Nuclear Physics B (Proc. Suppl.) 37A (1994) 51-58

North-Holland

I | lllIIIII If_q t | "4 |'&'l[l~1:1

PROCEEDINGS SUPPLEMENTS

c'/e: theoretical results and updated phenomenological analysis Laura, Reina. a~

aUniversit~ Libre de Bruxelles, Service de Physique Th~orique, Boulevard du Triomphe, CP 225, B-1050 Brussels, Belgium

We present the results for the Next-to-Leading Order effective hamiltonian for AS = 1 decays, in presence of both QCD and QED corrections, and update the existing theoretical predictions for ¢l/c.

1. I n t r o d u c t i o n

During the last few years a big effort has been made in order to state the theoretical predic- tions for K- and B-meson physics on a more solid ground. Our interest in K- and B-physics is due both to theoretical and experimental reasons. From the theoretical point of view, this kind of physics is the natural framework to analyze and test the mechanisms of flavour mixing and CP vi- olation proposed in the Standard Model and be- yond. A F = 2 (F=flavour) mixings, A F = 1 weak decays (and related asymmetries), rare K- and B- decays provide in principle all the necessary tools for a full-fledged analysis of this important as- pect of the present phenomenology of elementary particle physics. Several "still missing" param- eters of the Standard Model, tipically the ones related to the top quark - as the top mass (mr) and couplings (t~u,. . . ) - can be constrained by CP violation and flavour mixing results: a real alternative prediction to the pure electroweak de- terminations, moreover improvable in the near fu- ture.

On the other hand, the nowadays theoretical enthusiasm is supported by an exciting experi- mental scenario, where more and more precision is reached in single measurements or expected in first generation experiments. This is expecially the case of B-physics, so extensively discussed at this conference, and of e'/c, the direct CP vio- lation parameter in Kaon decays into two pions. The smallness of this CP violation parameter, due

*Work done in collaboration with M. Ciuchini, E. Franco and G. Martinelli of the University of Rome "La Sapienza".

both to its radiative origin and to its intrinsic de- pendence on the A I = 1/2 rule [18], is at the same time the origin of its interest and of its illness. e'/e is today still compatible with zero, both the- oretically and experimentally. At the same time it is crucial to state if it. vanishes, in order to trust the Standard Model or not. Thus, it is a real challenge to reduce more and more all the uncertainties in the problem.

Our contribution in this direction has been the calculation of the effective hamiltonian for AS = 1 weak decays at. the Next-to-Leading Order (NLO) in QCD and QED. The effective hamiltonian for- realism is the typical framework for the analytical calculation of the physical amplitudes for weak processes, i.e. in a theory where very large masses and mass gaps are present. Within this for- realism, the hamiltonian which describes a given physical process is expressed as a linear combina- tion of effective operators with certain Wilson co- efficients. In so doing, the perturbative and non- perturbative realms factorize and we can fully ex- ploit our analytical capabilities in calculating the Wilson coefficients at a given perturbative order, while taking the operator matrix elements from some non-perturbative results.

They are precisely the non-perturbative inputs which represent the "dark side" of the problem, being affected by very large uncertainties, which show up not only in the operator matrix elements, but also in some physical input parameters (like the CKM parameters, etc.) whose prediction re- lies on some "long-distance physics".

Taking "long-distance physics" from Lattice QCD (just a method among others, to be predic-

0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00585-J

52 L. Reina/et/e." Theoretical results and updated phenomenological analysis

tive from a quantitative point of view), our effort has been to state the perturbative calculation of the Wilson coefficients on a more solid ground. A NLO effective hamiltonian constitutes a more re- liable result in many respects: the p-dependence is greatly reduced (# being the scale of match- ing between Wilson coefficients and matrix ele- ments); AQCD can be properly taken from other experiments which use NLO results; the stability of the perturbative expansion can be tested; the dependence on the heavy masses of the theory can be precisely taken into account and enforced. Moreover, the results obtained in the calculation of the AS= 1 Effective Hamiltonian can be easily generalized to the case of other effective hamil- tonians, in the Standard Model or even beyond it.

