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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Anomaly and gaugino mediation}
\setcounter{equation}{0}
\setcounter{footnote}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{``Supergravity" mediation}
\label{Chap16:anommed:sec:sugra}


$X$ is in the hidden sector, $M_{Pl}$ suppressed couplings
\beq
W&=& W_{\rm hid}(X) + W_{\rm vis}(\psi)\\
f &=& \left(\delta_j^i - {{c_j^i}\over{M_{\rm Pl}^2}} X^\dagger X\right)
\psi^{j \dagger} e^V \psi_i + \ldots~\\
\tau &=& {{\theta_{\rm YM}}\over{2 \pi}} +i {{4 \pi}\over{g^2}} 
+ i {{k}\over{M_{\rm Pl}}} X + \ldots~
\eeq
SUSY breaking VEV
\beq
\langle X \rangle = M + {\mathcal F}_X \theta^2~,
\eeq
 induced squark and gluino masses:
\beq
(M^2_q)^i_j = c^i_j {{{\mathcal F}_X^2}\over{M_{\rm Pl}^2}}~,\,\,\,
M_\lambda = k {{{\mathcal F}_X}\over{M_{\rm Pl}}} ~
\eeq
no reason for the $c^i_j$ to 
respect flavor symmetries $\Rightarrow$ \Red{FCNCs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Naive Expectation}
 \Kahler function  might have 
flavor-blind form:
\beq
K = X^\dagger X + \psi^{i \dagger} e^V \psi_i~
\eeq
$\Rightarrow$
\beq
f= -3 + {{1}\over{M_{\rm Pl}^2}}\left[ X^\dagger X +
\left( 1 + {{X^\dagger X }\over{M_{\rm Pl}^2}} \right)\psi^{i \dagger} e^V \psi_i+\ldots
\right]
\eeq
interactions are flavor-blind but there are direct interactions
induced by Planck scale (string) states which have been integrated out\newline 
these
interactions should not be flavor-blind \newline \Red{ they must generate 
Yukawa couplings}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\myhead{Extra Dimensions and Pure Supergravity Mediation}
%\label{Chap16:anommed:sec:extradim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Extra Dimensions}
SUSY breaking sector 
separated by a distance $r$
 from the MSSM \newline  two sectors on  different 3-branes embedded in the 
 higher dimensional theory
Interactions  suppressed by
\beq
e^{-M r}~
\eeq
where $M$ is the higher dimensional Planck/string scale\newline   If \Red{only supergravity states propagate in  bulk}\newline  then setting $e^a_\mu=0$
and $\Sigma = M$ \Red{must decouple} the two sectors\newline  Lagrangian
must have the form
\beq
W&=& W_{\rm hid} + W_{\rm vis}~\\
f &=& c+f_{\rm hid} + f_{\rm vis}~\\
\tau W_\alpha^2 &=& \tau_{\rm hid} W_{\alpha^2 {\rm hid}}^2 + 
\tau_{\rm vis} W_{\alpha {\rm vis}}^2~
\label{Chap16:sequestered}
\eeq
 all  interactions between the two sectors due to 
supergravity

