A Simple Model of Neutrino Masses from Supersymmetry Breaking

Physics Letters B 593 (2004) 181–188 www.elsevier.com/locate/physletb

A simple model of neutrino masses from supersymmetry breaking

John March-Russell, Stephen West

Theoretical Physics, Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK Received 22 March 2004; received in revised form 19 April 2004; accepted 20 April 2004 Available online 25 May 2004 Editor: P.V. Landshoff

Abstract We analyze a class of supersymmetric models first introduced by Arkani-Hamed et al. and Borzumati et al. in which the light neutrino masses result from higher-dimensional supersymmetry-breaking terms in the MSSM super-potentials and Kahler potentials. The mechanism is closely related to the Giudice–Masiero mechanism for the MSSM µ parameter, and leads to TeV-scale right-handed neutrino and sneutrino states, that are in principle accessible to direct experimental study. The dominant contribution to the light neutrino (Majorana) mass matrix is a one-loop term induced by a lepton-number violating B-term for the sneutrino states that is naturally present. We focus upon the simplification and analysis of the flavour structure of this general class of models, finding that simple and novel origins for the light neutrino mass matrix are possible. We find that a subdominant tree-level ‘see-saw’ contribution may lead to interesting perturbations of the leading one-loop-induced flavour 2 2 structure, possibly generating the small ratio msolar/matm dynamically.  2004 Elsevier B.V. All rights reserved.

× −5 2 2 × −5 2 1. Introduction and 5.4 10 eV m21 9.5 10 eV and 2 0.23 sin θ21 0.39 from the solar data, while the 2 The case for the existence of small neutrino masses remaining real mixing angle is bounded by sin θ13 and associated physical neutrino mixing angles has 0.066. The phases that can appear in the neutrino mix- enormously strengthened in recent years as a conse- ing (MNS) matrix are currently unconstrained. quence of the now numerous experimental studies of The traditional and much studied explanation for atmospheric and solar neutrinos, and neutrinos from these small masses is the see-saw mechanism [6].This terrestrial sources. In particular, a recent global analy- hypothesises the existence of two or more standard- sis [1] which incorporates data from CHOOZ [2],SNO model-singlet right-handed (rhd) neutrino states Ni , [3], KamLAND [4] and Super-Kamiokande [5], leads with very large lepton-number violating Majorana to values for the mass difference squares, m2,and masses, which couple to the weak-SU(2) lepton dou- three real mixing angles θij of the neutrinos in the 3σ blets Lj via a conventional Yukawa coupling − − × 3 2 | 2 | × 3 2 λij Li Nj H involving the electroweak Higgs. The ranges, 1.4 10 eV m23 3.7 10 eV 2 Yukawa couplings are typically taken to be of size and 0.36 sin θ23 0.67 from the atmospheric data, comparable to that of either the charged lepton Yukawas or the quark Yukawas, depending on the pre- E-mail address: [email protected] (S. West). cise model. After the Higgs gains its vacuum expecta-

0370-2693/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.04.050 182 J. March-Russell, S. West / Physics Letters B 593 (2004) 181–188 tion value, v, this Yukawa coupling leads to the Dirac by no means the only way that such Majorana masses D = mass matrix mν λv. Finally, the light neutrino mass can be generated. matrix obtained by integrating out the heavy rhd neu- In this Letter we study a notably attractive alter- trinos is given by native, advocated by Arkani-Hamed et al. [7,8],and Borzumati et al. [9,10] (for related work see [11]), =− D T −1 D mν mν MR mν , (1) that links the light neutrino masses to TeV-scale su- where MR is the rhd Majorana mass matrix. If we persymmetry breaking physics, and which has the sig- take the heaviest Dirac mass to be of order either the nificant virtue, compared to the see-saw mechanism, τ lepton or bottom quark mass, then agreement with of being directly testable, at least in part, at the LHC the light neutrino mass inferred from the atmospheric and other proposed high-energy colliders. In overall 11 neutrino anomaly requires roughly MR ∼ 10 GeV, structure our models are similar to those previously while if the heaviest Dirac mass is taken to