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Home , C parity, Hadron, Hadron spectroscopy

Feng-Kun Guo

Institute of Theoretical , CAS

The Fourth Asia-Europe-Pacific School of High- Physics, 12–25 Sept. 2018, Quy Nhon, Vietnam

Feng-Kun Guo (ITP) 13–14.09.2018 1 / 72 Outline

1 Inroduction

2 Hadron resonances discovered since 2003 Open-flavor heavy XYZ states candidates

3 Theory ideas and applications Symmetries of QCD: chiral and heavy Applications to new heavy Threshold cusps and triangle singularties Compositeness and hadronic

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 2 / 72 Introduction

Two recent reviews: S. L. Olsen, T. Skwarnicki, Nonstandard heavy mesons and : • Experimental evidence, Rev. Mod. Phys. 90 (2018) 015003 [arXiv:1708.04012] experimental facts and interpretations F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, B.-S. Zou, Hadronic • molecules, Rev. Mod. Phys. 90 (2018) 015004 [arXiv:1705.00141] theoretical formalisms

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 3 / 72 Two facets ofQCD

2 Running of the αs = g /(4π) • s

High lectures by Gavin Salam • , perturbative degrees of freedom: and Low energies • nonperturbative, Λ 300 MeV = 1 fm−1 QCD ∼ O color confinement, detected : mesons and baryons challenge: how do hadrons emerge/how is QCD spectrum organized? ⇒ Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 4 / 72 Theoretical tools for studying nonperturbative QCD

Lattice QCD: numerical simulation in discretized Euclidean space-time • finite volume (L should be large) finite lattice spacing (a should be small)

often using mu,d larger than the physical values chiral extrapolation ⇒

Phenomenological models, such as , QCD sum rules , ... • Low-energy EFT: • mesons and baryons as effective degrees of freedom

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 5 / 72 Quarks and hadrons

Mesons and baryons in quark model

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 6 / 72 Light flavor symmetry

Light SU(3) [u, d, s] multiplets (octet + singlet): Vector mesons ds us • meson quark content (MeV) ρ+/ρ− ud¯ /du¯ 775 ρ0 (uu¯ dd¯)/√2 775 du uu dd ud − ss K∗+/K∗− us¯ /su¯ 892 K∗0/K¯ ∗0 ds¯ /sd¯ 896 ω (uu¯ + dd¯)/√2 783 su sd φ ss¯ 1019

approximate SU(3) symmetry

mρ mω, mφ mK∗ mK∗ mρ ' − ' − very good SU(2) symmetry

m 0 m ± = ( 0.7 0.8) MeV, m ∗0 m ∗± = (6.7 1.2) MeV ρ − ρ − ± K − K ±

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 7 / 72 Light flavor symmetry

Light meson SU(3) [u, d, s] multiplets (octet + singlet): Pseudoscalar mesons ds us • meson quark content mass (MeV) π+/π− ud¯ /du¯ 140 π0 (uu¯ dd¯)/√2 135 du uu dd ud − ss K+/K− us¯ /su¯ 494 K0/K¯ 0 ds¯ /sd¯ 498 η (uu¯ + dd¯ 2ss¯)/√6 548 ∼ − su sd η0 (uu¯ + dd¯+ ss¯)/√3 958 ∼ very good isospin SU(2) symmetry

m ± m 0 = (4.5936 0.0005) MeV, m 0 m ± = (3.937 0.028) MeV π − π ± K − K ± Q: Why are the so light?

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 7 / 72 What are exotic hadrons?

Quark model notation: • any hadron resonances beyond picture of qq¯ for a meson and qqq for a Gluoinc excitations: hybrids and

 Multiquark states A Schematic Model of Baryons and Mesons

M.Gell-Mann, Phys.Lett.8(1964)214-215

Hadronic molecules: • bound states of two or more hadrons, analogues of nuclei

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 8 / 72 J PC and exotic quantum numbers

PC J of regular qq¯ mesons s1 L • L+1 P = ( 1) q¯ − C = ( 1)L+S for flacor-neutral q − mesons s2 L: orbital S = (0, 1): total of q and q¯

 For S = 0, the meson spin J = L, one has P = (−1)J+1 and C = (−1)J . Hence, J PC = even−+ and odd+−  For S = 1, one has P = C = (−1)L+1. Hence, J PC = 1−−, {0, 1, 2}++, {1, 2, 3}−−,... Exotic J PC for mesons: • J PC = 0−−, even+− and odd−+

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 9 / 72 Additive quantum numbers

Some trivial facts about additive quantum numbers of regular mesons

Light-flavor mesons (here S =) ds us • — Nonstrange mesons: S = 0, I = 0, 1 — Strange mesons: S = ±1, I = 1 2 du uu dd ud Open-flavor heavy mesons • ss — Qq¯(q = u, d): S = 0, I = 1/2 — Qs¯: S = 1, I = 0 su sd Heavy quarkonia (QQ¯): S = 0,I = 0, neutral •

Charge, isospin, strangeness etc. which cannot be achieved in the qq¯ and qqq scheme would be a smoking gun for an exotic nature

more subtlety later...

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 10 / 72 Decay patterns: regular hadrons

SU(3) flavor symmetry is usually satisfied to 30% • Example

∗+  2  3 Γ(K ) 51 MeV 3 Mρ qKπ + = =0.34 [exp], =0.29 [SU(3)] Γ(ρ ) 149 MeV 4 MK∗ qππ Okubo–Zweig–Iizuka (OZI) rule: •

drawing only quark lines, the disconnected diagrams are strongly suppressed relative to the connected ones

Example ψ(3770): 40 MeV above the DD¯ threshold ∼ (DD¯) = (93+8)% (sum of any other modes) B −9 B Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 11 / 72 Godfrey–Isgur quark model

q q  m2 + p2 + m2 + p2 + V Ψ = E Ψ 1 2 | i | i Potential V : One- exchange + linear confinement + relativistic effects

Isospin-1 light meson spectroscopy

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 12 / 72 New discoveries since 2003

Many new hadron resonances observed in experiments since 2003

Inactive: BaBar, Belle, CDF, CLEO-c, D0, ... • Running: Belle-II, BESIII, COMPASS, LHCb, ... • Under construction/discussion: PANDA, EIC, EicC, ... •

Common strategy: search for peaks, fit with Breit–Wigner

1 ∝ (s M 2)2 + s Γ2(s) Γ − Lots of mysteries right now ... M

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 13 / 72 Open-flavor heavy mesons

