The particle origin of dark matter (DM) remains elusive despite decades of direct, indirect, and collider searches. Several groups have reported a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi></math>-ray excess toward the Galactic Center commonly referred to as the Galactic Center excess (GCE). Its spectrum is consistent with annihilation of weakly interacting massive particles (WIMPs) of mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>10</mn><mo>−</mo><mn>100</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> and a thermal-relic cross section. Although many concrete WIMP models reproduce the GCE spectrum, most are now excluded by direct detection experiments that are approaching the neutrino floor. We investigate a class of anomaly-free extensions of the Standard Model featuring gauged differences of lepton number, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><msub><mi>L</mi><mi>i</mi></msub><mo>−</mo><msub><mi>L</mi><mi>j</mi></msub></mrow></msub></math>, and gauged baryon minus lepton number, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub></math>. We show that these models can reproduce the GCE while remaining compatible with the observed relic abundance. We then impose collider and direct detection constraints, accounting for both tree-level and loop-induced kinetic mixing. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>e</mi></msub></math> model gives the best fit to the GCE: a DM mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>χ</mi></mrow></msub><mo>∼</mo><mn>40</mn><mo>−</mo><mn>50</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> remains consistent with the muon and electron magnetic moment anomalies, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>g</mi><mo>−</mo><mn>2</mn><msub><mo stretchy="false">)</mo><mrow><mi>μ</mi><mo>,</mo><mi>e</mi></mrow></msub></math>, as well as current collider and direct detection limits, for mediator masses in the range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>∼</mo><mn>70</mn><mo>−</mo><mn>86</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> and a DM mediator coupling of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math>. By contrast, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>e</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></math> models yield poorer fits; satisfying both the relic density and experimental bounds forces the DM mass to lie very close to resonance (i.e. approximately half the mediator mass). Finally, while the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> model also matches the GCE well, its parameter space is almost entirely ruled out by strong direct detection limits, except for the narrow resonance region where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>χ</mi></msub></math> should be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><msup><mi>A</mi><mo>′</mo></msup></msub><mo>/</mo><mn>2</mn></math> requiring a fine-tuning at the few-percent level.

Italy

Jordan Koechler

Mattia Di Mauro

Leptophilic dark matter in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><msub><mi>L</mi><mi>i</mi></msub><mo>−</mo><msub><mi>L</mi><mi>j</mi></msub></mrow></msub></math> models: A solution to the Fermi-LAT Galactic Center excess consistent with cosmological and laboratory observations

We present a comprehensive QCD analysis of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi></math> form factors within the framework of light-cone sum rules, employing two-pion light-cone distribution amplitudes (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>π</mi><mi>DA</mi></math>s). For the first time, we derive the twist-three <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>π</mi><mi>DA</mi></math>s explicitly by exploiting the conformal symmetry of massless QCD. This breakthrough enables the first systematic calculation of higher-power corrections to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi></math>-type transitions. Leveraging the recently measured partial decay fractions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>l</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>l</mi></msub></math> by the Belle detector, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>4.27</mn><mo>±</mo><mn>0.49</mn><msub><mo stretchy="false">|</mo><mrow><mi>data</mi></mrow></msub><mo>±</mo><mn>0.55</mn><msub><mo stretchy="false">|</mo><mtext>LCSRs</mtext></msub><mo stretchy="false">)</mo></mrow><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> within the dominant region of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math> resonance. We clarify that the relatively small value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> extracted from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>ρ</mi><mi>l</mi><mi>ν</mi></math>, when compared to the golden channel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>π</mi><mi>l</mi><mi>ν</mi></math>, arises due to the finite width and nonresonant effects. We anticipate future measurements of purely isovector and isoscalar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mrow><mi>l</mi><mn>4</mn></mrow></msub></math> decays, which will aid in reducing uncertainties by enhancing our understanding of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>π</mi></math> phase shifts in the two-pion distribution amplitudes through heavy flavor decays.

