diff --git "a/99E3T4oBgHgl3EQfSQn_/content/tmp_files/load_file.txt" "b/99E3T4oBgHgl3EQfSQn_/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/99E3T4oBgHgl3EQfSQn_/content/tmp_files/load_file.txt" @@ -0,0 +1,1152 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf,len=1151 +page_content='On the η1(1855), π1(1400) and π1(1600) as dynamically generated states and their SU(3) partners Mao-Jun Yan,1, ∗ Jorgivan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Dias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' † Adolfo Guevara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ‡ Feng-Kun Guo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' § and Bing-Song Zou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ¶ 1CAS Key Laboratory of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Institute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China 2School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China 3Peng Huanwu Collaborative Center for Research and Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China 4Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' China In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' we interpret the newly observed η1(1855) resonance with exotic JP C = 1−+ quantum numbers in the I = 0 sector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' reported by the BESIII Collaboration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' as a dynamically generated state from the interaction between the lightest pseudoscalar mesons and axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The interaction is derived from the lowest order chiral Lagrangian from which the Weinberg-Tomozawa term is obtained, describing the transition amplitudes among the relevant channels, which are then unitarized using the Bethe-Salpeter equation, according to the chiral unitary approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We evaluate the η1(1855) decays into the ηη′ and K ¯K∗π channels and find that the latter has a larger branching fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also investigate its SU(3) partners, and according to our findings, the π1(1400) and π1(1600) structures may correspond to dynamically generated states, with the former one coupled mostly to the b1π component and the latter one coupled to the K1(1270) ¯K channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, our result for the ratio Γ(π1(1600) → f1(1285)π)/Γ(π1(1600) → η′π) is consistent with the measured value, which supports our interpretation for the higher π1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also report two poles with a mass about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV in the I = 1/2 sector, which may be responsible for the K∗(1680).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We suggest searching for two additional η1 exotic mesons with masses around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the predicted η1(1700) is expected to have a width around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 GeV and can decay easily into K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ∗ yanmaojun@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn † jorgivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='mdias@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn ‡ aguevara@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn § fkguo@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn ¶ zoubs@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04432v1 [hep-ph] 11 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' INTRODUCTION Over the last two decades, the experimental observation of many new hadronic states is chal- lenging our current understanding of hadrons as conventional mesons and baryons with valence contents of quark-antiquark and three quarks, respectively, since most of them do not fit in the well-known quark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This difficulty brought back a long-standing discussion on the exotic hadronic structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', multiquark configurations that might have quantum numbers beyond those assigned to the conventional mesons and baryons [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Exotic quark configurations such as tetraquarks [3, 4], hadron-hadron molecules [5], glueballs, and hybrids [6, 7], among others, have been suggested to describe suitably most of the properties of these new states, such as the JPC quantum numbers, mass, and decay width, especially for those lying in the charmonium and bottomonium spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, distinguishing the exotic states from the conventional hadrons is a more complicated task in the light quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Many states have their masses close to each other, and the possibility of mixing brings additional difficulty to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The situation improves as the quantum numbers do not fall into those allowed by the conventional quark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It seems to be the case of the newly discovered state, dubbed η1(1855), by the BESIII Collaboration [8, 9], observed in the invariant mass distribution of the η η′ meson pair in the J/ψ → γ η η′ decay channel with a significance of 19σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Its mass and width reported by BESIII are 1855 ± 9+6 −1 MeV and 188 ± 18+3 −8 MeV, respectively, with likely JPC = 1−+ quantum numbers, which cannot be formed by a pair of quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η1(1855) is not the only state experimentally found with that set of quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As of today, three other hadronic structures, called π1(1400), π1(1600) and π1(2015), with JPC = 1−+, were observed by several collaborations [7, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From the theoretical point of view, the hybrid model has been used to investigate these exotic meson states, in particular the 1−+ ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Lattice quantum chromodynamics (QCD) calculations have pointed out hybrid supermultiplets with exotic JPC quantum numbers, including the 1−+ one [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this picture, however, the mass of the lightest 1−+ state and decay modes are in- consistent with the corresponding experimental results, while the π1(1600) and π1(2015) structures can fit into the nonets predicted by lattice QCD [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The newly observed η1(1855) state has also been the focus of some studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [17] proposed two hybrid nonet schemes in which the η1(1855) resonance can be either the lower or higher mass state with isospin I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [18], an effective Lagrangian respecting flavor, parity, and charge conjugation symmetries is used to study the hybrid nonet 3 decays into two-body meson states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The authors have fixed the couplings to those two-body meson states by performing a combined fit to the experimental and lattice results available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As a result, the decay width value estimated for the isoscalar member of the hybrid nonet agrees with the one observed for η1(1855) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Also addressing the same picture, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [19] applied the approach of QCD sum rules to describe the η1(1855) mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' By contrast, within the same approach, the η1(1855) resonance is described as a tetraquark state in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η1(1855) resonance also supports a meson-meson molecule interpretation due to its prox- imity to the K ¯K1(1400) threshold, as put forward by Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21] have investigated the K ¯K1(1400) interaction through the one-boson exchange model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' According to their findings, the