Physics Letters B 318 (1993) 212-216 PHYSICS LETTERS B North-Holland

An analytic formula and an upper bound for e'/e in the Standard Model*

A n d r z e j J. B u r a s a,b,1 a n d M a r k u s E. L a u t e n b a c h e r a,2

a Physik Department, Technische Universitdt Mfinchen, D-85748 Garching, Germany b Max-Planck-Institut J~r Physik - Werner-Heisenberg-lnstitut, FOhringer Ring 6, D-80805 Mfinchen, Germany

Received 6 September 1993 Editor: P.V. Landshoff

Using the idea of the penguin box expansion we find an analytic expression for e'/e in the Standard Model as a m t ms (mc) and two non-perturbative parameters B~ 1/2) and B (3/2) This formula includes next-to-leading function o f

QCD/QED short distance effects calculated recently by means of the operator product expansion and renormalization group techniques. We also derive an analytic expression for the upper bound on e'/e as a function of [Vcb], I Vub/Vcb I, BK and other relevant parameters. Numerical examples of the bound are given.

This year [1] we have analyzed the C P violating ratio d i e in the Standard Model including leading and next-to-leading logari thmic contr ibut ions to the Wilson coefficient functions of the relevant local op- erators. Another next-to-leading order analysis o f e ' / e can be found in ref. [2].

Imposing the constraints from the C P conserving K ---, rtn data on the hadronic matr ix elements of these operators we have given numerical results for e ' / e as a function of A~-g, mt and two non-perturbat ive pa-

rameters B~ 1/2) and B~ 3/2) which cannot be fixed by the C P conserving data at present. These two param- eters are defined by

__ R ( I / 2 ) . , ~ (vac) (Q6(mc))o =- ~6 (Q6~mcl)o ,

( aa (mc) )2 - B~ 3/2) ( a s " ,,(vac) (mc)) 2 ,

where

(1)

Q6 = (g~dp)v-,,i Z (dtp q,~)v+A, (2) q

* Supported by the German Bundesministerium fiir Forschung und Technologie under contract 06 TM 732 and by the CEC Science project SC1-CT91-0729.

1 E-mail: buras@feynman.t30.physik.tu-muenchen.de. 2 E-mail: lauten@feynman.t30.physik.tu-muenchen.de.

Qs = ~(g ,~dp)v - -AZeq(qpq ,~ )V+A (2 cont 'd) q

are the dominant QCD and electroweak penguin op- erators, respectively. The subscripts on the hadronic matr ix elements denote the isospin o f the final 7tri- state. The label "vac" stands for the vacuum insertion est imate of the hadronic matr ix elements in question for which B~ 1/2) = B~ 3/2) = 1. The same result is

found in the large N l imit [3,4]. Also lattice calcula- tions give similar results B~ 1/2) = 1.0 + 0.2 [5,6] and

B~ 3/2) = 1.0 -4- 0.2 [5-8] . We have demonstra ted in

ref. [ 1 ] that in QCD the parameters B(61/2) and B~ 3/2) depend only very weakly on the renormalizat ion scale /t when # > 1 GeV is considered. The # dependence for the matrix elements in ( 1 ) is then given to an ex- cellent accuracy by 1/m 2 (#) with ms (~) denoting the running strange quark mass.

In the present letter we would like to cast the numer- ical results of ref. [ 1 ] into an analytic formula which exhibits the mt-dependence of e ' / e together with the dependence on ms, B~ 1/2) and B~ 3/2). Such an analytic formula should be useful to those phenomenologists and experimentalists who are not interested in getting involved with the technicalit ies of ref. [ 1 ]. Combin- ing the analytic formula obtained here with the ana- lytic lower bound on mt derived recently by one of us [9] we will be able to find an analytic upper bound

212 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

Volume 318, number 1 PHYSICS LETTERS B 25 November 1993

on ~t/8,. In order to find an analytic expression for e ' /e which

exactly reproduces the results of ref. [1] we use the idea of the penguin-box expansion (PBE) suggested and developed in ref. [ 10]. This method allows to ex- press flavour-changing neutral current (FCNC) pro- cesses, in particular e ' /e , as linear combinations of universal m/-dependent functions resulting from var- ious penguin and box diagrams common to many FCNC processes. The mr-independent coefficients of these functions are # and renormalization scheme in- dependent. They depend in the case at hand only on A~g, m,, B~ '/2) and B~ 3/2).

