\documentclass[10pt]{beamer} %\documentclass[ignorenonframetext,handout]{beamer} %\usepackage{pgfpages} %\pgfpagesuselayout{4 on 1}[a4paper]%,border shrink=5mm] \usepackage{euler} \usepackage{graphics} \usepackage[english]{babel} \usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps} \usepackage{amsmath,amssymb}%,marvosym} \usepackage{marvosym} \usepackage{times}%,mathtime} %\usepackage{palatino} \usepackage{slashed,wasysym} %\usepackage[garamond]{mathdesign} %\usepackage[math]{anttor} \usepackage[latin1]{inputenc} \usefonttheme{structurebold} \def\colorize<#1>{% \temporal<#1>{\color{structure!50}}{\color{black}}% {\color{black!50}}} \def\link#1#2{\href{#1}{\underline{#2}}} \def\M{\textrm{mass}} \def\Lm{{\ell_{P}\Lambda}} \def\calV{\mathcal{V}} \def\cN{\mathcal{N}} \def\fM{\mathfrak{M}} \def\fA{\mathfrak{A}} \def\tr{\textrm{tr}} %\def\iconarticle{\includegraphics[height=.5cm]{articles.pdf}} %\def\iphtlogo{\link{http://ipht.cea.fr}{\includegraphics[height=1cm,width=1cm]{ipht.jpg}}} %\def\iheslogo{\link{http://www.ihes.fr}{\includegraphics[height=1cm,width=1.2cm]{ihes-texte.jpg}}} %\def\iconbook{\includegraphics[height=.5cm]{books.pdf}} % boxes \newcommand\simpleb[2]{ \setbeamercolor{lightstruct}{fg=black,bg=#1!9}% %\setbeamercolor{medstruct}{fg=black,bg=structure.fg!30}% \begin{beamerboxesrounded}[lower=lightstruct,shadow=true]{}{#2% }% \end{beamerboxesrounded}} \newcommand\simpleblock[1]{\begin{beamerboxesrounded}[shadow=true]{}{#1% }% \end{beamerboxesrounded}} \newcommand\inverseblock[2]{ \setbeamercolor{lightstruct}{fg=#1,bg=black!50}% %\setbeamercolor{medstruct}{fg=black,bg=structure.fg!30}% \begin{beamerboxesrounded}[lower=lightstruct,shadow=true]{}{#2% }% \end{beamerboxesrounded}} %--- \colorlet{darkgreen}{green!80!black} \newcommand{\Blue}[1]{{\color{blue} #1}} \newcommand{\Red}[1]{{\color{red} #1}} \newcommand{\Green}[1]{{\color{darkgreen} #1}} \newcommand{\Rose}[1]{{\color{purple} #1}} \newcommand{\Magenta}[1]{{\color{magenta} #1}} \newcommand{\myref}[1]{{\color{darkgreen}{\tt{\footnotesize [#1]}}}} %\newcommand{\myref}[1]{} \def\M{\textrm{mass}} \def\Lm{{\ell_{P}\Lambda}} \def\calV{\mathcal{V}} \def\cN{\mathcal{N}} \def\fM{\mathfrak{M}} \def\fA{\mathfrak{A}} \mode { \usetheme{Madrid} \setbeamercovered{transparent} } \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 $\Delta a/a \sim 10^{-15} \to 1-\gamma^{\rm PPN} \sim 10^{-11}$. } \end{frame} %----------------------------------------------------------------------------- \end{document}