Triangle singularities in $\bar{B}^0\to χ_{c1}K^-π^+$ relevant to $Z_1(4050)$ and $Z_2(4250)$
Satoshi X. Nakamura
备注：
6 pages, 5 figures, (v2) numerical results corrected; changes are small, discussion on B0 -> J/psi K pi added; (v3) published version: relativistic energy-momentum relation is used for all particles; study on cutoff-dependence added
$Z_1(4050)$ and $Z_2(4250)$ observed in $\bar{B}^0\toχ_{c1}K^-π^+$ by the Belle Collaboration are candidates of charged charmonium-like states that minimally includes two quarks and two antiquarks. While $Z_1(4050)$ and $Z_2(4250)$ have been interpreted as tetraquark states previously, we propose a completely different scenario based on a kinematical effect called the triangle singularity. We demonstrate that the triangle singularities cause in the $χ_{c1}π^+$ invariant mass distribution resonance-like bumps that fit very well the Belle data. If these bumps are simulated by the $Z_1(4050)$ and $Z_2(4250)$ resonance excitations, the spin-parity of them are predicted to be $1^-$ for $Z_1(4050)$ and $1^+$ or $1^-$ for $Z_2(4250)$. The bump corresponding to $Z_1(4050)$ has a highly asymmetric shape, which the Belle data exactly indicate. We show that the asymmetric shape originates from an interplay between the triangle singularity and the opening of the $X(3872)π^+$ channel near the triangle-singularity energy. This characteristic lineshape could be used to discriminate different interpretations of $Z_1(4050)$. An interesting prediction from interpretting $Z_1(4050)$ and $Z_2(4250)$ as the triangle singularities is that similar bumps caused by the same mechanisms possibly appear also in $\bar{B}^0\to J/ψK^-π^+$ data; the already observed $Z_c(4200)$ corresponds to $Z_2(4250)$ of $J^P=1^+$.
The spectrum of hadrons is important for understanding the confinement of quantum chromodynamics. Many new puzzles arose since 2003 due to the abundance of experimental discoveries with the $XYZ$ structures in the heavy quarkonium mass region being the outstanding examples. Hadronic resonances correspond to poles of the $S$-matrix, which has other types of singularities such as the triangle singularity due to the simultaneous on-shellness of three intermediate particles. Here we briefly discuss a few possible manifestations of triangle singularities in the $XYZ$ physics, paying particular atte…
The knowledge of hadron spectrum is based on experimental observations of hadronic resonances. The resonances are normally observed as peaks in certain invariant mass distributions. However, neither is a peak necessarily due to the presence of a resonance, nor does a resonance necessarily lead to a peak. Kinematic singularities can also produce peaks. Here, we discuss such possibilities and methods distinguishing genuine resonances from kinematic effects.
The spectrum of hadrons is the manifestation of color confinement of quantum chromodynamics. Hadronic resonances correspond to poles of the S-matrix. Since 2003, lots of new hadron resonant structures were discovered in the mass regions from light mesons to hadrons containing a pair of a heavy quark and an antiquark. Many of them are candidates of exotic hadrons, and they are usually observed as peaks in invariant mass distributions. However, the S-matrix also has kinematical singularities due to the on-shellness of intermediate particles for a process, such as two-body thresholds and triangle…
We investigate the anomalous triangle singularity (ATS) and its possible manifestations in various processes. We show that the ATS should have important impact on our understanding of the nature of some newly observed threshold states. Discussions on how to distinguish the ATS phenomena from genuine dynamic pole structures are presented.
