CP Invariance Research Articles

CP violation from finite groups

We discuss the origin of CP violation in settings with a discrete (flavor) symmetry G. We show that physical CP transformations always have to be class-inverting automorphisms of G. This allows us to categorize finite groups into three types: (i) Groups that do not exhibit such an automorphism and, therefore, in generic settings, explicitly violate CP. In settings based on such groups, CP violation can have pure group-theoretic origin and can be related to the complexity of some Clebsch–Gordan coefficients. (ii) Groups for which one can find a CP basis in which all the Clebsch–Gordan coefficients are real. For such groups, imposing CP invariance restricts the phases of coupling coefficients. (iii) Groups that do not admit real Clebsch–Gordan coefficients but possess a class-inverting automorphism that can be used to define a proper (generalized) CP transformation. For such groups, imposing CP invariance can lead to an additional symmetry that forbids certain couplings. We make use of the so-called twisted Frobenius–Schur indicator to distinguish between the three types of discrete groups. With Δ(27), T′, and Σ(72) we present one explicit example for each type of group, thereby illustrating the CP properties of models based on them. We also show that certain operations that have been dubbed generalized CP transformations in the recent literature do not lead to physical CP conservation.

One Antimatter— Two Possible Thermodynamics

Conventional thermodynamics, which is formulated for our world populated by radiation and matter, can be extended to describe physical properties of antimatter in two mutually exclusive ways: CP-invariant or CPT-invariant. Here we refer to invariance of physical laws under charge (C), parity (P) and time reversal (T) transformations. While in quantum field theory CPT invariance is a theorem confirmed by experiments, the symmetry principles applied to macroscopic phenomena or to the whole of the Universe represent only hypotheses. Since both versions of thermodynamics are different only in their treatment of antimatter, but are the same in describing our world dominated by matter, making a clear experimentally justified choice between CP invariance and CPT invariance in context of thermodynamics is not possible at present. This work investigates the comparative properties of the CP- and CPT-invariant extensions of thermodynamics (focusing on the latter, which is less conventional than the former) and examines conditions under which these extensions can be experimentally tested.

BRUNO PONTECORVO AND HIS VISION

The first time I met Bruno Pontecorvo was in 1964 at Dubna where the International High Energy Physics Conference was held. Never before the USSR allowed physicists from USSR to go and visit their colleagues in the East and vice versa. It was a great event (see Fig. 1), Bruno Pontecorvo having played a vital role for its establishment. He was eager to interact with the young fellows engaged in the physics frontiers of the time. During the conference, Cronin presented the discovery of CP violation in K-meson physics. Ten years before the celebrated invariance Laws of Charge Conjugation, C, and Parity, P, were proved to be violated. A theoretical proposal in 1957 by Landau on the conservation of CP invariance appeared to be the solution. On the same year Lee, Oehme and Yang proved that the existence of two neutral K-states K 1 and K 0 2 was not the proof of CP invariance. This paper was neglected by nearly all physicists but Bruno Pontecorvo. At CERN only John Bell and Victor Weisskopf knew this paper and were supporting the proposal to check CP invariance in K decays. Had Bruno Pontecorvo been the then CERN Research Director, the experiment would have been performed at CERN. It should be noticed that the experiment of Christenson et al. was not planned to search for the 2π

Search for CPT and Lorentz invariance violation with neutral kaons at KLOE

The neutral kaon system offers a unique possibility to perform fundamental tests of fundamental symmetry such CP and CPT invariance. In this contribution the KLOE search for CPT symmetry and Lorentz Invariance breaking in the Standard Model Extension framework is reported. Preliminary results are:

-odd fluctuations and long range order in heavy ion collisions. Deformed QCD as a toy model

We study the local violation of P and CP invariance in heavy ion collisions as observed at RHIC and LHC using a simple "deformed QCD" model. This model is a weakly coupled gauge theory, which however has all the relevant crucial elements allowing us to study difficult and nontrivial questions which are known to be present in real strongly coupled QCD. Essentially, we want to understand the physics of long range order in form of coherent low dimensional vacuum configurations observed in Monte Carlo lattice simulations. Apparently precisely such kind of configurations are responsible for sufficiently strong intensity of asymmetries observed in heavy ion collisions.

