---
address: |
     Department of Physics and Astronomy\
    Michigan State University\
    East Lansing, MI 48824 USA\
    E-mail: shri@pa.msu.edu, yuan@pa.msu.edu
author:
- 'Shrihari Gopalakrishna and C.–P. Yuan'
title: |
     B-physics Signature of a\
    Supersymmetric U(2) Flavor Model[^1]
---

Introduction
============

The Standard Model (SM) of high energy physics suffers from the gauge hierarchy problem and the flavor problem. Supersymmetry (SUSY) eliminates the gauge hierarchy problem, and a (horizontal) flavor symmetry in generation space could explain the flavor problem. A SUSY theory with a flavor symmetry might relate the quark/lepton flavor structure with that of the scalar quark/lepton sector. Such a theory would imply certain predictions for flavor changing neutral current (FCNC) processes that we wish to investigate in this work, along with the constraints from experimental FCNC data.

We do not assume an alignment of the quark/lepton flavor structure with that of the scalar quark/lepton sector, leading to a non-minimal flavor violation (NMFV) scenario. We consider a spontaneously broken U(2) flavor symmetry [@Pomarol:1995xc; @Barbieri:1995uv] in the framework of “effective supersymmetry” [@Cohen:1996vb], in which the first two generation scalars are relatively heavy (a few TeV mass), thereby satisfying neutron electric dipole moment constraint, etc., while still allowing large CP violating phases in the scalar sector. We analyze the implications of such a framework to B-physics observables. We will present details in a forthcoming paper [@shricp].

Consider that the first and second generation superfields ($\psi_a$, a=1,2) transform as a U(2) doublet while the third generation superfield ($\psi$) is a singlet [@Barbieri:1995uv]. The most general U(2) symmetric superpotential can be written as = \_1 H + \_2 H \_a + \_a \_3 H \_b + \_a \_4 H \_b\
+ \_a \_5 H \_b + H\_u H\_d  , where $M$ is the cutoff scale below which such an effective description is valid, the $\alpha_i$ are O(1) constants, $\phi^a$ is a U(2) doublet, $\phi^{ab}$ and $S^{ab}$ are second rank antisymmetric and symmetric U(2) tensors respectively. If U(2) is broken spontaneously by the Vacuum Expectation Values (VEV) =

0\
V

;    = v \^[ab]{};    = 0, = V , with $\frac{V}{M} \equiv \epsilon \sim 0.02$ and $\frac{v}{M} \equiv \epsilon ' \sim 0.004$, and if U(2) is broken below the SUSY breaking scale, the SUSY breaking masses would also have a structure dictated by U(2). The resulting quark and scalar down-type masses are $$\begin{aligned}
{\cal M}_d =  v_d \begin{pmatrix} O & -\lambda_1\epsilon ' & O \\ \lambda_1\epsilon '  & \lambda_2\epsilon & \lambda_4\epsilon \\  O & \lambda_4'\epsilon & \lambda_3 \end{pmatrix} \ &,& \quad 
{\cal M}^2_{RL} = v_d \begin{pmatrix}O & -A_1 \epsilon ' & O \\ A_1 \epsilon '  & A_2\epsilon & A_4 \epsilon \\  O & A_4'\epsilon & A_3 \end{pmatrix} \ , \\
{\cal M}^2_{LL} = \begin{pmatrix}m_1^2 & 0 & 0 \\
     0 & m_1^2+\epsilon^2 m_2^2 & \epsilon m_4^{2*} \\ 
    0 & \epsilon m_4^2 & m_3^2 \end{pmatrix}_{LL}&,& \quad 
{\cal M}^2_{RR} = \begin{pmatrix} m_1^2 & 0 & 0 \\
     0 & m_1^2+\epsilon^2 m_2^2 & \epsilon m_4^{2*} \\ 
    0 & \epsilon m_4^2 & m_3^2\end{pmatrix}_{RR}, \nonumber
\label{MSQ.EQ}\end{aligned}$$ where $v_d = \left< h_d \right>$ is the VEV of the Higgs field, the $\lambda_i$’s are O(1) coefficients, and, $m_i$ and $A_i$ (complex in general) are determined by the SUSY breaking mechanism. It has been shown [@Barbieri:1995uv] that such a pattern of the quark mass matrix explains the quark masses and CKM elements.

For our study, we consider the following values for the various SUSY parameters: ${m_{\tilde b_R,\tilde t_R}}=100$GeV, the other squark masses given by $m_0=1000$GeV, $A=1000$GeV, $\tan{\beta}=5$, $|\mu|=150$GeV, $M_2=250$GeV, $M_{\tilde g}=250$GeV and $m_{H^\pm}=250$GeV. ($m_0$ and $A$ denote generic SUSY breaking mass scales.)

Here, we consider processes that go through the $b\rightarrow s$ quark level transition, and in our framework the dominant SUSY contributions are due to $\delta_{32,23}^{RL,RR,LL} \equiv \frac{({\cal M}^2_{RL,RR,LL})_{32,23}}{m_0^2}$. For the chosen values of the parameters, we find $|\delta_{32,23}^{RL}| \sim \frac{v_d A\epsilon}{\tilde{m}_0^2}=6.8\times 10^{-4}$, and, $|\delta_{32}^{LL,RR}| \sim \epsilon \frac{m_4^2}{m_0^2} = 0.02$.

B-physics probes
================

Given such an effective SUSY theory we estimate the sizes of various B-physics observables that we expect are modified from their SM predictions. In addition to the SM contribution, we include the charged Higgs, chargino and gluino contributions. We analyze the $\Delta B=1$ FCNC processes, , , , ; and the $\Delta B=2$ processes  mixing and the dilepton asymmetry in $B_s$. We find regions in U(2) SUSY parameter space that are consistent with current experimental data and obtain expectations for measurements that are forthcoming. To illustrate the effects, we present here expectations for CP asymmetries in  and , and,  mixing. A more exhaustive analysis will be presented elsewhere [@shricp].

The CP asymmetry in  is given by A\_[CP]{}\^ &&  , \[ACPBSG.EQ\] and the expectation in the U(2) SUSY theory is shown in Fig. (\[BSGACPLR.FIG\]). We see that significant CP asymmetry is possible in the scenario we are considering, while satisfying experimental constraints.

The CP asymmetry in  is defined by A\_[CP]{}\^ &&\
&=& -C\_[K]{} + S\_[K]{}, and Fig. (\[BPHIKCPBS.FIG\] left) shows the CP asymmetry in  for a scan on $\delta_{32}^{RL}$ while satisfying all experimental constraints.

The  mixing parameter $\Delta m_{B_s}$ depends quite sensitively on $\delta_{32}^{RR}$ and can be significantly altered from the SM prediction as shown in Fig. (\[BPHIKCPBS.FIG\] right).

In conclusion, we note that similar results hold for flavor models that have the same order of magnitude for the 23 element in the squark mass matrix. In such cases, the prospects look exciting for discovering SUSY in B meson processes at current and upcoming colliders.

[0]{}

A. Pomarol and D. Tommasini, Nucl. Phys. B [**466**]{}, 3 (1996). R. Barbieri, G. R. Dvali and L. J. Hall, Phys. Lett. B [**377**]{}, 76 (1996); R. Barbieri, L. J. Hall and A. Romanino, Phys. Lett. B [**401**]{}, 47 (1997). A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B [**388**]{}, 588 (1996). S. Gopalakrishna and C.–P. Yuan, In preparation.

[^1]: alk presented by at [ *2003: upersymmetry in the esert*]{}, held at the niversity of rizona, ucson, , une 5-10, 2003. o appear in the roceedings.
