---
address: |
     Department of Physics, Korea Advanced Institute for Science and Technology, Daejeon, 305-701, Korea\
    [E-mail:abe@hep.kaist.ac.kr, kchoi@hep.kaist.ac.kr, ksjeong@hep.kaist.ac.kr, okumura@hep.kaist.ac.kr]{}
author:
- 'HIROYUKI ABE, KIWOON CHOI, KWANG-SIK JEONG AND KEN-ICHI OKUMURA'
title: ' Flavor Structure of Scherk-Schwarz Supersymmetry Breaking and Constraints from Low Energy Processes [^1] '
---

Localization of matter fields along the extra-dimension can naturally reproduce observed Yukawa hierarchy [@paper1], while the Scherk-Schwarz mechanism provides a simple alternative way to break supersymmetry [@paper2]. 5D orbifold $S^1/Z_2$ is a minimal realization of both of these attractive features of introducing new dimensions. Let us first define the model using 5D bulk action on $S^1$ in terms of 4D superfield representation [@paper3], $$\begin{aligned}
S_{bulk} &=&
\int d^4x \int dy \,\left[\,
\int d^4 \theta \,
\left\{
Re(T)\left( \Phi_I \Phi^*_I+\Phi^c_I
                  \Phi^{c*}_I\right)
+\frac{1}{g_{5 a}^2}
\frac{1}{Re(T)}
\left(
\partial_5 V -\sqrt{2}Re(\chi)
\right)^2
\right\}
\right.
\nonumber\\
&&
\left.\phantom{\int d^5x \,[}
+ \left(\,\int d^2\theta\,
\Phi^c_I\left(\partial_y-\frac{1}{\sqrt{2}}\chi+M_I T \right)\Phi_I
+\frac{1}{4g^2_{5a}}T W^{a\alpha} W^a_{\alpha} + {\rm h.c.}\right)
\,\right],
\label{eq1}\end{aligned}$$ where 4D chiral multiplets $(\Phi_I, \Phi^c_I)$ and 4D vector and chiral multiplets $(V, \chi)$ constitute 5D hyper and vector multiplets respectively. 5D supersymmetry forbids bulk Yukawa coupling. Here, radius modulus R of $S^1$, satisfying $x^5 = R y~~(0\le y<2 \pi)$, is promoted to the radion chiral multiplet $T$ so that 4D N=1 supersymmetry is manifest. In 5D supergravity, it is identified as a part of 5D supergravity multiplet [@paper2]. Because we are interested in 4D chiral theories, we orbifold $S_1$ by $Z_2$ imposing boundary conditions, $${V}^a(-y)={V}^a(y),\quad
{\chi}^a(-y)=-{\chi}^a(y),\quad
\Phi_I (-y) =\Phi_I (y),\quad
\Phi_I^c(-y) = -\Phi^c_I (y),
\label{eq2}$$ so that 4D $N=2$ supersymmetry is broken to $N=1$. After orbifolding, the surviving bulk mass should be regarded as a kink mass, $M_I \rightarrow M_I\epsilon(y)$ respecting $Z_2$. Then the zero mode equation for $\Phi_I$ fermion component, $\psi_I=
\chi(x)\tilde{\phi}_{0I}(y)$ is given by $
\big(\,\partial_y+M_IT\epsilon(y)\,\big)\tilde{\phi}_{0I}=0,
$ yielding the zero mode wavefunction $
\tilde{\phi}_{0I}\propto e^{-M_IT|y|},
$ which shows that it is quasi-localized at $y=0$ if $M_I>0$, and at $y=\pi$ if $M_I<0$. Inspired by this observation, we assume Higgs fields exist on the fixed point at $y=0$ and introduce the following brane action, $$S_{brane}=\int d^4 x \int d^2 \theta \, H
\int d y
\,\delta(y) 
\lambda_{IJ}^{(5)} \Phi_I \Phi_J  
+ {\rm h.c}\,,
\label{eq3}$$ After redefinition of the chiral fields, ${\Phi}_I \,\rightarrow\, e^{M_I T |y|} \Phi_I$,  ${\Phi}^c_I \,\rightarrow\, e^{-M_I T |y|} \Phi^c_I$, and appropriate renormalization, we can extract the 4D effective action for $\Phi_I$ zero mode, $$\begin{aligned}
&&S_{4D} = \int d^4x
\left[\int d^4\theta
\left(
\Phi_I \Phi_I^\ast+H H^\ast
\right)
+\left( \int d^2\theta\, y_{IJ} H \Phi_I \Phi_J
+{\rm H.c.}
\right)
\right],
\label{eq4}\end{aligned}$$ where we use the same symbol $\Phi_I$ for the corresponding canonical zero-modes and, $$y_{IJ}=\lambda_{IJ}^{(4)}
\sqrt{\frac{N_IN_J}{(\epsilon^{-2N_I}-1)(
\epsilon^{-2N_J}-1)}}
,~~{\rm for}~~
\lambda^{(4)}_{IJ}=\frac{\ln(1/\epsilon)}{\pi R}\lambda^{(5)}_{IJ}
,~~
N_I=-\frac{\pi R}{
\ln(1/\epsilon)}M_I.
\label{eq5}$$ Here, we choose $
\epsilon\,\equiv \,\mbox{Cabibbo angle}\,
\approx 0.2
$ for later convenience. Naive dimensional argument shows $\lambda^{(4)}_{IJ}\sim {\cal O}(1)$ while hierarchical $y_{IJ}$ is naturally realized for $N_I>0$.