China

Shan Cheng

QCD analysis of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi></math> form factors and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></mrow></math> extraction from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mrow><mi>l</mi><mn>4</mn></mrow></msub></math> decays

We study an alternative $$U(1)_{B-L}$$ model with singlet chiral fermions which can act as WIMP and FIMP type DM, which is anomaly-free by constructions in contrast to the well studied $$B-L$$ extension where three right handed neutrinos are needed for gauge anomaly cancellation. Fermion dark matter masses are generated after the $$U(1)_{B-L}$$ is broken spontaneously, so the Yukawa couplings for WIMP and FIMP components can be distinguished by the hierarchical values of the vacuum expectation values of the singlet scalar fields. Moreover, the $$U(1)_{B-L}$$ gauge boson receives a TeV-scale mass for a tiny extra gauge coupling, so it goes out of equilibrium from the rest of the model content in the early Universe. Both the $$U(1)_{B-L}$$ gauge boson and FIMP component are produced from the decays of the bath particles, and the former can decay into FIMP DM and/or WIMP DM before BBN. The WIMP component can reside in the resonance region of the Higgs bosons or dominantly annihilate into a pair of singlet-like scalars. Thus, there is a flexibility to choose a small mixing between the visible and dark sectors, thereby evading all the current direct and indirect detection bounds. Furthermore, we show that WIMP and FIMP components can coexist in suitable fractions, depending on the choice of model parameters, allowing for additional protection for WIMP DM against various experimental bounds. Finally, we identify the dimension-6 and dimension-7 operators for Majorana neutrino masses in our model, being consistent with the $$U(1)_{B-L}$$ gauge symmetry, and provide a possibility of extending the model with additional singlet fermions for neutrino masses.

South Korea

Sarif Khan

Hyun Lee

WIMP-FIMP option and neutrino masses via a novel anomaly-free $$B-L$$ symmetry

The near-singularity BKL dynamics of five dimensional gravity and supergravity (and also an extended four-dimensional supergravity) is known to be given by the billiard problem of a particle within a fundamental domain of the Bianchi groups PSL(2, 𝒪) ⊂ PSL(2, ℂ), acting on ℍ . Here 𝒪 are the Gaussian or Eisenstein integers, which define a square or triangular lattice in ℂ. The Wheeler-DeWitt wavefunctions near the singularity are, correspondingly, automorphic Maaß forms of PSL(2, 𝒪). We show how these wavefunctions are associated to certain $\textit{L}$-functions evaluated along their critical axis. Each of these $\textit{L}$-functions admits an Euler product representation over the complex primes 𝒫 ⊂ 𝒪. From this fact we write the $\textit{L}$-function as the trace over an auxiliary Hilbert space of charged harmonic oscillators, labeled by the complex primes 𝒫 . In this way we have constructed a ‘dual’ primon gas partition function for the wavefunction of the universe close to a five dimensional cosmological singularity.

United Kingdom

Marine Clerck

Sean Hartnoll

Ming Yang

Wheeler-DeWitt wavefunctions for 5d BKL dynamics, automorphic $\textit{L}$-functions and complex primon gases

We explore the potential to detect the <math altimg='si39.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math> model featuring triplet scalars &#916; at the <italic>&#956;</italic>TRISTAN collider. The new gauge boson <italic>Z</italic>&#8242;, arising from the spontaneous breaking of <math altimg='si39.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math>, can exhibit maximal flavor changing interactions under the exchange symmetry, while &#916; mediates the flavor conserving interactions. The absence of muon <math altimg='si6.svg'><msub><mrow><mo>(</mo><mi>g</mi><mo>&#8722;</mo><mn>2</mn><mo>)</mo></mrow><mi>&#956;</mi></msub></math> can be explained by interference effects arising from opposite contributions of <italic>Z</italic>&#8242; and &#916;, with similar interference patterns also manifesting in the tau decay process <math altimg='si7.svg'><mrow><mi>&#964;</mi><mo>&#8594;</mo><mi>&#956;</mi><mi>&#957;</mi><mover accent='true'><mi>&#957;</mi><mo>&#175;</mo></mover></mrow></math>. These counteracting effects render the model phenomenologically interesting and warrant further investigation. For the mass <math altimg='si22.svg'><msub><mi>m</mi><msup><mi>Z</mi><mo>&#8242;</mo></msup></msub></math> in the range of hundreds of GeV, we find that <math altimg='si30.svg'><mrow><msup><mi>&#956;</mi><mo>+</mo></msup><msup><mi>&#956;</mi><mo>+</mo></msup></mrow></math> and <math altimg='si35.svg'><mrow><msup><mi>&#956;</mi><mo>+</mo></msup><msup><mi>e</mi><mo>&#8722;</mo></msup></mrow></math> collider at the <italic>&#956;</italic>TRISTAN can probe many regions inaccessible to current experiments and offer greater projected sensitivity than opposite-sign muon colliders. This suggests that <italic>&#956;</italic>TRISTAN can serve as complementary exploration to the <math altimg='si39.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math> model, providing compelling motivation for the next generation of high-energy lepton colliders.

