Atomic Frontiers

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

# Atomic Frontiers in Fundamental Physics

### _Relativistic Effects in Atomic Scattering and Absorption_
          ### _for probing low-mass dark sector and neutrino physics_
          
          <br>

#### Sydney-CPPC seminar, UNSW
          #### 9 April 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>
              <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>
              <h2>Low-mass dark matter</h2>
              <h3> • Mass range: $\sim$10 GeV down to $10^{-23}$ eV</h3>
            </div>
            <br>
            <div>
              <h3>AMPSCI</h3>
              <h3> • Atomic structure calculations</h3>
              <h3>   • One- and two-valence systems</h3>
              <h3>   • Relativistic scattering</h3>
              <h3> • <a href="https://ampsci.dev">ampsci.dev</a></h3>
            </div>
          </div>

# The Standard Model + General Relativity

## Several unexplained mysteries...

<h2>Matter anti-matter asymmetry</h2>
              <h3> • CP-violation in the Standard Model is too small</h3>
            </div>

<h2>Nature of Dark Energy</h2>
              <h3> • Accelerated expansion of universe</h3>
                
            </div>

<h2>Nature of Dark Matter</h2>
              <h3> • Missing mass from the universe
                • Matter that interacts <em>only</em> gravitationally? </h3>

</div>

<br>

</div>

# How to break physics?

## Want to find direct evidence for physics _beyond_ standard model

<div style="text-align: left; float: right; width: 45%">
            <br>
            <div class="fragment" data-fragment-index="1">
              <h2> Particle Accelerators</h2>
              <h3> • Directly search for new particles</h3>
              <h3> • High energy, very expensive</h3>
            </div>

<div class="fragment" data-fragment-index="1">
              <br>
              <h2> CERN/LHC</h2>
              <h3> • Hoped to find evidence for something beyond</h3>
              <h3> • So far, no luck!</h3>
            </div>

<div class="fragment">
              <br>
              <h2> Alternative approach</h2>
              <h3> • Low-energy, high precision</h3>
              <h3> • Look for any violations of laws of physics</h3>
            </div>
          </div>

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

* BMR, Dzuba, Flambaum, [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="2" style="text-align: left;" data-markdown>
              <h2> Symmetry Violations:</h2>
              <h3> • Atomic parity violation (P)<br>
                 • Electric dipole moments (CP, P, T)</h3>
            </div>
            <br>

<div class="fragment" data-fragment-index="3" style="text-align: left;" data-markdown>
              <h2> Dark Sector + Neutrinos:</h2>
              <h3> • Scattering, 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 / MIT]</font>
            </div>
            <br>
            <br>
            <br>
            <img class="fragment" data-fragment-index="2" src="img/EDM.png" width="85%"><br>
            <font size="3">[BMR, Dzuba, Flambaum, Ann. Rev. Nuc. Part. Sci. (2015)]</font>
          </div>

<br>

# Low-Energy searches for new physics

<br>

<div class = "fragment" style="text-align: left; float: left; width: 48%">
            <h2>SM background + NP signal</h2>
            <h3> • Need high-sensitivity experiment</h3>
            <h3> • Need high-accuracy theory to disentangle</h3>
            <br>
            <div style="text-align: center">
              <img src="img/Qweak4.png" width="85%">
            </div>
            <br>
            <h3> • Atomic parity violation</h3>
            <h3> • Non-standard neutrino interactions</h3>
          </div>

<div class = "fragment" style="text-align: left; float: right; width: 48%">
            <h2>SM-forbidden signal</h2>
            <h3> • Still need high-sensitivity experiment</h3>
            <h3> • Order-of-magnitude theory enough</h3>
            <br>
            <div style="text-align: center">
              <img src="img/AusDipole2-Murphy12.png" width="75%"><br>
              <font size="5">[M. Murphy 2012]</font>
            </div>
            <br>
            <h3> • Electric Dipole Moments</h3>
            <h3> • CPT/Lorentz/EEP violation</h3>
            <h3> • Variation of fundamental constants</h3>
            <font size="5">Webb, Murphy, Flambaum, Dzuba <i>et al.</i> PRL 87, 091301 (2001).</font>

