QCD duality analysis of B0B0 mixing

Volume 206, n u m b e r 2

PHYSICS LETTERS B

19 May 1988

QCD DUALITY ANALYSIS OF B°-B ° MIXING

A. PICH CERN, CH-1211 Geneva 23, Switzerland Received 16 February 1988

Using finite energy sum rules, the hadronic matrix element of the A B = 2 operator governing the mixing between the B ° and 1~° mesons is evaluated. The result is very sensitive to the input value of the bottom quark mass. Allowing it to vary in the range mb = 4.70 + 0.14 GeV, one finds ~a-= IfB II x / q ~ aI = 0.13 + 0.05 GeV.

Important experimental progress in our knowledge of flavour changing phenomena has been achieved during the last year. Together with previous electroweak measurements, the new experimental results provide useful constraints on the parameters of the standard model. However, in order to completely pin down the remaining uncertainties in these parameters (or perhaps even to disprove the model), it is first necessary to reduce the size of the theoretical errors entering the analysis of weak transitions. With the expected increasing accuracy of forthcoming experiments, the improvement of our theoretical ability to make definite predictions for these processes is even more strongly called for. The B°-B ° system is a good example of this situation. While it is clear that the recent observation of a relatively large amount of B°-l) ° mixing [ 1 ] seems to require [ 2 ] a rather heavy top quark, mt >~50-100 GeV, the actual value of the extracted mt lower bound is quite uncertain. The biggest source of error originates from the matrix element of the AB= 2 operator OAB= 2 ~ (l~Ly~d L ) ( f i L y ~ d L )

(

1)

between the B° and ~o states, which is usually parametrized as 4 ( f3° I OAa = 21 B° > = -~S MS2 Ba •

(2)

Here, fB is the decay constant of the B° meson (the normalization corresponds tof. ~ 93.3 MeV) and the factor Ba takes into account the discrepancy between the true matrix element and the result ~ 4 a M a 2 obtained by inserting the vacuum in all possible ways. 322

At present, the best evaluation of the B o meson decay constant fa comes from QCD sum rules. Although calculations by different groups gave initially quite different results [ 3 ], the situation seems to be clarified by recent analyses [4,5 ] showing that various versions of QCD duality sum rules lead to a unique prediction [ 5 ] fB -~ 129_+ 13 MeV.

(3)

The error bar appears to be dominated by the input value of the bottom quark mass (fB decreases for increasing values of rnb) which has been taken to be ,1 rob=4.71 +0.13 GeV. Concerning the BB factor, one would naively expect that the vacuum insertion approximation becomes more accurate with the increase of the meson mass, so Ba = l is usually assumed. However, a quantitative analysis of this statement is still lacking. The purpose of this letter is to present a direct calculation of the B°-B ° matrix element, without relying on the vacuum insertion approximation ~2. This can be done by studying the two-point function ~xa=2 (q:) = i f d4x e iqx

(OIT(O,xa=2(x) OaB=2 (0)*) 10)

(4)

constructed with the AB = 2 operator of eq. ( 1 ), and writing a system of finite energy sum rules relating *l mb refers to the "on-shell" or "pole" mass mb =--mb ( s = m 2 ). ,2 For previous model-dependent estimates see ref. [6 ].

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Volume 206, number 2

PHYSICS LETTERSB

integrals of the corresponding hadronic spectral function to their QCD counterparts. A similar correlator was used in ref. [7] to compute the BK parameter governing the K°-I~ ° mixing. However, due to the higher mass scale involved, the analysis of ~'As=2 ( q 2 ) requires a somewhat different approach. Unlike the kaon case, the onset of the asymptotic continuum can be taken here very near the physical threshold sth=4M 2. One can then approximate the hadronic spectral function ( 1 / g ) I m ~urB=2(s) by the lowest intermediate state contribution, which is obviously related to the matrix element of eq. (2). The effect of higher mass intermediate states is taken into account by the QCD continuum. Using for convenience the effective realization

O/,B=2efr= ~gAB_ ° 20/a_B°O~'B

(5)

where

gAB=2 = ~ BB

(6)

one immediately gets 1 Im

-n

err

2 2lg~B=212 9(16n2 )

~AB=2(s)=O(s--4MB)

×s2(1 - - 2 M ~ / s ) 2 x / 1 - 4 M ~ / s .

