Neutrino assisted gauge mediation

Abstract

Recent observation shows that the Higgs mass is at around 125 GeV while the prediction of the minimal supersymmetric standard model is below 120 GeV for stop mass lighter than 2 TeV unless the top squark has a maximal mixing. We consider the right-handed neutrino supermultiplets as messengers in addition to the usual gauge mediation to obtain sizeable trilinear soft parameters A _t needed for the maximal stop mixing. Neutrino messengers can explain the observed Higgs mass for stop mass around 1 TeV. Neutrino assistance can also generate charged lepton flavor violation including μ→eγ as a possible signature of the neutrino messengers. We consider the S ₄ discrete flavor model and show the relation of the charged lepton flavor violation, θ ₁₃ of neutrino oscillation and the muon’s g−2.

Appendices

Appendix A: Sparticle spectrum sample point

Table 3 Sparticle spectrum at the point giving 125 GeV Higgs mass with the lowest B _N

Appendix B: Representations of S ₄ symmetry and tensor products

Table 4 Sparticle spectrum at the point giving 123 GeV Higgs mass with the lowest B _N

The S ₄ is a non-abelian discrete symmetry and consists of all permutations among four quantities. For a review, see [83]. Irreducible representations of S ₄ are two singlets 1,1′, one singlet 2, and two triplets 3,3′. Tensor products among them are given as follows:

$$\begin{aligned} &{\left ( \begin{array}{c} x_1 \\x_2 \\x_3 \end{array} \right )_\mathbf{3} \times \left ( \begin{array}{c} y_1 \\y_2 \\y_3 \end{array} \right )_\mathbf{3}} \\ &{\quad = (x_1y_1+x_2y_2+x_3y_3)_\mathbf{1}+ \left ( \begin{array}{c} x_1 y_1+\omega x_2y_2 +\omega^2 x_3y_3\\x_1 y_1+\omega^2 x_2y_2 +\omega x_3y_3 \end{array} \right )_{\mathbf{2}}} \\ &{\qquad {}+ \left ( \begin{array}{c} x_2y_3 + x_3y_2\\x_3 y_1+ x_1y_3 \\x_1y_2+x_2y_1 \end{array} \right )_\mathbf{3} + \left ( \begin{array}{c} x_2y_3 - x_3y_2\\x_3 y_1- x_1y_3 \\x_1y_2-x_2y_1 \end{array} \right )_{\mathbf{3}^\prime},} \end{aligned}$$

(B.1)

$$\begin{aligned} &{\left ( \begin{array}{c} x_1 \\x_2 \\x_3 \end{array} \right )_{\mathbf{3}^\prime} \times \left ( \begin{array}{c} y_1 \\y_2 \\y_3 \end{array} \right )_{\mathbf{3}^\prime}} \\ &{\quad = (x_1y_1+x_2y_2+x_3y_3)_\mathbf{1}+ \left ( \begin{array}{c} x_1 y_1+\omega x_2y_2 +\omega^2 x_3y_3\\x_1 y_1+\omega^2 x_2y_2 +\omega x_3y_3 \end{array} \right )_{\mathbf{2}}} \\ &{\qquad{}+ \left ( \begin{array}{c} x_2y_3 + x_3y_2\\x_3 y_1+ x_1y_3 \\x_1y_2+x_2y_1 \end{array} \right )_\mathbf{3} + \left ( \begin{array}{c} x_2y_3 - x_3y_2\\x_3 y_1- x_1y_3 \\x_1y_2-x_2y_1 \end{array} \right )_{\mathbf{3}^\prime},} \end{aligned}$$

