Tabletop Cosmology

<h1>
              Ultralight Dark Matter Search with Space-Time Separated Atomic Clocks and Cavities
            </h1>

<br>

<h3>
              <it>Tabletop-scale Cosmology at home</it>
            </h3>

<br>

<h4>

* M. Filzinger, A. Caddell, D. Jani, M. Steinel, L. Giani, N. Huntemann, BMR, [Phys. Rev. Lett. **134**, 031001 (2025)](https://doi.org/10.1103/PhysRevLett.134.031001) [[arXiv:2312.13723](https://arxiv.org/abs/2312.13723)]

* Savalle _et al._, [Phys. Rev. Lett. **126**, 051301 (2021)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.051301) [[arXiv:2006.07055](http://arxiv.org/abs/2006.07055)]
              * BMR _et al._, [New J. Phys. **22**, 093010 (2020)](https://iopscience.iop.org/article/10.1088/1367-2630/abaace) [[arXiv:1907.02661](http://arxiv.org/abs/1907.02661)]
              * BMR, Blewitt, Dailey, Murphy, Pospelov, Rollings, Sherman, Williams, Derevianko, [Nature Comms. **8**, 1195 (2017)](https://www.nature.com/articles/s41467-017-01440-4)

</h4>

### Benjamin M. Roberts

<br>

School of Mathematics and Physics,
            University of Queensland, Australia

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

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

# Dark Matter: What we know

<br>
            <h3> There's lots of it</h3>

<h3> It gravitationally clusters in halos around galaxies</h3>
            <h4> (really, galaxies gravitationally cluster around DM)</h4>

<div style="text-align: left; float: left; width: 50%"
              >
              <br><br>
              <h2>Very strong evidence:</h2>
              <br>
              <h3>• Rotation curves, gravitational lensing, CMB </h3>
              <br>
              <h3>• Also: large scale structure, BAO etc. </h3>
              <br>
              <h3>• It's not strange for particles to not interact with
                light (e.g., neutrinos)</h3>
              <br>
              <br>
              <h3 >• ΛCDM model works extremely well
                <br>  • Ωc =
                0.2589(57) [Planck] </h3>
              
            </div>

# 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!

<br>

<h2> Mass drops below nuclear mass:</h2>
                <h3> • No nuclear recoils
                   • Instead: electron recoils + ionisation
                   <font color="pink">• Ashlee Caddell (PhD student)</font></h3>
              </div>
              <br>

<div class="fragment" data-fragment-index="1"
                style="text-align: left;" data-markdown>
                <h2> Mass drops below electron mass:</h2>
                <h3> • No electron recoils<br>
                   • 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>
              <div class="fragment" data-fragment-index="0">
              <img src="img/WIMP-nucleon.png" width="70%"><br>
              <font size="3">[O'Hare - GitHub/DirectDetectionPlots]</font>
              </div>
              <br>
              <div class="r-stack">
                <div class="fragment" data-fragment-index="2">
                  <img
                    src="img/laser-JILA.jpg" width="70%"><br>
                  <font size="3">[JILA]</font>
                </div>
                
              </div>
            </div>

# Ultralight Dark Matter

$$ \Huge
            {\rho_{\rm DM}\simeq 0.3 ~ \frac{\rm GeV}{{\rm cm}^3}}
            $$

<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>
            </div>

# Atomic Clocks 101:

<br>

### Clock: frequency reference
            <div class="fragment fade-in">
              <h3>
                Best atomic clocks have incredible accuracy: $\delta f/f \sim
                10^{-18}$
              </h3>
            </div>

<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>
              <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, Science 2016</font>
              <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, Science 2010</font>
              <br>
              <h3>
                Lock on: Adjust local oscillator to maximum transition rate
              </h3>
            </div>

