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\mode<presentation>
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\setbeamertemplate{itemize item}[triangle]%bullet-symbol]
% \setbeamertemplate{itemize subitem}[$\star$]
 
\title[EP Violations and Light Dilaton]%
{Equivalence Principle Violations and Couplings of a Light Dilaton}

\author[Thibault Damour]{Thibault Damour \\ Institut des Hautes \'Etudes Scientifiques \\ { \ } \\ Second Microscope Colloquium, ONERA, Palaiseau 29-30 January 2013}
\institute[IHES]{}
\date[ONERA, Palaiseau 29-30/01/2013]{\\[2ex]
%based on work done with
%\begin{itemize}

%\item[\iconarticle] With 
%{\small S. Badger, N.  Berkovits, N.E.J.  Bjerrum-Bohr,}%\\[1ex]

%{\small P. Damgaard, M.B. Green, J. Russo}
%\item[\iconarticle] %\link{http://arxiv.org/abs/0802.0868}{0802.0868},
%  \link{http://arxiv.org/abs/0908.XXXX}{To appear},\\
% \link{http://arxiv.org/abs/0806.1726}{0806.1726},\\
%with N. Berkovits, M.B.~Green, J.G.~Russo

%\item[\iconarticle]  \link{http://arxiv.org/abs/hep-th/0610299}{hep-th/0610299},
% \link{http://arxiv.org/abs/hep-th/0611273}{hep-th/0611273} 
%\link{http://arxiv.org/abs/0807.0389}{0807.0389}
%+ work in progress\\
 %Ultraviolet properties of maximal supergravity. 
 % with M.B.~Green, J.G.~Russo%, Pierre Vanhove.

%\end{itemize}
}
 \subject{Talks}

\begin{document}
 \beamertemplatenavigationsymbolsempty
\begin{frame}
  \titlepage
\end{frame}
%------------------------------------------------------------------------------
\begin{frame}
\frametitle{(Einstein) Equivalence ``Principle'' (EP)}



% \vfill
\simpleb{blue}{$\bullet$ {\color{red} Not} a basic principle of physics}

\vfill

\simpleb{blue}{$\bullet$ A heuristic generalization of an experimental fact: ``hypothesis of equivalence'' (Einstein) $\longrightarrow \ $ very successful in building General Re\-lativity (GR)}

\vfill

Einstein's GR:
$$
{\color{red} \eta_{\mu\nu} \qquad \longrightarrow \qquad g_{\mu\nu} (x)}
$$

\hglue 35mm absolute, \hglue 15mm elastic spacetime,

\hglue 35mm rigid spacetime \hglue 5mm dynamically influenced 

\hglue 67mm by matter

\vfill

\simpleb{red}{BUT all the coupling constants of local (special relativistic) physics remain as {\color{red} absolute} and {\color{red} rigid} as in Special Relativity (SR):
$$
g_a , Y , \lambda_{\rm BEH} , \mu_{\rm BEH} \to  \ \mbox{{\color{red} non dynamical}} \ g_a , Y , \lambda_{\rm BEH} , \mu_{\rm BEH}
$$
}

\end{frame}


%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{What determines the coupling constants?}

\simpleb{blue}{$\bullet$ Very unsatisfactory to put them by hand: this is against the ``{\color{red} principle of reason}'' nihil est sine ratione (Leibniz)
}

\vfill

\simpleb{blue}{$\bullet$ The history of physics suggests that there are {\color{red} no absolute structures} in physics
}

\vfill 

\simpleb{red}{Kaluza-Klein's idea:
$$
{\color{red} g_1 \quad \mbox{or} \quad \alpha_{\rm em} \simeq \frac{3}{8} \, \frac{g_1^2}{4\pi \, \hbar c} \simeq \frac{1}{137} \quad \longrightarrow \quad g_{55} (x)}
$$

\hglue 80mm higher-dimensional

\hglue 80mm elastic spacetime
}

\vfill

\simpleb{blue}{Dynamical symmetry breaking: the vacuum state minimizes the energy $V(\phi)$ which dynamically determines
$$
{\color{red} \langle \phi \rangle \sim \frac{\mu}{\sqrt\lambda} \longrightarrow m_e \sim Y_e \langle \phi \rangle \sim Y_e \, \frac{\mu}{\sqrt\lambda}}
$$
}


\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Varying Coupling Constants and EP Violations}


