Enhanced μ−–e− conversion in nuclei in the inverse seesaw model

Nuclear Physics B 752 (2006) 80–92

Enhanced μ− –e− conversion in nuclei in the inverse seesaw model F. Deppisch a,∗ , T.S. Kosmas b , J.W.F. Valle c a Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany b Division of Theoretical Physics, University of Ioannina, GR-45110 Ioannina, Greece c AHEP Group, Instituto de Física Corpuscular—C.S.I.C./Universitat de València, Edificio Institutos de Paterna,

Apt 22085, E-46071 Valencia, Spain Received 18 January 2006; received in revised form 22 March 2006; accepted 16 June 2006 Available online 18 July 2006

Abstract We investigate nuclear μ− –e− conversion in the framework of an effective Lagrangian arising from the inverse seesaw model of neutrino masses. We consider lepton flavour violation interactions that arise /f from short range (non-photonic) as well as long range (photonic) contributions. Upper bounds for the L parameters characterizing μ− –e− conversion are derived in the inverse seesaw model Lagrangian using the available limits on the μ− –e− conversion branching ratio, as well as the expected sensitivities of upcoming experiments. We comment on the relative importance of these two types of contributions and their relationship with the measured solar neutrino mixing angle θ12 and the dependence on θ13 . Finally we show how /f μ− –e− conversion and the μ− → e− γ rates are strongly correlated in this model. the L © 2006 Elsevier B.V. All rights reserved. PACS: 12.60.Jv; 11.30.Er; 11.30.Fs; 23.40.Bw; 11.30.-j; 11.30.Hv; 26.65.+t; 13.15.+g; 14.60.Pq; 95.55.Vj Keywords: Neutrino mass and mixing; Lepton flavour violation; Exotic μ–e conversion in nuclei; Muon capture; Physics beyond the Standard Model

1. Introduction The discovery of neutrino oscillations [1–3] shows that neutrinos are massive [4] and that lepton flavour is violated in neutrino propagation. The violation of this conservation law could * Corresponding author.

E-mail addresses: [email protected] (F. Deppisch), [email protected] (J.W.F. Valle). 0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.06.032

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show up in other contexts, such as rare lepton flavour violating (LFV) decays of muons and taus, e.g. μ− → e− γ . In fact, there are strong indications from theory that this may be the case. Among the lepton flavour violating (L / f ) processes, the electron- and muon-flavour violating nuclear conversion μ− + (A, Z) → e− + (A, Z)∗ ,

(1)

is known to provide a very sensitive probe of lepton flavour violation [5–14]. This follows from the distinct feature of a coherent enhancement in nuclear μ− –e− conversion. From the experimental viewpoint, currently the best upper bound on the μ− –e− conversion branching ratio comes from the SINDRUM II experiment at PSI [15], using 197 Au as stopping target, Au Rμe  5.0 × 10−13

90% C.L.

(2)

The proposed aim of the MECO experiment, the μ− –e− conversion experiment at Brookhaven [16], with 27 Al as target is expected to reach [16] Al  2 × 10−17 Rμe

(3)

about three to four orders of magnitude better than the present best limit. An even better sensitivity is expected at the new μ− –e− conversion PRISM experiment at Tokyo, with 48 Ti as stopping target. This experiment aims at [17] Ti  10−18 . Rμe

(4)