3000 3000 2800 2800 2600 2600 D D D 2(2460) D s2(2573) 2400 1(2420) s1(2536) 2400 2200 Mass ( MeV ) Mass ( MeV ) 2200 Godfrey, Isgur, PRD32(1985)189 Godfrey, Isgur, PRD32(1985)189 2000 D* * mesons discovered before 2003 Ds mesons discovered before 2003 2000 1800 D Ds - - + + + 3+ ? - - + + + - ? 0 1 0 1 2 ? 1800 0 1 0 1 2 3 ? -nonstrange mesons Charm-strange mesons

Most quark-model predicted states were still missing before 2003

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 14 / 72 Charm-strange mesons (1) Discoveries in 2003 (both Belle and BaBar started data taking in 1999): ∗ + − + 0 D (2317): discovered in e e D π X BaBar, PRL90(2003)242001 [hep-ex/0304021] • s0 → s

J P = 0+,M = (2317.7 0.6) MeV, Γ< 3.8 MeV ± 0 I = 0, Dsπ : breaks isospin symmetry →

+ − ∗+ 0 Ds1(2460): discovered in e e D π X • → s CLEO, PRD68(2003)032002 [hep-ex/0305100] J P = 1+,M = (2459.5 0.6) MeV, Γ< 3.5 MeV ± I = 0, D∗π0: breaks isospin symmetry → s + + + − ∗ other decays: Ds γ, Ds π π , Ds0(2317)γ

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 15 / 72 Charm-strange mesons (2)

3000

2800

2600

D*K threshold D 2400 s1(2460) DK threshold D* (2317) Mass ( MeV ) s0 2200 Godfrey, Isgur, PRD32(1985)189 mesons discovered before 2003 2000 mesons discovered after 2003 - - + + + - ? 1800 0 1 0 1 2 3 ? Charm-strange mesons

∗ Ds0(2317) and Ds1(2460): the first established new hadrons ∗ Puzzle 1: Why are D (2317) and Ds1(2460) so light? • s0 Puzzle 2: Why M M ∗ M ∗± M ± ? Ds1(2460) Ds0(2317) D D • | {z− } ' | {z− } =(141.8±0.8) MeV =(140.67±0.08) MeV

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 16 / 72 Charm-nonstrange mesons (1)

Obervations of charm-nonstrange excited mesons in 2003 − (∗)+ − − B D π π Belle, PRD69(2004)112002 [hep-ex/0307021] → D∗(2400): J P = 0+ • 0 Γ = (267 40) MeV ± Mass (MeV):

2318 ± 29 PDG18 2297 ± 22 BaBar B decays 2308 ± 36 Belle B decays 2401 ± 41 FOCUS γA 2360 ± 34 LHCb B decays

P + D1(2430): J = 1 • +130 Γ = 384−110 MeV M = (2427 36) MeV ±

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 17 / 72 Notice: all these experiments used a Breit–Wigner to extract the resonance • (∗) (∗) (∗) D π-D η-Ds K¯ coupled-channel effects are absent chiral symmetry constraint on soft pions is absent

Charm-nonstrange mesons (2)

3000

2800

2600

2400 D 1(2430) D* (2400) 2200 0 Mass ( MeV ) Godfrey, Isgur, PRD32(1985)189 2000 mesons discovered before 2003 mesons discovered after 2003 1800 - 0- 1- 0+ 1+ 2+ 3 ?? Charm-nonstrange mesons

∗ ∗ Puzzle 3: Why MD (2400) & MD (2317) and MD1(2430) MDs1(2460)? • 0 s0 ∼

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 18 / 72 Most exotic and newest observation: X(5568)

X(5568) by D0 Collaboration (pp¯ collisions) PRL117(2016)022003; PRD97(2018)092004 •

M = 5567.8 2.9+0.9 MeV ± −1.9 Γ = 21.9 6.4+5.0 MeV ± −2.5

Estimate of isospin breaking decay (∗)0 + Observed in Bs π , sizeable width • width: I = 1:  2 ⇒ md−mu minimal quark contents is ¯bsdu¯ ! Λ Γ6 I  QCD  (100 MeV) ∼ 2 × O a favorite mulqituark candidate: α • explicitly flavor exotic, minimal number = (10 keV) O of quarks 4 ≥

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 19 / 72 XYZ states

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 20 / 72 XYZ states

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 20 / 72 Naming convention For states with properties in conflict with naive quark model (normally): X: I = 0, J PC other than 1−− or unknown • Y : I = 0, J PC = 1−− • Z: I = 1 • PDG2018 naming scheme:

“Young man, if I could remember the names of these particles, I would have been a botanist.” — Enrico Fermi Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 21 / 72 X(3872) (1)

Belle, PRL91(2003)262001 [hep-ex/0309032] The beginning of the XYZ story, • discovered in B± K±J/ψππ → MX = (3871.69 0.17) MeV ± Γ < 1.2 MeV Belle, PRD84(2011)052004 • Confirmed in many experiments: Belle, • BaBar, BESIII, CDF, CMS, D0, LHCb, ... 10 years later, J PC = 1++ • LHCb, PRL110(2013)222001 S-wave coupling to DD¯ ∗ ⇒ Mysterious properties: Mass coincides with the D0D¯ ∗0 threshold: • M 0 + M ∗0 MX = (0.00 0.18) MeV D D − ±

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 22 / 72 X(3872) (2) Mysterious properties (cont.): Large coupling to D0D¯ ∗0: • 0 ∗0 (X D D¯ ) > 30% Belle, PRD81(2010)031103 B → 0 0 0 (X D D¯ π ) > 40% Belle, PRL97(2006)162002 B → No isospin partner observed I = 0 • ⇒ but, large isospin breaking: (X ωJ/ψ) B → = 0.8 0.3 (X π+π−J/ψ) ± BaBar, PRD77(2008)011102 B → C(X) = +, C(J/ψ) = C(π+π−) = I(π+π−) = 1 − ⇒ − ⇒ Radiative decays: • (X γψ0) B → = 2.6 0.6 PDG18 average of BaBar(2009) and LHCb(2014) measurements (X γJ/ψ) ± B → Exercise: 1) Why is the isospin of the negative C- π+π− system equal to 1? + − PC ++ 2) Is Υπ π a good choice of final states for the search of Xb, the J = 1 bottom analogue of the X(3872)?