 form of $f$   implies  a  \Kahler function of the form
\beq
K =-3 M_{\rm Pl}^2  \ln \left( 1 - \frac{ f_{\rm hid} + f_{\rm vis} }{3 M_{\rm Pl}^2} \right) ~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Integrate out Hidden Sector}
dropping Planck suppressed interactions in effective
theory:
\beq
{\mathcal L}_{\rm eff}=
\int d^4 \theta \psi^\dagger e^V \psi {{\Sigma^\dagger \Sigma}
\over{M^2}}
 +\int d^2 \theta {{\Sigma^3}\over{M^3}} (m_0 \psi^2 + y \psi^3) 
 \nonumber \\
-{{i}\over{16 \pi}} \int d^2 \theta \tau W^\alpha W_\alpha +h.c.~
\eeq
where the conformal weights of the fields determine the $R$-charges to be
\beq
R[\Sigma]= {{2}\over{3}}, \,\, R[\psi] = 0~
\eeq
only  trace of the hidden sector is  in  compensator field $\Sigma$, rescale
\beq
{{\Sigma \psi}\over{M}} \rightarrow \psi ~,
R[\psi]= {{2}\over{3}}~
\eeq
\beq
{\mathcal L}_{\rm eff}&=&
\int d^4 \theta \psi^\dagger e^V \psi 
 +\int d^2 \theta ( {{\Sigma}\over{M}} m_0 \psi^2 + y \psi^3) 
 \nonumber \\
&&-{{i}\over{16 \pi}}\int d^2 \theta \tau W^\alpha W_\alpha +h.c.
\eeq
If $m_0=0$, then the theory is classically scaleand
conformally invariant\newline
 \Red{$\Sigma$ decouples classically }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Super-Weyl anomaly}
quantum corrections
break scale-invariance: 
couplings run\newline e.g.  a two-point function
has dependence on the cutoff $\Lambda$ \newline $\Rightarrow$ dependence on $\Sigma$ spurion of  conformal symmetry:
\beq
G = {{1}\over{p^2}} \,
h\left( \frac{p^2 M^2}{\Lambda^2 \Sigma^\dagger \Sigma} \right)
\eeq
$h$ can only depend on the combination $\Lambda \Sigma/M$ and  conjugate
because of the classical conformal invariance\newline since $\Lambda$ is real,  only the 
combination  $\Lambda^2 \Sigma^\dagger \Sigma/M^2$ appears\newline effects of the scaling anomaly determined by $\beta$ functions
and $\gamma$

\Blue{cutoff dependence only occurs in the \Kahler function and 
$\tau$}\newline if we  renormalize effective theory down to  scale $\mu$ we must have:
\beq
{\mathcal L}_{\rm eff}&=&
\int d^4 \theta \,Z\left( {{\mu M}\over{\Lambda \Sigma}}, 
{{\mu M}\over{\Lambda \Sigma^\dagger}} \right)  
 \psi^\dagger e^V \psi 
\nonumber \\
 &&+\int d^2 \theta \,  y \psi^3
-{{i}\over{16 \pi}}\int d^2 \theta\, \tau W^\alpha W_\alpha +h.c.
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Compensator Dependence}
$Z$ is real and $R$-symmetry-invariant,  must have
\beq
Z = Z\left( {{\mu M}\over{\Lambda |\Sigma|}} \right)
\label{Chap16:Zfunc}
\eeq
where
\beq
|\Sigma|= \left( \Sigma^\dagger \Sigma\right)^{1/2}~
\eeq
for global SUSY,  with $\Sigma = M_{\rm Pl}$,  axial symmetry is anomalous
$\theta_{\rm YM}$ shifts \newline when the $\psi$s are re-phased due to the chiral anomaly\newline  in superconformal gravity,  scale and axial anomalies vanish\newline 
\Red{$\Sigma$ dynamical, re-phased} $\Rightarrow$  shift in $\theta_{\rm YM}$  is canceled\newline since $\tau$ is  holomorphic we have 
\beq
\tau = i{{\wtilde{b}}\over{2 \pi}} \ln \left( 
{{\mu M}\over{\Lambda \Sigma }} \right) ~
\eeq
 $\mu$ dependence determines that $\wtilde{b}=b$ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY breaking: gaugino mass}
\label{Chap16:anommed:sec:break}

