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

Hadron spectrum and interactions

Hadron-hadron interactions are important for the hadron spectrum

• threshold efgects

Ds 2317 : contribution of DK continuum

• molecular states

Deuteron: bound state of proton and neutron Pc states reported at LHCb recently; Zb 10610 , Zb 10650

• other exotic states

X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects?

2

Hadron spectrum and interactions

Hadron-hadron interactions are important for the hadron spectrum

• threshold efgects

Ds(2317): contribution of DK continuum

• molecular states

Deuteron: bound state of proton and neutron Pc states reported at LHCb recently; Zb 10610 , Zb 10650

• other exotic states

X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects?

2

Hadron spectrum and interactions

Hadron-hadron interactions are important for the hadron spectrum

• threshold efgects

Ds(2317): contribution of DK continuum

• molecular states

Deuteron: bound state of proton and neutron Pc states reported at LHCb recently; Zb(10610), Zb(10650)

• other exotic states

X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects?

2

Hadron spectrum and interactions

Hadron-hadron interactions are important for the hadron spectrum

• threshold efgects

Ds(2317): contribution of DK continuum

• molecular states

Deuteron: bound state of proton and neutron Pc states reported at LHCb recently; Zb(10610), Zb(10650)

• other exotic states

X, Y, Z states, debate with difgerent interpretations: molecules? tetraquark? ordinary charmonium? two diquark? kinetic efgects?

2

Study of Interactions within chiral perturbation theory (ChPT)

• ChPT with respect on symmetries of QCD
• Power counting
• NOT in power series: αs, α2

s, α3 s, ...

• expanded with small momentum
• systematically study, order by order, error controlled
• check of standard model
• Natual extension

2-body force, 3-body force,...

• Wide applications

3

Nucleon-nucleon interaction

4

ChPT with heavy hadrons involved

• Dealing systems with light mesons

ChPT results can be expanded as power series of mφ/Λχ, q/Λχ, ...

• Power Counting Breaking (PCB) in systems with heavy hadrons

involved

large masses of heavy hadrons make qµ is never small again power counting can be recovered with the help of residual momentum ˜ qµ ˜ qµ = qµ − m(1,⃗ 0).

5

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

• Infrared regularization

relativistic Lagrangian, drop PCB terms at regularization good power counting and analyticity

• Extended on-mass-shell scheme

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

ChPT with few hadrons involved—new trouble

The amplitude of following 2-Particle-Reducible diagram contains 1 I ≡ i ∫ dl0 i l0 + P0 − ⃗ P2/(2mN) + iε i −l0 + P0 − ⃗ P2/(2mN) + iε ≈ −π ⃗ P2/(2mN) + iε . (1)

• naïve power counting scheme

→ I ∼ O(1/|⃗ P|)

• eq. (1)

→ I ∼ O(mN/|⃗ P|2) I is actually enhanced by a large factor mN/|⃗ P|.

l l P + l P − l

P P

Solid line for nucleon, dashed line for pion. (P represents the residual momentum) Box Diagram.

1we have not listed the parts preserving power counting

7

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∗D
• D∗D∗

Similar for B(∗)B(∗) and corresponding anti-meson pair system. We have not studied systems like D¯ D because there exist annihilation efgects.

9

Lagrangians

• Leading order vertice

contact terms: D(∗)D(∗)D(∗)D(∗) vertice D(∗)D(∗)π, D(∗)D(∗)ππ vertice

• Next-to-leading order vertice

they absorb divergences, provide fjnite higher-order corrections

L(0)

4H

= Da Tr [Hγµ ¯ H] Tr [Hγµ ¯ H] + Db Tr [Hγµγ5 ¯ H] Tr [Hγµγ5 ¯ H] +Ea Tr [Hγµλa ¯ H] Tr [Hγµλa ¯ H] + Eb Tr [Hγµγ5λa ¯ H] Tr [Hγµγ5λa ¯ H], L(1)

Hφ

= −⟨(iv · ∂H)¯ H⟩ − ⟨Hv · Γ¯ H⟩ + g⟨H̸uγ5 ¯ H⟩ − 1 8∆⟨Hσµν ¯ Hσµν⟩,

10

Lagrangians

• Leading order vertice

contact terms: D(∗)D(∗)D(∗)D(∗) vertice D(∗)D(∗)π, D(∗)D(∗)ππ vertice

• Next-to-leading order vertice

they absorb divergences, provide fjnite higher-order corrections

L(2)

4H

= Dh

a Tr [Hγµ ¯

H] Tr [Hγµ ¯ H] Tr (χ+) + ... +Dd

a Tr [Hγµ ˜

χ+ ¯ H] Tr [Hγµ ¯ H] + ... +Dq

1 Tr [(DµH)γµγ5(Dν ¯

H)] Tr [Hγνγ5 ¯ H] + ...