be studied in Refs. [7–10], in that the dominant contri- of order the top quark mass, as is commonly the bution to the light neutrino masses arises from a one- 15 case in explicit models, then MR ∼ 10 GeV. The loop diagram involving a supersymmetry breaking and comparative proximity of this second scale to the lepton-number violating B-term for the right-handed inferred supersymmetric grand unified (GUT) scale sneutrinos, but, by an alteration of the model, we have 16 MGUT 2 × 10 GeV is interpreted as evidence in been able to significantly simplify the way in which favour of the see-saw mechanism, as is the fact that rhd the flavour structure of the light neutrino mass matrix SU(3) × SU(2) × U(1) singlet neutrinos are naturally arises, and are thus able for the first time to study in included in many GUT theories. detail some of the consequences of this very attractive Certainly therefore the see-saw mechanism is an at- class of models. tractive explanation of why the light neutrino masses are so small. However, it is not without its faults. In particular, there is a tension between the strongly hi- 2. Outline of the model erarchical nature of the observed Yukawa couplings in the quark and charged lepton sectors, and the es- sentially hierarchy-free masses implied by the m2’s. Supersymmetric extensions of the standard model, Moreover, both the θ12 and θ23 mixing angles are large such as the minimal supersymmetric standard model while the angle θ13 is small which is in sharp contrast (MSSM), explain the origin of the weak scale in terms with the corresponding mixings in the quark sector of the scale of the coefficients of the supersymmetry which are all small. These problems can be solved in breaking soft operators that must be included within specific models, for example, the m2 values can be the MSSM. The usual assumption is that these susy- fitted by taking the spectrum of rhd neutrino masses to breaking operators arise from the (super)gravitational be hierarchical in such a way as to almost compensate mediation of susy breaking that occurs primordially for the hierarchical neutrino Yukawa couplings. But in some hidden sector at an intermediate scale mI ∼ this has the price of introducing a wide range of rhd 1010–1011 GeV, giving rise to soft susy breaking in 10 15 ∼ 2 neutrino masses MR ∼ 10 –10 which then require the MSSM of order TeV m√I /M.HereM is the 18 explanation. reduced Planck mass M = Mpl/ 8π = 2×10 GeV. In addition, there is the worrying question of the Infamously, there is one mass parameter in the MSSM, testability of the see-saw mechanism. Given the large the µHuHd superpotential interaction, that naively mass scale of the rhd neutrino states there is very appears to be independent of supersymmetry breaking. little prospect of their ever being a direct test of the If this were indeed true then we would lose completely correctness of the see-saw mechanism. We are forced our understanding of the origin of the weak scale to fall back on indirect, and sadly not definitive tests. in susy theories [12]. The realization of Giudice and For example, the discovery of neutrino-less double Masiero [13] was that the potentially Planck-scale beta decay would point to a Majorana nature for the µHuHd term in the superpotential can be forbidden light neutrinos as required by the see-saw mechanism, by a global symmetry, while an effective µ-term is but as we shall see below, the see-saw mechanism is generated from 1/M-suppressed terms in the Kahler J. March-Russell, S. West / Physics Letters B 593 (2004) 181–188 183 potential via supersymmetry breaking effects, thus R charge equal to 0, and the usual R-parity is naturally implying µ ∼ TeV. assumed, with in addition Rp(X) = Rp(Y ) =+1. All As emphasized by Arkani-Hamed et al. and Borzu- dimensionless couplings g,h, etc., are taken to be mati et al., the lesson of the Giudice–Masiero mecha- order one parameters. nism for neutrino masses in the context of susy theo- Let us now suppose that after supersymmetry is ries is that SM-singlet operators, such as the rhd neu- broken in the hidden sector at scale mI the field Y trino mass MRNN, or the neutrino Yukawa coupling acquires the following F -andA-component vacuum λLNHu, might only appear to be renormalizable su- expectation values, perpotential