Such an impressive sensitivity can place severe constraints on the underlying parameters of μ− –e− conversion. There are many mechanisms beyond the Standard Model that could lead to lepton flavour violation (see [5–7,9,10] and references therein). The corresponding Feynman diagrams can be classified according to their short-range or long-range character into two types: photonic and nonphotonic, as shown in Fig. 1. The long-distance photonic mechanisms in Fig. 1(a) are mediated by virtual photon exchange between nucleus and the μ–e lepton current. The hadronic vertex is characterized in this case by ordinary electromagnetic nuclear form factors. Contributions to μ–e conversion arising from virtual photon exchange are generically correlated to μ → eγ decay. The short-distance non-photonic mechanisms in Fig. 1(b) include effective 4-fermion quarklepton L / f interactions which couple the quarks and leptons via heavy intermediate particles (W, Z, Higgs bosons, supersymmetric particles, etc.) at the tree level, at the 1-loop level or via box diagrams. The various mechanisms can significantly differ in many respects, in particular, in what concerns nucleon and nuclear structure treatment. As a result, they must be treated on a case-by-case basis. In this paper we consider μ− –e− conversion in the context of a variant of the seesaw model [18], called inverse seesaw [19]. It differs from the standard one in that no large mass scale is necessary, providing a simple framework for enhanced L / f rates, unsuppressed by small neutrino masses [20,21]. The enhancement of L / f rates holds in this model even in the absence of supersymmetry and in the absence of neutrino masses. For this reason it plays a special role. For simplicity here we neglect possible supersymmetric contributions to the L / f rates that could exist in this model, see [22]. Other seesaw constructions with extended gauge groups have been considered recently, using either left–right gauge symmetry [23] or full SO(10) unification [24–26]. They, too, will lead to enhanced LFV rates. However, both for definiteness and simplicity, here we focus our discussion on the case of the simplest SU(2) ⊗ U(1) inverse seesaw model which

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Fig. 1. (a) Photonic (long-distance) and (b) non-photonic (short-distance) contributions to the nuclear μ− –e− conversion.

we take as a reference model. First, we derive a formula for the μ− –e− conversion branching ratio in terms of L / f parameters of the effective Lagrangian of the model. The transformation of this Lagrangian, first to the nucleon and then to the nuclear level, needs special attention to the effects of nucleon and nuclear structure. The nucleon structure is taken into account on the basis of the QCD picture of baryon masses and experimental data on certain hadronic parameters. The nuclear physics, which is involved in the muon-nucleus overlap integrals [10,27] is evaluated paying special attention on specific nuclei that are of current experimental interest like, 27 Al, 48 Ti and 197 Au. 2. Inverse seesaw mechanism The model extends minimally the particle content of the Standard Model by the sequential addition of a pair of two-component SU(2) ⊗ U(1) singlet leptons, as follows   νi νic , Si , , eic , (5) ei with i a generation index running over 1, 2, 3. In addition to the more familiar right-handed neutrinos characteristic of the standard seesaw model, the inverse seesaw scheme contains an equal number of gauge singlet neutrinos Si . In the original formulation of the model, these were superstring inspired E(6) singlets, in contrast to the right-handed neutrinos, members of the spinorial representation. Recently similar constructions have been considered in the framework of left-right symmetry [23] or SO(10) unified models [24–26]. In the ν, ν c , S basis, the 9 × 9 neutral leptons mass matrix M is given as ⎛ ⎞ 0 mTD 0 M = ⎝ mD (6) 0 MT ⎠ , 0

M

μ

where mD and M are arbitrary 3 × 3 complex matrices in flavour space, whereas μ is complex symmetric. The matrix M can be diagonalized by a unitary mixing matrix Uν , UνT MUν = diag(mi , M4 , . . . , M9 ),

(7)

yielding 9 mass eigenstates na . In the limit of small μ three of these correspond to the observed light neutrinos with masses mi , while the three pairs of two-component leptons (νic , Si ) combine to form three heavy leptons, of the quasi-Dirac type [28].

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The light neutrino mass states νi are given in terms of the flavour eigenstates via the unitary matrix Uν 9  νi = (Uν )ia na ,

(8)

a=1

which has been studied in earlier papers [20,21]. The diagonalization results in an effective Majorana mass matrix for the light neutrinos [29], mν = mTD M T

