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 the $Ξ(2120)$ are $J^P=1/2^+$, $1/2^-$ or $3/2^+$, the peak structure should be sharper than the other $J^P$ choices. This suggests that although the whole strength is unknown, we should pay attention to the contributions from the $Ξ^*$-charmonium-$Λ$ triangle diagram if structures are observed in the $J/ψΛ$ invariant mass spectrum experimentally. In addition, a few triangle diagrams with the $D_{s1}^*(2700)$ as one of the intermediate particles can also produce singularities in the $J/ψΛ$ distribution, but at higher energies above 4.9 GeV.
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.
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…
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 study effects of SUSY particles in the unitarity triangle analysis and the CP asymmetries in b -> s decays. We consider three different SUSY models, the minimal supergravity, SU(5) SUSY GUT with right-handed neutrinos, and MSSM with U(2) flavor symmetry, and show that the patterns and correlations of deviations from the standard model will be useful to discriminate the different SUSY models in future B experiments.
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 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 study effects of supersymmetric particles in various rare B decay processes as well as in the unitarity triangle analysis. We consider three different supersymmetric models, the minimal supergravity, SU(5) SUSY GUT with right-handed neutrinos, and the minimal supersymmetric standard model with U(2) flavor symmetry. In the SU(5) SUSY GUT with right-handed neutrinos, we consider two cases of the mass matrix of the right-handed neutrinos. We calculate direct and mixing-induced CP asymmetries in the b to s gamma decay and CP asymmetry in B_d to phi K_S as well as the B_s--anti-B_s mixing amplit…
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 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.
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…
Motivated by the possibility of the low value of sin2βin the measurements of BABAR and BELLE collaborations, we have explored the possibilty of construction of reference unitarity triangle using the unitarity of the CKM matrix, the existence of nonzero CP violating phase δand the experimental values of the well known CKM elements, without involving any inputs from the processes which might include the new physics effects. The angles of the reference triangle are evaluated by finding δthrough the Jarlskog's rephasing invariant parameter J. The present data and the unitarity of the CKM matrix gi…
In view of the latest T2K and MINOS observations regarding the mixing angle s_{13}, we have explored the possibility of existence of CP violation in the leptonic sector. Using hints from the construction of the 'db' unitarity triangle in the quark sector, we have made an attempt to construct the 'ν_1.ν_3' leptonic unitarity triangle, suggesting a good possibility of having non zero CP violation.
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.
$Z_c(4430)$ 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…
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 the determination of the CKM parameters from the forthcoming $CP$ violation observables in $B_s \to K^+ K^-$ decays. Combining the information on mixing induced CP violation in $B_s \to K^+ K^-$, with the $B_d \to J/ψK_s$ precision observable $\sin 2β$ and the $B^0_s$--$\bar{B^0_s}$ mixing phase $ϕ_s$, we propose a determination of the unitarity triangle $(\barρ, \barη)$. Computing the penguin parameters $(r, θ)$ within QCD factorization yield precise determination of $(\barρ, \barη)$, reflected by a weak dependence on the $θ$ which is shown as a second order effect. The impact of t…
$Z_c(4430)$ 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…
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 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 …
$Z_1(4050)$ 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…
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.
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…
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…
The B-meson system provides many strategies to perform stringent tests of the Standard-Model description of CP violation. In this brief review, we discuss implications of the currently available B-factory data on the angles alpha, beta and gamma of the unitarity triangle, emphasize the importance of Bs studies at hadronic B experiments, and discuss new, theoretically clean strategies to determine gamma.
