Making Neutrinos Massive with an Axion in Supersymmetry

UCRHEP-T300 February 2001 arxiv:hep-ph/0102008v1 1 Feb 2001 Making Neutrinos Massive with an Axion in Supersymmetry Ernest Ma Physics Department, University of California, Riverside, California 92521 Abstract The minimal supersymmetric standard model (MSSM) of particle interactions is extended to include three singlet (right-handed) neutrino superfields together with three other singlet superfields. The resulting theory is assumed to be invariant under an anomalous global U(1) (Peccei-Quinn) symmetry. The soft breaking of supersymmetry at the TeV scale is shown to generate an axion scale f a given by m 2 M SUSY where m 2 is a large fundamental scale such as the Planck scale or the string scale. Neutrino masses are generated by f a according to the usual seesaw mechanism.

The minimal supersymmetric standard model (MSSM) is a well-motivated extension of the standard model (SM) of particle interactions. Nevertheless, it is missing at least two important ingredients. There is no neutrino mass and the strong CP problem [1] is unresolved. Whereas separate remedies exist for both shortcomings, they are in general unrelated [2]. In the following, I start with a supersymmetric theory of just one large fundamental mass (m 2 ) which may be the Planck scale or the string scale. I assume it to be invariant under an anomalous global U(1) symmetry which is an extension of the well-known Peccei-Quinn symmetry [3] to include three singlet (right-handed) neutrino superfields ( ˆN c ) and three other singlet superfields (Ŝ). The supersymmetry is then softly broken at M SUSY of order 1 TeV. As a result of the assumed particle content of the theory, an axion scale f a of order m 2 M SUSY 10 11 GeV is generated [4], from which neutrinos obtain masses via the usual seesaw mechanism with m N f a. Thus the two vastly different magnitudes of f a and m ν are both naturally derived from the minimal assumption of the existence of only m 2 and M SUSY. Consider first the MSSM superpotential: Ŵ = µĥuĥd + h u Ĥ u ˆQû c + h d Ĥ d ˆQ ˆdc + h e Ĥ dˆlê c. (1) Under U(1) PQ, the quark ( ˆQ, û c, ˆd c ) and lepton (ˆL, ê c ) superfields have charges +1/2, whereas the Higgs (Ĥu, Ĥd) superfields have charges 1. Hence the µ term is forbidden. It is replaced here by h 2 Ŝ 2 Ĥ u Ĥ d, where Ŝ2 is a singlet superfield with PQ charge +2. Add three singlet superfields ˆN c with PQ charges 1/2. The term h N Ĥ uˆl ˆNc is then allowed with the new assignment of charge 3/2 for ˆL and 1/2 for ê c, but the usual Majorana mass term m N ˆNc ˆNc is forbidden. Instead, it is replaced by h 1 Ŝ 1 ˆNc ˆNc, where Ŝ1 has PQ charge +1. [The PQ charge assignments in the MSSM are not unique because baryon number B and lepton number L are also conserved U(1) symmetries. Hence Q PQ Q PQ +L changes (1/2, 1/2) for ( ˆQ, ê c ) to (3/2, 1/2). With the introduction of ˆN c, L may now be broken (to 2

be shown below), and its PQ charge assignment of 1/2 will be unique.] So far there is no mass scale in the superpotential of this theory. It is thus natural to introduce a third singlet superfield Ŝ0 with PQ charge 2 so that the complete superpotential of this theory is given by Ŵ = m 2 Ŝ 2 Ŝ 0 + fŝ1ŝ1ŝ0 + h 1 Ŝ 1 ˆNc ˆNc + h 2 Ŝ 2 Ĥ u Ĥ d + h N Ĥ uˆl ˆNc + h u Ĥ u ˆQû c + h d Ĥ d ˆQ ˆdc + h e Ĥ dˆlê c. (2) The mass m 2 is a large fundamental scale such as the Planck scale or the string scale. The term Ŝ1Ŝ1Ŝ0 is automatically present and will be the key to understanding how f a is generated from M SUSY. [If the PQ charge of ˆNc were +1/2 instead, a different term, i.e. Ŝ 2 Ŝ 1 Ŝ 1, would be there.] Consider next the spontaneous breaking of U(1) PQ by the vacuum expectation values v 2,1,0 of S 2,1,0 respectively. The µ term of the MSSM is then given by h 2 v 2 and the Majorana mass of the neutrino singlet is 2h 1 v 1. Hence v 1 should be many orders of magnitude greater than v 2. This is a hierarchy problem but it will be resolved naturally here with the realization of the condition [4, 5] v 2 fv2 1 m 2 M SUSY, (3) as shown below. With m N = 2h 1 v 1, the usual seesaw relationship m ν = h2 Nv 2 u m N (4) is also obtained. Now the axion scale f a is of order v 1 as well, thus m N f a. Whereas this relationship was also proposed previously [2], the value of v 1 was an arbitrary input, and not predicted as it is here from Eq. (3). The strong CP problem is the problem of having the instanton-induced term [1] g 2 s L θ = θ QCD 64π 2ǫ µναβg µν 3 a Gαβ a (5)