</div>

# Atomic Parity Violation
          
          <br>

<div class = "fragment" style="text-align: left; float: left; width: 50%">
            <h3>Forbidden Atomic Transition (parity)</h3>
              <h4> • Electron-nuclear weak interaction: Allows Mixing</h4>
              <div class="fragment" style="font-size: 1.15em; padding-left: 2em;">
              $$
                R_{\rm PNC} = k_{\rm atomic} ~~ Q_{\rm weak}
              $$
              </div>
              <br>

<div style="text-align: center">
              <img src="img/pnc.png" width="85%"><br>
            <font size="3">[BMR, Dzuba, Flambaum, Ann. Rev. Nuc. Part. Sci. (2015)]</font>
            </div>
            <br>

<div class = "fragment">
              <h4>• Parity-violating mixing: proportional to weak charge $Q_{\rm weak}$</h4>
              <h4>• Atomic theory: translate mixing to observable transition rate $k_{\rm atomic}$</h4>
              <h4>• Experiment: observe parity-violating amplitude $R_{\rm PNC}$</h4>
            
            <br>
            <font size="5">• Dzuba, Flambaum, Sushkov, Phys. Lett. A <b>141</b>, 147 (1989).</font><br>
            <font size="5">• Wiemen <i>et al.</i> Science <b>275</b>, 1759 (1997).</font><br>
            <font size="5">• Dzuba, Berengut, Flambaum, BMR, PRL <b>109</b>, 203003 (2012).</font>
            </div>

</div>

<div class = "fragment" style="text-align: left; float: right; width: 50%">
            <div style="text-align: center">
              <img src="img/Weinberg-RMP-edit.png" width="80%"><br>
            <font size="3">[Safronova <i>et al.</i>, RMP <b>90</b>, 025008 (2018)]</font>
            </div>
            <br>
            <br>
            <div class = "fragment">
              <h4>• $Q_{\rm weak}$ - calculated from Standard Model (+radiative corrections)</h4>
                <h5>   • Flambaum, Samsonov (2026), <a href="https://doi.org/10.48550/arXiv.2602.22466">[arXiv:2602.22466]</a>.</h5>
              <h4>• Probes extra Z-like boson at low energy. For $\Delta Q/Q\sim 0.1\%$:</h4>
            <br>
            <div class="fragment" style="font-size: 1.15em">
            $$
            m_\chi \lesssim m_Z \sqrt{Q_w/\Delta Q_w} \sim 3\,{\rm TeV} ~ !!
            $$

<br>
           <font size="5">• BMR, Dzuba, Flambaum, Ann. Rev. Nuc. Part. Sci. (2015)</font>
           </div>
           </div>

</div>

# Violation: EEP, Lorentz, CPT, ...
            <br>

<div style="text-align: left; float: left; width: 48%">
              <div class="fragment">
                <h3>Test fundamental symmetries of nature</h3>
                <h4>  • Background "cosmic" fields</h4>
                <h4>  • Ascociated with dark matter, dark energy, new physics</h4>
                <h4>  • Precision atomic sensors: clocks, interferometers</h4>
              </div>
              <br>
              <div style="text-align: center">
                <img src="img/CosmicField2.png" width="30%">
              </div>
              <br>
              <div class="fragment">
                <h3>Differential Measurement</h3>
                <h4>  • Oscillating signal (dark matter)</h4>
                <h4>  • Daily modulation (frame dependence)</h4>
                <h4>  • Annual modulation (position dependence)</h4>
                <br>
                <h4>  • Lahs,Comparat, Kirk, BMR, <a href="https://link.aps.org/doi/10.1103/j7s3-wjg6">PRD <b>113</b>, 015041 (2026)</a></h4>
              </div>
            </div>