terms, but in fact may arise from 1/M- = = 2 suppressed terms involving the supersymmetry break- Y F FY fY mI , ing scale m . I Y = A = 0, (4) Specifically, consider the usual MSSM Lagrangian A Y to be supplemented by a set of superpotential and while the field X acquires the F and A-component Kahler terms involving the rhd neutrino superfields Ni expectation values: (i = 1, 2, 3 is a generation index), and two Standard- Model-singlet chiral superfields X and Y which arise Xij F = FXij = 0, from the hidden sector. In general, the fields which communicate supersymmetry breaking to the neutri- Xij A = AXij = aXij mI. (5) nos can either be flavour singlets or flavour non- Here f and a are order one parameters, and singlets. Let us generically call the flavor non-singlet Y Xij the zero entries for A and F can be replaced by fields, X, which therefore carry generation indices i, j, Y X non-zero values A m ,andF m2 without and the flavour singlet fields, Y , and for simplicity sup- Y I X i significant change. We will often rewrite the scale m pose that there is just one X field and one Y field. I in terms of the gravitino mass m = m2/M,andthe Then the model we wish to study is defined by the 3/2 I reduced Planck mass M. Ni -dependent terms in the superpotential Before we proceed to analyze the consequences of Xij Y the above effective Lagrangian for the light neutrino LW = d2θ g L N H + g L N H +··· , N M i j u M i i u masses and mixings, a comment is in order concerning 2 (2) the assumption that FX ij mI .Asiswellknown, typical supergravity mediation of susy breaking has while the set of terms involving the rhd N fields in the i difficulties with FCNC and CP violation constraints Kahler potential are unless there is a high degree of degeneracy among the Y † Y †Y squark and slepton soft masses of different generations LK = d4θ h N N + h˜ N†N N M i i M2 i i (we here ignore the possibility of alignment mechanisms which are typically much harder to implement). † † Y YXij Such degeneracy is accommodated in our model when + h N N +··· . (3) B 3 i j 2 M FX ij mI . In this Letter we will not be concerned about the detailed origin of this high-level of degener- The ellipses in Eqs. (2) and (3) stand either for terms acy, but take it as a phenomenologically necessary as- involving the replacement of Y fields by X fields ij sumption. As we will argue below, in this case a simple (with obvious changes to N flavour indices), or for i prediction for the structure of the light neutrino mass terms higher order in the 1/M-expansion. It is simple matrix can result. to check that both types of additional term will lead to trivial or sub dominant contributions not relevant for our discussion. The Lagrangian displayed in (2) and (3) can be justified with an R-symmetry, where 3. Neutrino and sneutrino masses at tree level both hidden sector fields X and Y have R charge 4/3, N has R charge 2/3, E (rhd charged lepton superfield) Given Eqs. (2) and (3), the relevant terms in has R charge 2 and the remaining superfields have the Lagrangian after the X and Y fields gain their 184 J. March-Russell, S. West / Physics Letters B 593 (2004) 181–188 expectation values are thus Note, however, that since the operators that give rise to the Yukawa and rhd neutrino mass terms 2 L = d θ λij Li Nj Hu + MN Ni Ni are higher-dimension non-renormalizable operators, a more appropriate estimate of the couplings g and + ˜ ˜ + 2 ˜ ˜ +··· ALini hu Bij ninj , (6) h might be the values found by applying the so- called ‘naive-dimensional analysis’ (NDA) methodol- where n˜ are the rhd sneutrino fields, L˜ is the i i ogy which assumes that the cutoff M is bounded by the lhd slepton doublet, h is the up-type Higgs scalar u UV strong coupling scale of the non-renormalizable doublet, and the omitted terms include the usual theory, and in which geometrical factors of 1/(4π)2 soft scalar mass terms. First note that the effective which enter loop calculations are taken into account neutrino Yukawa coupling in the superpotential of [14]. (It is known that NDA works well for estimat- Eq. (6) is suppressed in magnitude by a factor of 1/2 −7 −8 ing the coefficients of the higher-dimensional opera- (m3/2/M) ∼ 10 –10 , tors in the low-energy chiral-Lagrangian description m3/2 of QCD.) A simple calculation