$Z_c(4430)$ discovered by the Belle and confirmed by the LHCb in $\bar{B}^0\toψ(2S)K^-π^+$ is generally considered to be a charged charmonium-like state that includes minimally two quarks and two antiquarks. $Z_c(4200)$ found in $\bar{B}^0\to J/ψK^-π^+$ by the Belle is also a good candidate of a charged charmonium-like state. We demonstrate that kinematical singularities in triangle loop diagrams induce a resonance-like behavior that can consistently explain the properties (mass, width, and Argand plot) of $Z_c(4430)$ and $Z_c(4200)$ from the experimental analyses. The triangle diagrams includ…
$Z_c(4430)$ discovered in $\bar{B}^0\toψ(2S)K^-π^+$, $Z_c(4200)$ found in $\bar{B}^0\to J/ψK^-π^+$, and $Z_1(4050)$ and $Z_2(4250)$ observed in $\bar{B}^0\toχ_{c1}K^-π^+$ are candidates of charged charmonium-like states. All surviving theoretical models interpreted these candidates as four-quark states, until we recently identified a compelling alternative. We discuss that kinematical singularities in triangle loop diagrams induce a resonance-like behavior that can consistently explain the properties (such as spin-parity, mass, width, and Argand plot) of $Z_c(4430)$, $Z_c(4200)$, $Z_1(4050)$ a…
The values of the masses of the particles involved in the decay of $T^+_{cc}\to D^{*+}D^0\toπ^+D^0D^0$ suggest that due to the final state interactions in the transition vertex $T^+_{cc}\to D^{*+}D^0 $ there may be triangle logarithmic singularities. We discuss their possible role and show that the tree approximation for calculating the decay widths $T^+_{cc}\to(D^{*+}D^0+D^{*0}D^+)\toπ^+D^0D^0$, $π^0D^0 D^+$, $γD^0D^+$ is quite sufficient at the current level of measurement accuracy.
We predict that triangle singularities of hadron spectroscopy can be strongly affected in heavy ion collisions. To do it we examine various effects on the singularity-inducing triangle loop of finite temperature in the terminal hadron phase. It appears that peaks seen in central heavy ion collisions are more likely to be hadrons than rescattering effects under two conditions. First, the flight-time of the intermediate hadron state must be comparable to the lifetime of the equilibrated fireball (else, the reaction mostly happens in vacuo after freeze out). Second, the medium effect over the tri…
We discuss properties of three-particle Dalitz distributions in coupled channel systems in presence of triangle singularities. The single channel case was discussed long ago where it was found that as a consequence of unitarity, effects of a triangle singularity seen in the Dalitz plot are not seen in Dalitz plot projections. In the coupled channel case we find the same is true for the sum of intensities of all interacting channels. Unlike the single channel case, however, triangle singularities do remain visible in Dalitz plot projections of individual channels.
It was proposed that the narrow $P_c(4450)$ structure observed by the LHCb Collaboration in the reaction $Λ_b\to J/ψp K$ might be due to a triangle singularity around the $χ_{c1}$--proton threshold at 4.45~GeV. We discuss the occurrence of a similar triangle singularity in the $J/ψp$ invariant mass distribution for the decay $Λ_b\to J/ψp π$, which could explain the bump around 4.45~GeV in the data. More precise measurements of this process would provide valuable information towards an understanding of the $P_c$ structures.
We analyze possible singularities in the $J/ψΛ$ invariant mass distribution of the $Ξ^-_{b}~\to~K^- J/ψΛ$ process via triangle loop diagrams. Triangle singularities in the physical region are found in 18 different triangle loop diagrams. Among those with $Ξ^*$-charmonium-$Λ$ intermediate states, the one from the $χ_{c1} Ξ(2120) Λ$ loop, which is located around 4628 MeV, is found the most likely to cause observable effects. One needs $S$- and $P$-waves in $χ_{c1} Λ$ and $J/ψΛ$ systems, respectively, when the quantum numbers of these systems are $1/2^+$ or $3/2^+$. When the quantum numbers of th…
