Direct CP Violation Research Articles

The \(\varepsilon '/\varepsilon \)-story: 1976–2021

The ratio $\epsilon'/\epsilon$ measures the size of the direct CP violation in $K_L\to\pi\pi$ decays $(\epsilon^\prime)$ relative to the indirect one described by $\epsilon$ and is very sensitive to new sources of CP violation. As such it played a prominent role in particle physics already for 45 years. Due to the smallness of $\epsilon'/\epsilon$ its measurement required heroic efforts in the 1980s and the 1990s on both sides of the Atlantic with final results presented by NA48 and KTeV collaborations 20 years ago. Unfortunately, even 45 years after the first calculation of $\epsilon'/\epsilon$ we do not know to which degree the Standard Model agrees with this data and how large is the room left for new physics contributions to this ratio. This is due to significant non-perturbative (hadronic) uncertainties accompanied by partial cancellation between the QCD penguin contributions and electroweak penguin contributions. While the significant control over the short distance perturbative effects has been achieved already in the early 1990s, with several improvements since then, different views on the non-perturbative contributions to $\epsilon'/\epsilon$ have been expressed by different authors over last thirty years. In fact even today the uncertainty in the room left for NP contributions to $\epsilon'/\epsilon$ is very significant. My own work on $\epsilon'/\epsilon$ started in 1983 and involved both perturbative and non-perturbative calculations. This writing is a non-technical recollection of the steps which led to the present status of $\epsilon'/\epsilon$ including several historical remarks not known to everybody. The present status of the $\Delta I=1/2$ rule is also summarized. This story is dedicated to Jean-Marc Gerard on the occasion of the 35th anniversary of our collaboration and his 64th birthday.

Bd,s0→f1f1 decays with f1(1285)−f1(1420) mixing in the perturbative QCD approach

We investigate the $B_{d,s}^0 \to f_1 f_1$ decays in the framework of perturbative QCD(PQCD) approach with a referenced value $\phi_{f_1} \sim 24^\circ$. Here, $f_1$ denotes the axial-vector mesons $f_1(1285)$ and $f_1(1420)$ with mixing angle $\phi_{f_1}$ in the quark-flavor basis. The observables such as branching ratios, direct CP violations, and polarization fractions of the $B_s^0 \to f_1 f_1$ decays are predicted for the first time. We find that: (i) the almost pure penguin modes $B_s^0 \to f_1 f_1$ have large branching ratios in the order of $10^{-6} \sim 10^{-5}$ due to the Cabibbo-Kobayashi-Maskawa enhancement and generally constructive interferences between the amplitudes of $B_s^0 \to f_n f_s$ and $B_s^0 \to f_s f_s$ with $f_n$ and $f_s$ being the quark-flavor states of $f_1$ mesons; (ii) The observables receive important contributions from the weak annihilation diagrams in the PQCD approach. In particular, without the annihilation contributions, the $B_s^0 \to f_1(1420) f_1(1420)$ branching ratio will decrease about 81% and its longitudinal polarization fraction will reduce around 43%. And (iii) the dependence of the $B_{d,s}^0 \to f_1 f_1$ decay rates on $\phi_{f_1}$ exhibits some interesting line shapes, whose confirmations would be helpful to constrain the determination of $\phi_{f_1}$ inversely. All the PQCD predictions await for the (near) future examinations at Large Hadron Collider beauty and/or Belle-II experiments to further understand the properties of the axial-vector mesons and the perturbative dynamics released from the considered decay modes.

