Search

The International Muon Collider Collaboration (IMCC) [1] was established in 2020 following the recommendations of the European Strategy for Particle Physics (ESPP) and the implementation of the European Strategy for Particle Physics-Accelerator R&D Roadmap by the Laboratory Directors Group [2], hereinafter referred to as the the European LDG roadmap. The Muon Collider Study (MuC) covers the accelerator complex, detectors and physics for a future muon collider. In 2023, European Commission support was obtained for a design study of a muon collider (MuCol) [3]. This project started on 1st March 2023, with work-packages aligned with the overall muon collider studies. In preparation of and during the 2021-22 U.S. Snowmass process, the muon collider project parameters, technical studies and physics performance studies were performed and presented in great detail. Recently, the P5 panel [4] in the U.S. recommended a muon collider R&D, proposed to join the IMCC and envisages that the U.S. should prepare to host a muon collider, calling this their "muon shot". In the past, the U.S. Muon Accelerator Programme (MAP) [5] has been instrumental in studies of concepts and technologies for a muon collider., Comment: This document summarises the International Muon Collider Collaboration (IMCC) progress and status of the Muon Collider R&D programme

The experimental data on the helicity amplitudes of charmonium decays allow us to measure entanglement in final state spin correlations and test possible violations of the Bell inequality. We find that the Bell inequality is violated with a significance of 5$\sigma$ or more in the decays $\eta_{c}, \, \chi_{c}^{0}, \, J/\psi \to \Lambda +\bar \Lambda$ $J/\psi \to \Xi^- +\bar \Xi^+,\; \Xi^0 +\bar \Xi^0, \;\Sigma^{-} +\bar \Sigma^{+},\; \Sigma^{0} +\bar \Sigma^{0}$, $\psi(3686) \to \Xi^- +\bar \Xi^+,\; \Sigma^- +\bar \Sigma^+,\; \Sigma^{0} +\bar \Sigma^{0}$, $\chi^{0}_{c}, \, \chi_{c}^{1} \to \phi+ \phi$ . The decays $\psi(3686) \to \Lambda +\bar \Lambda$ and $\Xi^0 +\bar \Xi^0$ show the same violation but with less significance. The decay $\psi(3686) \to \Omega^- + \bar \Omega^+$ displays entanglement. These results firmly establish the presence of entanglement and quantum non-separability at high energies, in a setting with particles of different spins and interacting through electroweak and strong interactions. In addition, the relatively long lifetime of some of the strange baryons produced in the decays provides a natural probe to test whether quantum spin correlations remain after the particles have interacted with the beam pipe and the first few layers of the detector., Comment: 31 pages, 12 figures, version sent to journal

The Future Circular Collider (FCC) -- in its first incarnation as a lepton collider -- will produce, according to the proposed design, more than 100 billion pairs of $\tau$ leptons after working for four years at the energy of the $Z$-boson resonance. The $\tau$ lepton is special because its relatively long lifetime makes it possible to reconstruct the momenta of neutrinos emitted in the single pion decay mode. The resulting large number of events is an ideal source for a full quantum tomography of the process that will test quantum entanglement and the violation of Bell inequality with unprecedented precision. In addition, the study of polarizations and spin correlations can provide a competitive determination of the Weinberg angle $ \theta_W$ and constrain possible anomalous couplings in the neutral electroweak current. We utilize analytic results and Monte Carlo simulations to explore to what extent these goals might be accomplished., Comment: 19 pages

The large number of $\tau$ leptons available at the Belle II experiment makes it possible to study their properties and the extent of their compositeness. Our strategy relies on three observables defined in terms of elements of the polarization density matrix of the produced $\tau$ pairs, which we obtain via quantum tomography. The observables are able to explore values of the magnetic dipole moment down to $6.3 \times 10^{-4}$, which is two order of magnitude better than the current experimental limit, constrain the electromagnetic radius below $4.3 \times 10^{-3}$ fm and exclude an electric dipole moment larger than $1.7 \times 10^{-17}$ e cm. The quoted limits are at the $95\%$ confidence level and are obtained for a benchmark integrated luminosity of 1 ab$^{-1}$., Comment: 9 pages, 2 figures