SUSY breaking will be
communicated to auxiliary  supergravity fields:
\beq
\langle \Sigma \rangle = M + {\mathcal F}_\Sigma \theta^2~
\eeq
induces a $\theta^2$ term in $\tau$ $\Rightarrow$ gaugino mass:
\beq
M_\lambda &=& {{i}\over{2 \tau}} 
{{\partial \tau}\over{\partial \Sigma}}  {\Big |}_{\Sigma=M} {\mathcal F}_\Sigma 
={{b g^2}\over{16 \pi^2}} {{{\mathcal F}_\Sigma}\over{M}}~.
\label{Chap16:anomMedgaugino}
\eeq
this SUSY breaking mass arises through the one-loop anomaly this mechanism
is known as \Red{anomaly mediation }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY breaking}

We can also Taylor expand $Z$ in superspace:
\beq
Z = \left[ Z -{{1}\over{2}}{{\partial Z}\over{\partial \ln \mu}} \left(
{{{\mathcal F}_\Sigma}\over{M}} \theta^2 +{{{\mathcal F}_\Sigma^\dagger}\over{M}}
{\bar \theta}^{2}\right) +{{1}\over{4}}{{\partial^2 Z}\over{\partial (\ln \mu)^2}} 
{{|{\mathcal F}_\Sigma|^2}\over{M^2}} \theta^2 
{\bar \theta}^{2} \right] {\Big |}_{\Sigma=M}~\,\,\,\,\,\,\,
\eeq
canonically normalize kinetic terms by rescaling: 
\beq
\psi^\prime &=& Z^{1/2}\left( 1 - {{1}\over{2}}\frac{\partial \ln Z} {\partial \ln \mu}
{{{\mathcal F}_\Sigma}\over{M}} \theta^2 \right)  {\Big |}_{\Sigma=M} \psi ~
\eeq
Using 
\beq
\gamma\equiv \frac{\partial \ln Z} {\partial \ln \mu}~\,,\,\, 
\beta_g \equiv \frac{\partial g} {\partial \ln \mu}~,\,\,\, 
\beta_y \equiv \frac{\partial y} {\partial \ln \mu}~
\eeq
we find 
\beq
Z \psi^\dagger e^V \psi &=& \left[1 +{{1}\over{4}}{{\partial \gamma}\over{\partial \ln \mu}}
{{|{\mathcal F}_\Sigma|^2}\over{M^2}} \theta^2 {\bar \theta}^{2} \right] 
\psi^{\prime \dagger} e^V \psi^\prime \nonumber \\
&=& \left[1 +{{1}\over{4}}\left(
{{\partial \gamma}\over{\partial g}} \beta_g +
{{\partial \gamma}\over{\partial y}} \beta_{y} \right)
{{|{\mathcal F}_\Sigma|^2}\over{M^2}} \theta^2 {\bar \theta}^{2} \right] 
\psi^{\prime \dagger} e^V \psi^\prime~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Squark and Slepton Masses}
\beq
M_{\wtilde \psi}^2 = -{{1}\over{4}} \left(
{{\partial \gamma}\over{\partial g}} \beta_g +
{{\partial \gamma}\over{\partial y}} \beta_{y} \right)
{{|{\mathcal F}_\Sigma|^2}\over{M^2}}~
\eeq
to leading order 
\beq
\gamma = \frac{1}{16 \pi^2} (4 \,C_2(r) g^2 - a y^2),\,\,\,
\beta_g = - \frac{b\,g^3 }{16 \pi^2}  ~,\,\,\,
\beta_y = \frac{y}{16 \pi^2} (e y^2 - f g^2)\,\,\,\,\,\,\,
\eeq
so 
\beq
M_{\wtilde \psi}^2 = \frac{1}{512 \pi^4}\left[
4 \,C_2(r) \,b \,g^4 +
a y^2 (e y^2 - f g^2) \right]
{{|{\mathcal F}_\Sigma|^2}\over{M^2}}~
\label{Chap16:squarksleptonmass}
\eeq
 first term is positive for asymptotically free gauge theories\newline \Red{negative mass squared for sleptons} since in the MSSM
the $U(1)_Y$ and $SU(2)_L$ gauge couplings are not asymptotically free