A gapless hard wall: magnetic catalysis in bulk and boundary

We study various aspects of fermions and their chiral condensates, both in the bulk of AdS4 spacetime and in the dual boundary theory. For the most part, we focus on a geometry with an infra-red hard wall. We show that, contrary to common lore, there exist boundary conditions in which the hard wall gives rise to a discrete, but gapless, fermionic spectrum. In such a setting, the presence of a magnetic field induces a bulk fermion condensate which spontaneously breaks CP invariance. We develop the holographic dictionary between composite operators and show that this bulk condensate has the interpretation of boundary magnetic catalysis involving a double-trace operator. Finally, we explain how one can replace the hard wall with bulk magnetic monopoles. In such a framework, magnetic catalysis can be viewed as a consequence of the Callan-Rubakov effect.

Local violation effects and thermalization in QCD: Views from quantum field theory and holography

We argue that the local violation of P and CP invariance in heavy ion collisions and the universal thermal aspects observed in high energy collisions are in fact two sides of the same coin, and both are related to quantum anomalies of QCD. We argue that the low energy relations representing the quantum anomalies of QCD are saturated by coherent low dimensional vacuum configurations as observed in Monte Carlo lattice studies. The thermal spectrum and approximate universality of the temperature with no dependence on energy of colliding particles in this framework is due to the fact that the emission results from the distortion of these low dimensional vacuum sheets rather than from the colliding particles themselves. The emergence of the long- range correlations of P odd domains (a feature which is apparently required for explanation of the asymmetry observed at RHIC and LHC) is also a result of the same distortion of the QCD vacuum configurations. We formulate the corresponding physics using the effective low energy effective Lagrangian. We also formulate the same physics in terms of the dual holographic picture when low-dimensional sheets of topological charge embedded in 4d space, as observed in Monte Carlo simulations, are identified with D2 branes. Finally, we argue that study of these long range correlations in heavy ion collisions could serve as a perfect test of a proposal that the observed dark energy in present epoch is a result of a tiny deviation of the QCD vacuum energy in expanding universe from its conventional value in Minkowski spacetime.

The Origin of Matter

“If we accept the view of complete symmetry between positive and negative electric charge so far as concerns the fundamental laws of Nature, we must regard it rather as an accident that the Earth (and presumably whole solar system), contains a preponderance of negative electrons and positive protons. . ..” This is a sentence in the Nobel Lecture given by Paul A. M. Dirac in 1933. 1) This is the first proposal for baryon-anti-baryon symmetric universe. However, astrophysical observations have ruled out the presence of antimatter in the present universe up to the scale of clusters of galaxies. If our universe indeed began baryon symmetric, those imbalances in numbers of baryons and anti-baryons must be generated by some physical processes in the early universe. I show, in this talk, that decays of the heavy (right-handed) Majorana neutrinos in the early universe generate the imbalances between baryon and anti-baryon numbers. The presence of super heavy (right-handed) Majorana neutrinos is an important prediction of unified theories beyond the standard model. An integration of the heavy Majorana neutrinos induces very small masses for neutrinos via the seesaw mechanism. 2) Thus, small neutrino masses is a reflection of unification at very high energy scale. It is quite natural to consider that the heavy Majorana neutrinos are produced thermally in the early universe. When the temperature of the universe cools down to the masses of the heavy Majorana neutrinos, the heavy neutrinos begin to decay into Higgs boson and lepton or Higgs boson and anti-lepton. The decay may produce the lepton-number asymmetry if CP invariance is violated in the decay processes. The CP invariance can be easily violated by phases in new complex Yukawa coupling of Higgs boson to the right-handed neutrinos and left-handed leptons. The decay processes can easily be out of thermal equilibrium. Thus, the decay processes of heavy Majorana neutrinos can satisfy three Sakharov conditions 3) for baryogenesis. This produced lepton-number asymmetry is converted into the baryon-number asymmetry in the present universe through non-perturbation effects of the electroweak gauge theory. 5) This is called “leptogenesis”. 4) I show that the neutrino masses observed in various neutrino-oscillation experiments are just in the range predicted by the thermal leptogenesis. 6)

CP violation in charged Higgs production and decays in the Complex Two Higgs Doublet Model

