Interactons between two heavy mesons within chiral efgective fjeld - PowerPoint PPT Presentation

Interactons between two heavy mesons within chiral efgective fjeld theory Zhan-Wei LIU Lanzhou University Collaborators: Ning LI, Xiang LIU, Bo WANG, Hao XU, Shi-Lin ZHU XVIII International Conference on Hadron Spectroscopy and Structure, Guilin, 20/8/2019

CONTENTS 1. Introduction 2. Efgective potentials between two heavy mesons 3. Possible molecular states 1

Introduction

D s 2317 : contribution of DK continuum P c states reported at LHCb recently; Z b 10610 , Z b 10650 Hadron spectrum and interactions Hadron-hadron interactions are important for the hadron spectrum • threshold efgects • molecular states Deuteron: bound state of proton and neutron • other exotic states X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects? 2

P c states reported at LHCb recently; Z b 10610 , Z b 10650 Hadron spectrum and interactions Hadron-hadron interactions are important for the hadron spectrum • threshold efgects • molecular states Deuteron: bound state of proton and neutron • other exotic states X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects? 2 D s ( 2317 ) : contribution of DK continuum

Hadron spectrum and interactions Hadron-hadron interactions are important for the hadron spectrum • threshold efgects • molecular states Deuteron: bound state of proton and neutron • other exotic states X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects? 2 D s ( 2317 ) : contribution of DK continuum Z b ( 10610 ) , Z b ( 10650 ) P c states reported at LHCb recently;

Hadron spectrum and interactions Hadron-hadron interactions are important for the hadron spectrum • threshold efgects • molecular states Deuteron: bound state of proton and neutron • other exotic states X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects? 2 D s ( 2317 ) : contribution of DK continuum Z b ( 10610 ) , Z b ( 10650 ) P c states reported at LHCb recently;

Study of Interactions within chiral perturbation theory (ChPT) • ChPT with respect on symmetries of QCD • Power counting s , ... • expanded with small momentum • systematically study, order by order, error controlled • check of standard model 2-body force, 3-body force,... • Wide applications 3 • NOT in power series: α s , α 2 s , α 3 • Natual extension

Nucleon-nucleon interaction 4

ChPT with heavy hadrons involved • Dealing systems with light mesons ChPT results can be expanded as power series of • Power Counting Breaking (PCB) in systems with heavy hadrons involved 5 m φ / Λ χ , q / Λ χ , ... large masses of heavy hadrons make q µ is never small again q µ power counting can be recovered with the help of residual momentum ˜ q µ = q µ − m ( 1 ,⃗ ˜ 0 ) .

Solutions for systems with one heavy hadron • Heavy hadron efgective fjeld theory (EFT) nonrelativistic reduction at Lagrangian level, breaking of analyticity. Simple and still correct if not analytically extending results too far away relativistic Lagrangian, drop PCB terms at regularization good power counting and analyticity relativistic Lagrangian, drop PCB terms at fjnal results good power counting and analyticity Results with three difgerent schemes will be same if • being summarized at ALL orders, or • the mass of heavy hadron becomes infjnitely large. 6 • Infrared regularization • Extended on-mass-shell scheme

ChPT with few hadrons involved—new trouble (1) 1 we have not listed the parts preserving power counting Box Diagram. ( P represents the residual momentum) Solid line for nucleon, dashed line for pion. • eq. (1) • naïve power counting scheme 7 i i dl 0 The amplitude of following 2-Particle-Reducible diagram contains 1 − π ∫ I ≡ i ≈ . l 0 + P 0 − ⃗ − l 0 + P 0 − ⃗ ⃗ P 2 / ( 2 m N ) + i ε P 2 / ( 2 m N ) + i ε P 2 / ( 2 m N ) + i ε → I ∼ O ( 1 / | ⃗ P | ) → I ∼ O ( m N / | ⃗ P | 2 ) I is actually enhanced by a large factor m N / | ⃗ P | . P − l P l l P P + l

Weinberg scheme • not directly calculate physical observables with perturbation theory • systematically study efgective potentials fjrst (without 2PR contribution) • solve the dynamical equation to get the physical observables (equivalent to recover the 2PR contributions) 8

Efgective potentials between two heavy mesons

With Heavy Meson EFT, we study the systems made up of • DD D because there exist annihilation efgects. 9 • D ∗ D • D ∗ D ∗ Similar for B ( ∗ ) B ( ∗ ) and corresponding anti-meson pair system. We have not studied systems like D ¯