Direct CP violation and the ΔI=1/2 rule in K→ππ decay from the standard model

We present a lattice QCD calculation of the $\Delta I=1/2$, $K\to\pi\pi$ decay amplitude $A_0$ and $\varepsilon'$, the measure of direct CP-violation in $K\to\pi\pi$ decay, improving our 2015 calculation of these quantities. Both calculations were performed with physical kinematics on a $32^3\times 64$ lattice with an inverse lattice spacing of $a^{-1}=1.3784(68)$ GeV. However, the current calculation includes nearly four times the statistics and numerous technical improvements allowing us to more reliably isolate the $\pi\pi$ ground-state and more accurately relate the lattice operators to those defined in the Standard Model. We find ${\rm Re}(A_0)=2.99(0.32)(0.59)\times 10^{-7}$ GeV and ${\rm Im}(A_0)=-6.98(0.62)(1.44)\times 10^{-11}$ GeV, where the errors are statistical and systematic, respectively. The former agrees well with the experimental result ${\rm Re}(A_0)=3.3201(18)\times 10^{-7}$ GeV. These results for $A_0$ can be combined with our earlier lattice calculation of $A_2$ to obtain ${\rm Re}(\varepsilon'/\varepsilon)=21.7(2.6)(6.2)(5.0) \times 10^{-4}$, where the third error represents omitted isospin breaking effects, and Re$(A_0)$/Re$(A_2) = 19.9(2.3)(4.4)$. The first agrees well with the experimental result of ${\rm Re}(\varepsilon'/\varepsilon)=16.6(2.3)\times 10^{-4}$. A comparison of the second with the observed ratio Re$(A_0)/$Re$(A_2) = 22.45(6)$, demonstrates the Standard Model origin of this "$\Delta I = 1/2$ rule" enhancement.

Probing the top Yukawa coupling at the LHC via associated production of single top and Higgs

We study Higgs boson production associated with single top or anti-top via t-channel weak boson exchange at the LHC. The process is an ideal probe of the top quark Yukawa coupling, because we can measure the relative phase of htt and hWW couplings, thanks to the significant interference between the two amplitudes. By choosing the emitted W momentum along the polar axis in the th ( overline{t}h ) rest frame, we obtain the helicity amplitudes for all the contributing subprocesses analytically, with possible CP phase of the Yukawa coupling. We study the azimuthal asymmetry between the W emission and the Wb ( overline{b} ) → t( overline{t} ) h scattering planes, as well as several t and overline{t} polarization asymmetries as a signal of CP violating phase in the htt coupling. Both the azimuthal asymmetry and the polarization perpendicular to the scattering plane are found to have the opposite sign between the top and anti-top events. We identify the origin of the sign of asymmetries, and propose the possibility of direct CP violation test in pp collisions by comparing the top and anti-top polarization at the LHC.

Studying the localized CP violation and the branching fraction of the decay * * Supported by National Natural Science Foundation of China (11575023, 11775024, 11947001, 11605150), the Fundamental Research Funds for the Provincial Universities of Zhejiang Province and Ningbo Natural Science Foundation (2019A610067)

In this work, we study the localized violation and the branching fraction of the four-body decay by employing a quasi-two-body QCD factorization approach. Considering the interference of and channels, we predict and , respectively, which shows that the interference mechanism of these two channels can induce the localized violation to this four-body decay. Meanwhile, within the two quark model framework for the scalar mesons and , we calculate the direct CP violations and branching fractions of the and decays, respectively. The corresponding results are , , and , indicating that the violations of these two-body decays are both positive and the branching fractions quite different. These studies provide a new way to investigate the aforementioned four-body decay and can be helpful in clarifying the configuration of the structure of the light scalar meson.

Dealing with T and CPT violations in mixing as well as direct and indirect CP violations for neutral mesons decaying to two vectors

A large number of observables can be constructed from differential decay rate based on the polarization of final state while considering decay of a neutral meson (P^{0} text { or } {bar{P}}^{0}) to two vector particles. But all of these observables are not independent to each other since there are only a few independent theoretical parameters controlling the whole dynamics and therefore various relations among observables emerge. In this paper, we have studied the behaviour of observables for neutral meson decaying to two vectors in presence of T and C!P T violations in mixing accompanied by both direct and indirect C!P violations. We have expressed all of the fourteen unknown theoretical parameters for this scenario in terms observables only and constructed the complete set of thirty four relations among observables whose violation would signify the existence of some new Physics involving direct violation of C!P T. In addition, using this formalism we have studied three special cases too: (a) SM scenario, (b) SM plus direct C!P violation (c) SM plus T and C!P T violation in mixing.

