Long Zeng, Xing-Gang Wu, Dan-Dan Hu, Hai-Bing Fu, Tao Zhong
{"title":"Longitudinal leading-twist distribution amplitude of the P11 -state b1(1235) meson and its implications on B→b1(1235)ℓ+νℓ decays","authors":"Long Zeng, Xing-Gang Wu, Dan-Dan Hu, Hai-Bing Fu, Tao Zhong","doi":"10.1103/physrevd.111.056030","DOIUrl":null,"url":null,"abstract":"In the paper, we derive the ξ</a:mi></a:math> moments <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mo stretchy=\"false\">⟨</c:mo><c:msubsup><c:mi>ξ</c:mi><c:mrow><c:mn>2</c:mn><c:mo>;</c:mo><c:msub><c:mi>b</c:mi><c:mn>1</c:mn></c:msub></c:mrow><c:mrow><c:mi>n</c:mi><c:mo>;</c:mo><c:mo stretchy=\"false\">∥</c:mo></c:mrow></c:msubsup><c:mo stretchy=\"false\">⟩</c:mo></c:math> of the longitudinal leading-twist distribution amplitude <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><h:msubsup><h:mi>ϕ</h:mi><h:mrow><h:mn>2</h:mn><h:mo>;</h:mo><h:msub><h:mi>b</h:mi><h:mn>1</h:mn></h:msub></h:mrow><h:mo stretchy=\"false\">∥</h:mo></h:msubsup></h:math> for the <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mrow><k:mmultiscripts><k:mrow><k:msub><k:mrow><k:mi>P</k:mi></k:mrow><k:mrow><k:mn>1</k:mn></k:mrow></k:msub></k:mrow><k:mprescripts/><k:none/><k:mrow><k:mn>1</k:mn></k:mrow></k:mmultiscripts></k:mrow></k:math>-state <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:msub><m:mi>b</m:mi><m:mn>1</m:mn></m:msub><m:mo stretchy=\"false\">(</m:mo><m:mn>1235</m:mn><m:mo stretchy=\"false\">)</m:mo></m:math> meson by using the QCD sum rules under the background field theory. Considering the contributions from the vacuum condensates up to dimension six, its first two nonzero <q:math xmlns:q=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><q:mi>ξ</q:mi></q:math> moments at the scale 1 GeV are <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:mo stretchy=\"false\">⟨</s:mo><s:msubsup><s:mi>ξ</s:mi><s:mrow><s:mn>2</s:mn><s:mo>;</s:mo><s:msub><s:mi>b</s:mi><s:mn>1</s:mn></s:msub></s:mrow><s:mrow><s:mn>1</s:mn><s:mo>;</s:mo><s:mo stretchy=\"false\">∥</s:mo></s:mrow></s:msubsup><s:mo stretchy=\"false\">⟩</s:mo><s:mo>=</s:mo><s:mo>−</s:mo><s:mn>0.64</s:mn><s:msubsup><s:mn>7</s:mn><s:mrow><s:mo>−</s:mo><s:mn>0.113</s:mn></s:mrow><s:mrow><s:mo>+</s:mo><s:mn>0.118</s:mn></s:mrow></s:msubsup></s:math> and <x:math xmlns:x=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><x:mo stretchy=\"false\">⟨</x:mo><x:msubsup><x:mi>ξ</x:mi><x:mrow><x:mn>2</x:mn><x:mo>;</x:mo><x:msub><x:mi>b</x:mi><x:mn>1</x:mn></x:msub></x:mrow><x:mrow><x:mn>3</x:mn><x:mo>;</x:mo><x:mo stretchy=\"false\">∥</x:mo></x:mrow></x:msubsup><x:mo stretchy=\"false\">⟩</x:mo><x:mo>=</x:mo><x:mo>−</x:mo><x:mn>0.32</x:mn><x:msubsup><x:mn>8</x:mn><x:mrow><x:mo>−</x:mo><x:mn>0.052</x:mn></x:mrow><x:mrow><x:mo>+</x:mo><x:mn>0.055</x:mn></x:mrow></x:msubsup></x:math>, respectively. Using those moments, we then fix the Gegenbauer expansion series of <cb:math