Hadronic bound states and effective field theories

Transcription

1 Hadronic bound states and effective field theories Feng-Kun Guo Institute of Theoretical Physics, CAS 55th Winter School for Theoretical Physics Feb. 13 Feb. 17, 2017, Admont, Austria Feng-Kun Guo (ITP) Bound states and EFTs / 89

2 Outline 1 Symmetries of QCD EFT and chiral symmetry in a nut shell Heavy quark symmetries Combining chiral symmetry and heavy quark symmetries 2 From HQS to new hadrons HQS and open-flavor heavy mesons HQS and quarkonium-like states 3 Hadronic molecules Hadronic molecules and compositeness Nonrelativistic effective field theories NREFT for transitions Triangle singularity Feng-Kun Guo (ITP) Bound states and EFTs / 89

3 Bibliogrphy Useful textbooks/monographs: H. Georgi, Weak Interactions and Modern Particle Physics (2009) J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model (1992) A. Dobado, A. Gómez-Nicola, A.L. Maroto, J.R. Peláez, Effective Lagrangians for the Standard Model (1997) S. Scherer, M.R. Schindler, A Primer for Chiral Perturbation Theory (2012) A.V. Manohar, M.B. Wise, Heavy Quark Physics (2000) A.A. Petrov, A.E. Blechman, Effective field theories (2016) and lectures by Jose! FKG, Hanhart, Meißner, Wang, Zhao, Zou, Hadronic molecules (2017) Feng-Kun Guo (ITP) Bound states and EFTs / 89

4 EFT and chiral symmetry in a nut shell Feng-Kun Guo (ITP) Bound states and EFTs / 89

5 Two facets of QCD Running of the coupling constant α s = g 2 s/(4π) High energies asymptotic freedom, perturbative degrees of freedom: quarks and gluons Low energies nonperturbative, Λ QCD 300 MeV = O ( 1 fm 1) color confinement, degrees of freedom: mesons and baryons theory of quarks and gluons? low-energy hadron spectrum Feng-Kun Guo (ITP) Bound states and EFTs / 89

6 Low-energy EFT in one page Things need to be remembered for an EFT: separation of energy scales: systematic expansion with a power counting symmetry constraints from the full theory For low-energy QCD, we will consider (approximate) chiral symmetry of light quarks CHiral Perturbation Theory (non-decoupling EFT) full theory EFT via spontaneous symmetry breaking (SSB) generation of new light degrees of freedom heavy quark symmetry: spin and flavor Pros and Cons: Pro: model-independent, controlled uncertainty Con: number of parameters increases fast when going to higher orders Q: Why is CHPT the EFT of QCD but not of some other theory? Feng-Kun Guo (ITP) Bound states and EFTs / 89

7 Low-energy EFT in one page Things need to be remembered for an EFT: separation of energy scales: systematic expansion with a power counting symmetry constraints from the full theory For low-energy QCD, we will consider (approximate) chiral symmetry of light quarks CHiral Perturbation Theory (non-decoupling EFT) full theory EFT via spontaneous symmetry breaking (SSB) generation of new light degrees of freedom heavy quark symmetry: spin and flavor Pros and Cons: Pro: model-independent, controlled uncertainty Con: number of parameters increases fast when going to higher orders Q: Why is CHPT the EFT of QCD but not of some other theory? Feng-Kun Guo (ITP) Bound states and EFTs / 89

8 QCD symmetries L QCD = f= u,d,s, c,b,t q f (i D m f )q f 1 4 Ga µνg µν,a + g2 sθ 64π 2 ɛµνρσ G a µνg a ρσ Exact: Lorentz-invariance, SU(3) c gauge, C (for θ = 0 w/ real m f, P, T as well) Hierachy of quark masses: m u,d,s Λ QCD ( 300 MeV) m c,b,t For light quarks (up, down, strange), SU(N f ) L SU(N f ) R SSB SU(N f ) V For heavy quarks (charm, bottom) in a hadron heavy quark spin symmetry (HQSS) heavy quark flavor symmetry (HQFS) heavy anti-quark diquark symmetry (HADS) Lectures by Jose Feng-Kun Guo (ITP) Bound states and EFTs / 89

9 QCD symmetries L QCD = f= u,d,s, c,b,t q f (i D m f )q f 1 4 Ga µνg µν,a + g2 sθ 64π 2 ɛµνρσ G a µνg a ρσ Exact: Lorentz-invariance, SU(3) c gauge, C (for θ = 0 w/ real m f, P, T as well) Hierachy of quark masses: m u,d,s Λ QCD ( 300 MeV) m c,b,t For light quarks (up, down, strange), SU(N f ) L SU(N f ) R SSB SU(N f ) V For heavy quarks (charm, bottom) in a hadron heavy quark spin symmetry (HQSS) heavy quark flavor symmetry (HQFS) heavy anti-quark diquark symmetry (HADS) Lectures by Jose Feng-Kun Guo (ITP) Bound states and EFTs / 89

10 Light flavor symmetry and chiral symmetry Light meson SU(3) [u, d, s] multiplets (octet + singlet): Vector mesons meson quark content mass (MeV) ρ + /ρ u d /dū 775 du uu dd ss ud ρ 0 (uū d d)/ K + /K u s /sū 892 K 0 / K 0 d s /s d 896 su sd ω (uū + d d)/ φ s s 1019 approximate SU(3) symmetry very good isospin SU(2) symmetry m ρ 0 m ρ ± = ( 0.7 ± 0.8) MeV, m K 0 m K ± = (6.7 ± 1.2) MeV Feng-Kun Guo (ITP) Bound states and EFTs / 89

11 Light flavor symmetry and chiral symmetry Light meson SU(3) [u, d, s] multiplets (octet + singlet): Pseudoscalar mesons meson quark content mass (MeV) π + /π u d /dū 140 du uu dd ss ud π 0 (uū d d)/ K + /K u s /sū 494 K 0 / K 0 d s /s d 498 su sd η (uū + d d 2s s)/ η (uū + d d + s s)/ very good isospin SU(2) symmetry m π ± m π 0 = ( ± ) MeV, m K 0 m K ± = (3.937 ± 0.028) MeV Q: Why are the pions so light? Feng-Kun Guo (ITP) Bound states and EFTs / 89

12 Summary of chiral symmetry (I) QCD chiral symmetry: QCD in the chiral limit m u = m d = m s = 0 has a global symmetry SU(3) L SU(3) R U(1) V U(1) A is anomalously broken by quantum effects Strong evidence for chiral symmetry spontaneous breaking SU(N f ) L SU(N f ) R SSB SU(N f ) V 3 (8) pseudoscalar Goldstone bosons: π ±, π 0 (, K ±, K 0, K0, η) Massless GBs couple derivatively Feng-Kun Guo (ITP) Bound states and EFTs / 89

13 Summary of chiral symmetry (II) Let g L SU(N f ) L, g R SU(N f ) R, GBs can be parameterized as ( U = g L g R = exp i φ ) φ = N 2 f 1 a=1 λ a φ a F ( ) π 0 2π + φ SU(2) =, φ 2π π 0 SU(3) = 2 2 φ 3 + φ8 K K0 2φ8 6 Chiral Lagrangian for GBs: L = L 2 + L 4 + L invariant under g = (L, R) = SU(N f ) L SU(N f ) R and C, P (for θ = 0) π U g L U R, U C U T, U P U 6 π + K + φ3 2 + φ8 6 K 0 at LO: L 2 = F 2 [ µ U µ U + 2B MU + M U ] with U = e iφ/f 4 GMOR relation: Mπ 2 = B(m u + m d ); m q = O ( p 2) F : pion decay constant in the chiral limit Exercise: Show that U µ U = 0, therefore U µ U U µ U is not present. Feng-Kun Guo (ITP) Bound states and EFTs / 89

14 Summary of chiral symmetry (III) CHPT as the low-energy ( Λ χ 1 GeV) EFT of QCDs most general Lagrangian with the same global QCD symmetries; hadrons as the dynamical degrees of freedom explicit breaking of chiral symmetry can be included perturbatively using the spurion technique double expansion in terms of low momentum and light quark masses GBs live in the coset space G/H [G SSB H]; free to choose the set of representative elements. G = SU(N f ) L SU(N f ) R, H = SU(N f ) V For instance, (g L, g R ) SU(N f ) L SU(N f ) R for g L SU(N f ) L, g R SU(N f ) R Choice of a representative element inside each left coset is free: (g L, g R )H = (g L g R, 1) (g R, g R ) }{{} H=SU(N f ) V H = (g L g R, 1) H U L U R or (g L, g R )H = (1, g R g L )(g L, g L )H = (1, g R g L ) H U R U L Feng-Kun Guo (ITP) Bound states and EFTs / 89

15 Explicit symmetry breaking: the spurion trick (I) Spurion in 3 steps: very useful trick for explicit symmetry breaking 1. Introduce a spurion field (e.g. quark mass, electric charge, γ µ,... ) with a transformation property so that the symmetry breaking term in the full theory is invariant 2. Write down invariant operators in EFT including the spurion field 3. Set the spurion field to the value which it should take Feng-Kun Guo (ITP) Bound states and EFTs / 89

16 Explicit symmetry breaking: the spurion trick (II) Apply the spurion trick to quark masses: Treat M as a spurion field L QCD = L 0 QCD q LMq R q R M q L M M = LMR Then construct Lagrangian invariant under SU(N f ) L SU(N f ) R L eff = L eff (U, U, 2 U,..., M) This procedure guarantees that chiral symmetry is broken in exactly the same way in the effective theory as it is in QCD L 2 = F 2 [ µ U µ U + 2B MU + M U ] with U L UR 4 The spurion trick is very useful to construct EFT operators with a given symmetry transformation property Feng-Kun Guo (ITP) Bound states and EFTs / 89

17 Heavy quark symmetries Feng-Kun Guo (ITP) Bound states and EFTs / 89

18 Heavy quark symmetries (I) For heavy quarks (charm, bottom) in a hadron, typical momentum transfer Λ QCD heavy quark spin symmetry (HQSS): chromomag. interaction σ B m Q spin of the heavy quark decouples heavy quark flavor symmetry (HQFS) for any hadron containing one heavy quark: velocity remains unchanged in the limit m Q : v = p = Λ QCD m Q m Q heavy quark is like a static color triplet source, m Q is irrelevant heavy anti-quark diquark symmetry Savage, Wise (1990) m Q v Λ QCD, the diquark serves as a point-like color- 3 source, like a heavy anti-quark. It relates doubly-heavy baryons to antiheavy mesons Feng-Kun Guo (ITP) Bound states and EFTs / 89

19 Heavy quark symmetries (I) For heavy quarks (charm, bottom) in a hadron, typical momentum transfer Λ QCD heavy quark spin symmetry (HQSS): chromomag. interaction σ B m Q spin of the heavy quark decouples heavy quark flavor symmetry (HQFS) for any hadron containing one heavy quark: velocity remains unchanged in the limit m Q : v = p = Λ QCD m Q m Q heavy quark is like a static color triplet source, m Q is irrelevant heavy anti-quark diquark symmetry Savage, Wise (1990) m Q v Λ QCD, the diquark serves as a point-like color- 3 source, like a heavy anti-quark. It relates doubly-heavy baryons to antiheavy mesons Feng-Kun Guo (ITP) Bound states and EFTs / 89

20 Heavy quark symmetries (I) For heavy quarks (charm, bottom) in a hadron, typical momentum transfer Λ QCD heavy quark spin symmetry (HQSS): chromomag. interaction σ B m Q spin of the heavy quark decouples heavy quark flavor symmetry (HQFS) for any hadron containing one heavy quark: velocity remains unchanged in the limit m Q : v = p = Λ QCD m Q m Q heavy quark is like a static color triplet source, m Q is irrelevant heavy anti-quark diquark symmetry Savage, Wise (1990) m Q v Λ QCD, the diquark serves as a point-like color- 3 source, like a heavy anti-quark. It relates doubly-heavy baryons to antiheavy mesons Feng-Kun Guo (ITP) Bound states and EFTs / 89