\vspace{12pt}\noindent $W(\psi)$ after  rescaling gives trilinear
interactions with  coefficient
\beq
A_{ijk} = {{1}\over{2}} \left( \gamma_i + \gamma_j + \gamma_k \right)
y_{ijk} {{{\mathcal F}_\Sigma}\over{M}}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Trilinear Terms}

gauge mediation:
messengers have masses

\beq
\langle X \rangle = M_X \left(1 + {{{\mathcal F}_X}\over{M_X}} \theta^2 \right)
\eeq
anomaly mediation: 
the cutoff is 
\beq
\Lambda {{\Sigma}\over{M}} = \Lambda
\left(1 + {{{\mathcal F}_\Sigma}\over{M}} \theta^2\right)
\eeq
mass of the regulator fields
with anomaly mediation\newline \Blue{the
regulator is the messenger }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Heavy SUSY Thresholds}
after rescaling,
the mass $m$ of a SUSY threshold becomes 
$m \Sigma/M$\newline low-energy  $Z$ and $\tau$
have the following dependence:
\beq
Z\left( {{\mu M}\over{\Lambda |\Sigma|}},
{{ |m| |\Sigma| }\over{\Lambda |\Sigma|}}  \right),\,\,\,
\tau\left( {{\mu M}\over{\Lambda \Sigma}},
{{ m \Sigma }\over{\Lambda \Sigma}}  \right)  ~
\eeq
gaugino and sfermion masses are independent of $m$ since $m/\Lambda$ has no
dependence on the spurion $\Sigma$ \newline   anomaly is insensitive to UV physics,
completely determined\newline by the low-energy effective theory\newline threshold and  regulator contribute with opposite signs and cancel \newline   SUSY breaking in the mass term $\to$  cancellation would not  complete

\vspace{24pt} \noindent \Blue{soft masses
only depend on $\beta_g$, $\beta_y$, $\partial \gamma / \partial g$, 
$\partial \gamma / \partial y$ at weak scale, $M_W$}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $\mu$ problem}
\label{Chap16:anommed:sec:mu}

in order to get EWSB in the MSSM
need $\mu$ and $b$ terms:
\beq
W= \mu \, H_u H_d~,\,\,\,
V= b \, H_u H_d~
\eeq
with
\beq
b \sim \mu^2~
\eeq
need $\mu \sim$  soft masses, so in anomaly-mediation require
\beq
\mu \sim {{\alpha}\over{4 \pi}} \frac{{\mathcal F}_\Sigma}{M}~
\eeq
 including a coupling to  spurion field $\Sigma$ that
directly gives a $\mu$ term:
\beq
W=  \mu {{\Sigma^3}\over{M^3}} H_u H_d~
\eeq
we also gives a tree-level $b$ term
\beq
b&=& 3 {{{\mathcal F}_\Sigma}\over{M}} \mu \sim {{12 \pi}\over{\alpha}} \mu^2~
\eeq
which is much too large
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $\mu$ problem}