We study the effects of CP violation in charged Higgs boson production pp → tH ± + X at the LHC, as well as in the charged Higgs boson decays H ± → tb and H ± → W ± H i 0, i = 1, 2, 3. The study is done in the framework of the type II complex Two Higgs Doublet Model (2HDM) with softly broken Z 2 symmetry. In this model violation of CP invariance is induced by the complex parameter m 12 2 of the tree-level Higgs potential. We calculate the CP violating rate asymmetries for H + and H − production and decays as well as for the combined processes at one-loop level and perform a detailed numerical analysis. All calculations are done with the automatic amplitude generator FeynArts and the calculational tool FormCalc, for which we have written a complete complex 2HDM model file and relevant fortran drivers. The implementation of the complex 2HDM in FeynArts and FormCalc is described. In comparison with the analogous results in the MSSM, all considered CP violating asymmetries are smaller by an order of magnitude and do not exceed 2 ÷ 3%.

Formula omitted] and [formula omitted] violation and new thermalization scenario in heavy ion collisions

The violation of local P and CP invariance in QCD has been a subject of intense discussions for the last couple of years as a result of very interesting ongoing results coming from RHIC. Separately, a new thermalization scenario for heavy ion collisions through the event horizon as a manifestation of the Unruh effect, has been also suggested. In this paper we argue that these two, naively unrelated phenomena, are actually two sides of the same coin as they are deeply rooted into the same fundamental physics related to some very nontrivial topological features of QCD. We formulate the universality conjecture for P and CP odd effects in heavy ion collisions analogous to the universal thermal behaviour observed in all other high energy interactions.

CP-odd weak basis invariants for neutrino mass matrices with a texture zero and a vanishing minor

We construct the CP-odd weak basis invariants in the flavor basis for all the phenomenologically viable neutrino mass matrices with a texture zero and a vanishing minor and, also, find the necessary and sufficient conditions for CP invariance. We examine the interrelationships between different CP-odd weak basis invariants for these texture structures and investigate their implications for Dirac- and Majorana-type CP violation.

Discrete symmetries and the propagator approach to coupled fermions in Quantum Field Theory. Generalities: The case of a single fermion–antifermion pair

Starting from Wigner’s symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightman’s definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest among coupled fermionic system, the one made with one fermion and its antifermion. Last, we put in correspondence the propagation of C eigenstates (Majorana fermions) and the criteria cast on their propagator by C and CP invariance.

Effects of minimal supersymmetry standard model in neutral K 0 meson mixing

Mixing of neutral K0 mesons is studied in the framework of the minimal supersymmetric model (MSSM) with a Yukawa type II sector and explicit violation of CP invariance in the Higgs potential. Mixing parameters ΔmLS and ɛ are calculated in the limit of the four-fermion approximation with charged Higgs boson exchange. This work specifies the data presented in [1] more precisely and, taking into account the influence of new contributions on the above observables, proves the small degree of supersymmetric effects for a system of K0 mesons.

Fermion masses and mixings in aμ-τsymmetricSO(10)

$\mu$-$\tau$ symmetry imposed on the neutrino mass matrix in the flavour basis is known to be quite predictive. We integrate this very specific neutrino symmetry into a more general framework based on the supersymmetric SO(10) grand unified theory. As in several other models, the fermion mass spectrum is determined by Hermitian mass matrices resulting from the renormalizable Yukawa couplings of the 16-plet of fermions with the Higgs fields transforming as $10, \bar{126},120$ representations of the SO(10) group. The $\mu$-$\tau$ symmetry is spontaneously broken through the 120-plet. Consequences of this scheme are considered for fermion masses using both type-I and type-II seesaw mechanism. This scenario is shown to lead to a generalized CP invariance of the mass matrices and vanishing CP violating phases if the Yukawa couplings are invariant under the $\mu$-$\tau$ symmetry. Small explicit breaking of the $\mu$-$\tau$ symmetry is then shown to provide a very good understanding of all the fermion masses and mixing. Detailed fits to the fermion spectrum are presented in several scenarios. One obtains a very good fit to all observables in the context of the type-I seesaw mechanism but type-II seesaw model also provides a good description except for the overall scale of the neutrino masses. Three major predictions on the leptonic mixing parameters in the type-I seesaw case are (1) the atmospheric mixing angle $\theta_{23}^{l}$ close to maximal, (2) $\theta_{13}^{l}$ close to the present upper bound and (3) negative but very small Dirac CP violating phase in the neutrino oscillations.