shows that the NDA λ = ga , (7) ij Xij M estimates for the sizes of the couplings h and g which set the rhd neutrino mass and lhd-rhd-Higgs Yukawa and gains its flavour structure from the Xij A expec- coupling are h 1andg 4π (and a UV strong cou- tation value, and in addition, the scale of the rhd neu- pling scale Λ related to the reduced Planck mass as trino masses is lowered to the TeV scale Λ 4πM). This leads to an improved estimate of the size of the tree (see-saw) contribution to the light neu- MN = hf Y m3/2. (8) trino masses Second, there exists a TeV-scale, but flavour diagonal, v2 2 β g2 trilinear scalar A-term tree − sin −3 mν ij 10 eV. (12) M h A = g f m . (9) Y 3/2 Alternatively, the large mass scale M might be the Finally, there is a small but significant rhd sneutrino string or susy-GUT scale, thus similarly increasing the tree lepton-number violating B-term with coefficient estimate for mν . In any case, we will argue in the next section that a small tree-level term of size Eq. (12) 5 m / can sometimes be an interesting perturbation to the B2 = h f 2as 3 2 (10) ij B Y Xij M dominant one-loop contribution to the light neutrino mass matrix. 2 ∼ × 2 of magnitude Bij (few 100 MeV) and flavour Turning to the sneutrino sector of the theory the structure related to that of the neutrino Yukawa cou- effective Lagrangian (6) implies, after electroweak s pling (aXij denotes the symmetric part of aXij ). symmetry breaking, a 12 by 12 mass matrix that mixes After electroweak symmetry breaking, with Higgs the lhd and rhd sneutrinos and their conjugates (here, 0= 0= 2 expectation values Hu v sin β and Hd v cosβ, for simplicity, assuming A and B real), the effective Lagrangian (6) implies that at tree level in  2  the neutrino sector, after integrating out the TeV-scale MLδij Av sin βδij 00  Av sinβδ M2 δ 0 B2  rhd neutrinos, there is generated a light neutrino mass  ij R ij ij  , 00M2 δ Av sin βδ matrix of see-saw type: L ij ij 2 2 0 Bij Av sinβδij MRδij 2 2 2 m tree =−v sin β T ∼ v 3/2 (13) mν ij λikλkj . (11) MN m3/2 M ˜ ∗ ˜ ˜ ˜∗ 2 = 2 + where we work in the basis (ν , n, ν,n ), ML mL 18 2 If one takes M = 2 × 10 GeV, v = 174 GeV, and mZ cos(2β)/2 is the lhd sneutrino mass arising from 2 all aXik, fY , g and h’s to be of order 1, this gives the usual soft mass mL and electroweak breaking 2 = 2 + 2 a contribution to the light neutrino mass matrix of terms, and MR mR MN is the total rhd sneutrino order 10−5 eV, certainly too small to generate the mass including the soft mass-squared. (Our notation is 2 2 ˜ required m23 and m12. that Latin indices run from 1 to 3, so, e.g., the n are J. March-Russell, S. West / Physics Letters B 593 (2004) 181–188 185 indexed as i + 3, and sneutrino mass eigenstates are labeled by Greek indices α, β = 1,...,12. ) This mass matrix is diagonalised by a unitary rotation of the form  √ √  V/ 20√ V/ 20√  0 V/ 20V/ 2  U =  √ √  −V/ 20√ V/ 20√ 0 −V/ 20V/ 2   cos φ sin φ 00  − sin φ cosφ 00 ×   , (14) Fig. 1. The dominant contribution to mν . 00cosφ sin φ − 00sin φ cosφ bottom of the loop. Thus only these states need to be = 2 − 2 expanded in terms of the 4 neutralino mass eigenstates, where tan 2φ 2Av sin β/(ML MR),andV is the 2 leading to 3 by 3 matrix that diagonalises Bij . The expression Eq. (14) is correct up to small terms of order 2 2M ∗ ∗ 2 B2/(M2 − M2 ). The corresponding sneutrino mass χ = Z N sin θ − N cosθ M , (16) L R x v2 x1 w x2 w χx eigenvalues fall into 2 groups of almost 3-fold degen- erate complex states. where NxI is the standard neutralino mixing matrix, and Mχx are the neutralino masses. A transparent and elegant form for the light neu- 4. Structure of light neutrino masses at 1-loop trino mass matrix results if we consider the situation in which the A-term mixing is small. This greatly simplifies the flavor structure of the result Eq. (15). The rhd sneutrino interaction Eq. (10) gives rise to (The V matrices in U of Eq. (14) cancel by virtue of a radiative contribution to the light neutrino masses, VV† = 1.) Moreover, in this limit the loop factor L illustrated in Fig. 1,that,forY ∼ m2 and X ∼ F I ij A also simplifies, mI, dominates over the tree level contribution arising from the mixing of the TeV-mass rhd neutrinos with 1 L = the lhd neutrino states. In detail, the contribution to