Nucl.Phys.B v.587

Nuclear Physics B 587 (2000) 324www.elsevier.nl/locate/npe

Flavour non-conservation in goldstino interactionsAndrea Brignole , Anna Rossi

Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Dipartimento di Fisica G. Galilei,Universit di Padova, Via Marzolo n. 8, I-35131 Padua, Italy

Received 13 June 2000; accepted 12 July 2000

Abstract

We point out that the interactions of goldstinos with matter supermultiplets are a potential sourceof flavour violation, if fermion and sfermion mass matrices are not aligned and supersymmetryis spontaneously broken at a low scale. We study the impact of those couplings on low-energyprocesses such as e , eee, K+,KSK transitions and analogous ones. Moreover,we address the issue of flavour violation in low-energy processes involving two goldstinos and twomatter fermions, generalizing earlier results obtained in the flavour-conserving case. 2000 ElsevierScience B.V. All rights reserved.

PACS: 11.30.Pb; 12.60.Jv; 13.25.-r; 13.20.-vKeywords: Supersymmetry; Goldstino; Flavour violation

1. Introduction

Contributions to flavour changing neutral currents (FCNC) are adequately suppressed inthe standard model (SM) [1,2]. Supersymmetric extensions of the SM generate additionalcontributions to FCNC, even in the case of minimal field content and conserved R-parity(MSSM) [38]. Such effects are due to both charged and neutral couplings. In particular,the interactions of matter multiplets with neutral gauginos, which have the form gf f ,are a potential source of flavour violation that has no counterpart in the SM. Indeed,if fermion and sfermion mass matrices are not diagonal in the same superfield basis,those couplings induce flavour changing effects, which have been extensively studied(see, e.g., [911] and references therein). On the other hand, if fermion and sfermionmass matrices are misaligned, FCNC receive additional contributions from another classof neutral couplings: the interactions of matter multiplets with goldstinos. This observationis the starting point of the investigation we would like to present in this paper.

Corresponding author.E-mail addresses: [email protected] (A. Brignole), [email protected] (A. Rossi).

0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0550-3213(00) 00 45 8- 2

4 A. Brignole, A. Rossi / Nuclear Physics B 587 (2000) 324

We recall that the supersymmetry breaking masses of the MSSM are expected tooriginate from the spontaneous breaking of supersymmetry in some underlying theory.This phenomenon entails the existence of a Goldstone fermion, the goldstino, whichcouples to matter and gauge supermultiplets in a characteristic way. For instance, theinteraction of a goldstino G with a fermion-sfermion pair has the form (m2/F ) f f G,where

F is the supersymmetry breaking scale and m2 denotes the mass splitting

in the sector under consideration [12]. The form of the interaction resembles that of aneutral gaugino, although the strength is m2/F instead of g. If the mass splitting haselectroweak size whilst

F is much larger, the goldstino is essentially decoupled. Here

we are interested in the opposite scenario, in whichF is not far from the Fermi scale

1/GF . In this case the interactions involving goldstinos are no longer negligible and

can have observable effects. 1 How to obtain low values ofF in concrete models is an

open issue, which we will not discuss. We only recall that such a possibility is not ruledout by present experiments, as shown for instance by recent studies on goldstino pair-production at e+e and hadron colliders [1518]. Therefore we believe that it is worthexploring this scenario also in connection to flavour changing phenomena, taking intoaccount the possible flavour structure of sfermion mass matrices (that is, of m2). Thisis the purpose of the present paper. Notice that we do not try to solve the supersymmetricflavour problem, in contrast to more fundamental approaches which directly address theorigin of flavour and/or supersymmetry breaking, or at least the mediation of that breaking.For instance, in models in which such mediation is due to ordinary gauge interactions (for areview see, e.g., [19])F is relatively low, but still two or three orders of magnitude largerthan 1/

GF , so the goldstino is very weakly coupled to matter; furthermore, the mediation

mechanism generates flavour-blind sfermion masses, so flavour violations through gauginocouplings are automatically suppressed in those models. Our approach here is somewhatorthogonal, i.e., phenomenological rather than model-based: we would like to study theflavour dependence of goldstino-matter interactions and the related implications for flavourchanging processes in a general way, by treating

F and sfermion mass matrices as free

parameters, without referring to specific supersymmetry breaking or flavour models. Onthe phenomenological side, we will be interested in low-energy processes that only involveordinary fermions and photons (and possibly goldstinos) as external particles. 2

The paper is organized as follows. In Section 2, we embed the MSSM in an effectiveLagrangian in which supersymmetry is linearly realized and spontaneously broken by theauxiliary component vev of some chiral superfield Z, as in [20]. The couplings of thegoldstino superfield Z to matter and gauge superfields generate both mass and interactionterms for the component fields. In Section 3, we use those interactions to computethe rate of flavour changing radiative decays, such as e and analogous ones. In

1 In the context under consideration, the words goldstino and light gravitino can be interchanged, thanks tothe equivalence theorem [1214]. We recall that the gravitino becomes massive by absorbing the goldstino. Itsmass is related to the supersymmetry breaking and Planck scales as m3/2=F/(

3MP ). For F ' O(G1F ),

m3/2 'O(105) eV.2 If sfermions are also allowed to be external, other processes could be considered. The flavour changing decays

fi fj G are obvious examples.

A. Brignole, A. Rossi / Nuclear Physics B 587 (2000) 324 5

particular, we compare the contributions with goldstino exchange with the conventionalones which do not involve the goldstino multiplet. In Section 4, we discuss the generationof flavour changing effective operators with four external matter fermions, and discussthe phenomenological implications for processes such as eee, KL + andKK transitions. In Section 5, we address the issue of flavour violation in processesinvolving two matter fermions and two goldstinos as external states. In this case, we alsoconsider a more general approach based on the non-linear realization of supersymmetry,generalizing earlier results [21,22]. We compare this approach with the linear one, alsoextending the latter to the case of mixed FD breaking. Section 6 is devoted to summaryand conclusions.

2. Supersymmetry breaking masses and goldstino couplings

The MSSM contains quark, lepton and Higgs supermultiplets interacting with theSU(3)cSU(2)LU(1)Y vector supermultiplets. For our purposes, it will be sufficient toconsider quark and charged lepton supermultiplets, coupled to the electromagnetic U(1)Qvector supermultiplet. The matter supermultiplets in each charge sector will be genericallydenoted by Ei = (f , f )i (charge Qf ) and Eci = (f c, f c)i (charge Qf c = Qf ), wheref = u,d, `, and i is a generation (i.e., flavour) index. The U(1)Q vector supermultipletcontains photon (A) and photino () fields. The associated Lagrangian for the componentfields reads 3

L0 =14FF + i + 12 (M+ h.c.)

+

f=u,d,`

[if Df + if c Df c +

(Df

)(Df )+

(Df c

)(Df c)

+ ge

2Qf(if f if cf c+ h.c.) (f cmf + h.c.)

( f f c )(mm+ m2LL m2LRm2RL mm

+ m2RR

)(f

f c

)]+ ,

(1)where ge is the electromagnetic gauge coupling, D = + igeAQ and the dots denotesfermion self-interactions. Note that m, m2LL, m2RR, m2RL, m2LR = (m2RL) are 3 3matrices in each charged sector and should be labeled by an index f . Both this index andgeneration indices are understood for simplicity. 4 If fermion and sfermion mass matricesare not diagonal in the same superfield basis, flavour changing effects arise through the

3 We use two component spinor notation, with (1, E), (1,E), 14 ( ), 14 (

) and g = diag(+1,1,1,1). We recall that f and f c correspond to the left and rightcomponents of the four component Dirac spinor f , whereas f and f c

correspond to the fields usually denoted

as fL and fR .4 The inclusion of neutrinos requires minor modifications. Consider the case in which only and are

present in the low-energy theory. Then, wherever a sumf appears or is understood, the neutrino contributions


gauginofermionsfermion vertices. Notice that we do not assume that the matrix m2RL isproportional to the matrixm. However, in order to simplify the power counting in the nextsections, we will make the reasonable assumption that both matrices have a common chiralsuppression. Thus, in the fermion mass basis, the diagonal entries of m2RL are expected tobe of order supersymmetry breaking mass appropriate fermion mass, and a further(model dependent) suppression factor can be expected in the off-diagonal entries.

Now we have to specify how the matter and gauge multiplets couple to the goldstino,without relying on some specific fundamental mechanism for supersymmetry breaking.This can be done in several ways. For instance, the interactions between one goldstinoand a fermionsfermion pair, which are model-independent, can easily be derived fromsupercurrent conservation [12]. For interactions involving more than one goldstino, othermethods have to be used. Here we follow the approach of Ref. [20], where the spontaneousbreaking of supersymmetry is described at an effective level. Therefore we will considerthe above Lagrangian (1) as part of an effective globally supersymmetric Lagrangian inwhich the matter and gauge superfields are also coupled to some neutral chiral superfieldZ. Supersymmetry, which is linearly realized, is assumed to be spontaneously broken bythe auxiliary component of Z, through a non-vanishing expectation value Z | = F .The mass parameter

F is the supersymmetry breaking scale. The physical components in

Z are a Weyl fermion, namely the goldstino G, and a complex scalar z, called sgoldstino. 5The effective couplings between the goldstino superfield Z and matter or gauge superfieldsgenerate not only the supersymmetry breaking masses shown in L0, through the above vev,but also closely related interactions, to be illustrated now.