Isospin-breaking in varepsilon '/varepsilon : impact of eta _0 at the dawn of the 2020s

For direct CP-violation in Krightarrow pi pi decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the Delta I=1/2 rule and conventionally encoded in Omega _mathrm{IB} parameters. The updated prediction Omega _mathrm{IB}^{(8)}=(15.9pm 8.2)times 10^{-2} of the Chiral Perturbation Theory for the strong isospin-breaking due to pi _3-eta _8 mixing confirms such a tendency but is quite sensitive to the theoretical input value of the low-energy constant corresponding to the flavour-singlet eta _0 exchange contribution in this truncated octet scheme. We rather exploit the phenomenological eta _8-eta _0 mixing as a probe for the non-negligible flavour-singlet component of the physical eta pole to find Omega _mathrm{IB}^{(9)}=(35pm 7)times 10^{-2} in a complete nonet scheme. A large central value in the nonet scheme is thus substituted for a large uncertainty in the octet one. Including the experimental pi ^+-pi ^0 mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio varepsilon '/varepsilon an improved result hat{Omega }_mathrm{eff}^{(9)}=(29pm 7)times 10^{-2} to be compared with hat{Omega }_mathrm{eff}^{(8)}=(17pm 9)times 10^{-2} used in recent analyses of varepsilon '/varepsilon . Accordingly, we get a reduction from (varepsilon '/varepsilon )_text {SM}^{(8)}=(17.4pm 6.1)times 10^{-4} to (varepsilon '/varepsilon )_text {SM}^{(9)}=(13.9pm 5.2)times 10^{-4} and thereby an effective suppression of (varepsilon '/varepsilon )_text {SM} by isospin-breaking corrections as large as 40% relative to the recent RBC-UKQCD value.

A Goldilocks Higgs

The Higgs may couple to a topological 4-form sector which yields a complex vacuum structure. In many of the resulting Higgs vacua electroweak symmetry is unbroken. In just as many it breaks when the 4-form flux is large enough. For a fixed value of flux, the symmetry breaking vacua have a smaller vacuum energy than the symmetric ones, where the difference is quantized because it is set by the 4-form flux. This leads to the possibility that there is a value of the 4-form flux for any UV contributions to the Higgs vev that automatically cancels it down to the right value, ∼ TeV, if the 4-form charges are quantized in the units of the electroweak scale. This would still leave the cosmological constant which could be selected anthropically. In addition, the 4-form couplings could lead to direct CP violation in a non-minimal Higgs sector.

Direct CP violation in D^+ rightarrow K^0({bar{K}}^0) pi ^+ decays as a probe for new physics

In this paper we investigate CP violation in charged decays of D meson. Particularly, we study the direct CP asymmetry of the Cabibbo favored non-leptonic D^+ rightarrow {bar{K}}^0 pi ^+ and the doubly Cabibbo-suppressed decay mode D^+ rightarrow K^0 pi ^+ within standard model, two Higgs doublet model with generic Yukawa structure and left right symmetric models. In the standard model, we first derive the contributions from box and di-penguin diagrams contributing to their amplitudes which are relevant to the generation of the weak phases essential for non-vanishing direct CP violation. Then, we show that the generated phases are so tiny leading to null direct CP asymmetries of both decay modes. Regarding the two Higgs doublet model with generic Yukawa structure, after taking into account all constraints on the parameter space of the model, we show that the weak phases of the amplitudes can be enhanced compared to the standard model ones. However, the enhancement is still not enough to have sizable direct CP asymmetries. Finally, within left right symmetric models, we find that |A^{SM+LR}_{CP} (D^+ rightarrow {bar{K}}^0 pi ^+)|lesssim mathcal {O}(10^{-3}) after respecting all relevant constraints on the parameter space of the model.