more complicated possibility:
\beq
 {\mathcal L}_{\rm int}=  \int d^4 \theta\,\delta\, {{X+X^\dagger}\over{M}}
 H_u H_d {{\Sigma \Sigma^\dagger}\over{M^2}}+h.c.~ ~~~(*)
 \label{Chap16:complInt}
\eeq
where $X$ is a SUSY breaking field, 
rescale
\beq
{{\Sigma H_i}\over{M}} \rightarrow H_i ~
\eeq
\beq
{\mathcal L}_{{\rm eff}\,\,{\rm int}}= 
\int d^4 \theta\,\delta\, {{X+X^\dagger}\over{M}}
 H_u H_d {{\Sigma^\dagger}\over{\Sigma }}+h.c.~
\eeq
assuming $\langle X\rangle=\theta^2 {\mathcal F}_X$, picking out the ${\bar \theta}^2$ and  $\theta^2 {\bar \theta}^2$ terms $\Rightarrow$
\beq
\mu &=& \delta \left({{{\mathcal F}_X^\dagger}\over{M}} +
{{{\mathcal F}_\Sigma^\dagger}\over{M}} \right)\\
b&=&  \delta \left({{{\mathcal F}_X}\over{M}} 
{{{\mathcal F}_\Sigma^\dagger}\over{M}} -{{{\mathcal F}_X^\dagger}\over{M}} 
{{{\mathcal F}_\Sigma}\over{M}} \right)
\eeq
\Blue{ $b$ vanishes at tree-level if ${\mathcal F}_\Sigma \propto {\mathcal F}_X$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $\mu$ problem}
$b$ term is generated at one-loop,  canonically normalize the
Higgs:
\beq
H_i^\prime &=& Z_i^{1/2}\left( 1 - {{1}\over{2}}\gamma_i
{{{\mathcal F}_\Sigma}\over{M}} \theta^2 \right)  {\Big |}_{\Sigma=M} H_i~
\eeq
if $\delta\sim \alpha/4\pi$, we find:
\beq
b&=&  \delta {{{\mathcal F}_X^\dagger}\over{2 M}} 
\left(\gamma_u
{{{\mathcal F}_\Sigma}\over{M}} + \gamma_d
{{{\mathcal F}_\Sigma}\over{M}} \right)= {\mathcal O}(\mu^2)
\eeq

this relies on the coefficients of $X$ and $X^\dagger$ in  (*)
being  equal\newline  \Blue{seems fine-tuned}

\vspace{24pt}
generated  in 5D toy model without fine-tuning
\newline  fifth (extra) dimension has a compactification radius $r_c$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{5D Gravity}
 for $r\ll r_c$ the gravitational potential is
\beq
{{1}\over{r^2 M^3}} ~
\eeq
rather than the 4D Newton potential
\beq
{{1}\over{r M_{\rm Pl}^2}} ~
\eeq
static potential given by the spatial Fourier transform of the graviton propagator with zero  energy exchange:
\beq
V(r)\sim \int d^{D-1} p\, \frac{e^{i {\vec p}.{\vec r}} }{\vec p^2}\sim \frac{1}{r^{D-3}}~
\eeq
Matching the potentials at $r=r_c$ we have
\beq
\Red{M_{\rm Pl}^2=r_c M^3}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{5D Vector Exchange}
introduce
a massive vector superfield $V$   which propagates in the 5D bulk (canonical dimension $3/2$)\newline Integrating over fifth dimension, assume the 4D effective theory has form:
\beq
{\mathcal L} = \int d^4 \theta\,  r_c m^2 V^2 +a V(X+X^\dagger)M^{1/2}
+{{b V}\over{M^{1/2}}} H_u H_d  {{\Sigma \Sigma^\dagger}\over{M^2}}
+h.c.~\,\,\,\,\,
\eeq
 first term is a mass term and $V$ is normalized to  dimension
$\sfrac{1}{2}$\newline 
Integrating out $V$ and performing the usual rescaling gives
\beq
{\mathcal L}_{\rm int} \sim \int d^4 \theta {{a b}\over{ r_c m^2}}(X+X^\dagger)
 H_u H_d {{\Sigma \Sigma^\dagger}\over{M^2}}
 +h.c.+\ldots~
\eeq
with 
 \beq
 r_c m \sim {\mathcal O}(1),\,\,\,
a b \sim {\mathcal O}\left(\frac{\alpha}{4 \pi}\right)
\eeq
$\rightarrow$ required interaction
\newline  
\Red{existence proof that  $\mu$ problem can be solved in anomaly-mediation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Slepton masses}
\label{Chap16:anommed:sec:slepton}


squark and slepton masses:  \beq
M_{\wtilde \psi}^2 = \frac{1}{512 \pi^4}\left[
4 \,C_2(r) \,b \,g^4 +
a y^2 (e y^2 - f g^2) \right]
{{|{\mathcal F}_\Sigma|^2}\over{M^2}}~
\eeq