−1

μM −1 mD ,

(9)

where we are assuming μ, mD  M. One sees that the neutrino masses vanish in the limit μ → 0 where lepton number conservation is restored. In models where lepton number is spontaneously broken by a vacuum expectation value σ  [29] one has μ = λσ . Typical parameter values may be estimated from the required values of the light neutrino masses indicated by oscillation data [4] as    2   −2 mν mD M μ (10) . = 0.1 eV 100 GeV 1 keV 104 GeV For typical Yukawas λ ∼ 10−3 one sees that μ = 1 keV corresponds to a low scale of L violation, σ  ∼ 1 MeV (for very low values of σ  this might lead to interesting signatures in neutrinoless double beta decays [30]).1 In contrast, in the conventional seesaw mechanism without the gauge singlet neutrinos Si one would have   0 mTD (11) , mD  MR ⇒ mν = mTD MR−1 mD . mD MR Note that in the “inverse seesaw” scheme the three pairs of singlet neutrinos have masses of the order of M and their admixture in the light neutrinos is suppressed as mMD . It is crucial to realize that the mass M of our heavy leptons can be much smaller than the MR characterizing the righthanded neutrinos in the conventional seesaw, since the suppression in Eq. (10) is quadratic in M −1 (as opposed to the linear dependence in MR−1 given by Eq. (11)), and since we have the independent small parameter μ characterizing the lepton number violation scale. As a result the value of M may be as low as the weak scale (if light enough, these neutral leptons could give signatures at accelerator experiments [31,32]). Without loss of generality one can assume μ to be diagonal, μ = diag μi ,

(12)

and using the diagonalizing matrix U of the effective light neutrino mass matrix mν , U T mν U = diag mi , Eq. (9) can be written as √ √ −1 T T T −1 −1 · diag μi · diag μi · M mD U · diag m−1 1 = diag mi · U mD M i . 1 Note that such a low scale is protected by gauge symmetry.

(13)

(14)

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In the basis where the charged lepton Yukawa couplings are diagonal the lepton mixing matrix is simply the rectangular matrix formed by the first three rows of Uν [33]. In analogy to the standard seesaw mechanism [34] it is thus possible to define a complex orthogonal matrix √ R = diag μi · M −1 mD U · diag m−1 (15) i with 6 real parameters. Using R, the neutrino Yukawa coupling matrix Yν = expressed as √ 1 † M · diag μ−1 Yν = i · R · diag mi · U . v sin β

1 v sin β mD

can be

(16)

To further simplify our discussion we make the assumption that the eigenvalues of both M and μ are degenerate and that R is real. This allows us to easily compare our results with those obtained previously in Refs. [35,36] for the case of the conventional seesaw mechanism. 3. The effective quark-level Lagrangian In our model the L / f arises from penguin photon and Z exchange as well as box diagrams, as illustrated in Fig. 2. The resulting effective Lagrangian can be expressed as [5] γ

box Leff = Leff + LZ eff + Leff , γ

Leff = −

e2 q2

×

(17)

  L

R β R e¯ q 2 γα AL 1 PL + A1 PR + mμ iσαβ q A2 PL + A2 PR μ



Qq qγ ¯ αq

(18)

q

  e2  1  L 1 R i L R V L A R α =− 2 A + A1 jα + A1 − A1 jα + A2 + A2 mμ jαβ q 2 2 q q 2 1 Vβ

× Qq J(q) ,

(19)

 gZ2  L R F μ × e ¯ γ P + F P qγ ¯ αq α L R 2 m2Z q   q q Z + ZR V α g2  1  L 1 F + F R jαV + F R − F L jαA L J(q) , = Z2 2 2 mZ q 2 

 L 2 R Qq qγ ¯ αq Lbox eff = e e¯ γα Dq PL + Dq PR μ × q ZL

LZ eff =

= e2

 1  q

q + ZR

(21) (22)

q

 1 Vα DqL + DqR jαV + DqR − DqL jαA Qq J(q) 2 2

with Qq the electric charge of quark q, and  q q ZL/R = T3 L/R − Qq sin2 θW .

(20)

(23)

(24)

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Fig. 2. Diagrams for μ–e conversion: photonic (a), Z-boson (a), (b) and W-boson (c) exchange.