10

Diagrams

• Leading order

contact, one-pion exchange

• Next-to-leading order

two-pion exchange, renormalization to D(∗)D(∗)π coupling, loop corrections to contact term, tree diagrams with NL vertice

(a) (b)

11

Diagrams

• Leading order

contact, one-pion exchange

• Next-to-leading order

two-pion exchange, renormalization to D(∗)D(∗)π coupling, loop corrections to contact term, tree diagrams with NL vertice

(c1) (c2) (c3) (c4) (c5) (c6) (c7) (c8) ( 9) ( 10)

11

Diagrams

• Leading order

contact, one-pion exchange

• Next-to-leading order

two-pion exchange, renormalization to D(∗)D(∗)π coupling, loop corrections to contact term, tree diagrams with NL vertice

(b1) (b2) (b3) (b4) (b5) (b6) (b7) (b8) 11

Diagrams

• Leading order

contact, one-pion exchange

• Next-to-leading order

two-pion exchange, renormalization to D(∗)D(∗)π coupling, loop corrections to contact term, tree diagrams with NL vertice

(a1) (a2) (a3) (a4) (a5) (a6) (a7) (a8) 11

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

• fjt to experimental data
• fjrst principle of QCD
• fjt to data of Lattice QCD
• phenomenological models

r(fm) 0.1 0.2 0.3 0.4 0.5 0.6 (MeV)

QLQCD

V 10 20 30 40 50 60 70 80 r(fm) 0.1 0.2 0.3 0.4 0.5 0.6 (MeV)

QLQCD

V

• 250
• 200
• 150
• 100
• 50

50 100

12

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

• 6
• 5
• 4
• 3
• 2
• 1

• 2

• 12
• 10
• 8
• 6
• 4
• 2

• 2

DD∗ : I = 1 DD∗ : I = 0

13

Possible molecular states

Search for new states

• Potentials→ partial waves, dynamical equation (momentum space)

→ T matrices → poles

• Potentials→ Fourier transform, dynamical equation (coordinate

space)

→ 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.

• Comparison with one-boson-exchange model

Li,Sun,Liu,Zhu,PRD88(2013),114008;I 0: 43 5 MeV; Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; I 0: 3

4 15.

/ /... contribution is covered not only by the two-pion-exchange part but also by contact terms.

14

Search for new states

• Potentials→ partial waves, dynamical equation (momentum space)

→ T matrices → poles

• Potentials→ Fourier transform, dynamical equation (coordinate

space)

→ 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.

• Comparison with one-boson-exchange model

Li,Sun,Liu,Zhu,PRD88(2013),114008;I 0: 43 5 MeV; Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; I 0: 3

4 15.

/ /... contribution is covered not only by the two-pion-exchange part but also by contact terms.

14

Search for new states

• Potentials→ partial waves, dynamical equation (momentum space)

→ T matrices → poles

• Potentials→ Fourier transform, dynamical equation (coordinate

space)

→ 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.

• Comparison with one-boson-exchange model

Li,Sun,Liu,Zhu,PRD88(2013),114008;I = 0: −43 ∼ −5 MeV; Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; I = 0: −3+4

−15.

ρ/ω/... contribution is covered not only by the two-pion-exchange part but also by contact terms.

14

¯ B(∗)¯ B(∗) systems

We use similar approaches to study the system of ¯ B(∗)¯ B(∗) in S wave

• ¯

B¯ B: I(JP) = 1(0+)

• ¯

B¯ B∗: I(JP) = 1(1+), I(JP) = 0(1+)

• ¯

B∗¯ B∗: I(JP) = 1(0+), I(JP) = 1(2+), I(JP) = 0(1+)

15

¯ B(∗)¯ B(∗) systems

We use similar approaches to study the system of ¯ B(∗)¯ B(∗) in S wave

• ¯

B¯ B: I(JP) = 1(0+)

• ¯

B¯ B∗: I(JP) = 1(1+), I(JP) = 0(1+)

• ¯

B∗¯ B∗: I(JP) = 1(0+), I(JP) = 1(2+), I(JP) = 0(1+)