21 Heavy quark symmetries (II) In the heavy quark limit m Q, consider the quark propagator /p + m Q i p 2 m 2 Q + iɛ = i m Q (/v + 1) + /k 2m Q v k + k 2 + iɛ m Q 1 + /v 2 here p = m Q v + k, with a residual momentum k Λ QCD. i v k + iɛ Velocity-dependent projector (in the rest frame v = 0 ) particle component ( ) 1 + /v = 1 + γ0 1 0 = (in Dirac basis) Decompose a heavy quark field into v-dep. fields Q(x) = e im Qv x [Q v (x) + q v (x)] with: Q v (x) = e im Qv x 1 + /v 2 Q(x), q v(x) = e im Qv x 1 /v 2 Q(x) Exercise: Let P ± = 1 ± /v 2, show P 2 ± = P ±, P + P = 0, and Q v γ µ Q v = Q v v µ Q v Feng-Kun Guo (ITP) Bound states and EFTs / 89

22 Heavy quark symmetries (III) At leading order of the 1/m Q expansion: L Q = Q(i /D m Q )Q = Q ( v (iv D)Q v + O Let total angular momentum J = s Q + s l, s Q : heavy quark spin, m 1 Q s l : spin of the light degrees of freedom (including orbital angular momentum) HQSS: s l is a good quantum number in heavy hadrons spin multiplets: for singly heavy mesons, e.g. {D, D } with s P l = 1 2, ) M D M D 140 MeV, M B M B 46 MeV for heavy quarkonia, e.g. {η c, J/ψ} (no flavor symmetry for 2 or more heavy quarks) Feng-Kun Guo (ITP) Bound states and EFTs / 89

23 Bispinor fields for heavy mesons (I) 1 2 (1 + /v) projects onto the particle component of the heavy quark spinor. Convenient to introduce heavy mesons as bispinors: [ P µ a γ µ P a γ 5 ], Ha = γ 0 H aγ 0 H a = 1 + /v 2 P = {Qū, Q d, Q s}: pseudoscalar heavy mesons, P : vector heavy mesons Charge conjugation: H a destroys mesons containing a Q, but does not create mesons with a Q Free to choose the phase convention for charge conjugation. If we use, e.g., P a C +P ( Q) a, P a,µ Pa,µ C C P ( Q) a,µ, then the fields annihilating mesons containing a Q is (C = iγ 2 γ 0 ) [ H ( Q) 1 + /v ( ) ] ( Q) a = C Pa,µ γ µ P ( Q) T a γ 2 }{{}}{{} 5 C 1 C P a ( = + P ( Q) a,µ γ µ P ( Q) a γ 5 ) 1 /v 2 Feng-Kun Guo (ITP) Bound states and EFTs / 89

24 Bispinor fields for heavy mesons (I) 1 2 (1 + /v) projects onto the particle component of the heavy quark spinor. Convenient to introduce heavy mesons as bispinors: [ P µ a γ µ P a γ 5 ], Ha = γ 0 H aγ 0 H a = 1 + /v 2 P = {Qū, Q d, Q s}: pseudoscalar heavy mesons, P : vector heavy mesons Charge conjugation: H a destroys mesons containing a Q, but does not create mesons with a Q Free to choose the phase convention for charge conjugation. If we use, e.g., P a C +P ( Q) a, P a,µ Pa,µ C C P ( Q) a,µ, then the fields annihilating mesons containing a Q is (C = iγ 2 γ 0 ) [ H ( Q) 1 + /v ( ) ] ( Q) a = C Pa,µ γ µ P ( Q) T a γ 2 }{{}}{{} 5 C 1 C P a ( = + P ( Q) a,µ γ µ P ( Q) a γ 5 ) 1 /v 2 Feng-Kun Guo (ITP) Bound states and EFTs / 89

25 Bispinor fields for heavy mesons (II) Free heavy-meson Lagrangian: L free = i Tr [ Ha v µ µ ] H a = 2i P a v µ µ P a 2i Pa νv µ µ Pa ν Tr: trace in the spinor space, a, b: indices in the light flavor space Notice that the mass dimension of H a is 3/2. Nonrelativistic normalization: H a M H H rel. a D-meson propagator: i 2v k + iɛ M H = i p 2 MH 2 }{{ + iɛ } i 2M H v k + iɛ [1+O(k2 /M 2 H)] (p = M H v + k) Feng-Kun Guo (ITP) Bound states and EFTs / 89

26 Applications of HQS: P -wave heavy mesons Examples of HQSS phenomenology: Widths of the two D 1 (J P = 1 + ) mesons Γ[D 1 (2420)] = (27.4 ± 2.5) MeV Γ[D 1 (2430)] = ( ) MeV s l = s q + L for P -wave charmed mesons: s P l = 1 2 for decays D 1 D π: in S-wave large width in D-wave small width + or thus, D 1 (2420): s l = 3 2, D 1(2430): s l = 1 2 Suppression of the S-wave production of heavy meson pairs in e + e annihilation Table VI in E.Eichten et al., PRD17(1978)3090; X. Li, M. Voloshin, PRD88(2013) Exercise: Try to understand this statement as a consequence of HQSS. Feng-Kun Guo (ITP) Bound states and EFTs / 89

27 Applications of HQS: P -wave heavy mesons Examples of HQSS phenomenology: Widths of the two D 1 (J P = 1 + ) mesons Γ[D 1 (2420)] = (27.4 ± 2.5) MeV Γ[D 1 (2430)] = ( ) MeV s l = s q + L for P -wave charmed mesons: s P l = 1 2 for decays D 1 D π: in S-wave large width in D-wave small width + or thus, D 1 (2420): s l = 3 2, D 1(2430): s l = 1 2 Suppression of the S-wave production of heavy meson pairs in e + e annihilation Table VI in E.Eichten et al., PRD17(1978)3090; X. Li, M. Voloshin, PRD88(2013) Exercise: Try to understand this statement as a consequence of HQSS. Feng-Kun Guo (ITP) Bound states and EFTs / 89

28 Applications of HQS: heavy meson decay constants Heavy quark mass scaling of heavy meson decay constants 0 qγ µ γ 5 Q(0) P rel. (p) = f P p µ, 0 qγ µ Q(0) P rel.(p, ɛ) = f P ɛ µ in HQET, qγ µ γ 5 Q qγ µ γ 5 Q v, (P rel., P rel. ) M H H thus, we can study MH 0 qγ µ Q v (0) H(v), with Γ µ = γ µ, γ µ γ 5 What is the operator in terms of heavy-meson field? spurion trick again transformation under HQSS: Q v SQ v, H a SH a, Note: S does not commute with γ-matrices! treat Γ µ as a spurion field: pretending Γ µ Γ µ S, qγ µ Q v (0) is invariant qγ µ Q v = c Tr[Γ µ H] Feng-Kun Guo (ITP) Bound states and EFTs / 89

29 Applications of HQS: heavy meson decay constants Heavy quark mass scaling of heavy meson decay constants 0 qγ µ γ 5 Q(0) P rel. (p) = f P p µ, 0 qγ µ Q(0) P rel.(p, ɛ) = f P ɛ µ in HQET, qγ µ γ 5 Q qγ µ γ 5 Q v, (P rel., P rel. ) M H H thus, we can study MH 0 qγ µ Q v (0) H(v), with Γ µ = γ µ, γ µ γ 5 What is the operator in terms of heavy-meson field? spurion trick again transformation under HQSS: Q v SQ v, H a SH a, Note: S does not commute with γ-matrices! treat Γ µ as a spurion field: pretending Γ µ Γ µ S, qγ µ Q v (0) is invariant qγ µ Q v = c Tr[Γ µ H] Feng-Kun Guo (ITP) Bound states and EFTs / 89

30 Applications of HQS (II) For M H 0 qγ µ Q v (0) H(v), with Γ µ = γ µ, γ µ γ 5, [ qγ µ Q v = c Tr[Γ µ H] = c Tr Γ µ 1 + /v ( P µ ) ] γ 2 µ P γ 5 { 2c v µ P, for Γ µ = γ µ γ 5 = 2c P µ, for Γ µ = γ µ thus, we get f P p µ = 0 qγ µ γ 5 Q(0) P rel. (p) = 2c pµ M P MP, f P ɛ µ = 0 qγ µ Q(0) P rel.(p, ɛ) = 2c ɛ µ M P f P 1/ M P, f P M P, f B mc 0.6 f D m b However, measured values indicate a large correction: f B + = (188 ± 17 ± 18) MeV, f D + = (203.7 ± 4.7 ± 0.6) MeV PDG2016 Feng-Kun Guo (ITP) Bound states and EFTs / 89

31 Applications of HQS (II) For M H 0 qγ µ Q v (0) H(v), with Γ µ = γ µ, γ µ γ 5, [ qγ µ Q v = c Tr[Γ µ H] = c Tr Γ µ 1 + /v ( P µ ) ] γ 2 µ P γ 5 { 2c v µ P, for Γ µ = γ µ γ 5 = 2c P µ, for Γ µ = γ µ thus, we get f P p µ = 0 qγ µ γ 5 Q(0) P rel. (p) = 2c pµ M P MP, f P ɛ µ = 0 qγ µ Q(0) P rel.(p, ɛ) = 2c ɛ µ M P f P 1/ M P, f P M P, f B mc 0.6 f D m b However, measured values indicate a large correction: f B + = (188 ± 17 ± 18) MeV, f D + = (203.7 ± 4.7 ± 0.6) MeV PDG2016 Feng-Kun Guo (ITP) Bound states and EFTs / 89

32 Combining chiral symmetry and heavy quark symmetries Feng-Kun Guo (ITP) Bound states and EFTs / 89

33 Including matter fields CHPT for (pseudo) Goldstone bosons in lectures by Jose Matter fields (fields which are not Goldstone bosons) can be included as well, e.g. baryon CHPT: nucleons [SU(2)] / baryon ground state octet [SU(3)] SU(2) kaon CHPT: kaons treated as matter fields rather than GBs heavy-hadron CHPT: heavy-flavor (charm, bottom) mesons and baryons At low-energies, 3-momenta remain small M π, derivative expansion is feasible Feng-Kun Guo (ITP) Bound states and EFTs / 89

34 Transformation of matter fields Proceed as before need to know how matter fields transform under SU(N f ) L SU(N f ) R construct effective Lagrangians according to increasing number of momenta Transformation properties of matter fields: well-defined transformation rule under the unbroken SU(N f ) V transformation of matter fields under SU(N f ) L SU(N f ) R not uniquely defined, related by field redefinition Example: heavy-meson CHPT Feng-Kun Guo (ITP) Bound states and EFTs / 89

35 Transformation of matter fields Proceed as before need to know how matter fields transform under SU(N f ) L SU(N f ) R construct effective Lagrangians according to increasing number of momenta Transformation properties of matter fields: well-defined transformation rule under the unbroken SU(N f ) V transformation of matter fields under SU(N f ) L SU(N f ) R not uniquely defined, related by field redefinition Example: heavy-meson CHPT Feng-Kun Guo (ITP) Bound states and EFTs / 89

36 Transformations of heavy meson fields (I) Consider the SU(3) case, the charmed meson ground state anti-triplet: H a = ( D 0, D +, D s + ) transform under the global unbroken SU(3) V as a H V SU(3) V H V [( cū, c d, c s )] Representation invariance: free to choose how H transforms under SU(3) L SU(3) R as long as it reduces to the above under SU(3) V Example: describe the heavy mesons by H 1 or H 2, under g = (L, R) SU(3) L SU(3) R H 1 g H1 L, H 2 g H2 R both transform as an anti-triplet under (V, V ) SU(3) V Feng-Kun Guo (ITP) Bound states and EFTs / 89