The basic ideas of PBE have been discussed at length in ref. [10] and will not be repeated here. The simplest method for transforming the results of ref. [ 1 ] into the analytic formula given below is presented at the end of section 3 of ref. [ 10 ]. In this letter we give only the final formula for e ' /e .

The resulting analytic expression for e ' /e is then given as follows:

[ Im2, ] g ' /g = 10-4 [1.7-X ]-0-4J F(xt ) , (3)

where

F ( x t ) = Po + Px X ( x t ) + Pr Y ( x t ) + Pz Z ( x t )

+ P E E ( x t ) . (4)

Next

Im2t = IVubl IV~bl sin~ = r/)].SA 2 (5)

in the standard parameterization of the CKM matrix [ 11 ] and in the Wolfenstein parameterization [ 12 ], respectively. Here

= IV~l = 0.22, I~1 = A2 (6)

and

1 ½ r / = R b s i n O , R b = ~ V~b " (7)

The basic mr-dependent functions are given by

xt [ xt + 2 X ( x t ) = -~ tx-/sT-1 + - -

xt [ xt - 4 Y ( x t ) = y Lx--~-I + - -

3xt - 6 ] (xt - 1)2 lnxt ,

3Xt In xt] (xt - 1 )2

(8)

Z (xt) = _1 lnxt

18x 4 - 163x 3 + 259x 2 - 108xt + 1 4 4 ( x t - 1)3

32xt 4 - 38xt 3 - 15x 2 + 18xt + In xt ,

72 (x¢ - 1 )4

2 x 2 ( 1 5 - 16x, + 4x 2) E ( x t ) = - ~ l n x t + 6(1 - x t ) 4 lnxt

Xt(18 -- 1 lxt - x 2) + (8 cont 'd)

12(1 - xt) 3 '

with xt = mE/M 2. In the range 100GeV ~< rnt ~< 300 GeV these functions can be approximated to bet- ter than 3% accuracy by the following expressions [10]:

X ( x t ) = 0.650xt 0"59, Y ( x t ) = 0.315x 0"78,

Z(x t ) = 0 . 1 7 5 x 0"93, E ( x t ) = 0.570x7 °'51 . (9)

The coefficients Pi contain the physics below the M w - scale. They are given in t e rms o fB~ 1/2) = B~ 1/2) (mc),

B~ 3/2) _ B~ 3/2) (me) and ms (me) as follows:

150MeV] 2j (r~6)a~ l/z' + r~8)B13/2)) . p, = +

(I0)

T h e Pi are /*-independent and renormalization scheme independent. They depend however on A~-g

- A (4). In table 1 we give the numerical values MS o f r~ 0), r~ 6) and re 8) for different values of A~-g at It = mc = 1.4 GeV and the N D R renormalization scheme. The coefficients re °), re 6) and re 8) do not depend on ms (mc) as this dependence has been fac- tored out. re °) does, however, depend on the partic- ular choice for the p a r a m e t e r B~ if2) in the parame- terization of the matrix element (Q2)o. The values given in the table correspond to the central value for Ro/2) = 6.7 ± 0.9 as determined from the CP ~2 conserving data in ref. [1]. Variation o f B~ I/2) in the full allowed range introduces an uncertainty of at most 18% in the r~ °) column of the table. Since the parameters r~ °) give only subdominant contributions to d / e keeping ROLE) and r~ °) at their central values ~2 is a very good approximation.

For different /t and renormalization schemes the numerical values in the table change without modify- ing the values of the Pi's as it should be. To this end

213

ilume 318, number 1

ble 1 = 1 PBE coefficients for various A~--g.

