The Belle II collaboration has recently observed the rare decay  $$B^+\rightarrow K^+ \nu \bar{\nu }$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  , finding an excess with respect to the Standard Model prediction. We explore the possibility that the data entails long-distance interactions induced by a massless dark photon,  $$\gamma _{\scriptscriptstyle {D}}$$  <math> <msub> <mi>γ</mi> <mstyle> <mi>D</mi> </mstyle> </msub> </math>  . This couples at the tree-level to an invisible, dark sector and to the Standard Model via higher-dimensional operators, such as the chromomagnetic-dipole coupling that we use to explain the excess. As the process  $$B^+\rightarrow K^+ \gamma _{\scriptscriptstyle {D}}$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msub> <mi>γ</mi> <mstyle> <mi>D</mi> </mstyle> </msub> </mrow> </math>   is forbidden by angular momentum conservation, the transition mediated by the off-shell dark photon yields a three-body final state comprising a pair of dark fermions that show as a missing energy continuum in the detector, faking the neutrino signature. We show that the Belle II data is explained for perturbative values of the parameters of the model. This scenario predicts new contributions to the neutral B meson decays  $$B^0\rightarrow K^* \gamma _{\scriptscriptstyle {D}}$$  <math> <mrow> <msup> <mi>B</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>∗</mo> </msup> <msub> <mi>γ</mi> <mstyle> <mi>D</mi> </mstyle> </msub> </mrow> </math>  , in which the emission of a on-shell dark photon is allowed, yielding a monochromatic missing energy signature. Analogously, an excess due to the emission of a dark photon is predicted for the  $$B^0_s\rightarrow \phi + E_\textrm{miss}$$  <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>→</mo> <mi>ϕ</mi> <mo>+</mo> <msub> <mi>E</mi> <mtext>miss</mtext> </msub> </mrow> </math>   decay that could be scrutinized next at the LHCb experiments.

Italy

Estonia

Emidio Gabrielli

Luca Marzola

Kristjan Müürsepp

Martti Raidal

Explaining the  $$B^+\rightarrow K^+ \nu \bar{\nu }$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mi>ν</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   excess via a massless dark photon