K ¯K1(1400) system binds for cutoff values above 2 GeV with a monopole form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, the comparison between their result for the branching fraction B(η1 → η η′) to the experimental one led them to conclude that the K ¯K1(1400) molecule can explain the η1(1855) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' An important point to be addressed is the meson-meson interaction around the K1(1400) ¯K threshold for the JPC = 1−+ quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this sector, many meson-meson pairs may contribute to that interaction, so a coupled-channel treatment seems appropriate to take these contributions into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, hadron-hadron interactions in coupled channels have been studied in many works to describe the properties of the new hadronic systems experimentally observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In those cases, these hadronic structures are called dynamically generated states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Following this approach, in this work, we aim to explore the η1(1855), π1(1400), and π1(1600) hadronic systems as dynamically generated states from pseudoscalar-axial vector meson interactions in coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Specifically, the low-energy interactions are given by the Weinberg-Tomozawa (WT) term from chiral Lagrangians at the leading order of the chiral expansion by treating the axial vector mesons as matter fields and the pseudoscalar mesons as the pseudo-Nambu-Goldstone bosons of the spontaneous breaking of chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Such Lagrangians have been used to study many hadron structures stemming from meson-meson and meson-baryon interactions in coupled channels in light and heavy sectors, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In our case, the amplitudes obtained from the WT term are unitarized via the Bethe-Salpeter equation from which bound states/resonances manifest as poles in the physical/unphysical Riemann sheets of the scattering matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The existence of a whole family of kaonic bound states has been pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [28] based on unitarizing the WT term for the scattering of the kaon off isospin-1/2 matter fields taking heavy mesons and doubly-charmed baryons as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As we shall show in this work, the newly observed η1(1855) structure may correspond to a dynamically generated state from the 4 pseudoscalar-axial vector interaction in the isospin I = 0 sector coupling strongly to the K1(1400) ¯K channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Moreover, the π1(1400) and π1(1600), may be assigned as the η1(1855) SU(3) partners which are also dynamically generated from the pseudoscalar-axial vector meson interactions in the I = 1 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The former resonance couples mainly to the b1π channel, and the latter has the K1(1270) ¯K as its main coupled channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we have also found two poles around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV in the I = 1/2 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' These poles are particularly interesting as they could be the origin of the K∗(1680) structure observed experi- mentally [10], which is the main component of the 1− contribution to the φK mass distribution in the B → J/ψφK decays recently measured by LHCb [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Section II, we discuss the relevant channels contributing to the pseudoscalar-axial vector meson interactions and the use of the chiral unitary approach (ChUA) for the evaluation of the transition amplitudes among those channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Sections III and IV, we investigate the dynamical generation of poles stemming from those interactions in the I = 0 and I = 1 sectors and discuss their possible decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Finally, in Section V, we also explore the dynamical generation of poles for I = 1/2 and their connection to the vector K∗(1680) structure observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Section VI gives a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' COUPLED CHANNEL SCATTERING IN CHIRAL UNITARY APPROACH We investigate the interactions between axial and pseudoscalar mesons in coupled channels in the 1300 ∼ 2000 MeV energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' First, we need to determine the space of states contributing to the interaction in this energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Tables I, II, III, and IV, we list all the relevant channels for the problem under consideration along with their corresponding mass thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The channels are organized from the lower to higher mass values and by the isospin, 0, 1 and 1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP C = 1−+ meson-meson channels with I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel a1π K1(1270) ¯K f1(1285)η K1(1400) ¯K f1(1420)η Threshold 1368 1748 1829 1898 1973 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP C = 1−+ meson-meson channels with I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel b1π f1(1285)π f1(1420)π K1(1270) ¯K a1η K1(1400) ¯K Threshold 1367 1419 1564 1748 1777 1895 5 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP = 1− meson-meson channels with I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial vector mesons have JP C = 1++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel a1K f1(1285)K+ K1(1270)η f1(1420)K K1(1400)η Threshold 1725 1777 1800 1921 1947 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' JP = 1− meson-meson channels with I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The threshold masses are in the units of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial vector mesons have JP C = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Channel h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η Threshold 1661 1725 1800 1911 1947 In what follows, we shall discuss the relevant scattering amplitudes among all those channels above for each isospin sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' These transitions can be written in the form of the WT term which then is unitarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that the channels displayed in Tables III and IV, in principle, should be grouped in the same space of states since they share identical isospin and JP quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, the relevant transitions among them arise only at the next-to-leading order in the chiral expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' see the discussion around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (17) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Thus, such transitions are of higher order than that of the WT term and will be neglected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The Weinberg-Tomozawa term In order to study the interactions among all the channels listed in the previous tables, we have to evaluate the interactions between the pseudoscalar and axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The latter are organized in two SU(3) octets according to their JPC quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' A1 = � � � � � a0 1 √ 2 + f8 1 √ 6 a+ 1 K+ 1A a− 1 − a0 1 √ 2 + f8 1 √ 6 K0 1A K− 1A ¯K0 1A − 2f8 1 √ 6 � � � � � (1) is the octet of resonances of axial-vector states with JPC = 1++ for the flavor-neutral mesons, and B1 = � � � � � b0 1 √ 2 + h8 1 √ 6 b+ 1 K+ 1B b− 1 − b0 1 √ 2 + h8 1 √ 6 K0 1B K− 1B K0 1B − 2 √ 6h8 1 � � � � � (2) describes the octet of axial-vector resonances with JPC = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The singlet and I = 0 octet flavor eigenstates are not mass eigenstates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' that is, the pairs of f1(1420), h1(1415) (also known as 6 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Two sets of