PHYSICS LETTERS B 25 November 1993

i A~--g = 0.20GeV A~--g = 0.25GeV A~--g = 0.30GeV A~--g = 0.35GeV

ri o) r~ 6) r~ 8) r~ 0) r~ 6) r~ 8) r~ 0) r~ 6) r~ 8) r~ 0) r~ 6) r~ 8,

3 -4.323 9.388 2.109 -4.406 10.660 1.957 0.988 0.016 0 0.958 0.019 0

Y 0.763 0.078 0 0.729 0.086 0 Z 0.235 -0.014 -10.072 0.297 -0.015 -10.899 E 0.359 - 1.207 0.411 0.339 - 1.327 0.462

-4.484 12.038 1.792 -4.557 13.573 1.609 0.933 0.022 0 0.910 0.026 0 0.701 0.094 0 0.676 0.103 0 0.361 -0.017 -11.779 0.427 -0.019 -12.737 0.319 -1.449 0.515 0.299 -1.579 0.573

;o B~ 1/2) a n d B~ 3/2) have to be modified as they de- nd on the renormalization scheme and albeit weakly , # .

Inspection of table 1 shows that the terms involving ;) and r~z s) dominate the ratio e ' /e. The function (xt) represents a gauge invariant combination of )- and 7-penguins [ 10 ] which increases rapidly with

and due to rz (8) < 0 suppresses e ' /e strongly for • ge mt [ 13,14]. The term r0 (°) which also plays some le represents the contributions of ( V - A) ® ( V - A) 2D penguins. The positive contributions of X (xt) d Y (xt) represent to large extent the ( V - A ) ® ( V -

electroweak penguins. Finally E (xt) describes the fidual rot-dependence of the QCD-penguins which only a very small correction to the full expression 1" et/~. Combining the formula (3) with the analytic lower ,und on mt from er derived recently in ref. [9] we n find an analytic upper bound on e'/e. Indeed the ~¢er bound of ref. [9] corresponds to sin~ = 1 at lich Im)~t is maximal. Since F (xt) decreases with zreasing xt we find

Rb2SA1 ] F((xt)min), (11) t/8]rna x = 10-4 r[ l~-x-]-6z4J

lere according to ref. [9]

1.316

:t'min = [~a2 (A-~K 1.2)1 , (12) Rb

th the renormalization group invariant BI¢ ring the size of the hadronic matrix element 7°1 (gd)v-A (gd)v-alK°). Formula (11) gives the aximal value for e ' /e in the Standard Model con- ~tent with the measured value of er as a function IVcbl, IVublVcbl, nr , B~ 11=), B~ 312), ms and A~--~. It

ould be noted that this formula does not involve

mt directly as mt has been eliminated by means o f the lower bound of ref. [9]. It is evident that the up- per bound increases with increasing I V~b], [V~b/Vcb[,

n (1/2) A~-g and with decreasing ms and B~ 3/2). B K , ~6 ,

In fig. 1 we show (e'/e)max as a function of [V~bl for different values of Bx, three choices of (B~ 1/2), B~3/2)), A~--g = 300MeV, ms(mc) = 150MeV or equivalently ms ( 1 GeV) = 175 MeV and [Vub/V~bl = 0.10. The range for [ V~ol has been chosen as in ref. [ 9 ] in accordance with the analyses of refs. [ 15-18 ] and the increased B-meson life-time ZB = 1.49 5:0.03 ps [191.

B~ i/2) ---- B~ 3/2) = 1 corresponds to the leading 1/N result or central values of lattice calculations. B~ ~/z) = 2, B~ 3/2) = 1 is representative for the case advocated in ref. [20]. Finally, B~ 1/2) -- B~ 3/2) --- 2 corresponds effectively to the first case with a smaller value of ms. Comparing fig. 1 with the most recent messages from NA31 and E731 collaborations [21,22 ],

R e ( e ' / e ) = ( ( 2 3 4 - 6 " 5 ) x 10 -4 (NA31) (13) (7 .45:6.0) × 10 -4 (E731) '

we observe that whereas the results of E731 are fully compatible with the cases considered here, the NA31 data lie above the bounds of fig. 1 if IV~bl ~ 0.040. This is in particular the case fo r B~ 1/2) -- B~ 3/2) = 1 which we advocate.