Matter fermion masses, as well as the associated supersymmetric sfermion masses, canbe derived from superpotential terms of the formEcmE. Consider now the supersymmetrybreaking mass matrices in the sfermion sector, i.e., m2LL, m2RR and m2RL. The LL (RR)mass terms can be derived from Khler potential terms of the form |Z|2EE (|Z|2EcEc ),suppressed by the square of some scale , whereas the RL mass terms can be derived fromsuperpotential terms of the form ZEcE. All such terms contain arbitrary dimensionlessflavour matrices. Since these matrices are in one-to-one correspondence with the matricesm2LL, m

2RR and m2RL, we will trade the former set of parameters (plus ) for the latter

set of physical parameters (including F ). Those Khler potential and superpotentialterms generate not only masses, but also several interactions (of dimension 4 or higher)involving the physical components of the matter and goldstino superfields. 6 In particular,the following cubic interactions emerge:

can be formally obtained by putting f = f c = , f = f c = , m2LL = m2RR and multiplying by a factor1/2. For instance, the neutrino and sneutrino mass terms in Eq. (1) would read as 12 (m + h.c.) and(mm+ m2LL) 12 (m2RL + h.c.).5 We work in field coordinates such that z = 0, Kzz = 1 and the parameters F and M are real and positive.

6 The effective supersymmetric Lagrangian could contain other superfield interactions besides those consideredhere. Some of them can be eliminated in favour of the existing ones through field redefinitions. Other ones dependon additional arbitrary parameters, not directly related to the mass spectrum. We will not discuss them, since wechoose to focus on couplings that are related to the mass spectrum.


1F

f

(f f c

)( m2LL m2LRm2RL m

2RR

)(f G

f cSG

)+ h.c. (2)

1F

f

zf cm2RLf + h.c. (3)

Among quartic interactions, the following ones will be relevant for our purposes:

12F 2

[ SGG+ i(zz zz)]f

(f m2LL

f f cm2RRf c). (4)

We stress again that the interactions of a goldstino with fermionsfermion bilinears in(2) are completely model independent, as one can easily check by using supercurrentconservation. As already mentioned, such interactions resemble those of a neutral gaugino,with m2/F playing the role of g [12]. Notice however that, in contrast to the case ofgauginos, here the coupling already has a flavour structure. However, this is not a newindependent structure: it is uniquely specified by the mass matrices, which also dictatethe subsequent rotation to the physical bases. So the interactions of neutral gauginos andthose of goldstinos are expected to have a similar impact on FCNC processes, once theparameter mapping is taken into account. Nevertheless, the fact that the goldstino is theGoldstone particle of spontaneously broken supersymmetry makes it quite special. Thepeculiar low-energy properties of goldstinos will especially emerge in our final section.

As regards the gaugino mass term in L0, one can effectively derive it from a superfieldcoupling of the form (/)ZWW (whereW is the gauge superfield strength), i.e., from alinear term in the gauge kinetic function. The associated interactions involving the physicalcomponents of the vector and goldstino superfields can be written in terms of M and F[20]:

M2FGF M4F z

(FF

+ iFF )+ h.c.+ , (5)

where F = 12F and the dots stand for terms which we will not need. Othersuperpotential and Khler potential terms generate kinetic, mass and interaction terms inthe Z sector [20]:

iSGG+ 12(SS m2SS2

)+ 12(PP m2PP 2

) 1

2

2F(m2SSGG im2PPGG+ h.c.

) m2S +m2P8F 2

GG SG SG+ . (6)

We have assumed for simplicity that the mass eigenstates in the sgoldstino sector coincidewith the real and imaginary parts of z= (S+ iP )/2. We will also assume that sgoldstinomasses are not much lighter than squark and slepton masses, as suggested by naturalnessconsiderations [23]. If this assumption is relaxed and sgoldstinos are allowed to be verylight, enhancement effects can appear in several processes (see, e.g., [2427]), includingsome FCNC processes to be discussed below. However, in such cases

F is typically


forced to be substantially larger than the electroweak scale, which is not the scenario wewould like to study. 7

3. Flavour changing radiative decays

In this section we will discuss flavour changing radiative decays. For definiteness, wefocus on the decay e , which violates individual lepton flavours. The effectiveoperator responsible for such a decay can be parametrized as

Leff = ige16pi2m(CRe

c +CLec)F. (7)

This leads to the branching ratio

BR( e )= 316piG2F

(|CR|2 + |CL|2)BR( ee), (8)where BR( ee)' 1.

The interactions involving goldstinos and sgoldstinos described in the previous sectiongenerate several one-loop contributions to CR and CL, through the diagrams schematicallyshown in Fig. 1. We disregard type (d) diagrams, since they are quadratic in m2LR and weare only working at first order in the muon mass. From the other diagrams, we obtain:

C(G)R =

12F 2

[1

6m2LL +

(m2M2

m2 M2(

1 M2

m2 M2 logm2

M2

))LL

+(

logm2Pm2S

)M

mm2LR

]e

, (9)

C(G)L =

12F 2

[1

6m2RR +

(m2M2

m2 M2(

1 M2

m2 M2 logm2

M2

))RR

+(

logm2Pm2S

)M

mm2RL

]e

, (10)

where m2 stands for the 66 slepton mass matrix, whose 33 blocks are m2LL, m2RR, m2LR,m2RL. These results generalize to flavour-changing transitions those obtained for diagonalmagnetic moments in the absence of flavour mixing [29,30,27]. The above expressionshold in the superfield basis in which leptons are mass eigenstates. The computation canbe performed by using matrix vertices and propagators in that basis. Alternatively, onecan diagonalize the slepton mass matrix as well and move back to the other basis in theend. The three terms in each expression originate from diagrams of type (a), type (b) andtype (c), respectively. Notice that type (a) contributions are simply proportional to the eelement of the matrices m2LL and m2RR. The latter result holds for arbitrary m2: it does notrely on any assumption on the size of the off-diagonal entries of m2LL and m2RR, or even onthe assumption that the entries of m2RL are linear in lepton masses. The matrix structure of

7 Collider signals of massive sgoldstinos in the case of lowF have been recently analysed in [28].


Fig. 1. Diagrams with goldstino or sgoldstino exchange contributing to e .

vertices and propagators just combine in the proper way, and the final result turns out tocoincide with what we would have obtained in the simple mass insertion approximation. 8Type (c) contributions are proportional to the e elements of m2LR and m2RL, but here thereason is more obvious. On the other hand, type (b) contributions are expressed througha non-trivial function of the matrix m2. In this case, in order to obtain an approximateexpression, we can expand m2 around the diagonal 9 and work to first order in the flavourchanging elements of m2, now assumed to be small. This corresponds to the mass-insertionapproximation and allows us to cast type (b) contributions in a form similar to the otherones. Under this approximation, we can rewrite C(G)R and C

(G)L as

C(G)R =

1F 2

[H1

(M2

m2

)(m2LL)e +

12

(log

m2Pm2S

)M(m2LR)e

m

], (11)

C(G)L =

1F 2

[H1

(M2

m2

)(m2RR)e +

12

(log

m2Pm2S

)M(m2RL)e

m

], (12)

where

H1(x)= 1+ 3x 15x2 + 13x3 6x2(1+ x) logx12(1 x)3 . (13)

The function H1(x) is negative for x < 1 and positive for x > 1. Notice that H1(1)= 0:when m2 =M2, type (a) and type (b) contributions cancel each other, to linear order in theflavour changing masses.

We would like to compare the above contributions, which we will simply call goldstinocontributions, to the more conventional non-goldstino contributions. In the full MSSM,the latter ones arise from both charged and neutral interactions. The reference LagrangianL0 only gives neutral contributions, from type (a) diagrams in which the goldstino isreplaced by a photino. In contrast to the goldstino, however, the photino propagatorcan either conserve or flip chirality, so several contributions arise. In the mass-insertionapproximation, we find (in agreement with [31,9]):

8 Incidentally, we remark that even the mass insertion method has new features in the present context. For type(a) diagrams, for instance, the flavour violating factor (m2LL)e (or (m2RR)e) can be inserted in either a sleptonpropagator or a leptonsleptongoldstino vertex. Moreover, the vertex contributions are twice as large as thepropagator contributions and have opposite sign.

9 For simplicity, we will consider the diagonal entries of m2 to have a common value m2. The generalizationis straightforward.


C(0)R =

2g2em4

[(H2

(M2

m2

)+H3

(M2

m2

)M(m2LR)

m2m

)(m2LL)e

H4(M2

m2

)M(m2LR)e

m

], (14)

C(0)L =

2g2em4

[(H2

(M2

m2

)+H3

(M2

m2

)M(m2RL)

m2m

)(m2RR)e

H4(M2

m2

)M(m2RL)e

m

], (15)

where the loop functions H2(x),H3(x),H4(x) are

H2(x)= 1 9x 9x2 + 17x3 6x2(3+ x) logx

12(1 x)5 , (16)

H3(x)= 1+ 9x 9x2 x3 + 6x(1+ x) logx

2(1 x)5 , (17)

H4(x)= 1+ 4x 5x2 + 2x(2+ x) logx

2(1 x)4 . (18)Goldstino and non-goldstino contributions exhibit a similar structure. They are propor-

tional to the flavour changing (e) elements of the slepton mass matrix, depend on dimen-sionless functions of superpartner masses and are suppressed by the fourth power of somescale. This scale is the supersymmetry breaking scale for the goldstino contributions anda supersymmetry breaking mass (e.g., the average slepton mass m) for the non-goldstinocontributions. If

F is much larger than the supersymmetry breaking masses, the gold-

stino contributions are negligible in comparison to the non-goldstino ones. On the otherhand, if

F and the supersymmetry breaking masses have a similar size, then goldstino

and non-goldstino diagrams give similar contributions to e . It is interesting to makea more quantitative comparison. For definiteness, we neglect LR terms (both flavour con-serving and flavour changing ones) and focus on the contributions proportional to (m2LL)e,i.e., we consider

C(G)R =

1F 2H1

(M2

m2

)(m2LL

)e, C

(0)R =

2g2em4

H2

(M2

m2

)(m2LL

)e. (19)

We recall that H1(x) can have either sign, whereas H2(x) is positive. To measure therelative importance of the goldstino contributions versus the non-goldstino ones, weintroduce the ratio

R = C(G)R

C(0)R

= m4

2g2eF 2H1(M2/m2)

H2(M2/m2), (20)

which does not depend on (m2LL)e. In the limit m M , for instance, that ratio isR = m4/(2g2eF 2), which becomes 1 when m ' 0.65

F . Contours of R in the

(m/F,M/

F ) plane are shown in Fig. 2. Goldstino contributions are smaller (larger)

than the other ones in the region with |R| < 1 (|R| > 1). We can also combine the twoclasses of contributions and study BR( e ) as a function of m, M , F and (LL)e


Fig. 2. The ratio of goldstino versus non-goldstino contributions to the e amplitude in the(m/F,M/

F) plane.

