Zheng-Shu Liu, Xu-Liang Chen, Ding-Kun Lian, Ning Li, Wei Chen
{"title":"K1(1270/1400) 的混合角和 η1(1855) 的 KK¯1(1400) 分子解释","authors":"Zheng-Shu Liu, Xu-Liang Chen, Ding-Kun Lian, Ning Li, Wei Chen","doi":"10.1103/physrevd.111.014014","DOIUrl":null,"url":null,"abstract":"Owing to the SU(3) symmetry breaking effect, the axial-vector kaons K</a:mi>1</a:mn></a:msub>(</a:mo>1270</a:mn>)</a:mo></a:math> and <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:msub><e:mi>K</e:mi><e:mn>1</e:mn></e:msub><e:mo stretchy=\"false\">(</e:mo><e:mn>1400</e:mn><e:mo stretchy=\"false\">)</e:mo></e:math> are established to be mixtures of two P-wave <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mrow><i:msub><i:mrow><i:mi>K</i:mi></i:mrow><i:mrow><i:mn>1</i:mn><i:mi>A</i:mi></i:mrow></i:msub><i:mo stretchy=\"false\">(</i:mo><i:mrow><i:msub><i:mrow><i:mmultiscripts><i:mrow><i:mi>P</i:mi></i:mrow><i:mprescripts/><i:none/><i:mrow><i:mn>3</i:mn></i:mrow></i:mmultiscripts></i:mrow><i:mrow><i:mn>1</i:mn></i:mrow></i:msub><i:mo stretchy=\"false\">)</i:mo></i:mrow></i:mrow></i:math> and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:msub><m:mi>K</m:mi><m:mrow><m:mn>1</m:mn><m:mi>B</m:mi></m:mrow></m:msub><m:mrow><m:mo stretchy=\"false\">(</m:mo><m:msub><m:mrow><m:mmultiscripts><m:mrow><m:mi>P</m:mi></m:mrow><m:mprescripts/><m:none/><m:mrow><m:mn>1</m:mn></m:mrow></m:mmultiscripts></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo stretchy=\"false\">)</m:mo></m:mrow></m:math> states. In QCD sum rules, we propose a new construction of the <q:math xmlns:q=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><q:msub><q:mi>K</q:mi><q:mn>1</q:mn></q:msub></q:math> current operators and calculate the two-point correlation functions by including the next-to-leading order four-quark condensates. The mixing angle is determined as <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:mrow><s:mi>θ</s:mi><s:mo>=</s:mo><s:mo stretchy=\"false\">(</s:mo><s:msubsup><s:mrow><s:mn>46.95</s:mn></s:mrow><s:mrow><s:mo>−</s:mo><s:mn>0.23</s:mn></s:mrow><s:mrow><s:mo>+</s:mo><s:mn>0.25</s:mn></s:mrow></s:msubsup><s:mo stretchy=\"false\">)</s:mo><s:mo>°</s:mo></s:mrow></s:math> by reproducing the masses of <w:math xmlns:w=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><w:msub><w:mi>K</w:mi><w:mn>1</w:mn></w:msub><w:mo stretchy=\"false\">(</w:mo><w:mn>1270</w:mn><w:mo stretchy=\"false\">)</w:mo></w:math> and K</ab:mi>1</ab:mn></ab:msub>(</ab:mo>1400</ab:mn>)</ab:mo></ab:math>. We further compose the <eb:math xmlns:eb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><eb:mi>K</eb:mi><eb:msub><eb:mover accent=\"true\"><eb:mi>K</eb:mi><eb:mo stretchy=\"false\">¯</eb:mo></eb:mover><eb:mn>1</eb:mn></eb:msub><eb:mrow><eb:mo stretchy=\"false\">(</eb:mo><eb:mn>1270</eb:mn><eb:mo stretchy=\"false\">)</eb:mo></eb:mrow></eb:math> and <kb:math xmlns:kb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><kb:mi>K</kb:mi><kb:msub><kb:mover accent=\"true\"><kb:mi>K</kb:mi><kb:mo stretchy=\"false\">¯</kb:mo></kb:mover><kb:mn>1</kb:mn></kb:msub><kb:mrow><kb:mo stretchy=\"false\">(</kb:mo><kb:mn>1400</kb:mn><kb:mo stretchy=\"false\">)</kb:mo></kb:mrow></kb:math> interpolating currents with exotic quantum numbers <qb:math xmlns:qb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><qb:msup><qb:mi>J</qb:mi><qb:mrow><qb:mi>P</qb:mi><qb:mi>C</qb:mi></qb:mrow></qb:msup><qb:mo>=</qb:mo><qb:msup><qb:mn>1</qb:mn><qb:mrow><qb:mo>−</qb:mo><qb:mo>+</qb:mo></qb:mrow></qb:msup></qb:math> to investigate the possible molecular interpretation of the recently observed <sb:math xmlns:sb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><sb:msub><sb:mi>η</sb:mi><sb:mn>1</sb:mn></sb:msub><sb:mo stretchy=\"false\">(</sb:mo><sb:mn>1855</sb:mn><sb:mo stretchy=\"false\">)</sb:mo></sb:math> state. We calculate the correlation functions and perform the QCD sum rule analyses for these two molecular systems. However, the spectral functions are found to be negative in physical regions so that they are not able to provide reliable investigations of the <wb:math xmlns:wb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><wb:mi>K</wb:mi><wb:msub><wb:mover accent=\"true\"><wb:mi>K</wb:mi><wb:mo stretchy=\"false\">¯</wb:mo></wb:mover><wb:mn>1</wb:mn></wb:msub></wb:math> molecular states. