Coupled-channel approach to spin partners of hadronic molecules

Transcription

1 Coupled-channel approach to spin partners of hadronic molecules A.V. Nefediev (MIPT, Moscow, Russia) in collaboration with V. Baru, E. Epelbaum, A. A. Filin, F.-K. Guo, C. Hanhart, Yu. S. Kalashnikova, U.-G. Meißner, R. Mizuk, Q. Wang Key refs: PRL115 (2015), ; PRD93 (2016), ; PLB763 (2016) 20; JHEP 1706 (2017) 158 QWG / 13

2 Introduction m[gev] 4.5 D D 2 D D 1 D D1 D D 4.0 D D D D η c(2s) 3.5 χ c0(3915) χ c0(1p) χ c2(2p) χ c2(1p) X(3872) χ c1(1p) Y(4660) ψ(4415) Y(4360) Y(4260) ψ(4160) ψ(4040) ψ(3770) ψ(2s) h c(1p) Z(4430) Z c(4020) Z c(3900) m[gev] 11.0 B B B B 10.5 η 10.0 b(2s) χ b0(2p) χ b0(1p) χ b2(2p) χ b2(1p) χ b1(3p) χ b1(2p) χ b1(1p) Υ(11020) Υ(5S) Υ(4S) Υ(3S) Υ(2S) Z b(10650) Z b(10610) h b(2p) h b(1p) 3.0 η c(1s) J/ψ(1S) 9.5 η b(1s) Υ(1S) J PC J PC Many hadronic states are found in spectrum of heavy quarks which do not fit into the quark model scheme reside near S-wave open-flavour thresholds have large decay branchings to nearby channels These are strong candidates to hadronic molecules (to be distinguished from tetraquarks, Esposito et al (2014)) 2 / 13

3 Heavy-quark spin symmetry Exotic XY Z states contain heavy quarks (HQ) In the limit m Q (m Q Λ QCD ) spin of HQ decouples = Heavy Quark Spin Symmetry (HQSS) For realistic m Q s HQSS is approximate but rather accurate symmetry of QCD HQSS is a tool to study properties of states with different HQ spin orientation = Spin partners Predictions of HQSS depend crucially on the nature of states under study Disclaimer: In this talk, only molecular scenario is discussed (Cleven et al (2015)) Quarkonium component of the w.f. (if exists) may impact the predictions (Cincioglu et al. (2016)) 3 / 13

4 Coupled-Channel Approach Unitary Parametrisation Fit for the Data Numerical Solution Parameters of Resonances Chiral Extrapolations Predictions for Spin Partners 4 / 13

5 Coupled-Channel Approach Unitary Parametrisation Fit for the Data Numerical Solution Parameters of Resonances Chiral Extrapolations Predictions for Spin Partners 4 / 13

6 Parameters and Input Short-range elastic interactions = Low-Energy Constants Transition potential between channels = Coupling constants Overall normalisation constants Bare poles (not necessary for Z b s) 5 / 13

7 Parameters and Input Short-range elastic interactions = Low-Energy Constants Transition potential between channels = Coupling constants Overall normalisation constants Bare poles (not necessary for Z b s) HQ limit = Reduced # of parameters Ways to proceed Proper way: combined coupled-channel fit for all measured channels Example of Z b (10610) and Z b (10650) (7 channels): = 7 parameters for profile of line shapes = 7 parameters for overall norms = CL 50% Simplified way: LEC s fixed to binding energies of known resonances 5 / 13

8 Parameters and Input Short-range elastic interactions = Low-Energy Constants Transition potential between channels = Coupling constants Overall normalisation constants Bare poles (not necessary for Z b s) HQ limit = Reduced # of parameters Ways to proceed Proper way: combined coupled-channel fit for all measured channels Example of Z b (10610) and Z b (10650) (7 channels): = 7 parameters for profile of line shapes = 7 parameters for overall norms = CL 50% Simplified way: LEC s fixed to binding energies of known resonances Binding energies of X(3872) and Z b (10610)/Z b (10650) will be used as input 5 / 13

9 Spin partners: Low-Energy Constants (LEC s) J P C states made of P (D or B) and V (D or B ) meson: 0 ++ : { P P ( 1 S 0 ), V V ( 1 S 0 ) } 1 + : { P V ( 3 S 1, ), V V ( 3 S 1 ) } 1 ++ : { P V ( 3 S 1, +) } 2 ++ : { V V ( 5 S 2 ) } In HQ limit, short-range elastic potentials depend on two LEC s (Grinstein et al. (1992), AlFiky et al. (2006), Nieves & Valderrama (2012)) V (0++ ) LO = 1 4 V (1+ ) LO = 1 2 ( 3C + C ) 3(C C ) 3(C C ) C + 3C ( ) C + C C C C C C + C V (1++ ) LO = V (2++ ) LO = C No symmetry relates C and C for different isospins HQ symmetry relates LEC s in c- and b-sector, but accuracy unclear 6 / 13