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 23 / 72 Y (4260)

Discovered by BaBar in 2005 PRL95(2005)142001 •

J PC = 1−−, confirmed by Belle, CLEO, BESIII

black: Godfrey,Isgur (1985) red: observed states

4500 3D 4S

Y(4260) 2D 3S 4000 BESIII, PRL118(2017)092001 1D M = (4230 8) MeV, Γ = (55 19) MeV PDG2018 2S

• ± ± Mass (MeV) Puzzles: 3500 • no slot in quark model ¯ ¯ 1S well above DD threshold, but not seen in DD 3000 (recall the OZI rule)

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 24 / 72 ± ± Zc and Zb (1) Z±, Z±: charged structures in heavy mass region, • c b excellent candidates: QQ¯du,¯ QQ¯ud¯ ± ± Zb(10610) and Zb(10650) : Belle, arXiv:1105.4583; PRL108(2012)122001 • ∓ ± observed in Υ(10860) π [π Υ(1S, 2S, 3S)/hb(1P, 2P )] →

∓ (∗) ∗ ± also in Υ(10860) π [B B¯ ] Belle, arXiv:1209.6450; PRL116(2016)212001 →

± ± ∗ ∗ ∗ Zb(10610) and Zb(10650) very close to BB¯ and B B¯ thresholds • Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 25 / 72 ± ± Zc and Zb (2)

± ± Zc(3900) : 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)252002 →∗ ± ∗ ∓ and in DD¯ by BESIII in Y (4260) π (DD¯ ) BESIII, PRD92(2015)092006 →

± ± ∗ ∗ ± Zc(4020) observed in hcπ and (D¯ D ) distributions • BESIII, PRL111(2013)242001; PRL112(2014)132001 ± ± ∗ ∗ ∗ Zc(3900) and Zc(4020) very close to DD¯ and D D¯ thresholds • Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 26 / 72 Charmonium spectrum: current status

Note: J PC of X(3915) might also be 2++

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 27 / 72 Bottomonium spectrum: current status

Bottomonium spectrum Charged 11020 11 000 Υ( ) Υ(10860)

* B*B Zb(10650) Υ(4S) B*B χb1 (3P) Zb(10610) 10 500 BB Υ(3S) h 2P 2P χb2 (2P) b( ) χb0 (2P) χb1 ( ) Υ2(1D) 2S ( MeV ) Mass ηb(2S) Υ( ) 10 000 1P hb(1P) χb1 (1P) χb2 ( ) χb0 (1P)

Godfrey-Isgur quark model 9500 Υ(1S) discovered before 2003 ηb(1S) discovered after 2003 0-+ 1-- 1+- 0++ 1++ 2++ 2-- 1+-

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 28 / 72 Pentaquark candidates

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 29 / 72 LHCb’s Pc (1)

LHCb, PRL115(2015)072001 [arXiv:1507.03414]

Two Breit–Wigner resonances needed:

M1 = (4380 8 29) MeV, Γ1 = (205 18 86) MeV, ± ± ± ± M2 = (4449.8 1.7 2.5) MeV, Γ2 = (39 5 19) MeV. ± ± ± ± Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 30 / 72 LHCb’s Pc (2)

In J/ψ p invariant mass distribution, with hidden charm • if they are hadron resonances ⇒ Quantum numbers not fully determined, for ( Pc(4380),Pc(4450) ): • (3/2−, 5/2+), (3/2+, 5/2−), (5/2+, 3/2−),... ∗ From a reanalysis using an extended Λ model: N. Jurik, CERN-THESIS-2016-086

Early prediction: • Prediction of narrow N ∗ and Λ∗ resonances with hidden charm above 4 GeV, J.-J. Wu, . Molina, E. Oset, B.-S. Zou, PRL105(2010)232001

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 31 / 72 Recent reviews on new hadrons

H.-X. Chen et al., The hidden-charm pentaquark and tetraquark states, Phys. • Rept. 639 (2016) 1 [arXiv:1601.02092] A. Hosaka et al., Exotic hadrons with heavy flavors — X, Y, Z and related states, • Prog. Theor. Exp. Phys. 2016, 062C01 [arXiv:1603.09229] R. F. Lebed, R. E. Mitchell, E. Swanson, Heavy-quark QCD exotica, Prog. Part. • Nucl. Phys. 93 (2017) 143, arXiv:1610.04528 [hep-ph] A. Esposito, A. Pilloni, A. D. Polosa, Multiquark resonances, Phys. Rept. 668 • (2017) 1 [arXiv:1611.07920] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, B.-S. Zou, Hadronic • molecules, Rev. Mod. Phys. 90 (2018) 015004 [arXiv:1705.00141] S. L. Olsen, T. Skwarnicki, Nonstandard heavy mesons and baryons: • Experimental evidence, Rev. Mod. Phys. 90 (2018) 015003 [arXiv:1708.04012] M. Karliner, J. L. Rosner, T. Skwarnicki, Multiquark states, Ann. Rev. Nucl. Part. • Sci. 68 (2018) 17 [arXiv:1711.10626] C.-Z. Yuan, The XYZ states revisited, Int. J. Mod. Phys. A 33 (2018) 1830018 • [arXiv:1808.01570]

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 32 / 72 Symmetries of QCD: chiral and heavy quark

Useful monographs: H. Georgi, Weak Interactions and Modern Physics (2009) • J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the • (1992) S. Scherer, M.R. Schindler, A Primer for Chiral Perturbation Theory (2012) • A.V. Manohar, M.B. Wise, Heavy Quark Physics (2000) •

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 33 / 72 Symmetries for different sectors

Different quark flavors: •

Spontaneously broken chiral Heavy quark spin symmetry symmetry: π, K and η as the Heavy quark flavor symmetry pseudo-Goldstone Heavy antiquark- symmetry

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 34 / 72 Chiral symmetry in a nut shell

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 35 / 72 Chiral symmetry (1)

QCD Lagrangian •  QCD = iq¯ Dq + iq¯ Dq q¯L qR +¯qR qL + ... L L 6 L R 6 R − M M 1 1 q = (1 γ5)q + (1 + γ5)q PLq + PRq = qL + qR 2 − 2 ≡ For mu,d,s = 0, invariant underU (3)L U(3)R transformations: • × 0 0 0 0 (Gµν , q ,Dµq ) = (Gµν , q, Dµq) LQCD LQCD 0 q = RP Rq + LP Lq = Rq R + Lq L

