Fill the Gap

Ben Roberts (UQ) — 2026 — broberts.io/slides

# Relativistic Effects in Atomic Scattering and Absorption:
          ## _Low-Mass Dark Matter and Neutrinos_

### Fill the Gap: Signatures of Electronic Excitations in Dark Matter
          ### and Neutrino Direct Detection Experiments
          
          <br>

#### Munich Institute for Astro-, Particle and BioPhysics
          #### 24 March 2026

### <u>Benjamin M. Roberts</u>, Narise Williams, Ashlee Caddell

<br>

_School of Mathematics and Physics,
          University of Queensland, Australia_

<img src="img/uq-logo.svg" width="25%">
                          Supported by: ARC and
                         
          <img src="img/BQI.png" width="20%">
          <br>

# About Us

<div style="text-align: left; float: left; width: 50%">
            <br>
            <br>
            <div class="fragment">
              <h2>High-precision atomic theory</h2>
              <h3> • Heavy atoms for precision physics</h3>
              <h3> • Atomic parity violation</h3>
              <h3> • Probing nuclear and fundamental physics</h3>
            </div>
            <br>
            <div class="fragment">
              <h2>Low-mass dark matter</h2>
              <h3> • Mass range: $\sim$10 GeV down to $10^{-23}$ eV</h3>
            </div>
            <br>
            <div class="fragment">
              <h3>AMPSCI</h3>
              <h3> • Open-source atomic structure code</h3>
              <h3> • All-order perturbation theory</h3>
              <h3>   • One- and two-valence systems</h3>
              <h3>   • Relativistic scattering</h3>
              <h3> • <a href="https://ampsci.dev">ampsci.dev</a></h3>
            </div>
          </div>

# Atomic Physics: low-energy, high-precision frontier

* BMR _et al._, [Annual Review of Nuclear and Particle Science (2015)](https://www.annualreviews.org/content/journals/10.1146/annurev-nucl-102014-022331)
          * Safronova _et al._, [Reviews of Modern Physics (2018)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.90.025008)

<div style="text-align: center; float: left; width: 50%">
            
            <div class="fragment" data-fragment-index="0" style="text-align: left;" data-markdown>
              <h2> Virtual particle exchange:</h2>
              <h3> • High-mass ⇒ short-range<br>
               • High-energy ⇒ high precision
              </h3>
            </div>

<br>

<div class="fragment" data-fragment-index="1" style="text-align: left;" data-markdown>
              <h2> Spectroscopy:</h2>
              <h3> • Fifth-force searches, isotope shift</h3>
            </div>
           
            <br>

<div class="fragment" data-fragment-index="1" style="text-align: left;" data-markdown>
              <h2> Symmetry Violations:</h2>
              <h3> • Atomic parity violation<br>
                 • Electric dipole moments</h3>
            </div>
            <br>

<div class="fragment" data-fragment-index="2" style="text-align: left;" data-markdown>
              <h2> Dark Matter + Neutrinos:</h2>
              <h3> • Direct detection?</h3>
            </div>
            <br>

</div>

<div style="text-align: center; float: right; width: 50%">
            <br>
            <div class="fragment" data-fragment-index="0" >
              <img src="img/fifth-force-ss.png" width="60%"><br>
              <font size="3">[J. Hur / Massachusetts Institute of Technology]</font>
            </div>
            <br>
            <br>
            <br>
            <img class="fragment" data-fragment-index="1" src="img/pnc.png" width="85%">
          </div>

<br>

# Dark Matter: What we don't know

<br>
          <div class="fragment" data-fragment-index="0">
            <h2> ... everything else</h2>
          </div>

<br>

<div class="r-stack">
            <img class="fragment" data-fragment-index="1" src="img/DM_masses2.png" width="80%">
            <img class="fragment" data-fragment-index="3" src="img/DM_masses3.png" width="80%">
          </div>
          <div class="fragment" data-fragment-index="1">
            <font size="3">US "Cosmic Visions" report [arXiv:1707.04591]</font>
          </div>

<div class="fragment" data-fragment-index="1">
            <h3>• Possible mass range: 90 orders of magnitude!</h3>
          </div>
          <div class="fragment" data-fragment-index="2">
            <h3>• Fundamental particle: 50 orders (de Broglie to Planck)</h3>
          </div>

<br>

<div class="fragment">
            <h3> • Vast majority of focus on WIMPs, but field is very wide</h3>
          </div>

# En*light*ening the search for Dark Matter?
          ## Atomic Physics!

<h2> Mass drops below nuclear mass:</h2>
              <h3> • No nuclear recoils. Instead: electron recoils + ionisation
              </h3>
            </div>

<h2> Mass drops below electron mass:</h2>
              <h3> • No electron recoils
                 • Instead: absorption (dark photoelectric effect)</h3>
            </div>
            <br>

<h2> Mass drops below eV:</h2>
              <h3> • Classical DM field
                 • Quantum sensing (atomic clocks)</h3>
            </div>

</div>

<div style="text-align: center; float: right; width: 40%">
            <br>
            <div class="fragment" data-fragment-index="0">
            <img  src="img/WIMP-nucleon.png" width="75%"><br>
                <div style="text-align: center"><font size="4">
                  [O'Hare - GitHub/DirectDetectionPlots]
                </font>
                </div>
            </div>
            <br>
            <div class="r-stack">
              <div class="fragment" data-fragment-index="2">
                <img src="img/laser-JILA.jpg" width="75%"><br>
                <font size="3">[JILA]</font>
              </div>
            </div>
          </div>

# Atomic Ionisation: Scattering and Absorption

## Goal: _Universal relativistic ionisation form-factors_

<h2>Excitation amplitudes</h2>
            <h3> • Plane wave for scatterer</h3>
            <h3> • Atomic states i,f</h3>

<div class="fragment" data-fragment-index="1" style="font-size: 1.35em">
              $$ \begin{align*} 
                \mathcal{M} &= 
                  -ig^e g^X
                  \left[{\color{skyblue}\bar\psi_f\Gamma_e^\mu\psi_i}\right]\,
                  {\color{orange} D_{\mu\nu}(q)}\,
                  \left[{\color{orange}\bar X'(k')\Gamma_X^\nu (k)}\right] \\~\\
                \mathcal{A} &= 
                  g^X
                  \left[{\color{skyblue}\bar\psi_f\Gamma_e^\mu\psi_i}\right]\,
                  {\color{orange}\chi_\mu(q)}
              \end{align*} $$
              <br>
              </div>
              <div class="fragment" data-fragment-index="2" style="font-size: 1.0em">
              $${\color{orange}
                \Gamma^\mu = \begin{cases}
                \gamma^\mu & \text{(V)}\\
                \gamma^\mu\gamma^5 & \text {(A)}
                \end{cases}
                \qquad
                \Gamma 
                = \begin{cases}
                1 & \text{(S)}\\
                i\gamma^5 & \text{(P)}\\
                \end{cases}}
              $$
            </div>
            <div class="fragment" data-fragment-index="4" style="font-size: 1.0em">
              <br>
              $${\color{white}
                \widetilde \gamma = \begin{cases}
                1 & \text{(V)}\\
                \gamma^5 & \text {(A)}
                \end{cases}
                \qquad
                \widetilde \gamma 
                = \begin{cases}
                \gamma^0 & \text{(S)}\\
                i\gamma^0\gamma^5 & \text{(P)}\\
                \end{cases}}
              $$
          </div>