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20167,"journal":{"name":"Physical Review D","volume":"118 1","pages":""},"PeriodicalIF":5.3000,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Mixing angle of K1(1270/1400) and the KK¯1(1400) molecular interpretation of η1(1855)\",\"authors\":\"Zheng-Shu Liu, Xu-Liang Chen, Ding-Kun Lian, Ning Li, Wei Chen\",\"doi\":\"10.1103/physrevd.111.014014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Owing to the SU(3) symmetry breaking effect, the axial-vector kaons K</a:mi>1</a:mn></a:msub>(</a:mo>1270</a:mn>)</a:mo></a:math> and <e:math xmlns:e=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><e:msub><e:mi>K</e:mi><e:mn>1</e:mn></e:msub><e:mo stretchy=\\\"false\\\">(</e:mo><e:mn>1400</e:mn><e:mo stretchy=\\\"false\\\">)</e:mo></e:math> are established to be mixtures of two P-wave <i:math xmlns:i=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><i:mrow><i:msub><i:mrow><i:mi>K</i:mi></i:mrow><i:mrow><i:mn>1</i:mn><i:mi>A</i:mi></i:mrow></i:msub><i:mo stretchy=\\\"false\\\">(</i:mo><i:mrow><i:msub><i:mrow><i:mmultiscripts><i:mrow><i:mi>P</i:mi></i:mrow><i:mprescripts/><i:none/><i:mrow><i:mn>3</i:mn></i:mrow></i:mmultiscripts></i:mrow><i:mrow><i:mn>1</i:mn></i:mrow></i:msub><i:mo stretchy=\\\"false\\\">)</i:mo></i:mrow></i:mrow></i:math> and <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><m:msub><m:mi>K</m:mi><m:mrow><m:mn>1</m:mn><m:mi>B</m:mi></m:mrow></m:msub><m:mrow><m:mo stretchy=\\\"false\\\">(</m:mo><m:msub><m:mrow><m:mmultiscripts><m:mrow><m:mi>P</m:mi></m:mrow><m:mprescripts/><m:none/><m:mrow><m:mn>1</m:mn></m:mrow></m:mmultiscripts></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo stretchy=\\\"false\\\">)</m:mo></m:mrow></m:math> states. In QCD sum rules, we propose a new construction of the <q:math xmlns:q=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><q:msub><q:mi>K</q:mi><q:mn>1</q:mn></q:msub></q:math> current operators and calculate the two-point correlation functions by including the next-to-leading order four-quark condensates. The mixing angle is determined as <s:math xmlns:s=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><s:mrow><s:mi>θ</s:mi><s:mo>=</s:mo><s:mo stretchy=\\\"false\\\">(</s:mo><s:msubsup><s:mrow><s:mn>46.95</s:mn></s:mrow><s:mrow><s:mo>−</s:mo><s:mn>0.23</s:mn></s:mrow><s:mrow><s:mo>+</s:mo><s:mn>0.25</s:mn></s:mrow></s:msubsup><s:mo stretchy=\\\"false\\\">)</s:mo><s:mo>°</s:mo></s:mrow></s:math> by reproducing the masses of <w:math xmlns:w=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><w:msub><w:mi>K</w:mi><w:mn>1</w:mn></w:msub><w:mo stretchy=\\\"false\\\">(</w:mo><w:mn>1270</w:mn><w:mo stretchy=\\\"false\\\">)</w:mo></w:math> and K</ab:mi>1</ab:mn></ab:msub>(</ab:mo>1400</ab:mn>)</ab:mo></ab:math>. We further compose the <eb:math xmlns:eb=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><eb:mi>K</eb:mi><eb:msub><eb:mover accent=\\\"true\\\"><eb:mi>K</eb:mi><eb:mo stretchy=\\\"false\\\">¯</eb:mo></eb:mover><eb:mn>1</eb:mn></eb:msub><eb:mrow><eb:mo stretchy=\\\"false\\\">(</eb:mo><eb:mn>1270</eb:mn><eb:mo stretchy=\\\"false\\\">)</eb:mo></eb:mrow></eb:math> and <kb:math xmlns:kb=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><kb:mi>K</kb:mi><kb:msub><kb:mover accent=\\\"true\\\"><kb:mi>K</kb:mi><kb:mo