R U(3)R,L U(3)L ∈ ∈ P Parity: q(t, ~x) γ0q(t, ~x) • → − P 0 0 0 q (t, ~x) PRγ q(t, ~x) = γ PLq(t, ~x) = γ qL(t, ~x) ⇒ R → − − − P 0 q (t, ~x) γ qR(t, ~x) L → − U(3)L U(3)R = SU(3)L SU(3)R U(1)L U(1)R • × × × ×

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 36 / 72 Chiral symmetry (2)

U(3)L U(3)R = SU(3)L SU(3)R U(1)V U(1)A • × × × | {z } × | {z } cons. broken by quantum anomaly Note: “SU(3)A” not a group

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 37 / 72 Chiral symmetry (3): Wigner–Weyl v.s. Nambu–Goldstone

Noether’s theorem: continuous symmetry conserved currents • ⇒ a a R 3 a,0 a,µ Let Q be symmetry charges: Q = d ~xJ (t, ~x), ∂µJ = 0 a a Qa is the symmetry generator: g = eiα Q , H: Hamiltonian, thus • gHg−1 = H [Qa,H] = 0, ⇒ [Qa,H] 0 = QaH 0 HQa 0 = 0 | i | {z| }i − | i =0 Wigner–Weyl mode: Qa 0 = 0 or equivalently g 0 = 0 • | i | i | i degeneracy in mass spectrum Nambu–Goldstone mode: g 0 = 0 , spontaneously broken (hidden) • | i 6 | i Qa 0 = 0 : new states degenerate with vacuum, massless Goldstone bosons | i 6 | i spontaneously broken continuous global symmetry massless GBs ⇒ the same quantum numbers as Qa 0 spinless | i ⇒ #(GBs) = #(broken generators)

A. Zee: “If you want to show off your mastery of mathematical jargon you can say that the Nambu–Goldstone bosons live in the coset space G/H.”

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 38 / 72 Chiral symmetry (4)

a −1 a PQ P = Q , if G = SU(Nf )L SU(Nf )R realized in Wigner-Weyl mode • A − A × parity doubling: hadrons have degenerate partners with opposite parity, but ⇒

mNucleon,P =+ = 939 MeV m ∗ = 1535 MeV,  N (1535),P =− mπ, P =− = 139 MeV m = 980 MeV  a0(980),P =+ a a vacuum invariant under H = SU(Nf )V : Q 0 = 0,Q 0 = 0 • V | i A| i 6 SSB massless pseudoscalar Goldstone bosons • ⇒ #(GBs) = dim(G) dim(H) = N 2 1 − f − ± 0 ± 0 0 for Nf = 3, 8 GBs: π , π ,K ,K , K¯ , η ± 0 for Nf = 2, 3 GBs: π , π Pions get a small mass due to explicit symmetry • breaking by tiny mu,d (a few MeV) 2 peudo-Goldstone bosons, Gell-Mann–Oakes–Renner: M (mu + md) π ∝ Mπ Mother hadron, also, ms mu,d MK Mπ   ⇒  Mechanism for SU(Nf )L SU(Nf )R SU(Nf )V in QCD not understood • × → Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 39 / 72 Heavy quark symmetries

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 40 / 72 Heavy quark symmetries (1)

For heavy quarks (charm, bottom) in a hadron, typical momentum transfer Λ • QCD heavy quark spin symmetry (HQSS): σ B chromomag. interaction · ∝ mQ spin of the heavy quark decouples

Let total angular momentum J = sQ + s`,

sQ: heavy quark spin, s`: spin of the light degrees of freedom (including orbital angular momentum)

mQ→+∞ ? angular momentum conservation s` is conserved ⇒ ? spin multiplets: − for singly-heavy mesons, e.g., D,D∗ with sP = 1 , { } ` 2 MD∗ MD 140 MeV, MB∗ MB 46 MeV − ' − ' for QQ¯, e.g., ηc, J/ψ , χc0, χc1, χc2, hc , ηb, Υ { } { } { }

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 41 / 72 Heavy quark symmetries (2)

For heavy quarks (charm, bottom) in a hadron, typical momentum transfer Λ • QCD heavy quark flavor symmetry (HQFS): for any hadron containing one heavy quark: velocity remains unchanged in the

limit mQ : → ∞ ∆p Λ ∆v = = QCD mQ mQ heavy quark is like a static color triplet source, mQ is irrelevant ⇒ heavy anti-quark–diquark symmetry M. Savage, M. Wise, PLB248(1990)177

if mQv ΛQCD,  the diquark serves as a point-like color-3¯ source, like a heavy anti-quark. It relates doubly-heavy baryons to anti- heavy mesons

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 42 / 72 Intermediate summary

Many new hadrons observed (in particular in the charm sector), lots of mysteries • Symmetries of QCD: • spontaneously broken chiral symmetry for light flavors • heavy quark spin and flavor symmetry for heavy flavors • next, applications of symmetries to the new hadrons ⇒

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 43 / 72 HQS for open-flavor heavy hadrons

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 44 / 72 Applications of HQS: P -wave heavy mesons

Examples of HQSS phenomenology: P + Widths of the two D1 (J = 1 ) mesons • +130 Γ[D1(2420)] = (27.4 2.5) MeV Γ[D1(2430)] = (384 ) MeV ±  −110 P 1 + 3 + s` = sq + L for P -wave charmed mesons: s = or ⇒ ` 2 2 ∗ for decays D1 D π: → 1 + 1 − − 2 2 + 0 in S-wave large width + → − ⇒ 3 1 + 0− in D-wave small width 2 → 2 ⇒ P 3 + P 1 + thus, D1(2420): s` = 2 , D1(2430): s` = 2 + − Suppression of the S-wave production of 3 + 1 heavy meson pairs in e+e− • 2 2 annihilation Table VI in E.Eichten et al., PRD17(1978)3090; X. Li, M. Voloshin, PRD88(2013)034012

Exercise: Try to understand this statement as a consequence of HQSS.