\simpleb{blue}{Then if {\color{red} any} of the coupling constants of local physics 

(e.g., $\alpha_{\rm em}$, $m_e / m_p$, $m_q / m_p$, $\ldots$) is {\color{red} $x$-dependent}

\bigskip

{\color{red} $\Longrightarrow$ violation of equivalence principle} (Dicke 1962)
}

\vfill

{\color{red} Notably violation of universality of free fall}
$$
S_{\rm mi} = - \int m_i [\alpha (x) , \ldots] \, \sqrt{-g_{\mu\nu} (x) \, dx^{\mu} \, dx^{\nu}}
$$

\vfill

\simpleb{blue}{
{\color{red} Composition-dependent acceleration}
$$
\vec a_i = \vec g - \vec\nabla \, \ell n \, m_i [\alpha (x) , \ldots] = \vec g - \frac{\partial \, \ell n \, m_i}{\partial \, \alpha} \, \vec\nabla \, \alpha - \ldots
$$
}

\end{frame} 

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{General dilaton-like model of EP violations}

\simpleb{green}{
Assume general dependence of coupling ``constants'' on some ``dilaton'' field $\varphi : \alpha_{EM} (\varphi) , (m_q/\Lambda_{QCD})(\varphi) , (m_e / \Lambda_{QCD})(\varphi),\ldots$. Then the dependence of $m_A$ upon fundamental coupling constants:
\vglue -1mm
$$
m_A = \Lambda_{QCD} \, \hat m_A \left( \alpha_{EM} , \frac{m_u}{\Lambda_{QCD}} , \frac{m_d}{\Lambda_{QCD}} , \frac{m_e}{\Lambda_{QCD}} \right)
$$
\vglue -1mm
$\to$ a $\varphi$ dependence of $m_A$ and a corresponding dilaton coupling to $m_A$
\vglue -1mm
{\color{red}
$$
\alpha_A = \frac{\partial \, \ln \, m_A (\varphi)}{\partial \, \varphi}
$$
}
\vglue -2mm
Composition-dependent modification of Newtonian interaction
\vglue -1mm
$$
V(r) = -G \, \frac{m_A \, m_B}{r} \, (1 + \alpha_A \, \alpha_B \, e^{-m_{\varphi} r})
$$
\vglue -1mm
In the following: inverse range of $\varphi : m_{\varphi} = 0$. $\to$ Weak EP violation
\vglue -1mm
$$
\eta_{AB} = \left( \frac{\Delta a}{a} \right)_{AB} \simeq (\alpha_A - \alpha_B) \, \alpha_E
$$
}


\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{General Dilaton Low-energy Couplings (Damour-Donoghue10)}

\vfill

\simpleb{blue}{Organizing principle: keep track of all the possible $\varphi$ couplings entering the effective action describing physics at the scale of nucleons. At this scale: heavy quarks ($c,b,t$; and, arguably, $s$) are integrated out.
$$
{\mathcal L}_{\rm eff} = - \frac{1}{4e^2} F_{\mu\nu}F^{\mu\nu} - \frac{1}{4} F^A_{\mu\nu}F^{A\mu\nu} + \sum_{i= e,u,d} \left[ i {\bar \psi}_i {\slashed{D}}(A, g_3 A^A) \psi_i - m_i{\bar \psi}_i\psi_i \right] 
$$
}

\vfill

\simpleb{yellow}{Five terms in ${\mathcal L}_{\rm eff} \to$ {\color{red} five} possible (dimensionless) $\varphi$ couplings: $d_e , d_g , d_{m_e}, d_{m_u} , d_{m_d}$
$$
{\cal L}_{{\rm int} \varphi} =  \varphi \left[ + \frac{d_e}{4e^2} F_{\mu\nu}F^{\mu\nu}
-\frac{d_g\beta_3}{2g_3} F^A_{\mu\nu}F^{A\mu\nu} - \sum_{i=e,u,d} (d_{m_i}+\gamma_{m_i}d_g) m_i{\bar \psi}_i\psi_i \right] \, .
$$
}


\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Relation between dilaton couplings $d_a$ and the ``constants of Nature''}

\simpleb{purple}{
The five possible dilaton couplings $d_a = \{ d_e , d_g , d_{m_e}, d_{m_u} , d_{m_d}\}$ are equivalent to:}