Fig. 3. Lines of constant BR( e ) = 1.2 1011 in the (F, |(LL)e|) plane, for differentchoices of (m,M).


(m2LL)e/m2. In Fig. 3 we have fixed some representative values of (m,M) and shown the

lines in the (F, |(LL)e|) plane along which BR( e ) saturates the present exper-

imental bound, which is 1.2 1011 [32]. In other words, the lines give the upper boundon |(LL)e| as a function of

F . When

F is large such bounds are determined by the

(conventional) non-goldstino contributions. When F decreases, the latter contributionsstart to interfere with the goldstino ones: for m > M the interference is destructive, thebound on |(LL)e| tends to disappear and the curves exhibit a peak, whereas for m < Mthe interference is constructive and the curves show a knee. For even smaller values ofF , i.e., to the left of the transition region, the goldstino contributions dominate and the

bounds on |(LL)e| become stronger than the conventional ones. In the limit mM , forinstance, the bound from goldstino contributions alone can be written as 10

(LL)e. 2 103(Fm

)4(m

300 GeV

)2. (21)

An identical discussion applies to the contributions that depend on (RR)e.It is straightforward to translate the above discussion to the decays e and ,

whose branching ratios are experimentally bounded by 2.7 106 [33] and 1.1 106[34], respectively. Notice that Fig. 2 applies to these cases as well, whereas in Fig. 3 onlythe scale of the vertical axis has to be changed: |(LL)e| has to be replaced by either103|(LL)e | or 1.4 103|(LL) |. Therefore, the qualitative description remains thesame as before, although the constraints on |(LL)e | and |(LL) | are of course muchweaker.

The above discussion can also be extended to flavour changing transitions in the quarksector, such as b s . This decay is potentially sensitive to the bs entries of the downsquark mass matrix, through both non-goldstino and goldstino contributions. The formerones are mainly due to squark-gluino (rather than squarkphotino) exchange [35,9]. Thelatter ones become comparable to those when m2/F gs , where gs is the strong couplingconstant and m is an average squark mass. However, neither contribution gives significantconstraints.

4. Flavour changing processes with four matter fermions

We now discuss the FCNC processes that involve four matter fermions as external states.In both the SM and the MSSM, the leading perturbative contributions to such processesgenerically arise at one-loop level, and are finite. If goldstino and sgoldstino couplings arealso taken into account, additional contributions arise.

10 Incidentally, we recall that perturbativity considerations require that the ratio m/F be smaller than 2 3

[20,23]: this should be understood everywhere. We also notice that the inequality (21) can be equivalently writtenas (m2LL)e . (13 GeV)

2(F/300 GeV)4, i.e., m2 drops out. This example shows that, in the case of goldstino

contributions, the parametrization in terms of (LL)e and m2 may be redundant. We adopt it to allow for aneasier comparison with the literature.


Some contributions arise already at tree level. Indeed, when two zff c vertices are joinedby a sgoldstino propagator, effective four fermion interactions are generated:

Leff = 14F 2[

1m2S

(f

f cm2RLf + h.c.)2+ 1m2P

(if

f cm2RLf + h.c.)2]

. (22)

However, owing to the assumed chiral suppression of m2RL (and the assumed sizeof sgoldstino masses), the coefficients of these four fermion operators are at mostO(m2f /F 2) . O(m2f G2F ), i.e., they are automatically suppressed by fermion masses. 11On the other hand, if the above mentioned assumptions are relaxed, enhancement effectscan appear: in this case, the required suppression should be provided by large values ofF and/or intrinsically small flavour violation in m2RL.We would like to discuss contributions to four fermion processes that are not chirally

suppressed, so we neglect m2RL and move to one-loop level. In analogy to the SM or theMSSM, several one-loop diagrams contribute, both 1PI (e.g., boxes) and 1PR (e.g., pen-guins). Some 1PI diagrams are skecthed in Fig. 4. Diagram (a) is a non-goldstino MSSMdiagram, which we show for comparison: it is a typical gauginosfermion box. By re-placing gauginofermionsfermion vertices with goldstinofermionsfermion vertices, weobtain box diagrams like (b) and (c). All these boxes give finite contributions. 12 In our su-persymmetric effective Lagrangian, however, goldstinos (and sgoldstinos) also couple tomatter through non-renormalizable couplings. For example, the theory contains quartic in-teractions of dimension six, like those in Eq. (4), whose coefficients are again determinedby sfermion masses and

F . These interactions cannot be simply dropped, since they play

a crucial role in the low-energy cancellations that take place in diagrams with externalgoldstinos (see next section). Such vertices can be used to build new (non box) 1PI di-agrams, like (d), (e), (f) in Fig. 4. These diagrams are not finite: the dependence on thecutoff scale is logarithmic for diagram (d) and quadratic for diagrams (e) and (f). Thus,in contrast to what happens in the SM or the MSSM, flavour changing four fermion in-teractions receive both finite and divergent contributions in the effective theory consideredhere. In principle, such divergences could be reabsorbed by introducing other terms in thesupersymmetric effective Lagrangian, such as (for instance) Khler potential terms quarticin the matter superfields. These new terms would not only act as counterterms, but alsogenerate new contributions. Strictly speaking, all this means that the coefficients of fourfermion interactions cannot be predicted in terms of the parameters already introduced. 13However, we can also adopt the milder point of view that the one-loop diagrams generated

11 For a similar reason, other interactions due to tree-level sgoldstino exchange are also suppressed. For instance,by connecting a zff c vertex with a z vertex, we obtain the effective (two-fermion)(two-photon) coupling(M/(4F 2))(

f f

cm2RLf )((1/m2S)FF

(i/m2P )FF) + h.c., which could contribute, e.g., to thedecay e .12 For instance, the coefficients of F = 1 four fermion operators induced by (a), (b) and (c) scale as(g4/m4)m2

ij, (g2/F 2)m2

ijand (m4/F 4)m2

ij, respectively, where m2

ijis the appropriate flavour changing

sfermion mass.13 The situation was slightly different in the computation of e presented in the previous section. In that

case the operator was different, and our choice to only focus on couplings related to the spectrum led us to obtaina finite result. If we had included other terms in the Khler potential or in the gauge kinetic function, however,


Fig. 4. Examples of one-particle-irreducible contributions to flavour changing four fermion operators.The symbols f and f generically denote matter fermions and sfermions. Flavour changes can occurin sfermion propagators, cubic vertices or quartic vertices.

by the interactions originally introduced give naturalness estimates of the coefficients offour-fermion interactions, once the cutoff scale is specified. 14 We continue the discussionin this spirit and focus on the contributions generated by diagrams (e) and (f) in Fig. 4. Byretaining only the quadratic dependence on the cutoff scale , we obtain:

Leff = 164pi22

F 4

[f

(f m2LL

f f cm2RRf c)]

[f

(f m2LLf f cm2RRf c

)]. (23)

These four fermion terms can alternatively be extracted from the general formula K =(2/(16pi2)) log detKmn, which summarizes the quadratically divergent contribution ofchiral supermultiplets to the Khler potential [36].

We will use the effective Lagrangian (23) to estimate the effect of flavour changinggoldstino (or sgoldstino) interactions on F = 1 processes such as eee, K0 ` `,K pi` `, . . . , or F = 2 processes such as KSK transitions. As far as sfermion massmatrices are concerned, we will assume for simplicity that, in the fermion mass basis, theflavour diagonal entries in the LL and RR blocks have a common value m2. We stressagain that our results below should be regarded as indicative, not only because we arefocusing on a specific class of contributions, but also because the quadratic sensitivity onthe cutoff scale introduces a further uncertainty. Indeed, in the absence of informationon the underlying theory, could either be a scale just above m or take larger values, up

we would also have obtained logarithmically divergent contributions, associated to a (supersymmetric) higherderivative term. For a more detailed discussion in the context of the flavour conserving magnetic moments, see[27].14 For a similar discussion about this and the previous points in the context of flavour conserving interactions,

see [23].


to the scale where unitarity breaks down. Hence is expected to lie somewhere in therange [min,max], where 2min m2 and 2max 16piF 2/m2 16pi2 [20,23]. In theexamples below, we will specialize our formulae to the two extreme values of . We notethat, for the conservative choice m, quadratically divergent contributions have similarparameter dependence and size as the other contributions (i.e., logarithmically divergentand finite ones), so in this case the former ones can also be interpreted as representativesof the latter ones.