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 45 / 72 ∗ Applications of HQS: Ds0(2317) and Ds1(2460) (1)

HQFS: for a singly-heavy hadron, •  Λ2  M = m + A + QCD with A independent of m HQ Q O mQ Q rough estimates of bottom analogues whatever the DsJ states are •    2 1 1 M ∗ = M ∗ + ∆ + Λ (5.65 0.15) GeV Bs0 Ds0(2317) b−c QCD O mc − mb ' ±    2 1 1 MBs1 = MDs1(2460) + ∆b−c + ΛQCD (5.79 0.15) GeV O mc − mb ' ±

here ∆b−c mb mc M B M D 3.33 GeV, where ≡ − ' s − s ' M Bs = 5.403 GeV, M Ds = 2.076 GeV: spin-averaged g.s. Qs¯ meson both to be discovered 1

Lattice QCD results: Lang, Mohler, Prelovsek, Woloshyn, PLB750(2015)17 • lat. MB∗ = (5.711 0.013 0.019) GeV s0 ± ± M lat. = (5.750 0.017 0.019) GeV Bs1 ± ± 1 The established meson Bs1(5830) is probably the bottom partner of Ds1(2536). Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 46 / 72 ∗ Applications of HQS: Ds0(2317) and Ds1(2460) (2)

∗ ∗ in hadronic molecular model: D (2317)[ DK], Ds1(2460)[ D K] • s0 ' ' Barnes, Close, Lipkin (2003); van Beveren, Rupp (2003); Kolomeitsev, Lutz (2004); FKG et al. (2006); . . .

D(∗)K bound states: poles of the T -matrix

HQSS similar binding energies MD + MK MD∗ 45 MeV • ⇒ − s0 ' MD (2460) MD∗ (2317) MD∗ MD is natural s1 − s0 ' − HQFS predicting the 0+ and 1+ bottom-partner masses • ⇒

MB∗ MB + MK 45 MeV 5.730 GeV s0 ' − ' MB MB∗ + MK 45 MeV 5.776 GeV s1 ' − '

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 47 / 72 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 bsud¯ : M MB + 200 MeV • ' s Mπ 140 MeV because pions are pseudo-Goldstone bosons ' 2 Gell-Mann–Oakes–Renner: M mq π ∝ For any field: MR Mπ; one expects Mud¯ MR Mσ  ∼ & M¯ MB + 500 MeV 5.9 GeV bsud¯ & s ∼ HQFS predicts an isovector Xc: •    2 1 1 MXc = MX(5568) ∆b−c + ΛQCD (2.24 0.15) GeV − O mc − mb ' ± ∗ but in Dsπ, only the isoscalar Ds0(2317) was observed! BaBar (2003) properties of X(5568) hard to understand

negative results reported by LHCb, LHCb, PRL117(2016)152003 • by CMS, CMS, PRL120(2018)202005 by CDF, CDF, PRL120(2018)202006 by ATLAS ATLAS, PRL120(2018)202007

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 48 / 72 From heavy baryons to doubly-heavy (1)

++ Development inspired by the LHCb discovery of the Ξcc(3620) Heavy antiquark-diquark symmetry (HADS): • repalcing Q¯ in Qq¯ by QQ QQq; ⇒ repalcing Q¯ in Q¯q¯q¯ by QQ QQq¯q¯; ⇒ Q¯q QQq, Q¯q¯q¯ QQq¯q¯ ⇒ ⇒ M : mQ + A mQQ + A,m Q + B mQQ + B ⇒ ⇒ Prediction: MQQq¯q¯ M ¯ MQQq M ¯ − Qq¯q¯ ' − Qq Doubly-charmed baryon discovered by LHCb PRL119(2017)112001 [arXiv:1707.01621] •

M ++ = (3621.40 0.78) MeV can be used as input Ξcc ± Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 49 / 72 From heavy baryons to doubly-heavy tetraquarks (2)

Eichten, Quigg, PRL119(2017)202002 HADS stable doubly-bottom tetraquarks (only decay weakly) are likely to exist • ⇒ see also Carlson, Heller, Tjon, PRD37(1988)744; Manohar, Wise, NPB399(1993)17; Karliner, Rosner,

PRL119(2017)202001; Czarnecki, Leng, Voloshin, PLB778(2018)233; ...

support from lattice QCD Francis, Hudspith, Lewis, Maltman, PRL118(2017)142001 • Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 50 / 72 HQS for XYZ states

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 51 / 72 HQSS for XYZ (1)

Assuming the X(3872) to be a DD¯ ∗ • − Consider S-wave interaction between a pair of sP = 1 (anti-)heavy mesons: • ` 2 0++ : DD,D¯ ∗D¯ ∗ 1 1+− : DD¯ ∗ + D∗D¯ ,D∗D¯ ∗ √2 1 1++ : DD¯ ∗ D∗D¯ √2 − 2++ : D∗D¯ ∗

C C here, charge conjugation phase convention: D +D,D¯ ∗ D¯ ∗ → →− Heavy quark spin irrelevant interaction matrix elements: • ⇒ D 0 0 E s1 `, s2 `,s L ˆ s , s ,s L H 1 ` 2 ` For each isospin,2 independent terms 1 1 1 1  1 1 1 1  , , 0 ˆ , , 0 , , , 1 ˆ , , 1 2 2 H 2 2 2 2 H 2 2

6 pairs grouped in 2 multiplets with sL = 0 and 1, respectively ⇒ Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 52 / 72 HQSS for XYZ (2)

PC PC For the HQSS consequences, convenient to use the basis of states: sL scc¯ • PC PC −+ −− ⊗ S-wave: sL ,s cc¯ = 0 or 1 multiplet with sL = 0:

0−+ 0−+ = 0++, 0−+ 1−− = 1+− L ⊗ cc¯ L ⊗ cc¯

multiplet with sL = 1:

1−− 0−+ = 1+−, 1−− 1−− = 0++ 1++ 2++ L ⊗ cc¯ L ⊗ cc¯ ⊕ ⊕ Multiplets in strict heavy quark limit: • X(3872) has three partners with 0++, 2++ and 1+−

Hidalgo-Duque et al., PLB727(2013)432; Baru et al., PLB763(2016)20 might be 6 I = 1 molecules: +− 0 +− ++ 0 ++ ++ ++ Zb[1 ],Zb[1 ] and Wb0[0 ], Wb0[0 ], Wb1[1 ] and Wb2[2 ] Bondar et al., PRD84(2011)054010; Voloshin, PRD84(2011)031502; Mehen, Powell, PRD84(2011)114013

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 53 / 72 HQSS for XYZ (3)

Recall the excercise in Lecture-1: • + − PC ++ Is Υπ π a good choice of final states for the search of Xb, the J = 1 bottom analogue of the X(3872)?