xmlns:cb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><cb:msubsup><cb:mi>ϕ</cb:mi><cb:mrow><cb:mn>2</cb:mn><cb:mo>;</cb:mo><cb:msub><cb:mi>b</cb:mi><cb:mn>1</cb:mn></cb:msub></cb:mrow><cb:mo stretchy=\"false\">∥</cb:mo></cb:msubsup></cb:math> and apply it to calculate <fb:math xmlns:fb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><fb:mi>B</fb:mi><fb:mo stretchy=\"false\">→</fb:mo><fb:msub><fb:mi>b</fb:mi><fb:mn>1</fb:mn></fb:msub><fb:mo stretchy=\"false\">(</fb:mo><fb:mn>1235</fb:mn><fb:mo stretchy=\"false\">)</fb:mo></fb:math> transition form factors (TFFs) that are derived by using the QCD light-cone sum rules. Those TFFs are then extrapolated to the physically allowable <kb:math xmlns:kb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><kb:msup><kb:mi>q</kb:mi><kb:mn>2</kb:mn></kb:msup></kb:math> range via the simplified series expansion. As for the branching fractions, we obtain <mb:math xmlns:mb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mb:mi mathvariant=\"script\">B</mb:mi><mb:mo stretchy=\"false\">(</mb:mo><mb:msup><mb:mover accent=\"true\"><mb:mi>B</mb:mi><mb:mo stretchy=\"false\">¯</mb:mo></mb:mover><mb:mn>0</mb:mn></mb:msup><mb:mo stretchy=\"false\">→</mb:mo><mb:msub><mb:mi>b</mb:mi><mb:mn>1</mb:mn></mb:msub><mb:mo stretchy=\"false\">(</mb:mo><mb:mn>1235</mb:mn><mb:msup><mb:mo stretchy=\"false\">)</mb:mo><mb:mo>+</mb:mo></mb:msup><mb:msup><mb:mi>e</mb:mi><mb:mo>−</mb:mo></mb:msup><mb:msub><mb:mover accent=\"true\"><mb:mi>ν</mb:mi><mb:mo stretchy=\"false\">¯</mb:mo></mb:mover><mb:mi>e</mb:mi></mb:msub><mb:mo stretchy=\"false\">)</mb:mo><mb:mo>=</mb:mo><mb:mn>2.17</mb:mn><mb:msubsup><mb:mn>9</mb:mn><mb:mrow><mb:mo>−</mb:mo><mb:mn>0.422</mb:mn></mb:mrow><mb:mrow><mb:mo>+</mb:mo><mb:mn>0.553</mb:mn></mb:mrow></mb:msubsup><mb:mo>×</mb:mo><mb:msup><mb:mn>10</mb:mn><mb:mrow><mb:mo>−</mb:mo><mb:mn>4</mb:mn></mb:mrow></mb:msup></mb:math>, <yb:math xmlns:yb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><yb:mi mathvariant=\"script\">B</yb:mi><yb:mo stretchy=\"false\">(</yb:mo><yb:msup><yb:mi>B</yb:mi><yb:mn>0</yb:mn></yb:msup><yb:mo stretchy=\"false\">→</yb:mo><yb:msub><yb:mi>b</yb:mi><yb:mn>1</yb:mn></yb:msub><yb:mo stretchy=\"false\">(</yb:mo><yb:mn>1235</yb:mn><yb:msup><yb:mo stretchy=\"false\">)</yb:mo><yb:mo>−</yb:mo></yb:msup><yb:msup><yb:mi>μ</yb:mi><yb:mo>+</yb:mo></yb:msup><yb:msub><yb:mi>ν</yb:mi><yb:mi>μ</yb:mi></yb:msub><yb:mo stretchy=\"false\">)</yb:mo><yb:mo>=</yb:mo><yb:mn>2.16</yb:mn><yb:msubsup><yb:mn>6</yb:mn><yb:mrow><yb:mo>−</yb:mo><yb:mn>0.415</yb:mn></yb:mrow><yb:mrow><yb:mo>+</yb:mo><yb:mn>0.544</yb:mn></yb:mrow></yb:msubsup><yb:mo>×</yb:mo><yb:msup><yb:mn>10</yb:mn><yb:mrow><yb:mo>−</yb:mo><yb:mn>4</yb:mn></yb:mrow></yb:msup></yb:math>, <gc:math xmlns:gc=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><gc:mi mathvariant=\"script\">B</gc:mi><gc:mo stretchy=\"false\">(</gc:mo><gc:msup><gc:mi>B</gc:mi><gc:mo>+</gc:mo></gc:msup><gc:mo