We discuss analytical properties of partial waves derived from projection of a 4-legged amplitude with crossed-channel exchanges in the kinematic region of the direct channel that corresponds to the XYZ peaks in charmonium and bottomonium. We show that in general partial waves can develop anomalous branch points in the vicinity of the direct channel physical region. In a specific case, when these branch points lie on the opposite side of the unitary cut they pinch the integration contour in a dispersion relation and if the pinch happens close to threshold, the normal threshold cusp is enhanced…
We discuss dispersion representations for the triangle diagram $F(p_1^2,p_2^2,q^2)$, the single dispersion representation in $q^2$ and the double dispersion representation in $p_1^2$ and $p_2^2$, with special emphasis on the appearance of the anomalous singularities and the anomalous cuts in these representations. For the double dispersion representation in $p_1^2$ and $p_2^2$, the appearance of the anomalous cut in the region $q^2>0$ is demonstrated, and a new derivation of the anomalous double spectral density is given. We point out that the double spectral representation is particularly sui…
We demonstrate that triangle singularity (TS) and box singularity (BS) mechanisms can produce unique narrow enhancements at the $Λ_c\bar{D}$ and $Λ_c\bar{D}^*$ thresholds in the invariant mass spectra of $J/ψp$ and $J/ψpπ$, respectively. Taking into account that such mechanisms only depend on the initial $Σ_c^{(*)}\bar{D}^{(*)}$ interactions near threshold within the TS or BS kinematic regimes, the $Λ_c\bar{D}$ and $Λ_c\bar{D}^*$ threshold enhancements can be regarded as a feed-down phenomenon originated from both the heavier pentaquark decays and the $Σ_c^{(*)}\bar{D}^{(*)}$ scatterings from …
The possible role of the triangle mechanism in the $B^-$ decay into $D^{*0}π^-π^0η$ and $D^{*0}π^-π^+π^-$ is investigated. In this process, the triangle singularity appears from the decay of $B^-$ into $D^{*0}K^-K^{*0}$ followed by the decay of $K^{*0}$ into $π^-K^+$ and the fusion of the $K^+K^-$ which forms the $a_0(980)$ or $f_0(980)$ which finally decay into $π^0η$ or $π^+π^-$ respectively. The triangle mechanism from the $\bar{K}^*K\bar{K}$ loop generates a peak around 1420 MeV in the invariant mass of $π^-a_0$ or $π^-f_0$, and gives sizable branching fractions $Br(B^-\rightarrow D^{*0}π^…
We have recently observed that hadron triangle singularities, that can mock new exotic hadrons, can be significanttly suppressed in relativistic heavy ion collisions, provided two conditions are met: these are, first, that the fireball lives long enough so that the triangle process has enough time to complete in the Norton-Coleman classical sense, and second, that the mass and/or width of the particles in the triangle diagram are sufficiently modified from their vacuum values. Here we add a very interesting example to the canon, which is $Y(4260) \to D_1 D \to πD^* D \to π+ J/ψπ$. This reactio…
We have analyzed the singularities of a triangle loop integral in detail and derived a formula for an easy evaluation of the triangle singularity on the physical boundary. It is applied to the $Λ_b \rightarrow J/ψK^{-} p$ process via $Λ^*$-charmonium-proton intermediate states. Although the evaluation of absolute rates is not possible, we identify the $χ_{c1}$ and the $ψ(2S)$ as the relatively most relevant states among all possible charmonia up to the $ψ(2S)$. The $Λ(1890)\, χ_{c1}\, p$ loop is very special as its normal threshold and triangle singularities merge at about 4.45 GeV, generating…
The BESIII Collaboration recently reported the observation of the $a_0(980)^0-f_0(980)$ mixing in the isospin breaking decay $J/ψ\to ηπ^0ϕ$. In the Dalitz plot for that decay with the $η$ reconstructed from two photons, there is a band around $1.4$~GeV on the $π^0 ϕ$ distribution. In general, this peak can be due to a resonance or a kinematic effect. In this paper, we study the effects of a set of $K^*K\bar K$ triangle diagrams, and show that due to triangle singularities such diagrams can lead to a peak around 1.4~GeV in the $π^0ϕ$ invariant mass distribution. The Dalitz plot induced by such …
