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Lattice QCD estimate of mixing parameters for heavy meson systems
Volume 23 l, number 1,2
PHYSICS LETTERS B
2 November 1989
L A T T I C E Q C D E S T I M A T E O F M I X I N G P A R A M E T E R S F O R HEAVY M E S O N S Y S T E M S Maurizio L U S I G N O L I a, Guido M A R T I N E L L I ~,b and Attilio M O R E L L I " Dipartimento di Fisica, Universiiz "La Sapienza" 1-00185 Rome, Italy and INFN - Sezione di Roma, 1-00185 Rome, Italy b CERN, CH-1211 Geneva 23, Switzerland Received 27 July 1989
The matrix elements of the two four-fermion operators relevant for the calculation of mixing in the D-I) system have been obtained by a lattice QCD calculation in the quenched approximation at fl= 6.0. The resulting B-parameters are close to each other and not very different from one. The physical implications are discussed, together with a possible extrapolation of the result to the more interesting B-I) case.
A theoretical estimate o f the difference in masses (AM) and widths (AF) for neutral heavy meson systems (D°-IS) °, Bd-Bo, o -o B~-Bs) o -o can be obtained by taking the matrix element o f an appropriate effective hamiltonian. In the effective hamiltonian the short-distance Q C D corrections are resummed in the coefficients multiplying local operators. The calculation o f the matrix elements o f these operators requires nonperturbative methods. In early times, the vacuum insertion approximation (VIA) was used as the simplest estimate for the matrix elements; nowadays, the VIA matrix element is still used as a reference value, to be multiplied by a (subtraction point dependent) factor, usually called the B parameter. Let us consider in particular the Bq-Bq o -o ( q = d , s) case. Including the short-distance perturbative Q C D contribution one has
where the O t ± ) are renormalized at a scale/t_~ m b. In eqs. ( 1 ), (2) the (i = V,~V*q are products o f elements of the Cabibbo-Kabayashi-Maskawa matrix, Gv is the Fermi constant, M the average mass of the mesons, qMand (qts) r/(a)) are Q C D correction factors, that are given in ref. [ 1 ] and refs. [2,3 ] respectively. In eq. ( 1 ) we have neglected further terms, that are suppressed at least by a factor (m~/m2t)X l o g ( m { / m 2 ) , and F ( m t / m w ) is a slowly varying function [4] ( F ( 0 ) = I , F ( 1 ) = 3 ) . In eq. (2), T~"(k) is a function of the external m o m e n t u m running in the loop of the box diagram and of the quark masses, whose explicit expression is given in ref. [ 3 ]; we only note here that it is symmetric under the interchange (p,--~u) and in the limit o f zero masses it is given by
Gz F 2 2 F ( m t / m w ), 32~-fM ~t//Mint
The four-fermion operators in eqs. ( 1 ), (2) are defined as follows (a, p are colour indices ):
M12~
(1)
O ( + ) - ~ ~r)(+)
X [ (]~t~) + ¼u(a)) (B° I n ( + )
] ,
(3)
O(~ ) ----~ [4a Y~( 1 -- ys)baqBy~( 1 - y s ) b B
+Cl,~7u(l-75)bP(lpy.(1-Ts)b'~+(p~u)],
r l ~ = G~ v ~ T ~ " ( k ) 4 M cj=~u,~
-
1
TU"(k) = - ~ ( k 2 g U " - k u k ~ ) .
(4)
(5)
It is easy to verify that (2)
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
O~+)=~gu,O (+),
gu~O~7)=O.
(6) 147
Volume 231, number 1,2
PHYSICS LETTERS B
As a consequence, one has only two independent B parameters, B ~÷ ) and B (-), defined by
(B°IOt+)If3°)=B~+)2(I+I/N).~BM2,
(7)
( B ° I o ~ ; ) Ia ° )
=B(-)½(1-1/N)fZ(4PuPv-M2gu~) ,
(8)
where fa is the B meson decay constant and N = 3 is the number of colours. A knowledge of B ~+ ) and B t - ) is therefore necessary to determine the physically interesting quantities A M = 2 Re ~/(M1z - ½iF12) (MT2 - AF=-4Im~/(M12
i 1 F 1 2 ) l•"
(9)
A viable method for the nonperturbative evaluation of matrix elements is provided by lattice QCD [ 5 ]. Unfortunately, present-day available values for the lattice spacing are not small enough to allow the propagation of a quark as heavy as the b. In this letter we present the results obtained in a Monte Carlo calculation of the B (+-) parameters for the A C = 2 operators, relevant for the calculation of AM and AF for D°-I7)°. Although the mixing parameters in this case are exceedingly small (in the standard model) we hope to obtain from this calculation also an insight for the more interesting case AB= 2. We present immediately the results and a short discussion of their physical implications, postponing the presentation of the details of the lattice calculation. At a scale # = a -1 (a is the lattice spacing) we have ~
1.~eV
GeV4'
(10)
( D (/3= 0)I Oti-)(a -1 ) 113(/3=0)) = (0.021 + - 0 . 0 0 7 ) (\ --'--a2 1.9 G e V ]~4 GeV4 '
( 11 )
(a,)
(12)
fD=(210+30)
1.9G-eV g e V ,
#t In order to minimize the systematic errors, the central value offD reported in ref. [6] was found by computing the ratio fD/f,~and by fixingf~ to its experimental value. It differs from the value in eq. ( 12 ) by about 8o/o.
148
1.9GeV
/
GeV.