Answer: No. Xb Υππ breaks isospin symmetry → FKG, Hidalgo-Duque, Nieves, Valderrama, PRD88(2013)054007; Karliner, Rosner, PRD91(2015)014014

M 0 M ± = (0.31 0.06) MeV [M ± M 0 = (4.822 0.015) MeV] B − B ± D − D ± Negative results: • CMS, Search for a new bottomonium state decaying to Υ(1S)π+π− in pp collisions at √ s = 8 TeV, PLB727(2013)57; + − ATLAS, Search for the Xb and other hidden-beauty states in the π π Υ(1S) channel at ATLAS, PLB740(2015)199 ++ The results can be reinterpreted as for the search of WbJ (I = 1,J ) •

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 54 / 72 HQSS for XYZ (4)

1−− 1−− = 0++ 1++ 2++ L ⊗ cc¯ ⊕ ⊕ Heavy quark spin selection rule for X(3872): Voloshin, PLB604(2004)69 • ++ ∗ for X(3872) being a 1 DD¯ molecule, sL = 1, scc¯ = 1 spin structure of QQ¯: • PC sL scc¯ J cc¯ −+ S-wave 000 ηc 011 −− J/ψ +− P -wave 101 hc ++ 11 (0 , 1, 2) χc0, χc1, χc2

allowed: X(3872) J/ψππ, X(3872) χcJ π, X(3872) χcJ ππ • → → → suppressed: X(3872) ηcππ, X(3872) hcππ →(0) → Interesting feature of Z : observed with similar rates in both Υππ[s ¯ = 1] and • b bb hbππ[sb¯b = 0] Bondar, Garmash, Milstein, Mizuk, Voloshin, PRD84(2011)054010 ∗ − − − − 0 ∗ ∗ − − − − Zb BB¯ 0 1 1 0 ,Z B B¯ 0 1 +1 0 ∼ ∼ b¯b ⊗ qq¯ − b¯b ⊗ qq¯ b ∼ ∼ b¯b ⊗ qq¯ b¯b ⊗ qq¯ Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 55 / 72 HQSS for XYZ (5)

Unitary transformation from two-meson basis to s1 c, s2 c,s cc¯; s1 `, s2 `,sL; J : | i X p s1 c,s 1 `, j1; s2 c,s 2 `, j2; J = (2j1 + 1)(2j2 + 1)(2scc¯ + 1)(2sL + 1) | i scc¯,sL   s1 c s2 c scc¯   s s s s1 c, s2 c,s cc¯; s1 `, s2 `,sL; J × 1 ` 2 ` L | i    j1 j2 J  j1,2: meson spins; J: the total angular momentum of the whole system 1 s = : spin of the heavy quark in meson 1 (2) 1 c(2 c) 2 1 s = : angular momentum of the light quarks in meson 1 (2) 1 `(2 `) 2 scc¯ = 0, 1: total spin of cc¯, conserved but decoupled • sL = 0, 1: total angular momentum of the light-quark system, conserved • ˆ 0 0 only two independent s`1, s`2,s L s`1, s`2,s L I terms for each isospin I: • D EH D E 1 1 ˆ 1 1 1 1 ˆ 1 1 FI0 = 2 , 2 , 0 2 , 2 , 0 ,FI1 = 2 , 2 , 1 2 , 2 , 1 H I H I

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 56 / 72 HQSS for XYZ (6)

! ! DD¯ ++ C √3C : V (0 ) = IA IB , ∗ ∗ D D¯ √3CIB CIA 2CIB − ¯ ∗ ! ! DD +− CIA CIB 2CIB : V (1 ) = − , ∗ ∗ D D¯ 2CIB CIA CIB − ∗ (1++) DD¯ : V = CIA + CIB, ∗ ∗ (2++) D D¯ : V = CIA + CIB,

1 1 here, CIA = (3FI1 + FI0), CIB = (FI1 FI0) 4 4 − This predicts a spin partner for X(3872): Nieves, Valderrama, PRD86(2012)056004; . . . • M M MD∗ MD X2(4013) − X(3872) ≈ −

ongoing efforts searching for X2, not found yet

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 57 / 72 Threshold cusps and triangle singularties

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 58 / 72 Peaks and resonances

Resonances do not always appear as peaks:

J. R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions

Peaks are not always due to resonances:

Dynamics poles in the S-matrix (resonances): genuine physical states. • ⇒ Kinematic effects branching points of S-matrix • ⇒ normal two-body threshold cusp triangle singularity ... tools/traps in hadron spectroscopy

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 59 / 72 Unitarity and threshold cusp

† † 4 4 Unitarity of the S-matrix: SS = S S = 1, Sfi = δfi i(2π) δ (pf pi)Tfi • † 4 X † − − T -matrix: Tfi Tfi = i(2π) δ(pn pi)TfnTni − − n − |{z} all physically accesible states assuming all intermediate states are two-body, partial-wave unitarity relation: P ∗ Im TL,fi(s) = T ρ (s) T − n L,fn n L,ni

2-body phase space factor: ρn(s) = qcm,n(s)/(2√s)θ(√s mn1 mn2), − − p 2 2 qcm,n(s) = [s (mn1 + mn2) ][s (mn1 mn2) ]/(2√s) − − −

There is always a cusp at an S-wave threshold •

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 60 / 72 Threshold cusp: a well-known example

Cusp effect as a useful tool for precise measurement: • example of the cusp in K± π±π0π0 → strength of the cusp measures the interaction strength!

Meißner, Müller, Steininger (1997); Cabibbo (2004); Colangelo, Gasser, Kubis, Rusetsky (2006); . . .

threshold, only sensitive to scattering length, (a0 a2)M + = 0.2571 0.0056 ∼ − π ± Very prominent cusp large scattering length likely a nearby pole • ⇒ ⇒ 1 effective range expansion (ERE): f(k) = 1/a+rk2/2−ik

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 61 / 72 Triangle singularity (TS)

q 1 2 2 2 ∗ ∗ λ(mA, m1, m2) p2,left = p2,right γ (β E2 p2) 2mA ≡ ≡ −

on-shell momentum of m2 at the left and right cuts in the A rest frame p 2 β = |~p23|/E23, γ = 1/ 1 − β Bayar et al., PRD94(2016)074039 ∗ ∗ p2 > 0, p3 = γ (β E + p ) > 0 m2 and m3 move in the same direction • 3 2 ⇒ velocities in the A rest frame: v3 > β > v2 • E∗ p∗/β E∗ + p∗/β v = β 2 − 2 < β,v = β 3 2 > β 2 E∗ β p∗ 3 E∗ + β p∗ 2 − 2 3 2

