G. Martinelli and G.C. Rossi for many useful suggestions and for help during all this work. We also thank M. Testa for discussions. One of us (A.B.) wishes to thank the TH-Division of CERN for kind hospitality.

Appendix A In this appendix we show explicitly, at one-loop, that using the improved lattice prescriptions described at the end of sect. 4, on-shell matrix elements of bilinear heavy—light operators (39) do not have terms of order a or g2a log a. The on-shell matrix elements of a generic heavy—light operator O~is obtained by adding all the diagrams contributing to the irreducible vertex correction A 1 and multiplying it by ~Z~Z~A1(p~,

p),

(A.i)

where Z~is the residue of the pole of the propagator of the improved light quark, and Z~that of the heavy quark. The one-shell conditions that we use in calculating (A.1) are

p~=0

(A.2)

for the heavy quark and ip’

=

(A.3)

—mR

for the light quark; here the renormalized light-quark mass is explicitly kept different from zero. For dimensional reasons, only graphs that involve light quarks in the loop give rise on-shell to terms of order a log a. The relevant calculations for light quarks in the improved theory have beenand done in const), ref. [81.thus, We keeping are not 2 const) (ag2 interested here in all the terms of order (g only the logarithmically divergent terms, one gets =

(1 —g2CFL)(1 —arm 0),

(A.4)

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A. Borrelli, C. Pittori

and BB mixing

where m0 is the “bare” mass of the light quark divergent term for a

and L is a logarithmically

*

—*

L

=

f

4q d

(2ir)4

1

1 =

—

q2(q2 + A2a2)

log A2a2 + const

(A.5)

16~,~2

where A is the infra-red regulator. For the heavy quark we find 1

=

2g2C~L,

+

(A.6)

As for the contributions of vertex diagrams, using the one-shell conditions (A.2, A.3), we get fig. la—*g2CFF[1

—

+ar(iil+m 0)]L,

(A.7)

0L),

(A.8)

2Cfarm fig. 2a

—*

F(

fig. 2b

—*

g

—

~ari~+ ~g

2CFF(

—

~arij,I 2arm —

0)L.

(A.9)

2, the relation g2m 2m~ holds, we have On-shell, noting that, at order g 0 g A 2C~L+g2CF( —armR)L1. (A.10) 1= F[1 + 2armR +g Finally, using the relation [81 =

mR =

m

2CFL)

0(1

+

(A.11)

3g

we obtain for eq. (A.i) ~Z~Z~A

2C~L), 1=F(i

+

(A.12)

~g

where the leading logs are the same as in the continuum effective theory, and we see that there are no terms of order g2a log a.

Appendix B In this appendix we will discuss with more detail the new graphs we have computed, and give the analytical expressions of their amplitudes and their *

We neglect here the g2/a divergent term coming from the lattice light self-energy, since it does not give rise to additional logarithmically divergent contributions.

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B-decay and BB mixing

519

numerical values as a function of the Wilson parameter r. For convenience, we will define the following quantities, which have been introduced in refs. [7,17],

4~=~sin2~q~,

~2

44—

~sin2q~

+4r2(4

2, 1)

~sinq~,

4~=~sin2q,~ sin2~q~, 41.

4— 6

cos 2q~, 2~q~cos6~q~.

47=

(B.1)

~sin

Similar quantities, which are summed over J 1, 2, 3, instead of ~ 0, 1, 2, 3, will be indicated with superscript (3). Let us consider the lattice renormalization of the improved two-fermion heavy—light operators O~!’~-,eq. (39). The contributions of the diagrams of fig. 1 to the lattice renormalization constant are =

=

fig. 1a~ ~ 2C~[—log A2a2+di+(d2=dI)G]F,

(B.2)

[forthe definition of GF see eq. (44) of the text] 2a2 fig. lb

16~2CF( —

fig. lc~

1

+

~e)F,

(B.3)

log A 2a2+ ~(f+f’)IF.