</div>

<div class="fragment" data-fragment-index="3" style="font-size: 1.5em">
              $$
              {\rm d}\sigma \propto |{\color{orange} W_\mu} {\color{skyblue} J^\mu}|^2
              $$
              <br>
              $${\color{orange} W_\mu} \, \langle {\color{skyblue}f} | 
              {\color{skyblue}\gamma^0\Gamma^\mu} {\color{violet}e^{i\mathbf{\vec q\cdot \vec r}}}| 
              {\color{skyblue}i} \rangle 
              $$
              <br>
              </div>
              <div class="fragment" data-fragment-index="4" style="font-size: 1.5em">
              $$
              = {\color{orange} W_T} \, \langle {\color{skyblue}f} | 
              {\color{white}\tilde\gamma} {\color{violet}e^{i\mathbf{\vec q\cdot \vec r}}}| 
              {\color{skyblue}i} \rangle 
              -
              {\color{orange} \mathbf{\vec W}}\!\cdot\!\langle {\color{skyblue}f} | 
              {\color{skyblue}\mathbf{\vec \alpha}}
              {\color{white}\tilde\gamma} {\color{violet}e^{i\mathbf{\vec q\cdot \vec r}}}| 
              {\color{skyblue}i} \rangle 
              $$
              <br>
              ${\color{skyblue}\mathbf{\vec \alpha}} = {\color{skyblue}\gamma^0\mathbf{\vec \gamma}}$
              $$
              \text{e.g.,}~~{\color{white}\tilde\gamma} = g_V - g_A\gamma^5
              $$
            </div>
          </div>

# Atomic Matrix elements

<br>
            <div style="font-size: 1.8em">
            $$
            e^{i\vec{q}\cdot\vec{r}}\qquad\qquad
            \text{No dipole approximation in general}
            $$
            </div>
            <br>

<h2>Wigner-Eckart theorem</h2>
              <h3> • Spherical tensor operators $T^k_\mu$</h3>
              <h3> • <span style="color: magenta"><i>Reduced</i> matrix element</span></h3>
              <h3>   • Independent of projections $m$, $\mu$</h3>

<br>

<div class="fragment" style="font-size: 1.2em">
              $$
              {\color{skyblue}\langle{bm_b}|{T^k_\mu}|{am_a}\rangle}
              =
              {\footnotesize\begin{pmatrix}
                  j_b & k & j_a \\
                  -m_b & \mu & m_a \\
              \end{pmatrix}}\,
              {\color{magenta}\langle{b}\|{T^k}\|{a}\rangle},
              $$
              </div>
              <br>
              <br>
              <div class="fragment">
              <h3> • Simplifies complicated sums over $m$</h3>
              <br> 
              <div style="font-size: 1.2em">
              $$
              \sum_{m_a m_b} {\color{skyblue}\langle{bm_b}|{T^K_Q}|{am_a}\rangle
              \langle{bm_b}|{S^{K'}_{Q'}}|{am_a}\rangle} \qquad\qquad\qquad\\
              \qquad\qquad\qquad
              =\frac{1}{2K+1}
              {\color{magenta}\langle{b}\|{T^K}\|{a}\rangle
              \langle{b}\|{S^{K}}\|{a}\rangle} \, \delta_{KK'}\delta_{QQ'}.
              $$
              </div>
              </div>

</div>

<h2>Vector Spherical Expansion</h2>
              <h3> • $e^{iqr}$, $\vec{\alpha}\,e^{iqr}$: <i>not</i> spherical tensors</h3>
              <h3> • $t^k$, $T_K$: <i>are</i> (multipole expansion)</h3>

<br>

<div class="fragment" style="font-size: 1.25em">
                $$
                \begin{align*}
                e^{i \mathbf{\vec q} \cdot \mathbf{\vec r}} 
                &\propto 
                \sum_{KQ} \, 
                 {\color{orange}{Y^*_{KQ}}{(\hat{\mathbf q})}} \,\, 
                 {\color{skyblue}t^K_{Q}(q,\mathbf{\vec r})}   
                \\
                e^{i\mathbf{\vec q}\cdot\mathbf{\vec r}}\gamma^0\mathbf{\vec \gamma}
                & \propto \sum_{KQ\,\sigma} \, 
                {\color{orange}\mathbf{\vec Y}_{KQ}^{(\sigma)*}(\hat{\mathbf{q}})} \,\, 
                {\color{skyblue}T_{KQ}^{(\sigma)}(q,\mathbf{\vec r})}
                \end{align*}
                $$
              </div>
              <div class="fragment">
              <br>
              <h3> • Spherical Harmonics/Bessel: ${\color{skyblue}T_K\propto Y_K(\hat r) j_K(qr)}$</h3>
              </div>
              <div class="fragment">
              <h4>  • "Electric" ($\sigma=1$), "magnetic" ($\sigma=0$), "Longitudanal" ($\sigma=-1$)</h4> 
              <br>
              <div style="font-size: 1.1em">
              $$
              \begin{equation*}
              \begin{split}
                  \hat{\mathbf{q}}\cdot{\color{orange}\mathbf{\vec{Y}}^{(1)}_{KQ}} &= \hat{\mathbf{q}}\cdot{\color{orange}\vec{Y}^{(0)}_{KQ}}  = 0\\
                  \hat{\mathbf{q}}\times{\color{orange}\vec{Y}^{(-1)}_{KQ}}  &= 0.
              \end{split}
              \end{equation*}
              $$
              </div>
              </div>

</div>

# Dimensionless Atomic Ionisation Formfactors

<h2>Factors</h2>
              <div class="fragment fade-in">
              <h3> • 10 reduced matrix elements (each K)</h3>
              <div style="font-size: 1.5em">
              $$
              \langle {\color{skyblue}f} \| 
               {\color{skyblue}t^k} {\color{red}\tilde \gamma} \| 
              {\color{skyblue}i} \rangle 
              ~,~
              \langle {\color{skyblue}f} \| 
              {\color{skyblue}T^{(\sigma)}_{K}}
              {\color{red}\tilde\gamma}\| 
              {\color{skyblue}i} \rangle 
              $$
              </div>
              
              <br>
              <div class="fragment fade-in">
              $$\begin{align*}
              |M_{ba}|^2 
              &= \sum_{m_f m_i}|\mathcal{M}_{ba}|^2\\
              &= |T_{ba}|^2 + |S_{ba}|^2 - 
              2 \Re[T^*_{ba}\, S_{ba}].
              \end{align*}$$
              </div>
              <br>
              <div class="fragment fade-in">
                <h3> • Spatial-temporal interference (longitudanal)</h3>
                <h3> • Vector-Axial interference (transverse)</h3>
                <h3> • $\implies$ 13 ionisation factors (each K)</h3>
              </div>
              </div>

<br>

<div class="fragment fade-in">
              <h3>Continuum Final State</h2>
              <h4> • Energy, <i>not</i> momentum, eigenstates</h4>
              <h4> • Energy-normalised: Extra $1/\sqrt{E}$ dimension</h4>
              <br>
              $$
              \int \langle\varepsilon'|\varepsilon\rangle \ {\rm d}\varepsilon = 1
              $$
              </div>