For all these reasons and many others I do not detail here, we think that this calculation will be important, even if, as we will show in the phe- nomenological analysis, for the time being the theoretical prediction for d/e is not adequate to be competitive with the improvements of the ex- perimental measurements in the near future.

2. The A S = 1 Effective Hami l ton ian : the- oretical cons t ruc t ion

I would like to report here about some main points in the theoretical calculation of the ef- fective hamiltonian which describes AS = 1 de- cays and d/e. I will recall the general construc- tion of the effective hamiltonian for the physical problem at hand, the importance of the match- ing between "short-distance" and "long-distance" physics, the independence of any physical result from the matching scale and its consequences. I will finally present the NLO solution of the evolu- tion equation for the Wilson coefficients and dis- cuss the regularization scheme independence of the final result. Full details and references about each single point can be found in [11].

The most general effective hamiltonian can be written, using the Operator Product Expansion (OPE) of the product of the two original weak currents as:

( FlU Sj--t lI) (1)

4 2 2 d zDw(x , Mw)(FIT (Jt,(x), J~(0)) II)

---. E(F]Oi(p)II)C,(p) = dr(p)d(p) i

where {0i(#)} is a complete basis of indepen- dent operators (which mix under renormalization, when QCD and QED corrections are taken into account), appropriate to the physical problem at hand, and {Ci(#)} the related Wilson coefficients. For the specific case of the AS=I effective hamil- tonian, we have used the following basis made of vertex-type and penguin-type operators:

01 = zdo)c, ,_A) =

q=u,d,s,...

04'6 ---- ('~adfl) (v-A) E (q~qo:)(W~:A, q=u,d,s,...

3 - d

q=u,d,s ,...

3 08'10 --~ -2 (gada)(v-a) E eq(gtt~q")(v+a)

q=u,d,s,...

= (2)

where the subscript (V ~: A) indicates the chiral structure and a and fl are colour indices. The sum is intended over those flavours which are ac- tive at the scale #.

The factorization of/-~as-~\ \,~eff ! in operator ma- trix elements and Wilson coefficients is com- pletely arbitrary. It depends on the scale # at which we decide to match the "short-distance" and "long-distance" physics. But the final phys- ical amplitude must be p-independent. Imposing this condition, a Renormalization Group Equa. tion (RGE) for the Wilson coefficients can be de- rived. When both QCD (c~.) and QED (c~.) radia- tive corrections are taken into account, the RGE for the coefficients is:

_ - o • 2

(z)

L. Reina / et /e : Theoretical results and updated phenomenological analysis 53

where (in the case of a weak interaction effec- tive theory) t = ln(M~v/p2), ff = (4 , ,4 . ) and in particular we ignore the running of a.. The solution of eq.(3) gives the evolution of the co- efficients with the mass scale # and depends on f~(a,) (the QCD B-function), ~(a.,a,.) (the QCD+QED anomalous dimension matrix) and the value of the coefficients at a given initial scale (initial conditions for the solution of the RGE). All these quantities are perturbative objects, ex- panded in powers of 4. and 4. (or in powers of 4k, 4h(4,t)n(4,t) m in a Leading Logarithm expan- sion). Thus the solution of eq.(3) has also to be specified at a given order, f.i. LO (Leading Or- der), NLO (Next-to-Leading Order), etc. Our re- sult concerns the NLO evolution of the Wilson coefficients. Assuming the initial value of the co- efficients to be C(Mw) [2,3,11], their value at the scale ~ will be given in term of the evolution ma- trix W~u, Mw] as:

C(I~) = IV[#, Mw]C( Mw) (4)

Consider the perturbative expansion of B(4.) and

= --041r - ~ 1 ~ + 0(42) (51

4 . ; j 0 ) a'.~.(0) + zf = 4 r "~ + 4 1 r 1e

(a' , ~2:~(1 ) + a.__~. 4__ z ^~,~

4r ~ "' 4r 4~7~-" (6)

~.(0,1) (10 x 10) matrices expressing the where/s,e are mixing under renormalization (at one and two loops respectively) among the operators in eq.(2). Then at the NLO in QCD and QED, when only one power of 4. is taken into account (4~ negli- gible for n > 1. This means that terms of order (4,t) n, 4.(4,t) n, (4,t)(4,t) n and 4o(a,t) '~ are re- summed in the Leading-Log expansion), the evo- lution matrix IV[#, Mw] is of the form:

IV[p, Mw] = M~]g][p, Mw]J~l'[Mw] (7)

with:

~?[~]

(8)

and

M'[M.,] ( "° ) = i 4.(Mw)P (9)

Here/~', J and P are recursively determined by a set of complicate equations I do not report here (see ref. [10,11] for more details) and results to be functions of .~(0) .~(1) ~0 and fll I $ 1 e , I $ ~ e ]

On the other hand, the NLO initial conditions C(Mw), defined by matching the full theory with the effective one at the Mw scale and at order 0 ( 4 . a,.), are of the form:

4 . ÷

4-" - s f ( ° ) ) 00) 4~r

where ~(1) and/~(1) represent the coefficients ob- tained calculating the one loop matching at the scale Mw, while ~ and g express the mixing in the operator matrix elements, at the Mw scale and at the given order (O(4, ,4,)) . Both ~(1), /~(1), ÷ and g depend on the external states chosen for the Feynman diagrams involved in their cal- culation. However in the final expression for the C(Mw) coefficients this dependence cancels. Any "unbalanced" modifications of the initial condi- tion C(Mw) will break this cancellation and the dependence on the external states must be recov- ered by the operator matrix elements at the Mw scale, as we have indeed verified.

Our contribution has been the calculation of ..}.(~) s,e widl respect to the operator basis in eq.( 2) and the development of the whole formalism summarized up to here in order to get the NLO answer for the Wilson coefficients.

We have worked in the modified Minimal Sub- traction renormalization scheme (MS) and in two different regularization schemes: Naive Dimen- sional Reduction (NDR) and 't Hooft-Veltman scheme (HV). The regularization scheme indepen- dence of the final result has been demonstrated and explicitely checked. On this point we agree with the only other similar calculation in the lit- erature [6,7], while a difference on two single di-

54 L. Reina let /e : Theoretical results and updated phenomenological analysis

agrams is still present (see [11] for further de- tails). The scheme-dependent objects in the cal- culation are those involving divergent parts of two-loop Feynman diagrams or finite parts of one- loop Fevnman diagrams, i.e..~(1) . ~,e on one side and fO) , /~O), ÷ and g on the other (for the list of the Feynman diagrams contributing at different levels see ref. [11]). However, they enter the final expression for the evoluted coefficients C(y) in a scheme independent combination.

The final results for the NLO coefficients can be given in any definite regularization scheme, provided that the relative operator matrix ele- ments are also calculated in the same scheme, as is affor~lable using Lattice QCD matrix ele- ments. One could also use scheme independent coefficients, but this would correspond to what we have called an "unbalanced" definition of the initial conditions C(Mw), i.e. to an additional dependence of C(y) on the external states, to be compensated by using matrix elements calculated between the same external states.

3. Re s u l t s for e'/e

Using the AS = l effective hamiltonian we have computed, we can calculate the NLO amplitudes for Kaon decays into two pions: A ( K ~ rc~r (I = 0)) and A ( K - - rTr (I = 2)), whose imaginary parts enter the expression for d:

e i7r[4 09 e' - [w- 1 (ImA2) ' -

x/2 ReA0

(1 - ftIB) ImA0] ( 1 1 )

where ~: = ReA,. /ReAo = 0.045 and:

ImA~ = (ImA2)' + ftlBImAo (12)

~rB = 0.25 -4- 0.10 denotes the isospin breaking contribution and we assume ReA0 = 3.3.10 -7.

Inu40 and (ImA2)' are expressed in terms of Wilson coefficients and operator matrix elements. We have implemented a numerical program for the calculation of the evolution of the Wilson coefficients at NLO and we have taken the ma- trix elements of the operators in eq.(2) from Lat- tice QCD. We have chosen as matching scale St = 2 GeV, because at this scale the Wilson co-

efficients can be safely computed in a pure per- turbative regime and operator matrix elements, when calculable, can be taken from Lattice QCD (this would not be true for other non-perturbative methods, a much lower scale # would be required, p --~ 0.8 - 1.0 GeV, where the Wilson coefficients show a quite strong instability - see Fig.(2)).