We investigate the Schmid theorem, which states that if one has a tree level mechanism with a particle decaying to two particles and one of them decaying posteriorly to two other particles, the possible triangle singularity developed by the mechanism of elastic rescattering of two of the three decay particles does not change the cross section provided by the tree level. We investigate the process in terms of the width of the unstable particle produced in the first decay and determine the limits of validity and violation of the theorem. One of the conclusions is that the theorem holds in the st…
We propose a novel scenario for the peak structures, usually interpreted as hidden charm pentaquark ($P_c$) contributions, in the LHCb's $Λ_b^0\to J/ψp K^-$ data. The key idea is to utilize leading or lower-order singularities from double triangle mechanisms. The singularities cause anomalous threshold cusps, which are significantly more singular than the ordinary ones, at the $Σ_c^{(*)}\bar{D}^{(*)}$ threshold. We demonstrate that the double triangle amplitudes interfere with other common mechanisms to create peak structure that fit well the $P_c(4312)^+$, $P_c(4380)^+$, and $P_c(4457)^+$ pea…
We study the appearance of structures in the decay of the $B^-$ into $K^- π^- D_{s0}^+(2317)$ and $K^- π^- D_{s1}^+(2460)$ final states by forming invariant mass distributions of $π^- D_{s0}^+$ and $π^- D_{s1}^+$ pairs, respectively. The structure in the distribution is associated to the kinematical triangle singularity that appears when the $B^- \to K^- K^{*\,0} D^0$ ($B^- \to K^- K^{*\,0} D^{*\,0}$) decay process is followed by the decay of the $K^{*\,0}$ into $π^- K^+$ and the subsequent rescattering of the $K^+ D^0$ ($K^+ D^{*\,0}$) pair forming the $D_{s0}^+(2317)$ ($D_{s1}^+(2460)$) reso…
Effects of Final State Interactions on Landau Singularities
Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn（波恩大学亥姆霍兹射线与核物理研究所（理论）和贝特理论物理中心）；Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern（伯尔尼大学爱因斯坦基础物理中心理论物理研究所）；Institute for Nuclear Studies and Department of Physics, The George Washington University（乔治华盛顿大学核物理研究所和物理系）；Tbilisi State University（第比利斯国立大学）；Theory Center, Thomas Jefferson National Accelerator Facility（托马斯·杰斐逊国家加速器设施理论中心）
We investigate the processes $e^+e^-$$\to$$γJ/ψϕ$, $γJ/ψω$ and $π^0 J/ψη$ to search for the charmnium-like states with hidden $s\bar{s}$, such as $Y(4140)$, $Y(4274)$, $X(4350)$ and $X(3915)$. These processes will receive contributions from the charmed-strange meson rescatterings. When the center-of-mass energies of the $e^+e^-$ scatterings are taken around the $D_{s0}(2317)D_s^{*}$, $D_{s1}(2460)D_s$ or $D_{s1}(2460)D_s^{*}$ threshold, the anomalous triangle singularities can be present in the rescattering amplitudes, which implies a non-resonance explanation about the resonance-like structur…
A $Σ^*$ resonance with spin-parity $J^P = 1/2^-$ and mass in the vicinity of the $\bar{K}N$ threshold has been predicted in the unitary chiral approach and inferred from the analysis of CLAS data on the $γp \to K^+ π^0 Σ^0$ reaction. In this work, based on the dominant Cabibbo favored weak decay mechanism, we perform a study of $Λ_c^+ \to π^+ π^0 Σ^*$ with the possible $Σ^*$ state decaying into $π^- Σ^+$ through a triangle diagram. This process is initiated by $ Λ_c^+ \to π^+ \bar{K}^*N $, then the $\bar{K}^*$ decays into $\bar{K} π$ and $\bar{K} N$ produce the $Σ^*$ through a triangle loop co…
We compare contributions from the triangle diagram and the $D\bar D^*$ bubble chain to the processes of $e^{+}e^{-}\rightarrow J/ψπ^{+}π^{-}$ and $e^{+}e^{-}\rightarrow (D\bar D^*)^\mpπ^{\pm}$. By fitting the $J/ψπ$ maximum spectrum and the $D\bar D^*$ spectrum, we find that the triangle diagram cannot explain the new data from BESIII Collaboration at center of mass at 4.23GeV and 4.26GeV, simultaneously. On the contrary, the molecular assignment of $Z_c(3900)$ gives a much better description.