The expressions in Eqs. (19), (21) and (23) correspond to the notation in Eq. (5) of Ref. [8]. L/R L/R L/R The coefficients A1 , A2 , F L/R , Dq which give rise to lepton flavour violation, are given by (for μ–e conversion, the indices i, j are always i = 2 (∼ μ) and j = 1 (∼ e), and are thus omitted for simplicity in the above formulae) [42]: AL 1 =

9 

∗ U2i U1i Fγ (λi ),

(25)

i=1

AR 1 = 0, AL 2 = AR 2 =

(26) 9 

me mμ 9 

∗ U2i U1i Gγ (λi ),

(27)

i=1 ∗ U2i U1i Gγ (λi ),

(28)

i=1

FL =

9 

 ∗ U2i U1i FZ (λi ) + Cij HZ (λi , λj ) + Cij ∗ GZ (λi , λj ) ,

(29)

i,j =1

F R = 0, DqL =

9  i,j =1

(30) 

∗ ∗ U2i U1i Fbox (λi , λj ) + U2i U1i Gbox (λi , λj ) ,

(31)

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DqR = 0,

(32)

with Cij =

3 

∗ Uki Ukj .

(33)

k=1

The corresponding form-factor functions in the above terms are given by [42] 7x 3 − x 2 − 12x x 4 − 10x 3 + 12x 2 − ln x, 12(1 − x)3 6(1 − x)4 2x 3 + 5x 2 − x 3x 3 Gγ (x) = − − ln x, 4(1 − x)3 2(1 − x)4 5x 5x 2 FZ (x) = − ln x, − 2(1 − x) 2(1 − x)2  2  1 y 2 (1 − x) x (1 − y) GZ (x, y) = − ln x − ln y , 2(x − y) 1−x 1−y  2  √ 2 xy y − 4y x − 4x ln x − ln y , HZ (x, y) = 4(x − y) 1 − x 1−y    1 1 xy x 2 ln x y 2 ln y 1 − 1+ + − Fbox (x, y) = x −y 4 1 − x (1 − x)2 1 − y (1 − y)2   1 1 x ln x y ln y − 2xy − + − , 2 1 − x (1 − x) 1 − y (1 − y)2   √  xy 1 x ln x y ln y 1 − (4 + xy) + − Fbox (x, y) = − x −y 1 − x (1 − x)2 1 − y (1 − y)2   1 x 2 ln x y 2 ln y 1 − + − . −2 1 − x (1 − x)2 1 − y (1 − y)2 Fγ (x) =

(34) (35) (36) (37) (38)

(39)

(40)

The effective Lagrangians for the μ–e diagrams of Eqs. (18), (20) and (22) can be compactly written as     (q)  (q) q (q) Bμ μν D Leff = Ga ηAB jμA J(q) + ηCD j C J(q) + ηT jμν J(q) , a = ph, nph, A,B;q

q

C,D;q

(41) where summation involves A, B = {A, V }; C, D = {S, P } and q = {u, d, s}. The coupling √F in the non-photonic and by Gph = 4πα in the phostrength factor Ga is given by Gnph = G q2 2

(q)

tonic case. The parameters ηi currents are jμV = eγ ¯ μ μ, Vμ J(q)

= qγ ¯ q, μ

depend on the specific L / f model assumed. The lepton and quark

jμA = eγ ¯ μ γ5 μ, Aμ J(q)

= qγ ¯ γ5 q, μ

j S = eμ, ¯ S J(q)

j P = eγ ¯ 5 μ,

= qq, ¯

P J(q)

jμν = eσ ¯ μν μ,

= qγ ¯ 5 q,

μν

J(q) = qσ ¯ μν q. (42)

In our model, the only nonvanishing contributions are (q)

ηV V =

(q) ηV V

and

 q 1  1 L q F + F R ZL + ZR + Qq DqL + DqR , 2 2

(q) ηAV ,

(43)

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 q 1  1 L q F − F R ZL + ZR + Qq DqL − DqL , 2 2 in the non-photonic case and (q)

ηAV =

(44)

1  (q) R ηV V = Qq AL 1 + A1 , 2 1  (q) R ηAV = Qq AL 1 − A1 , 2 in the photonic case.