(13)
For the B -+ parameters we give two different estimates, obtained with two equivalent methods to be discussed later on, B t + ) ( a - l ) = 0 . 7 3 + 0 . 0 5 (0.69_+0.05),
(14)
B ( - ) ( a - ~ ) =0.79_+0.05 (0.73 + 0 . 0 4 ) .
(15)
In eqs. ( 10 ) - ( 15 ) the errors are purely statistical. A quantity less subject to systematic uncertainties is given by the ratio of the two B parameters
f l ( a _ l ) _ B(+ B ~ - ))(a_l) ( a - ' ) - 1.08+_0.10,
(16)
which is directly obtained as a ratio of three-point correlation functions on the lattice. To allow a comparison with other possible calculations of the same quantities made at any scale greater than me, we also give the scale invariant B parameters [ 7 ] defined as B ~ -) = [ a s ( a - l ) ] -6/25B(+)(a-1 ) =0.95 +_0.07, B~/-) = [oq(a-~)]-4/2SB(-)(a-1)=0.94+_O.06,
flsi=[ot~(a-J)]+2/zsfl(a-l)=0.99+O.09.
(17)
The numerical values in eq. (17) have been obtained setting a - t = 1.9 GeV, A4=200 MeV and using the average of the two values of the B factors given in eqs. (14) and (15). The ratio f l ( a - t ) is interesting in order to compare our results with the estimate given by Voloshin et al.
[8].
( D I O ( + ) ( a - 1 ) 113) =(0.16+0.05)
a-I
roD=(1.51_+0.02)
,
l' • - ~IFt2)(MIE~t"' 1 2• ) -
(al)
2 November 1989
In their approach, they assume that the vacuum insertion approximation is valid at a very low scale, of the order of the inverse radius of the meson, #-~ 300 MeV. Therefore, they have studied the renormalization of the operators O ~+) at scales much less than the mass of the heavy quark rnQ. In this region, the anomalous dimensions of the operators are different from those given in eq. (17) since the heavy quark behaves as a static colour source. For/z < m o the B ( + ) parameter is scale invariant [9 ] (since the O ~+ ) operator's anomalous dimension is twice the axial current's one) and fl(mQ) is given by [ 3 ]
B(-)('I2) (O~s(mQ) ~ 8/(33- 2J) f l ( m Q ) ~ B ( + ) ( # ) \ a~(U) J
(18)
Volume 231, number 1,2
PHYSICS LETTERS B
From eq. (18) and the hypothesis B ( - ) (p) = B (+) (p) -~ 1 ( V I A ) one has evidently flViA(mQ) < 1 for/~< m o . In the particular case m Q = m~= 1.2 G e V , / t = 300 MeV one has flv~A(m~) = 0 . 7 6 (0.59)
(19)
f o r A 4 = 100 (200) MeV. On the other hand the evolution from a - ~ = l . 9 GeV to m e = 1.2 GeV of our result, eq. (16), gives fl(rn~) = 1.06 + 0.10.
(20)
The reason for the discrepancy between our result, eq. (20), and the vacuum insertion approximation at a low scale, eq. (19), should be clear from fig. 1, where we report the central values that would be obtained by rescaling with the appropriate factors the B parameters calculated at a - ~. Coming now to the physical quantities o f interest, the largest mixing parameters A M and A F are expected for the B~-Bs o -o system. One has
1.2
[
'
,
,
,
[
l
1.0
'
'
'
'
I
\
--
2 November 1989
2 -o ) - GFfaB, M3 Fl2 (B~o -B~ 8n ~ ( 1 --4X)t/2B~+)(mb)
× [~ (3q(s)+ q(")) (1 --2X) + ~ (3r/(s) -- I"/(a)) ( 1 + 2x)fl(rnb) ] .
(21 )
In eq. (21) x = m Z / m 2, we have made the approximation m b ~ M and Cabibbo suppressed contributions are neglected. If we assume that flBs(mb) ~ 1 (as it happens for fib(me) ) we obtain
where we have used the Wolfenstein [ 10 ] parametrization for ~, the experimental average B lifetime and A 4QcD = 200 MeV. This result essentially agrees with ref. [ 8 ]. In fact, the difference between the results of our calculation (B(+))-~0.71, f l ~ l ) and the values assumed by Voloshin et al. ( B ( + ) _~ 1, fl--0.6) does not show up in AF. It could be seen however in the ratio of the two mixing parameters AF/AM, for which we obtain a result larger by a factor ~ 1.3. With the same assumptions, the estimate for 0 -0 B d - B d is
FiB a
140 M e V / ~--.~ ( 0 . - ~ ) 2
0.8
B(÷)
0.6
1
,
I 0,5
. . . .
I
. . . .
1
I 1.5
. . . .
I
where GB (Z) = 1 - 9 . 5 8 Z + 22.7Z 2. We have actually calculated the mixing parameters for D ° - D °. With obvious modifications in eq. (2) and using eqs. (10) and (20) and the experimental D o lifetime we obtain
2
u(GeV)
(2F,2"]
\ - - F - ] D = (1"4+0"5) X 10 Fig. I. The running B-factors B (+) (p) and B (-) (p) at the scale p obtained by evolving B C±)(a -t ) calculated in the quenched lattice QCD at fl= 6.0. The results are obtained by using the leading logarithmic anomalous dimensions given in eq. (17) for mc~ 1.2 GeV ~
-3c:_ {VcbV~,u) \VcsVsuJ ~'JD
.