Consider for instance the terms in (23) that contribute to the lepton flavour violatingprocess ee+e:

Leff = 132pi2m22

F 4(e e ecec)

[(m2LL

)ee

(m2RR

)eec

c]. (24)

If we focus on the part proportional to (m2LL)e and neglect any other contribution to thisprocess, we obtain:

BR( ee+e)'[

6128pi2

m42

GFF 4(LL)e]2, (25)

where (LL)e = (m2LL)e/m2. Comparing the above expression with the experimentalupper bound 1012 [33] gives constraints on the parameters (LL)e, m and

F , for a

given . For the two extreme choices of , we obtain(LL)e. 5 104(Fm

)8( m300 GeV

)2(=min), (26)

(LL)e. 105(Fm

)4(m

300 GeV

)2(=max). (27)

If we take the smallest value of , the constraints on (LL)e are comparable or strongerthan those obtained in the previous section from the analysis of goldstino contributionsto e . This can be seen, for instance, by comparing Eq. (26) with Eq. (21) (alsonotice the different dependence on

F/m). If we take the largest possible value of , the

constraints on (LL)e are even stronger, for fixed values of the other parameters. We recallthat for large

F , i.e., when the non-goldstino contributions are the dominant ones, the

process e is more sensitive to (LL)e as compared to eee [31]. Here we havefound that when

F approaches m, i.e., when goldstino contributions become important,

the situation may be reversed. Similar considerations apply to (RR)e, of course.The lepton flavour violating decay pi0 e is another process that is sensitive to

(m2LL)e and (m2RR)e. This decay receives contributions from terms in (23) which couplea muon-electron current to up or down quark currents. The latter currents couple to thepion provided the up and down squark masses are non-degenerate. However, owing tothe rapidity of the dominant decay pi0 , no significant constraints are obtained on(LL)e and (RR)e. Another process potentially sensitive to the latter quantities is the e conversion on nuclei. The effective interactions in (23) also contribute to otherlepton flavour violating processes, such as decays into either three charged leptons or acharged lepton and a pi0. These processes are sensitive to flavour changing slepton massesinvolving the third generation, but no strong constraints are obtained.


In the final part of this section, we consider some examples of flavour violation inthe quark sector induced by the four-fermion terms in (23). In particular, we focus ontwo processes that are sensitive to the sd entries of the squark mass matrix, i.e., thedecay KL + (S = 1) and KK transitions (S = 2). Since we are dealing withorder-of-magnitude estimates, we neglect QCD corrections and use the vacuum insertionapproximation. The terms in (23) that contribute to the decay KL + are 15

Leff = 132pi2m22

F 4( cc)[(m2LL)dsds (m2RR)dsdcsc + h.c.].

(28)If we only consider the part proportional to (m2LL)ds , we obtain:

BR(KL+)' [

232pi2

m42

sin cGFF 4Re(LL)ds

]2, (29)

where (LL)ds = (m2LL)ds/m2, BR(K+ +) (KL)/(K+) ' 2.7 and c is theCabibbo angle. By imposing that the value of the above expression does not exceed theobserved value 7 109 [33], we can obtain combined constraints on |Re(LL)ds |, m andF , for a given. For the two extreme choices of , we obtainRe(LL)ds. 3 103(F

m

)8(m

300 GeV

)2(=min), (30)

Re(LL)ds. 6 105(Fm

)4(m

300 GeV

)2(=max). (31)

We now consider the terms in (23) that contribute to KK transitions:

Leff = 164pi22

F 4

[(m2LL

)dsd s (m2RR)dsdcsc]

[(m2LL

)dsds

(m2RR

)dsdcs

c]+ h.c. (32)

Retaining again only the LL contributions, we obtain:mKmK' 196pi2 f 2Km42F 4 Re(LL)2ds, (33)

where fK ' 160 MeV is the kaon decay constant. We can again find constraints onthe parameters by imposing that the value of the above expression does not exceed theexperimental value 7 1015. For the two extreme choices of , we obtain

|Re(LL)2ds |. 5 103(

F

m

)4(m

300 GeV

)(=min), (34)

|Re(LL)2ds |. 7 104(

F

m

)2(m

300 GeV

)(=max). (35)

15 These terms or analogous ones (with the muon replaced by another lepton or a quark) also contribute to otherS = 1 decays, such as K pi` ` or K pipi (hence ).


WhenF m, the above Eqs. (30), (31) and (34), (35) do not give significant bounds

on (LL)ds , which is instead constrained by the non-goldstino contributions. We recall that,in this case, the strongest bound comes from mK rather than KL +, since onlythe former quantity receives significant contributions from diagrams with gluino exchange.When

F approaches m, on the other hand, the goldstino contributions become more and

more relevant and the limit on (LL)ds obtained from KL + can be comparable oreven more stringent than that from mK .

The quantity (RR)ds is constrained in the same way as (LL)ds . The bounds frommK on the combination

|Re[(LL)ds(RR)ds]| are slightly more stringent than thosein Eqs. (34), (35). The S = 2 Lagrangian in Eq. (32) also contributes to the CPviolating parameter K . The resulting bounds on

|Im(LL)2ds |,

|Im(RR)2ds | and|Im[(LL)ds(RR)ds]| are about an order of magnitude smaller than those of the

corresponding real parts.Flavour violating processes involving B (D) mesons can be discussed along similar

lines. The bounds on (LL)db ((LL)uc) from BdBd (DD) mixing, for instance, areslightly weaker than the corresponding ones in theKK system. The effective Lagrangian(23) also contributes to flavour changing processes involving external top quarks, if theappropriate entries in m2LL or m2RR are non-vanishing. Moreover, the latter processes canalso be sensitive to the off-diagonal entries of m2RL related to the top, since the chiralsuppression is less effective. In this respect, even the effective interactions due to tree-levelsgoldstino exchange (see Eq. (22) and related paragraph) can play a role in decays such ast cf f or t c .

5. Flavour changing processes with external goldstinos

Up to now we have discussed flavour changing processes that have photons and ordinaryfermions (leptons and quarks) as external states, with goldstinos, sgoldstinos, sleptons,squarks and photinos present in internal lines only. Now we will consider the possibilitythat the external states also include goldstinos, but (again) not the other superpartners,which we integrate out. In particular, we would like to discuss whether flavour violationscan occur in low-energy processes involving two ordinary fermions and two goldstinos,i.e., whether transitions such as eGG or s dGG can take place.

We first study what happens when sfermions and sgoldstinos are integrated out at tree-level. Using the masses and couplings described in Section 2, we find that three types ofdiagrams contribute (see Fig. 5). We recall that vertices and sfermion propagators havea non-trivial flavour structure. However, such structures combine in a characteristic wayif we expand the scalar propagators around the heavy (supersymmetry breaking) scalarmasses and treat momenta and fermion masses (i.e., 2 and mm terms) as perturbations.Indeed, we obtain: 16

16 This holds for each fermion species f , of course: we omit the sumf for brevity. Notice that generation

(i.e., flavour) indices are still understood.


Fig. 5. Tree-level diagrams contributing to effective interactions between two goldstinos and twomatter fermions. The symbols f and f generically denote matter fermions and sfermions.

(a)H 1F 2

( SGf Gf c )( m2LL m2LRm2RL m

2RR

)(f G

f cSG

)+ , (36)

(b)H 1F 2

( SGf Gf c )( 0 m2LRm2RL 0

)(f G

f cSG

)+ , (37)

(c)H 1F 2

( SGf Gf c )( m2LL 00 m2RR)(

f G

f cSG

), (38)

where the dots denote terms suppressed by powers of momenta or fermion masses. Weimmediately see that, once we sum the leading terms from sfermion (a) and sgoldstino (b)exchange with the contact term (c), a complete cancellation takes place, as it should. Thefirst nonvanishing contributions arise at the next order in the expansion, and are quadraticin momenta or fermion masses. In particular, from sfermion diagrams we obtain

(a)H 1F 2

[SGf (2+mm)f G+ Gf c(2+mm)f cSG]= 2

F 2(SGf )(f G)+ (f f c), (39)

where we have used the equations of motion to write the second expression. Contributionsquadratic in momenta also come from sgoldstino diagrams (b). However, these operatorsalso contain factors like m2RL/m2S or m

2RL/m

2P , so they can be considered of higher order,

under the assumption that m2RL is chirally suppressed 17 (i.e., linear in fermion masses).So, according to this procedure, the leading non-vanishing interaction between on-shellgoldstinos and matter fermions is the dimension 8 operator in Eq. (39). Notice that thisoperator is manifestly flavour universal and does not depend on the superpartner spectrum.In particular, no trace remains of the flavour structure of sfermion mass matrices.

This result generalizes that obtained in the one-flavour case by a similar procedure [20,21]. On the other hand, the effective low-energy interactions between goldstinos and matterfermions can also be obtained by other methods, e.g., by direct non-linear realizations ofsupersymmetry [37,38], which do not require the explicit introduction of superpartners.In such frameworks, a more general result can be obtained. In the one-flavour case,

17 We recall that such a suppression could be rather mild if the top quark is involved. However, here (as before)we are mainly interested in low-energy processes, where only light fermions are involved.


for instance, it was recently shown [21,22] that the on-shell interactions between twogoldstinos and two f -type fermions are described by two independent operators:

Lnl = 1F 2

[(SGG)(f f ) (f G)C(f )(SGf )], (40)where C(f ) is an arbitrary dimensionless coefficient. 18 This result can be easilygeneralized to the multi-flavour case: we only need to reinterpret f as a collection offermions fi and the coefficient C(f ) as a matrix in flavour space, C(f ) = C(f )ij . Theoperators for the fermions f c are analogous, with a matrix C(f c). We recall that thefirst operator in (40) corresponds to the standard coupling of goldstinos to the energy-momentum tensor [37,38]. Notice that it is flavour universal, but differs from the operatorin Eq. (39). Flavour violations can only come from the second operator in (40), if thematrix C(f ) is not diagonal in the fermion mass basis. Before discussing this possibility,it is convenient to write Eq. (40) in two additional equivalent forms. Using Fierzrearrangements and the goldstino equations of motion, we obtain:

Lnl = 2F 2

[(SGf )(f G)+ (SGf )(1C(f ))(f G)] (41)= 2

F 2

[(SGf )(f G)+ 14 (f (1C(f )) f )2(SGG)], (42)where 1 is the unity matrix in flavour space. In particular, by comparing Eq. (41) withthe interaction found by integrating out superpartners, Eq. (39), we can see that the lattercorresponds to the special case C(f ) = 1. In the one-flavour case, an analogous result wasobtained in [21] by explicit computation and comparison of scattering amplitudes.