The violation of the Bell inequality is one of the hallmarks of quantum mechanics and can be used to rule out local deterministic alternative descriptions. We utilize the data analysis published by the LHCb collaboration on the helicity amplitudes for the decay $B^0\to J/\psi \,K^*(892)^0$ to compute the entanglement among the polarizations of the final vector mesons and the violation of the Bell inequality that it entails. We find that quantum entanglement can be detected with a significance well above 5$\sigma$ (nominally 84$\sigma$) and Bell inequality is violated with a significance well above 5$\sigma$ (nominally 36$\sigma$) -- thereby firmly establishing these distinguishing feature of quantum mechanics at high energies in a collider setting and in the presence of strong and weak interactions. Entanglement is also present and the Bell inequality is violated in other decays of the $B$ mesons into vector mesons, but with lesser significance., Comment: 5 pages, improved text, a few results updated, same as published version in PRD Letter

Quantum tomography provides the full reconstruction of the density matrix of a state. We use it to study the Higgs boson decay into weak gauge bosons. Anomalous couplings beyond the Standard Model can be constrained by means of observables easily defined in terms of the polarization density matrix. We describe a strategy based on three observables that together provide the most stringent limits. Two of these observables are linked to the entanglement between the polarizations of the two gauge bosons, the other is based on CP-odd combinations of one momentum and two polarizations. We find for the $Z$ channel that this strategy offers, already with the available LHC data, limits competitive with the best available bounds. We argue that the inclusion of these observables in routine experimental analyses can lead to more stringent global fit limits., Comment: 14 pages, 4 figures, typos corrected in equations: 2.1, 3.3, 3.4, 3.8, 3.11, and 3.14-3.20

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 $H\to W W^*$ and $H\to Z Z^*$, in which $W^*$ and $Z^*$ 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 $H\to Z Z^*$ 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 $\pi/2$ 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 $e^+e^-$ 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., Comment: 59 pages, 20 figures, same as published version

Entanglement between the spins of top-quark pairs produced at a collider can be used to test a (generalized) Bell inequality at energies never explored so far. We show how the measurement of a single observable can provide a test of the violation of the Bell inequality at the 98% CL with the data already collected at the Large Hadron Collider and at the 99.99% CL with the higher luminosity of the next run., Comment: 4 pages, 1 figure

We minimally extend the Standard Model field content by adding new vector-like fermions at the TeV scale to allow gauge coupling unification at a realistic scale. We embed the model into a $SU(5)$ grand unified theory that is asymptotically safe and features an interacting fixed point for the gauge coupling. There are no Landau poles of the $U(1)$ gauge and Higgs couplings. Gauge, Yukawa and Higgs couplings are retraced from the fixed point and matched at the grand unification scale to those of the Standard Model rescaled up to the same energy. All couplings, their fixed point values and critical exponents always remain in the perturbative regime., Comment: 25 pages, 3 figures, 6 tables

We study the decay of the $Z$ vector boson into a photon and a massless (invisible) dark photon in high-energy collisions. The photon can be used as trigger for the event, while the dark photon is detected indirectly as missing momentum in the event final state. We investigate the possibility of searching for such a dark photon at the LHC, HL-LHC and future lepton colliders, and compare the respective sensitivities. As expected, the best result is found for the lepton colliders running at the $Z$ mass, FCC-ee and CEPC, with a final sensitivity to branching ratios of order $O(10^{-11})$. We also discuss how to use the photon angular distribution of the events in lepton collisions to discriminate between the dark photon and a pseudo-scalar state like the axion., Comment: 12 pages, 8 figures, G. Panizzo corresponding author

We study the invisible decay of neutral hadrons in a representative model of the dark sector. The mesons $K_L$ and $B^0$ decay into the dark sector with branching rates that can be at the current experimental limits. The neutron decays with a rate that could either explain the neutron lifetime puzzle (although only for an extreme choice of the parameters and a fine tuned value of the masses) or be just above the current limit of its invisible decay ($\tau_N^{\rm inv} \ge 10^{29}$ years) if kinematically allowed. These invisible decays of ordinary matter provide a novel and promising window into new physics that should be vigorously pursued., Comment: 11 pages, 3 figures. Extended text to match published version

This note proposes common standards and prescriptions for the effective-field-theory interpretation of top-quark measurements at the LHC., Comment: LHC TOP WG note, 10 pages of main text, 5 appendices, 1 figure, 21 tables