(24) •
In eq. (24), we have denoted by GD(Z) a second degree polynomial whose coefficients are rapidly varying functions of the ratio x = mZ/mZ¢. We have in two extreme cases G D ( Z ) = { -1-7)< 1 0 - s } +~'7"25"~Z+42 5 Z 2 [ . 0) . 3 .
.
(25)
149
Volume 23 1, number
PHYSICS
1,2
The upper values in eq. (25 ) are obtained assuming m,=0.45 GeV, mc= 1.2 GeV; the lower ones correspond to m,= 0.15 GeV, m, = m,. We remind that 1VcbV~u/VccsV~u1=O( 10F3) so that for small x values the second term in GD dominates. To give an estimate for A&I we consider its expression, neglecting QCD corrections and next to leading terms: Ml2
x (D”]O~;‘-O;;)]Do)
.
(26)
The reason to keep all the terms in eq. (26) is that ]&,/&I -O( 10e3), and the term proportional to rn; is not at all dominant. A numerical estimate from eq. (26) gives the upper bound <2x
10-5.
2 November
B
1989
ping parameter for the heavy quark has been fixed to K=O. 1350, corresponding to a mass of the charm quark of about 1 GeV (in the lattice regularization scheme). We have used three values for the light quark masses corresponding to K=0.1515, 0.1530 and 0.1545 and then extrapolated all the results to the zero mass limit for the light quarks corresponding to K,=O.1564(2) [ 121. The renormalized operators (MS scheme [ 13 ] of which we have estimated the matrix elements are linear combinations of lattice bare operators. The coefficients of the lattice operators have been calculated at first order in perturbation theory, following the general prescription given in ref. [ 141. The renormalization of two fermion operators was already done inref. [15] andforO(+)inrefs. [16,17]. Following the same procedure of ref. [ 16 ] we have performed the analogous calculation for O,$; ) . The final result is
~o::Lmt.
= (1+ 6
+&
r2Z*(r)$gpv
Z+)Ol:’
(27)
This value has been obtained assuming for m, the “constituent” value ( N 450 MeV), and it is dominated by the
LETTERS
-N-l x
OSTP_ -2;osP
( 2N
[o$&.~,=(
+&2 x
(
N2+N-1 4N
-
1+ &-))
OVA ) > (28)
0:;’
r2Z*(r)
hNO::_
N2-N-l 4N
0
VA “’ >
(29)
o(+)=dg,“eYp(l-Ys)qQYp(l-Y,)q, PJ 01;)=f[ey,(l-y,)q&y,(l-y,)q+(~Uttv)l -0::’
)
OSTP=&&7+
Sea,“qeo~“q+&Y,qQY,q,
0SP=QqQq-12Y54QY54> 0VA=eYpqQY/14-~YrYs4~Y~Ys4,
(30)
Volume 231, number 1,2
PHYSICS LETTERS B
2 November 1989
Table 1 The results for the matrix elements ( D l O t + ) ( a -~ ) 113), (DlO[i-)(a -~ ) I D ) and f o r f ~ m ~ are given in units of the inverse lattice spacing a - ~, for three values of the hopping parameter of the light quarks, K. The fourth row of the table contains the linear extrapolation to the chiral limit, corresponding to K¢= 0.1564. Since the lattice calculations are performed in the euclidean theory, the signs in the second column and in eq. ( 11 ) are opposite. K
a4
a4(DlO}~')(a-')lf))
a4~fDrnD2 2
0.1515 0.1530 0.1545
(3.0_+0.8))< 10 -2 (2.5___0.6)>(10 -2 (1.9+0.5))<10 -2
- (4.0_+0.8))< 10 -3 -(3.3-+0.7))<10 -3 -(2.6-+0.6)×10 -3
( 14-+ 3) )< 10 -3
0.1564
(1.2_+0.4))<10 -2
- (1.6-1-0.5) x 10 -3
(7_+2)×10 -3
0 TT = ½ [ Q a p u q O . a ~ q + (p~-'u) ]
o.ao
' ' ' ' 1
(12_+3))<10 -3 (9_+2))<10 -3
. . . .
I . . . .
I . . . .
I . . . .
I . . . . *J
0.B5
V A
O75
O?0
(30 c o n t ' d )
-- I g u u O V A .
0.65
We h a v e calculated the matrix elements ( D I O ( + ) ] IS)) a n d ( D [ O t F ) 115) using the p l a t e a u m e t h o d d e s c r i b e d in ref. [ l 1 ] for the three v a l u e s o f the h o p p i n g p a r a m e t e r o f the light quarks. T h e m a trix e l e m e n t s in lattice units t o g e t h e r w i t h t h e i r ext r a p o l a t i o n to the chiral l i m i t ( f o r the light q u a r k s ) are r e p o r t e d in table 1. In the last c o l u m n o f the table the q u a n t i t y f E m 2 o a n d its e x t r a p o l a t i o n to the chiral l i m i t is also given. T h e B-factors h a v e b e e n e s t i m a t e d w i t h two differe n t m e t h o d s in o r d e r to e v i d e n t i a t e the s y s t e m a t i c s i n v o l v e d in the e x t r a p o l a t i o n a n d t h e a c c u r a c y o f the perturbative corrections. M e t h o d 1. F o r each v a l u e o f the h o p p i n g p a r a m e ter o f the light q u a r k s we h a v e c a l c u l a t e d the ratio b e t w e e n the m a t r i x e l e m e n t s a n d t h e i r v a c u u m insert i o n v a l u e o n t h e lattice a n d t h e n we h a v e e x t r a p o lated the results to the chiral l i m i t as it is s h o w n in fig. 2. M e t h o d 2. We h a v e m e a s u r e d a n d e x t r a p o l a t e d to the chiral l i m i t d i r e c t l y t h e ratio Zx,x2 ( P s ( t l ) O ( O ) P s ( t 2 )
)
B(o)
060
. . . .