values of the axial-vector meson mixing angles taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Set B is preferred in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η-η′ mixing angle θP is taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For more discussions about these mixing angles, we refer to the review of Quark Model in the Review of Particle Physics [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Angles θK1 θ3P1 θ1P1 θP Set A 57◦ 52◦ −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='5◦ −17◦ Set B 34◦ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1◦ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0◦ −17◦ h1(1380)) and f1(1285), h1(1170) mesons are mixtures of the singlet (1) and octet (8) mesons such that � � |f1(1285)⟩ |f1(1420)⟩ � � = � � cos θ3P1 sin θ3P1 − sin θ3P1 cos θ3P1 � � � � ��f1 1 � ��f8 1 � � � , (3) and � � |h1(1170)⟩ |h1(1415)⟩ � � = � � cos θ1P1 sin θ1P1 − sin θ1P1 cos θ1P1 � � � � ��h1 1 � ��h8 1 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (4) Furthermore, the K1A and K1B members of the multiplets in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (1) and (2) are the strange partners of the a1(1260) and b1(1235), and their mixture contributes to the physical K1(1270) and K1(1400) mesons, that is � � |K1(1270)⟩ |K1(1400)⟩ � � = � � sin θK1 cos θK1 cos θK1 − sin θK1 � � � � |K1A⟩ |K1B⟩ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (5) The corresponding values for the mixing angles in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (3), (4), and (5) are listed in Table V, where they are grouped into two sets, denoted by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Although set B is preferred in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [30], we will use both sets to have an estimate of the uncertainties caused by such an angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In order to determine the WT term we start with the Lagrangian (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [32]) L0 = −1 4 � VµνV µν − 2M2 V VµV µ� , (6) where ⟨, ⟩ takes trace in the SU(3) flavor space, Vµν = DµVν − DνVµ , (7) while Dµ is the chirally covariant derivative, which when acting on SU(3) octet matter fields reads as Dµ = ∂µ + [Γµ, ] , (8) 7 with [ , ] the usual commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, Γµ stands for the chiral connection, given by Γµ = 1 2 � u†∂µu + u∂µu†� , (9) with u = exp � i √ 2Fπ φ8 � , (10) where Fπ = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 MeV is the pion decay constant [10], and φ8 is the pseudoscalar SU(3) octet, that is φ8 = � � � � � π0 √ 2 + 1 √ 6η8 π+ K+ π− − 1 √ 2π0 + 1 √ 6η8 K0 K− ¯K0 − 2 √ 6η8 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (11) In addition, the physical η and η′ mesons are the mixtures of η8 and η1 � � |η⟩ |η′⟩ � � = � � − sin θP cos θP cos θP sin θP � � � � ��η1� ��η8� � � , (12) where η1 becomes the ninth pseudo-Goldstone boson in large Nc QCD [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The Goldstone boson nonet is written as φ9 = φ8 + 1 √ 3η1, (13) which leads to a relation in the commutator � φ9, ∂µφ9� = � φ8, ∂µφ8� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (14) Therefore, only the scattering of the octet Goldstone bosons off the axial-vector mesons in Weinberg-Tomozawa term contributes to JP(C) = 1−(+) spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The covariant derivative Dµ by means of the connection Γµ encodes the leading order interaction between the pseudoscalar mesons and the vector field Vµ [32, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, by replacing the Vµ field to the axial-vector field Aµ corresponding to either the A1 or B1 multiplet, the chiral tran- sition between φ8 (pseudoscalar) and A (1+) (axial-vector) is described by the following interaction Lagrangian LI = − 1 4F 2π � [Aµ, ∂νAµ] � φ8, ∂νφ8�� , (15) 8 which accounts for the WT interaction term for the PA → PA process, with P and A corresponding to the pseudoscalar and axial-vector mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From this Lagrangian we obtain the S- wave transition amplitude among the channels listed in Tables I, II, III and IV, that is Vij(s) = −ϵ · ϵ′ 8F 2π Cij � 3s − � M2 + m2 + M′2 + m′2� − 1 s � M2 − m2� � M′2 − m′2�� , (16) where ϵ (ϵ′) stands for the polarization four-vector of the incoming (outgoing) axial-vector me- son [25, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The masses M (M′) , m (m′) correspond to the initial (final) axial-vector mesons and initial (final) pseudoscalar mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The indices i and j represent the initial and final PA states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The coefficients Cij are given in Tables VI, VII, VIII, and IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave and I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij a1π K1(1270) ¯K f1(1285)η K1(1400) ¯K f1(1420)η a1π −4 � 3 2 sin θK1 0 � 3 2 cos θK1 0 K1(1270) ¯K −3 − 3 √ 2 sin θ3P1 sin θK1 0 − 3 √ 2 cos θ3P1 sin θK1 f1(1285)η 0 − 3 √ 2 cos θK1 sin θ3P1 0 K1(1400) ¯K −3 − 3 √ 2 cos θ3P1 cos θK1 f1(1420)η 0 TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP C = 1−+ in S-wave and I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij b1π f1(1285)π f1(1420)π K1(1270) ¯K a1η K1(1400) ¯K b1π −2 0 0 cos θK1 0 − sin θK1 f1(1285)π 0 0 � 3 2 sin θK1 sin θ3P1 0 � 3 2 cos θK1 sin θ3P1 f1(1420)π 0 � 3 2 cos θ3P1 sin θK1 0 � 3 2 cos θK1 cos θ3P1 K1(1270) ¯K −1 − � 3 2 sin θK1 0 a1η 0 − � 3 2 cos θK1 K1(1400) ¯K −1 Before proceeding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' let us discuss the A1φ8 → B1φ8 transitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' with A1 and B1 the two SU(3) octets of axial-vector mesons and φ8 the octet of the pseudo-Nambu-Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Let A1µ and B1µ denote the fields for the 1++ and 1+− axial-vector meson multiplets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Under parity transformation, we have A1µ → −Aµ 1 and B1µ → −Bµ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' under charge conjugation, we have A1µ → AT 1µ and B1µ → −BT 1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Then the A1φ8 → B1φ8 transitions can only arise at O � p2� with p 9 TABLE VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave and I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij a1K f1(1285)K K1(1270)η f1(1420)K K1(1400)η a1K −2 0 − 3 2 sin θK1 0 − 3 2 cos θK1 f1(1285)K 0 3 2 sin θK1 sin θ3P1 0 3 2 sin θK1 cos θK1 K1(1270)η 0 3 2 cos θ3P1 sin θK1 0 f1(1420)K 0 3 2 cos θ3P1 cos θK1 K1(1400)η 0 TABLE IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) for axial and pseudoscalar pairs coupled to JP = 1− in S-wave and I = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Cij h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η h1(1170)K 0 0 3 2 cos θK1 sin θ1P1 0 3 2 sin θK1 sin θ1P1 b1K −2 − 3 2 cos θK1 0 − 3 2 sin θK1 K1(1270)η 0 3 2 cos θK1 cos θ1P1 0 h1(1415)K 0 3 2 sin θK1 cos θ1P1 K1(1400)η 0 the momentum scale in the chiral power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' They are given by operators such as ⟨A1µ[B1ν, [uµ, uν]]⟩ , (17) with uµ = i � u†∂µu − u∂µu†� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (18) Such terms are one order higher in the chiral power counting than the WT terms describing the A1φ8 → A1φ8 and B1φ8 → B1φ8 transitions, and thus will be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Unitarization procedure The unitarization procedure we adopt follows ChUA in which the scattering amplitudes in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) are the elements of a matrix, denoted by V , such that it enters as an input to solve the Bethe-Salpeter equation, which in its on-shell factorization form, reads [23] T = (1 − V G)−1 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (19) 10 The V matrix describes the transition between the channels listed in Tables I, II, III, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, G is the diagonal loop function matrix whose diagonal matrix elements are given by Gl = i � d4q (2π)4 1 q2 − m2 l + iϵ 1 (q − P)2 − M2 l + iϵ , (20) with ml and Ml the masses of