\vfill

\simpleb{blue}{fine-structure constant $\quad \alpha = \frac{e^2}{4\pi} \simeq \frac{1}{137} \to \alpha (\varphi) = (1+d_e \, \varphi) \, \alpha$}

\vfill

\simpleb{green}{QCD energy scale $\quad \Lambda_3 \sim 100 \, {\rm MeV} \to \Lambda_3 (\varphi) = (1 + d_g \, \varphi) \, \Lambda_3$}

\vfill

\simpleb{blue}{electron mass $\quad m_e \to m_e (\varphi) = (1 + d_{m_e} \, \varphi) \, m_e$}

\vfill

\simpleb{green}{light-quark masses at QCD scale $\quad m_i (\Lambda_3) \to [m_i (\Lambda_3)](\varphi) = (1+ d_{m_i} \, \varphi) \, m_i (\Lambda_3)$, $i=u,d$}

\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Ratios of dimensional parameters}

\simpleb{blue}{As the Planck scale $1/\kappa = 1/\sqrt{4\pi \, G}$ does not directly enter physics at the QCD scale (besides its possible impact on $\Lambda_3$ via $\Lambda_{\hbox{\scriptsize cut-off}} \propto 1/\kappa$?) :
}

\vfill 

\simpleb{yellow}{Mass of an atom:
$$
m_A = \Lambda_3 \, M_A \left( \frac{m_u}{\Lambda_3} , \frac{m_d}{\Lambda_3} , \frac{m_e}{\Lambda_3} , \alpha \right)
$$
where $M_A$ is a dimensionless function of {\color{red} four} dimensionless quantities:
$$
k_a = (k_u , k_d , k_e , k_{\alpha}) \equiv \left( \frac{m_u}{\Lambda_3} , \frac{m_d}{\Lambda_3} , \frac{m_e}{\Lambda_3} , \alpha \right)
$$
}

\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Composition-dependence of $\varphi$ coupling to atom}

\simpleb{blue}{
$$
\alpha_A =  \frac{\partial \ln[\kappa m_A(\varphi)] }{\partial \varphi} =
\sum_a \frac{\partial \ln[\kappa m_A(k_a)] }{\partial k_a}  \frac{\partial k_a }{\partial \varphi} 
$$

\medskip

$$
\alpha_a = d_g + \bar\alpha_A
$$

\medskip

where $d_g = \frac{\partial \ln \Lambda_3}{\partial \, \varphi}$ is a universal (non EP-violating) contribution and

\bigskip

$$
\bar{\alpha}_A =  \frac{1}{M_A}\frac{\partial M_A}{\partial\varphi}
= \frac{1}{M_A}\left[ \sum_{a=u,d,e} (d_{m_a}-d_g)\frac{\partial M_A}{\partial \ln k_a}+ d_e \frac{\partial M_A}{ \partial \ln \alpha}\right].
$$
}



\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Analysis of scalar couplings to the binding energy of nuclei}

\simpleb{green}{Need to relate the various contributions to the nuclear binding energy
$$
E^{\rm bind} =- a_v A +a_s A^{2/3} +a_a \frac{(A-2Z)^2}{A} + a_c \frac{Z(Z-1)}{A^{1/3}}  - \delta \frac{a_p}{A^{1/2}} 
$$
to the variability of light quark masses $m_u , m_d$, or $\hat m = \frac{m_d + m_u}{2}$, $\delta m = m_d - m_u$.
}

\vfill

\simpleb{blue}{Possible by combining Walecka-type analysis of nuclei binding (parametrized by scalar and vector coupling strengths $G_S , G_V$) with recent work of Donoghue (2006) on the {\color{red} pion-mass} dependence of $G_S$ and $G_V$:
$$
\bar{\alpha}_A^{{\rm bind}} = -\frac{(d_{\hat{m}} -d_g)}{m_A}(120 A -{97}{A^{2/3}})
m^2_\pi \frac{\partial \eta_S}{\partial m_\pi^2}
$$

$$
\hat{m} \frac{\partial \eta_S }{\partial \hat{m}}=m^2_\pi \frac{\partial \eta_S}{\partial m_\pi^2} =-0.35\pm 0.10
$$
}


\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Implications for the Equivalence Principle}

\simpleb{purple}{
$$
{\alpha}_A = d_g +  \bar{\alpha}_A
$$
$$
\bar{\alpha}_A =  \left[ (d_{\hat m} - d_g) Q_{\hat m} + (d_{\delta m} -d_g) Q_{\delta m} + (d_{m_e} - d_g) Q_{m_e} + d_e Q_e \right]_A
$$
where the various ``dilaton charges'' $Q_{k_a}$ are given by 