090
.
,
I .... ,
0.05 ,
I .... 0
,
,
,
I ....
1
,
,
,
,
0.15 ,
i
I ,
,
0.2 ,
. . . . ,
,
I ,
0.25 ,
.
.
.
-.
0.3 ,
,
,
,
O.B5 080 0.75 0.?0 0.65 0.150 O
0.05
O. 1
0.15 (=m.I=
0.2
0.25
0.3
Fig. 2. The values of the B-factors are plotted versus the pion
squared masses corresponding to the Wilson hopping parameters K=0.1515, 0.1530, 0.1545 of the light quarks, the hopping parameter of the heavy quark being fixed to K--0.1350. The solid line corresponds to the linear extrapolation to the chiral limit. Table 2 Values of the B-factors B ( ÷ ) ( a - ~), B ( - ) ( a - ~), and of their ratio fl(a - ~) = B (-) (a - t ) / B ( ÷ ) (a - ~). The B-factors are calculated in the two equivalent methods described in the text, while fl(a-J) is obtained directly from the ratio of three-point correlation functions. Method
B(+)(a -I )
B(-)(a -l )
fl(a -1 )
I II
0.69_+0.05 0.73_+0.05
0.73_+0.04 0.79_+0.05
1.08_+0.10
Xx, ( A o ( O ) P s ( tl ) ) ~ 2 ( A o ( O ) P s ( t2) )
- - , large
22 (D[O(0)115> ffom~
'
(31)
time distances
w h e r e Z A = 0.87 is the axial c u r r e n t r e n o r m a l i z a t i o n
factor c a l c u l a t e d p e r t u r b a t i v e l y in ref. [ 14 ]. R e s u l t s are g i v e n in table 2. A p r e v i o u s c a l c u l a t i o n o f B ( ÷ ), rnD a n d f D with the s a m e s a m p l e o f gauge c o n f i g u r a t i o n s that we use, was 151
Volume 231, number 1,2
PHYSICS LETTERS B
r e p o r t e d in ref. [ 5 ], a n d agrees with the p r e s e n t results. We h a v e calculated fl (see eq. ( 1 6 ) ) directly f r o m the ratio
- 8 Zx'x: ( Ps( tl )O~F ) ( O)Ps( t2) ) Zx,~: ( Ps( t~ )O( + ) ( O )Ps( tz) )
,p. large time distances
(32) We stress that eq. ( 3 2 ) gives exactly the q u a n t i t y fl, w i t h o u t m u l t i p l y i n g or d i v i d i n g b y a n y other factor, so that the s y s t e m a t i c errors are reduced. I n this p a p e r we have calculated o n the lattice the m a t r i x e l e m e n t s o f the operators w h i c h are r e l e v a n t for D - l ) mixing. T h e results are in good a g r e e m e n t with the V I A at a scale a - l which is large with respect to the heavy q u a r k mass. T h i s is in c o n t r a s t with the e s t i m a t e s o f ref. [ 8 ] where the v a l u e o f V I A was ass u m e d to be v a l i d at a scale # = 300 M e V in the case o f the B - m e s o n . T h e p h e n o m e n o l o g i c a l i m p l i c a t i o n s o f the lattice results for the D - I ) m i x i n g a n d (possib l y ) for the B - B system h a v e also b e e n discussed.
152
2 November 1989
References [ 1 ] A.J. Buras et al., Nucl. Phys. B 245 (1984) 369; F. Gilman and M. Wise, Phys. Rev. D 27 (1983) 1128. [ 2 ] J.S. Hagelin, Nucl. Phys. B 193 ( 1981 ) 123; E. Franco et al., Nucl. Phys. B 194 (1982) 403. [3] M. Lusignoli, Z. Phys. C 41 (1989) 645. [ 4 ] T. Inami and C.S. Lim, Progr. Theor. Phys. 65 ( 1981 ) 297, 1772 (E). [5] K.G. Wilson, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York, 1977). [ 6 ] M.B. Gavela et al., Phys. Lett. B 206 ( 1988 ) 113. [7] A.J. Buras et al., Nucl. Phys. B 238 (1984) 529. [ 8 ] M. Voloshin et al., Sov. J. Nucl. Phys. 46 ( 1987 ) 112. [9] M. Voloshin and M. Shifman, Sov. J. Nucl. Phys. 45 (1987 ) 292; H. Politzer and M. Wise, Phys. Len. B 208 (1988) 504. [10] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [ 11 ] M.B. Gavela et al., Nucl. Phys. B 306 (1988) 677. [ 12] L. Maiani and G. Martinelli, Phys. Lett. B 178 ( 1986 ) 265. [ 13] W. Siegel, Phys. Lett. B 84 (1979) 193; D.M. Capper, D.R.T. Jones and P. Van Nieuwenhuizen, Nucl. Phys. B 167 (1980) 479. [ 14] M. Bochicchio et al., Nucl. Phys. B 262 ( 1985 ) 331; L. Maiani et al., Phys. Lett. B 176 ( 1986 ) 445; Nucl. Phys. B289 (1987) 505. [15] G. Martinelli and Y.C. Zhang, Phys. Lett. B 123 (1983) 433;B 125 (1983) 77. [ 16 ] G. Martinelli, Phys. Lett. B 141 (1984) 395. [ 17 ] C. Bernard, T. Draper and A. Soni, Phys. Rev. D 36 ( 1987 ) 3224.