A constellation of quarkonium-mass peaks has been reported in the last decade, opening what could be an entire new spectroscopy of nuclear-physics like complexity. Salient among these structures are the $Z_c$, much analyzed at BESIII as nicely summarised by J.~Zhao at this conference. Their stature as new hadron states has often been challenged by proposals that the peaks arise from triangle singularities instead, because they appear in three-body reactions and often satisfy the special kinematic conditions of the Coleman-Norton theorem. To experimentally decide, we should like to resort to di…
We investigate the $B^0\to π^- K^+ X(3872)$ decay via the $D_{s1}(2536)\bar{D} D^*$ rescattering diagram. The line shape of the $K^+X(3872)$ distribution curve around $D_{s1}(2536)\bar{D}$ threshold is very sensitive to the $X(3872)$ mass because the triangle singularity (TS) can be generated from the loop. By means of this characteristic, we can determine whether the $X(3872)$ mass is below or above the $D^{\ast 0}\bar{D}^0$ threshold with high precision. The narrowness of $D_{s1}(2536)$ in the loop is one of the key reasons why the TS mechanism of measuring the $X(3872)$ mass may work. The $…
With the help of the generalized Wolfenstein parametrization of quark flavor mixing and CP violation, we calculate fine differences between the twin $b$-flavored unitarity triangles defined by $V^{*}_{ub} V^{}_{ud} + V^{*}_{cb} V^{}_{cd} + V^{*}_{tb} V^{}_{td} = 0$ and $V^{*}_{ud} V^{}_{td} + V^{*}_{us} V^{}_{ts} + V^{*}_{ub} V^{}_{tb} = 0$ in the complex plane. We find that vertices of the rescaled versions of these two triangles, described respectively by $\barρ + {\rm i} \barη = -\left(V^{*}_{ub} V^{}_{ud}\right)/\left(V^{*}_{cb} V^{}_{cd}\right)$ and $\widetildeρ + {\rm i} \widetildeη = -\…
In this work, we study the spin density matrix element $ρ_{00}$ of the $ϕ$ in the decay $J/ψ\rightarrow ηπϕ$. In previous studies, a band around 1.4 GeV on the $π^0ϕ$ distribution in Dalitz plot was reported by the BESIII Collaboration. This structure may be caused by the production of a resonance or the triangle singularity mechanism. We find that the predictions of the spin density matrix elements of the final $ϕ$ based on these mechanisms show distinct features. Thus the measurement of the spin density matrix elements of the $ϕ$ in this reaction may offer an alternative way to study the tri…
Triangle mechanisms for $B^0\to (J/ψπ^+π^-) K^+π^-$ are studied. Experimentally, an $X(3872)$ peak has been observed in this process. When the final $(J/ψπ^+π^-)π$ invariant mass is around the $D^*\bar D^*$ threshold, one of the triangle mechanisms causes a triangle singularity and generates a sharp $X(3872)$-like peak in the $J/ψπ^+π^-$ invariant mass distribution. The Breit-Wigner mass and width fitted to the spectrum are 3871.68 MeV (a few keV above the $D^{*0}\bar{D}^0$ threshold) and $\sim$0.4 MeV, respectively.These Breit-Wigner parameters hardly depends on a choice of the model paramete…
The LHCb data on $B^+\to J/ψϕK^+$ show four peaks and three dips in the $J/ψϕ$ invariant mass distribution, and the peaks are interpreted as $X(4140)$, $X(4274)$, $X(4500)$ and $X(4685)/X(4700)$ resonance contributions. Interestingly, all the peaks and dips are located at (or close to) $D^*_{s}\bar{D}^{(*)}_{s}$, $D_{s0}^*(2317)\bar{D}^{(*)}_{s}$, $D_{s1}(2536)\bar{D}^{(*)}_{s}$, and $ψ'ϕ$ thresholds. These coincidences suggest a close connection between the structures and the thresholds, which however has not been seriously considered in previous theoretical studies on the $X$ structures. In …
We perform calculations for the $\bar B_s^0 \to J/ψπ^0 f_0(980)$ and $\bar B_s^0 \to J/ψπ^0 a_0(980)$ reactions, showing that the first one is isospin-suppressed while the second one is isospin-allowed. The reaction proceeds via a triangle mechanism, with $\bar B_s^0 \to J/ψK^* \bar K +c.c.$, followed by the decay $K^* \to Kπ$ and a further fusion of $K\bar K$ into the $f_0(980)$ or $a_0(980)$. We show that the mechanism develops a singularity around the $π^0 f_0(980)$ or $π^0 a_0(980)$ invariant mass of 1420 MeV where the $π^0 f_0$ and $π^0 a_0$ decay modes are magnified and also the ratio of…