(with $F_A \equiv A \, m_{amu} / m_A \simeq 1$)
$$
 Q_{\hat m} = F_A \left[ 0.093 -\frac{0.036}{A^{1/3}} - 0.020 \frac{(A-2Z)^2}{A^2}
- 1.4 \times 10^{-4} \, \frac{Z(Z-1)}{A^{4/3}} \right]  ,
$$

$$
Q_{\delta m} = F_A \left[0.0017   \, \frac{A-2Z}{A} \right] ,
$$

$$
Q_{m_e} = F_A \left[ 5.5 \times 10^{-4}  \, \frac{Z}{A} \right] ,
$$

$$
Q_e = F_A   \left[  -1.4 + 8.2 \frac{Z}{A} + 7.7 \frac{Z(Z-1)}{A^{4/3}}  \right]\times 10^{-4}.
$$
}

\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Simplified Parametrization of EP Violations}

\simpleb{blue}{Under plausible approximations, only two dilaton charges dominate: 

\smallskip

$Q'_{\hat m}$ linked to average quark-mass sensitivity to nuclear binding, and $Q'_{\alpha} \equiv Q'_e$ linked to the fine-structure constant:

$$
{\alpha}_A \simeq d_g^* + \left[ (d_{\hat m} - d_g) Q'_{\hat m}  + d_e Q'_e \right]_A
$$

\medskip

$$
Q'_{\hat m} = -\frac{0.036}{A^{1/3}} - 1.4 \times 10^{-4} \, \frac{Z(Z-1)}{A^{4/3}}
$$

\medskip

$$
Q'_{e} =  + 7.7 \times 10^{-4} \frac{Z(Z-1)}{A^{4/3}} . 
$$
}

\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Approximate EP-violating ``dilaton charges''}

\simpleb{green}{
\begin{table}[h]\centering
\caption{Approximate EP-violating `dilaton charges' for a sample of materials. These charges are averaged over the (isotopic or chemical, for SiO$_2$) composition.}
\begin{tabular}{ccccc}
\\
{\rm Material} &$A$ &$Z$ &$-Q'_{\hat m}$ &$Q'_e$ \\ \\
{\rm Li} &7 &3 &18.88 $\times 10^{-3}$ &0.345 $\times 10^{-3}$ \\
{\rm Be} &9 &4 &17.40 $\times 10^{-3}$ &0.494 $\times 10^{-3}$ \\
{\rm Al} &27 &13 &12.27 $\times 10^{-3}$ &1.48 $\times 10^{-3}$ \\
{\rm Si} &28.1 &14 &12.1 $\times 10^{-3}$ &1.64 $\times 10^{-3}$ \\
{\rm SiO$_2$} &... &... &13.39 $\times 10^{-3}$ &1.34 $\times 10^{-3}$ \\
{\rm Ti} &47.9 &22 &10.28 $\times 10^{-3}$ &2.04 $\times 10^{-3}$ \\
{\rm Fe} &56 &26 &9.83 $\times 10^{-3}$ &2.34 $\times 10^{-3}$ \\
{\rm Cu} &63.6 &29 &9.47 $\times 10^{-3}$ &2.46 $\times 10^{-3}$ \\
{\rm Cs} &133 &55 &7.67 $\times 10^{-3}$ &3.37 $\times 10^{-3}$ \\
{\rm Pt} &195.1 &78 &6.95 $\times 10^{-3}$ &4.09 $\times 10^{-3}$ \\
\end{tabular}
\end{table}
}

\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Composition-dependence of weak EP violations}

\simpleb{blue}{
General possible (dilaton-like) phenomenology (Damour-Polyakov'94, Dent'08, Damour-Donoghue'10): $A \equiv N + Z$
$$
\left( \frac{\Delta a}{a} \right)_{AB} = \left[ \frac{c_1}{A^{1/3}} + c_2 \, \frac{Z^2}{A^{4/3}} + c_3 \, \frac{A-2Z}{A} + c_4 \, \frac{(A-2Z)^2}{A^2} \right]_{AB}
$$
}
\simpleb{blue}{
Plausible simplified (dilaton-like) phenomenology (Damour-Donoghue2010)
$$
\left( \frac{\Delta a}{a} \right)_{AB} \simeq \left[ \frac{c_1}{A^{1/3}} + c_2 \, \frac{Z^2}{A^{4/3}} \right]_{AB}
$$
}