Exploiting CP asymmetries in rare charm decays

We analyze patterns from $CP$--violating new physics (NP) in hadronic and semileptonic rare charm $|\mathrm{\ensuremath{\Delta}}c|=|\mathrm{\ensuremath{\Delta}}u|=1$ transitions. Observation of direct $CP$ violation in hadronic decays, as in $\mathrm{\ensuremath{\Delta}}{A}_{CP}$, provides opportunities for $c\ensuremath{\rightarrow}u{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$, $\ensuremath{\ell}=e,\ensuremath{\mu}$ transitions, and vice versa. For the concrete case of flavorful, anomalyfree ${Z}^{\ensuremath{'}}$--models a NP--interpretation of $\mathrm{\ensuremath{\Delta}}{A}_{CP}$ suggests measurable $CP$ asymmetries in semileptonic decays such as $D\ensuremath{\rightarrow}\ensuremath{\pi}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ or $D\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$. Conversely, an observation of $CP$ violation in $c\ensuremath{\rightarrow}u{e}^{+}{e}^{\ensuremath{-}}$ or $c\ensuremath{\rightarrow}u{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ decays supports a NP--interpretation of $\mathrm{\ensuremath{\Delta}}{A}_{CP}$. Flavorful $U(1{)}^{\ensuremath{'}}$--extensions provide explicit U--spin and isospin breaking which can be probed in patterns of hadronic decays of charm mesons. We work out signatures for $CP$ asymmetries in ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ and ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$, ${D}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ decays, which can be probed in the future at LHCb and Belle II and provide further informative cross checks.

Direct CP violation in internal W-emission dominated baryonic B decays

The observation of CP violation has been experimentally verified in numerous B decays but is yet to be confirmed in final states with half-spin particles. We focus our attention on baryonic B-meson decays mediated dominantly through internal W-emission processes and show that they are promising processes to observe for the first time the CP violating effects in B decays to final states with half-spin particles. Specifically, we study the bar{B}^0rightarrow pbar{p}pi ^0(rho ^0) and bar{B}^0rightarrow pbar{p}pi ^+pi ^- decays. We obtain mathcal{B}(bar{B}^0rightarrow pbar{p}pi ^0)=(5.0pm 2.1)times 10^{-7}, in agreement with current data, and mathcal{B}(bar{B}^0rightarrow pbar{p}rho ^0)simeq mathcal{B}(bar{B}^0rightarrow pbar{p}pi ^0)/3. Furthermore, we find mathcal{A}_{CP}(bar{B}^0rightarrow pbar{p}pi ^0,pbar{p}rho ^0,pbar{p}pi ^+pi ^-) =(-16.8pm 5.4,-12.6pm 3.0,-11.4pm 1.9)%. With measured branching fractions mathcal{B}(bar{B}^0rightarrow pbar{p}pi ^0,pbar{p}pi ^+pi ^-)sim mathcal{O}(10^{-6}), we point out that mathcal{A}_{CP}sim -(10-20)% can be new observables for CP violation, accessible to the Belle II and/or LHCb experiments.

Z′ models with less-minimal flavor violation

We study the phenomenology of simplified $Z^\prime$ models with a global $U(2)^3$ flavour symmetry in the quark sector, broken solely by the Standard Model Yukawa couplings. This flavour symmetry, known as less-minimal flavour violation, protects $\Delta F=2$ processes from dangerously large new physics (NP) effects, and at the same time provides a free complex phase in $b\to s$ transitions, allowing for an explanation of the hints for additional direct CP violation in kaon decays ($\epsilon^\prime/\epsilon$) and in hadronic $B$-decays ($B\to K\pi$ puzzle). Furthermore, once the couplings of the $Z^\prime$ boson to the leptons are included, it is possible to address the intriguing hints for NP (above the 5$\,\sigma$ level) in $b\to s \ell^+\ell^-$ transitions. Taking into account all flavour observables in a global fit, we find that $\epsilon^\prime/\epsilon$, the $B\to K\pi$ puzzle and $b\to s \ell^+\ell^-$ data can be explained simultaneously. Sizeable CP violation in $b\to s \ell^+\ell^-$ observables, in particular $A_8$, is predicted, which can be tested in the near future, and an explanation of the $B\to K\pi$ and $\epsilon^\prime/\epsilon$ puzzles leads to effects in di-jet tails at the LHC, that are not far below the current limits. Once $b\to s \ell^+\ell^-$ is included, cancellations in di-muon tails, possibly by a second $Z^\prime$, are required by LHC data.