In particular, the matrix element of the generic operator Oi is given by:

(Oi> = Bi(Oi>vIA (13)

where the subscript "VSA" denotes the matrix ele- ment calculated in the Vacuum Insertion Approx- imation and Bi is the B-parameter of the oper- ator Oi, which quantitatively represents the de- viation of the physical matrix element from the VIA expectation. The VIA matrix elements are calculated as functions of three quantities:

X = A ( M ~ - - M ~ ) (14)

\ m,(u) + ) (15)

-~ 12X (0"15GeV~ 2 l

z = 4 - 1) v (16)

while the B-parameters are in part taken from Lattice QCD (with a given error) and, when this is not possible, they are allowed to vary in a suit- able range, consistent with some theoretical pre- dictions or experimental results.

Thus, in the effective hamiltonian formalism the expressions for ImA0 and (hm4,) ' result to be:

: - -~2 I ,n (V~: V~d) x (17) ImA0

Z CsB~I~Z ( 2 y + .~ + X ) _ ," B 1 / ' X

L. Reina let&: Theoretical results and updated phenomenological analysis 55

and

* /

As one can see from eqs,(17) and OS), de- pends on the analytical calculation of the coeffi- cients (as functions of the scale #, the top mass ms and AQCD) and on some experimental and theoretical input parameters, in particular: the B-parameters, the strange quark mass rn,, the CKM matr ix elements. Many other well known quantities enter the determination of d /e , but they are clearly not so relevant in the analysis of the uncertainties which still affect our theoretical analysis.

I would like to separate two main arguments. The first one concerns the discussion of e' /e in the more general contest in which we can understand the lnechanism of CP violation in the Standard Model (or even beyond it). The second one, on the other hand, is more specifically related to the determination of d / e itself, on the basis of the NLO calculation of the Wilson coefficients [10,8].

In a global analysis of CP-violation within the Standard Model framework, other physical quan- tities have to be taken into account, mainly: the e parameter of CP violation in the K0-/~'0 mixing and the xd parameter of B0-/30 mixing (related to fB or to the BB B-parameter). e and Xd, as d / c depend on some CKM matrix elements and moreover introduce a strong dependence on BK (the B-parameter for the K0-/~'0 mixing) and on the B-meson lifetime rt~. Many phenomenologi- cal analysis of this kind exists in the literature, f.i. [3,4,8,9]. We are presently updating the anal- ysis of ref.[9], both with respect to the values of the parameters used and to the use of the NLO expression for e ' /e [12]. The main idea of ref.[9] is to use the common dependence of e, Zd and d / e on cos5 (5 being the CP-violating phase in the CKM matrix) in order to select between positive and negative values of cos6. For a fixed value of m~ (we have taken m t = (160=1=30) GeV) the theo- retical value of e - obtained allowing B K , ./kQcD,

A and g (in the Wolfestein parametrization of the CKM matrix, see f.i. [9]) varying in a la-interval around their central values - is fitted with the experimental one ([e[ = 2 .268.10-3) . For the se- lected values of cos 5, fB and d / e are calculated (we assume Zd = 0.685 -4- 0.0786 and this time we vary also rB, f21B, m, and the B-parameters with the previous criterion) and from the knowledge of both of them a single region, cos 6 > 0 or cos 6 < 0 should be selected. Indeed the present proposed values for fB:

(205 + 40)MeV [13]

fB = (195 + 10 5= 30)MeV [14] (19) +53

( 1 6 0 + 6 + - 1 9 )MeV [15]

seem to select quite unambiguously the cos 6 > 0 region. On the other and, as we will discuss more extensively in the following, this is not the case for et/e, at the present level of theoretical and experimental uncertainty, see figs.(3)-(5).

Let us add a few comments on the values used A N ! = 4 for some relevant parameters. We use "'-QCD -~-

(340 + 120) MeV, derived from [16] where the • Ns=5 value for AQC D is given.