$b$ is negative for $SU(2)_L$ and $U(1)_Y$ $\Rightarrow$
sleptons are tachyonic
\newline possible solutions:
\begin{itemize}
\item{} new bulk fields which couple leptons and the SUSY breaking fields
\item{} new Higgs fields with large Yukawa couplings
\item{} new asymptotically free  gauge interactions for sleptons,
$\Rightarrow$  leptons and sleptons are composite 
\item{} \Blue{heavy SUSY violating threshold (messengers) with a light singlet}
\end{itemize}
consider the last possibility, sometimes known as \Blue{``anti-gauge mediation'' }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Anti-Gauge Mediation}

 consider  a singlet $X$ and $N_m$ messengers $\phi$ and 
$\overline{\phi}$ in $\fund\,$s and $\overline{\fund}\,$s of  $SU(5)$ GUT
with a superpotential
\beq
W= \lambda X \phi \overline{\phi}~
\eeq
$X$ is pseudo-flat:  it gets a mass through anomaly mediation \newline 
when we renormalize down to a scale $\sim  X$ we have a \Kahler term
\beq
\int d^4\theta\, Z\left(\frac{X X^\dagger M^2}{\Lambda^2 \Sigma \Sigma^\dagger}\right) X^\dagger X~
\label{Chap16:XKahler}
\eeq
 scalar potential
\beq
V(X) &=& m_X^2(X) |X|^2 
\nonumber\\
&=& {{N_m}\over{16 \pi^2}} \lambda^2(X)\left[A\lambda^2(X) -C^a g_a^2(X)\right]
{{|{\mathcal F}_\Sigma|^2}\over{M^2}} |X|^2~
\label{Chap16:Xpot}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Anti-Gauge Mediation}

If  messengers have  asymptotically free gauge interactions then   $m^2_X(X)$ 
can change sign, and $X$ is stabilized nearby (Coleman--Weinberg)
\beq
\langle X \rangle = m~
\eeq
then ${\mathcal F}$ component of $X$ is proportional to
$m {\mathcal F}_\Sigma$:
\beq
{\mathcal F}_X \sim \frac{N_m \lambda^2}{16\pi^2}
\frac{m \,{\mathcal F}_\Sigma }{M}~
\label{Chap16:FXtilde}
\eeq
splitting in the messenger masses is a loop effect
\newline threshold depends on light  VEV $\rightarrow$
extra contribution to  soft masses
\newline low-energy couplings  only depend on 
\beq
\wtilde{X} = X {{M}\over{\Sigma}}~,
{{{\mathcal F}_{\wtilde X}}\over{\langle \wtilde{X} \rangle}} = {{{\mathcal F}_X}\over{m}}-{{{\mathcal F}_\Sigma}\over{M}}
\approx
-{{{\mathcal F}_\Sigma}\over{M}}~
\eeq
because of the loop factor suppression 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Taylor Expansion in Superspace}
 gaugino mass:
\beq
M_\lambda &=& -{{1}\over{2 \tau}} 
{{\partial \tau}\over{\partial \ln \Sigma}}  {\Big |}_{\Sigma=M}
 {{{\mathcal F}_\Sigma}\over{M_{\rm Pl}}} \nonumber\\
&=& {{1}\over{2 \tau}} \left(
{{\partial \tau}\over{\partial \ln \mu}}+{{\partial \tau}\over{\partial \ln X}} \right)
 {{{\mathcal F}_\Sigma}\over{M_{\rm Pl}}} \nonumber\\
 &=&{{\alpha(\mu)}\over{4 \pi}} (b-N_m){{{\mathcal F}_\Sigma}\over{M_{\rm Pl}}}~
\eeq
first term is usual anomaly mediation 
\newline  second
term is minus the gauge mediation answer