</div>

<div style="text-align: left; float: right; width: 57%">
              <div class="fragment fade-in" style="font-size: 1.5em">
              $$
                K^{\rm ion}(q,\omega) \equiv E_H \sum_{n_i \kappa_i}\sum_{\kappa_f}\left|{\langle{\varepsilon_f\kappa_f}\|{T^K}\|{n_i\kappa_i}\rangle}\right|^2,
              $$
              </div>
              <br>
              <img class="fragment fade-in"  
                src="img/FormFactors-modified.png" width="99%">
            </div>

# Atomic Ionisation: Examples

<br>

_General framework:_

<br>

<div style="text-align: left; width: 80%; margin: auto">
            <h3 class="fragment">1.   <b>Non-relativistic scattering (WIMP)</b>   —   $\chi + A \to \chi + A^+ + e^-$</h3>
            <div class="fragment" style="font-size: 1.15em; padding-left: 2em;">
            $$
            {\color{orange}W^\mu = (F(q), \ \mathcal{O(v/c)})}
            $$
            </div>
            <br>
            <div class="fragment" style="font-size: 1.15em; padding-left: 2em;">
            $$
              \begin{align*}
              \frac{\langle {\rm d}\sigma \ v \rangle}{dE}
              &=
              \frac{ {\color{orange}\overline{\sigma}_e} \ c}{2 m_e c^2}
              \int dv \ {\color{green}\frac{f_\chi(v)}{v/c}}
              \int_{q_-}^{q_+} a_0^2 \  q \  dq \  {\color{orange}\left|F(q)\right|^2} \ {\color{skyblue}K(E,q)}\\
              {\color{skyblue}K(E,q)} & ~ {\color{skyblue} =  E_H \ \sum_{\kappa_f\kappa_i n_i}\sum_{k}(2k+1)
              \left|\langle \varepsilon_f \| t^{k} \| n \rangle\right|^2}
              \end{align*}
            $$
            </div>
            <br>
            <h3 class="fragment">2.   <b>Photo-absorption</b>   —   $\gamma + A \to A^+ + e^-$</h3>
            <br>
            <div class="fragment" style="font-size: 1.15em; padding-left: 2em;">
            $$
            {\color{orange}W^\mu = (0, \ \mathbf{\vec \epsilon})}
            $$
            </div>
            <br>
            <div class="fragment" style="font-size: 1.15em; padding-left: 2em;">
              $$\begin{align*}
              \sigma_\gamma
              &=  \frac{2\pi^2 a_0^2  }{\alpha } \frac{E_H  }{\hbar \omega } \
              {\color{skyblue}K(\omega, q=\omega)}\\
              {\color{skyblue} K(\omega)} & ~{\color{skyblue}  =
               E_H \ \sum_{k}(2k+1)  \sum_{\sigma=0,1}
              \left|\langle \varepsilon_f \| T_{k}^{(\sigma)} \| n \rangle\right|^2}
              \end{align*}$$
            </div>
            <h4 class="fragment"> • Orthogonality between transferse $\vec{\epsilon}$ and $\vec Y^{(-1)}$ removes $\sigma=-1$ longitudanal contribution</h4>
          </div>

# Atomic Physics

### _Approximation is important_

<div>
              <h3>Typical scale</h3>
              <h4> • $\Delta E \sim {\rm keV}$</h4>
              <h4> • $q\sim {\rm MeV}$</h4>
              <h4> • Atomic scale: $a_0^{-1} \sim 4\,{\rm keV}$</h4>
              </div>

<br>
              <div class="fragment fade-in" data-fragment-index="2">
              <h3> • Electron relativistic effects</h3>
                <h4>   • Significant at high $q$</h4>
              <br>
              <h3> • Continuum state potential </h3>
                <h4>   • Sommerfeld enhancement</h4>
                <h4>   • <i>Not</i> plane waves!</h4>
              <br>
              <h3> • Bound state potential</h3>
                <h4>   • $Z_{eff}$ not reliable</h4>
              </div>
              <br>

</div>

<div style="text-align: left; float: right; width: 66%">
              <img class="fragment fade-in" data-fragment-index="1" 
                src="img/K-approx2.png" width="70%">
              <br><br>
              <div class="fragment fade-in">
                <div style="display: flex; align-items: flex-start;">
                  <div style="flex: 1;">
                    <h3>Simple approximation:</h3>
                    <h4>  • Plane wave + $Z_{\rm eff}$ + non-relativistic</h4>
                    <h4>  • Many orders of magnitude off at $\sim$1 MeV!</h4>
                  </div>
                  <div style="flex: 1; text-align: center;" class="fragment fade-in">
                    <img src="https://broberts.io/images/students/AshleeCaddell2.jpg" width="50%">
                  </div>
                </div>
              </div>
              <br>
            </div>

* A. R. Caddell, **BMR**, [Phys. Rev. D **108**, 083030 (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030) -- _Led by PhD student: Ashlee Caddell_

# Exponential Suppression and lack thereof

<br>
          <div style="overflow: hidden">
          <div style="text-align: left; float: left; width: 50%">
            <h3>Typical scale</h3>
            <h4> • $\Delta E \sim {\rm keV}$</h4>
            <h4> • $q\sim {\rm MeV}$</h4>
            <h4> • Atomic scale: $a_0^{-1} \sim 4\,{\rm keV}$</h4>
          </div>

<div style="text-align: center; float: right; width: 50%; font-size: 1.5em">
          $$
            K = \left|\langle {\rm Xe} | e^{-i\mathbf{\vec{q}}\cdot\mathbf{\vec{r}}} | {\rm Xe}^{+}+e^-\rangle\right|^2
          $$
          <br>
          $$
            q_{\rm min} = m_\chi v - \sqrt{m_\chi^2 v^2 - 2m_\chi E}
          $$
          </div>
          </div>

<div style="overflow: hidden">
          <div class="fragment" style="text-align: left; float: left; width: 48%">
            <h2>Simple Estimate</h2>
            <h3> • $r\sim q^{-1}\sim 10^{-3}a_B$ — close to nucleus</h3>
            <h3> • $s$-states ($l=0$): non-zero $\psi(0)$</h3>
            <h3> • Oscillator-like: $\psi\sim A e^{-\beta r^2}$</h3>

<br>

<div style="font-size: 1.2em">
            $${\color{red}\langle f | e^{-i\mathbf{\vec{q}}\cdot\mathbf{\vec{r}}} | i \rangle \propto e^{-q^2/8\beta}}$$
            </div>
          </div>

<div class="fragment" style="text-align: left; float: right; width: 50%">
            <h2>Non-relativistic Coulomb</h2>

<div style="font-size: 1.05em">
            $$\psi\sim A r^l \left[1-\frac{Z}{l+1}r+\cdots\right]$$
            </div>

<h3> • Cusp — strongest for $l=0$</h3>
            <h3> • Lowest-order: $\sim\int r^{l+l'+2}j_L(qr)\,dr = 0$</h3>
            <div class="fragment">
            <h3> • Next term: $\propto Z\,q^{-(l+l'+4)}$</h3>
            <h3> • ${\color{green}d\sigma\sim q^{-8}}$  —  $s$-waves dominate</h3>
            </div>
          </div>
          </div>