PHYSICS LETTERS B
2 November 1989
L A T T I C E Q C D E S T I M A T E O F M I X I N G P A R A M E T E R S F O R HEAVY M E S O N S Y S T E M S Maurizio L U S I G N O L I a, Guido M A R T I N E L L I ~,b and Attilio M O R E L L I " Dipartimento di Fisica, Universiiz "La Sapienza" 1-00185 Rome, Italy and INFN - Sezione di Roma, 1-00185 Rome, Italy b CERN, CH-1211 Geneva 23, Switzerland Received 27 July 1989
The matrix elements of the two four-fermion operators relevant for the calculation of mixing in the D-I) system have been obtained by a lattice QCD calculation in the quenched approximation at fl= 6.0. The resulting B-parameters are close to each other and not very different from one. The physical implications are discussed, together with a possible extrapolation of the result to the more interesting B-I) case.
A theoretical estimate o f the difference in masses (AM) and widths (AF) for neutral heavy meson systems (D°-IS) °, Bd-Bo, o -o B~-Bs) o -o can be obtained by taking the matrix element o f an appropriate effective hamiltonian. In the effective hamiltonian the short-distance Q C D corrections are resummed in the coefficients multiplying local operators. The calculation o f the matrix elements o f these operators requires nonperturbative methods. In early times, the vacuum insertion approximation (VIA) was used as the simplest estimate for the matrix elements; nowadays, the VIA matrix element is still used as a reference value, to be multiplied by a (subtraction point dependent) factor, usually called the B parameter. Let us consider in particular the Bq-Bq o -o ( q = d , s) case. Including the short-distance perturbative Q C D contribution one has
where the O t ± ) are renormalized at a scale/t_~ m b. In eqs. ( 1 ), (2) the (i = V,~V*q are products o f elements of the Cabibbo-Kabayashi-Maskawa matrix, Gv is the Fermi constant, M the average mass of the mesons, qMand (qts) r/(a)) are Q C D correction factors, that are given in ref. [ 1 ] and refs. [2,3 ] respectively. In eq. ( 1 ) we have neglected further terms, that are suppressed at least by a factor (m~/m2t)X l o g ( m { / m 2 ) , and F ( m t / m w ) is a slowly varying function [4] ( F ( 0 ) = I , F ( 1 ) = 3 ) . In eq. (2), T~"(k) is a function of the external m o m e n t u m running in the loop of the box diagram and of the quark masses, whose explicit expression is given in ref. [ 3 ]; we only note here that it is symmetric under the interchange (p,--~u) and in the limit o f zero masses it is given by
Gz F 2 2 F ( m t / m w )
The four-fermion operators in eqs. ( 1 ), (2) are defined as follows (a, p are colour indices ):
M12~
(1)
O ( + ) - ~ ~r)(+)
X [ (]~t~) + ¼u(a)) (B° I n ( + )
(3)
O(~ ) ----~ [4a Y~( 1 -- ys)baqBy~( 1 - y s ) b B
+Cl,~7u(l-75)bP(lpy.(1-Ts)b'~+(p~u)],
r l ~ = G~ v ~ T ~ " ( k ) 4 M cj=~u,~
-
1
TU"(k) = - ~ ( k 2 g U " - k u k ~ ) .
(4)
(5)
It is easy to verify that (2)
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
O~+)=~gu,O (+),
gu~O~7)=O.
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Volume 231, number 1,2
PHYSICS LETTERS B
As a consequence, one has only two independent B parameters, B ~÷ ) and B (-), defined by
(B°IOt+)If3°)=B~+)2(I+I/N).~BM2,
(7)
( B ° I o ~ ; ) Ia ° )
=B(-)½(1-1/N)fZ(4PuPv-M2gu~) ,
(8)
where fa is the B meson decay constant and N = 3 is the number of colours. A knowledge of B ~+ ) and B t - ) is therefore necessary to determine the physically interesting quantities A M = 2 Re ~/(M1z - ½iF12) (MT2 - AF=-4Im~/(M12
i 1 F 1 2 ) l•"
(9)
A viable method for the nonperturbative evaluation of matrix elements is provided by lattice QCD [ 5 ]. Unfortunately, present-day available values for the lattice spacing are not small enough to allow the propagation of a quark as heavy as the b. In this letter we present the results obtained in a Monte Carlo calculation of the B (+-) parameters for the A C = 2 operators, relevant for the calculation of AM and AF for D°-I7)°. Although the mixing parameters in this case are exceedingly small (in the standard model) we hope to obtain from this calculation also an insight for the more interesting case AB= 2. We present immediately the results and a short discussion of their physical implications, postponing the presentation of the details of the lattice calculation. At a scale # = a -1 (a is the lattice spacing) we have ~
1.~eV
GeV4'
(10)
( D (/3= 0)I Oti-)(a -1 ) 113(/3=0)) = (0.021 + - 0 . 0 0 7 ) (\ --'--a2 1.9 G e V ]~4 GeV4 '
( 11 )
(a,)
(12)
fD=(210+30)
1.9G-eV g e V ,
#t In order to minimize the systematic errors, the central value offD reported in ref. [6] was found by computing the ratio fD/f,~and by fixingf~ to its experimental value. It differs from the value in eq. ( 12 ) by about 8o/o.
148
1.9GeV
/
GeV.