Lagrangians they absorb divergences, provide fjnite higher-order corrections • Leading order vertice 4 H 10 contact terms: D ( ∗ ) D ( ∗ ) D ( ∗ ) D ( ∗ ) vertice D ( ∗ ) D ( ∗ ) π , D ( ∗ ) D ( ∗ ) ππ vertice • Next-to-leading order vertice H ] Tr [ H γ µ ¯ L ( 0 ) D a Tr [ H γ µ ¯ H ] + D b Tr [ H γ µ γ 5 ¯ H ] Tr [ H γ µ γ 5 ¯ = H ] + E a Tr [ H γ µ λ a ¯ H ] + E b Tr [ H γ µ γ 5 λ a ¯ H ] Tr [ H γ µ λ a ¯ H ] Tr [ H γ µ γ 5 λ a ¯ H ] , 8 ∆ ⟨ H σ µν ¯ L ( 1 ) −⟨ ( iv · ∂ H )¯ H ⟩ − ⟨ Hv · Γ¯ H ⟩ + g ⟨ H ̸ u γ 5 ¯ = H ⟩ − 1 H σ µν ⟩ , H φ

Lagrangians 4 H • Leading order vertice D h 10 they absorb divergences, provide fjnite higher-order corrections contact terms: D ( ∗ ) D ( ∗ ) D ( ∗ ) D ( ∗ ) vertice D ( ∗ ) D ( ∗ ) π , D ( ∗ ) D ( ∗ ) ππ vertice • Next-to-leading order vertice H ] Tr [ H γ µ ¯ L ( 2 ) a Tr [ H γ µ ¯ = H ] Tr ( χ + ) + ... H ] Tr [ H γ µ ¯ χ + ¯ + D d a Tr [ H γ µ ˜ H ] + ... 1 Tr [( D µ H ) γ µ γ 5 ( D ν ¯ H )] Tr [ H γ ν γ 5 ¯ + D q H ] + ...

Diagrams • Leading order contact, one-pion exchange corrections to contact term, tree diagrams with NL vertice 11 • Next-to-leading order two-pion exchange, renormalization to D ( ∗ ) D ( ∗ ) π coupling, loop ( a ) ( b )

Diagrams corrections to contact term, tree diagrams with NL vertice • Leading order 11 contact, one-pion exchange • Next-to-leading order two-pion exchange, renormalization to D ( ∗ ) D ( ∗ ) π coupling, loop ( c 1) ( c 2) ( c 3) ( c 4) ( c 5) ( c 6) ( c 7) ( c 8) ( 9) ( 10)

Diagrams • Leading order contact, one-pion exchange corrections to contact term, tree diagrams with NL vertice 11 • Next-to-leading order two-pion exchange, renormalization to D ( ∗ ) D ( ∗ ) π coupling, loop ( b 1) ( b 2) ( b 3) ( b 4) ( b 5) ( b 6) ( b 7) ( b 8)

Diagrams • Leading order contact, one-pion exchange corrections to contact term, tree diagrams with NL vertice 11 • Next-to-leading order two-pion exchange, renormalization to D ( ∗ ) D ( ∗ ) π coupling, loop ( a 1) ( a 2) ( a 3) ( a 4) ( a 5) ( a 6) ( a 7) ( a 8)

Determination of low-energy constants • fjt to experimental data • fjrst principle of QCD • fjt to data of Lattice QCD • phenomenological models 12

Determination of low-energy constants • phenomenological models • fjt to experimental data 12 • fjt to data of Lattice QCD • fjrst principle of QCD 80 100 70 50 0 60 (MeV) (MeV) -50 50 QLQCD QLQCD 40 -100 V V -150 30 -200 20 -250 10 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 r(fm) r(fm)

Determination of low-energy constants • fjt to experimental data • fjrst principle of QCD • fjt to data of Lattice QCD • phenomenological models 12

Efgective potentials in momentum space 13 1 2 0 0 -1 -2 ) ) -2 -2 V ( q ) ( GeV V ( q ) ( GeV -2 -4 -3 -6 O ( 0 ) 1 -4 O ( 0 ) 1 -8 2 O ( ) 1 2 O ( ) 1 O ( 2 ) 2 -5 O ( 2 ) 2 -10 Total Total -6 -12 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 q ( GeV ) q ( GeV ) DD ∗ : I = 1 DD ∗ : I = 0

Possible molecular states

38 MeV. Search for new states • Comparison with one-boson-exchange model part but also by contact terms. / /... contribution is covered not only by the two-pion-exchange 15 . 4 3 0: Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; I 5 MeV; 43 0: Li,Sun,Liu,Zhu,PRD88(2013),114008; I 1: no bound state. I 19 21 0: bound state with around E • I as an example Taking DD space ) 14 • Potentials → partial waves, dynamical equation ( momentum space ) → T matrices → poles • Potentials → Fourier transform, dynamical equation ( coordinate → eigenvalues of bound states for difgerent partial waves

Search for new states 0: part but also by contact terms. / /... contribution is covered not only by the two-pion-exchange 15 . 4 3 0: Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; I 5 MeV; 43 Li,Sun,Liu,Zhu,PRD88(2013),114008; I • Comparison with one-boson-exchange model space ) 14 • Potentials → partial waves, dynamical equation ( momentum space ) → T matrices → poles • Potentials → Fourier transform, dynamical equation ( coordinate → eigenvalues of bound states for difgerent partial waves Taking DD ∗ as an example • I = 0: bound state with around E = − 21 + 19 − 38 MeV. I = 1: no bound state.