<div style="clear: both; text-align: center" class="fragment">
            <br>
            <h4>Eighth power is still eighth power ... but better than exponential</h4>
            <br>
            <div style="font-size: 0.85em; text-align: left; padding-left: 2em">
              • BMR, Flambaum, and Gribakin, <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.023201">Phys. Rev. Lett. 116, 023201</a> (2016), <a href="http://arxiv.org/abs/1509.09044">[arXiv:1509.09044]</a>
            </div>
          </div>

# Dirac Wavefunctions

<div style="text-align: left; width: 90%; margin: auto">
            <h2>Relativistic Case</h2>

<div style="font-size: 1.35em">
            $$\psi\sim A r^{\gamma-1} \left[\gamma-\kappa+Br+\cdots\right] \qquad \gamma=\sqrt{\kappa^2-(Z\alpha)^2}\approx1-(Z\alpha)^2$$
            </div>
            <br>

<h3> • $\kappa=-1$ for $s$-states, $\kappa=1$ for $p_{1/2}$</h3>
            <div class = "fragment">
            <h3> • Lowest-order term: $\sim\int r^{\gamma+\gamma'}j_L(qr)\,dr$ — <b>non-zero!</b></h3>
            </div>
            <br>
            <div class = "fragment" style="font-size: 1.85em">
            $$
              {\color{green} d\sigma\sim q^{-6+2(Z\alpha)^2} }
            $$
            </div>
            <br>
            
          </div>

<br>

<div class = "fragment" style="text-align: left; float: right; width: 60%">
            <div style="font-size: 1.2em">
            $$e^{-q^2} \to q^{-8} \to q^{-6} \to {\color{orange}q^{-6+2(Z\alpha)^2}} \approx q^{-5.7}$$
            </div>
            <h3> • Can be significant enhancement</h3>
            <h3> • cf: Shao-Feng Ge's talk</h3>
            <h3> • $s,p_{1/2}$-waves: $d\sigma\sim q^{-6+2(Z\alpha)^2} \simeq q^{-5.7...}$ for Xe, I</h3>
            <br>
            <br>
            <div style="font-size: 0.85em; text-align: left; padding-left: 2em">
              • BMR, Flambaum, and Gribakin, <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.023201">Phys. Rev. Lett. 116, 023201</a> (2016), <a href="http://arxiv.org/abs/1509.09044">[arXiv:1509.09044]</a>
            </div>
          </div>

# Sommerfeld Enhancement

For large $p$ ($|p|=\sqrt{2m_e\varepsilon}$), plane waves should be OK?

$$\langle\mathbf{\vec{r}}|\mathbf{\vec{p}}\rangle = e^{i\mathbf{\vec{p}}\cdot\mathbf{\vec{r}}/\hbar}$$

But high $q$ means low $r$ — close to nucleus. Continuum _energy_ eigenstates enhanced near origin:
          <div style="font-size: 1.1em">
          $${\color{red}\int_{\varepsilon-\delta\varepsilon}^{\varepsilon+\delta\varepsilon}\langle\varepsilon'jlm|\varepsilon jlm\rangle\,d\varepsilon' = 1}$$
          </div>
          </div>
          <div class="fragment" style="text-align: left; float: right; width: 48%">
          <br>
          <div style="font-size: 1.0em">
          $$\frac{K_{ns_{1/2}}}{K^{\rm pw}_{ns_{1/2}}}\Bigg|_{r\to0} \approx \frac{8\pi Z}{\left[1-e^{-2\pi Z/|p'|}\right]n^3|p'|}$$
          </div>
          <h3> • Orders of magnitude enhancement</h3>
          </div>
          </div>
          </div>

<div class="fragment fade-in">
          <br>
          <h2>Bound states: Low-$r$ Scaling</h2>

Similar enhancement for bound states
          <br>
          <br>

<div style="overflow: hidden">
          <div class="fragment" style="text-align: left; float: left; width: 50%">
          <h3> • Common: H-like wavefunctions with $Z_{\rm eff}$</h3>
          <div style="font-size: 1.25em">
          $$Z_{\rm eff} = n\sqrt{|E|/R_y}$$
          </div>
          <br>
          <h3> • Works well at intermediate to large $r$</h3>
          <h3> • <b>Fails</b> at low $r$</h3>
          </div>
          <div class="fragment" style="text-align: left; float: right; width: 48%">
          <h3> • H-like: $\psi(0)^2\sim Z_{\rm eff}^3$</h3>
          <h3> • True: $\psi_{\rm inner}(0)^2\sim Z^3$, $\psi_{\rm outer}(0)^2\sim Z^1$</h3>
          <h3> • Orders of magnitude "enhancement"</h3>
          </div>
          </div>
          </div>

# Solution: Relativistic Atomic Wavefunctions

<br>

<div class = "fragment" style="text-align: left; float: left; width: 52%">
            <h2>Relativistic Hartree-Fock</h2>
            <h3> • Dirac equation in self-consistent field</h3>
            <h3> • Continuum <em>energy</em> eigenstates</h3>
            <br>
            <div style="font-size: 1.1em">
            $$
              H^{\rm RHF} = c\gamma^0\boldsymbol{\vec \gamma}\!\cdot\!\mathbf{\vec p}
              + V_{\rm nuc} + (\gamma^0-1) c^2 + V^{\rm HF} + \delta V
            $$
            <br>
            $$
              \psi_{n\kappa m} = \frac{1}{r}
              \begin{pmatrix} f(r)\,|\kappa,m\rangle \\ i\,g(r)\,|\!-\!\kappa,m\rangle \end{pmatrix}
            $$
            <br>
            <div class = "fragment">
            <h4> • $f(r)$ -- "large" radial Dirac component</h4>
            <h4> • $g(r)$ -- "small" relativistic $\sim {\alpha}(\vec{\sigma}\!\cdot\!\vec{p}) f$</h4>
            <h4> • $|\kappa,m\rangle$ -- two-component spherical spinor</h4>
            <h4> • $V^{\rm HF}$ -- Non-local hartree-Fock potential</h4>
            <h4> • Correlation corrections: $\delta V$ (RPA)</h4>
            </div>
            </div>
          </div>

<div class="fragment" style="text-align: left; float: right; width: 35%">
            <br>
            <img src="img/github.png" width="85%"><br>
            <font size="6"><a href="https://github.com/benroberts999/ampsci">github.com/benroberts999/ampsci</a></font>
            <br><br>
            <h4> • Existing code: temporal part</h4>
            <h4> • <a href="https://ampsci.dev">ampsci.dev</a></h4>
            <h4> • Full form factors: coded, release soon</h4>
            