the pseudoscalar and axial-vector mesons, respectively, involved in the loop in the channel l, and P the total four-momentum of those mesons (P 2 = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' After the integration over the temporal component q0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (20) becomes Gl(s) = � d3q (2π)3 ω1 + ω2 2ω1ω2 1 (P 0)2 − (ω1 + ω2)2 + iϵ , (21) with ω1 = � Ml2 + |⃗q|2 and ω2 = � ml2 + |⃗q|2, and can be regularized by means of a cutoff in the three-momentum qmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, the function Gl can also be regularized using a subtraction constant as [40] GDR l (s) = 1 16π2 � αl(µ) + log M2 l µ2 + m2 l − M2 l + s 2s log m2 l M2 l + pl √s � log s − m2 l + M2 l + 2pl √s −s + m2 l − M2 l + 2pl √s + log s + m2 l − M2 l + 2pl √s −s − m2 l + M2 l + 2pl √s �� , (22) where pl is the three-momentum of the mesons in the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=') frame pl = �� s − (Ml + ml)2� � s − (Ml − ml)2� 2√s , (23) while µ is an arbitrary scale of the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Any changes in the µ scale can be absorbed by the subtraction constant αl(µ) such that the result is independent of the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We may determine the subtraction constant for each intermediate state of the scattering problem by comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (21), regularized using qmax, and (22) at the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The equivalence between the two prescriptions for the loop-function is discussed in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this work, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [44] and set µ = 1 GeV and α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35, which is obtained by matching to hard cutoff regularization with qmax ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV in the f1(1285)η channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This set of parameters are used for all channels, and a variation of the cutoff within qmax = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1) GeV, and correspondingly α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, will be used to show the dependence of the results on this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Searching for poles We move on to the complex energy plane to search for poles in the T-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Specifically, for a single-channel problem, there are two Riemann sheets for the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Bound states 11 show up as poles, below the threshold, in the transition matrix on the real energy axis on the first Riemann sheet, while virtual states manifest themselves below the threshold on the real axis on the second Riemann sheet, and resonances correspond to poles off the real axis on the second Riemann sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The Riemann sheets come about because the G loop function has a cut extending from the threshold to infinity which is usually chosen to be along the positive real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For n coupled channels, there are n cuts and thus 2n Riemann sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From unitarity and the Schwarz reflection principle, the discontinuity of the Gl function can be read off from its imaginary part, Im Gl(s) = − pl 8π√s , (24) which we can use to perform an analytic continuation to the entire complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this case, the Gl loop function on the “second” Riemann sheet with respect to the lth channel reads GII l (s) = GI l(s) + i pl 4π√s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (25) the lower half plane of this Riemann sheet is directly connected to the physical region when the lth channel is open, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Re(√s) ≥ m + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We will label the Riemann sheets according to the sign of the imaginary part of the corresponding c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' momentum for each channel (see the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Furthermore, it is also possible to determine the pole couplings to the lth channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Note that close to the pole singularity the T-matrix elements Tij(s) admit a Laurent expansion, Tij(s) = gi gj s − zp + regular terms, (26) where zp = (Mp −iΓ/2)2 is the pole location on the complex energy plane, with Mp and Γ standing for the pole mass and width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, the product of couplings gigj is the residue at the pole in Tij(s) which takes values on the Riemann sheet where the pole is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this way, the couplings can be evaluated straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For instance, for a diagonal transition it is given by g2 i = r 2π � 2π 0 Tii(z(θ))eiθdθ = lim s→zp(s − zp)Tii(s) = � d ds 1 Tii(s) �−1 s=zp , (27) where z(θ) = zp + i reiθ with r the radius of contour for the integral, and the two lines give two equivalent ways of computing residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12 TABLE X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The poles (in GeV) and their corresponding couplings (in GeV) to the channels contributing to the PA interaction with I = 0 and exotic quantum numbers JP C = 1−+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The corresponding Riemann sheet for each pole is listed below the pole position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The dominantly coupled channel is emphasized in boldface for each pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='21 + i3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='22 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='36 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='36 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='16 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='64 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='09 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='46 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− − − + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='07 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='68 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='33 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='16 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='06 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='21 + i3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='81 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='55 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='34 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='37 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='58 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 a1π K1(1270) ¯ K f1(1285)η K1(1400) ¯ K f1(1420)η (− − − + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='33 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='83 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='09 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='05 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='81 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='20 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' η1(1855) AND ITS DECAYS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Dynamical generation of the η1(1855) Following the unitarization procedure described previously, we seek dynamically generated states stemming from the S-wave interactions between pseudoscalar and axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For the I = 0 case, the transition amplitudes among the channels listed in Table I are determined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) with the Cij coefficients given in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Table X, we show the isoscalar poles with exotic quantum numbers JPC = 1−+ obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (19) using those coefficients as well as each set of mixing angles listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also show the couplings of these poles to the channels spanning the space of states in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Furthermore, in Table X we also highlight the Riemann sheets, the first and the second one for each channel, denoted by the + and − signs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We get three poles such that their 13 locations are barely affected by the change of the mixing angles from set A to set B listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The lower pole is at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 GeV with a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 GeV, which is above the a1π threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, this channel is open for decay, and the fact that it is this channel the one for which the pole couples mostly, as pointed out in Table X, explains why that pole has such a value for its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' By contrast, although the a1π channel is also open for decay, the pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 GeV has a much smaller width because its coupling to this channel is small compared to the one for K1(1400) ¯K, which is the dominant channel for that pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Similarly, the highest pole, located at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 GeV, couples mostly to the K1(1400) ¯K channel, and has a small imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we can also understand why the highest pole couples more to the K1(1400) ¯K than to the f1(1285)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The latter channel is closer to the pole than the former, but from Table VI, the diagonal f1(1285)η transition is not allowed since its WT term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nevertheless, the pole couples to f1(1285)η through the nondiagonal K1(1400) ¯K–f1(1285)η transition, which leads to a small coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Effects of the widths of the axial-vector mesons So far we have neglected the nonzero widths of the axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In order to investigate their effects on the results, we use complex masses for the intermediate resonances, that is, Mi → Mi − iΓi/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, by doing that, the analytic properties are lost such that the poles of the T matrix do not correspond to the masses and widths of the obtained resonances any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, we can see the impact of such nonzero widths on the lineshapes of the transition matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 1 we show a comparison between the lineshape for the T-matrix element corresponding to the elastic transition TK1(1400) ¯ K→K1(1400) ¯ K with and without including the widths for the inter- mediate particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This channel has the strongest coupling to the pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' therefore, we expect that any nontrivial structure should manifest most in its associated T-matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The dashed and solid lines are the TK1(1400) ¯ K→K1(1400) ¯ K with zero and nonzero width, respectively, for both sets A and B of mixing angles in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that, for the case of zero width approxima- tion, the TK1(1400) ¯ K→K1(1400) ¯ K lineshape has narrow peaks around 1845 MeV, right at the range of energy where we expect the η1(1855) manifests in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The inclusion of finite widths for the axial-vector mesons changes the sharp peak to a broad bump with a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='2 GeV, which is around the width of the K1(1400) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that the width matches nicely that of the η1(1855) measured by BESIII, � 188 ± 18+3 −8 � MeV [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In the following, we will continue to present 14 w/o Γ w/o Γ w/ Γ w/ Γ 1600 1700 1800 1900 2000 0 10 20 30 40 50 s [MeV] |T44 2 Set A Set B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The blue dashed and solid lines are, respectively, the modulus squared of the T-matrix element, cor- responding to the diagonal K1(1400) ¯K → K1(1400) ¯K transition, evaluated with and without the inclusion of the widths associated with the axial-vector mesons taking part in the loop function Gl (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' predictions neglecting the width effects of the axial-vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Let us briefly discuss the other two predicted isoscalar exotic η1 mesons in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The one with a mass of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='39 GeV, denoted as η1(1400), is expected to be rather broad due to the large width of the a1(1260) as it couples most strongly to the a1π channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It can be searched for in final states such as ρππ and K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The one with a mass around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV, denoted as η1(1700), couples most strongly to the K1(1270) ¯K and is expected to have a width similar to that of the K1(1270), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It can also be searched for in final states of K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The η1(1855) → η′η and K∗ ¯Kπ decays Let us first discuss the η1 → ηη′ decay, whose Feynman diagram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Within our approach the η1(1855) structure decays via its K1(1400) ¯K component, with the corresponding coupling constant listed in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also need to evaluate the K1(1400) ¯K → ηη′ transition, for which we use the resonance chiral theory (RχT) operators given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The RχT operators can be divided regarding the intrinsic-parity sector to which they contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Due to its nature, the odd-intrinsic parity sector will contain a Levi-Civita tensor [46–48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' for the η1 → ηη′ decay one cannot saturate the Lorentz indices in such tensor to get a nonzero contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Thus, only the even-intrinsic parity operators must give a nonvanishing contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Since the chiral O(p2) Lagrangian does not contribute to such processes [49], we will use the O(p4) Lagrangian given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From these operators, only three will contribute to this decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' To get the largest possible contribution from such operators, we use the upper bounds imposed from chiral counting 15 as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This amounts to making equal the three coupling constants and setting them to λA 1 = λA 2 = λA 3 = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='025 GeV−1, which gives a Lagrangian L = g � ⟨Aµν (uµuαhνα + hναuαuµ)⟩ + ⟨Aµν (uαuµhνα + hναuµuα)⟩ + ⟨Aµν (uµhναuα + uαhναuµ)⟩ � , (28) where uµ has been given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (18), hµν = D{µuν} is the symmetrized covariant derivative of uµ and the spin-1 resonance field is given in the antisymmetric tensor formalism [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, since the η1 → K1 ¯K transition is given in terms of Proca fields, we need to express the K1 as a Proca field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [49], the antisymmetric tensor field can be expressed in terms of the Proca one as follows, Rµ = 1 MR ∂νRνµ, (29) where MR is the mass of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Using the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (28) and expressing the axial resonance in the Proca representation, we get the η1 → ηη′ decay amplitude Mη1→ ηη′ = − 4m2 η1 