CERN

Carlotta Accettura

United Kingdom

Dean Adams

United States

Rohit Agarwal

Claudia Ahdida

Chiara Aimè

Nicola Amapane

David Amorim

Paolo Andreetto

Fabio Anulli

Robert Appleby

Artur Apresyan

Aram Apyan

France

Sergey Arsenyev

Pouya Asadi

Egypt

Mohammed Mahmoud

Aleksandr Azatov

John Back

Lorenzo Balconi

Laura Bandiera

Roger Barlow

Nazar Bartosik

Emanuela Barzi

Fabian Batsch

Matteo Bauce

J. Berg

Andrea Bersani

Alessandro Bertarelli

Alessandro Bertolin

Kevin Black

Fulvio Boattini

Alex Bogacz

Maurizio Bonesini

Bernardo Bordini

Salvatore Bottaro

Luca Bottura

Alessandro Braghieri

Marco Breschi

Natalie Bruhwiler

Xavier Buffat

Laura Buonincontri

Philip Burrows

Graeme Burt

Dario Buttazzo

Barbara Caiffi

Marco Calviani

Simone Calzaferri

Daniele Calzolari

Rodolfo Capdevilla

Christian Carli

Fausto Casaburo

Massimo Casarsa

Luca Castelli

Maria Catanesi

Lorenzo Cavallucci

Gianluca Cavoto

Spain

Francesco Celiberto

Luigi Celona

Alessandro Cerri

Gianmario Cesarini

Cari Cesarotti

Portugal

Grigorios Chachamis

Antoine Chance

Switzerland

Siyu Chen

Yang-Ting Chien

Mauro Chiesa

Anna Colaleo

Francesco Collamati

Gianmaria Collazuol

Marco Costa

Nathaniel Craig

Camilla Curatolo

Canada

David Curtin

Giacomo Molin

Germany

Magnus Dam

Heiko Damerau

Sridhara Dasu

Jorge Blas

Stefania Curtis

Ernesto Matteis

Stefania Rosa

Jean-Pierre Delahaye

Dmitri Denisov

Türkiye

Haluk Denizli

Christopher Densham

Radovan Dermisek

Luca Luzio

Elisa Meco

Biagio Micco

Keith Dienes

Eleonora Diociaiuti

Tommaso Dorigo

Alexey Dudarev

Robert Edgecock

Filippo Errico

Marco Fabbrichesi

Stefania Farinon

Anna Ferrari

Jose Somoza

Netherlands

Frank Filthaut

Davide Fiorina

Erratum: Towards a muon collider

A muon collider would enable the big jump ahead in energy reach that is needed for a fruitful exploration of fundamental interactions. The challenges of producing muon collisions at high luminosity and 10 TeV centre of mass energy are being investigated by the recently-formed International Muon Collider Collaboration. This Review summarises the status and the recent advances on muon colliders design, physics and detector studies. The aim is to provide a global perspective of the field and to outline directions for future work.

Towards a muon collider

Quantum entanglement of weak interaction gauge bosons produced at colliders can be explored by computing the corresponding polarization density matrix. To this end, we consider the Higgs boson decays  $$H\rightarrow W W^*$$  <math> <mrow> <mi>H</mi> <mo>→</mo> <mi>W</mi> <msup> <mi>W</mi> <mo>∗</mo> </msup> </mrow> </math>   and  $$H\rightarrow Z Z^*$$  <math> <mrow> <mi>H</mi> <mo>→</mo> <mi>Z</mi> <msup> <mi>Z</mi> <mo>∗</mo> </msup> </mrow> </math>  , in which  $$W^*$$  <math> <msup> <mi>W</mi> <mo>∗</mo> </msup> </math>   and  $$Z^*$$  <math> <msup> <mi>Z</mi> <mo>∗</mo> </msup> </math>   are off-shell states, and the WW, WZ and ZZ di-boson production in proton collisions. The polarization density matrix of the di-boson state is determined by the amplitude of the production process and can be experimentally reconstructed from the angular distribution of the momenta of the final states into which the gauge bosons decay. We show that a suitable instance of the Bell inequality is violated in  $$H\rightarrow Z Z^*$$  <math> <mrow> <mi>H</mi> <mo>→</mo> <mi>Z</mi> <msup> <mi>Z</mi> <mo>∗</mo> </msup> </mrow> </math>   to a degree that can be tested at the LHC with future data. The same Bell inequality is violated in the production of WW and ZZ boson pairs for invariant masses above 900 GeV and scattering angles close to  $$\pi /2$$  <math> <mrow> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </math>   in the center of mass frame. LHC data in this case are not sufficient to establish the violation of the Bell inequality. We also analyze the prospects for detecting Bell inequality violations in di-boson final states at future  $$e^+e^-$$  <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> </mrow> </math>   and muon colliders. A further observable that provides a lower bound on the amount of polarization entanglement in the di-boson system is computed for each of the examined processes. The analytic expressions for the polarization density matrices are presented in full in an Appendix. We also provide the unitary matrices required in the optimization procedure necessary in testing the Bell inequalities.

Roberto Floreanini

Bell inequalities and quantum entanglement in weak gauge boson production at the LHC and future colliders

The measurement of quantum entanglement can provide a new and most sensitive probe to physics beyond the Standard Model. We use the concurrence of the top-quark pair spin states produced at colliders to constrain the magnetic dipole term in the coupling between top quark and gluons, that of  $$\tau $$  <math> <mi>τ</mi> </math>  -lepton pair spin states to bound contact interactions and that of  $$\tau $$  <math> <mi>τ</mi> </math>  -lepton pairs or two-photons spin states from the decay of the Higgs boson in trying to distinguish between CP-even and odd couplings. These four examples show the power of the new approach as well as its limitations. We show that differences in the entanglement in the top-quark and  $$\tau $$  <math> <mi>τ</mi> </math>  -lepton pair production cross sections can provide constraints better than those previously estimated from total cross sections or classical correlations. Instead, the final states in the decays of the Higgs boson remain maximally entangled even in the presence of CP-odd couplings and cannot be used to set bounds on new physics. We discuss the violation of Bell inequalities featured in all four processes.