37 Transformations of heavy meson fields (I) Consider the SU(3) case, the charmed meson ground state anti-triplet: H a = ( D 0, D +, D s + ) transform under the global unbroken SU(3) V as a H V SU(3) V H V [( cū, c d, c s )] Representation invariance: free to choose how H transforms under SU(3) L SU(3) R as long as it reduces to the above under SU(3) V Example: describe the heavy mesons by H 1 or H 2, under g = (L, R) SU(3) L SU(3) R H 1 g H1 L, H 2 g H2 R both transform as an anti-triplet under (V, V ) SU(3) V Feng-Kun Guo (ITP) Bound states and EFTs / 89

38 Transformations of heavy meson fields (I) Consider the SU(3) case, the charmed meson ground state anti-triplet: H a = ( D 0, D +, D s + ) transform under the global unbroken SU(3) V as a H V SU(3) V H V [( cū, c d, c s )] Representation invariance: free to choose how H transforms under SU(3) L SU(3) R as long as it reduces to the above under SU(3) V Example: describe the heavy mesons by H 1 or H 2, under g = (L, R) SU(3) L SU(3) R H 1 g H1 L, H 2 g H2 R both transform as an anti-triplet under (V, V ) SU(3) V Feng-Kun Guo (ITP) Bound states and EFTs / 89

39 Transformations of heavy meson fields (II) Example: describe the heavy mesons by H 1 or H 2, under g = (L, R) SU(3) L SU(3) R H 1 g H1 L, H 2 g H2 R both transform as an anti-triplet under (V, V ) SU(3) V Related to each other through field redefinition: H 2 = H 1 U = H 1 + i F H 1φ +..., U = exp But H 1,2 are inconvenient: complicated parity transformation (P ) L(L ) needs to be replaced by R(R ) under parity ( ) i F φ g L U R H 1,a (t, x) P γ 0 H 1,b (t, x)γ 0 U ba g γ 0 H 1,b (t, x)γ 0 U ba R [recall for a spinor: ψ(t, x) P γ 0 ψ(t, x)] Feng-Kun Guo (ITP) Bound states and EFTs / 89

40 Transformations of heavy meson fields (II) Example: describe the heavy mesons by H 1 or H 2, under g = (L, R) SU(3) L SU(3) R H 1 g H1 L, H 2 g H2 R both transform as an anti-triplet under (V, V ) SU(3) V Related to each other through field redefinition: H 2 = H 1 U = H 1 + i F H 1φ +..., U = exp But H 1,2 are inconvenient: complicated parity transformation (P ) L(L ) needs to be replaced by R(R ) under parity ( ) i F φ g L U R H 1,a (t, x) P γ 0 H 1,b (t, x)γ 0 U ba g γ 0 H 1,b (t, x)γ 0 U ba R [recall for a spinor: ψ(t, x) P γ 0 ψ(t, x)] Feng-Kun Guo (ITP) Bound states and EFTs / 89

41 Transformations of heavy meson fields (III) An elegant/convenient way: introduce ( ) iφ u = exp 2F or u 2 = U it transforms under g SU(3) L SU(3) R as (recall U g L U R ) u g L u h (L, R, φ) = h(l, R, φ) u R h(l, R, φ): space-time dependent nonlinear function, called compensator field For for SU(3) V transformations (L = R = V ), reduces to h(l, R, φ) = V We can construct H = H 1 u or H = H 2 u, it transforms as H g H h Exercise: Show that under parity transformation h(t, x) H(t, x) P γ 0 H(t, x)γ 0. P h(t, x), and Feng-Kun Guo (ITP) Bound states and EFTs / 89

42 Transformations of heavy meson fields (III) An elegant/convenient way: introduce ( ) iφ u = exp 2F or u 2 = U it transforms under g SU(3) L SU(3) R as (recall U g L U R ) u g L u h (L, R, φ) = h(l, R, φ) u R h(l, R, φ): space-time dependent nonlinear function, called compensator field For for SU(3) V transformations (L = R = V ), reduces to h(l, R, φ) = V We can construct H = H 1 u or H = H 2 u, it transforms as H g H h Exercise: Show that under parity transformation h(t, x) H(t, x) P γ 0 H(t, x)γ 0. P h(t, x), and Feng-Kun Guo (ITP) Bound states and EFTs / 89

43 Building blocks of the chiral Lagrangian Useful to introduce combinations of u whose transformations only involve h: Γ µ = 1 ( u µ u + u µ u ), u µ = i ( u µ u u µ u ) 2 Γ µ : chiral connection, vector; u µ : chiral vielbein, axial vector Γ µ g h Γ µ h + h µ h, u µ g h u µ h Introduce a covariant derivative: D µ H = µ H H Γ µ which transform the same way as H under SU(N f ) L SU(N f ) R D µ H g D µ H h Include the quark mass term χ = 2BM +... by introducing χ + = u χu + uχ u, χ + h χ + h All fields transform in terms of h, convenient to construct the effective Lagrangians Feng-Kun Guo (ITP) Bound states and EFTs / 89

44 Building blocks of the chiral Lagrangian Useful to introduce combinations of u whose transformations only involve h: Γ µ = 1 ( u µ u + u µ u ), u µ = i ( u µ u u µ u ) 2 Γ µ : chiral connection, vector; u µ : chiral vielbein, axial vector Γ µ g h Γ µ h + h µ h, u µ g h u µ h Introduce a covariant derivative: D µ H = µ H H Γ µ which transform the same way as H under SU(N f ) L SU(N f ) R D µ H g D µ H h Include the quark mass term χ = 2BM +... by introducing χ + = u χu + uχ u, χ + h χ + h All fields transform in terms of h, convenient to construct the effective Lagrangians Feng-Kun Guo (ITP) Bound states and EFTs / 89

45 Building blocks of the chiral Lagrangian Useful to introduce combinations of u whose transformations only involve h: Γ µ = 1 ( u µ u + u µ u ), u µ = i ( u µ u u µ u ) 2 Γ µ : chiral connection, vector; u µ : chiral vielbein, axial vector Γ µ g h Γ µ h + h µ h, u µ g h u µ h Introduce a covariant derivative: D µ H = µ H H Γ µ which transform the same way as H under SU(N f ) L SU(N f ) R D µ H g D µ H h Include the quark mass term χ = 2BM +... by introducing χ + = u χu + uχ u, χ + h χ + h All fields transform in terms of h, convenient to construct the effective Lagrangians Feng-Kun Guo (ITP) Bound states and EFTs / 89

46 Heavy meson CHPT at LO LO Lagrangian [O (p)]: L (1) = i Tr [ HM Ha v µ (D µ ] H) a + g }{{} 2 Tr [ ] Ha H b γ µ γ 5 u µ ba }{{} kinetic term+dπ scattering+... terms for D Dπ+... invariant under Lorentz transformation, chiral symmetry, parity Tr: trace in the spinor space, a, b: indices in the light flavor space Notice that the mass dimension of H a is 3/2. Nonrelativistic normalization: H a M H H rel. a D-meson propagator: i 2v k + iɛ M H = i p 2 MH 2 }{{ + iɛ } i 2M H v k + iɛ [1+O(k2 /M 2 H)] (p = M H v + k) Feng-Kun Guo (ITP) Bound states and EFTs / 89

47 Simplified two-component notation (I) The superfield for pseudoscalar and vector heavy mesons: ( (4) means 4-component) H a (4) = 1 + /v [Pa µ γ µ P 2 a γ 5 ] In the rest frame of heavy meson, v µ = (1, 0). We take the Dirac basis ( ) ( ) ( ) σ γ 0 =, γ i i 0 1 =, γ σ i Simplifications: 1 + /v 2 H (4) a = = 1 + γ0 2 = ( ) 1 0 ( ) 0 (Pa + Pa σ) , H(4) a = Thus, it is convenient to simply use the two-component notation ( ) 0 0 ( P a + Pa σ ) 0 H a = P a + Pa σ, H a (4) H a, H(4) a H a Feng-Kun Guo (ITP) Bound states and EFTs / 89

48 Simplified two-component notation (II) In the simplified two-component notation: [ ] i Tr H(4) a v µ (D µ H) (4) a = i Tr [ H a ( 0 H a H b Γ 0 ba)] = 2i ( P a 0 P a + Pa i 0 Pa i ) i ( + P 4F 2 a P b + Pa Pb ) [ φ, 0 φ ] +... ba }{{} scattering between (D,D, B, B ) and GBs(π,K,η) The LO, O (p), scattering amplitudes are completely determined by chiral symmetry (strength in term of F )! The axial coupling term (u k = 1 F k φ +...): g [ 2 Tr H(4) a H (4) = g [ (P 2 Tr i a σ i + P a the Weinberg Tomozawa term ] b γ µ γ 5 u µ ba = g 2 Tr [ H ah b σ i] u i ba ) ( ) P j b σj + P b σ k] u k ba = 1 F P ap b i i φ ba + 1 }{{} F P a i P b i φ ba + i F ɛijk Pa i term for D Dπ P j b k φ ba +... Feng-Kun Guo (ITP) Bound states and EFTs / 89

49 Simplified two-component notation (II) In the simplified two-component notation: [ ] i Tr H(4) a v µ (D µ H) (4) a = i Tr [ H a ( 0 H a H b Γ 0 ba)] = 2i ( P a 0 P a + Pa i 0 Pa i ) i ( + P 4F 2 a P b + Pa Pb ) [ φ, 0 φ ] +... ba }{{} scattering between (D,D, B, B ) and GBs(π,K,η) The LO, O (p), scattering amplitudes are completely determined by chiral symmetry (strength in term of F )! The axial coupling term (u k = 1 F k φ +...): g [ 2 Tr H(4) a H (4) = g [ (P 2 Tr i a σ i + P a the Weinberg Tomozawa term ] b γ µ γ 5 u µ ba = g 2 Tr [ H ah b σ i] u i ba ) ( ) P j b σj + P b σ k] u k ba = 1 F P ap b i i φ ba + 1 }{{} F P a i P b i φ ba + i F ɛijk Pa i term for D Dπ P j b k φ ba +... Feng-Kun Guo (ITP) Bound states and EFTs / 89

50 Determination of g Measured D widths: Γ(D 0 ) < 2.1 MeV, Γ(D ± ) = (83.4 ± 1.8) kev PDG2016 B(D + D 0 π + ) = (67.7 ± 0.5)%, B(D + D + π 0 ) = (30.7 ± 0.5)% Amplitude from the LO Lagrangian: A(D + D 0 π + ) = i 2g F ε (λ) q π MD M }{{ D } and the two-body decay width accounts for NR normalization Γ(D + D 0 π + ) = 1 q π 1 8π MD 2 3 λ A 2 = g2 M D q π 3 12πF 2 M D we used λ εi (λ) εj (λ) = δ ij HQFS: g should be approximately the same in bottom sector with a relative uncertainty of O ( ΛQCD m c ΛQCD m b ) O (20%) g 0.57 Lattice QCD results: g b = ± ALPHA Collaboration, Phys. Lett. B 740 (2015) 278 g b = 0.56 ± 0.03 ± 0.07 RBC and UKQCD Collaborations, Phys. Rev. D 93 (2016) Feng-Kun Guo (ITP) Bound states and EFTs / 89