3F 3πmK1 ggK1(1400) ¯ KGK1 ¯ K �� αp2 η′ + 1 √ 2βp2 η � εη1 · pη + � pη ↔ pη′�� , (30) where Fπ is the pion decay constant, gK1 ¯ K is the coupling constant of the pole to the K1(1400) ¯K channel, GK1 ¯ K is the loop function for the K1 and ¯K mesons , εη1 is the η1 vector polarization, and pη(′) is the momentum of the η(′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here, α and β are given in terms of the η-η′ mixing angle α = cos 2θP + 2 √ 2 sin 2θP , (31a) β = 2 √ 2 cos 2θP − sin 2θP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (31b) K1 ¯K η η′ η1 (1855) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Diagram corresponding to the η1 → ηη′ decay through the K1 ¯K loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Although one might try to rely in a much simpler way to describe the direct coupling of one axial- vector and three pseudoscalar fields by means of the Hidden Local Symmetry (HLS) Lagrangian 16 [51–53], it is worth to notice that nonetheless, the total amplitude for this process given by the HLS Lagrangian vanishes, which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (30) in the chiral limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The decay of η1 state into ηη′ is given by Γ2B = 1 2J + 1 1 8πM2η1 |Mη1→ ηη′|2 q , (32) with the amplitude Mη1→ ηη′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (30), while J stands for the η1 spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Besides that, q reads q = 1 2Mη1 λ1/2 � M2 η1, m2 η′, m2 η � , (33) with Mη1, mη′, and mη the masses for the η1(1855), η′, and η mesons, respectively, where λ (x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2zx is the K¨all´en triangle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, we get the following results for the decay width in this channel Γ2B = � � � (19 ± 4) MeV (set A) , (7 ± 2) MeV (set B) , (34) where the error is from choosing subtraction constant to be in the range α(µ = 1GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, corresponding to the hard cutoff qmax = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1) GeV as discussed at the end of Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For set A, our result agrees with that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21], where the η1(1855) was assumed to be a K1 ¯K molecule and the same θK1 mixing angle was used for accounting for the K1A and K1B mixture contributing to the physical K1(1270) and K1(1400) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Feynman Diagram associated with the three-body decay of the pole through its main component K1 ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As for the η1 → ¯KK∗π three-body decay, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3 shows the Feynman diagrams contributing to this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the η1(1855) decays through its molecular components, that in our approach are the K1(1270) ¯K and K1(1400) ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this case, the contribution from the K1(1270) ¯K component can be ignored for the following reasons: 1) from Table X, we see that the relative coupling strength for the K1(1270) ¯K channel is much smaller than that for the K1(1400) ¯K one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 117 2) the branching ratio B[K1(1270) → K∗π] is only 16%, while 96% of the K1(1400) decays is dominated by the K∗π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Therefore, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 3 the η1(1855) → ¯KK∗π amplitude is written as M3B = gK1(1400) ¯ K � −gµν + pµpν M2 K1 � 1 p2 − M2 K1 + i MK1ΓK1 gK∗π εµ η1εν K∗ , (35) where gK1(1400) ¯ K is the coupling of the pole associated with the η1 state to the K1(1400) ¯K channel, gK∗π is the K1(1400)K∗π coupling extracted from the K1(1400) → K∗π reaction in the Review of Particle Physics (RPP) [10], and εµ η1 and εν K∗ are the polarization vectors of the η1 and K∗ mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The differential decay width for the η1 → ¯KK∗π process is given by dΓ dMK1 ¯ K = 1 (2π)3 pK ˜pπ 4M2η1 |M3B|2 1 2J + 1 , (36) where ˜pπ = 1 2MK1 λ1/2 � M2 K1, m2 K∗, m2 π � , (37) and pK = 1 2Mη1 λ1/2 � M2 η1, m2 K, M2 K1 � , (38) with MK1, mK∗, mπ being the masses of the K1(1400), K∗ and π mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (36) we obtain the following results for the η1 → ¯KK∗π decay width Γ3B = � 81+11 −24 MeV �A , Γ3B = � 74+12 −24 MeV �B , (39) where the uncertainties come from the subtraction constant (cutoff) used to regularize the loops in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (22) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (39), we obtain similar results whether we use the sets A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For the sake of comparison to other works, we evaluate the ratio Γ2B/Γ3B, and get Γ2B Γ3B = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='23−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='16 �A or � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 �B , (40) which is consistent to the results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21], where the η1 is also assumed to be a K1(1400) ¯K molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, adopting the same multiquark configuration than the present work and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [21], the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [22] have found a different result for the ratio, Γ2B/Γ3B ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nevertheless, in all the cases the results point out that the ¯KK∗π three-body channel is more likely than the ηη′ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 18 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' THE π1(1400/1600) DYNAMICAL GENERATION The WT amplitudes for the pseudoscalar-axial vector meson interactions with I = 1 are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16), with the corresponding Cij coefficients listed in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In this case, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (19), we get two π1 poles shown in Table XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles and their corresponding couplings to the channels contributing to the PA interaction with JP C = 1−+ and I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − + + ++) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='22 + i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='33 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − − + ++) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='10 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='73 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='84 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='49 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='65 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='53 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − + + ++) gl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 + i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='03 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='97 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='91 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 − i(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01) b1π f1(1285)π f1(1420)π K1(1270) ¯ K a1η K1(1400) ¯ K (− − − + ++) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='13 + i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='37 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='86 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='80 − i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='53 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='54 − i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='77 Similar to the previous section, we also provide the couplings of these dynamically generated states to the channels listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Table XI shows a broad π1 pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='47 GeV, and a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='12 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 This state is above the b1π and f1(1285)π thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Its large width stems from the large coupling to the b1π and the fact that this channel is open for decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The f1(1285)π channel is also open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, according to Table VII, the corresponding WT term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) is zero for the diagonal f1(1285)π transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' On the other hand, the next π1 pole in Table XI has a sizeable dependence on the mixing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Using set A, we find that pole at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='75 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It couples most strongly to the K1(1270) ¯K channel, which is closed for decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nonetheless, the state can decay into b1π and f1(1285)π, albeit their corresponding couplings are small compared to the K1(1270) ¯K one, but still large enough to provide a sizeable width for the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In contrast, when set B is adopted, the higher π1 pole is now located at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='77 GeV, above the f1(1420)π threshold, 1 As discussed in Section III B, the widths of the dynamically generated poles will be significantly increased once the width effects of the axial-vector mesons are taken into account;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' see also the discussions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 19 which is now open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' One might think that the width should increase since now three channels are open for decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, although the coupling to the f1(1420)π has increased in this case, at the same time the couplings to the other open channels have decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Hence, the overall effect leads to a smaller width compared to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' π f1(1285) π1 (1600) K1/a1 ¯K/η π η′ π1 (1600) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' a) Diagram corresponding to the π1(1600) → f1(1285)π reaction, and b) the π1(1600) → η′π decay also via the AP loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The filled circles represent the effective couplings of the π1 to the AP meson pairs calculated from the residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The rectangles are the AP → η′π transition amplitudes at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The lower pole mass is slightly higher than the mass of the π1(1400) state listed in RPP, (1354 ± 25) MeV [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that we use the same subtraction constant for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In principle, it can take different values and lead to a shift of the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we did not include in the loops the b1 width, that is relatively large and whose effects could influence the pole position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' However, it is expected to affect more the imaginary part of the pole than the real one (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5(a) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We can get a rough estimate of this change by adding the b1 width to the previous result for Im(z1), with z1 the lower π1 pole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=', Γb1 + 2Im(z1) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 GeV , (41) which is close to the π1(1400) width reported in RPP, (330 ± 35) MeV [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' From these results, we are led to claim that the lower π1 pole may explain the π1(1400) resonance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' in other words, the π1(1400) is suitably described in our approach as a dynamically generated state with the b1π as its main component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Alternatively, following the prescription used in Section III, we can also study the changes in the results caused by the inclusion of the finite widths for the axial-vector mesons by looking at the line shape for the relevant T-matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5(a) we show the line shapes for the T-matrix element corresponding to the elastic b1π → b1π transition, which is the one we would expect the lower pole in Table XI manifests most due to its large coupling to the b1π channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It becomes clear that the bumps become broader when the widths of axial-vector mesons are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' A similar behavior can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5(b) for the T-matrix element associated 20 with the scattering of K1 (1270) ¯K, which is the channel to which the higher π1 pole couples most strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' w/o Γ w/o Γ w/ Γ w/ Γ 1300 1400 1500 1600 1700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='0 s [MeV] |T11 2 Set A Set B (a) Modulus square of elastic b1π scattering w/o Γ w/o Γ w/ Γ w/Γ 1400 1500 1600 1700 1800 1900 0 1 2 3 4 s [MeV] |T44 2 Set A Set B (b) Modulus square of elastic K1 (1270) ¯K scattering FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The dashed and solid lines correspond to zero and full widths of the axial-vector mesons in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The higher π1 pole, denoted now by z2, has a mass consistent with that of the π1(1600), whose pole mass has been reported to be � 1623 ± 47+24 −75 � MeV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [54] and (1564 ± 24 ± 86) MeV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It can decay into the η′π and f1(1285)π channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The corresponding diagrams for both amplitudes are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 4, from which we have Mf1(1285)π = gf1(1285)πεη1 · εf1 , (42) and Mη′π = gK1 ¯ KGK1 ¯ KVK1 ¯ K,η′π · εη1 + ga1ηGa1ηVa1η,η′π · εη1 , (43) with εη1 and εf1 the polarization vectors of the η1 and f1 (1285) mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here gf1(1285)π, gK1 ¯ K and ga1η are the effective coupling of the z2 pole to the corresponding couplings, and GK1 ¯ K and Ga1η are the loops involving the K1 ¯K and a1η mesons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Notice that the effective couplings are computed from the residues of the T matrix elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' thus they contain contributions from all coupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In order to compare our findings with the experimental information, we evaluate the ratio R1 = |Mf1(1285)π|2 q |Mη′π|2 ˜q , (44) where q and ˜q are the momentum in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' frame of the f1(1285)π and η′π pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Numerically, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (44) gives R1 = � � � � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='6 �A , � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='3 �B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (45) 21 The ratio is slightly bigger for the mixing angles in the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Nevertheless, the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (45) is consistent to the corresponding ratio 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='78 reported by the E852 Collaboration [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This good agreement with the experimental data supports the molecular picture for the π1(1600) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' DYNAMICAL GENERATION IN I = 1/2 SECTOR In the I = 1/2 sector, the corresponding WT amplitudes are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (16) with the Cij coefficients given in Tables VIII and IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' For each case, we have found two