(B.2)

$$\begin{aligned} &{\left ( \begin{array}{c} x_1 \\x_2 \\x_3 \end{array} \right )_\mathbf{3} \times \left ( \begin{array}{c} y_1 \\y_2 \\y_3 \end{array} \right )_{\mathbf{3}^\prime}} \\ &{\quad = (x_1y_1+x_2y_2+x_3y_3)_{\mathbf{1}^\prime}} \\ &{\qquad{}+ \left ( \begin{array}{c} x_1 y_1+\omega x_2y_2 +\omega^2 x_3y_3\\-(x_1 y_1+\omega^2 x_2y_2 +\omega x_3y_3) \end{array} \right )_{\mathbf{2}}} \\ &{\qquad{} + \left ( \begin{array}{c} x_2y_3 + x_3y_2\\x_3 y_1+ x_1y_3 \\x_1y_2+x_2y_1 \end{array} \right )_{\mathbf{3}^\prime} + \left ( \begin{array}{c} x_2y_3 - x_3y_2\\x_3 y_1- x_1y_3 \\x_1y_2-x_2y_1 \end{array} \right )_\mathbf{3},} \end{aligned}$$

(B.3)

$$\begin{aligned} &{\left ( \begin{array}{c} x_1 \\x_2 \end{array} \right )_{\mathbf{2}} \times \left ( \begin{array}{c} y_1 \\y_2 \end{array} \right )_{\mathbf{2}}} \\&{\quad = (x_1y_2+x_2y_1)_\mathbf{1}+ (x_1y_2-x_2y_1)_{\mathbf{1}^\prime}+ \left ( \begin{array}{c} x_2y_2 \\x_1 y_1 \end{array} \right )_{\mathbf{2}},} \end{aligned}$$

(B.4)

$$\begin{aligned} &{\left ( \begin{array}{c} x_1 \\x_2 \end{array} \right )_{\mathbf{2}} \times \left ( \begin{array}{c} y_1 \\y_2 \\y_3 \end{array} \right )_\mathbf{3}} \\&{\quad = \left ( \begin{array}{c} (x_1+x_2)y_1\\(\omega^2 x_1+\omega x_2) y_2 \\(\omega x_1 + \omega^2 x_2)y_3 \end{array} \right )_\mathbf{3} + \left ( \begin{array}{c} (x_1-x_2)y_1\\(\omega^2 x_1-\omega x_2) y_2 \\(\omega x_1 - \omega^2 x_2)y_3 \end{array} \right )_{\mathbf{3}^\prime},} \end{aligned}$$

(B.5)

$$\begin{aligned} &{\left ( \begin{array}{c} x_1 \\x_2 \end{array} \right )_{\mathbf{2}} \times \left ( \begin{array}{c} y_1 \\y_2 \\y_3 \end{array} \right )_{\mathbf{3}^\prime}} \\&{\quad = \left ( \begin{array}{c} (x_1+x_2)y_1\\(\omega^2 x_1+\omega x_2) y_2 \\(\omega x_1 + \omega^2 x_2)y_3 \end{array} \right )_{\mathbf{3}^\prime} + \left ( \begin{array}{c} (x_1-x_2)y_1\\(\omega^2 x_1-\omega x_2) y_2 \\(\omega x_1 - \omega^2 x_2)y_3 \end{array} \right )_\mathbf{3}} \end{aligned}$$

(B.6)

and, trivially, we have 3×1′=3′, 3′×1′=3′, and 2×1′=2.

Appendix C: Remarks on the flavon vacuum stability

In [55], it was shown that the A ₄ triplet flavon vacuum in the direction of (1,1,1) and (1,0,0) ((0,1,0), (0,0,1) are the same) is favored compared to other directions, such as (1,1,0). Since A ₄ symmetry is the subgroup of the S ₄ composed of even permutations, similar arguments hold. In this appendix, we argue that triplet flavon directions favored in the A ₄ model are also favored in the S ₄ model and that the S ₄ doublet vacuum favors the (1,1) direction.