So far the 4D effective theory is supersymmetric, however, the remaining supersymmetry can be generally broken by non-vanishing radion F-term, $F^T$ [@paper4]. The resultant soft terms are easily obtained by standard calculus as follows, $$S_{soft} = -\int d^4x \, \Big(\,
\frac{1}{2}m^2_{IJ}\phi^*_I\phi_J+
A_{IJ}\phi_I\phi_J h+
\frac{1}{2}M_a\lambda^a\lambda^a+{\rm h.c.}\,\Big),
\label{eq6}$$ $$\begin{aligned}
&&
M_a= M_{1/2}=-\frac{F^T}{2R}\,,~~~~~
A_{IJ}=2y_{IJ}\ln(1/\epsilon)\frac{F^T}{2R}\left(\,
\frac{N_I}{1-\epsilon^{2N_I}}+
\frac{N_J}{1-\epsilon^{2N_J}}\,\right),
\nonumber \\
&&
m^2_{I\bar{J}}=\delta_{IJ}\left(\,
2\ln(1/\epsilon)\frac{N_I}{\epsilon^{N_I}-
\epsilon^{-N_I}}\frac{F^T}{2R}\,\right)^2,
\label{eq7}\end{aligned}$$ where lower-case fields denote scalar components of the chiral multiplets and $\lambda^a$ represents gaugino. In $F^T/R<<1$ limit, this mass spectrum exactly matches with that of the Scherk-Schwarz supersymmetry breaking (SS SUSY breaking), where twisted boundary condition is imposed on $SU(2)_R$ over $S^1/Z_2$ [@paper5]. It is highly dependent on the flavor structure given in (\[eq5\]), therefore becomes a source of dangerous flavor changing processes at the electro-weak scale. In the following, we present a brief summary of systematic analysis of this low energy constraint on the flavor structure of the SS SUSY breaking.