(45) (46)

4. The effective nucleon-level Lagrangian The nucleon level effective Lagrangian obtained through the reformulation of the quark level effective Lagrangian (41) can be written in terms of the effective nucleon fields and the nucleon isospin operators as    (0)  (0) Bμ (3) Bμ (3) D N D Leff = Ga jμA αAB J(0) + αAB J(3) + j C αCD J(0) + αCD J(3) A,B

C,D

   (0) μν (3) μν , + jμν αT J(0) + αT J(3)

a = ph, nph.

(47)

The isoscalar J(0) and isovector J(3) nucleon currents are defined as Vμ ¯ μ τk N, J(k) = Nγ

Aμ J(k) = N¯ γ μ γ5 τk N,

P ¯ 5 τk N, = Nγ J(k)

S J(k) = N¯ τk N,

J(k) = N¯ σ μν τk N, μν

(48)

where k = 0, 3 and τ0 ≡ Iˆ. The relationship between the coefficients α in Eq. (47) and the fundamental L / f parameters ηAB of the quark level Lagrangian (41) can be found as follows. We start from the equations (q,N) which relate the various nucleon form factors GK with matrix elements of the quark states and those of the nucleon states (q,N)

N|qΓ ¯ K q|N = GK

Ψ¯ N ΓK ΨN ,

(49)

with q = {u, d, s}, N = {p, n} and K = {V , A, S, P }, ΓK = {γμ , γμ γ5 , 1, γ5 }. Since the maximum momentum transfer in μ–e conversion is much smaller than the nucleon mass scale, we (q,N) may neglect the q2 -dependence of GK . Assuming isospin symmetry, we find (u,n)

GK

(d,p)

= GK

≡ GdK ,

(d,n)

GK

(u,p)

= GK

≡ GuK ,

(s,n)

GK

(s,p)

= GK

≡ GsK .

(50)

For the coherent nuclear μ− –e− conversion, only the vector and scalar nucleon form factors are needed (the axial and pseudoscalar nucleon currents couple to the nuclear spin and for spin zero nuclei they can contribute only to the incoherent transitions). The vector current form factors are determined through the assumption of conservation of vector current at the quark level which gives GuV = 2,

GdV = 1,

GsV = 0.

(51)

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The coefficients αAB of the nucleon level Lagrangian (47) can be expressed in terms of the L / f parameters ηAB of the quark level effective Lagrangian in Eq. (41) as 1  (u) 1  (u) (d)  (3) (d)  αI V = ηI V − ηI V GuV − GdV , ηI V + ηI V GuV + GdV , (52) 2 2 with I = A, V . From the Lagrangian (47), following standard procedure, we can derive a formula for the total μ–e conversion branching ratio. In this paper we restrict ourselves to the coherent process which is the dominant channel of μ–e conversion. For most experimentally interesting nuclei, this accounts for more than 90% of the total μ− –e− branching ratio [27]. To leading order of the non-relativistic reduction the coherent μ–e conversion branching ratio takes the form (0)

αI V =

coh Rμe = Qa G2a

pe E e M2a , 2π Γ (μ− → capture)

a = ph, nph,

(53)

where pe , Ee are the outgoing electron 3-momentum and energy and M2ph (M2nph ) represents the square of the nuclear matrix element for the photonic (non-photonic) mode of the process. In this work we do not discuss the interference between the photonic and non-photonic contribution, for which it is in general not trivial to write down an expression like (53) [6]. As will be seen in Section 5, the photonic contribution is dominant in our model, and the non-photonic and interference terms can be neglected. The quantity Qa is defined as  (0)  (0) (3) 2 (3) 2 Qa = αV V + αV V φ  + αAV + αAV φ  , (54) with the corresponding coefficients α for the photonic and non-photonic contributions and depends on the nuclear parameters through the factor φ = (Mp − Mn )/(Mp + Mn ). The Mp,n are given by  Mp,n = 4π (ge gμ + fe fμ )ρp,n (r)r 2 dr.