We study the $Z\rightarrow \gamma \bar \gamma$ process in which the $Z$ boson decays into a photon $\gamma$ and a massless dark photon $\bar \gamma$, when the latter couples to standard-model fermions via dipole moments. This is a simple yet nontrivial example of how the Landau-Yang theorem---ruling out the decay of a massive spin-1 particle into two photons---is evaded if the final particles can be distinguished. The striking signature of this process is a resonant monochromatic single photon in the $Z$-boson center of mass together with missing momentum. LEP experimental bounds allow a branching ratio up to about 10$^{-6}$ for such a decay. In a simplified model of the dark sector, the dark-photon dipole moments arise from one-loop exchange of heavy dark fermions and scalar messengers. The corresponding prediction for the rare $Z\to \gamma \bar \gamma$ decay width can be explored with the large samples of $Z$ bosons foreseen at future colliders., Comment: Same as published PRL version, minor comments added, 5 pages, 3 figures

If dark photons are massless, they couple to standard-model particles only via higher dimensional operators, while the kinetic mixing with photons, which motivates most of the current experimental searches, is absent. We consider the effect of possible flavor-changing magnetic-dipole couplings of massless dark photons in kaon physics. In particular, we study the branching ratio for the process $K^+\rightarrow \pi^+\pi^0 \bar \gamma$ with a simplified-model approach, assuming the chiral quark model to evaluate the hadronic matrix element. Possible effects in the $K^0$-$\bar K^0$ mixing are taken into account. We find that branching ratios up to $O(10^{-7})$ are allowed---depending on the dark-sector masses and couplings. Such large branching ratios for $K^+\rightarrow \pi^+\pi^0 \bar \gamma$ could be of interest for experiments dedicated to rare $K^+$ decays like NA62 at CERN, where $\bar \gamma$ can be detected as a massless invisible system., Comment: same as the published PRL version, 5 pages, 3 figures, a few comments and new references added

We identify the differential cross sections for $t\bar t$ production and the total cross section for Higgs production through gluon fusion as the processes in which the two effective operators describing the leading non-standard interactions of the top quark with the gluon can be disentangled and studied in an independent fashion. Current data on the Higgs production and the $ {\rm d}\sigma/{\rm d} {p^t_T}$ differential cross section provide limits comparable, but not more stringent, than those from the total $t\bar t$ cross sections measurements at the LHC and Tevatron, where however the two operators enter on the same footing and can only be constrained together. Given the present uncertainties, we find that the most stringent bounds are provided by a combination of data on the $t \bar t$ total cross sections together with those from the Higgs production. We conclude by stating the (modest) reduction in the uncertainties necessary to provide more stringent limits by means of the Higgs production and $t\bar t$ differential cross section observables at the LHC with the future luminosity of 300 and 3000 fb$^{-1}$., Comment: 12 pages, 9 figures, 3 tables. Discussion with the relation of the considered operators with the ones of the Warsaw basis extended. Missing contribution to the gluon fusion amplitude added. Main results unchanged

We review and update current limits on possible anomalous couplings of the top quark to gauge bosons. We consider data from top quark decay (as encoded in the $W$-boson helicity fractions) and single-top production (in the $t$-, $s$- and $Wt$-channels). We find improved limits with respect to previous results (in most cases of almost one order of magnitude) and extend the analysis to include four-quark operators. We find that new electroweak physics is constrained to live above an energy scale between 430 GeV and 3.2 TeV (depending on the form of its contribution); strongly interacting new physics is bounded by scales higher than 1.3 or 1.5 TeV (again depending on its contribution)., Comment: 9 pages, 4 figures, 1 table

If the top quark is a composite state made out of some constituents, its interaction with the gluon will be modified. We introduce the leading effective operators that contribute to the radius and anomalous magnetic moment of the top quark and study their effect on the cross section for top anti-top production at the Tevatron and the LHC. Current measurements of the cross sections set a stringent limit on the scale of compositeness. This limit is comparable to similar limits obtained for light quarks and those from electroweak precision measurements. It can be used to constrain the parameter space of some composite Higgs models., Comment: 18 pages, 9 figures, 1 table. Analysis extended to quadratic contributions of the effective couplings. Study of top spin polarizations added. New confidence intervals obtained. Figures of the bounds replaced. Bounds on C2 result weaker. Version accepted for publication in PRD

We discuss how two birds---the little hierarchy problem of low-scale type-I seesaw models and the search for a viable dark matter candidate---are (proverbially) killed by one stone: a new inert scalar state, Comment: 5 pages, 3 figures. arXiv admin note: text overlap with arXiv:1212.5065

We discuss two possible extensions to the standard model in which an inert singlet scalar state that only interacts with the Higgs boson is added together with some fermions. In one model the fermions provide for a see-saw mechanism for the neutrino masses, in the other model for grand unification of the gauge couplings. Masses and interaction strengths are fixed by the requirement of controlling the finite one-loop corrections to the Higgs boson mass thus addressing the little hierarchy problem. The inert scalar could provide a viable dark matter candidate. Direct detection of this scalar singlet in nuclear scattering experiments is possible with a cross section within reach of future experiments., Comment: arXiv admin note: text overlap with arXiv:1207.0951

