B-quark mediated neutrinoless $\mu^--e^-$ conversion in presence of R-parity violation

We found that in supersymmetric models with R-parity non-conservation the b-quarks may appreciably contribute to exotic neutrinoless muon-electron conversion in nuclei via the triangle diagram with two external gluons. This allowed us to extract previously overlooked constraints on the third generation trilinear R-parity violating parameters significantly more stringent than those existing in the literature.

In the standard model (SM) the lepton flavors (L f ) are conserved quantum numbers as an accidental consequence of gauge invariance and field content. Thus, observation of L f non-conservation would imply the presence of physics beyond the standard model. Nonconservation of muon lepton flavor L µ in neutrino oscillations has been recently established by the SuperKamiokande experiment in atmospheric neutrino measurements. In this case lepton flavor violation (L f / ) is generated by non-zero neutrino masses. Various sources of L f / can be probed by searching for certain exotic processes. Among them the neutrinoless muon-toelectron conversion in muonic atoms, µ − + (A, Z) −→ e − + (A, Z) * , is known to be one of the most powerful tools to constrain L f / interactions [1]- [6]. In particular, it allows setting stringent constraints on the L f / interactions of the supersymmetric models with R-parity violation (R p / SUSY). In the literature there have been obtained upper bounds on various products of the L f / trilinear R-parity violating (R p / ) couplings λλ ′ , λ ′ λ ′ [3,4,5,7]. In the present letter we derive new constraints on the products λλ ′ with some other combinations During the last decade R p / SUSY models have been extensively studied in the literature. For the minimal field content the R-parity violating part of the superpotential reads The definition of the couplingsλ ′ ,λ ′′ corresponds to the gauge basis for the quark fields. We setλ ′′ = 0, since these are irrelevant for our consideration. This "ad hoc" condition ensures proton stability and can be guaranteed by special discreet symmetries other than R-parity. The leading quark-level tree diagrams with the trilinear R p / -couplings contributing to the µ − e conversion are listed in Ref. [5] and are of the following three types: The photonic 1-loop diagrams can also significantly contribute to this process [4]. However they are irrelevant for the present case of heavy quark contribution and, therefore, are not included in our analysis. Integrating out the heavy intermediate SUSY-particles from the above mentioned diagrams and carrying out a Fierz rearrangement one obtains the following 4-fermion effective Lagrangian for µ − e conversion at the quark level [5] L q ef f = The index i denotes generation so that u i = u, c, t and (2) we neglected the terms with axialvector and pseudoscalar quark currents which do not contribute to the dominant coherent mode of µ − − e − conversion [6,8,9,10]. The coefficients η in Eq. (2) accumulate the dependence on R p / SUSY parameters as Herem q(n) ,m ν(n) are the squark and sneutrino masses. In Eq. (3) we introduced the cou- corresponding to the R p / interactions in the quark mass eigenstate basis, related to the gauge basis q ′ through q L, (2) to the corresponding nucleon currents can be parametrized in the form Isospin symmetry requires that G Furthermore, conservation of vector current requires the vector charge to be equal to the quark number of the nucleon. This allows fixing the vector nucleon constants as G u Thus the strange and heavy quarks can contribute only to the scalar nucleon current 2 . Since the scalar currents in Eq. (2) involve only down quarks d, s, b it follows that among the heavy c, b, t-quarks only the b-quark can contribute to the coherent µ − − e − conversion. The heavy quarks contribution to the scalar current is realized via the triangle diagram in Fig. 1 with the two gluon lines. The heavy quark q h scalar current induced by the diagrams of this type can be estimated using the heavy quark expansion [11] Here α s and µ are the QCD coupling constant and a typical hadronic scale respectively, and GG = G a µν G µν a where G a µν is the gluon field strength. The quark scalar currents and the gluon operator GG also contribute to the trace of the energy-momentum tensor where b = 11 − (2/3)n f and n f = 6 is the number of quark species. The scalar form factors G q S can be extracted from the baryon octet B mass spectrum M B , expressed as [12] B|θ µ µ |B = M BB B, and from the data on the pion-nucleon sigma term σ πN = (1/2)(m u + m d ) p|ūu +dd|p . Using Eqs. (5)-(7) in combination with SU(3) relations [12] for the matrix elements in (7) we obtain Here we used for σ πN = 48 MeV [13], and m u = 4.2 MeV, m d = 7.5 MeV, m s = 150 MeV, Having the couplings G q V , G q S determined we rewrite the Lagrangian (2) in terms of the nucleon currents Here we defined J µ =N γ µ (α is the nucleon isospin doublet. The isosclar α (0) and the isovector α (3) coefficients are Following the approach of Refs. [5,6] we derive from the Lagrangian (9) the coherent µ − e conversion branching ratio in the form where Γ(µ − → capture) is the total rate of ordinary muon capture and In this formula we neglected the contribution of isovector currents which is small for most of the experimentally interesting nuclei [5,6]. The numerical values of the nuclear matrix elements M p,n for the currently interesting have been calculated in Ref. [6]. The most stringent experimental bounds on the branching ratio R µe have been set by the SINDRUM2 experiment (PSI) with 197 Au and 48 Ti stopping targets: 197 Au → e − + 197 Au) Γ(µ − + 197 Au → ν µ + 197 P t) ≤ 5.0 × 10 −13 , (90% C.L.) [14], , (90% C.L.) [15].
Note that a 48 Ti target will be used in the future experiment planned at the muon factory at KEK (Japan) [16]. This experiment is going to increase the sensitivity up to R µe ≤ 10 −18 . In the near future new bounds are expected from the MECO (Brookhaven) experiment with an 27 Al target From these experimental limits it is straightforward to extract upper bounds on various products of the type λλ ′ , λ ′ λ ′ . Many of them have been previously derived in Refs. [3]- [5]. In Table 1 we present the new upper bounds that are associated with the b-quark contribution. We show the three cases, corresponding to the experimental limits in Eqs. (13)- (15).

Parameters Previous limits Present limits (Au) Present limits (Ti) Expected limits (Al)
197 In the derivation of these bounds we assumed, as usual, the dominance of only one of these products with the specific combination of generation indices. We also assumed that all the scalar masses in Eq. (3) are equalm uL(n) ≈m dL,R(n) ≈m ν(n) ≈m.
As can be seen from Table 1, our new limits(columns 3-5) are significantly more stringent than those previously known in the literature [7] (column 2). In Table 1 the quantities B 1,2,3 denote scaling factors defined as They allow recalculating limits given in Table 1 to the case corresponding to the experimental upper bounds on the branching ratio R exp µe other than in Eqs. (13)- (15). In summary, we found new important contribution to µ − −e − conversion originating from the b-quark sea of the nucleon. We have shown that among the heavy c, b, t-quarks only the bquark can contribute to the coherent µ − − e − conversion via the scalar interactions involving down quarks d, s, b. The heavy quarks contribution to the scalar current is materialized with the gluon exchange shown in the triangle diagram of Fig. 1. From the existing data and expected experimental constraints on the branching ratio R µe − we obtained new upper limits on the products of the trilinear R p / parameters of the type λ n12 λ ′ n33 , λ n21 λ ′ n33 which are significantly more stringent than those existing in the literature.
This work was supported in part by Fondecyt (Chile) under grant 8000017, by a Cátedra Presidencial (Chile) and by RFBR (Russia) under grant 00-02-17587.