(13)
For the B -+ parameters we give two different estimates, obtained with two equivalent methods to be discussed later on, B t + ) ( a - l ) = 0 . 7 3 + 0 . 0 5 (0.69_+0.05),
(14)
B ( - ) ( a - ~ ) =0.79_+0.05 (0.73 + 0 . 0 4 ) .
(15)
In eqs. ( 10 ) - ( 15 ) the errors are purely statistical. A quantity less subject to systematic uncertainties is given by the ratio of the two B parameters
f l ( a _ l ) _ B(+ B ~ - ))(a_l) ( a - ' ) - 1.08+_0.10,
(16)
which is directly obtained as a ratio of three-point correlation functions on the lattice. To allow a comparison with other possible calculations of the same quantities made at any scale greater than me, we also give the scale invariant B parameters [ 7 ] defined as B ~ -) = [ a s ( a - l ) ] -6/25B(+)(a-1 ) =0.95 +_0.07, B~/-) = [oq(a-~)]-4/2SB(-)(a-1)=0.94+_O.06,
flsi=[ot~(a-J)]+2/zsfl(a-l)=0.99+O.09.
(17)
The numerical values in eq. (17) have been obtained setting a - t = 1.9 GeV, A4=200 MeV and using the average of the two values of the B factors given in eqs. (14) and (15). The ratio f l ( a - t ) is interesting in order to compare our results with the estimate given by Voloshin et al.
[8].
( D I O ( + ) ( a - 1 ) 113) =(0.16+0.05)
a-I
roD=(1.51_+0.02)
,
l' • - ~IFt2)(MIE~t"' 1 2• ) -
(al)
2 November 1989
In their approach, they assume that the vacuum insertion approximation is valid at a very low scale, of the order of the inverse radius of the meson, #-~ 300 MeV. Therefore, they have studied the renormalization of the operators O ~+) at scales much less than the mass of the heavy quark rnQ. In this region, the anomalous dimensions of the operators are different from those given in eq. (17) since the heavy quark behaves as a static colour source. For/z < m o the B ( + ) parameter is scale invariant [9 ] (since the O ~+ ) operator's anomalous dimension is twice the axial current's one) and fl(mQ) is given by [ 3 ]
B(-)('I2) (O~s(mQ) ~ 8/(33- 2J) f l ( m Q ) ~ B ( + ) ( # ) \ a~(U) J
(18)
Volume 231, number 1,2
PHYSICS LETTERS B
From eq. (18) and the hypothesis B ( - ) (p) = B (+) (p) -~ 1 ( V I A ) one has evidently flViA(mQ) < 1 for/~< m o . In the particular case m Q = m~= 1.2 G e V , / t = 300 MeV one has flv~A(m~) = 0 . 7 6 (0.59)
(19)
f o r A 4 = 100 (200) MeV. On the other hand the evolution from a - ~ = l . 9 GeV to m e = 1.2 GeV of our result, eq. (16), gives fl(rn~) = 1.06 + 0.10.
(20)
The reason for the discrepancy between our result, eq. (20), and the vacuum insertion approximation at a low scale, eq. (19), should be clear from fig. 1, where we report the central values that would be obtained by rescaling with the appropriate factors the B parameters calculated at a - ~. Coming now to the physical quantities o f interest, the largest mixing parameters A M and A F are expected for the B~-Bs o -o system. One has
1.2
[
'
,
,
,
[
l
1.0
'
'
'
'
I
\
--
2 November 1989
2 -o ) - GFfaB, M3 Fl2 (B~o -B~ 8n ~ ( 1 --4X)t/2B~+)(mb)
× [~ (3q(s)+ q(")) (1 --2X) + ~ (3r/(s) -- I"/(a)) ( 1 + 2x)fl(rnb) ] .
(21 )
In eq. (21) x = m Z / m 2, we have made the approximation m b ~ M and Cabibbo suppressed contributions are neglected. If we assume that flBs(mb) ~ 1 (as it happens for fib(me) ) we obtain
where we have used the Wolfenstein [ 10 ] parametrization for ~, the experimental average B lifetime and A 4QcD = 200 MeV. This result essentially agrees with ref. [ 8 ]. In fact, the difference between the results of our calculation (B(+))-~0.71, f l ~ l ) and the values assumed by Voloshin et al. ( B ( + ) _~ 1, fl--0.6) does not show up in AF. It could be seen however in the ratio of the two mixing parameters AF/AM, for which we obtain a result larger by a factor ~ 1.3. With the same assumptions, the estimate for 0 -0 B d - B d is
FiB a
140 M e V / ~--.~ ( 0 . - ~ ) 2
0.8
B(÷)
0.6
1
,
I 0,5
. . . .
I
. . . .
1
I 1.5
. . . .
I
where GB (Z) = 1 - 9 . 5 8 Z + 22.7Z 2. We have actually calculated the mixing parameters for D ° - D °. With obvious modifications in eq. (2) and using eqs. (10) and (20) and the experimental D o lifetime we obtain
2
u(GeV)
(2F,2"]
\ - - F - ] D = (1"4+0"5) X 10 Fig. I. The running B-factors B (+) (p) and B (-) (p) at the scale p obtained by evolving B C±)(a -t ) calculated in the quenched lattice QCD at fl= 6.0. The results are obtained by using the leading logarithmic anomalous dimensions given in eq. (17) for mc~ 1.2 GeV ~
-3c:_ {VcbV~,u) \VcsVsuJ ~'JD
.
(24) •
In eq. (24), we have denoted by GD(Z) a second degree polynomial whose coefficients are rapidly varying functions of the ratio x = mZ/mZ¢. We have in two extreme cases G D ( Z ) = { -1-7)< 1 0 - s } +~'7"25"~Z+42 5 Z 2 [ . 0) . 3 .
