Rare kaon decays revisited

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  • Physics Letters B 595 (2004) 301308

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    WeQCD,The prspectruCollabdirect 200

    1. In

    Inwith lnon-leest orare ex

    exchal+tion Khowevvious

    E-1 U


    0370-2doi:10.present an updated discussion of K ll decays in a combined framework of chiral perturbation theory and large-Ncwhich assumes the dominance of a minimal narrow resonance structure in the invariant mass dependence of the ll pair.oposed picture reproduces very well, both the experimental K+ +e+e decay rate and the invariant e+e massm. The predicted Br(KS 0e+e) is, within errors, consistent with the recently reported result from the NA48oration. Predictions for the K + modes are also obtained. We find that the resulting interference between theand indirect CP-violation amplitudes in KL 0e+e is constructive.4 Elsevier B.V. All rights reserved.


    the Standard Model, transitions like K l+l,= e,, are governed by the interplay of weakptonic and electromagnetic interactions. To low-

    der in the electromagnetic coupling constant theypected to proceed, dominantly, via one-photonnge. This is certainly the case for the K l and KS 0l+l decays [1]. The transi-02 0 0l+l, via one virtual photon, iser forbidden by CP-invariance. It is then not ob-whether the physical decay KL 0l+l will

    mail address: greynat@cpt.univ-mrs.fr (D. Greynat).nit Mixte de Recherche (UMR 6207) du CNRS et dessits Aix Marseille 1, Aix Marseille 2 et sud Toulon-Var, la FRUMAM.

    still be dominated by the CP-suppressed -virtualtransition or whether a transition via two virtual pho-tons, which is of higher order in the electromagneticcoupling but CP-allowed, may dominate [2]. The pos-sibility of reaching branching ratios for the modeKL 0e+e as small as 1012 in the near futurededicated experiments of the NA48 Collaboration atCERN, is a strong motivation for an update of the the-oretical understanding of these modes.

    The CP-allowed transition K02 0 0e+e has been extensively studied in the literature(see Refs. [3,4] and references therein). We have noth-ing new to report on this mode. A recent estimate ofa conservative upper bound for this transition gives abranching ratio [5]

    (1.1)Br(KL 0e+e)CPC < 3 1012.693/$ see front matter 2004 Elsevier B.V. All rights reserved.1016/j.physletb.2004.05.069Rare kaon decay

    Samuel Friot a, David Greynat a

    a Centre de Physique Thorique, 1 CNRS-Luminy, Cab Grup de Fsica Terica and IFAE, Universitat Autn

    c Instituci Catalana de Recerca i Estu

    Received 19 April 2004; received in revised form

    Available online 22 J

    Editor: G.F. Giu



    uardo de Rafael a,b,c

    , F-13288 Marseille cedex 9, Francee Barcelona, 08193 Barcelona, Spainanats (ICREA), Spainay 2004; accepted 28 May 2004


  • 302 S. Friot et al. / Physics Letters B 595 (2004) 301308

    There are two sources of CP-violation in the tran-sition K0L 0 0l+l. The direct source isthe one induced by the electroweak penguin-like di-agramopera

    Q11 =

    Q12 =

    moduimagiof theCP-viof therametfore,transiCP-vilogicaducedsitiveindiretive in

    Ththe frwas fiorderampliloop goperaof thcontriL(xL(xwhereGoldstive L


    Here D is a covariant derivative which, in thepresence of an external electromagnetic field sourceA only, reduces to DU(x) = U(x) ieA(x)



    7 Matrixell-M



    he of noI =ctor


    ffectay. U

    == d

    nd in







    s which generate the effective local four-quarktors [6]


    dL) l=e,

    (lLlL) and

    (1.2)4(sL dL) l=e,


    lated by Wilson coefficients which have annary part induced by the CP-violation phase

    flavour mixing matrix. The indirect source ofolation is the one induced by the K01 -componentKL state which brings in the CP-violation pa-

    er . The problem in the indirect case is, there-reduced to the evaluation of the CP-conservingtion K01 0e+e. If the sizes of the twoolation sources are comparable, as phenomeno-l estimates seem to indicate [2,4,5,7], the in-branching ratio becomes, of course, rather sen-to the interference between the two direct andct amplitudes. Arguments in favor of a construc-terference have been recently suggested [5].e analysis of K l+l decays withinamework of chiral perturbation theory (PT)rst made in Refs. [1,2]. To lowest non-trivialin the chiral expansion, the corresponding decaytudes get contributions both from chiral oneraphs, and from tree level contributions of local

    tors of O(p4). In fact, only two local operatorse O(p4) effective Lagrangian with S = 1bute to the amplitudes of these decays. With) the 3 3 flavour matrix current field

    (1.3)) iF 20 U(x)DU(x),U(x) is the matrix field which collects the

    tone fields ( s, Ks and ), the relevant effec-agrangian as written in Ref. [1], is1(x)






    ) ieF 20


    + w2tr(QLL)]F

    }+ h.c.