K → ππ decay, ε ′ and the RBC-UKQCD kaon physics program

An overview of the kaon physics program of the RBC and UKQCD collaborations will be presented with a focus on the lattice calculation of K → ππ decay and the direct CP violation parameter ε ′. We will describe substantial improvements to our earlier 2015 K → ππ calculation, including the use of three independent ππ interpolating operators and the results we obtain for I = 0 ππ scattering for energies at and below the kaon mass. While the new result for ϵ ′ is not yet complete, the enhanced statistics and improved analysis that underlies our expanded calculation will be presented.

Charged Kaons semi-leptonic form factors from NA48/2

The NA48/2 experiment collected unprecedent statistics of charged kaon decays in 2003 and 2004. The main purpose of the experiment was to measure direct CP violation in kaon decays to three pions while an extensive physics program was carried out together with the main goal. The huge statistics collected allowed to test with a high accuracy the predictions of low energy QCD theories in several charged kaon decay processes. The precise measurement of the charged kaon semileptonic form factors are reported here, based on data collected in 2004 with a dedicated minimum bias trigger.

Measurement of CP violation in the {B}_s^0to phi phi decay and search for the B0\u2192 \u03d5\u03d5 decay

A measurement of the time-dependent CP-violating asymmetry in {B}_s^0to phi phi decays is presented. Using a sample of proton-proton collision data corresponding to an integrated luminosity of 5.0 fb−1 collected by the LHCb experiment at centre-of-mass energies sqrt{s} = 7 TeV in 2011, 8 TeV in 2012 and 13 TeV in 2015 and 2016, a signal yield of around 9000 {B}_s^0to phi phi decays is obtained. The CP-violating phase {phi}_s^{soverline{s}s} is measured to be −0.073 ± 0.115(stat) ± 0.027(syst) rad, under the assumption it is independent of the helicity of the ϕϕ decay. In addition, the CP-violating phases of the transverse polarisations under the assumption of CP conservation of the longitudinal phase are measured. The helicity-independent direct CP-violation parameter is also measured, and is found to be |λ| = 0.99 ± 0.05(stat) ± 0.01(syst). In addition, T-odd triple-product asymmetries are measured. The results obtained are consistent with the hypothesis of CP conservation in overline{b}to overline{s}soverline{s}s transitions. Finally, a limit on the branching fraction of the B0 → ϕϕ decay is determined to be mathcal{B}left({B}^0to phi phi right)<2.7times {10}^{-8} at 90 % confidence level.

Revisiting CP violation in D→PP and VP decays

Direct CP violation in the hadronic charm decays provides a good testing ground for the Kobayashi-Maskawa mechanism in the Standard Model. Any significant deviations from the expectation would be indirect evidence of physics beyond the Standard Model. In view of improved measurements from LHCb and BESIII experiments, we re-analyze the Cabibbo-favored $D \to P\!P$ and $V\!P$ decays in the topological diagram approach. By assuming certain SU(3)-breaking effects in the tree-type amplitudes, we make predictions for both branching fractions and CP asymmetries of the singly Cabibbo-suppressed decay modes. While the color-allowed and -suppressed amplitudes are preferred to scale by the factor dictated by factorization in the $P\!P$ modes, no such scaling is required in the $V\!P$ modes. The $W$-exchange amplitudes are found to change by 10\% to 50\% and depend on whether $d\overline{d}$ or $s\overline{s}$ pair directly emerges from $W$-exchange. The predictions of branching fractions are generally improved after these SU(3) symmetry breaking effects are taken into account. We show in detail how the tree-type, QCD-penguin, and weak penguin-annihilation diagrams contribute and modify CP asymmetry predictions. Future measurements of sufficiently many direct CP asymmetries will be very useful in removing a discrete ambiguity in the strong phases as well as discriminating among different theory approaches. In particular, we predict $a_{CP}(K^+K^-)-a_{CP}(\pi^+\pi^-) = (-1.14 \pm 0.26) \times 10^{-3}$ or $(-1.25 \pm 0.25) \times 10^{-3}$, consistent with the latest data, and $a_{CP}(K^+K^{*-})-a_{CP}(\pi^+\rho^-) = (-1.52 \pm 0.43) \times 10^{-3}$, an attractive and measurable observable in the near future. Moreover, we observe that such CP asymmetry differences are dominated by long-distance penguin-exchange through final-state rescattering.