\vspace{24pt}\noindent \Blue{hence the name Anti-Gauge Mediation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Taylor Expansion in Superspace}
squark or slepton mass squared:
\beq
M_{\wtilde \psi}^2 &=& - \left(
{{\partial }\over{\partial \ln \mu}} +
{{\partial }\over{\partial \ln |X|}}  \right)^2 \ln Z(\mu,|X|)
{{|{\mathcal F}_\Sigma|^2}\over{4 M_{\rm Pl}^2}} \\
&=&{{2 C_2(r) b}\over{(4 \pi)^2}} \left[\alpha^2(\mu) - \alpha^2(\mu){{N_m}\over{b}}
+(\alpha^2(\mu) - \alpha^2(m))\frac{N_m^2}{b^2} \right]
{{|{\mathcal F}_\Sigma|^2}\over{M_{\rm Pl}^2}}~\nonumber
\eeq
first term 
is  anomaly mediation term\newline  second term is minus the gauge mediation term\newline  final term is \Blue{RG running induced by  gaugino mass}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Slepton Masses}
\beq
M_{\wtilde \psi}^2 &=& - \left(
{{\partial }\over{\partial \ln \mu}} +
{{\partial }\over{\partial \ln |X|}}  \right)^2 \ln Z(\mu,|X|)
{{|{\mathcal F}_\Sigma|^2}\over{4 M_{\rm Pl}^2}} \\
&=&{{2 C_2(r) b}\over{(4 \pi)^2}} \left[\alpha^2(\mu) - \alpha^2(\mu){{N_m}\over{b}}
+(\alpha^2(\mu) - \alpha^2(m))\frac{N_m^2}{b^2} \right]
{{|{\mathcal F}_\Sigma|^2}\over{M_{\rm Pl}^2}}~\nonumber
\eeq
\noindent  \Red{for the sleptons $M_{\wtilde \psi}^2>0$  $\Rightarrow$  RG term dominates
 $\Leftrightarrow$  $N_m$ sufficiently large}\newline
cannot
 $m$ too large,  higher dimension operators
 dominate, e.g.
\beq
\int d^4 \theta {{X^\dagger X}\over{M_{\rm Pl}^2}} \psi^\dagger e^V \psi~
\eeq
would give 
\beq
M_{\wtilde \psi}^2 = -{{|{\mathcal F}_X|^2}\over{M_{\rm Pl}^2}} ~
\eeq
for $m\sim M_{\rm GUT}$ we need $N_m \ge 4$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$\mu$ Problem}
adding singlet  $S$ cangenerate 
$\mu$ and $b$ terms  
\beq
\int d^2 \theta \lambda^\prime S H_u H_d +{{k}\over{3}} S^3 + {{y}\over{2}} S^2 X~
\eeq
one-loop a kinetic mixing:
\beq
\int d^4 \theta \wtilde{Z} S X^\dagger + h.c.
\eeq
for $\langle X \rangle \ne 0$, $S$ is massive and can be integrated out:
\beq
S \sim -\frac{\lambda^\prime}{y}  {{H_u H_d}\over{X}}~
\eeq 
\beq
\rightarrow~~~{\mathcal L}_{\rm eff}=
-\frac{\lambda^\prime}{y} \int d^4 \theta {{X^\dagger}\over{X}}
 H_u H_d \wtilde{Z}\left({{|X| M_{\rm Pl}}\over{\Lambda |\Sigma| }}\right)+h.c.~
\eeq
produces  $\mu$ term at one-loop, $b$ term at two-loops:
\beq
\mu = -\frac{\lambda^\prime}{y} {{1}\over{2}} {{\partial \wtilde{Z}}\over{\partial \ln |X|}}~,~~
b= -\frac{\lambda^\prime}{y} {{1}\over{4}} {{\partial^2 \wtilde{Z}}\over{\partial (\ln |X|)^2}}~
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gaugino mediation}
\label{Chap16:anommed:sec:gaugino}

