Advances in High Energy Physics

Volume 2018, Article ID 7627308, 11 pages

https://doi.org/10.1155/2018/7627308

## Analysis of Violation in

^{1}School of Nuclear Science and Technology, University of South China, Hengyang, Hunan 421001, China^{2}Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany

Correspondence should be addressed to Bo Zheng; moc.361@csu_obgnehz and Zhen-Hua Zhang; nc.ude.csu@hzgnahz

Received 21 August 2018; Accepted 3 October 2018; Published 22 November 2018

Guest Editor: Tao Luo

Copyright © 2018 Hang Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the violation induced by the interference between two intermediate resonances and in the phase space of singly-Cabibbo-suppressed decay . We adopt the factorization-assisted topological approach in dealing with the decay amplitudes of . The asymmetries of two-body decays are predicted to be very tiny, which are and , respectively, for and , while the differential asymmetry of is enhanced because of the interference between the two intermediate resonances, which can reach as large as . For some NPs which have considerable impacts on the chromomagnetic dipole operator , the global asymmetries of and can be then increased to and , respectively. The regional asymmetry in the overlapped region of the phase space can be as large as .

#### 1. Introduction

Charge-Parity () violation, which was first discovered in meson system in 1964 [1], is one of the most important phenomena in particle physics. In the Standard Model (SM), violation originates from the weak phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2, 3] and the unitary phases which usually arise from strong interactions. One reason for the smallness of violation is that the unitary phase is usually small. Nevertheless, violation can be enhanced in three-body decays of heavy hadrons, when the corresponding decay amplitudes are dominated by overlapped intermediate resonances in certain regions of phase space. Owing to the overlapping, a regional asymmetry can be generated by a relative strong phase between amplitudes corresponding to different resonances. This relative strong phase has nonperturbative origin. As a result, the regional asymmetry can be larger than the global one. In fact, such kind of enhanced violation has been observed in several three-body decay channels of meson [4–7], which was followed by a number of theoretical works [8–19].

The study of violation in singly-Cabibbo-suppressed (SCS) meson decays provides an ideal test of the SM and exploration of New Physics (NP) [20–23]. In the SM, violation is predicted to be very small in charm system. Experimental researches have shown that there is no significant violation so far in charmed hadron decays [24–33]. asymmetry in SCS meson decay can be as small asor even less, due to the suppression of the penguin diagrams by the CKM matrix as well as the smallness of Wilson coefficients in penguin amplitudes. The SCS decays are sensitive to new contributions to the QCD penguin and chromomagnetic dipole operators, while such contributions can affect neither the Cabibbo-favored (CF) nor the doubly-Cabibbo-suppressed (DCS) decays [34]. Besides, the decays of charmed mesons offer a unique opportunity to probe violation in the up-type quark sector.

Several factorization approaches have been wildly used in nonleptonic decays. In the naive factorization approach [35, 36], the hadronic matrix elements were expressed as a product of a heavy to light transition form factor and a decay constant. Based on Heavy Quark Effect Theory, it is shown in the QCD factorization approach that the corrections to the hadronic matrix elements can be expressed in terms of short-distance coefficients and meson light-cone distribution amplitudes [37, 38]. Alternative factorization approach based on QCD factorization is often applied in study of quasi two-body hadronic decays [19, 39, 40], where they introduced unitary meson-meson form factors, from the perspective of unitarity, for the final state interactions. Other QCD-inspired approaches, such as the perturbative approach (pQCD) [41] and the soft-collinear effective theory (SCET) [42], are also wildly used in meson decays.

However, for meson decays, such QCD-inspired factorization approaches may not be reliable since the charm quark mass, which is just above 1 GeV, is not heavy enough for the heavy quark expansion [43, 44]. For this reason, several model-independent approaches for the charm meson decay amplitudes have been proposed, such as the flavor topological diagram approach based on the flavor symmetry [44–47] and the factorization-assisted topological-amplitude (FAT) approach with the inclusion of flavor breaking effect [48, 49]. One motivation of these aforementioned approaches is to identify as complete as possible the dominant sources of nonperturbative dynamics in the hadronic matrix elements.

In this paper, we study the violation of SCS meson decay in the FAT approach. Our attention will be mainly focused on the region of the phase space where two intermediate resonances, and , are overlapped. Before proceeding, it will be helpful to point out that direct asymmetry is hard to be isolated for decay process with -eigen-final-state. When the final state of the decay process is eigenstate, the time integrated violation for , which is defined ascan be expressed as [34]where , , and are the asymmetries in decay, in mixing, and in the interference of decay and mixing, respectively. As is shown in [34, 50, 51], the indirect violation is universal and channel-independent for two-body -eigenstate. This conclusion is easy to be generalized to decay processes with three-body -eigenstate in the final state, such as . In view of the universality of the indirect asymmetry, we will only consider the direct violations of the decay throughout this paper.

The remainder of this paper is organized as follows. In Section 2, we present the decay amplitudes for various decay channels, where the decay amplitudes of are formulated via the FAT approaches. In Section 3, we study the asymmetries of and the asymmetry of induced by the interference between different resonances in the phase space. Discussions and conclusions are given in Section 4. We list some useful formulas and input parameters in the Appendix.

#### 2. Decay Amplitude for

In the overlapped region of the intermediate resonances and in the phase space, the decay process is dominated by two cascade decays, and , respectively. Consequently, the decay amplitude of can be expressed asin the overlapped region, where and are the amplitudes for the two cascade decays and is the relative strong phase. Note that nonresonance contributions have been neglected in (4).

The decay amplitude for the cascade decay can be expressed aswhere and represent the amplitudes corresponding to the strong decay and weak decay , respectively, is the helicity index of , is the invariant mass square of system, and and are the mass and width of , respectively. The decay amplitude for the cascade decay, , is the same as (5) except replacing the subscripts and with and , respectively.

For the strong decays , one can express the decay amplitudes aswhere and represent the momentum for and mesons, respectively, and is the effective coupling constant for the strong interaction, which can be extracted from the experimental data viawithand . The isospin symmetry of the strong interaction implies that .

The decay amplitudes for the weak decays, and , will be handled with the aforementioned FAT approach [48, 49]. The relevant topological tree and penguin diagrams for are displayed in Figure 1, where and denote a light pseudoscalar and vector meson (representing and in this paper), respectively.