In fig. 1 we only show (e'/e)m,x >/ 0. We observe that for Br <~ 0.5 the upper bound is very low and for B~ 1/2) = B~ 3/2) = 1 and IVcb[ ~< 0.039 the ratio e ' /e becomes negative. This is simply the result of the high value o fmt required by eK (see eq. (12)) when BK and IV~b] are low. For BK = 0.7 5:0.2 obtained in the 1/N and lattice calculations, (e'/e)max is in the ball park of the E731 result. We note however that

Volume 318, number l PHYSICS LETTERS B 25 November 1993

20.0

v ' - ' 15.0 O v-

10.0 E

~ 5.o

0.0 20.0

4--" 15.0 O

10.0 E

5.0

0.0

7.5

4 - " o 5.0 v--

'7

2.5

0.0 3.4 4.4

(a)

t . " / ~ .." • / / .-"

/ / z ~ . / "

/ " / / , , '¢ ./"" / / / / / /

(b)

/ j / / . .

(c) "

/ " / /

/ , " / / / " /.-'"

/ / ," /" / / " / ." ./" /

/ ," / . / , , / I ~ " I ~" I , ~ i i

3.6 3.8 4,,.0 4.2 IV~l [1 0 3

Fig. 1. (e'/e)max as a function of I~b[ for IVub/Vcbl = 0.10 and various choices of BK. The three (e'/e)max plots correspond to hadronic parameters (a) B~l/2)(mc) = B~3/2)(mc) = 2, (b) B~t/2)(mc) = 2, B~3/2)(mc) = 1 and (c) B~l/2)(mc) = B~ 3/2) (mc) = l, respectively.

for IVcbl ~ 0.040, Br ~< 0.8 and B~ 1/2) = B~ 3/2) = 1 one has (e'/e)max ~< 5 × 10 -4. Since IVub/V~bl is ex- pected to be smaller than 0.10 [23,24] we should in- deed be prepared for (e'/e)ma~ ~ afew x 10 -4 in the Standard Model if l / N calculations and lattice results are the full story. This has been already emphasized in refs. [ 1,2 ] but the bounds in fig. 1 c make this point even stronger. On the other hand we should keep in mind the strong dependence of the upper bound ( I 1 ) on BK, B~ 1/2), B~ 3/2), ms, I~bl and [~/'ub/[/'cbl. Future improvements on these parameters may bring sur- prises.

Finally it should be mentioned that below the upper bound (sin ~ < 1 ) the dependence o f e ' / e on Br, I Vcbl and ] V, b / Vcbl for fixed mt is more complicated and dif- ferent than in ( 11 ). e ' /e increases with decreasing BK and I Vcbl at fixed ]Vub/Vcb[ until the bound is reached. It increases (decreases) with decreasing I Vub / Vcb I for 7t/2 ~< ~ ~< ~z (0 ~ ~ ~< 7~/2) at fixed rot, B r and

I~bl- We hope that the analytic formula for e ' /e and the

analytic upper bound on this important ratio pre- sented in eqs. (3) and ( 11 ), respectively, should facil- itate the future phenomenological analyses once the values for I~bl, IV.b/~bl, Br , B~ 1/2), B~ 3/2), ms, A~-~ and mt have been improved and most importantly e ' /e accurately measured.

We gratefully acknowledge the most pleasant col- laboration with Matthias Jamin on e' /e in ref. [ 1 ] for which we supplemented an analytic formula and an analytic upper bound in the present paper.

References

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Volume 318, number 1 PHYSICS LETTERS B 25 November 1993

[17] P. Ball, Phys. Lett. B 281 (1992) 133. [18] G. Burdman, Phys. Lett. B 284 (1992) 133. [ 19 ] D. Karlen, B-Hadron Lifetimes, talk presented at the

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