ICDMS: Enlightening the search for Dark Matter

<div class="r-stack">
            <h1 class="fragment fade-out" data-fragment-index="0">Enlightening
              the search for Dark Matter:
            </h1>
            <h1 class="fragment current-visible" data-fragment-index="0">En<i>light</i>ening
              the search for Dark Matter:
            </h1>
          </div>

<br>

<h2>Atomic Frontiers in Dark Matter Detection</h2>

### Dark Matter and Stars
          #### Queens University, Kingston, Canada
          #### 14 $-$ 16 July 2025

### Benjamin M. Roberts

<br>

School of Mathematics and Physics,
          University of Queensland, Australia

<br>

* Slides [broberts.io/slides](https://broberts.io/slides/)

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

# 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 _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>
              <br>
              <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
                 • Electric dipole moments</h3>
            </div>
            <br>

<div class="fragment" data-fragment-index="2" style="text-align: left;" data-markdown>
              <h2> Dark Matter:</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>

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

<br>

<h2> Mass drops below nuclear mass:</h2>
              <h3> • No nuclear recoils. Instead: electron recoils + ionisation
                 <font size="6">• e.g., Ashlee Caddell, Flambaum, BMR, Phys. Rev. D <b>108</b>, 083030 (2023)
                </font>
              </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>
            <br>

</div>

<div style="text-align: center; float: right; width: 40%">
            <br>
            <img class="fragment" data-fragment-index="0" src="img/atom.png" width="30%">
            <br><br>
            <div class="r-stack">
              <div class="fragment" data-fragment-index="2">
                <img src="img/laser-JILA.jpg" width="90%"><br>
                <font size="3">[JILA]</font>
              </div>
              <div class="fragment" data-fragment-index="3">
                <img src="img/Ye_UCBolder-SrLattice.jpg" width="85%"><br>
                <font size="3">[Ye/UCBolder]</font>
              </div>
            </div>
          </div>

# Light "WIMPs": electron ionisation

<div>
            <br>
            <div style="text-align: center; float: left; width: 40%">
              <img src="img/ionisation.svg" width="65%"><br>
              <font size="2">[nagwa.com/]</font>
            </div>

<div style="text-align: center; float: right; width: 60%">
              <br>
              <h2>Mass drops below nucleus: no nuclear recoils</h2>
              <br>
              <h2>Instead: electron recoils - ionisation $R\propto K$</h2>
              <br>
              <img class="fragment fade-in" src="img/K-approx2.png" width="65%">
            </div>
          </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_
          * **BMR**, V. V. Flambaum, [Phys. Rev. D **100** 063017
          (2019)](https://link.aps.org/doi/10.1103/PhysRevD.100.063017)
          * **BMR**, Flambaum, Gribakin, [Phys. Rev. Lett **116** 023201
          (2016)](http://link.aps.org/doi/10.1103/PhysRevLett.116.023201)

# Calculations difficult

<div style="text-align: center; float: right; width: 40%">
              <br>
              <h2 class="fragment fade-in">Can you test it?</h2>
              <div class="fragment fade-in">
                <h3>Yes! Electron-impact ionisation</h3>
                <img src="img/e-impact.png" width="95%">
                <h3>Better than dedicated calculations</h3><br>
              </div>
            </div>
          </div>

<br>

# Ultralight Dark Matter

$$ \Huge
          {\rho_{\rm DM}\simeq 0.4 ~ \frac{\rm GeV}{{\rm cm}^3}}
          $$

<br>

## Mass decreases $\implies$ number density increases:
          ## Classical boson field (e.g., axions, scalars)

<br>

<div style="text-align: left; float: right; width: 70%">
            <div class="fragment fade-in">
              <h3>
                1. No interaction: oscillation: $\phi = \phi_0 \cos(m_\phi
                t)$
              </h3>
              \[\phi_0^2\propto \rho_{\rm DM}\]
            </div>

<h3 class="fragment fade-in">
              2. Interactions: clumps
            </h3>
            <br>
            <h3 class="fragment fade-in">
              3. Also: constant build-up (local over-densities)
            </h3>
   
            <br>
            <h2 class="fragment fade-in">
              $\implies$ Classical field: quantum sensors (clocks) $\phantom{\implies}$
            </h2>

</div>

# Clock: a thing that ticks

### Predictable, period motation + method to count ticks

<br>
          <div>
            <div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
              <img src="img/StoneHenge.png" width="86%"><br>
              <font size="3">[Getty]</font>
              <br>
              <h3>
                Earth's orbit/rotation
              </h3>
            </div>

