Long-range effects in K0K 0 mixing

Volume 136B, number 5,6

PHYSICS LETTERS

15 March 1984

LONG-RANGE EFFECTS IN K o - ~ o MIXING ~ Neven BILLIC 1

Department of Physics, Brown University,Providence, R102 912 , USA and Branko GUBERINA

Rud/er Bo~kovi~ Institute, Yu-41001 Zagrev, Croatia, Yugosiavia Received 21 November 1983

We calculate the matrix element of the K° -1~° amplitude by saturating the product of currents by a set of one-particle intermediate states. We find that, although some cancellation takes place, the total contribution is comparable with that of the vacuum saturation. Our result is consistent with the bottom lifetime recently measured.

The matrix element o f the K 0 - K ° amplitude has been a subject o f great concern and uncertainty in the phenomenological analysis o f weak interactions. The first estimate o f the K 0 - K 0 matrix element based on the vacuum-saturation assumption led to a successful prediction o f the charged quark mass in the two-generation standard model [1]. Since then, a variety o f models have been used producing a variety o f results, unfortunately differing in both magnitude and sign. In the MIT bag model calculations, the value o f the matrix element in units o f the vacuum-saturation value ranges from - 0 . 4 2 to 0.34 [2,3]. The results obtained in the Isgur-Karl harmonic-oscillator type o f models, although stable in sign, range from 1.44 to 2.86 [2]. It has recently been shown [4] that hadronic sum rules set an upper bound o f (2.0 +-0.5) × (vacuum saturation) on the magnitude o f the K 0 - ~ 0 matrix element. An approach based on SU(3) and the PCAC assumption gives 0.33 in the same units [5 ]. Another type o f approach is to calculate the con-

Work supported by the US National Science Foundation and SIZ-1 of SR Croatia under grant No. YOR 82•051. 1 Supported in part by the US Department of Energy under Contract No. DE-AC02-76ER03130 A011 Task A. Permanent address: Rudjer Bo~'kovi~Institute, Yu-41001 Zagreb, Croatia, Yugoslavia. 440

tributions o f physical intermediate states other than the vacuum. The one-pion contribution is estimated to be less than 10% of the vacuum saturation and opposite in sign [6]. However, this result is strongly dependent on the scale/a, at which the integration over pion momentum has to be cut off in order to avoid double counting o f short-range effects. Shrock and Treiman found the ~r contribution to be o f the order of vacuum saturation [3]. It has been argued that the p and w contributions are suppressed and the 2rr contribution, having both negative and positive terms, might cancel at the scale/~ [7]. In view o f that, it would be o f interest to give a more precise estimate o f long-range effects associated with one-particle intermediate states including both pseudo-scalar- and vector-meson contributions. This is basically what we want to do in this paper. In our approach we neglect the 21r contribution and other possible continuum contributions, adopting the argument suggested by Regge phenomenology [8], and keep the one-particle dispersive contributions coming from quark-antiquark intermediate states [7]. The calculation o f the K 0 - K 0 amplitude is based on the effective current-current IASI = 2 lagrangian, which for two generations takes a simple form, ./9 effA S=2(x ) = (G2/4rr2)sin20c cos20c 2 X mc~l(1/mK)J ~6-i7 (x)J~6-i7 ( x ) ,

(1)

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PHYSICS LETTERS

as a result of the evaluation of box diagrams [1]. In (1) G F denotes the Fermi constant, 0 c is the Cabibbo angle, m K is the mass o f K 0 and 7/is a QCD coefficient which can be calculated in the leading log approximation [9]. To calculate the matrix element of the local current-current operator M = (~0 Ij6-i7(0) j 6 - i 7 ( 0 ) IK0),

(2)

we follow the procedure suggested in ref. [8]. We can formally write (~o I~(0) Su~(O)IK°) = lim

1

,if--.-} zo

X

fdn

6 (Inl - 1) (I(0 IT [J~u(x0/2)J~(-Xo/2)] IK° )

+31

fdn

8(Inl - 1)