We assume m, : (150 + 30) MeV, lower than in ref.[9], in order to better compare our results to other recent analyses [8], also if we still think that the status of uncertainty on ms is such that the our "old" rn, = (170 + 30) MeV would have been reasonably allowed.

Concerning the CKM matrix elements, the val- ues of A and cr (Wolfestein par.) are derived re- spectively from Vcb and IVub/Vcbl. The value of ~'~b used is:

t'~b = A,~ 2 = 0.043 =t= 0.005 (20)

as extracted from incluse and exclusive semiple- tonic B-decays data using a B-meson mean life- t ime of:

rB = (1.5 =!: 0.1)- 10 -12 sec

It is in agreement with refs. sponds to:

A = (0.9 + 0.1)

(21)

[8,17] and corre-

(22) The recent data presented at this conference [19, 20,18], propose a little lower value of V~b (Vcb =

56 L. Reina/e//e: Theoretical results and updated phenomenological analysis

(0.041+0.005), corresponding to A = (0.85~0.10)) and correspondentely a still higher value for the B-meson mean lifetime rB (rB = (1.535 :t: 0.025) ps). Finally, our choice of [V~,b/Vcbl:

ll~,~/I.~bl = aA = 0.085 + 0.015 (23)

corresponds to a central value which mediates be- tween the analysis of the inclusive semileptonic B- spectrum based on the ACCMM or ISGW models [17]. On the other hand, for the extimation of the error we trust much more the ACCMM model. The corrisponding value of c~ results to be:

~r = 0.39 ± 0.07 (24)

A much more detailed discussion will be given in ref.[12].

Let us now consider the NLO analysis of d/e . The new values assumed for the CKM parame- ters have clearly produced a change in the cen- tral value of d / e (at different mr) with respect to ref.[10]. The same holds also for the change in m,. For the time being we have not yet ultimate the analysis of the specific dependence on these parameters [12], but it will be clearly important to have it in view of a more general discussion (see previous argument).

Then, d /e mainly depends on the values of the B-parameters and on the NLO expression for the Wilson coefficients. The B-parameters are cer- tainly one of the major sources of uncertainty in the problem, being related to the poor knowl- edge we still have of the long-distance hadronic physics. A detailed discussion of the values used, see Table 1, is given in refs.[9,10]. I am not going to repeat it here, because no relevant improve- ment has been produced in the meanwhile.

On the other hand, considering the Wilson co- efficients, we may focus on two main points: their variation from LO to NLO and their dependence (once the NLO expression is assumed) on some peculiar quantities: mr, AQCD and the matching scale #. For the purpose of the following discus- sion, let us write e'/e in the following form:

R × c Bo(1- (25) /

i

where the contribution of 06 has been explicitely factorized. Indeed, at LO, d /e results to be dom- inated by 06, and we want to verify if it is still

the case at NLO, when electromagnetic penguin operators are expected to play a very important role. The leading fl's result always to be: fl~, f14 and fl3/2

7,8,9" First, let us fix mt = 160 GeV, # = 2 GeV,

A ( N s =4) _ QCD --340 MeV, and look at the effects of the NLO evolution (QCD+QED). The only relevant

o3/2 03/2 varies from 0.40 variations appear for "o7,8 : -7

(LO) to 0.66 (NLO) and 03/2 ,u s from-0.02 (LO) to ~3/2__,..,,3/2__,-,3/2 -0.09 (NLO). However, the sum ~t 7 -t-~t s -i-~t 9

does not change in a sensible way from LO to NLO (from 0.20 to 0.30), while the sum ~ + f14 is constant and equal to 0.10. Thus, although the contribution of each single electromagnetic pen- guin operator is relevant, their global effect is not and d / e continues to be dominated by the con- tribution of Os still at NLO level. Being Cs low- ered by the NLO corrections here considered (see fig.(1)), also the central value of e'/¢ is lowered going from LO to NLO, as figs.(3)-(4) show. The value measured by the Fermilab collaboration at E731 [21,22] seems to be in better agreement with our results.

-0.04

-0.06

- 0 . 0 6

' ' I . . . . I . . . . 1 . . . . 1 '

1 I $ 2 2 5

Figure 1. C6 as a function of p for AQco = 340 and m, = 160 GeV at LO and NLO.