(55)

(56)

In the latter equation, ρp,n (r) are the spherically symmetric proton (p) and neutron (n) nuclear densities normalized to the atomic number Z and neutron number N , respectively, of the target nucleus. Here gμ , fμ are the top and bottom components of the 1s muon wave function and ge , fe are the corresponding components of the Coulomb modified electron wave function [37,38]. In the present work, the matrix elements Mp,n , defined in Eq. (56), have been numerically calculated using proton densities ρp from Ref. [39] and neutron densities ρn from Ref. [40]. The muon wave functions fμ and gμ (and also ge , fe ) were obtained by solving the Dirac equation with the Coulomb potential produced by the densities ρp,n by using artificial neural network techniques. In this way, relativistic effects and vacuum polarization corrections have been taken into account [37,38]. The latter method has recently been applied for evaluating the τ − wave functions in a set of (medium-heavy and heavy) nuclei for obtaining the τ -capture rate by nuclei. 5. Results and discussion The results for Mph,nph corresponding to a set of nuclei throughout the periodic table including systems with good sensitivity to the μ–e conversion are shown in Table 1. In this table we also present the outgoing electron momentum (|pe |) and the experimental values for the total

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Table 1 Ingredients entering Eq. (53) which gives the branching ratio of the charged lepton flavour violating μ–e conversion for a set of nuclei throughout the periodic table. We note that by neglecting the electron mass we have Ee ≈ pe c Nucleus 12 C 27 Al 32 S 40 Ca 48 Ti 63 Cu 90 Zr 112 Cd 197 Au 208 Pb 238 U

|pe | (fm−1 )

Γμc (106 s−1 )

M2ph (fm−3 )

M2nph (fm−3 )

0.533 0.532 0.531 0.529 0.528 0.524 0.517 0.511 0.485 0.482 0.474

0.0388 0.705 1.352 2.557 2.590 5.676 8.660 10.610 13.070 13.450 13.100

0.00007 0.00204 0.00433 0.00982 0.01217 0.02883 0.06713 0.08416 0.15571 0.18012 0.19360

0.00029 0.00821 0.01656 0.03667 0.05560 0.12631 0.29713 0.37712 0.68691 0.80892 0.87797

φ 0.0000 −0.0022 0.0225 0.0350 −0.0645 −0.0445 −0.0493 −0.0552 −0.0478 −0.0563 −0.0608

Table 2 Upper bounds on the parameters Qa (see text, for definition) inferred from the SINDRUM II data on the μ− –e− conversion in 197 Au [Eq. (2)] as well as from the expected sensitivities of the current MECO (BNL) [Eq. (3)] and PRISM (KEK) experiments [Eq. (4)] with 27 Al and 48 Ti stopping targets respectively Parameter

Present limits (PSI) 197 Au

Expected limits (MECO) 27 Al

Expected limits (PRISM) 48 Ti

Qph Qnph

1.96×10−16

2.68×10−18

8.39×10−20 1.84×10−20

4.45×10−15

6.67×10−19

rate of the ordinary muon capture Γμc [41]. We give the ingredients required to determine the branching ratio Rμe for the three nuclei Al, Ti and Au, of current experimental interest. As has been discussed in Ref. [7], for the description of the long range photonic contribution only the proton matrix elements Mp are required. In the case of the non-photonic mechanisms (short-range contributions), however, both protons and neutrons contribute and therefore both Mp,n matrix elements are needed. The latter are obtained by using densities extracted from the data on pionic atoms or the pion–nucleus scattering [40]. Using the values of Table 1 and the existing [15] or expected [16,17] experimental sensitivities on Rμe , we can derive the corresponding sensitivity upper limits on the particle physics parameters characterizing the effective Lagrangians (41) and (47). The most straightforward limits can be set on the quantities Qa of Eq. (53) which are given in Table 2. In order to achieve limits on the particle physics leading to μ–e conversion, the quark level effective Lagrangian of the model is adjusted to the form of Eq. (41) and by identifying the (q) effective parameters ηAB with expressions in terms of model parameters. This way the upper bounds on Qa from Table 2 can be translated to restrictions on the model parameters present in these expressions. Fig. 3 shows the sensitivity bounds on the parameters M and μ characterizing the inverse seesaw model, that follow from experiments with Al, Au and Ti targets, respectively. In this and all subsequent figures, the neutrino oscillation parameters are fixed at their present best fit values, tan2 θ23 = 1.40,

tan2 θ13 = 0.005,

m212 = 3.30 × 10−5 eV2 ,

tan2 θ12 = 0.36,

m223 = 3.10 × 10−3 eV2 ,

(57)

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F. Deppisch et al. / Nuclear Physics B 752 (2006) 80–92

Fig. 3. Present and expected limits on the model parameters M and μ/M. The shaded areas are excluded by the bounds on Qnph (left panel) and Qph (right panel) given in Table 2. The light neutrino parameters used are given in (57).