We have seen that the non-linear formulation of spontaneously broken supersymmetryallows for a generic, flavour non-universal matrix C(f ). On the other hand, integratingout superpartners from an effective theory with linearly realized supersymmetry has ledus to find a specific, universal C(f ). Therefore, we can wonder whether a non-universalC(f ) could emerge also in the effective linear approach, by generalizing the decouplingprocedure and/or the theory itself. We will now mention a few such possibilities.

(i) One possibility could be to keep using the structure described in Section 2, andthen perform the decoupling of superpartners at one-loop level, rather than at tree-level.Since the full computation is quite involved, we could first focus on the quadraticallydivergent contributions only. If we do this, however, we find that the final result still hasthe form (39), which corresponds to a universal C(f ). This follows from the fact that theinclusion of such corrections amounts to use a corrected Khler potential. Thus, oncethe theory is expressed in terms of one-loop corrected fields, masses and couplings, thedecoupling procedure for the interactions under study is formally similar to the tree-levelone. However, this argument does not necessarily hold for logarithmically divergent andfinite corrections, where the flavour structure of sfermion mass matrices might survive andlead to non-universal contributions to C(f ). Notice that the superspace interpretation ofthose corrections corresponds to both Khler and non-Khler (i.e., higher derivative) terms.

18 The normalization we have chosen is such that C(f ) = 14 [21] = 12Cff [22].


These considerations also suggest a different (alternative) approach, in which computationsare done at the tree level, by starting however from an effective Lagrangian which alreadycontains higher derivative terms, besides the Khlerian ones.

(ii) Another possibility could be to relax the assumption of pure F -type supersymmetrybreaking used so far. We can consider a more general situation, with larger field contentand gauge structure, such that non-vanishing auxiliary component vevs appear in bothchiral and vector supermultiplets (mixed FD breaking). In this case, the goldstino is alinear combination of the fermions in those multiplets, and the sgoldstino sector includesthe bosonic components of such multiplets, i.e., both spin-0 fields (like z above) and spin-1fields (see e.g., [3943]). Sfermion masses receive both F -type andD-type supersymmetrybreaking contributions. Once all this is taken into account, one can again integrate outsfermions and sgoldstinos at tree level and find the effective interactions between twomatter fermions and two goldstinos. We have checked that the leading terms again cancel,as they should (see also [4143]). At first non-vanishing order we find dimension 8operators as in Eqs. (41),(42). Diagrams with sfermion exchange again give a universalcontribution to C(f ) (the same as before). Diagrams with spin-1 sgoldstino exchange givean additional model dependent contribution to C(f ). The latter contribution depends, e.g.,on the coupling (charges) of matter fermions with spin-1 sgoldstinos. If these charges areneither universal nor aligned with fermion masses, flavour changing effects can arise.In this case, however, one should also keep under control the contributions of spin-1sgoldstino exchange to (dimension 6) flavour changing operators involving four matterfermions.

Exploring in more detail the possibilities mentioned above, or other ones, lies beyondthe scope of the present paper. For the rest of our discussion, we will rely on the factthat supersymmetry in principle allows for the existence of effective (two goldstino)(twomatter fermion) operators with non-diagonal matrices C(f ), and ask what this could implyfor phenomenology. We will see that the high dimensionality of the effective operatorsimplies by itself a strong suppression, so that even low values of

F and large flavour

violating entries in C(f ) are allowed. Consider for instance the charged lepton sector, andassume that Ce and/or Cecc are non-vanishing, so that the flavour changing decay e GG can take place. Although this decay is flavour changing, the final state is verysimilar to that of the flavour conserving decay ee, which proceeds at leadingorder through the standard Fermi interaction. The corresponding operators can be cast intosimilar forms (see Eq. (42)):

12F 2

[Ce(e

)Ccec (ecc)]2(SGG), (43)

2

2GF (e )(e). (44)The presence of two derivatives in the first operator, however, gives a strong suppression.Indeed, the ratio of the two decay rates scales as:

( eGG) ( ee)

(|Ce|2 + |Ccec |2) m4G2FF

4 . (45)


Even if we take F ' G1F and |Ce|2 + |Cecc |2 = O(1), the branching ratio istiny, O(1013). Although the features of the electron emitted with the goldstinopair (polarization, energy and angular distributions) differ from those of the standardchannel, the numerical suppression is so strong that detection seems impossible. Similarconsiderations apply if we compare the decays eGG and GG to thecorresponding standard ones. Even though m4 above is replaced by m4 , the ratiosanalogous to (45) are still tiny, e.g., O(109) if F 'G1F and the Cij are O(1).

We finally consider flavour changing transitions with goldstino pair emission in thequark sector. Consider for instance the operator

12F 2

[Csd

(s d

)Cdcsc(scdc)]2(SGG). (46)This operator does not contribute to K0 GG, but does contribute, e.g., to K+ pi+GG. This is similar to the decay K+ pi+ , which is itself very suppressed whencompared to its charged counterpart. For instance, in the SM one expects (see, e.g., [44])

(K+ pi+) (K+ pi0e+e) 10

9. (47)

For K+ pi+GG, the corresponding ratio scales as (K+ pi+GG) (K+ pi0e+e)

(|Csd |2 + |Cdcsc |2) m4KG2FF

4 . (48)

Even in the extreme case in which F 'G1F and |Cds | (or |Cscdc |) is O(1), the latter ratiois O(1011), which is smaller than the ratio in (47). The rates for the analogous B decays(B piGG, BKGG) are also smaller than the corresponding ones with neutrino pairemission. To weaken the effect of the low-energy suppression, we could move to higherenergies and consider the top quark. Operators analogous to those discussed above couldinduce, for example, the flavour changing decay t cGG. The corresponding rate wouldbe strongly enhanced by the presence of mt : if F ' G1F and |Ctc| (or |Ctccc |) is O(1),BR(t cGG) could reach values as large as O(102).

6. Summary

In this paper we have pointed out and discussed a new source of flavour violationin supersymmetric models, namely the couplings of goldstinos with matter. Sincethose couplings are strictly related to the mass spectrum and are suppressed by thesupersymmetry breaking scale

F , significant effects on FCNC processes can be obtained

when the following two ingredients are present:(i) The sfermion mass matrices have a non-trivial flavour structure in the fermion mass

basis;(ii) The supersymmetry breaking scale F is not much larger than the electroweak

scale.


Notice that condition (i) is the same feature that is responsible for the well-known flavourchanging effects induced by gaugino-matter couplings. In the latter case, the effects areenhanced when the supersymmetry breaking masses m are close to the electroweak scale:point (ii) expresses the analogous property for goldstino contributions. In other words, forgiven flavour violating sfermion mass matrices, goldstino and non-goldstino contributionsto FCNC become comparable when

F and m have a similar size.

These considerations especially apply to the usual class of low-energy FCNC processes,which involve photons, leptons and quarks in the external states. In this case, goldstinos(or sgoldstinos) can contribute as virtual particles, as well as sleptons, squarks andgauginos. In Section 3 and Section 4 we have examined the sensitivity of several suchprocesses to the value of

F and to the amount of flavour violation in the sfermion

masses (parametrized by the popular quantities ij ), also making comparisons with theconventional non-goldstino contributions. In Section 3 we have discussed the decay e as a prototype of flavour changing radiative decays. The analysis confirms that, whenF and m (or M) have a similar size, the contributions from goldstinosleptonphotino

exchange become comparable to those from sleptonphotino exchange and can interfereeither constructively or destructively (see Fig. 2). When goldstino contributions dominate,the bounds on e become stronger than the conventional ones, for given (m,M) (seeFig. 3). A similar picture has emerged from the analysis of flavour violating processeswith four external matter fermions, discussed in Section 4. In this case we have focusedon a representative class of goldstino (sgoldstino) contributions, which we have used toobtain order-of-magnitude estimates, taking also into account the uncertainty due to thecutoff scale . In this context we have first considered the decay eee. Again, whenF m the contributions due to the goldstino multiplet become comparable or even more

important than those from photinoslepton exchange. In this situation, moreover, the decay eee seems to have similar or even stronger sensitivity to e as compared to e ,contrary to what happens in the conventional scenario (which corresponds toF m). Inthe quark sector, we have discussed processes such as KL + and KK transitions.Again, the goldstino contributions can become dominant for low values of

F . In this

limit, the bounds on the parameters ds from KL + can be comparable or evenmore stringent than those from mK .