51 Determination of g Measured D widths: Γ(D 0 ) < 2.1 MeV, Γ(D ± ) = (83.4 ± 1.8) kev PDG2016 B(D + D 0 π + ) = (67.7 ± 0.5)%, B(D + D + π 0 ) = (30.7 ± 0.5)% Amplitude from the LO Lagrangian: A(D + D 0 π + ) = i 2g F ε (λ) q π MD M }{{ D } and the two-body decay width accounts for NR normalization Γ(D + D 0 π + ) = 1 q π 1 8π MD 2 3 λ A 2 = g2 M D q π 3 12πF 2 M D we used λ εi (λ) εj (λ) = δ ij HQFS: g should be approximately the same in bottom sector with a relative uncertainty of O ( ΛQCD m c ΛQCD m b ) O (20%) g 0.57 Lattice QCD results: g b = ± ALPHA Collaboration, Phys. Lett. B 740 (2015) 278 g b = 0.56 ± 0.03 ± 0.07 RBC and UKQCD Collaborations, Phys. Rev. D 93 (2016) Feng-Kun Guo (ITP) Bound states and EFTs / 89

52 Mass splittings among heavy mesons (I) Spin-dependent term mass difference between vector and pseudoscalar mesons (σ µν = i 2 [γµ, γ ν ]) L = λ [ 2 Tr H(4) a σ m µν H a (4) σ µν] = 2λ 2 Tr [ H Q m aσ i H a σ i] Q = 4λ 2 m Q ( P a P a 3P ap a ) M P a M Pa = 8λ 2 m Q Thus, we expect M B M B M D M D m c m b 0.3 measured values: M D M D 140 MeV, H a = P a + P a σ M B M B 46 MeV Feng-Kun Guo (ITP) Bound states and EFTs / 89

53 Mass splittings among heavy mesons (II) Light quark mass-dependent terms in two-component notation: L χ = λ 1 Tr [ H ah b ] χ+,ba λ 1Tr [ H ah a ] χ+,bb here, χ + = u χu + uχ u = 4BM B { } φ, {φ, M} F 2 SU(3) mass differences: ( ) M D + M s D + = 4λ 1 B(m s m d ) = 4λ 1 M 2 K ± Mπ 2 ± λ GeV 1 Isospin splitting induced by m d m u : (M D 0 M D +) quark mass = 4λ 1 B(m u m d ) = 4λ 1 ( M 2 K ± M 2 K 0 M 2 π ± + M 2 π 0 ) = 2.3 MeV Exp. value: M D 0 M D + = (4.77 ± 0.08) MeV = ( ) M D 0 M + ( ) D + M quark mass D 0 M D + L χ also contributes to scattering between a heavy meson and the lightest pseudoscalar mesons (GBs) Feng-Kun Guo (ITP) Bound states and EFTs / 89 e.m.

54 Summary-1 Symmetries of QCD: spontaneously broken chiral symmetry for light flavors heavy quark spin and flavor symmetry for heavy flavors applications in this lecture to the research frontier: new hadrons observed in many experiments after 2002 Reviews in 2016: H.-X. Chen et al., The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639 (2016) 1 H.-X. Chen et al., A review of the open charm and open bottom mesons, arxiv: [hep-ph] A. Esposito, A. Pilloni, A. D. Polosa, Multiquark resonances, Phys. Rept. 668 (2017) 1 A. Hosaka et al., Exotic hadrons with heavy flavors X, Y, Z and Related States, Prog. Theor. Exp. Phys. 2016, 062C01 R. F. Lebed, R. E. Mitchell, E. Swanson, Heavy-quark QCD exotica, Prog. Part. Nucl. Phys., in print, arxiv: [hep-ph] Feng-Kun Guo (ITP) Bound states and EFTs / 89

55 Ordinary and exotic hadrons In quark model notation + Ordinary mesons and baryons + Exotic hadrons: multiquark states, hybrids and glueballs Hadronic molecules: extended, loosely bound states composed of asymptotic hadrons (distance hadron size), analogues of deuteron and other light nuclei Once the same quantum numbers, always mix source of difficulties/confusions Feng-Kun Guo (ITP) Bound states and EFTs / 89

56 Ordinary and exotic hadrons In quark model notation + Ordinary mesons and baryons + Exotic hadrons: multiquark states, hybrids and glueballs Hadronic molecules: extended, loosely bound states composed of asymptotic hadrons (distance hadron size), analogues of deuteron and other light nuclei Once the same quantum numbers, always mix source of difficulties/confusions Feng-Kun Guo (ITP) Bound states and EFTs / 89

57 Ordinary and exotic hadrons In quark model notation + Ordinary mesons and baryons + Exotic hadrons: multiquark states, hybrids and glueballs Hadronic molecules: extended, loosely bound states composed of asymptotic hadrons (distance hadron size), analogues of deuteron and other light nuclei Once the same quantum numbers, always mix source of difficulties/confusions Feng-Kun Guo (ITP) Bound states and EFTs / 89

58 HQS and open-flavor heavy hadrons Feng-Kun Guo (ITP) Bound states and EFTs / 89

59 Beginning of the story in 2003: discovery of D s0(2317) Charm-strange c s mesons D s0(2317) and D s1 (2460) D s0(2317): 0 + BaBar (2003) M = ( ± 0.6) MeV, Γ < 3.8 MeV The only hadronic decay: D s π D s1 (2460): 1 + CLEO (2003) M = ( ± 0.6) MeV, Γ < 3.5 MeV no isospin partner observed, tiny widths I = 0 M a s s (M e V ) E x p. d a ta G I q u a rk m o d e l T h re s h o ld 1 3 P 0 (2.4 8 ) D * s 0 ( ) P 1 (2.5 7 ) 1 1 P 1 (2.5 3 ) D s 1 ( ) 1 + D * K D K Notable features: masses are much lower than the quark model predictions for c s mesons M Ds1(2460) M D s0 (2317) M D M D +1 MeV Feng-Kun Guo (ITP) Bound states and EFTs / 89

60 Beginning of the story in 2003: discovery of D s0(2317) Charm-strange c s mesons D s0(2317) and D s1 (2460) D s0(2317): 0 + BaBar (2003) M = ( ± 0.6) MeV, Γ < 3.8 MeV The only hadronic decay: D s π D s1 (2460): 1 + CLEO (2003) M = ( ± 0.6) MeV, Γ < 3.5 MeV no isospin partner observed, tiny widths I = 0 M a s s (M e V ) E x p. d a ta G I q u a rk m o d e l T h re s h o ld 1 3 P 0 (2.4 8 ) D * s 0 ( ) P 1 (2.5 7 ) 1 1 P 1 (2.5 3 ) D s 1 ( ) 1 + D * K D K Notable features: masses are much lower than the quark model predictions for c s mesons M Ds1(2460) M D s0 (2317) M D M D +1 MeV Feng-Kun Guo (ITP) Bound states and EFTs / 89

61 Most exotic and newest observation: X(5568) X(5568) by D0 Collaboration (p p collisions) PRL117(2016) M = ( ± ) MeV Γ = ( 21.9 ± ) MeV Observed in B s ( )0 π +, sizeable width I = 1: minimal quark contents is bs du! a favorite mulqituark candidate: explicitly flavor exotic, minimal number of quarks 4 Estimate of isospin breaking decay width: ( ) 2 md m u Γ I Λ QCD O (100 MeV) α 2 = O (10 kev) Feng-Kun Guo (ITP) Bound states and EFTs / 89

62 Applications of HQS: D s0(2317) and D s1 (2460) HQFS: for a singly-heavy hadron, M HQ ( = m Q + A + O m 1 Q rough estimates of bottom analogues whatever the D sj states are ( ( 1 M B s0 = M D s0 (2317) + b c + O Λ 2 QCD 1 )) (5.65 ± 0.15) GeV m c m b ( ( 1 M Bs1 = M Ds1(2460) + b c + O Λ 2 QCD 1 )) (5.79 ± 0.15) GeV m c m b here b c m b m c M Bs M Ds 3.33 GeV, where M Bs = GeV, M Ds = GeV: spin-averaged g.s. Q s meson masses both to be discovered 1 more precise predictions can be made in a given model, e.g. hadronic molecules ) 1 The established meson B s1 (5830) is probably the bottom partner of D s1 (2536). Feng-Kun Guo (ITP) Bound states and EFTs / 89

63 Applications of HQS: D s0(2317) and D s1 (2460) in hadronic molecular model: D s0(2317)[dk], D s1 (2460)[D K] Barnes, Close, Lipkin (2003); van Beveren, Rupp (2003); Kolomeitsev, Lutz (2004); FKG et al. (2006);... D ( ) K bound states: poles of the T -matrix Lectures by Jose recall the LO DK and D K interactions from the Weinberg Tomozawa term: [ ] i Tr H(4) a v µ (D µ H) (4) a = i Tr [ H a ( 0 H a H b Γ 0 ba)] = 2i ( P a 0 P a + Pa i 0 Pa i ) i ( + P 4F 2 a P b + Pa Pb ) [ φ, 0 φ ] +... ba }{{} provides V, independent of heavy meson mass G is independent of heavy quark mass as well (recall the propagator i 2v k + iɛ ) Feng-Kun Guo (ITP) Bound states and EFTs / 89

64 Applications of HQS: D s0(2317) and D s1 (2460) a natural consequence of HQSS: similar binding energies M D + M K M D s0 45 MeV M Ds1(2460) M D s0 (2317) M D M D is understood predicting the bottom-partner masses in one minute: M B s0 M B + M K 45 MeV GeV M Bs1 M B + M K 45 MeV GeV to be compared with lattice results for the lowest positive-parity bottom-strange mesons: MB lat. s0 = (5.711 ± ± 0.019) GeV MB lat. s1 = (5.750 ± ± 0.019) GeV Lang, Mohler, Prelovsek, Woloshyn, PLB750(2015)17 Feng-Kun Guo (ITP) Bound states and EFTs / 89

65 Applications of HQS: X(5568) FKG, Meißner, Zou, How the X(5568) challenges our understanding of QCD, Commun.Theor.Phys. 65 (2016) 593 mass too low for X(5568) to be a bsūd: M M Bs MeV M π 140 MeV because pions are pseudo-goldstone bosons Gell-Mann Oakes Renner: M 2 π m q ; chiral counting: M π = O (p) For any matter field: M R = O ( p 0) M π ; we expect M qq M R M σ M bsūd M Bs MeV 5.9 GeV HQFS predicts an isovector X c : M Xc = M X(5568) b c + O ( ( 1 Λ 2 QCD 1 )) (2.24 ± 0.15) GeV m c m b but in D s π, only the isoscalar D s0(2317) was observed! BaBar (2003) immediately negative results reported by LHCb and by CMS LHCb-CONF CMS-PAS-BPH Feng-Kun Guo (ITP) Bound states and EFTs / 89

66 Applications of HQS: X(5568) FKG, Meißner, Zou, How the X(5568) challenges our understanding of QCD, Commun.Theor.Phys. 65 (2016) 593 mass too low for X(5568) to be a bsūd: M M Bs MeV M π 140 MeV because pions are pseudo-goldstone bosons Gell-Mann Oakes Renner: M 2 π m q ; chiral counting: M π = O (p) For any matter field: M R = O ( p 0) M π ; we expect M qq M R M σ M bsūd M Bs MeV 5.9 GeV HQFS predicts an isovector X c : M Xc = M X(5568) b c + O ( ( 1 Λ 2 QCD 1 )) (2.24 ± 0.15) GeV m c m b but in D s π, only the isoscalar D s0(2317) was observed! BaBar (2003) immediately negative results reported by LHCb and by CMS LHCb-CONF CMS-PAS-BPH Feng-Kun Guo (ITP) Bound states and EFTs / 89