Rigid SUSY makes the discussion more simple, because the potential V is minimized at 〈V〉=0. On the other hand, extra symmetries like Z ₄ and U(1) _L more restrict possible terms in the superpotential. Suppose that U(1) _L symmetry is discretized to, for example, Z ₈ symmetry. In this case, only quartic terms Φ ⁴ and χ ⁴ are allowed. Let us assume that breaking of extra symmetries introduces quadratic term, like m ₁ Φ ² or m ₂ χ ². To achieve this, let us consider ‘Z ₄ breaking singlets’ ψ ₁, $\bar{\psi}_{1}$ and ‘lepton number breaking singlets’ ψ ₂, $\bar{\psi}_{2}$ with S ₄×Z ₄×U(1) _L quantum numbers

(C.1)

They do not combine with $\bar{E}LH_{d}$, NLH _u and NN to make singlets under all symmetries imposed. Then, they can couple to Φ ² or χ ² such that a superpotential is given by

$$\begin{aligned} &{W(\psi_1, \bar{\psi}_1, \psi_2, \bar{\psi}_2)} \\&{\quad =\frac {1}{\varLambda}\bigl[ \varPhi^2\bar{\psi}_1\psi_1 + \chi^2\bar{\psi}_2\psi _2\bigr]} \\&{\qquad{}-M_1 \bar{\psi}_1\psi_1 + \frac{1}{\varLambda}\bigl[\kappa_1(\bar {\psi}_1 \psi_1)^2+\kappa_2 (\psi_1)^4 + \kappa_3 (\bar{\psi }_1)^4\bigr]} \\&{\qquad{} -M_2 \bar{\psi}_2\psi_2 +\frac{1}{\varLambda} \bigl[\kappa _1^\prime(\bar{\psi}_2 \psi_2)^2+\kappa_2^\prime( \psi_2)^4 + \kappa_3^\prime(\bar{ \psi}_2)^4\bigr] . } \\ \end{aligned}$$

(C.2)

In this superpotential, $\bar{\psi_{1}}{\psi}_{1}$ and $\bar{\psi _{2}}{\psi}_{2}$ pairs have VEVs and they provide m ₁ Φ ²+m ₂ χ ² terms. With this setup, the triplet superpotential has the form of

$$\begin{aligned} W =&mS^2+\frac{\lambda_1}{\varLambda} \bigl(x^2+y^2+z^2\bigr)^2 \\&{}+ \frac {\lambda_2}{\varLambda}\bigl(x^2+\omega y^2+ \omega^2 z^2\bigr) \bigl(x^2+ \omega^2 y^2+\omega z^2\bigr) \\&{}+\frac{\lambda_3}{\varLambda}(xy+yz+zx)^2, \end{aligned}$$

(C.3)

where S=(x,y,z) represents the generic S ₄ triplet such as Φ or χ. Note also that the superpotential has an accidental Z ₂ symmetry under which ψ _1,2 and $\bar{\psi}_{1,2}$ are odd whereas other fields are even. If this Z ₂ symmetry is imposed, $(\varPhi^{2}\psi_{1}/\varLambda^{3})\bar{E}LH_{d}$ and (Φ ² ψ ₂/Λ ²)NN terms, which change the flavor structure in the subleading orders are forbidden. In this case, charged lepton Yukawa coupling structure in dimension-4 operator is preserved up to dimension-6 operator whereas Majorana mass structure in dimension-3 operator is preserved up to dimension-5 operator so corrections to them are highly suppressed.

Each term of the F-term potential V=|∂W/∂x|²+|∂W/∂y|²+|∂W/∂z|² is given by

(C.4)

A stable vacuum requires that these three terms should be zero simultaneously. For vacuum 〈S〉=v(1,1,1), three terms give the same condition,

$$ 12(\lambda_1+\lambda_3) \biggl(\frac{v^3}{\varLambda} \biggr)+mv=0 $$

(C.5)

so the vacuum is stabilized at v ²=−mΛ/[12(λ ₁+λ ₃)]. For vacuum 〈S〉=v(1,0,0), the second and third terms vanish trivially and the first term gives

(C.6)

so the vacuum is stabilized at v ²=−mΛ/[4(λ ₁+λ ₃)]. The vacuum in the direction (0,1,0) and (0,0,1) gives the same result by permutational property of S ₄. On the other hand, vacuum 〈S〉=v(1,1,0) gives two conditions,

(C.7)

If λ ₃ is not forbidden by another symmetry, v=0 is the only solution and nontrivial vacuum cannot be developed.

The S ₄ doublet stabilization can be discussed in the same way. The renormalizable superpotential for doublet (x,y) is written as

(C.8)

and the stabilization condition

(C.9)

requires that x=y. So the vacuum choice for Eq. (33) is stable.