          </div>

# Important Many-Body effects

<br>

<div class = "fragment" data-fragment-index="1" style="text-align: left; float: left; width: 52%">
            <h2>Hole-Particle Interaction</h2>
            <h4> • Ionised electron "sees" $V^{N-1}$ potential</h4>
            <h4> • (Exact cancellation for bound states)</h4>
            <br>
            <div style="font-size: 1.25em">
            $$
              V = V^N_{\rm HF} - (1 - P_c) \, {\color{yellow}V_{\rm self}} \, (1 - P_c)
            $$
            <br>
            $$
              V^N_{\rm HF}|{\color{skyblue}\phi_a}\rangle = \sum_{\color{violet}c} \left[\langle {\color{violet}\phi_c}|Q_{12}|{\color{violet}\phi_c}\rangle|{\color{skyblue}\phi_a}\rangle - \langle {\color{violet}\phi_c}|Q_{12}|{\color{skyblue}\phi_a}\rangle|{\color{violet}\phi_c}\rangle\right]
            $$
            </div>
            <br>
            <h4> • $\Large {V^N_{\rm HF}}$ -- Self-consistant Dirac-Hartree-Fock + exchange</h4>
            <h4> • $\Large {\color{yellow}V_{\rm self}}$ -- Contribution of ionised state to Hartree-Fock potential</h4>
            <h4> • $\Large P_c$ -- Core projection operator; ensures orthogonality, Hermicity</h4>
            <br>
            <div style="font-size: 1.25em">
            $$
              P_c = \sum_{\color{violet}c}  |{\color{violet}\phi_c}\rangle \langle {\color{violet}\phi_c}|
            $$
            </div>
            </div>

<div style="text-align: left; float: right; width: 45%">
            <div class="fragment" data-fragment-index="2">
            <br>
            <h2>Core Polarisation (RPA)</h2>
            <h4> • Second-order (Field+Coulomb)</h4>
            <h4> • Usually small correction</h4>
            <h4> • Mainly important at lower energy</h4>
            <h4> • Can allow energetically-forbidden states to participate</h4>
            </div>
            <br>
            <br>
            <img class="fragment" data-fragment-index="1" src="img/hf-2.png" width="75%"><br>
            <img class="fragment" data-fragment-index="2" src="img/rpa-2.png" width="60%"><br>
          </div>

# Orthogonality and low-q relativistic effects

<br>
          <h2>Small momentum transfer</h2>
          <h3> • $e^{iqr}\to 1$</h3>
          
          <br>
          
          <div class="fragment" style="font-size: 1.5em">
          $$\begin{align*}
          \langle f |{\color{red}t^K}| i \rangle
          \overset{\rm low\ q}{\propto}
          \langle f |{\color{red}1}| i \rangle
          &= 
          \int_0^{\infty} \left[ f_{w}(r) f_{v}(r) + g_{w}(r) g_{v}(r) \right]  {\rm d} r ~ {\color{red} = 0}
          \\
          \langle f |{\color{orange}t^K\gamma^0}| i \rangle
          \overset{\rm low\ q}{\propto}
          \langle f |{\color{orange}\gamma^0}| i \rangle
          &= 
          \int_0^{\infty} \left[ f_{w}(r) f_{v}(r) ~{\color{orange}-}~ g_{w}(r) g_{v}(r) \right]  {\rm d} r ~ {\color{orange}\sim \alpha^2}
          \end{align*}$$
          </div>
          <br> 
          <br>

<div class="fragment">
              <h2>Vector</h2>
              <h3>• Goes to zero by orthogonality</h3>
              <h3>• Numerically challenging, well known</h3>
            </div>
            <br>
            <div class="fragment">
            <h3>• Common solution:</h3>
            <div style="font-size: 1.5em">
              $${\color{red}\langle f |t^K| i \rangle \to \langle f |t^K\!-\!1| i \rangle}$$
            </div>
            </div>
            <br><br>

</div>

<div style="text-align: center; float: right; width: 43%">
            <div class="fragment">
              <h2>Scalar</h2>
              <h3>• Relativistically suppressed</h3>
            <br>
            </div>
            <div class="fragment" style="font-size: 1.5em">
              $${\color{orange}\langle f |\gamma^0| i \rangle = -2\int_0^{\infty} g_{w}(r) g_{v}(r)\,{\rm d}r \sim \alpha^2}$$
            </div>
            <br>
            <div class="fragment">
            <h3>• Pseudoscalar/axial: even worse</h3>
            </div>
          </div>

# $\gamma^5$ matrix tomfoolery

### Even worse: _Naive calculations can be very unstable_

<br><br>
          <div style="text-align: left; width: 80%" class="fragment">
            <h3>• For axial-vector (temporal), and pseudo-scalar</h3>
            <h3>• Low $q$: Matrix elements of "raw" ${\color{orange}\gamma^5}$, ${\color{coral}\gamma^0\gamma^5}$</h3>
          </div>
          
          <div style="clear: both; text-align: center; width: 100%" class="fragment">
            <br>
            <div style="font-size: 1.5em">
            $$
            \begin{align*}
            {\color{orange}
            \gamma^5} ~ & {\color{orange}= i\,\alpha\,[H,\vec{\sigma}\sdot\vec{r}] - 2\gamma^5(1+\vec\sigma\!\cdot\!\vec l)} \\
            {\color{coral}\gamma^0\gamma^5} ~ & {\color{coral} = \frac{\alpha^2}{2}[H,\gamma^5]}
            \end{align*}
            $$
            <br>
            </div>
            <div class="fragment" style="display: flex; align-items: flex-start;">
              <div style="text-align: left; width: 60%;">
                <h3>• Suppressed by $\color{orange}\alpha$ and $\color{coral}\alpha^3$, respectively</h3>
                <h3>• In practice: large cancellation between order 1 terms!</h3>
                <h3>  • Pseudoscalar ($\sim\gamma^0\gamma^5$) particularly dangerous</h3>
                <h3>• Exact relations [<a href="https://arxiv.org/abs/1404.2723">1</a>]: replace, or use as check</h3>
              </div>
              <div class="fragment" style="font-size: 1.3em" style="flex: 1; text-align: center;">
              <br> 
                $$\begin{align*}
                 {\color{orange}\langle{w}|{\gamma^5}|{v}\rangle}
                &~{\color{orange}= \frac{i\alpha\,\omega_{wv}\langle{w}|{ {\hat \sigma}\!\cdot\!\vec{r}}|{v}\rangle}{1+\alpha^2\kappa_w\omega_{wv}}}
                \\~\\
                {\color{coral}\langle{w}|{i\gamma^0\gamma^5}|{v}\rangle }
                &~{\color{coral}= \frac{-\alpha^3\,\omega_{wv}^2\langle{w}|{ {\hat \sigma}\!\cdot\!\vec{r}}|{v}\rangle}{2(1+\alpha^2\kappa_w\omega_{wv})}}
                \end{align*}$$
              </div>
            </div>
            <br>
            
           
          </div>

* [1] BMR, Stadnik, Dzuba, Flambaum, Leefer, Budker, [PRL **113**, 081601](https://link.aps.org/doi/10.1103/PhysRevLett.113.081601) (2014), [[arXiv:1404.2723](https://arxiv.org/abs/1404.2723)]

# Calculations difficult
          ## Can you test it?

<div>
            <div style="text-align: left; float: left; width: 50%">
              <div class="fragment fade-in">
                <br>
                <h3>Photo-absorption</h3>
                <img src="img/xe-photoel.png" width="85%">
                <h4> • Just the simplest calculations (no many-body/RPA)</h4>
                <h4> • Perfect agreement at relevant high energies</h4>
              </div>
            </div>