Mass splitting of neutral B d,s 0 mesons in the framework of the minimal supersymmetric model

The effects of a minimal supersymmetric model with the Yukawa sector of the second type and explicit violation of CP invariance in the Higgs potential occurring during the mixing of physical states in systems of neutral Bd,s0 mesons are considered. The mixing parameter ΔmLS is calculated in the limit of four-fermion approximation with the exchange of charged Higgs bosons (H±). This study specifies data presented in [1] and establishes new constraints on \( m_{H^ \pm } \) in this model.

Chiral vacuum alignment and spontaneousCPviolation by four-Fermi operators

In models where there is a global chiral symmetry which spontaneously breaks to its vectorial subgroup, the introduction of an explicit symmetry breaking perturbation will define the true vacuum of the theory. This true vacuum is found via the minimization of the expectation value of the perturbing Hamiltonian between different vacua as prescribed by Dashen. The procedure of finding the correct vacuum of the theory may result in the spontaneous breaking of CP symmetry even if one initially demands CP invariance on the perturbation. In this work we study, in detail, models where the perturbation is provided by four-Fermi operators. We present the exact treatment for models with two fermion flavors and study the three flavor case in depth numerically. We show that after the Dashen procedure is applied the solutions for the true vacuum fall in three classes with different CP breaking patterns. Critical transitions are possible between these classes as one varies the parameters of the perturbation. We rigorously show that at these transitions a pseudo-Goldstone boson mass vanishes. We also advocate, and substantiate with numerical statistical analysis for various types of models, that if one imposes CP invariance on the perturbation before solving the vacuum alignment, themore » resulting vacuum structure will have a sizable probability for a light pseudo-Goldstone boson mass. That is a statistical variant of Peccei-Quinn mechanism can be speculated to operate.« less

A new type of CP symmetry, family replication and fermion mass hierarchies

We study a two-Higgs doublet model with four generalised CP symmetries in the scalar sector. Electroweak symmetry breaking leads automatically to spontaneous breaking of two of them. We require that these four CP symmetries can be extended from the scalar sector to the full Lagrangian and call this requirement the principle of maximal CP invariance. The Yukawa interactions of the fermions are severely restricted by this requirement. In particular, a single fermion family cannot be coupled to the Higgs fields. For two fermion families, however, this is possible. Enforcing the absence of flavour-changing neutral currents, we find degenerate masses in both families or one family massless and one massive. In the latter case the Lagrangian is highly symmetric, with the mass hierarchy being generated by electroweak symmetry breaking. Adding a third family uncoupled to the Higgs fields and thus keeping it massless we get a model which gives a rough approximation of some features of the fermions observed in Nature. We discuss a number of predictions of the model which may be checked in future experiments at the LHC.

Discrete accidental symmetry for a particle in a constant magnetic field on a torus

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the 1 / r and r 2 potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge–Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the θ -vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters θ x and θ y explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.

Mixing of K 0 and B 0 neutral mesons within the minimum supersymmetry model

Mixing of K0 and B0 mesons is studied in the scope of the minimum supersymmetry model (MSSM) with a type II Yukawa sector and explicit violation of CP invariance in the Higgs potential. The mixing parameters ΔmLS and ɛ are calculated in the limit of the low-energy four-fermion approximation with a charged Higgs boson exchange. It is shown that supersymmetric effects are very small for K0 mesons and may be quite significant for Bs0 and Bd0 mesons, which imposes constraints on the MSSM parameter space.

Erratum:CPviolation conditions inN-Higgs-doublet potentials [Phys. Rev. D74, 036003 (2006)

Conditions for CP violation in the scalar potential sector of general N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a minimum set of conditions for explicit and spontaneous CP violation is presented. The conditions can be given a clear geometrical interpretation in terms of quantities in the adjoint representation of the basis transformation group for the two doublets. Such conditions depend on CP-odd pseudoscalar invariants. When the potential is CP invariant, the explicit procedure to reach the real CP-basis and the explicit CP transformation can also be obtained. The procedure to find the real basis and the conditions for CP violation are then extended to general NHDM potentials. The analysis becomes more involved and only a formal procedure to reach the real basis is found. Necessary conditions for CP invariance can still be formulated in terms of group invariants: the CP-odd generalized pseudoscalars. The problem can be completely solved for three Higgs-doublets.