The light axial vector mesons can couple to $K^*\bar{K}+c.c.$ in an $S$ wave at the tree level. Due to the near-threshold $S$-wave interactions the couplings can be affected by the final state interactions. It is of peculiar interest that the pion exchange between $K^*\bar{K}+c.c.$ can go through a triangle diagram which is within the kinematics of the triangle singularity (TS). This mechanism will introduce energy dependence and a $D$-wave amplitude to the vertex couplings. Hence, it should be necessary to investigate the role played by the $K^*\bar{K}+c.c.$ threshold in order to have a bette…
We consider a triangle diagram for $B^0\to (J/ψπ^+π^-) K^+π^-$ where an $X(3872)$ peak has been observed experimentally. We demonstrate that a triangle singularity inherent in the triangle diagram creates a sharp peak in the $J/ψπ^+π^-$ invariant mass distribution when the final $(J/ψπ^+π^-)π$ invariant mass is at and around the $D^*\bar D^*$ threshold. The position and width of the peak is 3871.68 MeV (a few keV above the $D^{*0}\bar{D}^0$ threshold) and $\sim$0.4 MeV, respectively, in perfect agreement with the precisely measured $X(3872)$ mass and width: $3871.69\pm 0.17$ MeV and $< 1.2$ Me…
The LHCb collaboration reported the observation of a narrow peak in the $D^- K^+$ invariant mass distributions from the $B^+\to D^+ D^- K^+$ decay. The peak is parameterized in terms of two resonances $X_0(2900)$ and $X_1(2900)$ with the quark contents $\bar{c}\bar{s}ud$, and their spin-parity quantum numbers are $0^+$ and $1^-$, respectively. We investigate the rescattering processes which may contribute to the $B^+\to D^+ D^- K^+$ decays. It is shown that the $D^{*-}K^{*+}$ rescattering via the $χ_{c1}K^{*+}D^{*-}$ loop or the $\bar{D}_{1}^{0}K^{0}$ rescattering via the $D_{sJ}^{+}\bar{D}_{1…
We discuss the possibility that production of final states with bottomonium and light mesons at the peak $Υ(6S)$ in the $e^+e^-$ annihilation at approximately 11.00 GeV is in fact due to a triangular singularity at the threshold of the heavy meson pair production $B_1(5721) \bar B + c.c.$ through the process $e^+e^- \to B_1(5721) \bar B \to Z_b(10610) \, π$. The presence of the hidden-bottom resonance $Z_b(10610)$ then explains the observed enhanced production of the final channels with both ortho- and para- bottomonium states, $Υ(nS) ππ$ and $h_b(kP) ππ$. The discussed mechanism also predicts…
Understanding the nature of the hidden charm pentaquark(like) signals in the LHCb data for $Λ_b^0\to J/ψp K^-$ is a central problem of hadron spectroscopy. We propose a scenario completely different from previous ones such as hadron molecules and compact pentaquarks. We identify relevant double triangle mechanisms with leading or lower-order singularities. The associated anomalous threshold cusps at the $Σ_c^{(*)}\bar{D}^{(*)}$ thresholds are significantly more singular than the ordinary ones. Then we demonstrate that the double triangle amplitudes reproduce the peak structures of $P_c(4312)^+…
We analyze in the four-generation model the first measurement of the branching ratio of rare kaon decay K^+\toπ^+ν\barν, using the constraints from Δm_K, ε_K, B_d-\bar{B_d} mixing, Γ(b\to sγ), B_s-\bar{B_s} mixing, D^0-\bar{D^0} mixing, B(K_L\toπ^0ν\barν) and B(K_L\toμ\barμ), and study its effects on the unitarity triangle. With the results of searching for the maximum mixing of the fourth generation, we predict that D^0-\bar{D^0} mixing Δm_D and the branching ratio of "direct" CP-violating decay process K_L\toπ^0ν\barν could attain the values 1-2 orders of magnitude larger than the prediction…