<div style="text-align: center; float: right; width: 50%">
              <div class="fragment fade-in">
                <br>
                <h3>Electron-impact ionisation</h3>
                <img src="img/e-impact.png" width="85%">
                <h4> • More complicated (not weak coupling!)</h4>
                <h4> • Better than dedicated calculations</h4>
              <br>
              • Ashlee Caddell, <b>BMR</b>,
          <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030">Phys. Rev. D <b>108</b>, 083030 (2023)</a>
            </div>
            </div>
          </div>

<br>

<br>

## Vector form-factors for Xe

<div class="fragment fade-in">
            <h3>Outer shell:</h3>
            <h4>  •  $E \sim R_y \sim 10\ {\rm eV}$</h4>
            <h4>  •  $q \sim a_0^{-1} \sim 4\ {\rm keV}$</h4>
            </div>

<br><br>
            <div class="fragment fade-in">
            <h3>Inner shell:</h3>
            <h4>  •  $E \sim Z^2 R_y \sim 30\ {\rm keV}$</h4>
            <h4>  •  $q \sim Z a_0^{-1} \sim 200\ {\rm keV}$</h4> 
            </div>

<br>
            
            <div class="fragment fade-in">
            <h4>PhD student: Narise Williams</h4>
              <img src="https://broberts.io/images/students/Narise-20251113-IMG_6699.jpg" width="60%">
            </div>

</div>

# Conclusion

### Dark matter with atomic phenomena:

* Electron scattering, absorption, fifth-force searches
            * Fundamental symmetry violations (EDM, PNC)
            * Ultralight DM: atomic clocks, quantum sensors

### Atomic Ionisation Formfactors

* Relativistic effects important
            * Atomic structure important
            * [ampsci.dev](https://ampsci.dev)
              * Full code + form factors coming soon

<br>

### Fill the Gap, Munich, 2026

<br>
  
            ### Benjamin M. Roberts
    
            _University of Queensland, Australia_

* Caddell, Flambaum, BMR, [Phys. Rev. D **108**, 083030 (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030), [[arXiv:2305.05125]](http://arxiv.org/abs/2305.05125)
            <br>
            * BMR, Flambaum [Phys. Rev. D **100**, 063017 (2019)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.100.063017), [[arXiv:1904.07127]](http://arxiv.org/abs/1904.07127)
            <br>
            * BMR, Flambaum, Grabakin [Phys. Rev. Lett. **116**, 023201 (2016)](doi.org/10.1103/PhysRevLett.116.023201), [[arXiv:1509.09044]](http://arxiv.org/abs/1509.09044)
            <br>
            * Hees, Do, BMR, Ghez _et al._, [Phys. Rev. Lett. **124**, 081101 (2020)](https://link.aps.org/doi/10.1103/PhysRevLett.124.081101), [[arXiv:2002.11567]](http://arxiv.org/abs/2002.11567)
            <br>
            * Filzinger, Caddell _et al._, [Phys. Rev. Lett. **134**, 031001 (2025)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.134.031001), [[arXiv:2312.13723]](http://arxiv.org/abs/2312.13723)

# Explicit formulas: Form Factors

<br>
            <div style="font-size: 1.1em">
            $$
            \begin{align*}
            \sum_{if}|{\color{orange}W_\mu}{\color{skyblue}\mathcal{J}^\mu_{fi}}|^2 
            = 
            &|{\color{orange}W_T}|^2\left(c_V^2{\color{cornflowerblue}|\Phi|^2}
            + c_A^2{\color{turquoise}|\Theta|^2} \right)
            + \frac{\color{orange}|\vec{W}_\perp|^2}{2}\sum_{\sigma=1,0}\left(
            c_V^2{\color{cornflowerblue}|V^{(\sigma)}|^2}
            + c_A^2{\color{turquoise}|A^{(\sigma)}|^2}\right)
            + |{\color{orange}\hat{\vec{q}}\!\cdot\!\vec{W}}|^2\left(
            c_V^2{\color{cornflowerblue}|V^{(-1)}|^2}
            + c_A^2{\color{turquoise}|A^{(-1)}|^2}\right) \\
            &
            + 2\Re
            [{\color{orange}{W_T}^*\,\hat{\vec{q}}\sdot\vec{W}}]
            \left(c_V^2 {\color{palevioletred}X}
            -c_A^2
            {\color{palevioletred}Y}
            \right)
            +\Im\{ {\color{orange}
            \hat{\vec{q}}\!\cdot\![\vec{W}\!\times\!\vec{W}^*]}\}\,{c_V c_A}\,{\color{palevioletred}Z}
            \end{align*}
            $$
            </div>
            
            <br>

<div style="text-align: left; float: left; width: 50%">
            <h3>Vector</h3><br>
            <div style="font-size: 1.0em">
            $$\color{cornflowerblue}
            \begin{align*}
            |\Phi_K|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^K}\|{n_i\kappa_i}\rangle}|^2 \\
            |{V}^{(\sigma)}|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{T^{(\sigma)}_{K}}\|{n_i\kappa_i}\rangle}|^2
            \end{align*}
            $$
            </div>
            <h3>Axial</h3><br>
            <div style="font-size: 1.0em">
            $$\color{turquoise}
            \begin{align*}
            |\Theta|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^K\gamma^5}\|{n_i\kappa_i}\rangle}|^2\\
            |{A}^{(\sigma)}|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{T^{(\sigma)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle}|^2,
            \end{align*}
            $$
            </div>
            </div>

<div style="text-align: left; float: right; width: 50%">
            <h3>Scalar/Pseudoscalar</h3><br>
            <div style="font-size: 1.0em">
            $${\color{cornflowerblue}
            \begin{align*}
                |S|^2
                    &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^K\gamma^0}\|{n_i\kappa_i}}\rangle|^2
            \end{align*}}
            $$
            $${\color{turquoise}
            \begin{align*}
                |P|^2 
                    &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^Ki \gamma^0\gamma^5}\|{n_i\kappa_i}}\rangle|^2 .
            \end{align*}}
            $$
            </div>
            <h3>Interference</h3><br>
            <div style="font-size: 1.0em">
            $${\color{palevioletred}
            \begin{align*}
            Z
            &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}
            \Im\Big\{
            \langle{\varepsilon_f\kappa_f}\|{T^{(1)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(0)}_{K}}\|{n_i\kappa_i}\rangle
            \notag\\[-0.85em]
            &\qquad\qquad\qquad\qquad\qquad
            -\langle{\varepsilon_f\kappa_f}\|{T^{(1)}_{K}}\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(0)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \Big\} \\
            X &\equiv
            E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}
            \Im\left\{
            \langle{\varepsilon_f\kappa_f}\|{t^{K}}\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(-1)}_{K}}\|{n_i\kappa_i}\rangle
            \right\}\\
            Y &\equiv
            E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}
            \Im\left\{
            \langle{\varepsilon_f\kappa_f}\|{t^{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(-1)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \right\}.
            \end{align*}}
            $$
            </div>
            </div>