We discuss a minimal extension to the standard model in which two singlet scalar states that only interacts with the Higgs boson is added. Their masses and interaction strengths are fixed by the two requirements of canceling the one-loop quadratic corrections to the Higgs boson mass and providing a viable dark matter candidate. Direct detection of the lightest of these new states in nuclear scattering experiments is possible with a cross section within reach of future experiments., Comment: Finite corrections included. Model modified. Conclusion unchanged

Quark contact interactions are an important signal of new physics. We introduce a model in which the presence of a symmetry protects these new interactions from giving large corrections in flavor changing processes at low energies. This minimal model provides the basic set of operators which must be considered to contribute to the high-energy processes. To discuss their experimental signature in jet pairs produced in proton-proton colllisions, we simplify the number of possible operators down to two. We show (for a representative integrated luminosity of 200 pb^-1 at \surd s = 7 TeV) how the presence of two operators significantly modifies the bound on the characteristic energy scale of the contact interactions which is obtained by keeping a single operator., Comment: 8 pages, 2 figures

We consider a model in which Goldstone bosons, described by a SU(N) chiral nonlinear sigma model, are coupled to an N-plet of colored fermions by means of a Yukawa interaction. We study the one-loop renormalization group flow and show that the non-Gaussian UV fixed point, which is present in the purely bosonic model, is lost because of fermion loop effects unless N is sufficiently large. We then add four-fermion contact interactions to the lagrangian and show that in this case there exist several non-Gaussian fixed points. The strength of the contact interactions, predicted by the requirement that the theory flows towards a fixed point in the UV, is compared to the current experimental bounds. This toy model could provide an important building block of an asymptotically safe model of the weak interactions., Comment: 7 pages, 3 figures, 1 table

We assume that the Higgs boson or a possible resonance---playing its role in strongly interacting models of electroweak symmetry breaking---has been discovered at the LHC and propose a search strategy to determine its spin based on two simple asymmetries in the ZZ, W+W- and t t-bar decays channels. We consider some benchmark values for its mass (in the interval from 182 GeV/c^2 to 1 TeV/c^2) and discuss the relative advantages of the different decay processes. A full analysis, including the background, is given. For a center-of-mass energy of 14 TeV, we find that the lowest integrated luminosity required to discriminate between the different spins is, depending on the process and the resonance mass, between 40 fb^{-1} and 250 fb^{-1}., Comment: 17 pages, 3 figures

We consider the standard model without the Higgs boson, where the Goldstone modes are described by a nonlinear sigma model. We study the renormalization group flow of the sigma model coupling f and of the electroweak parameters S and T. The condition that the couplings reach a fixed point at high energy leaves the low energy values of f and T arbitrary (to be determined experimentally) and fixes S to a value compatible with electroweak precision data., Comment: 4 pages, 2 figures

We study the beta functions of the leading, two-derivative terms of the left-gauged SU(N) nonlinear sigma-model in d dimensions. In d>2, we find the usual Gaussian ultraviolet fixed point for the gauge coupling and an attractive non-Gaussian fixed point for the Goldstone boson coupling. The position of the latter fixed point controls the chiral expansion, unitarity and the strength of the tree-level Goldstone boson scattering amplitudes. Attention is paid to the gauge- and scheme-dependence of the results., Comment: REVTeX4, 11 pages, no figures, v2. Revised version accepted for publication in PRD where comments on large N have been removed

These proceedings collect the presentations given at the first three meetings of the INFN "Workshop on Monte Carlo's, Physics and Simulations at the LHC", held at the Frascati National Laboratories in 2006. The first part of these proceedings contains pedagogical introductions to several basic topics of both theoretical and experimental high pT LHC physics. The second part collects more specialised presentations., Comment: 239 pages, 222 figures

If electro-weak symmetry is broken by a new strongly interacting sector, new physics will probably manifest itself in gauge boson scattering at the LHC. The relevant dynamics is well described in terms of an effective lagrangian. We discuss the probable size of the coefficients of the relevant operators under a combination of model-independent constraints and reasonable assumptions based on two models of the strongly interacting sector. We compare these values with LHC sensitivity and argue that they will be too small to be seen. Therefore, the presence of vector and scalar resonances required by unitarity will be the only characteristic signature. We analyze the most likely masses and widths of these resonances., Comment: 14 pages, pdftex, 5 figures, improved discussion of bounds, 1 footnote removed

We discuss a minimal extension to the standard model in which there are two Higgs bosons and, in addition to the usual fermion content, two fermion doublets and one fermion singlet. The little hierachy problem is solved by the vanishing of the one-loop corrections to the quadratic terms of the scalar potential. The electro-weak ground state is therefore stable for values of the cut off up to 10 TeV. The Higgs boson mass can take values significantly larger than the current LEP bounds and still be consistent with electro-weak precision measurements.