It is expounded the approach to the Z\toγΨand Z\toγΥdecay study, based on the sum rules for the Z\to c\bar c\toγγ^{\ast} and Z\to b\bar b\toγγ^{\ast} amplitudes and their derivatives. The branching ratios of the Z\toγψand Z\toγΥdecays are calculated for different guesses as to saturation of the sum rules. The lower bounds of \sum_ψ BR(Z\toγψ) = 1.95\cdot 10^{-7} and \sum_Υ BR(Z\toγΥ) = 7.23\cdot 10^{-7} are found. Deviations from the lower bounds are discussed, among them the possibility of BR(Z\toγJ/ψ(1S))\sim BR(Z\toγΥ(1S))\sim 10^{-6}, that could be probably measured in LHC. The angle distr…
We calculate the cross section for $e^+ e^-$ annihilation into $D^{*0} \bar D^0 +γ$ at center-of-mass energies near the $D^{*0} \bar D^{*0}$ threshold under the assumption that $X(3872)$ is a weakly bound charm meson molecule. The Dalitz plot has a $\bar D^{*0}$ resonance band in the squared invariant mass $t$ of $\bar D^0 γ$. In the limit as the decay width of the $D^{*0}$ goes to 0, the Dalitz plot also has a narrow band in the squared invariant mass $u$ of $D^{*0} \bar D^0$ from a charm-meson triangle singularity. At the physical value of the $D^{*0}$ width, the narrow band reduces to a sho…
In this work, the triangle singularity mechanism is investigated in the $ψ(2S) \to p \bar{p} η/ p \bar{p} π^0$ process. The triangle loop composed by $J/ψ$, $η$ and $p$ has a singularity in the physical kinematic range for the $ψ(2S) \to p \bar{p} η/ p \bar{p} π^0$ process, and it would generate a very narrow peak in the invariant mass spectrum of $pη(π)$ around $1.56387$ GeV, which is far away from both the threshold and relative resonances. In these processes, all the involved vertices are constrained by the experimental data. Thus, we can make a precise model independent prediction here. It…
The decay of $Λ_c^+$ into $π^+π^0Λ(1405)$ with the $Λ(1405)$ decay into $π^0Σ^0$ through a triangle diagram is studied. This process is initiated by $ Λ_c^+ \to π^+ \bar{K}^*N $, then the $\bar{K}^*$ decays into $\bar{K} π$ and $\bar{K} N$ produce the $Λ(1405)$ through a triangle loop containing $\bar{K}^*N\bar{K}$ which develops a singularity around $1890$ MeV. This process is prohibited by the isospin symmetry, but the decay into this channel is enhanced by the contribution of the triangle diagram, which is sensitive to the mass of the internal particles. We find a narrow peak in the $π^0Σ^0…
We show that the $τ^-$ decay into $f_1(1285)π^-ν_τ$ is dominated by a triangle loop mechanism with $K^*$, $\bar K^*$ and $K$ (or $\bar K$) as internal lines, which manifests a strong enhancement reminiscent of a nearby singularity present in the narrow $K^*$ limit. The $f_1(1285)$ is then produced by its coupling to the $K^*\bar K$ and $\bar K^*K$ which is obtained from a previous model where this resonance was dynamically generated as a molecular $K^*\bar K$ (or $\bar K^*K$) state using the techniques of the chiral unitary approach. We make predictions for the $f_1 π$ mass distribution which …
We show that loop-induced processes involving new physics particles can readily satisfy Landau Equation and trigger triangular singularities at high energy colliders, leading to fully visible Standard Model final states. Four-particle vertices in new physics allow triangular singularity diagrams to evade large virtuality suppression. In addition, a $t$-channel triangular singularity can also occur in particle scattering processes that may extend to low momentum exchange. We discuss several typical scenarios in supersymmetric and extended Higgs models, then identify the singular component in th…
First Evidence for an Unambiguous Triangle Singularity from $ψ(2S) \to p\bar{p}η$
Nanjing Normal University（南京师范大学）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）
Phenomenological issues of the CP violation in the quark sector of the Standard Model are discussed. We consider quark mixing in the SM, standard and Wolfenstein parametrization of the $CKM$ mixing matrix and unitarity triangle. We discuss the phenomenology of the CP violation in $K^{0}_{L}$ and $B_{d}^{0} (\bar B_{d}^{0})$-decays. The standard unitarity triangle fit of the existing data is discussed. In appendix A we compare the $K^{0}\leftrightarrows \bar K^{0}$, $B_{d,s}^{0}\leftrightarrows \bar B^{0}_{d,s}$ etc oscillations with neutrino oscillations. In Appendix B we derive the evolution …