# "Listen" for DM field using atomic clocks

<br>

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

<div style="text-align: left; float: right; width: 50%">
              <br>
              <div>
                <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>
              <div class="fragment fade-in">
                <h3>Observable shift:</h3>
                <br>
                \[{\large
                \frac{\delta f}{f} = \kappa \, \phi_{\rm DM}(\vec{r},t)}
                \]
              <br>
                <h3>
                   • $\kappa$ - model dependent parameter: <br> 
                      <font color="pink">$\implies$ Different for different transitions</font>
                </h3>
                <h3>
                   • Calculated: high-precision atomic theory
                </h3>
              </div>

<br>
              <div class="fragment fade-in">
                <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>

# Variation of fundamental constants

<br>

<div>
        \[\large
          \omega^{A} = \underbrace{F_A(\alpha)}_{\text{Transition-specific}} 
          \times  \underbrace{m_e c^2\alpha^2}_{\text{Units}} 
        \]
        </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: left; 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="5">
            <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="6">
            <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>

# 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: 35%">
                <br>
                <h3>• 30 Cs, Rb atomic clocks</h3>
                <br>
                <h3>• Over 20 years of high-quality data</h3>
                <br>
                <h3>• Publicly available (JPL)</h3>
                <br>
                <h3>• 50,000 km Dark Matter observatory</h3>
                <br>
                <div>
                  <video width="75%" 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: 65%">
                <img div src="img/gps-data.png"
                  width="45%">
                <br>
                <br>
                <br>
                <div class="fragment fade-in">
                  <img src="img/GPSDM-results.png" width="98%">
                  <b>BMR</b>, Blewitt, Dailey, Murphy, Pospelov, Rollings,
                  Sherman,
                  Williams, Derevianko, <br>
                  <a
                    href="http://www.nature.com/articles/s41467-017-01440-4">
                    Nature Comms.
                    (2017)</a>

</div>

# European Fibre-linked network

<br>
              <div style="text-align: center; float: left; width: 50%">
                <img src="img/FibreLinkDiagram.png" width="70%">
                <br>
              </div>
              <div style="text-align: left; float: right; width: 50%">
                <h3>• Laboratory optical clocks</h3>
                <h3>• ~1000km, direct optical fibre connection</h3>
                <br>
                <h3>• Orders-of-magnitude higher precision: $10^{-17}$ level!!</h3>
                <br>
                <h3>• Orders-of-magnitude less data</h3>
                <h3>   • (hours, not decades)</h3>
              </div>

<br>

# 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>Constraints from clock-clock comparisons</h2><br>
                <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 class="fragment fade-in">
                <h3>• Dimensionful couplings always cancel in these ratios</h3>
              </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>

# $d_{m_e}$ alone?

* Usually, appears in ratios from $m_e/m_p$
        * True in general for any 'dimensionful' constant
        * Potential issue: universal couplings

$$\large
            \delta(m_e/m_p) ~ \propto ~ (d_{m_e} - d_g)
        $$

## Fast oscillations:

* Damped vs. undamped oscillator

<img src="img/Geraci.png" width="20%"><br>
        <font size="3">[Geraci (2019)]</font>

<br>

* Free vs. fixed optical cavity -- Geraci _et al._,  [Phys.Rev.Lett.123.031304 (2019)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.031304)
        * Atom vs (damped) solid -- Antypas _et al._, [Phys.Rev.Lett.123.141102 (2019)](https://doi.org/10.1103/PhysRevLett.123.141102)
        * Time-delay interferometry -- Savalle _et al._, [Phys.Rev.Lett.126.051301 (2021)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.051301)
         * <font color="pink">See talk by Ettiene Savalle</font>
        * Overview: Antypas _et al._, [Ann. Phys. 532 (2020)](https://onlinelibrary.wiley.com/doi/10.1002/andp.201900566)

</div>

# Oscillating Signals
      
                * 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 class="fragment fade-in" data-fragment-index="3">
          <img src="img/dailymod.png" width="90%"><br>
          </div>

</div>

# Conclusion

### Ultralight dark matter

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

### Space-time separated sensors
  
            * Access to otherwise unconstrained couplings
            * Unique signals: scale with network size, daily modulations
  