Appendix D: Comment on Kähler potential corrections

In our setup, Yukawa couplings are constructed from non-renormalizable dimension-4 superpotential with several flavons. These flavons also appear in the non-renormalizable Kähler potential and kinetic terms are written in the form of

(D.1)

where ϕ and ψ represent bosonic and fermionic fields, respectively. The Kähler potential of charged lepton supermultiplet L is given by

(D.2)

and similar terms can be written for other fields, $\bar{E}^{\dagger}\bar{E}$, N ^† N, $H_{u,d}^{\dagger}H_{u,d}$, etc. Then we have quite complicate terms. For example, from $(\varPhi_{\mathbf{3}}^{\dagger}\varPhi _{\mathbf{3}}/\varLambda^{2}) L^{\dagger}L$ where Φ ₃ vacuum is given by v ₂(1,1,1), we have

$$\begin{aligned} &{a_1\frac{\varPhi_\mathbf{3}^\dagger\varPhi_\mathbf{3}}{\varLambda^2} L^\dagger L \bigg|_{S_4~{\mathrm{singlets}}}} \\&{\quad = a_{1,1}\frac{v_2^2}{\varLambda ^2} \bigl(L_1^\dagger L_1 + L_2^\dagger L_2 + L_3^\dagger L_3\bigr)} \\&{\qquad{}+a_{1,2}\frac{v_2^2}{\varLambda^2} \bigl[ L_2^\dagger L_3 + L_3^\dagger L_2 + L_3^\dagger L_1+ L_1^\dagger L_3 + L_1^\dagger L_2} \\&{\qquad{} + L_2^\dagger L_1 \bigr].} \end{aligned}$$

(D.3)

Since 〈Φ〉/Λ=v ₂/Λ is responsible for charged lepton Yukawa couplings, we see $4\pi v_{2}/\varLambda\gtrsim Y_{\tau}= m_{\tau}/ [(v/\sqrt{2})\cos\beta] \sim0.1$ for tanβ=10. On the other hand, χ ₃ has another vacuum direction, w ₂(0,1,0). Then

$$\begin{aligned} &{a_2\frac{\chi^\dagger\chi}{\varLambda^2}L^\dagger L \bigg|_{S_4~{\mathrm{singlets}}}} \\&{\quad =a_{2,1}\frac{w_2^2}{\varLambda^2}\bigl(L_1^\dagger L_1 + L_2^\dagger L_2 + L_3^\dagger L_3\bigr)} \\&{\qquad{}+a_{2,2}\frac {w_2^2}{\varLambda^2}\bigl(-L_1^\dagger L_1 + L_2^\dagger L_2 - L_3^\dagger L_3\bigr)} \end{aligned}$$

(D.4)

so it just rescales the fields. Moreover, since See-Saw scale is about 10¹⁴ GeV, we have suppressed effect, 4πχ/Λ∼0.01 with Λ is the GUT scale. In the same way, doublet and singlet flavons in the Kähler potential just contribute to the field rescalings.

Physical fields are defined with canonical kinetic terms, so we should make field redefinitions and they affect flavor structures in principle. In our work, however, such effects are not considered by assuming small coefficients a _1,2. For example, diagonalization of Y _E demonstrated above is not affected if $a_{1} ( v_{2}^{2}/\varLambda^{2}) \lesssim(m_{e}/m_{\tau}) \sim3 \times10^{-4}$, i.e. a ₁≲3.

On the other hand, mixings in the Kähler potential between flavons can be dangerous. For example, kinetic mixing between flavons such as $\bar{\psi}_{1}^{\dagger}\psi_{2}^{\dagger}\varPhi_{\mathbf{3}}^{\dagger}\chi _{\mathbf{3}}/\varLambda^{2}$ can introduce small correction to Y _E or M _N with unwanted S ₄ triplet vacuum direction. Such an effect is suppressed by $\bar{\psi}_{1}^{\dagger}\psi_{2}^{\dagger}/\varLambda^{2}$ and can be more suppressed with a tiny coefficient.