Finally, in Section 5, we have considered processes with two matter fermions and twogoldstinos as external states. The corresponding four-fermion operators have effectivedimension 8 rather than 6, due to the special low-energy properties of goldstinos. Whenwe have obtained such operators by integrating out heavy superpartners at tree-level,that feature has emerged because of mutual cancellations among the (otherwise leading)dimension 6 terms. By using this procedure, however, we have found that the resultingoperators do not exhibit any flavour structure. The latter result is by itself quite remarkable.On the other hand, it cannot be regarded as completely general. Indeed, we have notedthat the more general context of non-linearly realized supersymmetry in principle allowsfor operators with a non trivial flavour dependence. We have discussed how this resultmight be recovered in the linear approach, e.g., by considering the case of mixed FDsupersymmetry breaking rather than pure F -breaking. As regards the phenomenological


implications, we have seen that, even in the presence of flavour violating couplings,the high dimensionality of the effective operators automatically suppresses potentiallyinteresting transitions with goldstino pair-emission. For instance, even in the case ofmaximal flavour violation and

F (GF )1/2, meson decays such as K piGG have

smaller rates as compared to K pi .In conclusion, our general analysis shows that low-energy FCNC processes are sensitive

probes of supersymmetric scenarios that involve both a low supersymmetry breaking scaleand some amount of flavour violation in the sfermion sector. It would be very interestingto see how both features could emerge in concrete models at a more fundamental level.

Acknowledgements

We thank F. Feruglio, A. Masiero and F. Zwirner for discussions.

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Nuclear Physics B 587 (2000) 2544www.elsevier.nl/locate/npe

Exotic e conversion in nucleiand R-parity violating supersymmetry

Amand Faessler a, T.S. Kosmas b,, Sergey Kovalenko c,1, J.D. Vergados ba Institut fr Theoretische Physik der Universitt Tbingen, D-72076 Tbingen, Germany

b Division of Theoretical Physics, University of Ioannina, GR-45110 Ioannina, Greecec Departamento de Fsica, Universidad Tcnica Federico Santa Mara, Casilla 110-V, Valparaso, Chile

Abstract

The flavor violating e conversion in nuclei is studied within the minimal supersymmetricstandard model. We focus on the R-parity violating contributions at tree level including the trilinearand the bilinear terms in the superpotential as well as in the soft supersymmetry breaking sector.The nucleon and nuclear structure have consistently been taken into account in the expression ofthe e conversion branching ratio constructed in this framework. We have found that thecontribution of the strange quark sea of the nucleon is comparable with that of the valence quarks.From the available experimental data on e conversion in 48Ti and 208Pb and the expectedsensitivity of the MECO experiment for 27Al we have extracted new stringent limits on the R-parityviolating parameters. 2000 Elsevier Science B.V. All rights reserved.

PACS: 12.60.Jv; 11.30.Er; 11.30.Fs; 23.40.BwKeywords: Lepton flavor violation; Exotic e conversion in nuclei; Supersymmetry; R-parity violation; Muoncapture

1. Introduction

The lepton flavor violating process of neutrinoless muon-to-electron ( e) conversionin a nucleus, represented by

+ (A,Z) e + (A,Z), (1)is an exotic process very sensitive to a plethora of new-physics extensions of thestandard model (SM) [17]. In addition, experimentally it is accessible with incomparablesensitivity. Long time ago Marciano and Sanda [1,2] has proposed it as one of the best

Corresponding author.E-mail address: [email protected] (T.S. Kosmas).

1 On leave of absence from the Joint Institut for Nuclear Research, Dubna, Russia

0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0550-3213(00) 00 44 6- 6

26 A. Faessler et al. / Nuclear Physics B 587 (2000) 2544

probes to search for lepton flavor violation beyond the standard model. Recently, in viewof the indications for neutrino oscillations in super-Kamiokande, solar neutrino and LSNDdata, new hope has revived among the experimentalists of nuclear and particle physics todetect other signals for physics beyond the SM. A prominent probe in this spirit is thisexotic process (1).

The fact that the upper limits on the branching ratio of the e conversion relativeto the ordinary muon capture,

Re = ( e)/ ( ), (2)offer the lowest constraints compared to any purely leptonic rare process motivated a new e conversion experiment, the so-called MECO experiment at Brookhaven [811],which got recently scientific approval and is planned to start soon. The MECO experimentis going to use a new very intense beam and a new detector operating at the AlternatingGradient Synchrotron (AGS). The basic feature of this experiment is the use of a pulsed beam to significantly reduce the prompt background from pi and e contaminations.For technical reasons the MECO target has been chosen to be the light nucleus 27Al.Traditionally the e conversion process was searched by employing medium heavy(like 48Ti and 63Cu) [1214] or very heavy (like 208Pb and 197Au) [12,13,15,16] targets(for a historical review on such experiments see Ref. [17]). The best upper limits on Reset up to the present have been extracted at PSI by the SINDRUM II experiments resultingin the values

Re 6 6.1 1013, for 48Ti target [12,13], (3)Re 6 4.6 1011, for 208Pb target [15] (4)

(at 90% confidence level). The experimental sensitivity of the Brookhaven experiment isexpected to be roughly

Re 6 2 1017, for 27Al target [8,9], (5)i.e., three to four orders of magnitude below the existing experimental limits.

It is well known that the process (1) is a very good example of the interplay betweenparticle and nuclear physics attracting significant efforts from both sides. The underlyingnuclear physics of the e conversion has been comprehensively studied in Refs. [1722]. From the particle physics point of view, processes like e conversion, is forbiddenin the SM by muon and electron quantum number conservation. Therefore it has longbeen recognized as an important probe of the flavor changing neutral currents and possiblephysics beyond the SM [17].

On the particle physics side there are many mechanisms of the e conversionconstructed in the literature (see [7,2325] and references therein). All these mechanismsfall into two different categories: photonic and non-photonic. Mechanisms from differentcategories significantly differ from the point of view of the nucleon and nuclear structurecalculations. This stems from the fact that they proceed at different distances and, therefore,involve different details of the structure. The long-distance photonic mechanisms aremediated by the photon exchange between the quark and the e-lepton currents.

A. Faessler et al. / Nuclear Physics B 587 (2000) 2544 27

These mechanisms resort to the lepton-flavor non-diagonal electromagnetic vertex which ispresumably induced by the non-standard model physics at the loop level. The contributionsto the e conversion via virtual photon exchange exist in all models which allow e decay. The short-distance non-photonic mechanisms contain heavy particles inintermediate states and can be realized at the tree level, at the 1-loop level or at the level ofbox diagrams.

The non-photonic mechanisms are mediated by various particles in intermediate statessuch as W,Z-bosons [17], Higgs bosons [3,4,7], supersymmetric particles (squarks,sleptons, gauginos, higgsinos etc.) with and without R-parity conservation in the vertices.

In the supersymmetric (SUSY) extensions of the SM with conserved R-parity (RpSUSY) e conversion has been studied in Refs. [22,26,27]. In this case the SUSYcontributions appear only at the loop or box level and, therefore, they are suppressed by theloop factor. The situation is different in the SUSY models with R-parity non-conservation(/Rp SUSY). In this framework there exist the tree level non-photonic contributions [23,25]and the 1-loop photonic contributions significantly enhanced by the large logarithms [24].

The primary purpose of this work is to offer a theoretical background for the running andplannede conversion experiments. We consider all the possible tree level contributionsto e conversion in the framework of the minimal SUSY model with most general formof R-parity violation (/Rp MSSM) including the trilinear /Rp couplings and the bilinear/Rp lepton-Higgs terms. We also examine some non-SUSY and Rp SUSY mechanismspreviously studied in [1922,26,27]. We develop a formalism of calculating the econversion rate for the quark level Lagrangian with all these terms.

In our study we pay special attention to the effect of nucleon and nuclear structuredependence of the e conversion branching ratio Re . In particular, we take intoaccount the contribution of the strange nucleon sea which, as we will see, gives acontribution comparable to the contribution of the valence quarks of a nucleon.

Thus, we apply our formalism to the case of nuclei 48Ti and 208Pb by calculatingnumerically the muonnucleus overlap integral and solving the Dirac equations withmodern neural networks techniques and using the PSI experimental data. A similarapplication is done for the 27Al target by employing the sensitivity of the designed MECOexperiment.

Our final goal is to derive on this theoretical basis the experimental constraints on theRp Yukawa couplings, the lepton-Higgs mixing parameters and on the sneutrino VEVs.Towards this end we use the experimental upper limits on the branching ratio Re givenabove and derive the new stringent constraints on the R-parity violating parameters.

2. The effective e conversion Lagrangian

It is well known that, the lepton flavors (Li) and the total baryon (B) number areconserved by the standard model interactions in all orders of perturbation theory. Asmentioned above, this is an accidental consequence of the SM field content and gaugeinvariance. Thus e conversion is forbidden in the SM. In contrast to Li the individual


quark flavors Bi are not conserved in the SM by the charged current interactions dueto the presence of non-trivial CabibboKobayashiMaskawa (CKM) mixing matrix. Liare conserved since the analogous mixing matrix can be rotated away by the neutrinofields redefinition. The latter is possible while the neutrino fields have no mass term and,therefore, are defined up to an arbitrary unitary rotation. As soon as the model is extendedby inclusion of the right-handed neutrinos lepton flavor violation can occur since neutrinosmay acquire a non-trivial mass matrix.

In the supersymmetric extensions of SM Li and B conservation laws are in generalviolated. As a result potentially dangerous total lepton (/L) and baryon (/B) number violatingprocesses become open.

One may easily eliminate /L and /B interactions from a SUSY model by introducing adiscrete symmetry known as R-parity. This is a multiplicative Z2 symmetry defined asRp = (1)3B+L+2S , where S is the spin quantum number. In this framework neutrinos aremassless. However the flavor violation in the lepton sector can occur at the 1-loop level viathe Li -violation in the slepton sector. Thus, e conversion is allowed in the SUSYmodels with R-parity conservation.

