Rare kaon decays revisited

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  • Physics Letters B 595 (2004) 301308

    s r

    , Ed

    se 907oma ddis Av

    26 M

    une 2

    dice

    Abstr

    WeQCD,The prspectruCollabdirect 200

    1. In

    Inwith lnon-leest orare ex

    exchal+tion Khowevvious

    E-1 U

    Univeraffilie

    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.

    troduction

    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

    actwww.elsevier.com/locate/physletb

    evisited

    uardo de Rafael a,b,c

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

    004

  • 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

    LS=eff.=

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

    ,U

    nsor

    7 Matrixell-M

    =(

    =(

    he of noI =ctor

    owevFo

    ffectay. U

    == d

    nd in

    (x)

    here

    (x)

    (x)

    Eq.

    S=eff.=

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

    4(sL

    dL) l=e,

    (lLlL) and

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

    (lRlR)

    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)

    GF2VudV

    usg8

    (1.4)

    {

    tr(LL

    ) ieF 20

    [w1tr(QLL)

    + w2tr(QLL)]F

    }+ h.c.

    [Qte8m

    G

    Q

    To

    fah

    e

    w

    Q

    Q

    a

    Lw

    L

    in

    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

    3I,

    (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)

    GF2VudV

    usg8

    (1.9)

    {tr(LL

    ) 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

    =wherecientssultinWilsocance

    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

    lectro(4)em(x

    ful+l+ale-

    + =

    heret theetermdiushe codepe

    redicf w+araboom t

    r(K

    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

    4[C11

    (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

    w

    w

    e

    L

    In

    sc

    w

    w

    a

    dra

    Tinpo

    pfr

    B

    go

    w

    Ustddbst

    w

    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

    13(4)2

    [w1 w2 + 3(w2 4L9)

    ](2.2) 1

    6log

    M2Km2

    4,

    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) 107,the two phenomenological solutions (for a valueoverall constant g8 = 3.3):

    (2.4)1.69 0.03 and w+ = 1.10 0.03.rtunately, this twofold determination of the con-

    + does not help to predict the KS 0e+erate. This is due to the fact that, to the same or-the chiral expansion, this transition amplitudein another scale-invariant combination of con-

    :

    (2.5)13(4)2[w1 w2] 13 log

    M2K2

    .

    edicted decay rate (KS 0e+e) as a func-f ws is also a parabola. From the recent results mode, reported by the NA48 Collaboration at

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

    Fig. 1. oupliwith th nd L9value g 0) and

    CERN

    Br(K

    =one o

    ws =At

    brancinduc

    Br(K

    =

    Here,sourc

    third(1.36for th

    Thdefinecouplillustr

    f theeore

    . Th

    .1. T

    In1 an

    s theL 0e+e)

    CPV

    (2.8)

    [(2.4 0.2)

    (Imt104

    )2+ (3.9 0.1)

    (13

    ws)2

    + (3.1 0.2) Imt104

    (13

    ws)]

    1012.

    the first term is the one induced by the directe, the second one by the indirect source and theone the interference term. With [13] Imt = 0.12) 104, the interference is constructivee negative solution in Eq. (2.7).e four solutions obtained in Eqs. (2.4) and (2.7),four different straight lines in the plane of the

    ing constants w2 4L9 and w (= w1 w2), asated in Fig. 1. We next want to discuss which

    calculation. This selects the octet channel in thetransition amplitudes as the only possible channel andleads to the relation

    (3.1)w2 = 4L9, octet dominance hypothesis (ODH).

    We now want to show how this hypothesis can infact be justified within a simple dynamical frameworkof resonance dominance, rooted in large-Nc QCD. Forthat, let us examine the field content of the Lagrangianin Eq. (1.9). For processes with at most one pion in thefinal state, it is sufficient to restrict and L to theirminimum of one Goldstone field component:

    = i

    2F0

    [,Q] + , and(3.2)L =

    2F0 + ,The four intersections in this figure define the possible values of the ce experimental input of Eqs. (2.3) and (2.6). The couplings w1, w2, a8 = 3.3. The cross in this figure corresponds to the values in Eqs. (3.2

    [12]:

    S 0e+e)

    (2.6)[5.8+2.82.3(stat.) 0.8(syst.)] 109,btains the two solutions for ws

    (2.7)2.56+0.500.53 and ws = 1.90+0.530.50.the same O(p4) in the chiral expansion, the

    hing ratio for the KL 0e+e transitioned by CP-violation reads as follows

    o

    th

    3

    3

    w

    angs which, at O(p4) in the chiral expansion, are compatiblehave been fixed at the = M scale and correspond to the(3.21) discussed in the text.