stretchy=\\\"false\\\">¯</kb:mo></kb:mover><kb:mn>1</kb:mn></kb:msub><kb:mrow><kb:mo stretchy=\\\"false\\\">(</kb:mo><kb:mn>1400</kb:mn><kb:mo stretchy=\\\"false\\\">)</kb:mo></kb:mrow></kb:math> interpolating currents with exotic quantum numbers <qb:math xmlns:qb=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><qb:msup><qb:mi>J</qb:mi><qb:mrow><qb:mi>P</qb:mi><qb:mi>C</qb:mi></qb:mrow></qb:msup><qb:mo>=</qb:mo><qb:msup><qb:mn>1</qb:mn><qb:mrow><qb:mo>−</qb:mo><qb:mo>+</qb:mo></qb:mrow></qb:msup></qb:math> to investigate the possible molecular interpretation of the recently observed <sb:math xmlns:sb=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><sb:msub><sb:mi>η</sb:mi><sb:mn>1</sb:mn></sb:msub><sb:mo stretchy=\\\"false\\\">(</sb:mo><sb:mn>1855</sb:mn><sb:mo stretchy=\\\"false\\\">)</sb:mo></sb:math> state. We calculate the correlation functions and perform the QCD sum rule analyses for these two molecular systems. However, the spectral functions are found to be negative in physical regions so that they are not able to provide reliable investigations of the <wb:math xmlns:wb=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><wb:mi>K</wb:mi><wb:msub><wb:mover accent=\\\"true\\\"><wb:mi>K</wb:mi><wb:mo stretchy=\\\"false\\\">¯</wb:mo></wb:mover><wb:mn>1</wb:mn></wb:msub></wb:math> molecular states. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>\",\"PeriodicalId\":20167,\"journal\":{\"name\":\"Physical Review D\",\"volume\":\"118 1\",\"pages\":\"\"},\"PeriodicalIF\":5.3000,\"publicationDate\":\"2025-01-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Physical Review D\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1103/physrevd.111.014014\",\"RegionNum\":2,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Physics and Astronomy\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review D","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevd.111.014014","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
Mixing angle of K1(1270/1400) and the KK¯1(1400) molecular interpretation of η1(1855)
Owing to the SU(3) symmetry breaking effect, the axial-vector kaons K1(1270) and K1(1400) are established to be mixtures of two P-wave K1A(P31) and K1B(P11) states. In QCD sum rules, we propose a new construction of the K1 current operators and calculate the two-point correlation functions by including the next-to-leading order four-quark condensates. The mixing angle is determined as θ=(46.95−0.23+0.25)° by reproducing the masses of K1(1270) and K1(1400). We further compose the KK¯1(1270) and KK¯1(1400) interpolating currents with exotic quantum numbers JPC=1−+ to investigate the possible molecular interpretation of the recently observed η1(1855) state. We calculate the correlation functions and perform the QCD sum rule analyses for these two molecular systems. However, the spectral functions are found to be negative in physical regions so that they are not able to provide reliable investigations of the KK¯1 molecular states. Published by the American Physical Society2025
期刊介绍：
Physical Review D (PRD) is a leading journal in elementary particle physics, field theory, gravitation, and cosmology and is one of the top-cited journals in high-energy physics.
PRD covers experimental and theoretical results in all aspects of particle physics, field theory, gravitation and cosmology, including:
Particle physics experiments,
Electroweak interactions,
Strong interactions,
Lattice field theories, lattice QCD,
Beyond the standard model physics,
Phenomenological aspects of field theory, general methods,
Gravity, cosmology, cosmic rays,
Astrophysics and astroparticle physics,
General relativity,
Formal aspects of field theory, field theory in curved space,
String theory, quantum gravity, gauge/gravity duality.
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