<div class="fragment" style="text-align: left; float: right; width: 48%">
              <h3 style="text-align: center">Example: Variation of fundamental constants</h3>
              <br>
              <div class="fragment" style="font-size: 1.15em">
              $$
              \frac{\Delta\omega}{\omega} = K\frac{\Delta\alpha}{\alpha}
              $$
              </div>
              <div class="fragment">

<br>
              <div style="text-align: center">
                <img src="img/AlphaSensitivity.png" width="65%"><br>
                <font size="4">[M. Murphy 2012]</font>
              </div>
              <br>

* In general, $K$ calculated from atomic theory
            * Difference transitions: different sensitivity

<br>

* Dzuba, Flambaum, Webb, [Phys Rev Lett **82**, 888
            (1999)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.888)

</div>
            </div>

# Super-massive black hole at the Galactic Centre

<div style="text-align: left; float: left; width: 45%">
            <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: 55%">
            <img src="img/Galactic_centre_pillars.jpg" width="100%">
            <font size="4">[ESA / C. Carreau]</font>
            <br>
            <br>
            
            <br>
          </div>

# New physics in high gravity?
          <br>

<div class="fragment fade-in">
              <img src="img/equation.png" width="80%">
              <br>
              <div class="fragment" style="font-size: 1.15em;">
              $$
              \Delta \alpha / \alpha \lesssim 10^{-5}
              $$
              </div>
            </div>
          </div>

<div style="text-align: center; float: right; width: 40%">
            <img src="img/keck-ao.jpg" width="100%">
            <font size="3">[Keck Observatory]</font>
          </div>

<div style="clear: both;"></div>
          <div style="float: left; text-align: left; width: 60%">
            <br>
            <div class="fragment fade-in">
              <h4>• Constrain models for physics beyond GR</h4>
              <h4>• Less sensitive than other searches</h4>
              <h4>• Position-dependent: First measurement near a black hole</h4>
              <h4>• Future dedicated observations: large improvements expected</h4>
            </div>
            <br>
              <font size="6">Hees, Do, <b>BMR</b>, Ghez <i>et al.</i>,
              <a href="https://link.aps.org/doi/10.1103/PhysRevLett.124.081101">Phys. Rev. Lett. <b>124</b>, 081101 (2020)</a></font>
              <br>
              <font size="6">Benergut <i>et al.</i>,
              <a href="10.1103/PhysRevLett.111.010801">Phys. Rev. Lett. <b>111</b> (2013)</a>; Ashby Nat. Phys <b>14</b> (2018)</font>
          </div>
          <div class="fragment" style="float: right; text-align: center; width: 40%">
            <img src="img/Compactness.png" width="75%">
          </div>

# Motivation: Large DM detection experiments

<br>
            <h2>Challenge: Extend to Low Masses</h2>
            <div class="fragment" data-fragment-index="0">
              <h3> Mass drops below nuclear mass:</h3>
              <h4> • No nuclear recoils: electron recoils + ionisation</h4>
            </div>
            <br>
            <div class="fragment" data-fragment-index="1">
              <h3> Mass drops below electron mass:</h3>
              <h4> • No electron recoils: absorption (dark photoelectric effect)</h4>
            </div>
            <br>
            <div class="fragment" data-fragment-index="2">
              <h3> Mass drops below eV:</h3>
              <h4> • Classical DM field: Quantum sensing (atomic clocks)</h4>
            </div>

<br>
            <div class="fragment" data-fragment-index="3">
              <div style="text-align: center">
              <img src="img/Weinberg.png" width="50%"><br>
              <font size="4">[Adapted: Maity, Bœhm, PRD <b>112</b>, 053001 (2025)]</font>
              </div>
            </div>

</div>

<div style="text-align: left; float: right; width: 50%">
            <br>
            <div style="text-align: center">
              <img  src="img/WIMP-nucleon.png" width="60%"><br>
              <font size="5">
                [O'Hare - GitHub/DirectDetectionPlots]
              </font>
            </div>
            <br>
            <div class="fragment" data-fragment-index="3">
            <h2>Oportunity</h2>
              <h3> Neutrino interactions:</h3>
              