.
(25)
149
Volume 23 1, number
PHYSICS
1,2
The upper values in eq. (25 ) are obtained assuming m,=0.45 GeV, mc= 1.2 GeV; the lower ones correspond to m,= 0.15 GeV, m, = m,. We remind that 1VcbV~u/VccsV~u1=O( 10F3) so that for small x values the second term in GD dominates. To give an estimate for A&I we consider its expression, neglecting QCD corrections and next to leading terms: Ml2
x (D”]O~;‘-O;;)]Do)
.
(26)
The reason to keep all the terms in eq. (26) is that ]&,/&I -O( 10e3), and the term proportional to rn; is not at all dominant. A numerical estimate from eq. (26) gives the upper bound <2x
10-5.
2 November
B
1989
ping parameter for the heavy quark has been fixed to K=O. 1350, corresponding to a mass of the charm quark of about 1 GeV (in the lattice regularization scheme). We have used three values for the light quark masses corresponding to K=0.1515, 0.1530 and 0.1545 and then extrapolated all the results to the zero mass limit for the light quarks corresponding to K,=O.1564(2) [ 121. The renormalized operators (MS scheme [ 13 ] of which we have estimated the matrix elements are linear combinations of lattice bare operators. The coefficients of the lattice operators have been calculated at first order in perturbation theory, following the general prescription given in ref. [ 141. The renormalization of two fermion operators was already done inref. [15] andforO(+)inrefs. [16,17]. Following the same procedure of ref. [ 16 ] we have performed the analogous calculation for O,$; ) . The final result is
~o::Lmt.
= (1+ 6
+&
r2Z*(r)$gpv
Z+)Ol:’
(27)
This value has been obtained assuming for m, the “constituent” value ( N 450 MeV), and it is dominated by the
LETTERS
-N-l x
OSTP_ -2;osP
( 2N
[o$&.~,=(
+&2 x
(
N2+N-1 4N
-
1+ &-))
OVA ) > (28)
0:;’
r2Z*(r)
hNO::_
N2-N-l 4N
0
VA “’ >
(29)
o(+)=dg,“eYp(l-Ys)qQYp(l-Y,)q, PJ 01;)=f[ey,(l-y,)q&y,(l-y,)q+(~Uttv)l -0::’
)
OSTP=&&7+
Sea,“qeo~“q+&Y,qQY,q,
0SP=QqQq-12Y54QY54> 0VA=eYpqQY/14-~YrYs4~Y~Ys4,
(30)
Volume 231, number 1,2
PHYSICS LETTERS B
2 November 1989
Table 1 The results for the matrix elements ( D l O t + ) ( a -~ ) 113), (DlO[i-)(a -~ ) I D ) and f o r f ~ m ~ are given in units of the inverse lattice spacing a - ~, for three values of the hopping parameter of the light quarks, K. The fourth row of the table contains the linear extrapolation to the chiral limit, corresponding to K¢= 0.1564. Since the lattice calculations are performed in the euclidean theory, the signs in the second column and in eq. ( 11 ) are opposite. K
a4
a4(DlO}~')(a-')lf))
a4~fDrnD2 2
0.1515 0.1530 0.1545
(3.0_+0.8))< 10 -2 (2.5___0.6)>(10 -2 (1.9+0.5))<10 -2
- (4.0_+0.8))< 10 -3 -(3.3-+0.7))<10 -3 -(2.6-+0.6)×10 -3
( 14-+ 3) )< 10 -3
0.1564
(1.2_+0.4))<10 -2
- (1.6-1-0.5) x 10 -3
(7_+2)×10 -3
0 TT = ½ [ Q a p u q O . a ~ q + (p~-'u) ]
o.ao
' ' ' ' 1
(12_+3))<10 -3 (9_+2))<10 -3
. . . .
I . . . .
I . . . .
I . . . .
I . . . . *J
0.B5
V A
O75
O?0
(30 c o n t ' d )
-- I g u u O V A .
0.65
We h a v e calculated the matrix elements ( D I O ( + ) ] IS)) a n d ( D [ O t F ) 115) using the p l a t e a u m e t h o d d e s c r i b e d in ref. [ l 1 ] for the three v a l u e s o f the h o p p i n g p a r a m e t e r o f the light quarks. T h e m a trix e l e m e n t s in lattice units t o g e t h e r w i t h t h e i r ext r a p o l a t i o n to the chiral l i m i t ( f o r the light q u a r k s ) are r e p o r t e d in table 1. In the last c o l u m n o f the table the q u a n t i t y f E m 2 o a n d its e x t r a p o l a t i o n to the chiral l i m i t is also given. T h e B-factors h a v e b e e n e s t i m a t e d w i t h two differe n t m e t h o d s in o r d e r to e v i d e n t i a t e the s y s t e m a t i c s i n v o l v e d in the e x t r a p o l a t i o n a n d t h e a c c u r a c y o f the perturbative corrections. M e t h o d 1. F o r each v a l u e o f the h o p p i n g p a r a m e ter o f the light q u a r k s we h a v e c a l c u l a t e d the ratio b e t w e e n the m a t r i x e l e m e n t s a n d t h e i r v a c u u m insert i o n v a l u e o n t h e lattice a n d t h e n we h a v e e x t r a p o lated the results to the chiral l i m i t as it is s h o w n in fig. 2. M e t h o d 2. We h a v e m e a s u r e d a n d e x t r a p o l a t e d to the chiral l i m i t d i r e c t l y t h e ratio Zx,x2 ( P s ( t l ) O ( O ) P s ( t 2 )
)
B(o)
060
. . . .