Fig. 4. Relation of branching ratios for μ− –e− conversion (left panel) and μ− → e− γ (right panel) with the solar neutrino mixing angle, for different values of θ13 . The inverse seesaw parameters are given by: M = 1 TeV, μ = 30 eV. The light neutrino parameters used are given in (57), except for θ13 and θ12 , which are varied as shown in the plots.

the light neutrino mass spectrum is assumed to be hierarchical, mν1 = 0 eV, and all CP violating phases are set to zero. A comparison between the two panels of Fig. 3 shows that the limits from the photonic contribution are much more stringent than from the non-photonic contribution. The photonic contribution is thus strongly dominant in our model, justifying our omission of the interference between the two modes. It is also interesting to explore how these rates depend on the relevant neutrino mixing angles which are probed in solar neutrino experiments. This is shown in Fig. 4, where the relation between R(μ− Ti → e− Ti) (Br(μ → eγ )) and the solar neutrino angle θ12 for fixed M and μ is displayed. As can be seen in the figure, there is also a strong dependence on the small neutrino mixing angle θ13 . In designing future experiments testing for L / f it is instructive to determine how the branching ratios for μ− –e− conversion and the μ− → e− γ are related. In Fig. 5 we show explicitly that, in the inverse seesaw model, the rates for μ− –e− conversion and that for the μ− → e− γ decay are strongly correlated, indicating the dominance of the photonic diagram in Fig. 1(a).

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Fig. 5. Correlation between nuclear μ− –e− conversion and μ− → e− γ decay in the inverse seesaw model. The light neutrino parameters used are given in (57).

6. Summary and conclusions In the present paper we constructed an effective Lagrangian describing the photonic and non/f photonic μ− –e− conversion in the context of the inverse seesaw model and specified the L parameters characterizing this process. We focused on the simplest inverse seesaw model. The interest in the model is that it accounts for the observed masses and mixings indicated by current oscillation data in such a way that the underlying physics “does not decouple” and can be /f phenomenologically probed experimentally. The model provides a framework for enhanced L rates with a rather simple, almost minimalistic, particle content. In contrast to the conventional seesaw, this is achieved without need of supersymmetrization. We derived a general formula for / f parameters of the above quark the coherent μ− –e− conversion branching ratio in terms of the L level effective Lagrangian and we calculated the corresponding nuclear matrix elements of currently interesting nuclear targets like 197 Au (the current SINDRUM II target), 27 Al (the target of the ongoing MECO experiment) and 48 Ti (the target for the upcoming PRISM experiment). These results are given in Table 2 and can be used to obtain sensitivity limits from existing or / f parameters in a variety of particle physics models. We have considplanned experiments on L ered in detail the new important contributions to μ− –e− conversion present in the inverse seesaw model that come from the exchange of the relatively light SU(2) ⊗ U(1) singlet neutral leptons. Fig. 3 shows the sensitivity bounds on the parameters M and μ characterizing the inverse seesaw model, that follow from experiments with Al, Au and Ti, respectively. On the other hand Fig. 4 displays the relation of the L / f rates with the relevant neutrino mixing angles, while Fig. 5 establishes that, in the inverse seesaw model, the rates for μ− –e− conversion and that for the μ− → e− γ decay are highly correlated. Acknowledgements This work was supported by Spanish grants BFM2002-00345, FPA2005-01269 and BFM200200345, and by the European Commission Human Potential Program RTN network MRTN-CT2004-503369. T.S.K. and F.D. would like to express their appreciation to IFIC for hospitality. References [1] Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Rev. Lett. 81 (1998) 1562, hep-ex/9807003.

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