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
              <img src="img/H1.png" width="96%"><br>
              <font size="3">[H1, Royal Museums Greenwich]</font>
              <br>
              <h3>
                Swinging pendulum
              </h3>
            </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 32%">
              <img src="img/laser-JILA.jpg" width="100%">
              <font size="3"> [JILA]</font>
              <br>
              <h3>
                Oscillating electromagnetic wave
              </h3>
            </div>
          </div>

<div class="fragment fade-in" style="text-align: left; float: left; width: 100%">
            <br>
            <br>
            <br>
            <h3>
              • Pendulums (+all kinematic clocks): depend on materials, location, temperature etc.
              <br>
              • Earth rotation: unstable at ~5 ms per day $10^{-7}$
              <br>
              • Atomic clock: universal timekeeping based on fundamental physics
            </h3>
          </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>19</sup><br>
               • <1 s over lifetime of the universe!
              </h3>
              </div>

</div>
          </div>

# Example: gravitational redshift

<br>

$$ \Huge {\color{orange}
          \frac{\Delta f}{f} = -\frac{\Delta U}{c^2} \sim 10^{-16} \frac{\Delta r}{{\rm m}}
          }
          $$

<br>

<div class="fragment fade-in" style="text-align: center; float: left; width: 60%">
            <img src="img/Chou-A.png" width="50%">
            <img src="img/Chou-B.png" width="45%"><br>
            <font size="4">30 cm [Chou et al., Science <b>329</b>, 1630 (2010)]</font><br><br>

* $\sim$ 30 cm
          </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 40%">
            <img src="img/Ye-Redshift.png" width="85%">
            <br>
            <font size="4">[Bothwell <i>et al.</i>, Nature <b>602</b>, 420 (2022)]
            </font><br><br>

* $\sim$ 1 mm
          </div>

* Gravitational red-shift is one of the biggest systematic which must accounted for in clock comparisons --
            even within the same lab!
          </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: space-time separated sensors
      
                * Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [PRL **134**, 031001 (2025)](https://doi.org/10.1103/PhysRevLett.134.031001)
      
                <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>

<br>
      
                  <img src="img/Local.png" width="70%"><br>
                </div>
                
                <div style="text-align: left; float: right; width: 50%" class="fragment fade-in">
      
                  ### 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
                  </div>
      
                  <img src="img/Separated.png" width="70%"><br>
                </div>

# Space-time separated sensors: initial results

* Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [PRL **134**, 031001 (2025)](https://doi.org/10.1103/PhysRevLett.134.031001)
        <br>

<br>

* 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>
          
          <br>
          <div>
          <img src="img/dailymod.png" width="90%"><br>
          </div>

</div>

# Transient Signals

Derevianko + Pospelov, [Nature Physics
              (2014)](http://www.nature.com/doifinder/10.1038/nphys3137)
            </div>

<div style="text-align: left; float: right; width: 50%">
              <h3>• Scalar DM with small self-interaction:</h3>
              <br>
              <h3>• May form "clumps": topological defects, Q-balls</h3>
              <br>
              <h3>   • Topological defects: size $d\sim 1/m_\phi$</h3>
              <br>
              <h3>   • $m=10^{-10}$ eV $\implies$ $d\simeq 1$ km</h3>
              <br>
              <h3>• Wait until one passes through Earth</h3>
              <br>
              <h3 class = "fragment">• How to distinguish from noise?</h3>
              <br>
              <h3 class = "fragment">   • Correlated signal across global network</h3>
            </div>

# GPS.DM

<div style="text-align: left; float: left; width: 30%">
            <br>
            <h3>• 30 Cs, Rb atomic clocks</h3>
            <br>
            <h3>• 20+ years of high-quality data</h3>
            <br>
            <h3>• Publicly available (JPL)</h3>
            <br>
            <h3>• 50,000 km DM Observatory</h3>
            <br>
            <div class="fragment fade-in">
              <video width="70%" data-autoplay muted>
                <source src="img/GPSDM.mp4">
              </video><br>
              <font size="3">[Video: Conner Dailey]</font>
              
            </div>
          </div>

<div style="text-align: center; float: right; width: 70%">
            <img div class="fragment fade-in" src="img/gps-data.png" width="40%">
            <br>
            <div class="fragment fade-in">
              <img src="img/GPSDM-results.png" width="90%"><br>
              • <b>BMR</b>, <i>et al.</i>, Derevianko,
              <a href="http://www.nature.com/articles/s41467-017-01440-4"> Nature Comms. (2017)</a>
              <br><br>
              Also lab clocks:<br>
              • <b>BMR</b>, <i>et al.</i>, (SYRTE, PTB, NPL)
            <a href="https://iopscience.iop.org/article/10.1088/1367-2630/abaace">
              N. J. Phys. (2020)</a>