(7)

The short distance calculation of ~'AB=2(q2) is rather involved. As shown in fig. 1, the lowest order contribution contains four internal quark lines. Working in the ma = 0 limit, we have still to take care of the exact mb dependence coming from the two heavy quark propagators. Nevertheless, it is possible to do analytically all the momentum integrations in order to get a compact expression in parametric form:

d

d

19 May 1988 S4

1n Im ~r8=2 = O ( s - 4 m 2 ) ( 16n2)3 A ( s ) ,

(8)

where ( 1 __ x/~")2

A(s)=2(I+~)

f

( I __ ~ - ) 2

dz

f

du21/2(1, z,u)

8

× ( 1-c~/z)2(1-8/u) 2 × [ ( 1 - z - u ) 2 ( 1 + 2c~/z) ( 1 + 2~/u) +8zu-2O(z+u) - 4 8 2 ] .

(9)

Here, cS=m2/s and 2(x, y, z ) - x Z + y Z + z 2 - 2 x y

-

2 x z - 2yz. The leading non-perturbative corrections come from dimension four operators appearing in the operator product expansion of the T-product ofeq. (4). The heavy quark condensate contribution, m b ( 0 [ b1310), is already taken into account by the Wilson coefficient of the ( 0l G2I 0) term, and a possible m b( 0 ] dd ]0 ) correction is absent because of the helicity projectors appearing in the O68=2 operator. The &B=2 spectral function can be written, to this order, as S4

1 Im ~Q~Cff2(s) = O(s--4m 2 ) - (167r2) 3

The gluon condensate coefficient B (s) is more easily computed by writing the quark propagators in an external background gluonic field and using the standard fixed-point gauge techniques [ 8 ]. The result one finds is I

y+

B ( s ) = f dx f d y { - ( A / Z y 2 ) [ A - y ( 1 - y ) ] x

y_

× [2xy+ ( 1 - x ) 2 ( 1 - y ) ]

+ (Sx/3y 3) ( 1 --x)2( 1 _ y ) 3 [2A--y( 1 --y) ]} (a)

(b)

Fig. 1. Lowestorder contributions to the AB= 2 correlator of eq. (4).

--

t

dzz(1-~/z)Z21/2(l'z'~)'

(11)

6

323

Volume 206, number 2

PHYSICS LETTERS B

where

19 May 1988 t

1 0/~

A= 6(y/x+ 1--y) --y( 1--y)

(12)

I

1.02

% = 4.56 fieV "".......

~

• -,,~...,.

"'....

and the p a r a m e t r i c integration limits are given by

~ - ~ .

:;.L.:.~_.,..___ . . . . .

" =Z2

1 00

x = 6 / (1 - , i S ) ~ ,

y+_=½[l+fi(1-1/x)+_2~/E(1,6,~/x)].

I

....................

X-8

(13)

0.98 I

I

I

b

Let us consider now the integrals SO

1.0~,

Gn(so)- f dssnlImctzxB=2(s) o

G. (So)eft= G. (So)QCD

( 15 )

which relate the G. (So) m o m e n t s c o m p u t e d with the hadronic p a r a m e t r i z a t i o n o f eq. ( 7 ), G. (so)elf, to the corresponding short-distance calculation, Gn (So) QCD, using eq. (10). The ratio

Rn (So) =-Gn+l (So)/G. (So)

(16)

does not d e p e n d on g~8=2 a n d therefore can be used to fix the o p t i m a l duality region on the So variable. F o r numerical convenience, we normalize it to the a s y m p t o t i c freedom b e h a v i o u r R . ( s o ) af, o b t a i n e d with the spectral function o f e q . ( 8 ) ,

(So)/R~ (So)af

(17)

and use the eigenvalue c o n d i t i o n Xn (So)effD)~n ( S o ) Q C D .

( 18 )

Inserting the value o f So thus o b t a i n e d in eq. ( 15 ) resuits then in a p r e d i c t i o n for [gAB=2I a n d hence for the matrix element ( 2 ) that we are looking for. W i t h the input value MB0 = 5.275 GeV, the functions Z. (so) are plotted in fig. 2, for n = - 8 a n d using three different values o f b o t t o m quark mass which cover the range [ 9 ] r o b = 4 . 7 0 + 0.14 G e V .