# Explicit formulas: Matrix elements

<br>

<div style="font-size: 1.3em">
            $$
            \begin{align*}
            T^{(+1)}_{KQ} 
                &= \left[(K+1)\tfrac{j_{K}(qr)}{qr} - j_{K+1}(qr)\right]\vec{\alpha}\!\cdot\!\vec{C}^{(+1)}_{KQ}
                +
                \tfrac{j_{K}(qr)}{qr}\,\sqrt{K(K+1)}\,\vec{\alpha}\!\cdot\!\vec{C}^{(-1)}_{KQ} 
                \\
            T^{(0)}_{KQ} &= j_K(qr)\, \vec{\alpha}\!\cdot\!\vec{C}^{(0)}_{KQ} \\
            T^{(-1)}_{KQ} 
                &= -\tfrac{j_K(qr)}{qr}\,\sqrt{K(K+1)}\,\vec{\alpha}\!\cdot\!\vec{C}^{(+1)}_{KQ}
                -
                \left[K\tfrac{j_K(qr)}{qr}-j_{K+1}(qr)\right]\vec{\alpha}\!\cdot\!\vec{C}^{(-1)}_{KQ} .
            \end{align*}
            $$
            </div>

<br>

<div style="text-align: left; float: left; width: 40%">
            <div style="font-size: 1.0em">
              $$\begin{align*}
              \langle n_w\kappa_w \| t^K \| n_v\kappa_v \rangle
              &= R^{(+)}_{wv}[j_K] \, \langle \kappa_w \| C^K \| \kappa_v \rangle\\
              \langle n_w\kappa_w \| t^K \gamma^0 \| n_v\kappa_v \rangle
              &= R^{(-)}_{wv}[j_K] \, \langle \kappa_w \| C^K \| \kappa_v \rangle\\
              \langle n_w\kappa_w \| t^K \gamma^5 \| n_v\kappa_v \rangle
              &= iP^{(-)}_{wv}[j_K] \, \langle \kappa_w \| C^K \| {-\kappa_v} \rangle\\
              \langle n_w\kappa_w \| t^K i\gamma^0\gamma^5 \| n_v\kappa_v \rangle
              &= -P^{(+)}_{wv}[j_K] \, \langle \kappa_w \| C^K \| {-\kappa_v} \rangle
              \end{align*}$$
            </div>
            <br>
            <div style="font-size: 1.0em">
              $$\begin{align*}
              R_{wv}^{(\pm)}[h] &\equiv
              \int_0^{\infty} \left[ f_{w} f_{v} \pm g_{w} g_{v} \right] h \, dr \\
              P_{wv}^{(\pm)}[h] &\equiv
              \int_0^{\infty} \left[ f_{w} g_{v} \pm g_{w} f_{v} \right] h \, dr
              \end{align*}$$
            </div>
            <br>
            </div>

<div style="text-align: left; float: right; width: 60%">
            <div style="font-size: 0.95em">
              $$\begin{align*}
              \langle n_w\kappa_w \| T^{\,(+1)}_{K} \| n_v\kappa_v \rangle
              &= i\sqrt{\tfrac{K+1}{K}}\left[
              (\kappa_w-\kappa_v)
              P^{(+)}_{wv}\!\left[\tfrac{j_K}{qr}-\tfrac{j_{K+1}}{K+1}\right]
              - KP^{(-)}_{wv}\!\left[\tfrac{j_K}{qr}\right]
              \right]C^K_{wv}\\
              \langle n_b\kappa_b \| T^{\,(0)}_{K} \| n_a\kappa_a \rangle
              &= \frac{-i}{\sqrt{K(K+1)}}\,(\kappa_w+\kappa_v)\,P^{(+)}_{ba}[j_K]\,C^K_{w,-v}\\
              \langle n_w\kappa_w \| T^{\,(-1)}_{K} \| n_v\kappa_v \rangle
              &= -i\left[
              (\kappa_w-\kappa_v)\,P^{(+)}_{wv}\!\left[\tfrac{j_K}{qr}\right]
              - P^{(-)}_{wv}\!\left[K\tfrac{j_K}{qr}-j_{K+1}\right]
              \right]C^K_{wv}
              \end{align*}$$
              </div>
              <div style="font-size: 0.95em">
              $$\begin{align*}
              \langle n_w\kappa_w \| T^{\,(+1)}_{K}\gamma^5 \| n_v\kappa_v \rangle
              &= \sqrt{\tfrac{K+1}{K}}\left[
              (\kappa_w+\kappa_v)
              R^{(-)}_{wv}\!\left[\tfrac{j_K}{qr}-\tfrac{j_{K+1}}{K+1}\right]
              - KR^{(+)}_{wv}\!\left[\tfrac{j_K}{qr}\right]
              \right]C^K_{w,-v}\\
              \langle n_b\kappa_b \| T^{\,(0)}_{K}\gamma^5 \| n_a\kappa_a \rangle
              &= \frac{-1}{\sqrt{K(K+1)}}\,(\kappa_w-\kappa_v)\,R^{(-)}_{ba}[j_K]\,C^K_{wv}\\
              \langle n_w\kappa_w \| T^{\,(-1)}_{K}\gamma^5 \| n_v\kappa_v \rangle
              &= -\left[
              (\kappa_w+\kappa_v)\,R^{(-)}_{wv}\!\left[\tfrac{j_K}{qr}\right]
              - R^{(+)}_{wv}\!\left[K\tfrac{j_K}{qr}-j_{K+1}\right]
              \right]C^K_{w,-v}
              \end{align*}$$
            </div>
            </div>

<div style="clear: both; font-size: 1.0em">
            <br>
            $$C^K_{ab} \equiv \langle \kappa_a \| C^K \| \kappa_b \rangle
              = (-1)^{j_a+1/2}\sqrt{(2j_a{+}1)(2j_b{+}1)}
              \begin{pmatrix} j_a & K & j_b \\ \tfrac{1}{2} & 0 & -\tfrac{1}{2} \end{pmatrix}
              \pi(l_a{+}K{+}l_b)$$
            </div>

# Atomic Clocks 101:

<br>

### Frequency reference: atomic transition
          ### Local oscillator: EM field (laser, microwave)

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
            <img src="img/Bohr.png" width="70%"><br>
            <font size="3"> JabberWok [Wikimedia Commons]</font>
            <br><br><br>
            <h3>
              Atomic transition: "perfect*" frequency reference
            </h3>
          </div>

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
            <img src="img/Wcislo2016.png" width="95%">
            <font size="3">[Wcislo, Nature Astronomy (2016)]</font>
            <br><br><br>
            <h3>
              Compare frequency of oscillator to atomic transition
            </h3>
          </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 32%">
            <img src="img/Al+Choi2010.png" width="95%">
            <font size="3">[Chou et al., Science 329, 1630 (2010)]</font>
            <br><br><br>
            <h3>
              Lock on: Adjust local oscillator to maximum transition rate
            </h3>
          </div>

# Moore's law for atomic clocks

<div style="text-align: center; float: left; width: 60%">
            <br>
            <div class="r-stack">
              <div class="fragment fade-out" data-fragment-index="0">
                <img src="img/MooreClock2020-MW.png" width="100%">
              </div>
              <div class="fragment fade-in" data-fragment-index="0">
                <img src="img/MooreClock2020.png" width="100%"><br>
              </div>
            </div>
            <br>
            <font size="4"> [Pacome Delva <i>et al.</i>, Chronometric Geodesy: Methods and Applications (2019)]
            </font>
          </div>

<div style="text-align: center; float: right; width: 40%">
            <br><br>
            
            <br><br><br>
            <div class="fragment fade-in" data-fragment-index="0">
              <img src="img/Stability.png" width="100%"><br>
              <font size="3"> [M. Safronova]</font>
              <br><br>