\noindent RG running from  gaugino mass $\rightarrow$ $+$ mass$^2$ for  squarks and sleptons
\newline consider models where  only gauginos get masses at leading order\newline 
squarks have  strong gauge coupling $\Rightarrow$ heavier than sleptons\newline  large top Yukawa coupling and  heavy stops $\Rightarrow$ \Blue{radiative EWSB}


simple set up: compact extra dimension with radius
\beq
r_c \sim {{1}\over{M_{\rm GUT}}}~
\eeq
\Blue{gauge fields propagate in  bulk}
\newline
SUSY breaking on brane at the other end of the
fifth dimension\newline 
Yukawa couplings  only source of flavor violation\newline  \Red{GIM mechanism suppresses FCNCs}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gaugino mediation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\hsize]{slideimages/gauginoMed}
\end{center}
\label{Chap16:fig:gauginoMed}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
gaugino propagating in bulk mediates SUSY breaking 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gaugino mediation}
4D gauge coupling related to the 5D coupling
by
\beq
\frac{1}{g_4^2} F^a_{\mu\nu}F^{a \mu\nu} = \frac{1}{g_5^2} \int dx^5\, F^a_{\mu\nu}F^{a \mu\nu} ~,\,\,\,
g_4^2= \frac{g_5^2}{r_c}~
\eeq
no chirality in 5D, minimal SUSY theory has ${\mathcal N}=2$\newline 
vector supermultiplet $\to$ 4D vector  $+$ adjoint chiral:
\beq
(A_N,\lambda_L,\lambda_R,\phi) \rightarrow (A_\nu,\lambda_L)+(\phi+ iA_5,\lambda_R)
\eeq
 fifth component of  gauge field is a scalar \newline 
choose boundary conditions so: \newline  \Blue{adjoint chiral supermultiplet vanishes on one  3-brane} \newline \Blue{vector multiplet does not}\newline only 
vector supermultiplet has  massless mode (independent of $x_5$) \newline 
$\Rightarrow$ \Red{breaks SUSY
$\to$ ${\mathcal N}=1$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gaugino mediation}
SUSY breaking on one brane  communicated  by local interactions
\beq
{\mathcal L} &\propto& \int d x^5 \int d^2 \theta \left(1 +\delta(x_5-r_c) {{X}\over{M^2}}
\right)  W^\alpha W_\alpha +h.c.\nonumber\\
&\propto& r_c \lambda^\dagger \overline{\sigma}^\mu D_\mu \lambda + 
{{{\mathcal F}_X}\over{M^2}} \lambda^\dagger \lambda + \ldots~
\eeq
gaugino mass  generated by auxiliary fields on SUSY breaking brane
\beq
\Red{M_\lambda = {{1}\over{r_c M}} {{{\mathcal F}_X}\over{M}}~}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Gaugino mediation}

bulk gluino loops  with two mass insertions give the largest contribution to the  squark/slepton masses:
\beq
M_{\wtilde \psi}^2 \sim {{g_5^2}\over{16 \pi^2}} \left({{{\mathcal F}_X}\over{M^2}} \right)^2
 {{1}\over{r_c^3}}={{g_4^2}\over{16 \pi^2}} M_\lambda^2~,
 \label{Chap16:5Dloop}
 \eeq
 suppressed relative to  gluino mass squared\newline for 
 $r_c \ll M_W^{-1}$  4D RG running 
 \beq
\mu {{d}\over{d \mu}} m_Q^2 \propto -g^2 M_\lambda^2 +c g^4 
{\rm Tr}\left((-1)^{2F} m_{i}^2 \right)~
\eeq
 dominates by a large logarithm, $\ln r_c M_W$, over the 5D loop contribution
 
 \vspace{24pt}\noindent
\Red{all the soft masses are determined by gaugino masses  and $r_c$}\newline  very predictive scenario





\end{document}

 
 