Taking now the NLO values for the Ci and looking to specific dependences, we observe first of all a remarkable variation with mr, see again figs.(3)-(4). Higher values of rn, correspond to

L. Reina/E//e: Theoretical results and updated phenomenological analysis 5 7

Table 1 Values of the B-parameters. Entries with a (*) are reasonable extimates; the others are taken from Lattice QCD calculations.

BK B~ 3/2) B c B3,4 B5,6 1 : ~ ( 1 / 2 ) / : ~ ( 3 / 2 ) : 1 - 2 ~ - ~ 7 - 8 - 9 " 7 - 8

0.8 -4- 0.2 0 - 0.15(*) 1 - 6(*) 1.0 5= 0.2 1(*) 1.0 -4- 0.2

AQCD-220-460 M e V

, . " 1 . . . . I . . . . I . . . . I '

/

- , . to c e O " )

-0,15 I i

, , I . . . . l , l l l ! 2 S 4

#

mt=160 d e V N L 0

i l . . . . I . . . . I . . . . I '

o.ooz I i ce(p. ) +

• ~ O.~l '. % ~,

• .. %

+ , l . . . . t . . . . I . . . . l l 2 3 4

Figure 2. C6 and Cs as a function of for AQCD : 220 (dotted),340 (solid) and 460 (dashed) MeV.

lower values of e'/e. The present "tendence" of mt seems to point towards higher values instead then smaller ones and this is another point in favour of a small value of e'/¢. Moreover, looking at fig.(2), we can see both the dependence on AQCD and the dependence on p for different values of AQcD. What is quite clear here is just the fact (already stressed in justifying the necessity of a high matching scale for the effective hamiltonian) that the coefficients really "blow up" for low val- ues of #, let us say below 1 GeV, indicating that the perturbative approach is nomore reliable at that scale.

4. Conc lus ions

From the previous phenomenological analysis, it is clear that for the time being the value of e'/e is till compatible with zero and nothing defini- tive can be stated neither experimentally nor the- oretically. Thus, the Standar Model prediction of a small but definitely non-zero e'/e could still be confirmed or not. The experimental scenario seems at the moment to be much more promis- ing [22] than the theoretical one. What we have done put the calculation of the perturbative part of the AS = 1 effective hamiltonian on a more solid ground, while some sensible improvements in the calculation of the long-distance physics of the problem is mandatory.

o.oo4 ~_1 ' 'N~.O' I . . . . I . . . . I . . . .

~1/c 0.000

, - o . o o 2 I . . . . I . . . . I . . . . I . . . .

+.°"

I l t o p = 1 3 0 .

. - o . ~ . . . . I . . . . I . . . . I . . . . -1 -0~ 0 0.8

c o s

= :

I -

Figure 3. LO and NLO predictions for die: the band of the allowed values is shown, for m~ = 130 GeV. The experimental values of NA31 and E731 are indicated (dashed lines).

58 L. Reina / et /e: Theoretical results and updated phenomenological analysis

Cl/E

0.004

0,002

0.000

- - 0 . 0 ~

0.004

0.000

. - 0 . ~

L N^al . . . . . . . . . -~

- , . . . . , . . . . , . . . . , . . . .

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

, . . . . , . . . ' ? ° . , . . . . , : - 1 . -0.6 0 o.ls

c o s

Figure 4. LO and NLO predictions for e#/e: the band of the allowed values is shown, for ms = 160 GeV. The experimental values of NA31 and E731 are indicated (dashed lines).

o°. [- Fm -E HI4141+IH J - - a . ~ . . . . I . . . . I . . . . I . . . . I ~ 1 ' ' ' I . . . . I . . . . I . . . . 1 0.004

° ' ~ " b . . . . . . . -NA-E . . . . . . . . .

bito p= 190. " ° ' l ~ l l F I . . . . I . . . . I . . . . I , , , , I -

- 1 -0.15 0 0.6

c o e

Figure 5. LO and NLO predictions for e#/e: the band of the allowed values is shown, for mt = 190 GeV. The experimental values of NA31 and E731 are indicated (dashed lines).

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