              <h4> • Weinberg angle at extremely low energy [Maity, Bœhm (2025)]</h4>
              <h4> • Non-standard neutrino couplings</h4>
              <br>
              <div class="fragment" data-fragment-index="4">
                <h3> Solar-produced low-mass dark sector</h3>
                <h4> • Axions, dark photons, dark Fermions</h4>
              </div>
            </div>

</div>

<h1><span style="color: rgb(235, 114, 15)">Require Atomic Scattering Cross-Sections</span></h1>

<br>
            <h2>Challenge: Extend to Low Masses</h2>
            <div  data-fragment-index="0">
              <h3> Mass drops below nuclear mass:</h3>
              <h4> • No nuclear recoils: <span style="color: rgb(235, 114, 15)">electron recoils + ionisation</span></h4>
            </div>
            <br>
            <div  data-fragment-index="1">
              <h3> Mass drops below electron mass:</h3>
              <h4> • No electron recoils: <span style="color: rgb(235, 114, 15)">absorption (dark photoelectric effect)</span></h4>
            </div>
            <br>
            <div  data-fragment-index="2">
              <h3> Mass drops below eV:</h3>
              <h4> • Classical DM field: Quantum sensing (atomic clocks)</h4>
            </div>

<br>
            <div  data-fragment-index="3">
              <div style="text-align: center">
              <img src="img/Weinberg.png" width="50%"><br>
              <font size="4">[Adapted: Maity, Bœhm, PRD <b>112</b>, 053001 (2025)]</font>
              </div>
            </div>

</div>

<div style="text-align: left; float: right; width: 50%">
            <br>
            <div style="text-align: center">
              <img  src="img/WIMP-nucleon.png" width="60%"><br>
              <font size="5">
                [O'Hare - GitHub/DirectDetectionPlots]
              </font>
            </div>
            <br>
            <div  data-fragment-index="3">
            <h2>Oportunity</h2>
              <h3> Neutrino interactions:</h3>
              
              <h4> • <span style="color: rgb(235, 114, 15)">Weinberg angle at extremely low energy</span> [Maity, Bœhm (2025)]</h4>
              <h4> • <span style="color: rgb(235, 114, 15)">Non-standard neutrino couplings</span></h4>
              <br>
              <div>
                <h3> <span style="color: rgb(235, 114, 15)">Solar-produced low-mass dark sector</span></h3>
                <h4> • <span style="color: rgb(235, 114, 15)">Axions, dark photons, dark Fermions</span></h4>
              </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="3" style="font-size: 1.0em">
                   <h3> • <span style="color: rgb(105, 175, 240)">Atomic Current</span></h3>
                  <h3> •  <span style="color: rgb(245, 144, 61)">"Externel" current</span></h3>
              </div>
              <div class="fragment" data-fragment-index="3" 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>

<div class="fragment" data-fragment-index="4" 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="5" style="font-size: 1.5em">
              $$
              = {\color{orange} W_0} \, \langle {\color{skyblue}f} | 
              {\color{white}\tilde\gamma} {\color{violet}e^{i\mathbf{\vec q\cdot \vec r} }}| 
              {\color{skyblue}i} \rangle 
              -
              {\color{orange} \mathbf{\vec W} }\!\cdot\!\langle {\color{skyblue}f} | 
              {\color{skyblue}\mathbf{\vec \alpha} }
              {\color{white}\tilde\gamma} {\color{violet}e^{i\mathbf{\vec q\cdot \vec r} }}| 
              {\color{skyblue}i} \rangle 
              $$
              <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</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">
              $$
                K^{\rm ion}(q,\omega) \equiv E_H \sum_K\sum_{n_i \kappa_i}\sum_{\kappa_f}\left|{\langle{\varepsilon_f\kappa_f}\|{h^K}\|{n_i\kappa_i}\rangle}\right|^2,
              $$
              </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: 1.0em; 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>
            <h4 class="fragment"> • Only temporal component: spatial parts $(v/c)^2$ suppressed</h4>
            <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: 1.0em; 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>