16

Example: potentials for ¯ B∗¯ B∗ with I(JP) = 0(1+) in momentum space

| q| (GeV)

0.05 0.1 0.15 0.2 0.25 0.3

V ( q) (GeV−2)

• 10
• 5

5 10 15

O(ǫ0) 1-π O(ǫ2) 1-π O(ǫ2) 2-π Total

(a)

17

Example: potentials for ¯ B∗¯ B∗ with I(JP) = 0(1+) in coordinate space

r (GeV−1)

2 4 6 8 10 12 14 16

V (r) (GeV)

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1

O(ǫ0)+O(ǫ2) 1-π O(ǫ2) 2-π O(ǫ0)+O(ǫ2) Contact Total

(b)

18

Results for ¯ B(∗)¯ B(∗) systems

we fjnd two bound states in the channels of 0(1+) ¯ B¯ B∗ and ¯ B∗¯ B∗

• binding energies: ∆E¯

B¯ B∗ ≃ −12.6+9.2 −12.9 MeV, ∆E¯ B∗ ¯ B∗ ≃ −23.8+16.3 −21.5

MeV masses: m¯

B¯ B∗ ≃ 10591.4+9.2 −12.9 MeV,

m¯

B∗ ¯ B∗ ≃ 10625.5+16.3 −21.5 MeV

• strong decays are forbidden because of phase space

they can be searched in ¯ B¯ Bγ or ¯ B¯ Bγγ Bound states bbqq with I JP 0 1 also existed in other framework

Eichten,Quigg„PRL.119(2017),202002; Karliner,Rosner,PRL.119(2017)202001; Bicudo,Scheunert,Wagner,PRD95(2017)034502; Wang,Acta Phys.Polon.B49(2018)1781; Park,Noh,Lee,Nucl.Phys.A983(2019)1–19; Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; Francis,Hudspith,Lewis,Maltman,PRL.118(2017)142001; Junnarkar,Mathur,Padmanath,PRD99(2019)034507; Leskovec,Meinel,Pfmaumer,Wagner,PRD100(2019),014503

19

Results for ¯ B(∗)¯ B(∗) systems

we fjnd two bound states in the channels of 0(1+) ¯ B¯ B∗ and ¯ B∗¯ B∗

• binding energies: ∆E¯

B¯ B∗ ≃ −12.6+9.2 −12.9 MeV, ∆E¯ B∗ ¯ B∗ ≃ −23.8+16.3 −21.5

MeV masses: m¯

B¯ B∗ ≃ 10591.4+9.2 −12.9 MeV,

m¯

B∗ ¯ B∗ ≃ 10625.5+16.3 −21.5 MeV

• strong decays are forbidden because of phase space

they can be searched in ¯ B¯ Bγ or ¯ B¯ Bγγ Bound states [bb¯ q¯ q] with I(JP) = 0(1+) also existed in other framework

Eichten,Quigg„PRL.119(2017),202002; Karliner,Rosner,PRL.119(2017)202001; Bicudo,Scheunert,Wagner,PRD95(2017)034502; Wang,Acta Phys.Polon.B49(2018)1781; Park,Noh,Lee,Nucl.Phys.A983(2019)1–19; Liu,Wu,Valderrama,Xie,Geng,PRD99(2019),094018; Francis,Hudspith,Lewis,Maltman,PRL.118(2017)142001; Junnarkar,Mathur,Padmanath,PRD99(2019)034507; Leskovec,Meinel,Pfmaumer,Wagner,PRD100(2019),014503

19

Uncertainty of low-energy constants

Table 1: The binding energies of 0(1+) ¯ B¯ B∗ and ¯ B∗¯ B∗ states obtained with difgerent strategies in units of MeV.

Binding energy No O(ϵ2) LECs Strategy A Strategy B ∆E¯

B¯ B∗

−12.6+9.2

−12.9

−10.4+7.2

−9.7

−15.9+9.7

−12.7

∆E¯

B∗ ¯ B∗

−23.8+16.3

−21.5

−20.1+14.5

−20.0

−28.2+18.6

−23.6 20

Summary

We have studied the potentials upto one-loop level between two heavy mesons within chiral perturbation theory. By solving the Schrodinger equations, we found some bound states in some channels. With the wavefunctions obtained, we can further study other properties

• f these new states.

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Thanks!

22

Thanks!

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This report is mainly based on the following articles

• Phys.Rev. D99 (2019) no.3, 036007
• Phys.Rev. D99 (2019) no.1, 014027
• Phys.Rev. D89 (2014) no.7, 074015

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