The above Yukawa couplings are quite similar to the Yukawa couplings in Frogatt-Nielsen models with $N_I$ being identified as the $U(1)_F$ charges. More explicitly, assuming that mass spectrum below the compactification scale is given by that of the minimal supersymmetric standard model (MSSM), $$\begin{aligned}
y^u_{ij}\,\simeq \,
\lambda^{u\,(4)}_{ij}\epsilon^{X^u_i+X^q_j}\,,~~
y^d_{ij}\,\simeq \,
\lambda^{d(4)}_{ij}\epsilon^{X^d_i+X^q_j}\,,~~
y^\ell_{ij}\,\simeq \,
\lambda^{\ell(4)}_{ij}\epsilon^{X^e_i+X^\ell_j}\,,
\label{eq8}\end{aligned}$$ where $u_i$, $d_i$ ($e_i$) denote the $SU(2)_L$ singlet quarks (leptons) and $q_i$ ($\ell_i$) represents the doublet quarks (leptons). The effective flavor charge $X_I$ is given by $X_I = N_I~(X_I = 0 )$ for $N_I \ge 0~ (N_I\le 0)$. The above soft terms also can be well approximated by $X_I$ and $N_I$, $$\begin{aligned}
&&
A^u_{ij} \simeq  M_0 (X^u_i+X^q_j)\,y^u_{ij},~~
A^d_{ij} \simeq  M_0 (X^d_i+X^q_j)\,y^d_{ij},~~
A^\ell_{ij} \simeq M_0(X^e_i+X^\ell_j)\,y^\ell_{ij},
\nonumber \\
&&
m^{2(\tilde{\psi})}_{i\bar{j}} \simeq
\delta_{i\bar{j}}
M_0^2 \left\{
\begin{array}{ll}
|N^\psi_i|^2 \epsilon^{2|N^\psi_i|}&~(\,N^{\psi}_i \neq 0\,)\\
1/[2\ln(1/\epsilon)]^2&~(\,N^{\psi}_i = 0\,)
\end{array}
\right.
~~ {\rm for}~~~ M_0=-2{M_{1/2}}\ln (1/\epsilon).
\label{sssoftmssm}\end{aligned}$$

To discuss the low energy observables, it is convenient to move from the above definition basis to the SCKM basis where quarks and leptons have diagonal mass matrices, [*e.g.*]{} $
y^\ell_{ij}
 \longrightarrow (\overline{V}_e)_{ik} y^\ell_{k\ell}
 (\overline{V}_\ell^\dag)_{\ell j} = \hat{y^\ell}_i \delta_{ij}. 
$ In which the structure of the Yukawa coupling given in (\[eq8\]) is well reproduced by imposing constraints on the unitary matrices, $
|(\overline{V}_{e,\ell})_{ij}| {\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
    \raise1pt\hbox{$<$}}}\epsilon^{|X^{e,\ell}_i-X^{e,\ell}_j|}
$ for given diagonal Yukawa couplings, $\hat{y^\ell}_i \sim {\cal O}(\epsilon^{X^e_i+X^\ell_i})$. Similar discussion is applied for quarks. These constraints are directly translated into the probable magnitudes of mass-insertion parameters at the electro-weak scale defined like, $(\delta^\ell_{RL})_{ij}\equiv A^\ell_{ij}
 v_d/\sqrt{m^{2(\widetilde{e})}_{ii} m^{2(\widetilde{\ell})}_{jj}}$ , $(\delta^d_{RR}) \equiv 
 m^{2(\widetilde{d})}_{ij}/\sqrt{m^{2(\widetilde{d})}_{ii}m^{2(\widetilde{d})}_{jj}}$ and $(\delta^d_{LL}) \equiv 
 m^{2(\widetilde{q})}_{ij}/\sqrt{m^{2(\widetilde{q})}_{ii}m^{2(\widetilde{q})}_{jj}}$ in the SCKM basis [@paper5]. Note that if some of the effective charges are degenerate, unitarity relation like $
(\overline{V}_e)_{ik} X^e_k (\overline{V}_e^\dag)_{kj} =
 (\overline{V}_e)_{i1} (\overline{V}_e^*)_{j1} (X^e_1-X^e_2)
-(\overline{V}_e)_{i3} (\overline{V}_e^*)_{j3} (X^e_2-X^e_3)
+\delta_{ij} X^e_2  \nonumber\\
$ can dramatically suppresses the mixing from the naive estimation. This suppression mechanism is quite natural if underlying physics quantizes the kink masses in some unit, which originate from $U(1)_{FI}$ or graviphoton charges in the supergravity formulation [@paper6].