Constraining new physics in entangled two-qubit systems: top-quark, tau-lepton and photon pairs

We explore the electroweak vacuum stability in the framework of a recently proposed paradigm for the origin of Yukawa couplings. These arise as low energy effective couplings radiatively generated by portal interactions with a hidden, or dark, sector at the one-loop level. Possible tree-level Yukawa couplings are forbidden by a new underlying symmetry, assumed to be spontaneously broken by the vacuum expectation value of a new scalar field above the electroweak scale. As a consequence, the top Yukawa interaction ceases to behave as a local operator at energies above the new sector scale and, therefore, cannot contribute to the running of the quartic Higgs coupling at higher energies. By studying two complementary scenarios, we explicitly show that the framework can achieve the stability of the electroweak vacuum without particular tuning of parameters. The proposed mechanism requires the existence of a dark sector and new portal messenger scalar interactions that, connecting the Standard Model to the dark sector fields, could be tested at the LHC and future collider experiments.

Ruiwen Ouyang

Vacuum stability with radiative Yukawa couplings

We explore the possibility of probing flavor violations in the charged-lepton sector by means of high-luminosity lepton-photon and electron-muon collisions, by inverting initial and final states in a variety of decay channels presently used to bound such violations. In particular, we analyze the resonant lepton, <math><mi>γ</mi><mspace width="0.2em"></mspace><mi>ℓ</mi><mo>→</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>′</mo></mrow></msup></math>, and neutral-meson, <math><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>→</mo><mi>ϕ</mi><mo>,</mo><mi>η</mi><mo>,</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></math>, scattering channels, whose cross sections are critically dependent on the colliding-beams energy spread, being particularly demanding in the case of leptonic processes. For these processes, we compute upper bounds to the cross-section corresponding to present limits on the inverse decay channel rates. In order to circumvent the beam energy spread limitations we extend the analysis to processes in which a photon accompanies the resonance in the final state, compensating the off-shellness effects by radiative return. These processes might be studied at future facilities with moderate energies, in case lepton-photon and electron-muon collisions with sufficiently high luminosity will be available.

Barbara Mele

Roberto Onofrio

Bounds on charged-lepton flavor violations via resonant scattering

In this work we evaluate the long-distance QED contributions, induced by the magnetic-dipole corrections to the final charged leptons, on the B meson decay widths B → (K, K*)ℓ + ℓ  −  and ratios R  K,K*  = Γ(B → (K, K*)μ + μ  − )/Γ(B → (K, K*)e + e  − ), as well as on   <math> <msubsup> <mi>R</mi> <mrow> <mi>K</mi> <mo>,</mo> <mi>K</mi> <mo>∗</mo> </mrow> <mi>τ</mi> </msubsup> </math>  $$ {R}_{K,K\ast}^{\tau } $$  (with μ replaced by the τ lepton). QED long-distance contributions induced by the Coulomb potential corrections (Fermi-Sommerfeld factors) were also included. Corresponding corrections to the inclusive decay widths of B → X  s  ℓ + ℓ  − , with ℓ = e, μ, τ, are also analyzed for completeness. The magnetic-dipole corrections, which are manifestly Lepton Flavor Universality violating and gauge-invariant, are expected to be particularly enhanced in R  K*  for the dilepton mass region close to the threshold. However, we find that the largest contribution of all these corrections to the R  K,K* observables do not exceed a few per mille effect, thus reinforcing the validity of previous estimates about the leading QED corrections to R  K,K* . Finally, viable new physics contributions to R  K,K*  induced by the exchange of a massless dark-photon via magnetic-dipole interactions, which provide the leading contribution to the corresponding B → (K, K*)ℓ + ℓ  −  amplitudes in this scenario, are analyzed in light of the present R  K,K*  anomalies.

