Confirming $$U(1)_{L_\\mu -L_{\\tau }}$$ as a solution for $$(g-2)_\\mu $$ with neutrinos

Abstract

<jats:p>The recent measurement of the muon anomalous magnetic moment by the Fermilab E989 experiment, when combined with the previous result at BNL, has confirmed the tension with the SM prediction at <jats:inline-formula><jats:alternatives><jats:tex-math>$$4.2\\,\\sigma $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>4.2</mml:mn>\n <mml:mspace />\n <mml:mi>σ</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>\xa0CL, strengthening the motivation for new physics in the leptonic sector. Among the different particle physics models that could account for such an excess, a gauged <jats:inline-formula><jats:alternatives><jats:tex-math>$$U(1)_{L_\\mu -L_{\\tau }}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n <mml:mo>-</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>τ</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> stands out for its simplicity. In this article, we explore how the combination of data from different future probes can help identify the nature of the new physics behind the muon anomalous magnetic moment. In particular, we contrast <jats:inline-formula><jats:alternatives><jats:tex-math>$$U(1)_{L_\\mu -L_{\\tau }}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n <mml:mo>-</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>τ</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> with an effective <jats:inline-formula><jats:alternatives><jats:tex-math>$$U(1)_{L_\\mu }$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>-type model. We first show that muon fixed target experiments (such as NA64<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>μ</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>) will be able to measure the coupling of the hidden photon to the muon sector in the region compatible with <jats:inline-formula><jats:alternatives><jats:tex-math>$$(g-2)_\\mu $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>g</mml:mi>\n <mml:mo>-</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n </mml:math></jats:alternatives></jats:inline-formula>, and will have some sensitivity to the hidden photon’s mass. We then study how experiments looking for coherent elastic neutrino-nucleus scattering (CE<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nu $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>ν</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>NS) at spallation sources will provide crucial additional information on the kinetic mixing of the hidden photon. When combined with NA64<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>μ</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula> results, the exclusion limits (or reconstructed regions) of future CE<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nu $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>ν</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>NS detectors will also allow for a better measurement of the mediator mass. Finally, the observation of nuclear recoils from solar neutrinos in dark matter direct detection experiments will provide unique information about the coupling of the hidden photon to the tau sector. The signal expected for <jats:inline-formula><jats:alternatives><jats:tex-math>$$U(1)_{L_\\mu -L_{\\tau }}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n <mml:mo>-</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>τ</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> is larger than for <jats:inline-formula><jats:alternatives><jats:tex-math>$$U(1)_{L_\\mu }$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> with the same kinetic mixing, and future multi-ton liquid xenon proposals (such as DARWIN) have the potential to confirm the former over the latter. We determine the necessary exposure and energy threshold for a potential <jats:inline-formula><jats:alternatives><jats:tex-math>$$5\\,\\sigma $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>5</mml:mn>\n <mml:mspace />\n <mml:mi>σ</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> discovery of a <jats:inline-formula><jats:alternatives><jats:tex-math>$$U(1)_{L_\\mu -L_{\\tau }}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>U</mml:mi>\n <mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mrow>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>μ</mml:mi>\n </mml:msub>\n <mml:mo>-</mml:mo>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>τ</mml:mi>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> boson, and we conclude that the future DARWIN observatory will be able to carry out this measurement if the experimental threshold is lowered to <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\,{\\mathrm {keV}}_{\\mathrm {nr}} $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mspace />\n <mml:msub>\n