(19)

In view o f the recent claims [ 10] that the gluon condensate has been usually u n d e r e s t i m a t e d in the literature, we have allowed it to vary within a factor o f two, i.e., we have taken

( (as/~Z)G 2) = (w= 1-2), 324

% = 4.70 GeV

....... ~ 1.02

for different values o f n, and write the duality constraints

Zn (So) -Rn

\.

(14)

(0.012 GeV 4) × w (20)

~

i

U

"

.............

1.00

"""'"""

.....,........

0.98 I

1.0~.

"

I

t

\

I n~ = 4.84 GeV

•~ .

........ ~

1.02

I

~....._.~

•.

.

1.00

.

.

.

~

"'"'.,,......

0.98

................... 1.00

I

I

1.05

1.10

I

1.15

......

r

"q

1.20

Fig. 2. The ratio Z,,(So) defined in eq. (17) is plotted versus r=-so/4M 2, for n = - 8 and three different values of the bottom

quark mass. The continuous (dot-dashed) curves represent the QCD predictions obtained with the parametrization ofeq. (10) and taking the value w= 1 (w=2) for the strength of the gluon condensate. They approach asymptotically the value one - the dashed lines in the figure - at large r. The lines of dots are the values obtained using the hadronic spectral function given in eq. (7).

where w = 1 corresponds to the so-called s t a n d a r d value. In the absence o f Q C D corrections Zn (So) would be one (the d a s h e d lines in fig. 2). W i t h inclusion o f the calculated ( ( a s / z Q G 2) contributions, these lines become the continuous ( w = l ) a n d d o t - d a s h e d ( w = 2) c u r v e s which approach asymptotically the value one at large So. The curves corresponding to the hadronic p a r a m e t r i z a t i o n o f the spectral function, given in eq. ( 7 ) , are the lines o f dots shown in the figure. The duality test o f eq. (18) turns out to be quite

Volume 206, number 2

PHYSICS LETTERS B

sensitive to the actual value of the gluon condensate. With the standard value w = 1, the eigenvalue solutions are found to be around r=- S o / 4 M 2 ~ 1.15, 1.10 and 1.05, for mb=4.56, 4.70 and 4.84 GeV respectively. Taking w = 2 and the same values o f mb, the best duality regions move to r ~ 1.13, 1.05 and 1.10. Note that for the biggest value of mb the duality behaviour is better with w = 1, while smaller quark masses seem to prefer higher values of the gluon condensate. The central mass value m b = 4.70 GeV shows indeed quite a good matching between the hadronic and Q C D curves for w = 2. From eq. ( 15 ) one can obtain

CB=x/lg,~=21 = Ifnl ~

(21)

as a function ofr. With n = - 8 , this is shown in fig. 3 ~B

I

(GeV)

1

19 May 1988

for w = 1 (continuous curves) and w = 2 (dot-dashed ones). When r approaches the value one, the Q C D calculation is unable to reproduce the physical threshold sth = 4M~o and I ~B I grows to infinity. On the other hand, if one chooses too high values for the onset of the asymptotic continuum So, the resulting ~n starts growing because the effect of higher mass intermediate states, not contained in the parametrization of eq. (7), becomes important. To study the stability of ~B versus the electron of the power n, we have solved the eigenvalue equation (18) and inserted the value of So thus obtained in eq. ( 15 ) for different values for n. The resulting predictions for ~B are plotted in fig. 4, which shows a remarkable stability for the central value of the bottom quark mass. The cases ( m b = 4 . 5 6 GeV, w = 1 ) and (mb = 4.84 GeV, w = 2 ) look however worse.

I I

!

I

I

I

I

I

~B (GeV)

rob=

\.

/+.56

0.2

GeV

mb=/+7OGeV

~.~ mB : ~.70 GeV

mb =/*8/+ GeV ~ ' / "

0.1

_

0.1

./"

mB

/ 1.00

:

/..84 GeV

/

1.05

1.10

1.15

r

Fig. 3. The quantity CB=- IfB I I , , ~ I obtained from eq. (15) is plotted versus r~So/4M~ for n = - 8 and three different values of the bottom quark mass. The continuous and dot-dashed curves correspond to the values w= 1 and w=2, respectively, for the strength of the gluon condensate.

I

0 -20

I

-10

I

I

0

1

;

1

n

Fig. 4. Behaviour o f the ~B value, obtained at the optimal duality so point, versus n, for different bottom quark masses. The continuous and dot-dashed curves correspond to the values w = 1 and w--2, respectively, for the strength o f the gluon condensate.