,

(3)

where

x o = (o, n/u). The limiting procedure in (3) corresponding to short distances has to be truncated at some value of/a, of the order of 0 5 GeV, at which strong interactions become weak. This is necessary in order to avoid double counting of the short-distance contribution, which is already taken care of by evaluating box diagrams together with leading QCD corrections. By making use of the Fourier transform and integrating over dn in (3), we Find that d4 M~t3 = (~0 Dr~j~ IK0) = f ( 2 ~ 4/o(k / i//a)c//Z~ ( q ) ,

(4)

where "Tg c~#(q)= .~f

d4x exp(iqx)

X (~0 [T[J~(x/2)J~(-x/2)]

i(2")3 n

(

2

8(q -- Pn)

8(q +Pn)

)

X q0 +(mK---~n) + i e - q 0 - ( m K - E n ) + i e

X (K°lJ~(O)ln>(nlJ~(O)lK°>+ [~ ~/3]

,

(7)

where (En, Pn) is the momentum of the state In ). Consider cFf~a#(q)as a function of q0 for fixed q. From eq. (7), ~ a#(q) can be seen to be an analytic function in the entire q0 complex plane except for poles and cuts on the real axis corresponding to oneparticle intermediate states and the continuum, respectively. By assuming that ~ a# is Regge behaved for large values of q0, it follows that the continuum contribution will be suppressed [8] owing to the fact that Regge meson trajectories with AI = 3/2 are "exotic" [10]. Therefore, we may approximate the value of our amplitude with low-lying meson intermediate states only. For further reference, let us first consider the vacuum insertion. Replacing the complete set of intermediate states in (7) by 10><01and combining with (4), we Fred that "dq0" _f o.

= f (2rr) J

1r3

8(q)

) qo+mK+ie

X (I(0 lS6-i710) (01J6-i7 IK0) .2 2 -_ f~m K = 0.0067 GeM4

(8)

which, apart from the colour factor 8/3, is the standard vacuum-saturation result [ 1]. Next, we consider one-particle contributions. For these contributions, the sum in (7) reads

:dap In, p)(n, pl

(9)

(5)

n~ (27r)3 a2En

(6)

Inserting (7) into (4) and performing the contour integration in the complex q0 plane, we obtain the f'mal expression for the one-particle contribution

and the function

]o(X) = (sin x)/x,

=

1

+ [~ ~+~] IKO),

15 March 1984

provides a smooth cut-off of the integral (4). By inserting a complete set of intermediate states En [n)(n I between the two currents in (5), we obtain

M~n~ = f

d3q (2~r)32En

]o(iql//a)(~Olj~uln,q)~n,ql4lKO), (10)

where

En = (m 2 +q2)I/2. 441

Volume 136B, number 5,6

15 March 1984

PHYSICS LETTERS

The evaluation o f M~na is straightforward once the relevant current-single meson matrix elements are known. The general expressions for matrix elements in (10) contain form factors depending on transfer momentum defined as

and FV(t) is the magnetic nucleon form factor given by

t=(p _q)2 ,

Fn = ~n [ - 3 f 2 ( t ) +

(11)

where p - (mK, 0) is the rest frame m o m e n t u m of K 0 . Therefore, it will be convenient to change the variables and transform the integral in (10) into the form _

Mn

1

(inK-m) 2

2M 2

f

X1/2

dt (2rr)2Jo(Xl/2/2mK#)Fn(t,m, mK)

-**

(12)

FV (t) = (1 - tim 2 )-2 ( 1 - t /aM 2 ) - 1 ,

(20)

where m 2 = 0.71 GeV 2 and M 2 = 0.88 GeV 2. (c) Vector mesons, axial-vector current:

+ (tf2(t) - 2 f l ( t ) f 3 ( t ) ) X(t, m, mK)/4m2].

(21)

The t dependence of the form factors f l and f3 can be obtained by assuming simple pole-dominated dispersion relations [8]. This gives the following approximated expressions:

where m is the mass of the intermediate meson and X is a function o f t defined as

f l (t) = f l (0)/( 1 - tim 2 ) ,

;k(t, m, mK) = 4 m 2 [q [2 = (t - m E - m 2 ) 2 - 4mEm2K .

f3(t) = f l (t)/( t - m 2 ) .