090
.
,
I .... ,
0.05 ,
I .... 0
,
,
,
I ....
1
,
,
,
,
0.15 ,
i
I ,
,
0.2 ,
. . . . ,
,
I ,
0.25 ,
.
.
.
-.
0.3 ,
,
,
,
O.B5 080 0.75 0.?0 0.65 0.150 O
0.05
O. 1
0.15 (=m.I=
0.2
0.25
0.3
Fig. 2. The values of the B-factors are plotted versus the pion
squared masses corresponding to the Wilson hopping parameters K=0.1515, 0.1530, 0.1545 of the light quarks, the hopping parameter of the heavy quark being fixed to K--0.1350. The solid line corresponds to the linear extrapolation to the chiral limit. Table 2 Values of the B-factors B ( ÷ ) ( a - ~), B ( - ) ( a - ~), and of their ratio fl(a - ~) = B (-) (a - t ) / B ( ÷ ) (a - ~). The B-factors are calculated in the two equivalent methods described in the text, while fl(a-J) is obtained directly from the ratio of three-point correlation functions. Method
B(+)(a -I )
B(-)(a -l )
fl(a -1 )
I II
0.69_+0.05 0.73_+0.05
0.73_+0.04 0.79_+0.05
1.08_+0.10
Xx, ( A o ( O ) P s ( tl ) ) ~ 2 ( A o ( O ) P s ( t2) )
- - , large
22 (D[O(0)115> ffom~
'
(31)
time distances
w h e r e Z A = 0.87 is the axial c u r r e n t r e n o r m a l i z a t i o n
factor c a l c u l a t e d p e r t u r b a t i v e l y in ref. [ 14 ]. R e s u l t s are g i v e n in table 2. A p r e v i o u s c a l c u l a t i o n o f B ( ÷ ), rnD a n d f D with the s a m e s a m p l e o f gauge c o n f i g u r a t i o n s that we use, was 151
Volume 231, number 1,2
PHYSICS LETTERS B
r e p o r t e d in ref. [ 5 ], a n d agrees with the p r e s e n t results. We h a v e calculated fl (see eq. ( 1 6 ) ) directly f r o m the ratio
- 8 Zx'x: ( Ps( tl )O~F ) ( O)Ps( t2) ) Zx,~: ( Ps( t~ )O( + ) ( O )Ps( tz) )
,p. large time distances
(32) We stress that eq. ( 3 2 ) gives exactly the q u a n t i t y fl, w i t h o u t m u l t i p l y i n g or d i v i d i n g b y a n y other factor, so that the s y s t e m a t i c errors are reduced. I n this p a p e r we have calculated o n the lattice the m a t r i x e l e m e n t s o f the operators w h i c h are r e l e v a n t for D - l ) mixing. T h e results are in good a g r e e m e n t with the V I A at a scale a - l which is large with respect to the heavy q u a r k mass. T h i s is in c o n t r a s t with the e s t i m a t e s o f ref. [ 8 ] where the v a l u e o f V I A was ass u m e d to be v a l i d at a scale # = 300 M e V in the case o f the B - m e s o n . T h e p h e n o m e n o l o g i c a l i m p l i c a t i o n s o f the lattice results for the D - I ) m i x i n g a n d (possib l y ) for the B - B system h a v e also b e e n discussed.
152
2 November 1989
References [ 1 ] A.J. Buras et al., Nucl. Phys. B 245 (1984) 369; F. Gilman and M. Wise, Phys. Rev. D 27 (1983) 1128. [ 2 ] J.S. Hagelin, Nucl. Phys. B 193 ( 1981 ) 123; E. Franco et al., Nucl. Phys. B 194 (1982) 403. [3] M. Lusignoli, Z. Phys. C 41 (1989) 645. [ 4 ] T. Inami and C.S. Lim, Progr. Theor. Phys. 65 ( 1981 ) 297, 1772 (E). [5] K.G. Wilson, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York, 1977). [ 6 ] M.B. Gavela et al., Phys. Lett. B 206 ( 1988 ) 113. [7] A.J. Buras et al., Nucl. Phys. B 238 (1984) 529. [ 8 ] M. Voloshin et al., Sov. J. Nucl. Phys. 46 ( 1987 ) 112. [9] M. Voloshin and M. Shifman, Sov. J. Nucl. Phys. 45 (1987 ) 292; H. Politzer and M. Wise, Phys. Len. B 208 (1988) 504. [10] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [ 11 ] M.B. Gavela et al., Nucl. Phys. B 306 (1988) 677. [ 12] L. Maiani and G. Martinelli, Phys. Lett. B 178 ( 1986 ) 265. [ 13] W. Siegel, Phys. Lett. B 84 (1979) 193; D.M. Capper, D.R.T. Jones and P. Van Nieuwenhuizen, Nucl. Phys. B 167 (1980) 479. [ 14] M. Bochicchio et al., Nucl. Phys. B 262 ( 1985 ) 331; L. Maiani et al., Phys. Lett. B 176 ( 1986 ) 445; Nucl. Phys. B289 (1987) 505. [15] G. Martinelli and Y.C. Zhang, Phys. Lett. B 123 (1983) 433;B 125 (1983) 77. [ 16 ] G. Martinelli, Phys. Lett. B 141 (1984) 395. [ 17 ] C. Bernard, T. Draper and A. Soni, Phys. Rev. D 36 ( 1987 ) 3224.