\vfill

\simpleb{yellow}{
{\color{red} Two} dominant EP signals, linked to {\color{red} nuclear physics} (variation of $m_q / \Lambda_{\rm QCD}$) and {\color{red} Coulomb effects} (variation of $\alpha_{\rm EM} = e^2 / \hbar c$)}
\simpleb{yellow}{
Two material pairs suffice to constrain the two dominant EP parameters $c_1 , c_2$
}
\simpleb{yellow}{
Dilaton-like models allow to a priori compare the sensitivity of various EP tests: e.g. the ``dilaton charge vector'' of the pair $Rb^{85}$, $Rb^{87}$ can be compared to that of $Pt$, $Ti$ and is found to be $\sim 10^{-2}$ smaller.
}


\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Present Experimental Bounds}

\simpleb{green}{
Using the two current EP experiments that have reached the $10^{-13}$ level, namely E\"otWash (Schlamminger et al. 2008) 
$$
\left(\frac{\Delta a}{a}\right)_{\rm Be \, Ti} = (\alpha_{\rm Be}-\alpha_{\rm Ti})\alpha_{\rm Earth} = (0.3 \pm 1.8)\times 10^{-13}
$$
and Lunar Laser Ranging (Williams et al. 2004, 2009) 
$$
\left( \frac{\Delta a}{a}\right)_{\rm Earth \, Moon} = (\alpha_{\rm Earth}-\alpha_{\rm Moon})\alpha_{\rm Sun} = ( -1.0 \pm 1.4)\times 10^{-13}
$$
one can get constraints on the two dilaton parameters
$$
D_{\hat m} = d^*_g \, (d_{\hat m} - d_g) \, , \qquad D_e = d^*_g \, d_e \, .
$$
Namely, at the $2\sigma$ level
$$
D_{\hat m} = \pm 0.87 \times 10^{-9}, \, \quad D_e = \pm 4.0 \times 10^{-9} \, .
$$
}

\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Comparing the Experimental Sensitivities of EP Experiments}

\simpleb{blue}{
The simplified dilaton framework contains three independent parameters, $d_g$ (composition-independent) and $d_q \equiv d_{\hat m} - d_g , d_e$ (composition-dependent). It is quite predictive and can be used as a guideline for comparing and/or planning EP experiments. Examples:
}

\vfill

\simpleb{yellow}{
Comparing composition-independent (Eddington's $\gamma$-parameter) and composition-dependent
$$
1- \gamma \simeq 2 d_g^2
$$
$\bullet$ In dilaton models: $\exists$ also link EP and tests of (PN) gravity 

\medskip

$$
\frac{\Delta a}{a} \sim 10^{-2} \, \frac{d_q}{d_g} \ \frac{1-\gamma^{\rm PPN}}{2}
$$

\medskip

where $d_q \equiv \partial \, \ell n (m_q / \Lambda_{\rm QCD}) / \partial \varphi$, $d_g \equiv \partial \, \ell n (\Lambda_{\rm QCD} / m_{\rm Planck})/\partial \varphi$ and either $d_q \sim d_g$ or $d_q \sim d_g / 40$. In the ``worst case'' $1 - \gamma^{\rm PPN} \sim 10^4 \, \Delta a/a$ so that $\Delta a/a \sim 10^{-15} \to 1-\gamma^{\rm PPN} \sim 10^{-11}$.
}


\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{}

\simpleb{green}{
Comparing the vectors of dilaton-charge differences 
$$
(Q'_{\hat m} , Q'_e)_{\rm Pt \, Ti} = (3.33 , 2.04) \times 10^{-3} 
$$

$$
{\rm {\color{red}vs}} \quad (Q_{\hat m} , Q_{\delta_m} , Q_{m_e} \, Q_e) _{^{87}Rb \, ^{85}Rb} \nonumber \\
= (-3.3,3.4,-0.55,-9.2) \times 10^{-5} \, . \nonumber
$$
}