There is no as yet convincing theoretical motivation for R-symmetry of the low energyLagrangian and, therefore, SUSY models with (Rp SUSY) and without (/Rp SUSY) R-parity conservation are a priori both plausible.

Despite the above mentioned problems with /L, /B interactions /Rp SUSY looks ratherattractive, since it may offer a clue to the solution of some long standing problems ofparticle physics, such as neutrino mass problem. In the /Rp SUSY framework neutrinosacquire Majorana masses at the weak-scale via mixing with the gauginos and higgsinosas well as via /L loop effects [2834]. Furthermore, /Rp SUSY models admit non-trivialcontributions to the lepton flavor violating processes. During the last few years the/Rp SUSY models have been extensively studied in the literature (for a recent review seeRefs. [3538]).

We analyze possible mechanisms for process (1) existing at the tree level in the minimal/Rp SUSY model with a most general form of R-parity violation.

A most general gauge invariant form of the R-parity violating part of the superpotentialat the level of renormalizable operators reads

W6Rp = ijkLiLjEck + ijkLiQjDck +jLjH2 + ijkUci DcjDck, (6)where L, Q stand for lepton and quark doublet left-handed superfields while Ec, Uc,Dc for lepton and up, down quark singlet superfields; H1 and H2 are the Higgs doubletsuperfields with a weak hypercharge Y = 1,+1, respectively. Summation over thegenerations is implied. The coupling constants () are antisymmetric in the first (last)two indices. The bar sign in , denotes that all the definitions are given in the gaugebasis for the quark fields. Later on we will change to the mass basis and drop the bars.Henceforth we set = 0 which are irrelevant for our consideration. This conditionensures the proton stability and can be guaranteed by special discreet symmetries otherthan R-parity.


The R-parity non-conservation brings into the SUSY phenomenology the lepton number(/L) and lepton flavor (/Li ) violation originating from the two different sources. One is givenby the /L trilinear couplings in the superpotential W6Rp of Eq. (6). Another is related tothe bilinear terms in W6Rp and in soft SUSY breaking sector. Presence of these bilinearterms leads to the terms linear in the sneutrino fields i in the scalar potential. The linearterms drive these fields to non-zero vacuum expectation values i 6= 0 at the minimumof the scalar potential. At this ground state the MSSM vertices Z and We produce thegauginolepton mixing mass terms Z, W e (with W , Z being wino and zino fields).These terms taken along with the leptonhiggsinoiLiH1 mixing from the superpotentialof Eq. (6) form 7 7 neutral fermion and 5 5 charged fermion mass matrices. For theconsidered case of e conversion the only charged fermion mixing is essential. Thecharged fermion mass term takes the form

L()mass = T()M (+) H.c. (7)in the basis of two component Weyl spinors corresponding to the weak eigenstate fields

T() =(eL ,

L,

L ,i, H1

), T(+) =

(e+L ,

+L,

+L ,i+, H+2

). (8)

Here are the SU2L gauginos while the higgsinos are denoted as H1,2. These fields arerelated to the mass eigenstate fields () by the rotation

()i =ij ()j . (9)The unitary mixing matrices diagonalize the chargino-charged lepton mass matrix as

()M(+) =Diag{m(l)i ,mk

}, (10)

where m(l)i and mk are the physical charged lepton and chargino masses. In the presentpaper we use the notations of Refs. [3941].

Rotating the MSSM Lagrangian to the mass eigenstate basis according to Eq. (9) oneobtains the new lepton number and lepton flavor violating interactions in addition to thosewhich are present in the superpotential Eq. (6). Note that the mixing between the chargedleptons (e+L ,

+L) and the chargino components (i+, H+2 ), described by the off diagonal

blocks of the +, is proportional to the small factor me,/MSUSY [3941] and is,therefore, neglected in our analysis.

Let us write down the MSSM terms generating by the rotation (9) the new lepton flavorviolating interactions relevant for the e conversion. In the two component form theycan be written as [42,43]

LMSSM = g22 cosW Z S iAij j + ig2uLdL, (11)

whereAij = (12 sin2 W )ij + i44j . Rotating this equation to the mass eigenstate basiswe write down in the four-component Dirac notation

LMSSM = g22 cosW aZZePL+ g2i ukPReci dLk. (12)

Here ei = (e,, ), PL,R = (1 5)/2. The lepton flavor violating parameters in thisformula are given by


aZ =1424 1121, i =i4 i1. (13)The approximate expressions for these parameters were found by the method of Refs. [39,40] expanding the mixing matrix in the small matrix parameter

i1 =g2(i H1i)2 (M2 sin 2M2W)

, (14)

where ,M2 are the ordinary MSSM mass parameters from the superpotential termH1H2 and from the SU2 gaugino soft mass term M2WkW k . The MSSM mixing angle isdefined as tan = H2/H1. The other lepton flavor violating interaction contributing tothe e conversion come from the superpotential (6).

The leading diagrams describing possible tree level /Rp MSSM contributions to theeconversion are presented in Fig. 1. The vertex operators encountered in these diagrams are

Le = 2i21Li ePL+ ijj Li dj PLdjijkVnj

(uLneiPRdk + dRk unPR eci

)+ g2iVnk unPReci dLk+1

2g2

cosWaZZ

ePLg2

cos WZq(LPL + RPR )q, (15)

where the first three terms originate from the superpotential (6), the fourth and fifth termscorrespond to the chargino-charged lepton mixing terms in Eq. (12) and the last one is theordinary SM neutral current interaction.

In Eq. (15) the SM neutral current parameters are defined as usualL = T3 sin2 WQ, R = sin2 WQ

with T3 and Q being the 3rd component of the weak isospin and the electric charge.

Fig. 1. Leading /Rp MSSM diagrams contributing to e conversion at the tree level. (i) Theupper diagrams originate from the trilinear , terms in the superpotential Eq. (6). (ii) The lowerdiagrams originate from the chargino-charged lepton mixing schematically denoted by crosses (X)on the lepton lines.


The Lagrangian (15) is given in the quark mass eigenstate basis which is related to theflavor basis q0 through

qL,R = V qL,R q0.For convenience we introduced the new couplings

ijk = imn(V dL)jm

(V dR)kn.

The CKM matrix is defined in the standard way as V = V uLV dL .Integrating out the heavy fields from the diagrams in Fig. 1 and carrying out Fierz

reshuffling we obtain the 4-fermion effective Lagrangian which describes the econversion at the quark level in the first order of perturbation theory. It takes the form

Lqeff =GF

2j[uiL J

uL(i) + uiR JuR(i) + diL JdL(i) + diR JdR(i)

]+ GF

2[diR JdR(i)jL + diL JdL(i)jR

]. (16)

The index i denotes generation so that ui = u, c, t and di = d, s, b. The coefficients accumulate dependence on the /Rp SUSY parameters in the form

uiL =12

l,m,n

2ln1mnGF m

2dR(n)

V il Vim + 2(

1 43

sin2 W)aZ

+4n

12M2Wm2dL(n)

|Vin|2,

diR =12

l,m,n

2mi1liGF m

2uL(n)

V nmVnl +43

sin2 WaZ,

diL =2(

1 23

sin2 W)aZ,

uiR =

83

sin2 WaZ,

diL =

2n

niin12GFm

2(n)

, diR =

2n

niin21GF m

2(n)

. (17)

Here mq(n), m(n) are the squark and sneutrino masses. In Eq. (16) we introduced the colorsinglet currents

J

qL/R(i)= qi PL/Rqi, JdL/R(i) = diPL/R di,

j = e , jL/R= eP

L/R, (18)

where qi = (ui, di).Since in the next sections we report the new results for the nuclear matrix elements

of 48Ti, 27Al and 208Pb we are going also to update e constraints on theeffective lepton flavor violating parameters corresponding to certain non-SUSY andRp SUSY mechanisms shown in Fig. 2. These mechanisms were previously studied inRefs. [22,26,27]. Let us shortly summarize these mechanisms for completeness. Thelong-distance photonic mechanisms mediated by the photon exchange between the quark


Fig. 2. Photonic and non-photonic mechanisms of the e conversion within some extensionsof the standard model occuring by introducing massive neutrinos (ac), as well as supersymmetricextensions (d), (e) with R-parity conservation.

and the e-lepton currents is realized at the 1-loop level as the W loop [Fig.2(a)] with the massive neutrinos i and the loop with the supersymmetric particlessuch as the neutralino(chargino)slepton(sneutrino) [Fig. 2(d)]. In the R-parity violatingSUSY models there are also leptonslepton and quarksquark loops generated by thesuperpotential couplings LLEc and LQDc , respectively, [24]. The short-distance non-photonic mechanisms in Fig. 2 contain heavy particles in intermediate states and is realizedat the 1-loop level [Fig. 2(a,b,d,e)] or at the level of box diagrams [Fig. 2(c)]. The 1-loop


diagrams of the non-photonic mechanisms include the diagrams similar to those for thephotonic mechanisms but with the Z-boson instead of the photon [Fig. 2(a,d)] as wellas additional Z-boson couplings to the neutrinos and neutralinos [Fig. 2(b,e)]. The boxdiagrams are constructed of the W -bosons and massive neutrinos [Fig. 2(c)] as well assimilar boxes with neutralinos and sleptons or charginos and sneutrinos. The branchingratio formula for these mechanisms is given in the next section.