Fei Huang

Jin Sun

Probing maximal flavor changing <italic>Z</italic>&#8242; in <math altimg='si39.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math> at <italic>&#956;</italic>TRISTAN

We discuss the interplay between dark matter, leptogenesis and a new gauge boson $$Z^\prime $$ in the $$B-L$$ model. A fermion dark matter $$\chi $$ carrying $$U(1)_{B-L}$$ charge is introduced to the model but not coupling with other particles. We consider the Freeze-out and Freeze-in mechanisms, and obtain the correct relic density respectively. We have scanned the feasible parameter space and found that the dark matter direct detection experiments imposed the most stringent constraints on the parameter space. The constraint on the parameter space places a limit on the mass of dark matter with $$m_{\chi } \approx 1/2 M_{Z'}$$ within a narrow region in the case of the Freeze-out scenario, and we can obtain the right baryon asymmetry result in the case of $$m_{\chi }\subset [654$$ GeV, 664 GeV]. For the Freeze-in scenario, we have a much broader parameter space for $$m_{\chi }$$ and $$M_{Z^\prime }$$ but $$g_{BL}$$ is restricted at $$10^{-5}$$ level for $$Q=0.1$$ . In both scenarios, $$g_{BL}-M_{Z^\prime }$$ is limited within a narrow region by the dark matter relic density, direct detection and baryon asymmetry constraints.

XinXin Qi

Hao Sun

Dark matter, leptogenesis and $$Z^\prime $$ in the $$B-L$$ model

We construct a realistic supersymmetric model for superheavy metastable cosmic strings (CSs) that can be investigated in the current pulsar timing array (PTA) experiments. It is based on shifted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi></math>-hybrid inflation in which the symmetry breaking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>4</mn><msub><mo stretchy="false">)</mo><mi>c</mi></msub><mo>×</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo><mi>L</mi></msub><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>R</mi></msub><mo stretchy="false">→</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><msub><mo stretchy="false">)</mo><mi>c</mi></msub><mo>×</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo><mi>L</mi></msub><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>R</mi></msub></math> proceeds along an inflationary trajectory such that the topologically unstable primordial monopoles are inflated away. The breaking of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>R</mi></msub><mo stretchy="false">→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>Y</mi></msub></math> after inflation ends yields the superheavy metastable CSs carrying the right handed neutrino zero modes that generate the stochastic gravitational wave background, which we show is consistent with the current PTA dataset and the LIGO O3 run. The scalar spectral index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>n</mi><mi>s</mi></msub></math> and the tensor-to-scalar ratio <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi></math> are compatible with the Planck 2018 and Atacama Cosmology Telescope measurements. Both reheating and leptogenesis are briefly discussed.