The function

(13)

Fn(t, m, mK) = (t(O ~16~-i7 ln, q) (n, q [j6~-iT IKO),

(14)

The value f l ( 0 ) may be determined from (22) by using the value f l (m2) obtained through soft-pion theorems from the electromagnetic coupling of the p meson:

will be determined for each case of intermediate meson using the expressions for the matrix elements given in ref. [8]. Pseudoscalar mesons Or0 , ~7) will contribute only to VV-type amplitudes (case a), whereas vector mesons (~0, co, ~) will contribute to both VV (case b) and AA (case c) amplitudes. (a) Pseudoscalar mesons:

Fn = ~n [f2(t)(2m2K + m2 - t ) --f2_(t)t],

(23)

f l ( m 2) = (2/f~r)f o = 1.8 G e V .

(24)

The values of the Clebsch-Gordan coefficients ~n in (15), (17) and (21) are given in table 1. For co and ~, we take the "ideal mixing": ICO) = (_~)1/216~0 ) + (1)1/31608), (25) I¢) = _(½)1/216%) + (2) 1/2 [6o8).

(16)

where m 2 = 0.65 GeV 2,f+(0) = 1 and f _ ( 0 ) ranges experimentally in the interval 0 - 0 . 5 . For simplicity, we have chosen f _ ( 0 ) = 0. (b) Vector mesons, vector current:

F n = - ~ n g 2 ( t ) X(t, m, m K ) ,

where m2B = m~l = 1.61 GeV 2, and

(15)

where ~n is the Clebsch-Gordan coefficient. The form factors]'_+ may be parametrized as [3]

f+_(t) = f_+(0)/(1 - t / m 2 ) ,

(22 )

(17)

By using expressions (15), (17) and (21), we can evaluate all relevant one-particle contributions given by (12). The results are given in table 2 for a reasonable range of the cut-off parameter/~. In order to reTable 1 Clebsch-Gordan coefficients ~n appearing in the product of matrix elements (14) given for each intermediate meson for both vector and axial-vector currents.

with g v approximated by [11] gv(t) = gv(0) FV(t).

(18)

Here g v ( 0 ) is obtained from the knowledge o f the decay co -+ 7r0~,, gv(O) = 2.59 GeV -1 , 442

(19)

meson

VV

AA

no n o° to

-1/2 -3/2 1/2 1/2 1

0 0 -1/2 -1/2 -1

Volume 136B, number 5,6

PHYSICS LETTERS

Table 2 The contributions of pseudoscalar (0-) and vector-meson (1-) poles to the matrix element of the K ° _go amplitude given by (12) with (13)-(21) for different values of u. Units are 10-2 GeV4" VV

AA

XMn

S

0.17 0.25 0.43 0.75 1.20

1.25 1.37 1.64 2.12 2.79

(GeV)

0.3 0.4 0.5 0.6 0.7

0-

1-

1-

-0.23 -0.46 --0.72 -1.01 -1.31

-0.21 -0.73 -1.34 -1.92 -2.45

0.61 1.44 2.49 3.68 4.96

late our results to the vacuum-saturation estimate, we also give the ratio

where Mvac is given b y (8) and ZnMn is the sum o f all one-particle contributions. F r o m table 2 we may draw the following conclusions: (i) The sum o f all one-particle contributions is positive and o f the order o f vacuum saturation. (!i) The results are rather sensitive to the variation o f the cut-off ~. (iii) The cancellation o f relatively large numbers takes place, which brings in an additional uncertainty to our conclusions. In order to give a more precise estimate, we shall set the scale # by fitting the A I = ~ amplitude in the process K 0 ~ rr+rr - , which was calculated b y Nardulli et al. [8] using the same m e t h o d . They concluded that ~u should be equal to 1. However, the long-range contribution suffers from uncertainty concerning the way o f continuation from the softopion limit (SPL) to the physcial amplitude. If pions are treated symmetrically, as it was done in ref. [8], then the SPL amplitude has to be multiplied b y a continuation factor