the 6 − 3 − 2 − 2 MR(r 1) (r x) (1 x) light Majorana neutrino mass matrix from the diagram × (1 − r)(1 + r − 2x)(1 − x)(r − x) in Fig. 1 is given by + 2r2 − x3(1 + r)+ 2x2(1 + r2 + r) logr 2 2 2 2 χxA B v sin β mloop = p − x log(r/x) + r log(x3r) ν,ij 16π2 − 2 3 4 + 3 × † † † r log(x /r ) r log x , (17) Uα,i+6Uα,k+6Uβ,k+3Uβ,+3Uγ,p+3Uγ,n+3 2 2 2 2 † where r ≡ M /M and x ≡ M /M . Therefore, × U + U L(α, β, γ, δ, x), (15) L R χx R δ,n 6 δ,j+6 in this limit of small sneutrino mixing the 1-loop where all repeated indices are summed over. In this contribution to the light neutrino mass matrix becomes expression, χx is a factor that depends on the ex- 2 2 2 2 loop χx A Bij v sin β changed neutralino, U is the sneutrino mixing matrix, mν = x = 1,...,4 denote the neutralino mass eigenstates, ij (4π)2 x and L(α, β, γ, δ, x), is a totally symmetric function of × L M2 ,M2 ,M2 ,M2 ,M2 . (18) the sneutrino and neutralino masses that arises from L R R L χx the momentum integral in Fig. 1. The most important feature of the result Eq. (18) loop Because of the small effective Yukawa coupling, is that the overall scale of the contribution mν is Eq. (7), between the higgsinos, lhd neutrinos, and naturally of the correct size to account for atmospheric rhd sneutrinos, in practice only the bino and zino neutrino oscillations (as is also true of the models components of the neutralino are exchanged across the of Refs. [8,10]). To see this explicitly it is useful to 186 J. March-Russell, S. West / Physics Letters B 593 (2004) 181–188 consider the simple case in which all the lhd and rhd with the scale set by µ ∼ 0.1–0.01 eV, and in the sneutrinos and the neutralinos are approximately equal range ∼ 10−2–10−4. An attractive feature of this to a common mass scale, msusy,giving structure is that it allows us in a simple way to account 1 for the hierarchy between the solar and atmospheric L − (19) neutrino mass-squared splittings. The atmospheric m6 12 susy m2 can arise from the one-loop contribution, while 2 which leads to a one-loop contribution of magnitude the tree-level correction leads to the small msolar 9 2 splitting, the hierarchy being entirely due to the α m v − loop ∼ ≡ w I −2 −1 small dynamical parameter ∼ 10 2. Of course, it mν µ 5 5 10 –10 eV (20) 96π M msusy is possible that the hierarchy instead arises entirely s depending on the precise magnitude of the A and B from the flavour structure of the dominant µaX term, terms. In addition, in our model, the flavour struc- the tree-level perturbation being insignificant, but this ture of this dominant one-loop contribution to the light leads to quite traditional neutrino flavour models, so neutrino mass matrix is determined directly and en- we here focus upon the new possibility. tirely by the rhd sneutrino lepton-number violating As a simple example of a model along these 2 lines consider the situation in which the matrix a B-term, Bij , which is in turn generated by the Xij A X expectation value. Moreover, as claimed earlier, the corresponds to the leading-order form for an inverted one loop result dominates over the tree level see-saw hierarchy model contribution. It is useful to define the (small) parameter as the ratio of magnitudes of the tree-level see- 01a = saw contribution Eq. (12) to the above 1-loop contri- aX 100. (24) bution. a 00 It is also interesting to consider the regime in which Therefore, in this case the full light neutrino mass the rhd sneutrino states are heavy compared to the matrix including both loop and tree contributions has neutralino and lhd sneutrino states, r x 1. In this the form case the loop factor is approximated by the expression (1 + a2) 1 a (x − r)+ x log(r/x) 1 tot = L − . (21) mν µ 1 a. (25) 6 − 2 4 2 2 MR(r x) 2MRML aaa ∼ ∼ Taking ML Mχ msusy and scaling the A and B This mass matrix has one zero√ eigenvalue, and two 2 ∼ 2 terms relative to their natural values, A0 msusy,and massive eigenvalues m± µ( a2 + 1 ± ).There- 2 ∼ 2 1/2 B0 msusy(msusy/M) given by Eqs. (9) and (10), fore, to accommodate the oscillation data requires leads to (1 + a2)µ2 2 × 10−3 eV2, while the solar oscilla- 2 1/2 2 −5 2 7/2 2 2 2 tion data requires 4(1 + a ) µ 7 × 10 eV . msusyM A B mloop Z . (22) For a ∼ 1thisgives ν 2 4 1/2 2 2 16π MRM A0 B0 2 −3 2 −2 Assuming msusy ∼ 300 GeV, and MR ∼ 1 