We study the exact (one-loop) effective potential of the littlest Higgs model and determine the dependence of physical quantities, such as the vacuum expectation value v_W and mass m_h of the Higgs boson, on the fundamental parameters of the Lagrangian--masses, couplings of new states, the fundamental scale f of the sigma model, and the coefficients of operators quadratically sensitive to the cutoff of the theory. On the one hand, we show that it is possible to have the electroweak ground state and a relatively large cutoff \Lambda = 4\pi f with f in the 2 TeV range without requiring unnaturally small coefficients for quadratically divergent quantities, and with only moderate cancellations between the contribution of different sectors to the effective potential of the Higgs. On the other hand, this cannot be achieved while at the same time keeping m_h close to its (bantamweight) current lower bound of 114.4 GeV. The natural expectation for m_h is O(f), mainly because of large logarithmically divergent contributions to the effective potential of the top-quark sector. Even a fine-tuning at the level of O(10^{-2}) in the coefficients of the quadratic divergences is not enough to produce small physical Higgs masses, and the natural expectation is in the 800 GeV range (cruiserweight) for f \sim 2 TeV. We conclude that the littlest Higgs model is a solution of the little hierarchy problem, in the sense that it stabilizes the electroweak symmetry breaking scale to be a factor of 100 less than the cutoff of the theory, but this requires a quite large physical mass for the Higgs, and hence precision electroweak studies should be redone accordingly. We also study finite temperature corrections., Comment: 18 pages, 9 figures, RevTex4. Final version accepted for publication in Phys. Rev. D

We present a unified picture of flavor and electroweak symmetry breaking at the TeV scale. Flavor and Higgs bosons arise as pseudo-Goldstone modes in a nonlinear sigma model. Explicit collective symmetry breaking yields stable vacuum expectation values and masses protected at one loop by the little-Higgs mechanism. The coupling to the fermions through a Yukawa lagrangian with a U(1) global flavor symmetry generates well-definite mass textures that correctly reproduce the mass hierarchies and mixings of quarks and leptons. The model is more constrained than usual little- Higgs models because of bounds on weak and flavor physics. The main experimental signatures testable at the LHC are a rather large mass mh0 = 317+/-80 GeV for the (lightest) Higgs boson and a characteristic spectrum of new bosons and fermions with masses around the TeV scale.

We present a complete analysis of the fermion masses and mixing matrices in the framework of the little flavon model. In this model textures are generated by coupling the fermions to scalar fields, the little flavons, that are pseudo-Goldstone bosons of the breaking of a global SU(6) symmetry. The Yukawa couplings arise from the vacuum expectation values of the flavon fields, their sizes controlled by a potential a la Coleman-Weinberg. Quark and lepton mass hierarchies and mixing angles are accomodated within the effective approach in a natural manner., Comment: 11 pages, RevTeX4, version to appear on Phys. Rev. D

Fermion masses and mixing matrices can be described in terms of spontaneously broken (global or gauge) flavor symmetries. We propose a little-Higgs inspired scenario in which an SU(2)xU(1) gauge flavor symmetry is spontaneously (and completely) broken by the vacuum of the dynamically induced potential for two scalar doublets (the flavons) which are pseudo-Goldstone bosons remaining after the spontaneous breaking--at a scale between 10 and 100 TeV--of an approximate SU(6) global symmetry. The vacuum expectation values of the flavons give rise to the texture in the fermion mass matrices. We discuss in detail the case of leptons. Light-neutrino masses arise by means of a see-saw-like mechanism that takes place at the same scale at which the SU(6) global symmetry is broken. We show that without any fine tuning of the parameters the experimental values of the charged-lepton masses,the neutrino square mass differences and the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix are reproduced., Comment: 13 pages, revTeX4. Version to be published in PRD

We discuss the generalization to global gauge anomalies of the familiar procedure for the cancellation of local gauge anomalies in effective theories of spontaneously broken symmetries. We illustrate this mechanism in a recently proposed six-dimensional extension of the standard model., Comment: 5 pages; v2: version to appear in Phys. Rev. D