Pakistan

Adeela Afzal

Maria Mehmood

Saudi Arabia

Mansoor Ur Rehman

United States

Qaisar Shafi

Supersymmetric hybrid inflation and current-carrying metastable cosmic strings in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>4</mn><msub><mo stretchy="false">)</mo><mi>c</mi></msub><mo>×</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo><mi>L</mi></msub><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>R</mi></msub></math>

Thermal dark matter is a compelling setup that has been probed by a multitude of experiments, mostly in the GeV–TeV mass range. The thermal paradigm in the sub-GeV range is about to experience the same experimental test with the next generation of low-energy accelerators and light dark matter detectors. Motivated by this, we investigate thermal dark matter in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></math> and assess how the introduction of a matter-dominated era impacts the parameter that yields the correct relic density. Interestingly, we show that the projected experiments, such as the Muon (Synchrotron) Ion Collider, Future Circular Collider-ee, and Light Dark Matter eXperiment, will probe a large region of the viable parameter space that yields the correct relic density. In the GeV–TeV mass regime, the usual large-scale detectors push the sensitivity. Our work highlights the rich interplay between early-Universe dynamics, dark matter phenomenology, and the discovery potential of next-generation experiments.

United Arab Emirates

Nicolás Bernal

Brazil

Jacinto P. Neto

Peru

Javier Silva-Malpartida

Chile

Farinaldo S. Queiroz

Enabling thermal dark matter within the vanilla <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></math> model

The Muon <math altimg='si4.svg'><mrow><mi>g</mi><mo linebreak='goodbreak'>&#8722;</mo><mn>2</mn></mrow></math> collaboration at Fermilab has announced their final result of the anomalous magnetic moment of the muon. By adopting the lattice-QCD evaluation of the leading-order hadronic-vacuum-polarization, this result is now in agreement with the latest theoretical prediction to the <math altimg='si15.svg'><mrow><mn>1</mn><mi>&#963;</mi></mrow></math> level. This new result further constrains the allowed parameter space, but does not rule out all possible new physics contributions the muon <math altimg='si4.svg'><mrow><mi>g</mi><mo linebreak='goodbreak'>&#8722;</mo><mn>2</mn></mrow></math>. We study the implications for one of the relevant models, the gauged <math altimg='si12.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math>. When using this model to resolve the previous <math altimg='si16.svg'><mrow><mn>4</mn><mi>&#963;</mi></mrow></math> tension, results from muon neutrino trident (MNT) scattering experiments would restrict the mass of the new gauge boson ( <math altimg='si17.svg'><msup><mi>Z</mi><mo>&#8242;</mo></msup></math>) to be less than 300 MeV. Since the theory and experimental data difference for muon <math altimg='si4.svg'><mrow><mi>g</mi><mo linebreak='goodbreak'>&#8722;</mo><mn>2</mn></mrow></math> is lowered down to <math altimg='si15.svg'><mrow><mn>1</mn><mi>&#963;</mi></mrow></math>, the requirement for <math altimg='si18.svg'><mrow><msub><mi>m</mi><msup><mi>Z</mi><mo>&#8242;</mo></msup></msub><mo>&#8818;</mo><mn>300</mn><mspace width='0.16em' /><mrow><mrow><mi mathvariant='normal'>M</mi></mrow><mi>e</mi><mi>V</mi></mrow></mrow></math> is much relaxed. Within the updated allowed range of <math altimg='si17.svg'><msup><mi>Z</mi><mo>&#8242;</mo></msup></math> boson mass, we study the models implications for electron and tauon <math altimg='si4.svg'><mrow><mi>g</mi><mo linebreak='goodbreak'>&#8722;</mo><mn>2</mn></mrow></math> as well as future muon colliders. We find that muon collider can effectively probe the <math altimg='si12.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math>.

Ming-Wei Li

Xiao-Gang He

Andrew Cheek

Xinhui Chu

The <math altimg='si12.svg'><mrow><mi>U</mi><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mi>L</mi><mi>&#956;</mi></msub><mo>&#8722;</mo><msub><mi>L</mi><mi>&#964;</mi></msub></mrow></msub></mrow></math> model meets the new <math altimg='si14.svg'><msub><mrow><mo>(</mo><mi>g</mi><mo>&#8722;</mo><mn>2</mn><mo>)</mo></mrow><mi>&#956;</mi></msub></math> data and muon neutrino trident scattering