poles for parameter sets A and B, as shown in Table XII and XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' TABLE XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles and their corresponding couplings to the channels contributing to the PA interaction with JP = 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 a1K f1(1285)K K1(1270)η f1(1420)K K1(1400)η (+ + + + +) gl 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='10 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 a1K f1(1285)K K1(1270)η f1(1420)K K1(1400)η (+ + + + +) gl 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='15 TABLE XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles and their corresponding couplings to the channels contributing to the PA interaction with JP = 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Here the flavor-neutral axial mesons have JP C = 1+−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The errors of the poles are from varying the subtraction constant within α(µ = 1 GeV) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17, and only the central values of the couplings are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Poles (Set A) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='38 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='21 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 Poles (Set B) Channels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 h1(1170)K b1K K1(1270)η h1(1415)K K1(1400)η (− + + + +) gl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='55 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='78 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='69 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='83 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='17 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='04 Similarly to the previous cases, the poles are located on the same Riemann sheets in both sets of mixing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The interactions in the a1K and b1K channels are strong to generate a bound 22 state in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The existence of a lower h1 (1170) K channel below the b1K threshold moves the pole in Table XIII to Riemann sheet (− + + + +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' It has a nonzero imaginary part of a few MeV, which is not shown in the table due to precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' As discussed before, the I = 1/2 poles in Tables XII and XIII will receive sizeable widths once the width effects of the axial-vector mesons are taken into account, and it is expected that the widths are of the order of a few hundred MeV, like those of the b1 and a1 mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Although we neglected the transitions between the A1P and B1P sectors as discussed around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (17) in Section II, strange mesons are not C-parity eigenstates and the two dynamically generated I = 1/2 1− states will inevitably mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The two mixed states together could correspond to the 1− K∗ (1680) structure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' CONCLUSIONS We have studied the interactions between the pseudoscalar and axial-vector mesons in coupled channels with JPC = 1−(+) quantum numbers for the isospin 0, 1, and 1/2 sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Using the chiral unitary approach, we describe the interaction with the Weinberg-Tomozawa term derived from chiral Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The transition amplitudes among all the relevant channels are unitarized using the Bethe-Salpeter equation from which resonances (bound states) manifest themselves as poles on the (un)physical Riemann sheets of the complex energy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We consider the physical isoscalar axial-vector states as mixtures of the corresponding SU(3) singlets and octets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, the K1(1270) and K1(1400) physical states are also mixtures of the K1A and K1B mesons, which are the strange partners of the a1 and b1 resonances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We group into two sets, called A and B, the mixing angles accounting for such mechanisms and investigate their influence on the pole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' According to our findings, we obtain poles with JP(C) = 1−(+) quantum numbers in the energy range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='30 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='00 GeV, in each isospin sector studied (I = 0, 1, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The 1−+ quantum numbers are exotic in the sense that they cannot be formed from a pair of quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, we have found an isoscalar state that may correspond to the η1(1855) state, newly observed by the BESIII Collaboration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we have also found two dynamically gener- ated isovector states that we assign to be the π1(1400) and π1(1600) resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Hence, within our formalism, they are dynamically generated through the pseudoscalar-axial vector meson interac- tions, with the η1(1855) state coupling mostly to K1(1400) ¯K channel, while the π1(1400) couples strongly to the b1π, and π1(1600) structure couples most strongly to the K1(1270) ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We also 23 find two I = 1/2 JP = 1− states with a mass around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' They combined together could be responsible to the observed K∗(1680) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In addition, we also evaluate the decays of the η1(1855) and the π1(1600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We find that the three-body decay channel ¯KK∗π has a significantly larger branching fraction than the η′η, which is the channel where the observation of the η1(1855) was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' The obtained ratio between the π1(1600) → f1(1285)π and π1(1600) → η′π decays, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' (45), is consistent with the corresponding experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' We suggest searching for two additional η1 exotic mesons with masses of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='7 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' In particular, the latter should be relatively narrow with a width around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content='1 GeV and one of its main decay channels is K ¯Kππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' ACKNOWLEDGMENTS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Y is grateful to Shuang-Shi Fang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Valderrama for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' This project is supported in part by the National Natural Science Foundation of China (NSFC) under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12125507, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 11835015, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12047503;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' by the China Postdoctoral Science Foun- dation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 2022M713229;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' by the NSFC and the Deutsche Forschungsgemeinschaft (DFG) through the funds provided to the Sino-German Collaborative Research Center TRR110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' 12070131001, DFG Project-ID 196253076);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' and by the Chinese Academy of Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' XDB34030000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Chodos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Jaffe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfSQn_/content/2301.04432v1.pdf'} +page_content=' Johnson, C.' metadata={'source': 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