\vfill

\simpleb{blue}{
$\exists$ also link between WEP and clock tests of EEP (e.g. grav. redshift) (see, e.g., TD gr-qc/9904032). When comparing frequencies of atomic transitions $A^* \to A$ at two different locations $r_1,r_2$:
$$
\frac{\nu_A^{A^*} (r_1)}{\nu_A^{A^*} (r_2)} \simeq 1 + (1+ \alpha_A^{A^*} \, \alpha_E)(U_E (r_1) - U_E (r_2))
$$
where
$$
\alpha_A^{A^*} = \frac{\partial \ln E_A^{A^*}}{\partial \varphi}
$$
computable from coupling-constant dependence of $E_A^{A^*}$. E.g. for hyperfine transition $E_A^{A^*} \propto m_e \, e^4 \, g_l \frac{m_e}{m_p} \, e^4 \, F_{\rm rel} (Ze^2)$.}


\end{frame}
%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Anthropic-type argument for EP violation (Damour-Donoghue2010)}


\simpleb{blue}{
Independently of any specific theoretical model  one might argue (along the ``anthropic'' approach to the vast ``multiverse'' of cosmological and/or string backgrounds) that:
}

\vfill

\simpleb{yellow}{
\begin{enumerate}
\item[(i)] the EP is not a fundamental symmetry principle of Nature
\item[(ii)] the level $\eta \sim \Delta a/a$ of EP violation can be expected to vary, quasi-randomly, within some range of order unity over the full multiverse
\item[(iii)] as there is probably a maximal level of EP-violation, say $\eta_* \ne 0$, which is compatible with the development of life (and physicists), one should a priori expect to observe, in our local environment, an EP violation $\eta$ of order $\eta_*$.
\end{enumerate}

}

\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Conclusions (I)}


\simpleb{blue}{
$\bullet$ EP is {\color{red} intimately connected} with some of the basic aspects of modern physics, and of the {\color{red} unification of gravity with particle physics.}
}

\vfill 

\simpleb{blue}{
$\bullet$ The historical tendency of physics to {\color{red} discard any absolute structures}, as well as the generalized Kaluza-Klein aspects (moduli) of string theory a priori suggests there could exist EP violations.
}

\vfill 

\simpleb{blue}{
$\bullet$ The recent observation of {\color{red} $\rho_{\rm vac} \sim 10^{-123} \, m_{\rm Planck}^4$} poses a challenge to physics which suggests that we are missing some key understanding of IR gravity. This might {\color{red} provide additional motivation} for EP violation (either via some Nambu-Goldstone mode, or via anthropic arguments).
}

\vfill 

\simpleb{blue}{
$\bullet$ Even within the ``majority view'' of the ``moduli stabilization'' issue, EP experiments are {\color{red} testing a key assumption} of current string models.
}




\end{frame}

%-----------------------------------------------------------------------------
\begin{frame}
\frametitle{Conclusions (II)}

\simpleb{blue}{
$\bullet$ $\exists$ {\color{red} no firm prediction for level of EP violation}, but some phenomenological models show that the violation could naturally be just below the currently tested level.
}

\vfill

\simpleb{blue}{
$\bullet$ In dilaton-like models, the composition-dependence of EP signals is (probably) dominated by {\color{red} two} signals, depending on $A^{-1/3}$ and $Z^2 \, A^{-4/3}$.
}

\vfill

\simpleb{blue}{
$\bullet$ In such dilaton-like models, there exist correlated modifications of gravity ($\Delta a/a$, $\gamma^{\rm PPN} - 1 \ne 0$, $\dot\alpha_a \ne 0$, $d\alpha_a / dU \ne 0$, $\ldots$) but EP tests {\color{red} stand out as our deepest probe of new physics}, when compared to, e.g., solar-system ($\gamma^{\rm PPN}$) or clock tests ($\dot\alpha_a$ or $d \alpha_a / dU$). Indeed,
$$
\frac{\Delta a}{a} \sim 10^{-2} \, \frac{d_q}{d_g} \ \frac{1-\gamma^{\rm PPN}}{2}
$$
where $d_q \equiv \partial \, \ell n (m_q / \Lambda_{\rm QCD}) / \partial \varphi$, $d_g \equiv \partial \, \ell n (\Lambda_{\rm QCD} / m_{\rm Planck})/\partial \varphi$ and either $d_q \sim d_g$ or $d_q \sim d_g / 40$. In the ``worst case'' $1 - \gamma^{\rm PPN} \sim 10^4 \, \Delta a/a$ so that $\Delta a/a \sim 10^{-15} \to 1-\gamma^{\rm PPN} \sim 10^{-11}$.
}


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