Using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> annihilation data corresponding to a total integrated luminosity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>7.33</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fb</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector, we provide the first amplitude analysis and absolute branching fraction measurement of the hadronic decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>. The branching fraction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> is determined to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>1.86</mn><mo>±</mo><mn>0.0</mn><msub><mrow><mn>6</mn></mrow><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>0.0</mn><msub><mrow><mn>3</mn></mrow><mrow><mi>syst</mi></mrow></msub><mo stretchy="false">)</mo><mo>%</mo></mrow></math>. Combining the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo stretchy="false">)</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">]</mo></mrow></math> obtained in this Letter and the world average of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mo stretchy="false">→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">]</mo></mrow></math>, we measure the relative branching fraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0.593</mn><mo>±</mo><mn>0.02</mn><msub><mrow><mn>3</mn></mrow><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mspace linebreak="goodbreak"></mspace><mn>0.01</mn><msub><mrow><mn>4</mn></mrow><mrow><mi>syst</mi></mrow></msub><mo>±</mo><mn>0.01</mn><msub><mrow><mn>6</mn></mrow><mrow><mi>ϕ</mi><mi>π</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math>, where the third error is due to the uncertainty of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mi>ϕ</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><mi>ϕ</mi><mo stretchy="false">→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math>. Our result deviates from the Particle Data Group value by more than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3</mn><mi>σ</mi></mrow></math>. Furthermore, the asymmetry of the branching fractions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo mathvariant="bold" stretchy="false">(</mo><mo stretchy="false">{</mo><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">+</mo></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">−</mo><mspace linebreak="goodbreak"></mspace><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">+</mo></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>/</mo><mo stretchy="false">{</mo><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">+</mo></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">+</mo><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">*</mo></mrow></msup><mo stretchy="false">(</mo><mn>892</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">+</mo></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo mathvariant="bold" stretchy="false">)</mo></mrow></math>, is determined to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mo>−</mo><mn>14.5</mn><mo>±</mo><msub><mn>5.1</mn><mrow><mi>stat</mi></mrow></msub><mo>±</mo><msub><mn>1.8</mn><mrow><mi>syst</mi></mrow></msub><mo stretchy="false">)</mo><mo>%</mo></math>.

M. Ablikim

Russia

M.N. Achasov

Sweden

P. Adlarson

X.C. Ai

Germany

R. Aliberti

A. Amoroso

Q. An

Y. Bai

JINR

O. Bakina

Y. Ban

H.R. Bao

Poland

V. Batozskaya

Mongolia

K. Begzsuren

N. Berger

M. Berlowski

M. Bertani

D. Bettoni

F. Bianchi

E. Bianco

A. Bortone

I. Boyko

R.A. Briere

A. Brueggemann

H. Cai

M.H. Cai

X. Cai

A. Calcaterra

G.F. Cao

N. Cao

Türkiye

S.A. Cetin

X.Y. Chai

J.F. Chang

G.R. Che

Y.Z. Che

G. Chelkov

C.H. Chen

Chao Chen

G. Chen

H.S. Chen

H.Y. Chen

M.L. Chen

S.J. Chen

S.L. Chen

S.M. Chen

T. Chen

X.R. Chen

X.T. Chen

X.Y. Chen

Y.B. Chen

Y.Q. Chen

Z.J. Chen

Z.K. Chen

S.K. Choi

X. Chu

G. Cibinetto

F. Cossio

J. Cottee-Meldrum

J.J. Cui

H.L. Dai

J.P. Dai

A. Dbeyssi

R.E. de Boer

D. Dedovich

C.Q. Deng

Z.Y. Deng

A. Denig

I. Denysenko

M. Destefanis

F. De Mori

B. Ding

X.X. Ding

Y. Ding

Y.X. Ding

J. Dong

L.Y. Dong

M.Y. Dong

X. Dong

M.C. Du

S.X. Du

Y.Y. Duan

Z.H. Duan

P. Egorov

G.F. Fan

J.J. Fan

Y.H. Fan

J. Fang

S.S. Fang

W.X. Fang

Y.Q. Fang

R. Farinelli

L. Fava

F. Feldbauer

G. Felici

C.Q. Feng

J.H. Feng

L. Feng

Study of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ϕ</mi><mo stretchy="false">→</mo><mi>K</mi><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></math> Asymmetry in the Amplitude Analysis of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msubsup><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> Decays