Modifications of the $π^o\to 2γ$ decay amplitude by thermal effects have already been considered by several authors, leading to quite different results. I consider in this paper the triangle diagram connecting a neutral pion to two photons in a constituent quark model, within the real-time formulation of thermal field theory and study the zero external momentum limit of this diagram. It appears that this limit is not unique and depends strongly on the kinematical configuration of the external particles. This non-uniqueness is shown to explain the contradiction between existing results. I end w…
We explore the would-be consequences of a low value of the CP-violating phase $\sin2β_{ψK}$. The importance of a reference triangle obtained from measurements that are independent of $B$--$\bar B$ and $K$--$\bar K$ mixing is stressed. It can be used to extract separately potential New Physics contributions to mixing in the $B_d$, $B_s$ and $K$ systems. We discuss several constructions of this triangle, which will be feasible in the near future. The discrete ambiguity is at most two-fold and eventually can be completely removed. Simultaneously, it will be possible to probe for New Physics in lo…
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）
We have chosen the reactions $D_s^+ \to π^+ π^0 a_0(980) (f_0(980))$ investigating the isospin violating channel $D_s^+ \to π^+ π^0 f_0(980)$. The reaction was chosen because by varying the $π^0 a_0(980) (f_0(980))$ invariant mass one goes through the peak of a triangle singularity emerging from $D_s^+ \to π^+\bar K^* K$, followed by $\bar K^* \to \bar K π^0$ and the further merging of $K \bar K$ to produce the $a_0(980)$ or $f_0(980)$. We found that the amount of isospin violation had its peak precisely at the value of the $π^0 a_0(980) (f_0(980))$ invariant mass where the singularity has its…
We point out that the pure ``tree'' decays $B_c^\pm\to D^\pm_s D$ are particularly well suited to extract the CKM angle $γ$ through amplitude relations. In contrast to conceptually similar strategies using $B^\pm\to K^\pm D$ or $B_d\to K^{\ast0} D$ decays, the advantage of the $B_c$ approach is that the corresponding triangles have three sides of comparable length and do not involve small amplitudes. Decays of the type $B_c^\pm\to D^\pm D$ -- the $U$-spin counterparts of $B_c^\pm\to D^\pm_s D$ -- can be added to the analysis, as well as channels, where the $D^\pm_s$- and $D^\pm$-mesons are rep…
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…
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 (re)examine CP violation in the decays B_d -> D K_s, where D represents D^0, D(bar), or one of their excited states. The quantity $\sin^2(2β+ γ)$ can be extracted from the time-dependent rates for $B_d(t) -> {\bar D}^{**0} K_s$ and $B_d(t) -> D^{**0} K_s$, where the $D^{**0}$ decays to $D^{(*)+}π^-$. If one considers a non-CP-eigenstate hadronic final state to which both D(bar) and D^0 can decay (e.g. $K^+π^-$), then one can obtain two of the angles of the unitarity triangle from measurements of the time-dependent rates for $B_d(t) -> (K^+π^-)_{D K_s}$ and $B_d(t) -> (K^-π^+)_{D K_s}$. Ther…
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…
An explicit example is presented (a one-loop triangle graph) where dimensional regularization fails to regulate the infra-red singularities that emerge at intermediate steps of studying large-$Q^2$ Sudakov factorization. The mathematical nature of the phenomenon is explained within the framework of the theory of the $As$-operation.
By assuming that Y(4260) either is a $D_1\bar{D}$ molecular state, or has sizeable couplings with $D_0 \bar{D}^*$ and $D_1^\prime \bar{D}$, we investigate several decay modes of Y(4260). Under the special kinematic configurations, triangle singularity (TS) may occur in the rescattering amplitude, which will significantly change the threshold behavior. Obvious threshold enhancements or narrow cusp structures appear quite naturally without introducing a genuine resonance. We emphasize that the radiative decay modes may be useful for studying $D^{(*)}\bar{D}$ $S$-wave scattering.
Analytic decomposition of two-body electroweak processes with left-hand cuts
Ruhr-Universität Bochum（波鸿鲁尔大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；CSIC‐Universitat de Valéncia（西班牙国家研究委员会-瓦伦西亚大学）
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…
Joint Institute for Nuclear Research（联合核研究所）；D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University（莫斯科国立大学 D.V. 斯科别尔钦核物理研究所）；Faculty of Physics, University of Vienna（维也纳大学物理学院）；INFN, Roma Tre（罗马第三大学意大利国家核物理研究所）
Uncovering Singularities in Feynman Integrals via Machine Learning
School of Physical Sciences, University of Science and Technology of China（物理科学学院，中国科学技术大学）；Institute for Theoretical Particle Physics, KIT（理论粒子物理研究所，卡尔斯鲁厄理工学院）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（交叉学科理论研究中心，中国科学技术大学）；Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China（彭桓武理论研究中心，安徽合肥230026，中国）；Center for High Energy Physics, Peking University（高能物理中心，北京大学）