in the effective Lagrangian of quantum chromodynamics (QCD), where g s is the strong coupling constant, and G µν a = µ G ν a ν G µ a + g sf abc G µ b Gν c (6) is the gluonic field strength. If θ QCD is of order unity, the neutron electric dipole moment is expected [6] to be 10 10 times its present experimental upper limit (0.63 10 25 e cm) [7]. This conundrum is most elegantly resolved by invoking a dynamical mechanism [3] to relax the above θ QCD parameter (including all contributions from colored fermions) to zero. However, this necessarily results [8] in a very light pseudoscalar particle called the axion, which has not yet been observed [9]. To reconcile the nonobservation of an axion in present experiments and the constraints from astrophysics and cosmology [10], three types of invisible axions have been discussed. The DFSZ solution [11] introduces a heavy singlet scalar field as the source of the axion but its mixing with the doublet scalar fields (which couple to the usual quarks) is very much suppressed. The KSVZ solution [12] also has a heavy singlet scalar field but it couples only to new heavy colored fermions. The gluino solution [13] identifies the U(1) R of superfield transformations with U(1) PQ and thus the axion is a dynamical phase attached to the gluino as well as all other superparticles. The present model is of the DFSZ type, but the crucial implementation of Eq. (3) follows from Refs.[4, 5]. Before discussing the spontaneous breaking of U(1) PQ in the context of soft supersymmetry breaking, consider Ŵ of Eq. (2) in terms of baryon number and lepton number. It is clear that the former is conserved as a global symmetry ( ˆQ has B = 1/3, û c and ˆd c have B = 1/3, all others have B = 0), whereas the latter is conserved only as a discrete symmetry (ˆL, ê c, and ˆN c are odd, all others are even). Thus the usual R parity of the MSSM is also conserved. The three ˆN c superfields are well-motivated because they allow small seesaw neutrino masses for neutrino oscillations [14, 15, 16]. The Peccei-Quinn symmetry is 4

well-motivated as the most attractive solution of the strong CP problem. Hence S 1 and S 2 are both well-motivated. Finally, S 0 is well-motivated because Ŵ should have a large fundamental mass scale. Given all these well-motivated inputs, Eq. (2) is uniquely determined and the two crucial extra terms m 2 Ŝ 2 Ŝ 0 and fŝ1ŝ1ŝ0 are automatically present [17]. Let Ŝ2,1,0 Ŝ2,1,0 + v 2,1,0, then the part of Ŵ affected by this change is given by Ŵ = (m 2 v 2 + fv 2 1 )v 0 + m 2 v 0 Ŝ 2 + 2fv 0 v 1 Ŝ 1 + (m 2 v 2 + fv 2 1 )Ŝ0 + fv 0 Ŝ 1 Ŝ 1 + (m 2 Ŝ 2 + 2fv 1 Ŝ 1 )Ŝ0 + h 1 v 1 ˆNc ˆNc + h 2 v 2 Ĥ u Ĥ d + fŝ1ŝ1ŝ0 + h 1 Ŝ 1 ˆNc ˆNc + h 2 Ŝ 2 Ĥ u Ĥ d. (7) There are two supersymmetric solutions: the trivial one with v 0 = v 1 = v 2 = 0, and the much more interesting one with v 0 = 0, v 2 = fv2 1 m 2, (8) where v 2 << v 1 unless v 1 m 2. The latter breaks U(1) PQ spontaneously and Ŵ becomes Ŵ = m 2 v 1 (v 1 Ŝ 2 2v 2 Ŝ 1 )Ŝ0 + h 1 v 1 ˆNc ˆNc + h 2 v 2 Ĥ u Ĥ d + fŝ1ŝ1ŝ0 + h 1 Ŝ 1 ˆNc ˆNc + h 2 Ŝ 2 Ĥ u Ĥ d, (9) which shows clearly that the linear combination v 1 Ŝ 1 + 2v 2 Ŝ 2 v 1 2 + 4 v 2 2 (10) is a massless superfield. Hence the axion is mostly contained in Ŝ1, and its effective coupling through Ŝ2 is suppressed by (2v 2 /v 1 )v 1 2 = 2v 1 1 as desired. At this point, the individual values of v 1 and v 2 are not determined. This is because the vacuum is invaraint not only under a phase rotation but also under a scale transformation as a result of the unbroken supersymmetry [18]. As such, it is unstable and the soft breaking 5