We present a model for the $J/ψΛ$ spectrum in $B^-\to J/ψΛ\bar{p}$ decays, including the $P_{ψs}^Λ(4338)$ baryon recently observed by the LHCb collaboration. We assume production via triangle diagrams which couple to the final state via non-perturbative interactions which are constrained by heavy-quark and $SU_3$-flavor symmetry. The bulk of the distribution is described by a triangle diagram with a color-favored electroweak vertex, while the sharp $P_{ψs}^Λ(4338)$ enhancement is due to the triangle singularity in another diagram featuring a $1/2^-$ baryon consistent with $Σ_c(2800)$. We predi…
We have studied the $γp \to p π^0 η$ reaction paying attention to the two main mechanisms at low energies, the $γp \to Δ(1700) \to ηΔ(1232)$ and the $γp \to Δ(1700) \to πN(1535)$. Both of them are driven by the photoexcitation of the $Δ(1700)$ and the second one involves a mechanism that leads to a triangle singularity. We are able to evaluate quantitatively the cross section for this process and show that it agrees with the experimental determination. Yet, there are some differences with the standard partial wave analysis which does not include explicitly the triangle singularity. The exercis…
We evaluate the contribution to the $X(3872)$ width from a triangle mechanism in which the $X$ decays into $D^{*0}\bar{D}^0 -cc$, then the $D^{*0} (\bar{D}^{*0})$ decays into $D^0 π^0$ ($\bar{D}^0 π^0$) and the $D^0 \bar{D}^0$ fuse to produce $π^+ π^-$. This mechanism produces an asymmetric peak from a triangle singularity in the $π^+ π^-$ invariant mass with a shape very sensitive to the $X$ mass. We evaluate the branching ratios for a reaction where this effect can be seen in the $B^- \to K^- π^0 π^+ π^-$ reaction and show that the determination of the peak in the invariant mass distribution…
The role of triangle singularity in isospin breaking process $J/ψ\to Λ\barΛ π$ and the possible evidence of $Σ^*(\frac{1}{2}^-)$ states
School of Physics and Technology, Nanjing Normal University（南京师范大学物理科学与技术学院）；University of Chinese Academy of Sciences (UCAS)（中国科学院大学）；Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所南方核科学理论中心）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）
We summarize the present knowledge on the non-perturbative hadronic inputs needed in the analysis of $B^0$-$\bar B^0$ mixing and the CP-violating parameter $\varepsilon$ of the $K^0$-$\bar K^0$ system. Using this information, together with the recently determined value of the top-quark mass, we update the phenomenological constraints on the unitarity triangle.
We perform calculations showing that a source producing $K^* \bar{K}^*$ in $J = 2$ and $L=0$ gives rise to a triangle singularity at $1810$ MeV with a width of about $200$ MeV from the mechanism $K^* \to πK$ and then $K\bar{K}^*$ merging into the $a_1(1260)$ resonance. We suggest that this is the origin of the present $f_2(1810)$ resonance and propose to look at the $πa_1(1260)$ mode in several reactions to clarify the issue.
The study of charmless hadronic two-body decays of B mesons is one of the most fascinating topics in B physics. A construction of the unitarity triangle based on such decays is presented, which is independent of B-B(bar) and K-K(bar) mixing. It provides stringent tests of the Standard Model with small theoretical uncertainties.
I examine a possibility to extract the angle $γ$ of the Cabibbo-Kobayashi-Maskawa unitarity triangle by the inclusive $B \to X_d ł^+ ł^- $ decay. An independent information for the angle is expected from the non-trivial contribution induced in $u \bar{u}$ and $c \bar{c}$ loops. The contributions induce CP asymmetry which has high sensitivity to the angle $γ$ in the whole dilepton invariant mass region. Particularly, in the low dilepton invariant mass region, the sensitivity is also recognized in the branching-ratio with any dilepton final states and the lepton polarization asymmetry with dimuo…