3. Nucleon and nuclear structure dependence of the e conversion rates

One of the main goals of this paper is the calculation of the e conversion rateusing realistic form factors of the participating nucleus (A,Z). This can be achieved byapplying the conventional approach based on the well known non-relativistic impulseapproximation, i.e., treating the nucleus as a system of free nucleons [6]. To follow thisapproach as a first step one has to reformulate the e conversion effective Lagrangian(16) specified at the quark level in terms of the nucleon degrees of freedom.

The transformation of the quark level effective Lagrangian, Lqeff, to the effectiveLagrangian at the nucleon level, LNeff, is usually done by utilizing the on-mass-shellmatching condition [44,45]

F |Lqeff|I F |LNeff|I , (19)where |I and F | are the initial and final nucleon states. In order to solve this equationwe use various relations for the matrix elements of the quark operators between the nucleonstates

N |qKq|N =G(q,N)K SNKN, (20)with q = {u,d, s}, N = {p,n} and K = {V,A,S,P }, K = {, 5,1, 5}. Since themaximum momentum transfer q2 in e conversion, i.e., |q| m/c with m = 105.6MeV the muon mass, is much smaller than the typical scale of nucleon structure we cansafely neglect the q2-dependence of the nucleon form factors G(q,N)K . For the same reasonwe drop the weak magnetism and the induced pseudoscalar terms proportional to the smallmomentum transfer.

The isospin symmetry requires that

G(u,n)K =G(d,p)K GdK, G(d,n)K =G(u,p)K GuK, G(s,n)K =G(s,p)K GsK, (21)

with K = V,A,S,P . Conservation of vector current postulates the vector charge to beequal to the quark number of the nucleon. This allows fixing of the vector nucleon constants

GuV = 2, GdV = 1, GsV = 0. (22)The axial-vector form factorsGA can be extracted from the experimental data on polarizednucleon structure functions [46,47] combined with the data on hyperon semileptonicdecays [48]. The result is


GuA 0.78, GdA 0.47, GsA 0.19. (23)The scalar form factors are extracted from the baryon octet B mass spectrum MB incombination with the data on the pionnucleon sigma term

piN = (1/2)(mu+md)p|uu+ dd|p. (24)Towards this end we follow the QCD picture of the baryon masses which is based on therelation [4952]

B| |B =MBSBB (25)and on the well known representation for the trace of the energymomentum tensor [49]

=muuu+mddd +msss (bs/8pi)GaGa , (26)where Ga is the gluon field strength, s is the QCD coupling constant and b is thereduced Gell-MannLow function with the heavy quark contribution subtracted. Usingthese relations in combination with SU(3) relations [5052] for the matrix elementsB| |B as well as the experimental data on MB and piN we derive

GuS 5.1, GdS 4.3, GsS 2.5. (27)The nucleon matrix elements of the pseudoscalar quark currents can be related to the

divergence of the baryon octet axial vector currents [5052]. Utilizing this fact we find thepseudoscalar form factors

GuP 103, GdP 100, GsP 3.3. (28)Note that the strange quarks of the nucleon sea significantly contribute to the nucleon

form factors GA, GP and GS . This result dramatically differs from the nave quark modeland the MIT bag model where GsA,S,P = 0. The contribution of the strange nucleon seawill allow us to extract additional constraints on the second generation /Rp parameters.

Now we can solve Eq. (19) and write the effective e conversion Lagrangian at thenucleon level as

LNeff =GF

2[e(1 5) J + e J+ + e5 J

], (29)

where we have introduced the nucleon currents

J = SN[((0)V + (3)V 3)+ ((0)A + (3)A 3)5]N,J = SN[((0)S + (3)S3)+ ((0)P + (3)P 3)5]N, (30)

for nucleon isospin doublet NT = (p,n). The coefficients in Eq. (30) are defined as

(0)V =

18(GuV +GdV

)(u1R + u1L + d1R + d1L

),

(3)V =

18(GuV GdV

)(u1R + u1L d1R d1L

),

(0)A =

18(GuA +GdA

)(u1R u1L + d1R d1L

)+ 14GsA

(d2R d2L

),


(3)A =

18(GuA GdA

)(u1R u1L d1R + d1L

),

(0)S =

116(GuS +GdS

)(d1L d1R

)+ 18GsS(d2L d2R

),

(3)S =

116(GuS GdS

)(d1L d1R

),

(0)P =

116(GuP +GdP

)(d1L d1R

) 18GsP

(d2L d2R

),

(3)P =

116(GuP GdP

)(d1L d1R

). (31)

Starting from the Lagrangian (29) it is straightforward to deduce the formula for thetotal e conversion rate. In the present paper we focus on the coherent process, i.e.,ground state to ground state transitions. This is a dominant channel of e conversionwhich, in most of the experimentally interesting nuclei, exhausts more than 90% of the total e branching ratio [1821]. To the leading order of the non-relativistic expansionthe coherent e conversion rate takes the form

cohe =

G2FpeEe

2piQ(Mp +Mn)2, (32)

where

Q= 2(0)V + (3)V 2 + (0)+S + (3)+S2 + (0)S + (3)S2+2Re{((0)V + (3)V )[(0)+S + (0)S + ((3)+S + (3)S)]}. (33)

The transition matrix elementsMp,n in Eq. (32) depend on the final nuclear state populatedduring the e conversion. We should stress that, after computing the nuclear matrixelementsMp,n the data provide constraints on the quantity Q of Eq. (33). For the groundstate to ground state transitions in spherically symmetric nuclei the following integralrepresentation is valid

Mp,n = 4pij0(per)(r)p,n(r)r

2 dr, (34)

where j0(x) the zero order spherical Bessel function and p,n the proton (p), neutron(n) nuclear density normalized to the atomic number Z and neutron number N of theparticipating nucleus, respectively. The space dependent part of the muon wave function is a spherically symmetric function which in our calculations (see Section 4) wasobtained by solving numerically the Shrndinger and Dirac equations with the Coulombpotential.

In defining Q, Eq. (33), we introduced the ratio = (Mp Mn)/(Mp +Mn) (A 2Z)/A, (35)

where A and Z are the atomic weight and the total charge of the nucleus. The quantityQ depends weakly on the nuclear parameters through the factor . In fact, the termsdepending on are small since < 1 (see Table 1) and GuS , GdS as well as GuV , GdVhave the same sign. In practice the nuclear dependence of Q can be neglected and thus,


Table 1The variation of the quantity of Eq. (33) through the periodic table. For comparison its approximateexpression (A 2Z)/A is also shown

A Z (A,Z) (A,Z)

12. 6. 0.000 0.00024. 12. 0.014 0.00027. 13. 0.000 0.03732. 16. 0.023 0.00040. 20. 0.037 0.00044. 20. 0.063 0.09148. 22. 0.083 0.08363. 29. 0.056 0.07990. 40. 0.054 0.111

112. 48. 0.108 0.143208. 82. 0.152 0.212238. 92. 0.175 0.227

Q coincides with the value of 2 where is defined below. It can be considered asa universal effective /Rp parameter measuring the /Rp SUSY contribution to the econversion. It also represents a suitable characteristic which allows comparison of econversion experiments on different targets treating the corresponding upper bounds on Qas their sensitivities to the /Rp SUSY signal.

For completeness, in Section 4 the limits for some non-SUSY [22,26,27] as well as RpSUSY contributions to the e nuclear conversion (see in Fig. 2) are updated. Thecorresponding expression for Re is written as [17]

Re = , (36)where is nearly independent of nuclear physics [1921] and contains the lepton flavorviolating parameters corresponding to the contributions in Fig. 2. Thus, e.g., for photon-exchange mode is given by

= (4pi)2 |fM1 + fE0|2 + |fE1 + fM0|2

(GFm2)2 , (37)

where the four electromagnetic form factors fE0, fE1, fM0, fM1 are parametrized in aspecific elementary model [1921].

The factor (A,Z) in Eq. (36) accumulates about all the nuclear structure dependenceof the branching ratio Re . Assuming that the total rate of the ordinary muon capture isgiven by the GoulardPrimakoff function, fGP, the nuclear structure factor (A,Z) takesthe form

(A,Z) = Eepem2

M2

G2ZfGP(A,Z), (38)


where G2 6. Thus, a non-trivial nuclear structure dependence of the econversion branching ratio Re is mainly concentrated in the nuclear matrix elementsM2 [7]. In the protonneutron representation one can write down

M2 = [Mp +QMn]2, (39)whereQ takes the values of Eq. (32) of Ref. [19] andMp,n are given by writing the matrixelements of Eq. (34) in terms of an effective muon wavefunction as

Mp,n = Mp,n. (40)In our present approach the role of in Eq. (36) is played by Q, since Q = 2, and thecorresponding function defined in Eq. (38) for R-parity violating interactions, 6Rp , isobtained from Eq. (38) by putting Q = 1 in Eq. (39). The separation of nuclear physicsfrom the elementary particle parameters is not complete but we have seen that is quitesmall. In any case we present in Table 1 the values of for the various nuclear systems.

4. Results and discussion

The pure nuclear physics calculations needed for the e conversion studies refer tothe integrals of Eq. (34). The results of Mp and Mn for the currently interesting nucleiAl, Ti and Pb are shown in Table 2. They have been calculated using proton densities pfrom the electron scattering data [53] and neutron densities n from the analysis of pionicatom data [54,55]. We employed an analytic form for the muon wave function (r)obtained by solving the Schrdinger equation using the Coulomb potential produced bythe charge densities discussed above. This way the finite size of a nucleus was taken intoconsideration. Moreover, we included vacuum polarization corrections as in Ref. [18]. Insolving the Schrdinger equation we have used modern neural networks techniques [56]which give the wave function (r) in the analytic form of a sum over sigmoid functions.Thus, in Eq. (34)

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