  ----------------- ----------------------- ----------------------------- ----------------------------- ----------------- ----------------------- ----------------------------- -----------------------------
   $X^e_{i}-X^e_3$   $X^\ell_{i}-X^\ell_3$   $|(\Delta^\ell_{RL})_{12}|$   $|(\Delta^\ell_{LR})_{12}|$   $X^e_{i}-X^e_3$   $X^\ell_{i}-X^\ell_3$   $|(\Delta^\ell_{RL})_{12}|$   $|(\Delta^\ell_{LR})_{12}|$
      $(5,1,0)$            $(0,0,0)$                     3.2                          0.015                 $(6,1,0)$            $(0,0,0)$                    0.80                          0.040
      $(0,0,0)$            $(5,1,0)$                    0.015                          3.2                  $(0,0,0)$            $(6,1,0)$                    0.040                         0.80
                                                                                                                                                                                
   $X^e_{i}-X^e_3$   $X^\ell_{i}-X^\ell_3$   $|(\Delta^\ell_{RL})_{12}|$   $|(\Delta^\ell_{LR})_{12}|$   $X^e_{i}-X^e_3$   $X^\ell_{i}-X^\ell_3$   $|(\Delta^\ell_{RL})_{12}|$   $|(\Delta^\ell_{LR})_{12}|$
                           $(6,2,0)$                  $(0,0,0)$                        3.2                    0.60                                                              
                           $(2,2,0)$                  $(4,0,0)$                        1.6                     3.2                                                              
                           $(4,0,0)$                  $(2,2,0)$                        3.2                     1.6                                                              
                           $(0,0,0)$                  $(6,2,0)$                       0.60                     3.2                                                              
  ----------------- ----------------------- ----------------------------- ----------------------------- ----------------- ----------------------- ----------------------------- -----------------------------

  :  Lepton mass hierarchy vs constraint from $\mu \to e\gamma$. $(\Delta^\ell_{RL(LR)})_{12}\equiv (\delta^\ell_{RL(LR)})_{12}/4.8\times 10^{-6}$. \[table1\]

We have explored various lepton flavor violating (LFV) and FCNC processes and find $\mu \to e,\gamma$ and $\epsilon_K$ give the most stringent constraint on the flavor structure of the SS SUSY breaking. Table \[table1\] lists surviving set of $X^{e,\ell}_i$ from $\mu \to
 e,\gamma$ with $(\delta^\ell_{RL(LR)})_{12}$ divided by a value which saturates the current upper-bound, $BR(\mu \to e,\gamma) = 1.2 \times 10^{-11}$ for $|M_{1/2}|=500$ GeV. The table shows that at least either $X^e$ or $X^\ell$ should have degenerate charges to satisfy the bound. Table \[table2\],\[table3\] show typical predictions of the models including other LFV processes. Note that degenerate charges of $\ell$ ($e$) yields the chirality of the processes opposite (similar) to the seesaw induced neutrino mass models [@paper7].

    $N^e_i$     $N^\ell_i$       $BR(\mu^+_R \to e^+_L,\gamma)$       $BR(\tau^+_R \to e^+_L,\gamma)$     $BR(\tau^+_R \to \mu^+_L,\gamma)$
  ------------ ------------ ---------------------------------------- --------------------------------- ---------------------------------------
    $6,1,-2$     $2,2,2$     $9.2(1+0.11t_\beta)^2\times 10^{-12}$                  $-$                 $2.4(1+0.091t_\beta)^2\times 10^{-8}$
    $8,3,2$     $-1,-1,-1$            $7.8\times 10^{-12}$                          $-$                 $2.4(1+0.091t_\beta)^2\times 10^{-8}$
    $N^e_i$     $N^\ell_i$       $BR(\mu^+_L \to e^+_R,\gamma)$       $BR(\tau^+_L \to e^+_R,\gamma)$     $BR(\tau^+_L \to \mu^+_R,\gamma)$
   $-1,-1,-1$    $8,3,2$              $7.7\times 10^{-12}$                          $-$                 $2.1(1+0.060t_\beta)^2\times 10^{-8}$
    $2,2,2$      $6,1,-2$    $7.7(1+0.075t_\beta)^2\times 10^{-12}$                 $-$                 $2.1(1+0.060t_\beta)^2\times 10^{-8}$

  : Predictions of LFV rates for $y^\ell_i={\cal O}(\epsilon^8,\epsilon^3,\epsilon^2)$ and $|{M_{1/2}}|=500
   {\rm GeV}$. $t_\beta\equiv \tan\beta$.\[table2\]