    L(x)]; F is the electromagnetic field strength; F0 is the pion decay coupling constant (F0 eV) in the chiral limit; Q the electric charge; and a short-hand notation for the SU(3)ann matrix (6 i7)/2:2/3 0 00 1/3 00 0 1/3


    (1.5)0 0 00 0 00 1 0


    verall constant g8 is the dominant couplingn-leptonic weak transitions with S = 1 and1/2 to lowest order in the chiral expansion. Theization of g8 in the two couplings w1 and w2 is,er, a convention.

    r the purposes of this Letter, we shall rewrite theive Lagrangian in Eq. (1.4) in a more convenient

    sing the relations

    Q = 13 and Q = Q 1


    (1.6)iag(1,0,0), I = diag(1,1,1),serting the current field decomposition

    (1.7)= L(x) eF 20 A(x)(x),

    = iF 20 U(x)U(x) and(1.8)= U(x)[Q,U(x)],

    (1.4), results in the Lagrangian1(x)





    ) eF 20 A tr[(L +L)]

    + ie3F 20

    F[(w1 w2) tr(LL)

    + 3w2 tr(LQL)]}+ h.c.

  • S. Friot et al. / Physics Letters B 595 (2004) 301308 303

    The Q11 and Q12 operators in Eq. (1.2) are pro-portional to the quark current density (sL dL) and,therefore, their effective chiral realization can be di-rectly[(sLof modoingthe efducesonly,w1 g8(w


    short-tegratwhenuationquarkfrompresenble, thw2 coin muhave brefereprogrenolominatcal arcombare alconclviolat

    2. Kexpan

    Asexpanand w

    level contribution to the K+ +e+e amplitudeinduced by the combination of the lowest O(p2)weak S = 1 Lagrangian (the first term in Eq. (1.4))

    ith thich



    + =

    heret theetermdiushe codepe

    redicf w+araboom t


    ivesf the

    + =nfo

    ant wecayer inringsants

    s =

    he pron on thiobtained from the strong chiral LagrangiandL) (L)23 to O(p)]. Using the equationstion for the leptonic fields F = ell, anda partial integration in the action, it follows thatfect of the electroweak penguin operators in-a contribution to the coupling constant w1 w2which from here onwards we shall denote w =w2; more precisely

    = w1 w2)|Q11,Q12(1.10)3


    (2)+ C12(2)],

    C11(2) and C12(2) are the Wilson coeffi-of the Q11 and Q12 operators. There is a re-

    g -scale dependence in the real part of then coefficient C11 + C12 due to an incompletellation of the GIM-mechanism because, in thedistance evaluation, the u-quark has not been in-ed out. This -dependence should be canceleddoing the matching with the long-distance eval-of the weak matrix elements of the other four-operators; in particular, with the contribution

    the unfactorized pattern of the Q2 operator in thece of electromagnetism. It is in principle possi-ough not straightforward, to evaluate the w anduplings within the framework of large-Nc QCD,ch the same way as other low-energy constantseen recently determined (see, e.g., Ref. [8] and

    nces therein). While awaiting the results of thisam, we propose in this Letter a more phenom-gical approach. Here we shall discuss the deter-ion of the couplings w and w2 using theoreti-guments inspired from large-Nc considerations,ined with some of the experimental results whichready available at present. As we shall see, ourusions have interesting implications for the CP-ing contribution to the KL 0e+e mode.

    ll decays to O(p4) in the chiralsion

    discussed in Ref. [1], at O(p4) in the chiralsion, besides the contributions from the w12 terms in Eq. (1.4) there also appears a tree


















    Ttiohe L9-coupling of the O(p4) chiral Lagrangiandescribes strong interactions in the presence ofmagnetism [9]:

    ).= ieL9F(x)

    (2.1) tr{QDU(x)DU(x)

    + QDU(x)DU(x)}.

    l generality, one can then predict the K+ l decay rates (l = e,) as a function of theinvariant combination of coupling constants


    [w1 w2 + 3(w2 4L9)

    ](2.2) 1




    w1, w2 and L9 are renormalized couplingsscale . The coupling constant L9 can be

    ined from the electromagnetic mean squaredof the pion [10]: L9(M) = (6.9 0.7) 103.mbination of constants w2 4L9 is in fact scalendent. To that order in the chiral expansion, theted decay rate (K+ +e+e) as a functiondescribes a parabola. The intersection of thisla with the experimental decay rate obtained

    he branching ratio [11]

    (2.3)+ +e+e)= (2.88 0.13)