of supersymmetry at the TeV scale will determine v 1 and v 2, and v 0 will become nonzero. However, as shown already in Ref.[4], Eq. (8) will still be valid to a very good approximation. Specifically, the supersymmetry of this theory is assumed broken by all possible soft terms which are invariant under U(1) PQ. In particular, all the usual MSSM soft terms are present except for the µbh u H d term. However, the term m 2 S 0 + h 2 H u H d 2 is there, hence µb = h 2 m 2 v 0. Recall that the µ parameter of the MSSM is replaced here by h 2 v 2. Hence v 2 should be of order M SUSY and m 2 v 0 of order MSUSY 2, and that is exactly what will be shown in the following. These results also explain why both Ŝ2 and Ŝ0 are required. Add now the other soft terms of the scalar potential: V soft = µ 2 0 S 0 2 + µ 2 1 S 1 2 + µ 2 2 S 2 2 + [µ 2 20 S 2S 0 + µ 10 S1 2 S 0 + µ 12 S1 2 S 2 + h.c.], (11) where all parameters are of order M SUSY 1 TeV. Consider then the minimum of the scalar potential of S 2,1,0 with the addition of V soft, i.e. V min = (m 2 2 + µ 2 0)v 2 0 + µ 2 1v 2 1 + (m 2 2 + µ 2 2)v 2 2 + 2µ 2 20v 2 v 0 + 2(m 2 f + µ 12 )v 2 1 v 2 + 2µ 10 v 2 1 v 0 + 4f 2 v 2 0 v2 1 + f2 v 4 1, (12) where every quantity has been assumed real for simplicity. The conditions on v 0,1,2 are (m 2 2 + µ2 0 + 4f2 v 2 1 )v 0 + µ 2 20 v 2 + µ 10 v 2 1 = 0, (13) (m 2 f + µ 12 )v 2 + 1 2 µ2 1 + µ 10v 0 + 2f 2 v 2 0 + f2 v 2 1 = 0, (14) (m 2 2 + µ 2 2)v 2 + (m 2 f + µ 12 )v 2 1 + µ 2 20v 0 = 0. (15) From the above, the following solution (with µ 2 1 < 0 and fµ 12 < 0) is obtained: v 0 µ 10v 2 1 m 2 2 v 2 1 m 2v 2 f, (16) ( 6 1 + µ 12 m 2 f ) µ2 1 2f 2 (17)

v 2 µ2 1 4µ 12. (18) As promised, v 2 is indeed of order M SUSY and Eq. (8) is maintained to high accuracy. Thus v 1 is of order m 2 M SUSY and v 0 is of order M 2 SUSY /m 2. To understand why v 1 >> M SUSY is possible, consider the superfield of Eq. (10). The phase of the corresponding scalar field is the axion, but the magnitude is a physical scalar particle of mass 2µ 2 1 M SUSY, and the associated Majorana fermion (axino) has a very small mass, i.e. 2fv 0. [Because the soft supersymmetry breaking mass term S 1 S1 is not invariant under U(1) PQ, S1 does not pick up a mass of order M SUSY as in Ref.[4].] Hence v 1 0 does not necessarily imply that supersymmetry is broken at that scale. For example, the superfield N c has the large mass 2h 1 v 1, and M SUSY accounts only for the relative small splitting between its scalar and fermion components. As the electroweak SU(2) L U(1) Y gauge symmetry is broken by the vacuum expectation values v u,d of H u,d, the observed doublet neutrinos acquire naturally small Majorana masses given by m ν = h 2 N v2 u /(2fv 1) via the usual seesaw mechanism. Since H u,d have PQ charges as well, the axion field is now given by a = v 1θ 1 + 2v 2 θ 2 2v 0 θ 0 v u θ u v d θ d, (19) v1 2 + 4v2 2 + 4v0 2 + vu 2 + vd 2 where θ i are the various properly normalized angular fields, from the decomposition of a complex scalar field φ = (1/ 2)(v + ρ) exp(iθ/v) with the kinetic energy term µ φ µ φ = 1 2 ( µρ) 2 + 1 2 ( µθ) 2 ( 1 + ρ v ) 2. (20) The axionic coupling to quarks is thus [ ( ) 1 vu 1 ( µ a) ūγ µ γ 5 u + 1 ( ] vd 1 dγ 2 v 1 v u 2 v 1) µ γ 5 d = 1 ( µ a) qγ µ γ 5 q, (21) v d 2v 1 q=u,d as in the DFSZ model. 7