It has been recently shown that the requirement of anomaly cancellation in a (non-supersymmetric) six-dimensional version of the standard model fixes the field content to the known three generations. We discuss the phenomenological consequences of the cancellation of the local anomalies: the strong CP problem is solved and the fundamental scale of the theory is bounded by the physics of the axion. Neutrinos acquire a mass in the range suggested by atmospheric experiments., Comment: 9 pages, RevTeX4

We discuss the interplay between direct constraints on non-Newtonian gravity and particle-physics bounds in models with large extra dimensions. Existing and future bounds and the most effective ways of further testing these models in gravitational experiments are compared and discussed., Comment: 5 pages, 4 figures, revTeX 4

Whenever fields are allowed to propagate in different portions of space-time, the four dimensional theory exhibits an effective violation of the principle of equivalence. We discuss the conditions under which such an effect is relevant for neutrino physics. In the simplest case of compactification on a flat manifold, the effect of gravity is many orders of magnitude too weak and plays no role for solar neutrino oscillation. Instead, it could be important in the study of ultra high-energy neutrinos in cosmic rays. Gravity could also be relevant for lower energy neutrino processes involving bulk sterile states, if the mechanism of compactification is more subtle than that on torii., Comment: Final version, published in The European Physical Journal C. Includes also preprint hep-ph/0103249. Comments added and typos corrected

I discuss the estimate of the CP-violating parameter eps'/eps based on hadronic matrix elements computed in the chiral quark model. This estimate suggested, before the current experimental results, that the favored value of eps'/eps in the standard model is of the order of 10^{-3}. I briefly review the physical effects on which this result is based and summarize current estimates., Comment: 3 pages, LaTeX, conference proceedings, ICHEP2000, Osaka (July 27 - August 2, 2000), and IVth Rencontres du Vietnam, Hanoi (July 19-25, 2000)

I briefly review recent work on the comparison between two and three graviton scattering in supergravity and matrix theory, Comment: Talk given at the TMR meeting, Kerkyria, September 1998, to appear in the proceedings

The real part of epsilon'/epsilon measures direct CP violation in the decays of the neutral kaons in two pions. It is a fundamental quantity which has justly attracted a great deal of theoretical as well as experimental work. Its determination may answer the question of whether CP violation is present only in the mass matrix of neutral kaons (the superweak scenario) or also at work directly in the decays. After a brief historical summary, we discuss the present and expected experimental sensitivities. In the light of these, we come to the problem of estimating epsilon'/epsilon in the standard model. We review the present (circa 1998) status of the theoretical predictions of epsilon'/epsilon. The short-distance part of the computation is now known to the next-to-leading order in QCD and QED and therefore well under control. On the other hand, the evaluation of the hadronic matrix element of the relevant operators is where most of the theoretical uncertainty still resides. We analyze the results of the currently most developed calculations. The values of the B_i parameters in the various approaches are discussed, together with the allowed range of the relevant combination of the Cabibbo-Kobayashi-Maskawa entries Im V_{td}V^*_{ts}. We conclude by summarizing and comparing all up-to-date predictions of epsilon'/epsilon. Because of the intrinsic uncertainties of the long-distance computations, values ranging from 10^{-4} to a few times 10^{-3} can be accounted for in the standard model. Since this range covers most of the present experimental uncertainty, it is unlikely that new physics effects can be disentangled from the standard model prediction. For updates on the review and additional material see http://www.he.sissa.it/review/., Comment: 42 pages, 13 figures. To appear in Reviews of Modern Physics

After updating the determination of the combination of Kobayashi-Maskawa elements $ \Im V_{td}V^*_{ts}$ according to our new estimate of the parameter $\hat B_K$, we study the CP-violating ratio $\epsilon'/\epsilon$ by means of hadronic matrix elements computed to $O(p^4)$ in the chiral expansion. It is the first time that this order in chiral perturbation theory is included. We also discuss the relevance of some $O(p^2)$ terms that are generally neglected in the calculation of the electroweak penguin matrix elements. The most important effect of this improved analysis is the substantial reduction (20%) of the leading electroweak penguin matrix element and accordingly a reduced cancelation between the electroweak and gluon penguin contributions. The ratio $\epsilon'/\epsilon$ is thus larger than previously estimated and its predicted value enjoys a smaller uncertainty. We find $\epsilon'/\epsilon = 1.7 ^{+1.4}_{-1.0} \times 10^{-3}$. Values positive and of the order of $10^{-3}$ are therefore preferred., Comment: Latex2e, JHEP.cls, 22 pages, 8 figures. Short-distance analysis updated. Introduction and conclusions expanded