Marco Palmiotto

Magnetic-dipole corrections to R  K  and R  K*  in the Standard Model and dark photon scenarios

We analyze the <math><mi>Z</mi></math>-boson decay <math><mi>Z</mi><mo>→</mo><mi>γ</mi><mi>X</mi></math> into a photon (<math><mi>γ</mi></math>) plus a hypothetical light boson (<math><mi>X</mi></math>) belonging to a dark or secluded sector. Because of its feeble interactions with Standard Model fields, this dark boson behaves as missing energy in the detector. We consider for <math><mi>X</mi></math> the cases of spin-1 (massless dark-photon), spin-0 (axionlike), and spin-2 (gravitonlike) particles and explore the way to untangle its spin origin. All these scenarios predict a universal signature for this decay, characterized by a single monochromatic photon in the <math><mi>Z</mi></math> center of mass, with energy about half of the <math><mi>Z</mi></math> mass, plus a neutrinolike missing energy associated with the <math><mi>X</mi></math> boson. We show that if the <math><mi>Z</mi><mo>→</mo><mi>γ</mi><mi>X</mi></math> signal should be discovered at <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> colliders, the angular distribution of the monochromatic photon in <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>Z</mi><mo>→</mo><mi>γ</mi><mi>X</mi></math> can provide a clean probe to discriminate between the <math><mi>J</mi><mo>=</mo><mn>1</mn></math> and alternative <math><mi>J</mi><mo>=</mo><mn>0</mn><mo>/</mo><mn>2</mn></math> spin nature of the dark boson.

Andrea Comelato

Untangling the spin of a dark boson in <math><mrow><mi>Z</mi></mrow></math> decays

We compute the contribution of the decays $$K_L \rightarrow \pi ^0 Q {\bar{Q}}$$ <math><mrow><msub><mi>K</mi><mi>L</mi></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><mi>Q</mi><mover><mrow><mi>Q</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math>  and $$K^+ \rightarrow \pi ^+ Q {\bar{Q}}$$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><mi>Q</mi><mover><mrow><mi>Q</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> , where Q is a dark fermion of the dark sector, to the measured widths for the rare decays $$K^+\rightarrow \pi ^+ \nu {\bar{\nu }}$$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><mi>ν</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math>  and $$K_L\rightarrow \pi ^0 \nu {\bar{\nu }}$$ <math><mrow><msub><mi>K</mi><mi>L</mi></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><mi>ν</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> . The recent experimental limit for $$\varGamma (K^+ \rightarrow \pi ^+ \nu {\bar{\nu }})$$ <math><mrow><mi>Γ</mi><mo>(</mo><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><mi>ν</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></mrow></math>  from NA62 sets a new and very strict bound on the dark-sector parameters. A branching ratio for $$K_L \rightarrow \pi ^0 Q {\bar{Q}}$$ <math><mrow><msub><mi>K</mi><mi>L</mi></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><mi>Q</mi><mover><mrow><mi>Q</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math>  within the reach of the KOTO sensitivity is possible. The Grossman–Nir bound is weakened by the asymmetric effect of the different kinematic cuts enforced by the NA62 and KOTO experiments. This last feature holds true for all models where the decay into invisible states takes place through a light or massless intermediate state.

Dark-sector physics in the search for the rare decays $$K^+\rightarrow \pi ^+ \nu {\bar{\nu }}$$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><mi>ν</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math>  and $$K_L\rightarrow \pi ^0 \nu {\bar{\nu }}$$ <math><mrow><msub><mi>K</mi><mi>L</mi></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><mi>ν</mi><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math>