67 Applications of HQS: X(5568) FKG, Meißner, Zou, How the X(5568) challenges our understanding of QCD, Commun.Theor.Phys. 65 (2016) 593 mass too low for X(5568) to be a bsūd: M M Bs MeV M π 140 MeV because pions are pseudo-goldstone bosons Gell-Mann Oakes Renner: M 2 π m q ; chiral counting: M π = O (p) For any matter field: M R = O ( p 0) M π ; we expect M qq M R M σ M bsūd M Bs MeV 5.9 GeV HQFS predicts an isovector X c : M Xc = M X(5568) b c + O ( ( 1 Λ 2 QCD 1 )) (2.24 ± 0.15) GeV m c m b but in D s π, only the isoscalar D s0(2317) was observed! BaBar (2003) immediately negative results reported by LHCb and by CMS LHCb-CONF CMS-PAS-BPH Feng-Kun Guo (ITP) Bound states and EFTs / 89

68 HQS and quarkonium-like states Feng-Kun Guo (ITP) Bound states and EFTs / 89

69 Beginning of the story in 2003: discovery of X(3872) X(3872) Belle, PRL91(2003) Discovered in B ± K ± J/ψππ, mass extremely close to the D 0 D 0 threshold M X = ( ± 0.17) MeV M D 0 + M D 0 M X = (0.00 ± 0.20) MeV Γ < 1.2 MeV J P C = 1 ++ S-wave coupling to D D Belle, PRD84(2011) LHCb PRL110(2013) Observed in the D 0 D 0 mode as well BaBar, PRD77(2008) Large coupling to D 0 D 0 : B(X D 0 D 0 ) > 24% PDG2016 Large isospin breaking: B(X ωj/ψ) B(X π + π = 0.8 ± 0.3 J/ψ) indicating a hadronic molecular structure Törnqvist (2004);... Feng-Kun Guo (ITP) Bound states and EFTs / 89

70 Beginning of the story in 2003: discovery of X(3872) X(3872) Belle, PRL91(2003) Discovered in B ± K ± J/ψππ, mass extremely close to the D 0 D 0 threshold M X = ( ± 0.17) MeV M D 0 + M D 0 M X = (0.00 ± 0.20) MeV Γ < 1.2 MeV J P C = 1 ++ S-wave coupling to D D Belle, PRD84(2011) LHCb PRL110(2013) Observed in the D 0 D 0 mode as well BaBar, PRD77(2008) Large coupling to D 0 D 0 : B(X D 0 D 0 ) > 24% PDG2016 Large isospin breaking: B(X ωj/ψ) B(X π + π = 0.8 ± 0.3 J/ψ) indicating a hadronic molecular structure Törnqvist (2004);... Feng-Kun Guo (ITP) Bound states and EFTs / 89

71 Beginning of the story in 2003: discovery of X(3872) X(3872) Belle, PRL91(2003) Discovered in B ± K ± J/ψππ, mass extremely close to the D 0 D 0 threshold M X = ( ± 0.17) MeV M D 0 + M D 0 M X = (0.00 ± 0.20) MeV Γ < 1.2 MeV J P C = 1 ++ S-wave coupling to D D Belle, PRD84(2011) LHCb PRL110(2013) Observed in the D 0 D 0 mode as well BaBar, PRD77(2008) Large coupling to D 0 D 0 : B(X D 0 D 0 ) > 24% PDG2016 Large isospin breaking: B(X ωj/ψ) B(X π + π = 0.8 ± 0.3 J/ψ) indicating a hadronic molecular structure Törnqvist (2004);... Feng-Kun Guo (ITP) Bound states and EFTs / 89

72 More exotic structures: Z ± c and Z± b with hidden Q Q Z c ±, Z± b : charged structures in heavy quarkonium mass region, Q Q du, Q Qūd Z c (3900), Z c (4020), Z c (4200), Z c (4430),... Z b (10610) and Z b (10650): observed in Υ(10860) π [π ± Υ(1S, 2S, 3S)/h b (1P, 2P )] Belle, arxiv: ; PRL108(2012) also in Υ(10860) π [B ( ) B ] ± Belle, arxiv: ; PRL116(2016) Feng-Kun Guo (ITP) Bound states and EFTs / 89

73 More exotic structures: Z ± c and Z± b with hidden Q Q Z c ±, Z± b : charged structures in heavy quarkonium mass region, Q Q du, Q Qūd Z c (3900), Z c (4020), Z c (4200), Z c (4430),... Z b (10610) and Z b (10650): observed in Υ(10860) π [π ± Υ(1S, 2S, 3S)/h b (1P, 2P )] Belle, arxiv: ; PRL108(2012) also in Υ(10860) π [B ( ) B ] ± Belle, arxiv: ; PRL116(2016) Feng-Kun Guo (ITP) Bound states and EFTs / 89

74 Z ± c and Z± b with hidden Q Q (II) Z c (3900/3885) ± : structure around 3.9 GeV seen in J/ψπ by BESIII and Belle in Y (4260) J/ψπ + π, BESIII, PRL110(2013)252001; Belle, PRL110(2013) and in D D by BESIII in Y (4260) π ± (D D ) BESIII, PRD92(2015) Z c (4020) ± observed in h c π ± and ( D D ) ± distributions BESIII, PRL111(2013)242001; PRL112(2014) Feng-Kun Guo (ITP) Bound states and EFTs / 89

75 Charmonium spectrum Feng-Kun Guo (ITP) Bound states and EFTs / 89

76 Charmonium spectrum Feng-Kun Guo (ITP) Bound states and EFTs / 89

77 Charmonium spectrum Note: X(3915) is probably just the χ c2 (2P ) with 2 ++ Z.-Y. Zhou et al., PRL115(2015) Feng-Kun Guo (ITP) Bound states and EFTs / 89

78 Charmonium spectrum Note: X(3915) is probably just the χ c2 (2P ) with 2 ++ Z.-Y. Zhou et al., PRL115(2015) Feng-Kun Guo (ITP) Bound states and EFTs / 89

79 HQSS for XY Z (I) Assuming the X(3872) to be a D D molecule Consider S-wave interaction between a pair of s P l = 1 2 (anti-)heavy mesons: 0 ++ : D D, D D ( : D D + D D), 2 D D 1 ++ : 1 2 ( D D D D) 2 ++ : D D here, phase convention: D C + D, D C D Heavy quark spin irrelevant interaction matrix elements: s 1 l, s 2 l, s L Ĥ s 1 l, s 2 l, s L For each isospin, 2 independent terms 1 2, 1 2, 0 Ĥ 1 2, 1 1 2, 0, 2, 1 2, 1 Ĥ 1 2, 1 2, 1 6 pairs grouped in 2 multiplets with s L = 0 and 1, respectively Feng-Kun Guo (ITP) Bound states and EFTs / 89

80 HQSS for XY Z (I) Assuming the X(3872) to be a D D molecule Consider S-wave interaction between a pair of s P l = 1 2 (anti-)heavy mesons: 0 ++ : D D, D D ( : D D + D D), 2 D D 1 ++ : 1 2 ( D D D D) 2 ++ : D D here, phase convention: D C + D, D C D Heavy quark spin irrelevant interaction matrix elements: s 1 l, s 2 l, s L Ĥ s 1 l, s 2 l, s L For each isospin, 2 independent terms 1 2, 1 2, 0 Ĥ 1 2, 1 1 2, 0, 2, 1 2, 1 Ĥ 1 2, 1 2, 1 6 pairs grouped in 2 multiplets with s L = 0 and 1, respectively Feng-Kun Guo (ITP) Bound states and EFTs / 89

81 HQSS for XY Z (II) For the HQSS consequences, convenient to use the basis of states: s P L C S-wave: s P L C, sp c c C = 0 + or 1 multiplet with s L = 0: multiplet with s L = 1: 0 + L 0 + c c = 0 ++, 0 + L 1 c c = L 0 + c c = 1 +, 1 L 1 c c = if X(3872) is a 1 ++ D D molecule, then its s L = 1 Multiplets in strict heavy quark limit: X(3872) has three partners with 0 ++, 2 ++ and 1 + sp C c c Voloshin, PLB604(2004)69 Hidalgo-Duque et al., PLB727(2013)432; Baru et al., PLB763(2016)20 might be 6 molecules: Z b, Z b and W b0, W b0, W b1 and W b2 (for I = 1) Bondar et al., PRD84(2011)054010; Voloshin, PRD84(2011)031502; Mehen, Powell, PRD84(2011) Feng-Kun Guo (ITP) Bound states and EFTs / 89

82 HQSS for XY Z (II) For the HQSS consequences, convenient to use the basis of states: s P L C S-wave: s P L C, sp c c C = 0 + or 1 multiplet with s L = 0: multiplet with s L = 1: 0 + L 0 + c c = 0 ++, 0 + L 1 c c = L 0 + c c = 1 +, 1 L 1 c c = if X(3872) is a 1 ++ D D molecule, then its s L = 1 Multiplets in strict heavy quark limit: X(3872) has three partners with 0 ++, 2 ++ and 1 + sp C c c Voloshin, PLB604(2004)69 Hidalgo-Duque et al., PLB727(2013)432; Baru et al., PLB763(2016)20 might be 6 molecules: Z b, Z b and W b0, W b0, W b1 and W b2 (for I = 1) Bondar et al., PRD84(2011)054010; Voloshin, PRD84(2011)031502; Mehen, Powell, PRD84(2011) Feng-Kun Guo (ITP) Bound states and EFTs / 89

83 HQSS for XY Z (III) unitary transformation from two-meson basis to s 1 c, s 2 c, s c c ; s 1 l, s 2 l,s L ; J : s 1 c, s 1 l, j 1 ; s 2 c, s 2 l, j 2 ; J = (2j1 + 1)(2j 2 + 1)(2s c c + 1)(2s L + 1) s c c,s L j 1,2 : meson spins; J: the total angular momentum of the whole system s 1 c(2 c) = 1 : spin of the heavy quark in meson 1 (2) 2 s 1 c s 2 c s c c s 1 l s 2 l s L s 1 c, s 2 c, s c c ; s 1 l, s 2 l,s L ; J j 1 j 2 J s 1 l(2 l) = 1 : angular momentum of the light quarks in meson 1 (2) 2 s c c = 0, 1: total spin of c c, conserved but decoupled s L = 0, 1: total angular momentum of the light-quark system, conserved only two independent s l1, s l2, s L Ĥ s l1, s l2, s L terms for each isospin I: I 1 F I0 = 2, 1 2, 0 Ĥ 1 2, 1 2, 0 1, F I1 = 2, 1 2, 1 Ĥ 1 2, 1 2, 1 I Feng-Kun Guo (ITP) Bound states and EFTs / 89 I

84 HQSS for XY Z (IV) ( ) ( ) D D : V (0++) CIA 3CIB =, D D 3CIB C IA 2C IB ( ) ( ) D D : V (1+ ) CIA C IB 2C IB =, D D 2C IB C IA C IB D D : V (1++) = C IA + C IB, D D : V (2++) = C IA + C IB, here, C IA = 1 4 (3F I1 + F I0 ), C IB = 1 4 (F I1 F I0 ) This would suggest spin multiplets. Good candidates: X(3872) and X 2 (4013) (not observed yet!); Z c (3900) and Z c (4020) Nieves, Valderrama, PRD86(2012)056004;... M X2(4013) M X(3872) M Zc(4020) M Zc(3900) M D M D Z b (10610) and Z b (10650) : Bondar et al., PRD84(2011)054010;... M Zb (10650) M Zb (10610) M B M B Z c and Z b states need a suppression of coupled-channel effect Feng-Kun Guo (ITP) Bound states and EFTs / 89