We calculate the hadronic matrix elements to $O(p^4)$ in the chiral expansion for the ($\Delta S =1$) $K^0 \to 2 \pi$ decays and the ($\Delta S=2$) $\bar K^0$-$K^0$ oscillation. This is done within the framework of the chiral quark model. The chiral coefficients thus determined depend on the values of the quark and gluon condensates and the constituent quark mass. We show that it is possible to fit the $\Delta I =1/2$ rule of kaon decays with values of the condensates close to those obtained by QCD sum rules. The renormalization invariant amplitudes are obtained by matching the hadronic matrix elements and their chiral corrections to the short-distance NLO Wilson coefficients. For the same input values, we study the parameter $\hat B_K$ of kaon oscillation and find $\hat B_K = 1.1 \pm 0.2$. As an independent check, we determine $\hat B_K$ from the experimental value of the $K_L$-$K_S$ mass difference by using our calculation of the long-distance contributions. The destructive interplay between the short- and long-distance amplitudes yields $\hat B_K = 1.2 \pm 0.1$, in agreement with the direct calculation., Comment: Latex2e, JHEP.cls, 32 pages, 9 figures. Short-distance analysis updated. Parts of Sects. 3 and 6 revised

We compute the $\widehat B_K$ parameter and the mass difference $\Delta M_{LS}$ of the $K^0-\bar K^0$ system by means of the chiral quark model. The chiral coefficients of the relevant $\Delta S=2$ and $\Delta S=1$ chiral lagrangians are computed via quark-loop integration. We include the relevant effects of one-loop corrections in chiral perturbation theory. The final result is very sensitive to non-factorizable corrections of $O(\alpha_S N)$ coming from gluon condensation. The size of the gluon condensate is determined by fitting the experimental value of the amplitude $K^+ \to \pi^+\pi^0$. By varying all the relevant parameters we obtain $\widehat{B}_K= 0.87 \pm 0.33\ .$ We evaluate within the model the long-distance contributions to $\Delta M_{LS}$ induced by the double insertion of the $\Delta S = 1$ chiral lagrangian and study the interplay between short- and long-distance amplitudes. By varying all parameters we obtain $\Delta M_{LS}^{th}/\Delta M_{LS}^{exp} = 0.76 ^{+0.64}_{-0.34} .$ Finally, we investigate the phenomenological constraints on the Kobayashi-Maskawa parameter Im $\lambda _t$ entering the determination of $\epsilon'/\epsilon$., Comment: 31 pages, Latex file including 7 eps figures. Revised version to appear in Nucl. Phys. B

I review a recent attempt to reproduce the isospin $I= 0$ and 2 amplitudes for the decay of a kaon into two pions by estimating the relevant hadronic matrix elements in the chiral quark model. The results are parametrized in terms of the quark and gluon condensates and of the constituent quark mass $M$. The latter is a parameter characteristic of the model. For values of these parameters within the current determinations, the $\Delta I= 1/2$ selection rule is well reproduced by means of the cumulative effects of short-distance NLO Wilson coefficients, penguin diagrams, non-factorizable soft-gluon corrections and meson-loop renormalization., Comment: to appear in the proceedings of the Workshop on K Physics, Orsay, May 30-June 4, 1996

We give the complete $O(p^2)$ bosonization of the electroweak Penguin operators $Q_{7,8}$ and compare the result with that of the gluon Penguin operators $Q_{5,6}$. We find that, in addition to the usual (constant and current-current) parts, there are three new terms not discussed previously in the literature. Two of these are present in the factorization approximation and should be included in the standard definition of the $B_{7,8}$-factors. The impact of these corrections on the direct $CP$-violating parameter $\varepsilon '/\varepsilon$ is briefly discussed., Comment: 9 pages, uuencoded Latex + eps files