[121 Aphys = [2(m 2 _ m~r)/K 2 2 ] A ssym PL,

(27)

where K2 is usually taken to be m l [13] or l (ml + m 2) [14]. Hence, the long-range amplitude a3/2 calculated in ref. [8] should be multiplied b y at least a factor o f 2. Adding the vacuum contribution and

15 March 1984

comparing the total result thus obtained with the experimental value [15] , w e find that/a must be in the range 0 . 3 - 0 . 5 . This sets the ratio B between 1 . 2 5 1.64. Thus, our result supports the llarmonic-oscillator model [2] and is below the upper bound [4]. Other types o f models [2,3,5] give smaller values o f B, ~ 0 =3-0.4. The phenomenological analysis [ 16] based on these values and the short B-meson lifetime, 7"B ~ ( 0 . 2 - 1 . 5 ) × 10 -13 s, sets the bound on the tquark mass between 2 0 - 5 0 GeV. However, it has already been pointed out that only large values o f B in the range 1 . 4 - 2 are consistent with large r B ~ 10 -12 s, recently measured in SLAC experiments [ 17]. Finally, we note that our results are consistent with the value B = 1.1 - I .3 obtained b y fitting the e parameter in K°-decay [18]. One o f us (N.B.) would like to acknowledge the hospitality o f the Physics Department o f Brown University, where part o f this work was completed. We also acknowledge useful discussions with K. Kang, D. Tadi6 and J. Trampeti6.

References [ 1 ] M.K.Gaillard and B.W. Lee, Phys. Rev. D10 (1974) 897. [2] P. ColiE, B. Guberina, D. Tadi~ and J. TrampetiS, Nucl. Phys. B221 (1983) 141. [3] R.E. Shrock and S.B. Treiman, Phys. Rev. D19 (1979) 2148. [4] B. Guberina, B. Machet and E. de Rafael, Phys. Lett. 128B (1983) 269. [5] J.F. Donoghue, E. Golowich and B.R. Holstein, Phys. Lett. l19B (1982) 412. [6] M.I. Vysotsky, Yad. Fiz. 31 (1980) 1535; Soy. J. Nucl. Phys. 31 (1980) 797; K. Kang and J.E. Kim, Phys, Rev. D14 (1976) 1903. [71 L. Woffenstein, Nucl. Phys. BI60 (1979) 501; C.T. Hill, Phys. Lett. 97B (1980) 275. [8] G. Nardulli and G. Preparata, Phys. Lett. 104B (1981) 399; G. NarduUi, G. Preparata and D. Rotondi, Phys. Rev. D27 (1983) 557. [9] F.J. Gilman and M.B. Wise, Phys. Lett. 83B (1979) 83. [10] V. DeAlfaro, S. Fubini, G. Furlan and G. Rossetti, Phys. Lett. 21 (1966) 576. [11] P.A. Zucker, Phys. Rev. I)4 (1971) 3350. [12] M. Milo~eviS,D. Tadi~ and J. Trampeti~, Nucl. Phys. B187 (1981) 514. [13] M. Bonvin and C. Schmid, Preprint Institut ftlr Theoretische Physik, EidgenSssische Technische Hochschule (Zllrich, September, 1980). 443

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PHYSICS LETTERS

[14] J.F. Donoghue, E. Golowich, W.R. Ponce and B.R. Holstein, Phys. Rev. D21 (1980) 186. [15] T.J. Devlin and J.O. Dickey, Rev. Mod. Phys. 51 (1979) 237. [16] L. BergstrSm, E. Masso, P. Singer and D. Wyler, Phys. Lett. 134B (1984) 373.

444

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[17] E. Fernandez et al., Phys. Rev. Lett. 51 (1983) 1022; MARK II CoUab., G. Hanson, paper given at EPS Europhysics Conf. on High energy physics (Brighton, July 1983), SLAC-PUB-3187 (1983). [18] E.A. Paschos, B. Stech and U. Tllrke, Phys. Lett. 128B (1983) 240.