<div class="fragment fade-in" data-fragment-index="1">
              <h3> • Accurate to parts in 10<sup>18</sup><br>
               • 1 s over lifetime of the universe!
              </h3>
              </div>

</div>
          </div>

# "Listen" for DM field using atomic clocks

<br>

<div style="text-align: left; float: left; width: 50%">
            <br> <br>
            <div class="fragment fade-in" data-fragment-index="0">
            <img src="img/NHanacek_NIST.png" width="90%"><br>
            <font size="3">[N Hanacek/NIST]</font>
            </div>
          </div>

<div style="text-align: left; float: right; width: 50%">
            <br>
            <div class="fragment fade-in" data-fragment-index="0">
              <h3>
                 • Exotic field may have small interaction with matter<br>
                 • Shift atomic energy levels and frequencies
              </h3>
              <h3>
                    • Monitor with atomic clocks!
              </h3>
            </div>
            <br>
            <br>
            <div class="fragment fade-in" data-fragment-index="1">
              <h3>Observable shift:</h3>
              <br>
              \[{\large
              \frac{\delta f}{f} = \kappa \, \phi_{\rm DM}(\vec{r},t)}
              \]
              <br>
              <br>
              <h3>
                 • $\kappa$ - model dependent parameter
              </h3>
              <h3>
                 • Calculated: high-precision atomic theory
              </h3>
            </div>

<br>
            <div class="fragment fade-in" data-fragment-index="2">
              <h3>Example: variation of constants</h3>
              <br>
              \[{\large
              {\mathcal{L} = \phi F_{\mu\nu}F^{\mu\nu}}
              {\implies \alpha\to\alpha+\phi_{\rm DM}(\vec{r},t)}}
              \]
              $${\large
              {\kappa = \frac{ {\rm d} (f/f_0)}{ {\rm d} (\alpha/\alpha_0)}}}
              $$
            </div>
            <br>
          </div>

# Oscillating signals

<div style="text-align: left; float: left; width: 50%">
            <br>
            <h3>Case: oscillating field:</h3>
            \[{\large
            \phi_{\rm DM} = \phi_0 \cos({m_\phi t})}
            \]
            <br>
            \[\large
            \frac{\delta f}{f} = \kappa \phi_0 \cos({m_\phi t})
            \]
            <br>
            <br>
            <div class="fragment fade-in">
              <h2>Clock Comparisons</h2><br>
              <h3>• Require two <b>different</b> clocks</h3>
              <h3>• Different systems: probe different couplings</h3>
              <br>
              <h4>• Optical-Optical: $d_e$ ($\delta\alpha)$</h4>
              <br>
              <h4>• Optical-Microwave: $d_{m_e}-d_{m_p}$ ($\delta(m_e/m_p)$)</h4>
              <br>
            </div>
          </div>

<div style="text-align: center; float: right; width: 50%">
            <br> <br>
            <div class="fragment fade-in" data-fragment-index="3">
              <h3>Typical: Compare two local clocks, different $\kappa$</h3>
              <img src="img/scalarphoton2.png" width="100%"><br>
              <p>[O'Hare - GitHub/AxionLimits]</p><br>
            </div>
          </div>

# Space-time separated sensors
  
            * Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, **Phys. Rev. Lett.** (in press), [arXiv:2312.13723](https://arxiv.org/abs/2312.13723)
  
            <br>
            <br>
  
            <div style="text-align: left; float: left; width: 50%">
  
              ### Typical local experiment:
  
              <br>
  
              $$\large
                \frac{\delta(\nu_a/\nu_b)}{(\nu_a/\nu_b)} = \phi_0 \\, ( \kappa_{a} - \kappa_b) \\, \cos(\omega t)
              $$
  
              <br>

<div class="fragment fade-in">
              
              * Non-zero effect from two _different_ transitions only
              * Certain couplings always cancel 
              * e.g., Universal couplings

</div>

### Separated (identical) sensors:

<br>

<div>
              \[\large
                \frac{\delta(\nu_a/\nu_b)}{(\nu_a/\nu_b)} = \phi_0 \, \kappa \left(\cos(\omega t) -\cos(\omega t-\delta)\right) \\
                  \delta = \omega\Delta t - k_z D
              \]
              </div>
              <br>
              <div class="fragment fade-in">

* Spatial and temporal component (depends on experiment)
              *  <font color="pink">Allows access to unconstrained couplings</font>
              * Unique signals: scales with network size, daily modulation
            
              <img src="img/Separated.png" style="width: 70%"><br>
            </div>

</div>
          </div>

# Space-time separated sensors: initial results

* Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [**Phys. Rev. Lett. 134**, 031001](https://link.aps.org/doi/10.1103/PhysRevLett.134.031001) (2025), [arXiv:2312.13723](https://arxiv.org/abs/2312.13723)
          <br>

<div style="text-align: center; float: left; width: 50%">
            <br>
            <br>
            <div class="r-stack">
              <div class="fragment fade-out" data-fragment-index="1">
                <h3>Constraints on <i>any</i> Fermion coupling (d<sub>eff</sub>)
                <img src="img/deff-inv.png" width="90%">
              </div>
              <br>
              <div class="fragment fade-in" data-fragment-index="1">
                <h3>Constraints on electron coupling (d<sub>me</sub>)
                <img src="img/dme2-inv.png" width="90%"><br>
              </div>
            </div>
            <font size="4"><a href="http://arxiv.org/abs/2312.13723">[arXiv:2312.13723]</a></font>
          </div>

* Existing data: [Schioppo _et al._, Nat.Comms 13
            (2022)](https://www.nature.com/articles/s41467-021-27884-3)
            * Comparison of ultra-stable cavity lasers
            * 750 km apart (2220 km Fibre cable: PTB to NPL via SYRTE)
            * Sensitive to _spatial_ phase shift

<br>

* Also: Data from GPS satellites (Rb clocks)
            * Clocks less accurate, but setup sensitive to _temporal_ term

<br>

* First Constraints on $d_{m_e}$ alone at low-frequency<br>
            * Sensitivity to _spatial_ fluctions of DM, daily oscillation<br>

</div>

# Super-massive black hole at the Galactic Centre

<div style="text-align: left; float: left; width: 50%">
            <h3>• with UCLA Galactic Centre Group</h3>
            <h4>   • Observations led by Tuan Do</h4>
            <h4>   • Andrea Ghez: 2020 Nobel
              prize for discovery of black hole</h4>
            <h4>   • Analysis led by Aurelien Hees</h4>
            <br>
            <h3>• Keck telescope in Hawaii</h3>
            <br>
            <h3>• Motion of ∼1000 stars tracked</h3>
            <h4>   • Precise spectroscopy for many stars</h4>
            <br>
            <br>
            <div class="fragment fade-in">
              <h3>• High gravitational potential</h3>
              <h3>• Possibly large concentration of dark
                matter </h3>
              <h3>• (also other "exotic" physics)</h3>
            </div>
          </div>

<div style="text-align: center; float: right; width: 50%">
            <img src="img/keck-ao.jpg" width="95%">
            <font size="3">[Keck Observatory | Ethan Tweedie Photography]</font>
            <br>
            <br>
            
            <br>
          </div>