(B.4)

2CF[~log A

In eqs. (B.2)—(B.4) the quantities d,, d 2, e and f are the same as those of ref. [4]; for their analytical expressions see also the appendix of ref. [7]. The new quantity d’ is defined as

d1

=

(— 16~r2)~r

3q

4~

(2~~-)~ 44(3)4(3)~ d

(B.5)

520

A. Borrelli, C. Pittori

/ B-decay and

BR mixing

TABLE 3 t for several values of the Wilson parameter r Numerical values of the finite quantities d’, f’, v’ and w

r d’

f’ vt wt

1.00 —4.04 —3.63 —6.69 0.82

0.75 —3.66 —2.84 —5.73 0.80

0.50 —3.07 —1.92 —4.44 0.72

0.25 —2.02 —0.84 —2.34 0.45

0 0 0 0 0

The quantity f’, coming from the improved light-quark self-energy (see also ref. [18]), is defined as

(l6~2)f(2~)~ [4142

~

4~1~4~i(4~1)

—44(4—4k) — ~4~(44 14

+

(~[4,(4=4i)2

(4

—45) + ~(44

4)2k

fr ~

+ ~4,44(4—4,)

+(44—45)

— 45)(4

—

4~)

~4446—47]

-~)

-

_4s)1)~.

(B.6)

The numerical values of d’ and f’ as functions of r are given in the first and second row of table 3. From the diagrams of fig. 2 we get

fig. 2a 12CF[

fig.

— (d’ + q)G + n] F,

2b~ 12CF[—(l+m)]F,

fig. 2c—*

(B.7)

(B.8)

2 g 16ir

2CF(hG)F.

(B.9)

/

A. Borrelli, C. Pittori

B-decay and B~mixing

521

TABLE 4 Numerical values of the finite quantities h, 1, m, n, q and s for several values of the Wilson parameter r

r

1.00 —9.88 —3.42 7.35 0.73 —2.02 3.41

h m n q s

0.75 —7.41 —2.52 5.92 0.63 —1.20 3.20

0.50 —4.94 —1.54 4.08 0.47 —0.54 2.79

0.25 —2.47

—0.55 1.74 0.22 —0.11 1.72

0 0 0 0 0 0 0

The analytical expressions of these new quantities are d3q

1

r2 d4q ~ 1= (_16~2)~f(2~)~

r2

d4q (2~)~

~44 + 84, — 24~

—

m

=

(16~~2)_J

n

=

(16~2)~f

q

=

~

d4q

r2

r3

(B.iO)

~

~

d3q

4(3)

(2~)~44(3)’

(B.11)

while their numerical values are given in table 4. Thus for A 1 and A~..,eqs. (45), (47), we get A1= d,

+

d2G

+ ~(e

+f),

A~-=—(2d’ +q—h)G+ ~f’ —i—rn +n. In the case F

=

7oY~’cf.

(B.i2)

table 1, we have A~~=d1—d2+ ~(e+f), 1 =

2d

+

~f’

—

h

—

1

— rn + n +

q.

(B.i3)

522

A. Borrelli, C. Pittori

/

B-decay and BR mixing

Finally, we give the analytical expression of i10 defined in eq. (51) 3q 1 iZ d 3 84~~~’ 0 (l6~~r2)f (2ir)

(B.14)

for its numerical value, see eq. (52). We now discuss the diagrams referring to the four-fermion operator OL, eq. (40). The graphs in fig. 3a, computed with the improved light-quark—gluon vertex, give

16~~2 [~(—log A2a2) +

~d 1

+

~e +

~CI(O~)latt + l6~2[2(—d2 +dI)JO~tt. (B.15)

The graphs in fig. 3b, which involve only light quarks in the loops, give (see also ref. [18])

2a2 + 4(f+f’) + +(v + v’)l (OL)~’t+ 16~2~(w + wt)O~tt, (B.16) 16~2[flog A where f, f’ come from the improved light-quarks self-energy (see the previous discussion of two-fermion operators). The quantities v and w are the same of ref. [7], while the analytical expressions of the quantities v’ and w’ are

=

(—

i6~r~)f

d4q

r2 ~ 84~4~ {441(44

4~)+

—4,)

4~144(4

+r2[_2(4 4

w~ (i6~2)f =

—

4~)4~ + 2444~(4—

~

{ —4~1(~4

—

4~)+ 4~1~4(~

—

~

—

~4~(4 —4,)

(B.17) We give the numerical values of v’ and w’ in the third and fourth row of table 3. Now we compute the new graphs of figs. 4 and 5, corresponding to the rotated part of the operator, eq. (40). The graphs in figs. 4a and 5a give tt+ 12 2( —h i6~2(~n)(oL)ia

+q +

d’)O~tt.