85 HQSS for XY Z (IV) ( ) ( ) D D : V (0++) CIA 3CIB =, D D 3CIB C IA 2C IB ( ) ( ) D D : V (1+ ) CIA C IB 2C IB =, D D 2C IB C IA C IB D D : V (1++) = C IA + C IB, D D : V (2++) = C IA + C IB, here, C IA = 1 4 (3F I1 + F I0 ), C IB = 1 4 (F I1 F I0 ) This would suggest spin multiplets. Good candidates: X(3872) and X 2 (4013) (not observed yet!); Z c (3900) and Z c (4020) Nieves, Valderrama, PRD86(2012)056004;... M X2(4013) M X(3872) M Zc(4020) M Zc(3900) M D M D Z b (10610) and Z b (10650) : Bondar et al., PRD84(2011)054010;... M Zb (10650) M Zb (10610) M B M B Z c and Z b states need a suppression of coupled-channel effect Feng-Kun Guo (ITP) Bound states and EFTs / 89

86 HQSS for XY Z: Lagrangian could also be derived by constructing the effective Lagrangian L 4H = +C (τ) A [ ] [ H(Q) a H a (Q) γ µ Tr C A Tr [ ] [ Tr H(Q) a τ ab H (Q) b γ µ Tr [ ] +C B Tr H(Q) a H a (Q) γ µ γ 5 Tr +C (τ) B H ( Q) b H ( Q) b γ µ] H ( Q) c [ H ( Q) b [ [ ] Tr H(Q) a τ ab H (Q) b γ µ γ 5 Tr τ : Pauli matrices in isospin space; Isospin I = 0 or 1 4 independent terms: C 0A, C 0B ; C 1A, C 1B : linear combinations of C (τ) A,B AlFiky et al., PLB640(2006) τ cd H( Q) d γ µ] H ( Q) b γ µ γ 5 ] H ( Q) c τ cd H( Q) d γ µ γ 5 ] H a (Q) : D, D ; H ( Q) a C 0φ = C φ + 3C (τ) φ, C 1φ = C φ C (τ) φ, for φ = A, B : D, D Feng-Kun Guo (ITP) Bound states and EFTs / 89

87 Hadronic molecules and compositeness Feng-Kun Guo (ITP) Bound states and EFTs / 89

88 Hadronic molecules (I) Hadronic molecule: dominant component is a composite state of 2 or more hadrons Concept at large distances, so that can be approximated by system of multi-hadrons at low energies Consider a 2-body bound state with a mass M = m 1 + m 2 E B size: R 1 2µEB r hadron scale separation (nonrelativistic) EFT applicable! Only narrow hadrons can be considered as components of hadronic molecules, Γ h 1/r, r: range of forces Filin et al., PRL105(2010)019101; FKG, Meißner, PRD84(2011) Feng-Kun Guo (ITP) Bound states and EFTs / 89

89 Hadronic molecules (II) Why are hadronic molecules interesting? one realization of color-neutral objects, analogue of light nuclei important information for hadron-hadron interaction understanding the XY Z states EFT applicable; model-independent statements can be made for S-wave, compositeness (1 Z) related to measurable quantities compositeness: probability of the physical state being a 2-body bound state Weinberg, PR137(1965); Baru et al., PLB586(2004); Hyodo, IJMPA28(2013) ;... see also, e.g., Weinberg s books: QFT Vol.I, Lectures on QM g NR 2 (1 Z) 2π 2µEB µ 2 2π 2µEB µ 2 2(1 Z) a (2 Z) Z, r e 2µE B (1 Z) 2µE B Feng-Kun Guo (ITP) Bound states and EFTs / 89

90 Compositeness (I) Model-independent result for S-wave loosely bound composite states: Consider a system with Hamiltonian H = H 0 + V H 0 : free Hamiltonian, V : interaction potential Compositeness: the probability of finding the physical state B in the 2-body continuum q d 3 q 1 Z = (2π) 3 q B 2 Z = B 0 B 2, 0 (1 Z) 1 Z = 0: pure bound (composite) state Z = 1: pure elementary state Feng-Kun Guo (ITP) Bound states and EFTs / 89

91 Compositeness (II) d 3 q Compositeness : 1 Z = (2π) 3 q B 2 Schrödinger equation (H 0 + V ) B = E B B multiplying by q and using H 0 q = q2 2µ q : momentum-space wave function: q V B q B = E B + q 2 /(2µ) S-wave, small binding energy so that R = 1/ 2µE B r, r: range of forces q V B = g NR [1 + O(r/R)] Compositeness: d 3 q g NR 2 [ ( r )] 1 Z = (2π) 3 [E B + q 2 /(2µ)] O = µ2 g NR 2 [ ( r )] R 2π 1 + O 2µE B R Feng-Kun Guo (ITP) Bound states and EFTs / 89

92 Compositeness (III) Coupling constant measures the compositeness for an S-wave shallow bound state g NR 2 (1 Z) 2π 2µEB µ 2 2π 2µEB µ 2 bounded from the above Exercise: Show that g NR 2 is the residue of the T -matrix element at the pole E = E B : g NR 2 = lim E E B (E + E B ) k T k 1 Hint: use the Lippmann Schwinger equation T = V + V E H 0+iɛT and the completeness relation B B + d 3 q (2π) q 3 (+) q (+) = 1 to derive the Low equation (noticing T q = V q (+) ): k T k = k V k + k V B B V k E + E B + iɛ + d 3 q k T q q T k (2π) 3 E q 2 /(2µ) + iɛ Feng-Kun Guo (ITP) Bound states and EFTs / 89

93 Compositeness (IV) Z can be related to scattering length a and effective range r e Weinberg (1965) 2R(1 Z) [ ( r )] a = 1 + O, r e = RZ 2 Z R 1 Z [ ( r )] 1 + O R Effective range expansion: f 1 (k) = 1/a + r e k 2 /2 ik + O ( k 4) Derivation: T (E) k T k = 2π µ f(k) Im T 1 (E) = µ 2µE θ(e) 2π Twice-subtracted dispersion relation for t 1 (E) T 1 (E) = E + E B g NR 2 + (E + E B) 2 π = E + E B g NR 2 + µ R 4π + 0 ( 1 R 2µE iɛ Im T 1 (w) dw (w E iɛ)(w + E B ) 2 Example: deuteron as pn bound state. Exp.: E B = 2.2 MeV, a3 S 1 = 5.4 fm a Z=1 = 0 fm, ) 2 a Z=0 = ( 4.3 ± 1.4) fm Feng-Kun Guo (ITP) Bound states and EFTs / 89

94 Nonrelativistic effective field theories Feng-Kun Guo (ITP) Bound states and EFTs / 89

95 NREFT at LO (I) We consider a system of two particles of masses m 1, m 2 in the near-threshold region, a momentum expansion for the interactions with the LO being a constant L = ) φ i (i 0 m i + 2 φ i C 0 φ 1 2m φ 2 φ 1φ i=1,2 i i nonrelativistic propagator: p 0 m i p 2 /(2m i ) + iɛ to have a near-threshold bound state (hadronic molecule) T (E) = C 0 + C 0 G NR (E) C 0 + C 0 G NR (E) C 0 G NR (E) C = C 1 0 G NR (E) Feng-Kun Guo (ITP) Bound states and EFTs / 89

96 NREFT at LO (II) The loop integral is linearly divergent (E defined relative to m 1 + m 2 ), regularized with, e.g., a sharp cut d 3 kdk 0 ) )] 1 G NR (E) = i [(k 0 (2π) 4 k2 + iɛ (E k 0 k2 + iɛ 2m 1 2m 2 Λ d 3 k 1 = i2µ(2πi) (2π) 4 2µE k 2 + iɛ = µ (Λ π 2 ) Λ 2µE iɛ arctan 2µE iɛ for real E, = µ π 2 Λ + µ ( 2µE iɛ + O Λ 1 ) 2π 2µE iɛ = 2µE θ( E) i 2µE θ(e) Feng-Kun Guo (ITP) Bound states and EFTs / 89

97 NREFT at LO (III) Renormalization: T is Λ-independent, 1 T (E) = C0 1 G NR ( 1 = + µ C 0 π 2 Λ µ ) 1 2µE iɛ 2π }{{} 1/C0 r 2π/µ = 2π/(µ C0 r) 2µE iɛ Other regularization can be used as well, equiavalent to the sharp cutoff up to 1/Λ suppressed terms, e.g. with a Gaussian regulator exp ( k 2 /Λ 2 G), ΛG = 2/πΛ with the power divergence subtraction (PDS) scheme in dimensional regularization by letting, Λ PDS = 2Λ/π Kaplan, Savage, Wise (1998) Feng-Kun Guo (ITP) Bound states and EFTs / 89

98 NREFT at LO (III) Renormalization: T is Λ-independent, 1 T (E) = C0 1 G NR ( 1 = + µ C 0 π 2 Λ µ ) 1 2µE iɛ 2π }{{} 1/C0 r 2π/µ = 2π/(µ C0 r) 2µE iɛ Other regularization can be used as well, equiavalent to the sharp cutoff up to 1/Λ suppressed terms, e.g. with a Gaussian regulator exp ( k 2 /Λ 2 G), ΛG = 2/πΛ with the power divergence subtraction (PDS) scheme in dimensional regularization by letting, Λ PDS = 2Λ/π Kaplan, Savage, Wise (1998) Feng-Kun Guo (ITP) Bound states and EFTs / 89

99 NREFT at LO (IV) T (E) = 2π/µ 2π/(µ C0 r) 2µE iɛ = 2π/µ 2π/(µ C0 r) + i k from matching to effective range expansion, f 1 (k) = 2π µ T 1 = 1 a r ek 2 i k + O ( k 4) 2π/(µC r 0) = 1/a; higher terms are necessary to match both a and r e pole below threshold at E = E B with E B > 0 κ 2µE B bound state pole, in the 1st Riemann sheet 2π/(µC r 0) = κ virtual state pole, in the 2nd Riemann sheet 2π/(µC r 0) = κ unable to get a resonance pole at LO with a single channel Feng-Kun Guo (ITP) Bound states and EFTs / 89

100 Bound state and virtual state If the same binding energy, bound and virtual states cannot be distinguished above threshold (E > 0): T (E) 2 1 ±κ + i 2µE 2 = 1 κ 2 + 2µE Bound state and virtual state are different below threshold (E < 0): bound state: peaked below threshold Events / 4MeV virtual state bound state md 0 D *- [GeV] T (E) 2 1 (κ 2µE) 2 virtual state: a sharp cusp at threshold T (E) 2 1 (κ + 2µE) 2 Feng-Kun Guo (ITP) Bound states and EFTs / 89

101 Bound state and virtual state If the same binding energy, bound and virtual states cannot be distinguished above threshold (E > 0): T (E) 2 1 ±κ + i 2µE 2 = 1 κ 2 + 2µE Bound state and virtual state are different below threshold (E < 0): bound state: peaked below threshold T (E) 2 1 (κ 2µE) 2 virtual state: a sharp cusp at threshold T (E) 2 1 (κ + 2µE) 2 Events / 4MeV GZ E GeV virtual state bound state md 0 D *- [GeV] E MeV Lower Fig.: bound state and virtual state with E B = 5 MeV and a small width to the inelastic channel Cleven et al., EPJA47(2011)120 Feng-Kun Guo (ITP) Bound states and EFTs / 89

102 Resonance and virtual state options for Z c (3900) Albaladejo et al., PLB755(2016)(337) Feng-Kun Guo (ITP) Bound states and EFTs / 89

103 Predictions With the HQS relations for the interaction potential, one can make predictions using NREFT on loosely bound hadronic molecules Nieves, Pavon Valderrama, Voloshin, Mehen, FKG,... J P C States Thresholds (MeV) Masses (MeV) (D D D D) (input) 2 ++ D D (B B B B) B B D B LO predictions on the masses of various partners of the X(3872) using a Gaussian cutoff with Λ = 0.5 GeV. FKG, Hidalgo-Duque, Nieves, Pavon Valderrama (2013) Feng-Kun Guo (ITP) Bound states and EFTs / 89