# 3. Relaltivistic Scattering: Vector + Axial
            ### e.g., neutrino

#### Narise Williams + BMR, with Yevgeny Stadnik, Andrew Cheek

<div>
            <div style="font-size: 1.15em">
            $$
            { {\color{orange}
            W^\mu 
              \equiv \frac{1}{\sqrt{EE'} } ~ \bar u_\nu(k')\,\gamma^\mu(1-\gamma^5)\,u_\nu(k)
              \qquad-\qquad
              \text{Depends only on kinematics}~E,~q,~\omega\equiv E-E'
            } }
            $$
            </div>
            <br>
            <div class="fragment" style="font-size: 1.0em">
            $$
            \begin{align*}
            \sum_{if}|{\color{orange}W_\mu}{\color{skyblue}\mathcal{J}^\mu_{fi} }|^2 
            = 
            &|{\color{orange}W_0}|^2\left(c_V^2{\color{cornflowerblue}|\Phi|^2}
            + c_A^2{\color{turquoise}|\Theta|^2} \right)
            + \frac{\color{orange}|\vec{W}_\perp|^2}{2}\sum_{\sigma=1,0}\left(
            c_V^2{\color{cornflowerblue}|V^{(\sigma)}|^2}
            + c_A^2{\color{turquoise}|A^{(\sigma)}|^2}\right)
            + |{\color{orange}\hat{\vec{q} }\!\cdot\!\vec{W} }|^2\left(
            c_V^2{\color{cornflowerblue}|V^{(-1)}|^2}
            + c_A^2{\color{turquoise}|A^{(-1)}|^2}\right) \\
            &
            + 2\Re
            [{\color{orange}{W_0}^*\,\hat{\vec{q} }\sdot\vec{W} }]
            \left(c_V^2 {\color{palevioletred}X}
            -c_A^2
            {\color{palevioletred}Y}
            \right)
            +\Im\{ {\color{orange}
            \hat{\vec{q} }\!\cdot\![\vec{W}\!\times\!\vec{W}^*]}\}\,{c_V c_A}\,{\color{palevioletred}Z}
            \end{align*}
            $$
            </div>
            
            <br>

<div class="fragment">
            <div style="text-align: left; float: left; width: 50%">
            <h4>Vector</h4><br>
            <div style="font-size: 0.75em">
            $${ {\color{cornflowerblue}
            \begin{align*}
            |\Phi_K|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^K}\|{n_i\kappa_i}\rangle}|^2 \\
            |{V}^{(\sigma)}|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{T^{(\sigma)}_{K} }\|{n_i\kappa_i}\rangle}|^2
            \end{align*} } }
            $$
            </div>
            <h4>Axial</h4><br>
            <div style="font-size: 0.75em">
            $${ {\color{turquoise}
            \begin{align*}
            |\Theta|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^K\gamma^5}\|{n_i\kappa_i}\rangle}|^2\\
            |{A}^{(\sigma)}|^2 
                &\equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{T^{(\sigma)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle}|^2,
            \end{align*} } }
            $$
            </div>
            </div>