    $N^e_i$    $N^\ell_i$       $BR(\mu^+_R \to e^+_L,\gamma)$           $BR(\tau^+_R \to e^+_L,\gamma)$          $BR(\tau^+_R \to \mu^+_L,\gamma)$
  ----------- ------------ ----------------------------------------- ---------------------------------------- -----------------------------------------
   $2,2,-2$     $6,2,2$              $3.1\times 10^{-11}$                      $3.4\times 10^{-9}$                      $3.3 \times 10^{-9}$
    $4,4,2$    $4,-2,-2$    $3.4(1+0.053t_\beta)^2\times 10^{-11}$    $3.7(1+0.047t_\beta)^2 \times 10^{-9}$   $3.6(1+0.047t_\beta)^2 \times 10^{-9}$
    $N^e_i$    $N^\ell_i$       $BR(\mu^+_L \to e^+_R,\gamma)$           $BR(\tau^+_L \to e^+_R,\gamma)$          $BR(\tau^+_L \to \mu^+_R,\gamma)$
   $4,-2,-2$    $4,4,2$     $3.1(1+0.032 t_\beta)^2\times 10^{-11}$   $3.4(1+0.030t_\beta)^2 \times 10^{-9}$   $3.3(1+0.030 t_\beta)^2 \times 10^{-9}$
    $6,2,2$     $2,2,-2$             $3.1\times 10^{-11}$                      $3.4\times 10^{-9}$                       $3.3\times 10^{-9}$

  : Same as table 2 for $y^\ell_i={\cal O}(\epsilon^8,\epsilon^4,\epsilon^2)$. \[table3\]

   $N^q_i$    $N^d_i$   $\Im(\delta^d_{LL})^2_{12}/(1.5\times 10^{-2})^2$   $\Im(\delta^d_{RR})^2_{12}/(1.5\times 10^{-2})^2$   $\Im[(\delta^d_{RR})_{12}(\delta^d_{LL})_{12}]/(2.2\times 10^{-4})^2$
  ---------- --------- --------------------------------------------------- --------------------------------------------------- -----------------------------------------------------------------------
   $3,2,-1$   $3,2,2$                  $1.5\times 10^{-2}$                                 $7.1\times 10^{-1}$                                                  $466$
   $3,2,-1$   $4,2,2$                  $1.5\times 10^{-2}$                                 $2.8\times 10^{-2}$                                                  $93$
   $3,2,-1$   $3,3,2$                  $1.5\times 10^{-2}$                                 $2.8\times 10^{-2}$                                                  $93$
   $3,2,-1$   $4,3,2$                  $1.5\times 10^{-2}$                                 $5.7\times 10^{-3}$                                                  $42$

  : Quark mass hierarchy vs $\epsilon_K$. \[table4\]

On the other hand, table \[table4\] shows the constraints from the CP violating parameter of $\overline{K}^0$-$K^0$ mixing, $\epsilon_K$, where the mass-insertion parameters are divided by values saturating the observation $\epsilon_K=2.282 \times 10^{-3}$ for $|M_{1/2}|=500$ GeV. Models in the table can reproduce the quark masses and CKM angles with moderate tuning of the ${\cal O}(1)$ parameters, however, the $\epsilon_K$ constraint requires further ${\cal O}(10^{1-2})$ fine-tuning. However, if we allow abnormally large or small ${\cal O}(1)$ parameters by one order of $\epsilon \approx 0.2$, we can eliminate dangerous right-handed mixing by choosing degenerate $N^d_i$ and safely satisfy the constraint. We have also examined $\Delta M_K$, $\Delta M_{B_{d,s}}$, $\epsilon'/\epsilon$ , $b \to s,\gamma$, and CP asymmetry in $B_d \to \phi K_S$, however, can not find any meaningful constraints or predictions beyond the standard model.

Acknowledgments
===============

This work is supported by KRF PBRG 2002-070-C00022.

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[^1]: Talk given by K.O. at the 12th International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY04), June 17-23, 2004, Epochal Tsukuba, Tsukuba