104 Coupling constant for S-wave bound state T (E) = 2π/µ 2π/(µ C r 0 ) 2µE iɛ At LO, effective coupling strength for bound state g NR 2 = lim E E B (E + E B ) T (E) = 2π µ = 2π µ 2 2µEB Recall the compositeness formula: g NR 2 = (1 Z) 2π µ 2 2µEB ( ) d 1 2µE iɛ de E= E B Feng-Kun Guo (ITP) Bound states and EFTs / 89

105 Coupling constant for S-wave bound state T (E) = 2π/µ 2π/(µ C r 0 ) 2µE iɛ At LO, effective coupling strength for bound state g NR 2 = lim E E B (E + E B ) T (E) = 2π µ = 2π µ 2 2µEB Recall the compositeness formula: g NR 2 = (1 Z) 2π µ 2 2µEB ( ) d 1 2µE iɛ de E= E B Feng-Kun Guo (ITP) Bound states and EFTs / 89

106 NREFT for transitions Feng-Kun Guo (ITP) Bound states and EFTs / 89

107 NREFT for transitions (I) coupling constant contains important information; needs to be measured in decays or productions long-distance processes can be calculated in NREFT nonrelativistic power counting in terms of velocity v (m 1 + m 2 M)/m 1 d 3 kdk 0 ) )] 1 G NR (E) = i [(k 0 (2π) 4 k2 + iɛ (E k 0 k2 + iɛ 2m 1 2m 2 k = O (v), k 0 = O ( v 2) loop measure: O ( v 5) ; each propagator: O ( v 2) G NR (E) = O (v) Feng-Kun Guo (ITP) Bound states and EFTs / 89

108 NREFT for transitions (II) Considering a two-body decay A B C, considering C is light (such as pion) and couples to m 1 m 3 in P -wave, both B and C are heavy (such as hadronic molecules), m 1 m 2 m 3 m considering the case that A couples to m 1, m 2 in S-wave, B couples to m 2, m 3 in S-wave, ( ) v 5 ( q ) A O v 6 q = O v two heavy particles, two relevant cuts, how to estimate v? Feng-Kun Guo (ITP) Bound states and EFTs / 89

109 NREFT for transitions (III) cut-1 cut-2 The scalar three-point loop integral d 4 l 1 I = i (2π) 4 (l 2 m iɛ) [(P l)2 m iɛ] [(l q)2 m iɛ] i dl 0 d 3 l 1 N m (2π) 4 [l 0 T 1 ( l ) + iɛ] [P 0 l 0 T 2 ( l ) + iɛ] [l 0 E C T 3 ( l q ) + iɛ] here, N m 8m 1 m 2 m 3, and we have rewritten a propagator into two poles: 1 l 2 m iɛ = 1 (l 0 ω 1 + iɛ) (l 0 + ω 1 iɛ) 1 2m 2 [l 0 m 1 T 1 ( l ) + iɛ] with ω 1 = m l2, T 1 ( l ) = l 2 /(2m 1 ) Feng-Kun Guo (ITP) Bound states and EFTs / 89

110 NREFT for transitions (III) cut-1 cut-2 The scalar three-point loop integral I 4µ 12µ 23 d 3 [ l (l 2 N m (2π) 3 + c 1 iɛ ) ( l 2 + c 2 2µ )] 1 23 l q iɛ m 3 = N 1 [ ( ) ( )] c2 c 1 2a arctan arctan, a c 2 c 1 2 a c 1 iɛ }{{} cut-1 2 a c 2 a iɛ }{{} cut-2 here, N = µ 12 µ 23 /(2πm 1 m 2 m 3 ), and a = (µ 23 /m 3 ) 2 q 2, µ ij = m i m j /(m i + m j ), c 1 = 2µ 12 (m 1 + m 2 m A ), c 2 = 2µ 23 (m 2 + m 3 + E C m B ) + µ 23 m 3 q 2 Feng-Kun Guo (ITP) Bound states and EFTs / 89

111 NREFT for transitions (IV) Three-point loop integral can be rewritten as ( ) N c 2 c 1 [arcsin arcsin a (c2 c 1 ) 2 + 4ac 1 iɛ ( )] c 2 c 1 2a (c2 c 1 ) 2 + 4ac 1 iɛ logarithmically divergent at (c 2 c 1 ) 2 + 4ac 1 = 0 nonrelativistic version of the triangle singularity expansion in the region 4ac 1 < (c 2 c 1 ) 2 (away from the triangle singularity) [ (π I(q) N a 2 2 ) ( ac 1 π c 2 c ) ( ) ] ac2 (4ac) 3/2 + O c 2 c 1 (c 2 c 1 ) 3 = N 2 c2 + c N m two momentum scales: c 1 and c 2 2 v 1 + v 2 two velocities: v 1 = c 1 /(2µ 12 ), v 2 = c 2 a/(2µ 23 ). Feng-Kun Guo (ITP) Bound states and EFTs / 89

112 NREFT for transitions (IV) Three-point loop integral can be rewritten as ( ) N c 2 c 1 [arcsin arcsin a (c2 c 1 ) 2 + 4ac 1 iɛ ( )] c 2 c 1 2a (c2 c 1 ) 2 + 4ac 1 iɛ logarithmically divergent at (c 2 c 1 ) 2 + 4ac 1 = 0 nonrelativistic version of the triangle singularity expansion in the region 4ac 1 < (c 2 c 1 ) 2 (away from the triangle singularity) [ (π I(q) N a 2 2 ) ( ac 1 π c 2 c ) ( ) ] ac2 (4ac) 3/2 + O c 2 c 1 (c 2 c 1 ) 3 = N 2 c2 + c N m 2 v 1 + v 2 two momentum scales: c 1 and c 2 two velocities: v 1 = c 1 /(2µ 12 ), v 2 = c 2 a/(2µ 23 ). cut-1 cut-2 Feng-Kun Guo (ITP) Bound states and EFTs / 89

113 NREFT for transitions (V) 2 I(q) N c1 + N 2 c 2 m v 1 + v 2 NREFT I : counting v 1 v 2 = O (v), v = (v 1 + v 2 )/2, I(q) = O ( v 1) explicit calculation of the 3-point loop v.s. power counting Feng-Kun Guo (ITP) Bound states and EFTs / 89

114 NREFT for transitions (V) 2 I(q) N c1 + N 2 c 2 m v 1 + v 2 NREFT I : counting v 1 v 2 = O (v), v = (v 1 + v 2 )/2, I(q) = O ( v 1) explicit calculation of the 3-point loop v.s. power counting loop 1/v (a) 30 loop 1/v (b) normalized at m A = 4.22 GeV M A [GeV] M A [GeV] used masses: m 1 = GeV, m 2 = GeV, m 3 = GeV; (a) m B = GeV, m C = 0.14 GeV; (b) m B = GeV, m C = 0 GeV Feng-Kun Guo (ITP) Bound states and EFTs / 89

115 NREFT for transitions (VI) A m 1 C B cut-1 cut-2 m 2 cut 1 NREFT II : if v 1 and v 2 are well seperated, the larger one can be integrated out an EFT of NREFT I suppose v 2 v 1, i.e., c 2 c 1, I(q) 4µ 12µ 23 N m 4µ 12µ 23 N m c 2 µ 12µ 23 πn m c 2 d 3 [ l (l 2 (2π) 3 + c 1 iɛ ) ( l 2 + c 2 2µ 23 l q iɛ m 3 Λ d 3 [ ( )] l 1 c1 (2π) 3 l O + c 1 iɛ c 2 ( 2Λ π ) c 1 iɛ }{{} cut-1 )] 1 Feng-Kun Guo (ITP) Bound states and EFTs / 89

116 NREFT II and XEFT A m 1 C B cut-1 cut-2 m 2 cut 1 From NREFT I to NREFT II convergent I(q) in NREFT I but divergent in NREFT II less predictive power Λ should be smaller than c 2 smaller uncertainty: v 1 < v = (v 1 + v 2 )/2 XEFT as a NREFT for the X(3872) Fleming, Kusunoki, Mehen, van Kolck (2007) tiny binding energy: B X0 = M D 0 + M D 0 M X = (0.00 ± 0.20) MeV binding momentum in the D 0 D 0 : κ 0 20 MeV D + D integrated out: κ c 126 MeV κ 0 perturbative pion g 2 µ 0 µ π 8πF 2 π µ π = XEFT as NREFT II for decays like X(3872) χ cj π 0 (M D 0 M D 0) 2 m 2 π 0 44 MeV Mehen, PRD92(2015) Feng-Kun Guo (ITP) Bound states and EFTs / 89

117 NREFT II and XEFT A m 1 C B cut-1 cut-2 m 2 cut 1 From NREFT I to NREFT II convergent I(q) in NREFT I but divergent in NREFT II less predictive power Λ should be smaller than c 2 smaller uncertainty: v 1 < v = (v 1 + v 2 )/2 XEFT as a NREFT for the X(3872) Fleming, Kusunoki, Mehen, van Kolck (2007) tiny binding energy: B X0 = M D 0 + M D 0 M X = (0.00 ± 0.20) MeV binding momentum in the D 0 D 0 : κ 0 20 MeV D + D integrated out: κ c 126 MeV κ 0 perturbative pion g 2 µ 0 µ π 8πF 2 π µ π = XEFT as NREFT II for decays like X(3872) χ cj π 0 (M D 0 M D 0) 2 m 2 π 0 44 MeV Mehen, PRD92(2015) Feng-Kun Guo (ITP) Bound states and EFTs / 89

118 XEFT prediction Long-distance processes such as X(3872) D 0 D0 π 0, D 0 D0 γ are sensitive to the hadronic molecular structure XEFT predictions on the decay width Γ(X D 0 D0 π 0 ) Fleming et al (2007) Feng-Kun Guo (ITP) Bound states and EFTs / 89

119 Triangle singularity Feng-Kun Guo (ITP) Bound states and EFTs / 89

120 TS: some details (I) Consider the scalar three-point loop integral d 4 q 1 I = i (2π) 4 [(P q) 2 m iɛ] (q2 m iɛ) [(p 23 q) 2 m iɛ] Rewriting a propagator into two poles: 1 q 2 m iɛ = 1 (q 0 ω 2 + iɛ) (q 0 + ω 2 iɛ) with ω 2 = m q 2 Nonrelativistically, on the positive-energy poles i dq 0 d 3 q 1 I 8m 1 m 2 m 3 (2π) 4 (P 0 q 0 ω 1 + iɛ) (q 0 ω 2 + iɛ) (p 0 23 q0 ω 3 + iɛ) Feng-Kun Guo (ITP) Bound states and EFTs / 89

121 TS: some details (II) I 0 d 3 q 1 (2π) 3 [P 0 ω 1 (q) ω 2 (q) + i ɛ][e 23 ω 2 (q) ω 3 (p 23 q ) + i ɛ] dq The second cut: f(q) = q 2 P 0 ω 1 (q) ω 2 (q) + i ɛ f(q) dz E 23 ω 2 (q) m q2 + p p 23 qz + i ɛ Feng-Kun Guo (ITP) Bound states and EFTs / 89

122 TS: some details (III) Relation between singularities of integrand and integral singularity of integrand does not necessarily give a singularity of integral: integral contour can be deformed to avoid the singularity Two cases that a singularity cannot be avoided: endpoint singularity pinch singularity Feng-Kun Guo (ITP) Bound states and EFTs / 89

123 TS: some details (III) Relation between singularities of integrand and integral singularity of integrand does not necessarily give a singularity of integral: integral contour can be deformed to avoid the singularity Two cases that a singularity cannot be avoided: endpoint singularity pinch singularity Feng-Kun Guo (ITP) Bound states and EFTs / 89