325

Volume 206, number 2

PHYSICS LETTERS B

F r o m figs. 2 - 4 one then gets the results ~B-~ 0.17-0.18 GeV

(mb = 4 . 5 6 G e V ) ,

-~ 0.12-0.13 GeV

(mb = 4 . 7 0 GeV) ,

-~ 0.075-0.085 GeV

(mb = 4 . 8 4 GeV) .

de Rafael are gratefully acknowledged. This work has been partly supported by CAICYT, Spain, u n d e r " P l a n Movilizador de la Fisica de Altas Energias".

(22)

The allowed changes on the value of ( ( a s / ~ z ) G 2) have produced an effect of no more than 10% in these numbers. Due to the anomalous d i m e n s i o n of the A B = 2 operator, the parameter ~B is renormalization scale dependent. The n u m b e r s given in eq. (22) correspond to the scale/~2=So~4M~. Since the Wilson coefficient of OaB= 2 is usually computed at/./2 = m 2, the results (22) should be rescaled by a factor [as (m~,)~as (So) ]3/23. However, this a m o u n t s to a 3% effect only which is negligible. The largest source of uncertainty in our results is by far the input value of the b o t t o m quark mass. Allowing it to vary in the range mb = 4.70 _+0.14 GeV, we can give as a final result ~B =0.13_+0.05 G e V .

(23)

Comparing our prediction for ~B with the fB values given in eq. (3), one finds a good agreement for the central value of the quark mass m b = 4.70 GeV, which could give some support to the validity of the vacu u m saturation approximation in this case. However, the BB parameter resulting from this comparison is strongly m b dependent and therefore no firm conclusion can be extracted. Our analysis could obviously be improved by computing as corrections and higher order condensate contributions to the two-point function (4). However, the expected effect of these additional terms is smaller than the mb uncertainty already reflected in the error bar ofeq. (23). In order to reduce this large error, a better d e t e r m i n a t i o n of the b o t t o m quark mass is then mandatory. Discussions with G. Martinelli, S. Narison and E.

326

19 May 1988

References [ 1] ARGUSCollab.,H. Albrechtet al., Phys. Lett. B 192 ( 1987) 245; see also UAI Collab., C. Albajar et al., Phys. Len. B 186 (1987) 247; MAC Collab., H.R. Band et al., Phys. Lett. B 200 (1988) 221. [2] G. Altarelli, Proc. Intern. Europhys. Conf. on High energy physics, Vol. II (Uppsala, June-July 1987), ed. O. Batner (Uppsala University) p. 1002, and referencestherein. [3 ] T.M. Aliev and V.L. Eletskii, Sov. J. Nucl. Phys. 38 ( 1983 ) 936; V.S. Mathur and M.T. Yamawaki, Phys. Rev. D 29 (1984) 2057; L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Lett. B 104 (1981) 305; Phys. Rep. 127 (1985) 1; D.J. Broadhurst and S.C. Generalis, Open University report OUT-4102-8 ( 1982); S.G. Generalis,Ph.D. Thesis, Open Universityreport OUT4102-13 (1983); E.V. Shuryak, Nucl. Phys. B 198 (1982) 83; A.R. Zhitniskii, I.R. Zhitnitskii and V.L. Chernyak, Sov. J. Nucl. Phys. 38 (1983) 775; M.A. Shifman and M.B. Voloshin, Sov. J. Nucl. Phys. 45 (1987) 292. [4] C.A. Dominguez and N. Paver, Phys. Lett. B 197 (1987) 423. [5 ] S. Narison, Phys. Lett. B 198 ( 1987 ) 104. [6] I.I. Bigiand A.I. Sanda, Phys. Rev. D 29 (1984) 1393. [7 ] A. Pich and E. de Rafael, Phys. Lett. B 158 ( 1985 ) 477. [8] A.V. Smilga, Sov. J. Nucl. Phys. 35 (1982) 271; E.V. Shuryakand A.I. Vainshtein,Nucl. Phys. B 201 (1982) 141; V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Fortschr. Phys. 32 (1984) 585. [9] S. Narison, Phys. Len. B 197 (1987) 405. [10] J. Marrow, J. Parker and G. Shaw, Z. Phys. C 37 (1987) 103; R.A. Bertlmann, C.A. Dominguez, M. Loewe, M. Perrottet and E. de Rafael, CERN preprint TH.4898/87, Z. Phys. C, to be published.