</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>

# Conclusion

### Dark matter with atomic phenomena:

* Electron scattering, absorption, fifth-force searches
            * Fundamental symmetry violations (EDM, PNC)

### Ultralight dark matter

* Scalar coupling DM-SM: shift in atomic levels
            * Monitor frequencies with atomic clocks

<br>

### Dark Matter and Stars, Queens University, 2025

<br>
  
            ### Benjamin M. Roberts
    
            _University of Queensland, Australia_
  
            <br>
  
            * Slides [broberts.io/slides](https://broberts.io/slides/)

* Filzinger, Caddell _et al._, [Phys. Rev. Lett. **134**, 031001 (2025)](https://doi.org/10.1103/PhysRevLett.134.031001)
            <br>
            * Caddell, Flambaum, BMR, [Phys. Rev. D **108**, 083030 
            (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030)
            <br>
            * Savalle _et al._, [Phys. Rev. Lett. **126**, 051301 
            (2021)](https://link.aps.org/doi/10.1103/PhysRevLett.126.051301)
            <br>
            * Hees, Do, BMR, Ghez _et al._, [Phys. Rev. Lett. **124** 081101 
            (2020)](https://link.aps.org/doi/10.1103/PhysRevLett.124.081101)
            <br>
            * BMR _et al._, [N. J. Phys. **22**, 093010 
            (2020)](https://iopscience.iop.org/article/10.1088/1367-2630/abaace)
            <br>
            * BMR _et al._, [Nature Comms. **8**, 1195 
            (2017)](http://www.nature.com/articles/s41467-017-01440-4)
            <br>
            * BMR, Gribakin, Flambaum, [Phys. Rev. Lett. **116**, 023201 
            (2016)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.023201)

# Spatial variation

* Observe spectra from distant stars
            * Compare to measurements on Earth

</div>

### Possible non-zero result

<img src="img/AusDipole2-Murphy12.png" width="90%"><br>
            <font size="3">[M. Murphy 2012]</font>
            <br>
            <br>

* 3 $\sigma$ result: hard to quantify systematics
            * Webb _et al._, PRL **87**, 091301 (2001); **107**, 191101 (2011)

* Over year, earth moves through gradiant

$$
              \delta\alpha/\alpha_0 \sim 10^{-20} / \text{year}
              $$

* Very near-future clocks can probe this

</div>
          </div>

# Optical atomic clocks

<div class="fragment fade-in" style="text-align: left; float: left; width: 55%">
            <img src="img/Stability.png" width="95%"><br>
            <font size="3"> [M. Safronova]</font>
            <br>
            <br>
            <br>

* Requires ultra-stable laser ("see" narrow line)
            * Frequency comb ("listen" to high frequency)

</div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 45%">
            <img src="img/basicclock2.png" width="95%">
            <font size="3">[Centre for Quantum Technologies | U. Singapore]</font>
            <br>
            <br>
            <br>
            <img src="img/Diddams-Comb.png" width="95%">
            <font size="3">[Diddams <i>et al.</i>, Science <b>369</b>, 3676 (2020)]</font>
          </div>

* Accurate to parts in $10^{18}$
            * better than 1 s over lifetime of the universe!
          </div>

# Variation of fundamental constants
          ### Example: fine structure constant: $\alpha$

<div>
            \[\large
            \omega^{A} = \underbrace{F_A(\alpha)}_{\text{Transition-specific}}
            \times \underbrace{m_e c^2\alpha^2}_{\text{Units}}
            \qquad\qquad
            \alpha = \frac{e^2}{4\pi\epsilon_0\,\hbar c}
            \]
          </div>

<div class="fragment" style="text-align: center; float: left; width: 50%">
            <br><br>
            <img src="img/Bohr.png" width="30%"><br>
            <font size="3">[JabberWok/Wikipedia]</font>
            <br><br><br>
            \[
            \frac{\Delta\omega}{\omega} = K\frac{\Delta\alpha}{\alpha}
            \qquad\qquad\qquad
            \left(
            {\color{red}K \equiv \frac{\partial \omega}{\partial \alpha}\frac{\alpha_0}{\omega}}
            \right)
            \]

<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 class="fragment" style="text-align: center; float: right; width: 50%">
            <br> <br>
            <img src="img/AlphaSensitivity.png" width="90%"><br>
            <font size="3">[M. Murphy 2012]</font>
          </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>