124 TS: some details (IV) I f(q) = dq dz q 2 P 0 ω 1 (q) ω 2 (q) + i ɛ f(q) 1 A(q, z) dz E 23 ω 2 (q) m q2 + p p 23 qz + i ɛ Singularities of the integrand in the rest frame of initial particle: First cut: M ω 1 (l) ω 2 (l) + i ɛ = 0 q on+ 1 λ(m 2, m 2 1 2M, m2 2 ) + i ɛ Second cut: A(q, ±1) = 0 endpoint singularities of f(q) z = +1 : q a+ = γ (β E 2 + p 2) + i ɛ, q a = γ (β E 2 p 2) i ɛ, z = 1 : q b+ = γ ( β E 2 + p 2) + i ɛ, q b = γ (β E 2 + p 2) i ɛ β = p 23 /E 23, γ = 1/ 1 β 2 = E 23 /m 23 E 2(p 2): energy (momentum) of particle-2 in the cmf of the (2,3) system Feng-Kun Guo (ITP) Bound states and EFTs / 89

125 TS: some details (IV) I f(q) = dq dz q 2 P 0 ω 1 (q) ω 2 (q) + i ɛ f(q) 1 A(q, z) dz E 23 ω 2 (q) m q2 + p p 23 qz + i ɛ Singularities of the integrand in the rest frame of initial particle: First cut: M ω 1 (l) ω 2 (l) + i ɛ = 0 q on+ 1 λ(m 2, m 2 1 2M, m2 2 ) + i ɛ Second cut: A(q, ±1) = 0 endpoint singularities of f(q) z = +1 : q a+ = γ (β E 2 + p 2) + i ɛ, q a = γ (β E 2 p 2) i ɛ, z = 1 : q b+ = γ ( β E 2 + p 2) + i ɛ, q b = γ (β E 2 + p 2) i ɛ β = p 23 /E 23, γ = 1/ 1 β 2 = E 23 /m 23 E 2(p 2): energy (momentum) of particle-2 in the cmf of the (2,3) system Feng-Kun Guo (ITP) Bound states and EFTs / 89

126 TS: some details (V) All singularities of the integrand: q on+, q a+ = γ (β E 2 + p 2) + i ɛ, q a = γ (β E 2 p 2) i ɛ, q b+ = q a, q b = q a+ < 0 (for ɛ = 0) Im q Im q Im q q on+ q a+ q on+ q a+ q on+ q a+ 0 q a Re q 0 q a Re q 0 q a Re q 0 Im q q on+ (a) (b) q b+ q a+ singularity at q on+ = q a 2-body threshold triangle singularity at Re q m 23 = m 2 + m 3 (c) Feng-Kun Guo (ITP) Bound states and EFTs / 89

127 TS: some details (VI) Rewrite q a = p 2 i ɛ, p 2 γ (β E 2 p 2) Kinematics for p 2 > 0, which is relevant to triangle singularity: p 3 = γ (β E 3 + p 2) > 0 particles 2 and 3 move in the same direction in the rest frame of initial particle velocities in the rest frame of the initial particle: v 3 > β > v 2 v 2 = β E 2 p 2/β E 2 β p 2 < β, v 3 = β E 3 + p 2/β E3 + β > β p 2 particle 3 moves faster than particle 2 in the rest frame of initial particle Feng-Kun Guo (ITP) Bound states and EFTs / 89

128 TS: some details (VI) Rewrite q a = p 2 i ɛ, p 2 γ (β E 2 p 2) Kinematics for p 2 > 0, which is relevant to triangle singularity: p 3 = γ (β E 3 + p 2) > 0 particles 2 and 3 move in the same direction in the rest frame of initial particle velocities in the rest frame of the initial particle: v 3 > β > v 2 v 2 = β E 2 p 2/β E 2 β p 2 < β, v 3 = β E 3 + p 2/β E3 + β > β p 2 particle 3 moves faster than particle 2 in the rest frame of initial particle Feng-Kun Guo (ITP) Bound states and EFTs / 89

129 TS: Coleman-Norton theorem Coleman Norton theorem: S. Coleman and R. E. Norton, Nuovo Cim. 38 (1965) 438 The singularity is on the physical boundary if and only if the diagram can be interpreted as a classical process in space-time. physical boundary: upper edge (lower edge) of the unitary cut in the first (second) Riemann sheet Translation: all three intermediate states can go on shell p 2 p 3, m 3 can catch up with the m 2 to rescatter Feng-Kun Guo (ITP) Bound states and EFTs / 89

130 TS: Coleman-Norton theorem Coleman Norton theorem: S. Coleman and R. E. Norton, Nuovo Cim. 38 (1965) 438 The singularity is on the physical boundary if and only if the diagram can be interpreted as a classical process in space-time. physical boundary: upper edge (lower edge) of the unitary cut in the first (second) Riemann sheet Translation: all three intermediate states can go on shell p 2 p 3, m 3 can catch up with the m 2 to rescatter Feng-Kun Guo (ITP) Bound states and EFTs / 89

131 TS: phenomenology LHCb pentaquark-like structures P c (4380), P c (4450) PRL115(2015) M 1 = (4380 ± 8 ± 29) MeV, M 2 = ( ± 1.7 ± 2.5) MeV, Γ 1 = (205 ± 18 ± 86) MeV, Γ 2 = (39 ± 5 ± 19) MeV. Feng-Kun Guo (ITP) Bound states and EFTs / 89

132 TS: phenomenology Contribution from Λ χ c1 p triangle dagram to Λ b J/ψ p K From q on+ = q a, easy to find when M Λ = 1.89 GeV, a triangle singularity located exactly at the χ c1 p threshold, GeV Coincidentally, four-star baryon Λ(1890): J P = 3/2 +, Γ : MeV triangle loop with S-wave χ c1 p: Λ 0 b Λ p K p χ c1 J/ψ impossible to produce a narrow peak for χ c1 p in other partial waves does not exclude the existence of a pentaquark; careful analysis is necessary Feng-Kun Guo (ITP) Bound states and EFTs / 89

133 TS: phenomenology Contribution from Λ χ c1 p triangle dagram to Λ b J/ψ p K From q on+ = q a, easy to find when M Λ = 1.89 GeV, a triangle singularity located exactly at the χ c1 p threshold, GeV Coincidentally, four-star baryon Λ(1890): J P = 3/2 +, Γ : MeV triangle loop with S-wave χ c1 p: 2 [a.u.] ΓΛ *=60 MeV ΓΛ *=100 MeV Events/(15 MeV) s [GeV] m J/ p [GeV] impossible to produce a narrow peak for χ c1 p in other partial waves does not exclude the existence of a pentaquark; careful analysis is necessary Feng-Kun Guo (ITP) Bound states and EFTs / 89

134 TS: phenomenology Contribution from Λ χ c1 p triangle dagram to Λ b J/ψ p K From q on+ = q a, easy to find when M Λ = 1.89 GeV, a triangle singularity located exactly at the χ c1 p threshold, GeV Coincidentally, four-star baryon Λ(1890): J P = 3/2 +, Γ : MeV triangle loop with S-wave χ c1 p: 2 [a.u.] ΓΛ *=60 MeV ΓΛ *=100 MeV Events/(15 MeV) s [GeV] m J/ p [GeV] impossible to produce a narrow peak for χ c1 p in other partial waves does not exclude the existence of a pentaquark; careful analysis is necessary Feng-Kun Guo (ITP) Bound states and EFTs / 89

135 Backup slides Feng-Kun Guo (ITP) Bound states and EFTs / 13

136 Low-energy effective field theory (EFT) S. Weinberg, Phenomenological Lagrangians, Physica 96A (1979) 327 Translation: The most general effective Lagrangian, up to a given order, consistent with the symmetries of the underlying theory results consistent with the underlying theory! The degrees of freedom can be different from those of the underlining theory we can work with hadrons directly for low-energy QCD Feng-Kun Guo (ITP) Bound states and EFTs / 13

137 Low-energy EFT (II) However, the most general means an infinite number of parameters intractable (?) nonrenormalizable (in contrast to, e.g., QED and QCD) Solution: systematic expansion with a power counting only a finite number of parameters at a given order, can be determined from experiments lattice calculations for QCD renormalize order by order existence of a small (dimensionless) quantity, e.g., separation of energy scales, E Λ expansion in powers of (E/Λ) Neutron decay (n pe v e ): weak interactions for q 2 MW 2 (decoupling EFT) Feng-Kun Guo (ITP) Bound states and EFTs / 13

138 Low-energy EFT (II) However, the most general means an infinite number of parameters intractable (?) nonrenormalizable (in contrast to, e.g., QED and QCD) Solution: systematic expansion with a power counting only a finite number of parameters at a given order, can be determined from experiments lattice calculations for QCD renormalize order by order existence of a small (dimensionless) quantity, e.g., separation of energy scales, E Λ expansion in powers of (E/Λ) Neutron decay (n pe v e ): weak interactions for q 2 MW 2 (decoupling EFT) Feng-Kun Guo (ITP) Bound states and EFTs / 13

139 Low-energy EFT (II) However, the most general means an infinite number of parameters intractable (?) nonrenormalizable (in contrast to, e.g., QED and QCD) Solution: systematic expansion with a power counting only a finite number of parameters at a given order, can be determined from experiments lattice calculations for QCD renormalize order by order existence of a small (dimensionless) quantity, e.g., separation of energy scales, E Λ expansion in powers of (E/Λ) Neutron decay (n pe v e ): weak interactions for q 2 MW 2 (decoupling EFT) e sin θ W MW 2 q2 e 2 = (1 8MW 2 sin θ + q2 W MW 2 = G ( ) F q 2 + O 2 M 2 W ) +... Feng-Kun Guo (ITP) Bound states and EFTs / 13

140 Derivative coupling Symmetry implies a derivative coupling for GBs, i.e., GBs do not interact at vanishing momenta Consider GB π a : π a Q a A 0 = d 3 x π a A a 0(x) 0 0 Lorentz invariance π a (q) A a µ(0) 0 = iq µ F π Consider the matrix element ψ 1 A a µ(0) ψ 2 = + = R a µ + F π q µ 1 q 2 T a Current conservation q µ A a µ = 0, thus GBs couple in a derivative form!! q µ R a µ + F π T a = 0 lim q µ 0 T a = 0 Feng-Kun Guo (ITP) Bound states and EFTs / 13

141 Derivative coupling Symmetry implies a derivative coupling for GBs, i.e., GBs do not interact at vanishing momenta Consider GB π a : π a Q a A 0 = d 3 x π a A a 0(x) 0 0 Lorentz invariance π a (q) A a µ(0) 0 = iq µ F π Consider the matrix element A a µ ψ 1 A a µ(0) ψ 2 = A a µ ψ 1 ψ 2 ψ π a 1 ψ 2 + T = R a µ + F π q µ 1 q 2 T a Current conservation q µ A a µ = 0, thus GBs couple in a derivative form!! q µ R a µ + F π T a = 0 lim q µ 0 T a = 0 Feng-Kun Guo (ITP) Bound states and EFTs / 13

142 CSSB in QCD Hamiltonian invariant under a group G = SU(N f ) L SU(N f ) R, vacuum invariant under its vector subgroup H = SU(N f ) V. Q a V 0 = 0, Q a A 0 0 SSB massless pseudoscalar Goldstone bosons #(GBs) = dim(g) dim(h) = Nf 2 1 for N f = 3, 8 GBs: π ±, π 0, K ±, K 0, K 0, η for N f = 2, 3 GBs: π ±, π 0 Pions get a small mass due to explicit symmetry breaking by tiny m u,d (a few MeV) M π M other hadron also, m s m u,d M K M π Mechanism for SU(N f ) L SU(N f ) R SU(N f ) V in QCD not well understood Feng-Kun Guo (ITP) Bound states and EFTs / 13