A U-spin prediction for the CP-forbidden transition e+e−→ D0D̄0 → (K+K−)D(π+π−)D

The LHCb Collaboration has recently reported the discovery of direct CP violation in combined [Formula: see text] and [Formula: see text] decay modes at the [Formula: see text] level. Assuming U-spin symmetry (i.e. [Formula: see text] interchange symmetry) for the strong-interaction parts of these two channels, we find that their corresponding direct CP-violating asymmetries are [Formula: see text] and [Formula: see text]. The CP-forbidden transition [Formula: see text] on the [Formula: see text] resonance is therefore expected to have a branching fraction of [Formula: see text] or smaller under U-spin symmetry, and it can be observed at a high-luminosity super-[Formula: see text]-charm factory if at least [Formula: see text] pairs of coherent [Formula: see text] and [Formula: see text] events are accumulated.

The direct CP violation in a general two Higgs doublet model

In this paper, we study the CP violating processes in a general two-Higgs-doublet model (2HDM) with tree-level flavor changing neutral currents. In this model, sizable Yukawa couplings involving top and charm quarks are still allowed by the collider and flavor experiments, while the other couplings are strongly constrained experimentally. The sizable couplings, in general, have imaginary parts and could largely contribute to the CP violating observables concerned with the B and K mesons. In particular, the contribution may be so large that it affects the direct CP violating K meson decay, where the discrepancy between the experimental result and the Standard Model prediction is reported. We discuss how well the anomaly is resolved in the 2HDM, based on study of the other flavor observables. We also propose the way to test our 2HDM at the LHC.

Studies on the decays in the perturbative QCD approach**Research exercises for excellent undergraduate students.

The decays are investigated for the first time in the perturbative QCD formalism based on the kT factorization theorem, where the light scalar κ is assumed as a two-quark state. Our numerical results and phenomenological analyses on the CP-averaged branching ratios and CP-violating asymmetries show that: (a) the and decays have large decay rates around , which could be examined by the upgraded large hadron collider beauty and/or Belle-II experiments in the near future; (b) a large decay rate of about appears in the pure annihilation channel, which could provide more evidence to help distinguish different QCD-inspired factorization approaches, even understand the annihilation decay mechanism; (c) the pure penguin modes and would provide a promising ground to search for the possible new physics because of their zero direct and mixing-induced CP violations in the standard model. The examinations with good precision from the future experiments will help to further study the perturbative and/or nonperturbative QCD dynamics involved in these considered decay modes.

Left-handed color-sextet diquark in the kaon system

We investigate whether a color-sextet scalar diquark (${\bf H}_6$) coupling to the left-handed quarks contributes to the $\Delta S=2$ process. It is found that the box diagrams mediated by $W$ and ${\bf H}_6$ bosons have no contributions to $\Delta S=2$ when the limit of $m_t=0$ is used, and the flavor mixing matrices for diagonalizing quark mass matrices are introduced at the same time. When the heavy top-quark mass effects are taken into account, it is found that in addition to the $W-{\bf H}_6$ box diagrams significantly contributing to $\Delta S=2$, their effects can be as large as those from the ${\bf H}_6-{\bf H}_6$ box diagrams. Using the parameters that are constrained by the $K^0-\bar K^0$ mixing parameter $\Delta M_K$ and the Kaon indirect CP violation $\epsilon_K$, we find that the left-handed color-sextet diquark can lead to the Kaon direct CP violation being $Re(\epsilon'/\epsilon) \sim 0.4 \times 10^{-3}$. In the chosen scheme, although the diquark contribution to $K_L\to \pi^0 \nu \bar\nu$ is small, the branching ratio of $K^+ \to \pi^+ \nu \bar\nu$ can reach the current experimental upper bound.