<div style="text-align: left; float: right; width: 50%">
            <h4>Interference</h4><br>
            <div style="font-size: 0.75em">
            $${ {\color{palevioletred}
            {\begin{align*}
            Z
            \equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}
            \Im\Big\{
            \langle{\varepsilon_f\kappa_f}\|{T^{(1)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(0)}_{K} }\|{n_i\kappa_i}\rangle
            \notag\\[-0.85em]
            \qquad\qquad\qquad\qquad\qquad
            -\langle{\varepsilon_f\kappa_f}\|{T^{(1)}_{K} }\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(0)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \Big\} \\
            X \equiv
            E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}
            \Im\left\{
            \langle{\varepsilon_f\kappa_f}\|{t^{K} }\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(-1)}_{K} }\|{n_i\kappa_i}\rangle
            \right\}\\
            Y \equiv
            E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}
            \Im\left\{
            \langle{\varepsilon_f\kappa_f}\|{t^{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \langle{\varepsilon_f\kappa_f}\|{T^{(-1)}_{K}\gamma^5}\|{n_i\kappa_i}\rangle
            \right\}.
            \end{align*} } }
            }$$
            </div>
            <h4>(OR Scalar/Pseudoscalar)</h4>
            <div style="font-size: 0.75em">
            $$
            { {\color{cornflowerblue}
                |S|^2
                    \equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^K\gamma^0}\|{n_i\kappa_i} }\rangle|^2
            } }
            $$
            $$
            { {\color{turquoise}
                |P|^2 
                    \equiv E_H \sum_K(2K+1)\sum_{n_i \kappa_i}\sum_{\kappa_f}|{\langle{\varepsilon_f\kappa_f}\|{t^Ki \gamma^0\gamma^5}\|{n_i\kappa_i} }\rangle|^2 .
             } }
            $$
            </div>
            </div>
            </div>
          </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, Flambaum, **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">
          $${\color{white}
            K = \left|\langle {\rm Xe} | e^{-i\mathbf{\vec{q} }\cdot\mathbf{\vec{r} }} | {\rm Xe}^{+}+e^-\rangle\right|^2
          }$$
          <br>
          $${\color{white}
            q_{\rm min} = m_\chi v - \sqrt{m_\chi^2 v^2 - 2m_\chi E}
          }$$
          </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">
            $${\color{white}\psi\sim A r^l \left[1-\frac{Z}{l+1}r+\cdots\right]}$$
            </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">
            $${\color{white}\psi\sim A r^{\gamma-1} \left[\gamma-\kappa+Br+\cdots\right] \qquad \gamma=\sqrt{\kappa^2-(Z\alpha)^2}\approx1-(Z\alpha)^2}$$
            </div>
            <br>

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

<br>

<div  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>
            <br>
            <h3> • Can be significant enhancement</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><br>
              • BMR, Dzuba, Flambaum, Pospelov, Stadnik, PRD <b>93</b>, 115037 (2016), [1604.04559]
            </div>
          </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

### Funamental Physics: Atomic Frontiers

* Fundamental symmetry violations (EDM, PNC)
            * Violations of Lorentz/CPT/EEP 
            * Variation of fundamental constants
            * Ultralight DM: atomic clocks, quantum sensors

### Atomic Ionisation form factors

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

<br>

### CPPC Seminar, UNSW, 2026

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

</div>
          
          <div style="text-align: left; float: right; width: 60%">

* Lahs, Comparat, Kirk, BMR, [Phys. Rev. D **108**, 015041 (2026)](https://link.aps.org/doi/10.1103/j7s3-wjg6), [[2510.08007]](http://arxiv.org/abs/2510.08007)
            <br>
            * Caddell, Flambaum, BMR, [Phys. Rev. D **108**, 083030 (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030), [[2305.05125]](http://arxiv.org/abs/2305.05125)
            <br>
            * BMR, Dzuba, Flambaum, Pospelov, Stadnik, [PRD **93**, 115037 (2016)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.93.115037), [[1604.04559]](http://arxiv.org/abs/1604.04559)
            <br>
            * BMR, Flambaum [Phys. Rev. D **100**, 063017 (2019)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.100.063017), [[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), [[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), [[2002.11567]](http://arxiv.org/abs/2002.11567)
            <br>
            * Williams, Caddell, BMR (Coming soon)

# 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>

# Scalar-SM?

* Overview: Hees _et al._, [PhysRevD.98.064051
          (2018)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.98.064051)
          <br>