          </div>  
            <br>

<br>
  
            ### Benjamin M. Roberts
    
            _School of Mathematics and Physics,_

_University of Queensland, Australia_
  
            <br>
  
            * Slides [broberts.io/talks](https://broberts.io/talks)

* 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>
            * BMR _et al._, [Nature Comms. **8**, 1195 (2017)](http://www.nature.com/articles/s41467-017-01440-4)
            <br>
            * BMR _et al._, [N. J. Phys. **22**, 093010 (2020)](https://iopscience.iop.org/article/10.1088/1367-2630/abaace)
            <br>
            * Hees, Do, BMR, Ghez _et al._, [Phys. Rev. Lett. **124** 081101
            (2020)](https://link.aps.org/doi/10.1103/PhysRevLett.124.081101)
            <br>
            * Filzinger, Caddell _et al._, [Phys. Rev. Lett. **134**, 031001 (2025)](https://doi.org/10.1103/PhysRevLett.134.031001)

<br><br>
            <br><br>
            Supported by: ARC and       <img src="img/BQI.png" width="55%">
          </div>

</div>

# Complication + Resolution

### Example: H-like

<br>

<div style="text-align: left; float: left; width: 50%">
          \[
            \omega_{\rm SI} = \frac{m_e c^2Z^2{\color{red}\alpha^2}}{2}
            \left(\frac{1}{n^2}-\frac{1}{n'^2}\right)F(Z\alpha)
          \]
          <br>
          \[
            K = {\color{red}2} + K_{\rm rel}
          \]
        </div>

<div style="text-align: left; float: right; width: 50%" class="fragment">
          \[
              \omega_{\rm atomic} = \frac{Z^2}{2}
              \left(\frac{1}{n^2}-\frac{1}{n'^2}\right)F(Z\alpha)
          \]
          <br>
          \[
            K = K_{\rm rel}
          \]
        </div>

<div class="fragment">
        
          <br><br>
      
        * $K$ _and_ $\alpha$ both dimensionless, issue remains
        * $K$ only defined up to additive constant 
        * (constants only defined up to multiplicative constant)

</div>

### Resolution: ratios

$$
          \frac{\delta\left(\omega_A/\omega_B\right)}{\left(\omega_A/\omega_B\right)}
          = 
          K_A \frac{\delta\alpha}{\alpha}
          -K_B \frac{\delta\alpha}{\alpha}
          =
          (K_A  -K_B )\frac{\delta\alpha}{\alpha}
          $$

</div>

<div style="text-align: left; float: right; width: 50%">
           
               •  Unit ambiguity cancels in ratios
               •  Must have two lines at each spacetime location
          
          </div>

</div>

* Nice review: Kozlov, Budker [Ann. Phys. 1800254 (2018)](https://onlinelibrary.wiley.com/doi/10.1002/andp.201800254)

</div>

# Wait, variation of dimensionful constants?

* Almost always: appear as dimensionless ratios, so no issue
          * However, in some cases, appear alone

$$\large
          \frac{\delta\left(\omega_A/\omega_B\right)}{\left(\omega_A/\omega_B\right)}
          = 
          K_A \frac{\delta\alpha}{\alpha}(x_1)
          -K_B \frac{\delta\alpha}{\alpha}(x_2)
          ~{\color{red}\cancel{=
          (K_A  -K_B )\frac{\delta\alpha}{\alpha}}}
          $$

<br>

* Is this OK?

* Yes, so long as specific interaction is considered
            * Specific interaction Lagrangian removes ambiguity
          
          * Some care must be taken: correct $K$ factors
            * Interpretation as variation in constant may be unit dependent
            * Of course, physics/observables are not!

<br>

* Savalle, BMR, _et al._ [arXiv:1902.07192](https://arxiv.org/abs/1902.07192)
          * Savalle _et al._, [Phys.Rev.Lett.126.051301 (2021)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.051301)
          * Antypas _et al._, [Ann. Phys. 532 (2020)](https://onlinelibrary.wiley.com/doi/10.1002/andp.201900566)

</div>