    se four solutions, if any, may be favored bytical arguments.

    eoretical considerations

    he octet dominance hypothesis

    Ref. [1], it was suggested that the couplingsd w2 may satisfy the same symmetry properties

    chiral logarithms generated by the one loop

  • S. Friot et al. / Physics Letters B 595 (2004) 301308 305

    with the result (using partial integration in the termproportional to ieg8w2)LS=eff

    .=

    showimodusame

    A inRef. [this Othe colowesthe prThis cbetwemome

    of mLagrathe reto a firelatio

    Wielectrin Eq.

    Lem(x

    The nslopeGoldsa chan

    1

    In thelarge-

    saturated by the lowest order pole, i.e., the (770):

    (3.6)1 M2

    , which implies L = F20 .

    isbserv

    ByEq

    lectro

    ew(x

    mo

    wes

    ere,

    ructuontri

    ecaune by

    2

    F 20=

    , fur =Eq

    ig. 1egationstr

    .2. B

    A rnt mortedumady+ caum oromp1(x)

    GF2VudV

    usg8

    (3.3)

    {

    2F 20 tr(

    )

    + ie2F 20 A tr[(Q Q)

    ] iew2F tr

    [(Q Q)

    ]+ ie2

    3wF tr()

    }+ h.c.,

    ng that the two-field content which in the termlated by w2 couples to F is exactly theas the one which couples to the gauge fieldthe lowest O(p2) Lagrangian. As explained in

    1], the contribution to K+ + (virtual) from(p2) term, cancels with the one resulting frommbination of the first term in Eq. (3.3) with thet order hadronic electromagnetic interaction, inesence of mass terms for the Goldstone fields.ancellation is expected because of the mismatchen the minimum number of powers of externalnta required by gauge invariance and the powers

    omenta that the lowest order effective chiralngian can provide. As we shall next explain, it isflect of the dynamics of this cancellation which,rst approximation, is also at the origin of then w2 = 4L9.th two explicit Goldstone fields, the hadronicomagnetic interaction in the presence of the term(2.1) reads as follows

    ) = ie(A 2L9

    F 20F

    )

    (3.4) tr(Q )+ .et effect of the L9-coupling is to provide theof an electromagnetic form factor to the chargedtone bosons. In momentum space this results inge from the lowest order point like coupling to

    (3.5)1 2L9F 20

    Q2.

    minimal hadronic approximation (MHA) toNc QCD [14], the form factor in question is

    Ito

    ine

    L

    Inlo

    1

    Hstc

    1

    bo

    w

    2IfM

    inFn

    c

    3

    a

    ptrre

    w

    trpM2 + Q2 9 2M2well known [15,16] that this reproduces theed slope rather well.the same argument, the term proportional to w2

    . (3.3) provides the slope of the lowest orderweak coupling of two Goldstone bosons:

    ) = ieGF2VudV

    usg82F

    20

    (A w2

    2F 20F

    )

    (3.7) tr[(Q Q)]+ .mentum space this results in a change from thet order point like coupling to

    (3.8)1 w22F 20

    Q2.

    however, the underlying S = 1 form factorre in the same MHA as applied to L9, can have

    butions both from the and the K(892):

    (3.9)

    M2M2 + Q2

    + M2K

    M2K + Q2, with + = 1,

    se at Q2 0 the form factor is normalized togauge invariance. This fixes the slope to

    (3.10)(

    M2+

    M2K

    ).

    thermore, one assumes the chiral limit whereMK , there follows then the ODH relation

    . (3.1); a result which, as can be seen in, favors the solution where both w+ and ws areve, and the interference term in Eq. (2.8) is thenuctive.

    eyond the O(p4) in PT

    ather detailed measurement of the e+e invari-ass spectrum in K+ +e+e decays was re-a few years ago in Ref. [17]. The observed spec-

    confirmed an earlier result [18] which had al-claimed that a parameterization in terms of onlynnot accommodate both the rate and the spec-f this decay mode. It is this observation whichted the phenomenological analyses reported in

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

    Fig. 2. us theThe cro leadiEq. (2.4 9) be

    Refs.to undMHAO(p4als inparam

    We+e+form

    d

    dz=

    wheree+e

    G8 =

    r =

    The rein Fig

    ig. 2

    V (z

    Th

    (z)ith t

    4 1 5 1 4

    G28

    2M5K12(4)4

    3/...

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