<div>

</div>

<div class="r-stack">
            \[\huge
            \mathcal{L}_{\rm int.} = \phi \left[ {d_e \, F_{\mu\nu}F^{\mu\nu} } + {d_{m_f} \, \bar\psi\psi} + {d_g \,
            G^a_{\mu\nu}G^{a\mu\nu} }\right]
            \]
            <div class="fragment fade-in-then-out" data-fragment-index="1">
              \[\huge
              \mathcal{L}_{\rm int.} = \phi \left[ {\color{yellow}d_e \, F_{\mu\nu}F^{\mu\nu} } + {d_{m_f} \,
              \bar\psi\psi} + {d_g \, G^a_{\mu\nu}G^{a\mu\nu} }\right]
              \]
            </div>
            <div class="fragment fade-in-then-out" data-fragment-index="2">
              \[\huge
              \mathcal{L}_{\rm int.} = \phi \left[ {\color{yellow}d_e \, F_{\mu\nu}F^{\mu\nu} } + {\color{orange}d_{m_f}
              \, \bar\psi\psi} + {d_g \, G^a_{\mu\nu}G^{a\mu\nu} }\right]
              \]
            </div>
            <div class="fragment fade-in" data-fragment-index="3">
              \[\huge
              \mathcal{L}_{\rm int.} = \phi \left[ {\color{yellow}d_e \, F_{\mu\nu}F^{\mu\nu} } + {\color{orange}d_{m_f}
              \, \bar\psi\psi} + {\color{red}d_g \, G^a_{\mu\nu}G^{a\mu\nu} }\right]
              \]
            </div>
          </div>

<div style="text-align: left; float: left; width: 50%">
            <br>
            <div class="fragment fade-in" data-fragment-index="1">
              \[\large
              {\color{yellow}\alpha \to \alpha(1 + d_e \phi(r,t))}
              \]
              <h3>
                 • nb: $d_e = d_\gamma = d_\alpha = 1/\Lambda_\gamma$
              </h3>
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="2">
              \[\large
              {\color{orange}m_e \to m_e(1 + d_{m_e} \phi(r,t))}
              \]
              <h3>
                 • electron and quark masses
              </h3>
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="3">
              \[\large
              {\color{red}m_p \to m_p(1 + d_{g} \phi(r,t))}
              \]
              <h3>
                 • Proton mass: binding energy: QCD scale $\Lambda_{\rm QCD}$<br>
                 • Nuclear moments (+ radius): depend on $\Lambda_{\rm QCD}$, $m_q$
              </h3>
            </div>
          </div>

<div class="fragment fade-in" data-fragment-index="4">
              <h3>
                 • Optical clock
              </h3>
              \[
              \begin{align*}\large
              \omega &\propto R_y \, F_{\rm rel}(Z\alpha) \sim \alpha^{(2+K_{\rm rel})}m_e \\
              \frac{\delta\omega}{\omega} &= (2+K_{\rm rel}){\color{yellow}d_e} +{\color{orange}d_{m_e} }
              \end{align*}
              \]
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="4">
              <h3>
                 • Microwave (hyperfine) clock
              </h3>
              \[
              \begin{align*}\large
              \omega &\propto R_y \, [\alpha^2 F_{\rm rel}(Z\alpha)] \, (\mu \, m_e/m_p) \\
              \frac{\delta\omega}{\omega} &= (4+K_{\rm rel}'){\color{yellow}d_e} + {\color{orange}d_{m_e} } +
              ({\color{orange}d_{m_e} } - {\color{red}d_g}) + \kappa\, ({\color{orange}d_{m_q} } - {\color{red}d_g})
              \end{align*}
              \]
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="4">
              <h3>
                 • Cavity-stabilised laser
              </h3>
              \[
              \begin{align*}\large
              \omega &\propto 1/a_0 \\
              \frac{\delta\omega}{\omega} &= {\color{yellow}d_e} + {\color{orange}d_{m_e} }
              \end{align*}
              \]
            </div>
          </div>