Conditions (Coleman–Norton theorem): Coleman, Norton (1965); Bronzan (1964) • all three intermediate particles can go on shell simultaneously

~p2 ~p3, particle-3 can catch up with particle-2 (as a classical process) k needs very special kinematics process dependent! (contrary to pole position) • ⇒ Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 62 / 72 Coincidence of Pc(4450) with kinematic singularities

Mass: M = (4449.8 1.7 2.5) MeV • Pc(4450) ± ± Our trivial observation: Pc(4450) coincides with the χc1p threshold: •

M Mχ Mp = (0.9 3.1) MeV Pc(4450) − c1 − ± Our non-trivial observation: there is a triangle singularity at the same time! • Solving the equation p2,left = p2,right ⇒ to have a TS at M = Mχ + Mp, we need MΛ∗ 1.89 GeV J/ψp c1 ' On shell Λ∗ must be unstable, the TS is then a finite peak ⇒

K−

Λ∗ Λ0 p b p

χc1 J/ψ

More possible relevant TSs, see X.-H. Liu, Q. Wang, Q. Zhao, PLB757(2015)231

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 63 / 72 Compositeness and hadronic molecules

FKG, Hanhart, Meißner, Wang, Zhao, Zou, Hadronic molecules, Rev. Mod. Phys. 90 (2018) 015004

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 64 / 72 Hadronic molecules (1)

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 with a mass M = m1 + m2 EB −

1 size: R rhadron ∼ √2µEB 

scale separation power expansion in p/Λ, (nonrelativistic) EFT applicable! • ⇒

Only narrow hadrons can be considered as components of hadronic molecules, • Γh 1/r, r: range of  Filin et al., PRL105(2010)019101; FKG, Meißner, PRD84(2011)014013

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 65 / 72 Hadronic molecules (2)

Why are hadronic molecules interesting? • one realization of color-neutral objects, analogue of light nuclei important information for hadron-hadron interaction understanding the XYZ 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)1330045; . . .

see also, e.g., Weinberg’s books: QFT Vol.I, Lectures on QM

2 2π p 2π p gNR (1 Z) 2µEB 2µEB | | ≈ − µ2 ≤ µ2 2(1 Z) Z a − ,r e ≈− (2 Z)√2µEB ≈ (1 Z)√2µEB − −

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 66 / 72 Compositeness (1)

Model-independent result for S-wave loosely bound composite states:

Consider a system with Hamiltonian

= 0 + V H H 0: free Hamiltonian, V : interaction potential H

Compositeness: • the probability of finding the physical state B in the 2-body continuum q | i | i Z d3q 1 Z = q B 2 − (2π)3 |h | i| 2 Z = B0 B , 0 (1 Z) 1 • |h | i| ≤ − ≤ Z = 0: pure bound (composite) state Z = 1: pure elementary state

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 67 / 72 Compositeness (2)

Z 3 d q 2 Compositeness : 1 Z = q B − (2π)3 |h | i| Schrödinger equation • ( 0 + V ) B = EB B H | i − | i q2 multiplying by q and using 0 q = q : h | H | i 2µ| i momentum-space wave function: ⇒ q V B q B = h | 2| i h | i −EB + q /(2µ)

S-wave, small so that R = 1/√2µEB r, r: range of forces •  q V B = g [1 + (r/R)] h | | i NR O Compositeness: • Z 3 2 2 2 d q gNR h  r i µ gNR h  r i 1 Z = | | 1 + = | | 1 + 3 2 2 − (2π) [EB + q /(2µ)] O R 2π√2µEB O R

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 68 / 72 Compositeness (3)

Coupling constant measures the compositeness for an S-wave shallow bound • state 2 2π p 2π p gNR (1 Z) 2µEB 2µEB | | ≈ − µ2 ≤ µ2 bounded from the above

Exercise: 2 Show that gNR is the residue of the T -matrix element at the pole E = EB: | | − 2 gNR = lim (E + EB) k T k | | E→−EB h | | i

Hint: use the Lippmann–Schwinger equation T = V + V 1 T and the E−H0+i 3 completeness relation B B + R d q q q = 1 to derive the Low | ih | (2π)3 | (+)ih (+)| equation (noticing T q = V q ): | i | (+)i 0 Z 3 0 † 0 0 k V B B V k d q k T q q T k k T k = k V k + h | | ih | | i + 3 h | | 2 ih | | i h | | i h | | i E + EB + i (2π) E q /(2µ) + i −

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 69 / 72 Compositeness (4)

Z can be related to scattering length a and effective range re Weinberg (1965) • 2R(1 Z) h  r i RZ h  r i a = − 1 + ,r e = 1 + − 2 Z O R 1 Z O R − −1 2 − 4 Effective range expansion: f (k) = 1/a + rek /2 ik + k − O Derivation: 2π µ T (E) k T k = f(k) Im T −1(E) = √2µE θ(E) ≡ h | | i − µ ⇒ 2π Twice-subtracted dispersion relation for T −1(E)

2 Z +∞ −1 −1 E + EB (E + EB) Im T (w) T (E) = 2 + dw 2 gNR π (w E i)(w + EB) | | 0 − −  2 E + EB µ R 1 p = 2 + 2µE i gNR 4π R − − − | |

Example: deuteron as pn bound state. Exp.: EB = 2.2 MeV, a3 = 5.4 fm • S1 − aZ=1 = 0 fm, aZ=0 = ( 4.3 1.4) fm − ± Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 70 / 72 Applications

Coupling constant fixed by binding energy, long-distance processes such as • X(3872) D0D¯ 0π0,D0D¯ 0γ calculable → 0 0 0 E.g., XEFT prediction of Γ(X D D¯ π ) Fleming et al., PRD76(2007)034006 →

compositeness from scattering length: • scattering lengths calculable using the Lüscher formalism in lattice QCD E.g., from DKI = 0 scattering length D∗ (2317) contains 70%DK ⇒ s0 & Liu et al., PRD86(2013)014508; Martínez Torres et al., JHEP1505,053; Bali et al., PRD96(2017)074501

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 71 / 72 Summary

Lots of resonances or resonance-like structures observed in recent years, • many puzzles QCD symmetries (chiral, heavy quark) prove to be useful tools • Many more data needed, lots of work needs to be done •

Thank you for your attention!