stretchy=\"false\">→</gc:mo><gc:mspace linebreak=\"goodbreak\"/><gc:msub><gc:mi>b</gc:mi><gc:mn>1</gc:mn></gc:msub><gc:mo stretchy=\"false\">(</gc:mo><gc:mn>1235</gc:mn><gc:msup><gc:mo stretchy=\"false\">)</gc:mo><gc:mn>0</gc:mn></gc:msup><gc:msup><gc:mi>e</gc:mi><gc:mo>+</gc:mo></gc:msup><gc:msub><gc:mi>ν</gc:mi><gc:mi>e</gc:mi></gc:msub><gc:mo stretchy=\"false\">)</gc:mo><gc:mo>=</gc:mo><gc:mn>2.35</gc:mn><gc:msubsup><gc:mn>3</gc:mn><gc:mrow><gc:mo>−</gc:mo><gc:mn>0.456</gc:mn></gc:mrow><gc:mrow><gc:mo>+</gc:mo><gc:mn>0.597</gc:mn></gc:mrow></gc:msubsup><gc:mo>×</gc:mo><gc:msup><gc:mn>10</gc:mn><gc:mrow><gc:mo>−</gc:mo><gc:mn>4</gc:mn></gc:mrow></gc:msup></gc:math>, and <pc:math xmlns:pc=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><pc:mi mathvariant=\"script\">B</pc:mi><pc:mo stretchy=\"false\">(</pc:mo><pc:msup><pc:mi>B</pc:mi><pc:mo>+</pc:mo></pc:msup><pc:mo stretchy=\"false\">→</pc:mo><pc:msub><pc:mi>b</pc:mi><pc:mn>1</pc:mn></pc:msub><pc:mo stretchy=\"false\">(</pc:mo><pc:mn>1235</pc:mn><pc:msup><pc:mo stretchy=\"false\">)</pc:mo><pc:mn>0</pc:mn></pc:msup><pc:msup><pc:mi>μ</pc:mi><pc:mo>+</pc:mo></pc:msup><pc:msub><pc:mi>ν</pc:mi><pc:mi>μ</pc:mi></pc:msub><pc:mo stretchy=\"false\">)</pc:mo><pc:mo>=</pc:mo><pc:mn>2.33</pc:mn><pc:msubsup><pc:mn>9</pc:mn><pc:mrow><pc:mo>−</pc:mo><pc:mn>0.448</pc:mn></pc:mrow><pc:mrow><pc:mo>+</pc:mo><pc:mn>0.587</pc:mn></pc:mrow></pc:msubsup><pc:mo>×</pc:mo><pc:msup><pc:mn>10</pc:mn><pc:mrow><pc:mo>−</pc:mo><pc:mn>4</pc:mn></pc:mrow></pc:msup></pc:math>, respectively. <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":"42 1","pages":""},"PeriodicalIF":5.0000,"publicationDate":"2025-03-31","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.111.056030","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}
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Abstract
In the paper, we derive the ξ moments ⟨ξ2;b1n;∥⟩ of the longitudinal leading-twist distribution amplitude ϕ2;b1∥ for the P11-state b1(1235) meson by using the QCD sum rules under the background field theory. Considering the contributions from the vacuum condensates up to dimension six, its first two nonzero ξ moments at the scale 1 GeV are ⟨ξ2;b11;∥⟩=−0.647−0.113+0.118 and ⟨ξ2;b13;∥⟩=−0.328−0.052+0.055, respectively. Using those moments, we then fix the Gegenbauer expansion series of ϕ2;b1∥ and apply it to calculate B→b1(1235) transition form factors (TFFs) that are derived by using the QCD light-cone sum rules. Those TFFs are then extrapolated to the physically allowable q2 range via the simplified series expansion. As for the branching fractions, we obtain B(B¯0→b1(1235)+e−ν¯e)=2.179−0.422+0.553×10−4, B(B0→b1(1235)−μ+νμ)=2.166−0.415+0.544×10−4, B(B+→b1(1235)0e+νe)=2.353−0.456+0.597×10−4, and B(B+→b1(1235)0μ+νμ)=2.339−0.448+0.587×10−4, respectively. Published by the American Physical Society2025
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