We discuss direct $CP$ violation in the standard model by giving a new estimate of $\varepsilon'/\varepsilon$ in kaon decays. Our analysis is based on the evaluation of the hadronic matrix elements of the \mbox{$\Delta S =1$} effective quark lagrangian by means of the chiral quark model, with the inclusion of meson one-loop renormalization and NLO Wilson coefficients. Our estimate is fully consistent with the $\Delta I =1/2$ selection rule in $K\to \pi\pi$ decays which is well reproduced within the same framework. By varying all parameters in the allowed ranges and, in particular, taking the quark condensate---which is the major source of uncertainty---between $(-200\ {\rm MeV})^3$ and $(-280\ {\rm MeV})^3$ we find $$ -5.0 \times 10^{-3}\ <\varepsilon'/\varepsilon <\ 1.4 \times 10^{-3}\ .$$ Assuming for the quark condensate the improved PCAC result \mbox{$\vev{\bar qq} = -(221\: \pm 17\ {\rm MeV})^3$} and fixing $\Lambda_{\rm QCD}^{(4)}$ to its central value, we find the more restrictive prediction $$\varepsilon '/\varepsilon = ( 4 \pm 5 ) \,\times \,10^{-4}\ , $$ where the central value is defined as the average over the allowed values of Im $\lambda_t$ in the first and second quadrants. In these estimates the relevant mixing parameter Im $\lambda_t$ is self-consistently obtained from $\varepsilon$ and we take $m_t^{\rm pole} = 180 \pm 12$ GeV. Our result is, to a very good approximation, renormalization-scale and $\gamma_5$-scheme independent., Comment: 40 pages, uuencoded LATEX2e file including 13 eps figures, revised version to appear in Nucl. Phys. B

We compute the isospin $I= 0$ and 2 amplitudes for the decay of a kaon into two pions by estimating the relevant hadronic matrix elements in the chiral quark model. The results are parametrized in terms of the quark and gluon condensates and of the constituent quark mass $M$. The latter is a parameter characteristic of the model, that we restrict by matching the results in the two $\gamma_5$-schemes (HV and NDR) of dimensional regularization. We find that, for values of these parameters within the current determinations, the $\Delta I= 1/2$ selection rule is well reproduced by means of the cumulative effects of short-distance NLO Wilson coefficients, penguin diagrams, non-factorizable soft-gluon corrections and meson-loop renormalization., Comment: Revised version to be published in Nuclear Physics B

We use the chiral quark model to estimate the coefficients of the weak chiral lagrangian as obtained from the bosonization of the ten relevant operators of the $\Delta S = 1$ effective quark lagrangian. All contributions of order $N_c^2$ as well as $N_c$ and $\alpha_s N_c$ are included. The chiral coefficients are given in terms of $f_{\pi}$, the quark and gluon condensates and the scale-dependent NLO Wilson coefficients of the corresponding operators. In addition, they depend on the constituent quark mass $M$, a parameter characteristic of the model. The $\gamma_5$-scheme dependence of the chiral coefficients, computed via dimensional regularization, and the Fierz transformation properties of the operator basis are discussed in detail. We apply our result to the evaluation of the hadronic matrix elements for the decay $K^0 \rightarrow 2 \pi$, consistently including the renormalization induced by the meson loops. The effect of this renormalization is sizable and introduces a long-distance scale dependence that matches in the physical amplitudes the short-distance scale dependence of the Wilson coefficients., Comment: Revised version to appear in Nucl. Phys. B 48 pages, uuencoded LATEX file including 4 eps figures

We study the $CP$-violating, $|\Delta S| =1$ parameter $\varepsilon '/\varepsilon$ by computing the hadronic matrix elements in the chiral quark model. We estimate the coefficients of the next-to-leading order terms in the chiral expansion that correspond to the operators $Q_6$ and $Q_{11}$. We consider the impact of these corrections on the value of $\varepsilon '/\varepsilon$. We also investigate the possibility that the chiral quark model might drastically reduce the dimensional regularization $\gamma_5$-scheme dependence of current evaluations of $\varepsilon '/\varepsilon$., Comment: 38 pages, LATEX, including 4 eps figures

The standard model contribution to $\epsilon'/\epsilon$ of the magnetic- and electric-dipole penguin operators $Q_{11} = \frac{g_s}{16\pi^2} \, m_s \: \bar{s}\, \sigma_{\mu\nu} t^a G^{\mu\nu}_a (1-\gamma_5 )\, d $ and $Q_{12} = \frac{eQ_d}{16\pi^2}\, m_s \: \bar{s}\, \sigma_{\mu\nu} F^{\mu\nu} (1-\gamma_5 )\, d $ is discussed. While the electromagnetic penguin $Q_{12}$ seems to have a vanishingly small matrix element, we find that the gluonic dipole operator $Q_{11}$ may give a contribution to $\epsilon'/\epsilon$ comparable to that of other operators so far considered, and should therefore be consistently included in the analysis., Comment: 14 pages, LATEX, CERN-TH.7097/93 and SISSA 174/93/EP, (alternative evaluation of the hadronic matrix element and new tables)