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 72 / 72 Backup slides

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 1 / 10 Chiral symmetry (5): Derivative coupling

Symmetry implies a derivative coupling for GBs, i.e.,

GBs do not interact at vanishing momenta

Consider GB πa: πa Qa 0 = R d3x πa Aa(x) 0 = 0 • h | A| i h | 0 | i 6 a a Lorentz invariance π (q) A (0) 0 = iqµFπ ⇒ | µ | − Consider the matrix element Aa • µ a Aµ πa ψ ψ ψ1 ψ2 a 1 2 T ψ1 Aµ(0) ψ2 = + h | | i 1 = Ra + F qµ T a µ π q2 Current conservation qµAa = 0, thus ⇒ µ µ a a a q Rµ + FπT = 0 lim T = 0 ⇒ qµ→0

GBs couple in a derivative form !! •⇒

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 2 / 10 TS: some details (1)

Consider the scalar three-point loop integral Z d4q 1 I = i 4 2 2 2 2 2 2 (2π) [(P q) m + i](q m + i) [(p23 q) m + i] − − 1 − 2 − − 3 Rewriting a propagator into two poles: q 1 1 2 2 2 2 = 0 0 with ω2 = m2 + ~q q m + i (q ω2 + i)( q + ω2 i) − 2 − − focus on the positive-energy poles i Z dq0d3~q 1 I 4 0 0 0 0 0 ' 8m1m2m3 (2π) (P q ω1 + i)(q ω2 + i)(p q ω3 + i) − − − 23 − −

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 3 / 10 TS: some details (2)

Contour integral over q0 cut-1 cut-2 ⇒ Z d3~q 1 I 3 0 0 ∝ (2π) [P ω1(q) ω2(q) + i ][p ω2(q) ω3(~p23 ~q ) + i ] − − 23 − − − Z ∞ q2 dq 0 f(q) ∝ P ω1(q) ω2(q) + i  0 − − The second cut: Z 1 1 f(q)= dz 0 p 2 2 2 −1 p23 ω2(q) m3 + q + p23 2p23qz + i  Feng-Kun Guo (ITP) − Hadron− spectroscopy − 13–14.09.2018 4 / 10 TS: some details (3)

Relation between singularities of integrand and integral

singularity of integrand does not necessarily give • a singularity of integral: integral contour may be deformed to avoid the singularity Two cases that a singularity cannot be avoided: • endpoint singularity pinch singularity

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 5 / 10 TS: some details (4)

Z ∞ q2 I dq 0 f(q) ∝ P ω1(q) ω2(q) + i  0 − − Z 1 1 Z 1 1 f(q)= dz dz 0 p 2 2 2 −1 A(q, z) ≡ −1 p ω2(q) m + q + p 2p qz + i  23 − − 3 23 − 23 Singularities of the integrand of I in the rest frame of initial particle (P 0 = M):

1st cut: M ω1(l) ω2(l) + i  = 0 • − −  ⇒  1 p 2 2 2 qon± λ(M , m , m ) + i  ≡± 2M 1 2 2nd cut: A(q, 1) = 0 endpoint singularities of f(q) • ± ⇒ ∗ ∗ ∗ ∗ z = +1: qa+ = γ (β E + p ) + i  ,q a− = γ (β E p ) i , 2 2 2 − 2 − ∗ ∗ ∗ ∗ z = 1 : qb+ = γ ( β E + p ) + i  , qb− = γ (β E + p ) i  − − 2 2 − 2 2 − p 2 β = ~p23 /E23, γ = 1/ 1 β = E23/m23 | | − ∗ ∗ E2 (p2): energy (momentum) of particle-2 in the cmf of the (2,3) system

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 6 / 10 TS: some details (5)

All singularities of the integrand of I: ∗ ∗ ∗ ∗ qon+, qa+ = γ (β E + p ) + i ,q a− = γ (β E p ) i , 2 2 2 − 2 − qon− < 0, qb− = qa+ < 0 (for  = 0),q b+ = qa−, − − Im q Im q Im q

qon+ qa+ qon+ qa+ qon+ qa+

0 Re q 0 Re q 0 Re q qa qa qa − − −

(a) (b) (c) Im q 2-body threshold triangle singularity at

qon+ qb+ qa+ singularity at p2,left = p2,right 0 Re q m23 = m2 + m3

here p2,left = qon+, p2,right = qa−

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 7 / 10 Compositeness (5)

We may also start from a QFT (for very small EB, nonrelativistic)

Free field Interacting field i i E−E0 E−E0−Σ(E)

Here E0 = M0 m1 m2 with M0 the bare mass, Σ(E) is the self-energy (g0: bare − − coupling constant) Z 4  2   2 −1 2 d k 0 k 0 k Σ(E) = ig0 4 k + i E k + i (2π) − 2m1 − − 2m2 Z Λ d3k 1 = i2µg2(2πi) − 0 (2π)4 2µE k2 + i µ p − = g2 2µE i+ constant 0 2π − − µ hp p i = g2 2µEθ( E) i 2µEθ(E) + constant 0 2π − − −

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 8 / 10 Compositeness (6)

The physical mass is M = m1 + m2 + EB(EB 0) with EB the solution of ≥ − E E0 Σ(E) = 0, i.e. − − EB = E0 Σ( EB) − − − Expanding the self-energy around the pole, we rewrite the propagator i i = h i E E0 Σ(E) 0 E E0 Σ( EB) + (E + EB)Σ ( EB) + Σ(˜ E) − − − − − − i = 0 E + EB (E + EB)Σ ( EB) Σ(˜ E) − − − iZ = E + EB ZΣ(˜ E) − Z is the wave function renormalization constant

 2 2 −1 1 g0µ Z = 0 = 1 + 1 Σ ( EB) 2π√2µE − − B

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 9 / 10 Compositeness (7)

The physical coupling constant 1 2π g˜2= Zg2 = = (1 Z) p2µE 0 1 µ2 2 B 2 + √ − µ g0 2π 2µEB Taking into account the nonrel. normalization, we get the one in rel. QFT s 2 2 2 2EB g = 8m1m2(m1 + m2)˜g = 16π(1 Z)(m1 + m2) − µ

If the ERE is dominated by the scattering length (when the pole is extremely close to threshold), 2π/µ T (E) = 1/a √ 2µE i − − − − AtLO, effective coupling strength for bound state  −1 2 2π d p gNR = lim (E + EB) T (E) = 2µE i E→−EB µ dE | | − − − E=−EB 2π p = 2µEB Z = 0 at this leading order approximation µ2 ⇒

Feng-Kun Guo (ITP) Hadron spectroscopy 13–14.09.2018 10 / 10