TeV this µ 10 eV , 10 (26) gives comfortably of the sizes expected in this model. More- 2 2 loop A B mtot m ∼ (0.02 eV) over, since ν is diagonalised by first performing a 23 ν 2 2 = −1 A0 B0 rotation, R23(θ), with angle θ tan (a),andthen a 12 rotation, R (φ), with φ = π/4, we see that for showing that M cannot be much heavier than 1 TeV 12 R a ∼ 1 the atmospheric and solar angles will both be unless the scale of the MSSM superpartners is uncom- close to maximal. Specifically, recalling that the phys- fortably high. ical neutrino mixing (MNS) matrix includes a uni- In either case the final structure of the light neutrino V mass matrix is in total tary matrix L from the rotation of the charged leptons to a basis where the charged lepton mass ma- tot = s + T mν ij µ aX aXaX ij (23) trix is diagonal and real we find a MNS matrix of the J. March-Russell, S. West / Physics Letters B 593 (2004) 181–188 187 form the MSSM Higgs µ parameter, and in particular leads to TeV-scale rhd neutrino and sneutrino states, that are V = V RT (θ)RT (φ) MNS L 23 12 in principle accessible to direct experimental study, cφ sφ 0 unlike traditional see-saw mechanisms. A second dif- = VL −cθ sφ cθ cφ sθ . (27) ference is that the dominant contribution to the light sθ sφ −sθ cφ cθ neutrino mass matrix (which is of Majorana type) is Under the not unreasonable assumption that the mix- a one-loop term induced by a lepton-number violating ing angles in from the charged lepton sector are and supersymmetry breaking B-term for the sneutrino small, VL only slightly perturbs the above struc- states that is naturally present in the model. In this ture, thus leading to almost maximal physical at- Letter we have focused upon the simplification and mospheric and solar mixing angles, and a small θe3 analysis of the flavour structure of this general class of angle. models, and have found that simple predictions for the Alternatively, light neutrino masses with a normal light neutrino mass matrix are possible. In addition we hierarchy can be easily generated if aX contains an have found that the subdominant tree-level ‘see-saw’ a contribution may lead to interesting perturbations of antisymmetric piece, aX. In particular, consider the forms the leading one-loop-induced flavour structure, possibly generating the smaller m2 . 000 solar In this Letter we have not explored the important as = 0 bb, X issues of the possible collider and cosmological tests 0 bb of our models. In broad structure the implications of 0 cd our models are similar to those already analyzed in a = − aX c 0 e . (28) Refs. [7–10]. In particular, one expects the A-term − − d e 0 interactions in our model to lead to interesting pos- Substitution of these matrices into Eq. (23) with rea- sibilities for production and decay of the TeV-scale sonable (and not fine-tuned) O(1) values for the pa- rhd sneutrino states, including the possibility of anom- rameters b, c, d,ande, and values of the dynami- alous Higgs decays. Another intriguing possibility is − cally determined parameters µ2 few × 10 3 eV2, that the rhd sneutrino states could be the dark matter − and 10 2, leads to a light neutrino mass matrix, [15], or that, when CP violation in the neutrino sec- which when diagonalised produces mass squared dif- tor is taken into account, dark matter with dominant ferences within the experimental bounds for a normal inelastic interactions with matter results [16].Inafu- hierarchy. Moreover, again under the assumption that ture publication, [17], we show that the class of mod- the real mixing angles in VL are small this leads to els analyzed here naturally lead to a very attractive and large physical atmospheric and solar mixing angles, successful theory of TeV-scale resonant leptogenesis, and a small θe3 angle. More generally, if the matrix aX developing from the earlier work of [18]. is not real as in the above examples, but contains large phases then it is simple to generate successful mod- 2 els in which the one-loop term gives matm while the Acknowledgements 2 perturbing tree term leads to msolar. We wish to thank Thomas Hambye, Graham Ross and Lotfi Boubekeur for discussions. S.W. is supported 5. Comments and conclusions by PPARC Studentship Award PPA/S/S/2002/03530.

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