Consider now all the physical particles of this theory. (1) There is a superheavy Dirac fermion of mass m 2, formed out of S 0 and a field which is mostly S 2. The two associated scalars also have mass m 2 but S 0 = v 0 MSUSY 2 /m 2 whereas S 2 = v 2 M SUSY. (2) There are three heavy N c superfields with mass of order S 1 = v 1 m 2 M SUSY. They provide seesaw masses for the neutrinos and generate a primordial lepton asymmetry through their decays [19]. This gets converted into the present observed baryon asymmetry of the Universe through the B + L violating electroweak sphalerons [20]. (3) The particles of the MSSM and their interactions are all present, but with the µ parameter given by h 2 v 2 M SUSY and the µb parameter by h 2 m 2 v 0 MSUSY 2, thus solving the µ problem (i.e. why µ M SUSY and not m 2 ) without causing a µb problem. (4) Whereas the spontaneous breaking of U(1) PQ generates an axion (which is mostly the phase of S 1 ) at the scale v 1, thus solving the strong CP problem, the physical scalar field corresponding to the magnitude of S 1 has a mass M SUSY. It is effectively unobservable [4] because its couplings to all MSSM particles are suppressed by at least v 2,u,d /v 1. (5) What distinguishes this supersymmetric theory from all others is the existence of a very light axino of mass 2fv 0 MSUSY 2 /m 2. Since R parity is conserved and the axino has R = 1, it is the stable LSP (lightest supersymmetric particle) of this theory. In contrast to other axion proposals [21], there is now the exciting possibility of experimental verification at future colliders. The 7 7 mass matrix spanning the R = 1 neutral fermions of this theory in the basis ( B, W 3, H 0 u, H 0 d, S 2, S 0, S 1 ) is given by M = m 1 0 sm 3 sm 4 0 0 0 0 m 2 cm 3 cm 4 0 0 0 sm 3 cm 3 0 h 2 v 2 h 2 v d 0 0 sm 4 cm 4 h 2 v 2 0 h 2 v u 0 0 0 0 h 2 v d h 2 v u 0 m 2 0 0 0 0 0 m 2 0 fv 1 0 0 0 0 0 fv 1 2fv 0, (22) 8

where s = sin θ W, c = cos θ W, m 3 = M Z cosβ, m 4 = M Z sin β, with tan β = v u /v d. The mixings of S 1 with H u,d 0 and S 2 are then given by v d,u /v 1 and 2v 2 /v 1 respectively. Without S 2,1,0, the above is just the neutralino mass matrix of the MSSM and the LSP is a linear combination of the two gauginos and the two Higgsinos. In this theory, that combination is still a mass eigenstate but it is not the LSP and no longer stable. It will decay into the axino and the Z boson or a neutral Higgs boson. Assuming that the dominant decay is through the Ŝ2ĤuĤd term of Eq. (9), its lifetime is estimated to be of order 10 7 (v 1 /10 11 GeV) 2 s, so it will decay within a typical detector if v 1 is 10 10 GeV or less. The mass of the axion a is given by [10] m a = 1.3 10 4 (10 11 GeV/v 1 ) ev, (23) thus it is of the same order of magnitude as that of the axino S 1, i.e. MSUSY 2 /m 2 MSUSY 3 /v2 1 10 4 (10 11 GeV/v 1 ) 2 ev. The decay a S 1 S1 may be kinematically allowed and dominates over a γγ, but its lifetime is still very much longer than the age of the Universe. Hence both the axion and the axino are candidates for dark matter in this theory. In conclusion, a desirable extension of the MSSM has been presented which has only two input scales, i.e. the large fundamental scale m 2 and the soft supersymmetry breaking scale M SUSY. Assuming the validity of U(1) PQ and its implementation in terms of Eqs. (2) and (11), an axion scale m 2 M SUSY is generated, which solves the strong CP problem and makes neutrinos massive via the usual seesaw mechanism. The baryon asymmetry of the Universe is accommodated as well as the existence of dark matter. The µ problem of the MSSM is solved without causing a µb problem. Above all, a very light axino is predicted to be the true LSP and the decay of the MSSM LSP to it and a neutral Higgs boson at future colliders offers a direct experimental verification of this proposal. This work was supported in part by the U. S. Department of Energy under Grant No. DE- FG03-94ER40837. 9

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