(B.18)

A. Borrelli, C. Pittori

/

B-decay and BR mixing

Figs. 4b and Sb contain only light fermions in the loop they give, respectively,

2~

(~~s)(OL)

16~2

+ i6~

—

523

see also ref. [18]

~s)O~tt

—

and

(B.l9)

and

l6~2[-~(l+m)J

+

[~(1+rn)]O~~tt.

~2

(B.20)

The analytical expression for s is s

=

(16~2)f d4q (2ir)

— -~-~_

~

(~44(4

—

~

+

~r2[(4 4

—

4~)

—

44(4

—

4k)] (+~4

—

4~)

2

(B.21)

[4~(4—4,) + 44— (44_45)1)~

we give numerical values for s in the last row of table 4. For comparison with ref. [18], we get s

=

8~~2I6,

(B.22)

and 21 i+m=’,r

7,

(B.23)

where ‘6 and 17 are defined in ref. [18]. The contributions of the graphs in fig. 4c are zero. Finally we find, cf. eq. (68), DL= ~(1—d,) =

DR

—~c—~e+~(+—f)+~(—5—v),

2d2, (B.24)

=

and

t+ ~(m+1) D~= —~n— 4f D~=2(h —2d’—q), =

~[s

—

4w’

—

2(1

+ m)J.

—

~v’+ 4s,

(B.25)

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A. Borrelli, C. Pittori

/

B-decay and BR mixing

References [1] E. Eichten and F. Feinberg, Phys. Rev. D23 (1981) 2724 [2] E. Eichten and B. Hill, Phys. Lett. B234 (1990) 511 [31MB. Voloshin and MA. Shifman, Soy. J. NucI. Phys. 45 (1987) 292; G.P. Lepage and B.A. Thacker, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 199; H.D. Politzer and MB. Wise, Phys. Lett. B208 (1988) 504; N. Isgur and MB. Wise, Phys. Lett. B232 (1989) 113; B237 (1990) 527; H. Georgi, Phys. Lett. B240 (1990) 447; B. Grinstein, Nucl. Phys. B339 (1990) 253; A.F. Falk et al., NucI. Phys. B343 (1990) 1; L. Maiani, G. Martinelli and CT. Sachrajda, NucI. Phys. B368 (1992) 281 [41E. Eichten and B. Hill, Phys. Lett. B240 (1990) 193 [5] Ph. Boucaud, CL. Lin and 0. Pène, Phys. Rev. D40 (1989) 1529 [61 Ph. Boucaud et al., LPTHE-92/45 and CERN-TH. 6599/92 [71J. Flynn, 0. Hernández and B. Hill, Phys. Rev. D43 (1991) 3709 [81 G. Heatlie et al. Nucl. Phys. B352 (1991) 266 [91G. Heatlie et al., NucI. Phys. B (Proc. Suppl.) 17 (1990) 607; E. Gabrielli et al., NucI. Phys. B (Proc. Suppl.) 20 (1991) 448 [101 A. Borrelli and C. Pittori, NucI. Phys. B (Proc. Suppl.) 26 (1992) 398 [11] G.P. Lepage and P. Mackenzie, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 173 [121 Particle Data Group, Phys. Lett. B239 (1990) 111.50 [13] G. Altarelli et al., Phys. Lett. B99 (1981) 141 [14] V. Giménez, Rome Preprint no. 813, july 1991 [15] B. Sheikholeslami and R. Wholert, NucI. Phys. B259 (1985) 572 [16] J. Mandula et al., Nuci. Phys. B228 (1983) 9 [17] E. Gabrielli et al., Nucl. Phys. B362 (1991) 475 [181 R. Frezzotti et al., Nucl. Phys. B373 (1992) 781 [19] G. Martinelli and G.C. Rossi, private communication [20] CR. Allton et al., NucI. Phys. B349 (1991) 598 [21] A.J. Buras, M. Jamin and PH. Weisz, Nucl. Phys. B347 (1990) 491