10 Spin partners: Predictions of contact theory HQSS limit: δ = m V m P E B m = two decoupled families E (0) 2 ++ = E (0) 1 ++ = E (0) 1 + = E (0) 0 ++ E (0) 1 + = E (0) 0 ++ In actuality: E B δ m = Expansion parameter E B /δ (Hidalgo-Duque et al. (2013), our work (2016)) V (1++ ) (Bondar et al. (2011), Voloshin (2011), Mehen & Powell (2011), our work (2016)) LO = V (2++ ) LO (only one input needed) = 1 ++ bound state at D D threshold (X(3872)) implies 2 ++ bound state at D D threshold (X c2 ) (Nieves & Valderama (2013)) 7 / 13

11 Spin partners: Predictions of contact theory HQSS limit: δ = m V m P E B m = two decoupled families E (0) 2 ++ = E (0) 1 ++ = E (0) 1 + = E (0) 0 ++ E (0) 1 + = E (0) 0 ++ In actuality: E B δ m = Expansion parameter E B /δ (Hidalgo-Duque et al. (2013), our work (2016)) V (1++ ) (Bondar et al. (2011), Voloshin (2011), Mehen & Powell (2011), our work (2016)) LO = V (2++ ) LO (only one input needed) = 1 ++ bound state at D D threshold (X(3872)) implies 2 ++ bound state at D D threshold (X c2 ) 2 ++ state is uncoupled and has zero width (Nieves & Valderama (2013)) 7 / 13

12 Pionic Lagrangian One-pion exchange L = g Q 2f π ( V π a τ a P + P τ a π a V + i[v V ] π a τ a) OPE potential m m p p m π q = p + p P V P V (p, p ) = g2 Q (4πf π ) 2 (τ τ c q i q j ) D 3 (p, p ) V ij p p E π = q 2 + m 2 π m m [( ) ] D 3 (p, p ) = 2E π m + p2 2m + m + p 2 2m + E π (m + m + E) When m > m + m π = Three-body cut When recoil terms neglected = Static OPE When q i q j 1 3 q2 δ ij = Central (S-wave) OPE 8 / 13

13 Spin partners: OPE included on top of LEC s P -wave V P (V )π vertices = Extended basis 0 ++ : {P P ( 1 S 0 ), V V ( 1 S 0 ), V V ( 5 D 0 )} 1 + : {P V ( 3 S 1, ), P V ( 3 D 1, ), V V ( 3 S 1 ), V V ( 3 D 1 )} 1 ++ : {P V ( 3 S 1, +), P V ( 3 D 1, +), V V ( 5 D 1 )} 2 ++ : {P P ( 1 D 2 ), P V ( 3 D 2 ), V V ( 5 S 2 ), V V ( 1 D 2 ), V V ( 5 D 2 ), V V ( 5 G 2 )} Important and cannot be ignored: Coupled-channel dynamics High momenta (q 500 MeV) D waves (q 2 /m 2 π is large) Three-body dynamics (c-sector) 9 / 13

14 Spin partners: OPE included on top of LEC s P -wave V P (V )π vertices = Extended basis 0 ++ : {P P ( 1 S 0 ), V V ( 1 S 0 ), V V ( 5 D 0 )} 1 + : {P V ( 3 S 1, ), P V ( 3 D 1, ), V V ( 3 S 1 ), V V ( 3 D 1 )} 1 ++ : {P V ( 3 S 1, +), P V ( 3 D 1, +), V V ( 5 D 1 )} 2 ++ : {P P ( 1 D 2 ), P V ( 3 D 2 ), V V ( 5 S 2 ), V V ( 1 D 2 ), V V ( 5 D 2 ), V V ( 5 G 2 )} Important and cannot be ignored: Coupled-channel dynamics High momenta (q 500 MeV) D waves (q 2 /m 2 π is large) Three-body dynamics (c-sector) OPE couples 2 ++ channel to other channels = finite width 9 / 13

15 Spin partners: Results for the 2 ++ state c-sector; I = 0 b-sector; I = 1 (V.Baru et al. PLB763 (2016) 20) (V.Baru et al. JHEP 1706 (2017) 158) E Xc2 Γ Xc2 D D 40 ± 20 MeV Xc2(2 ++ ) 45 ± 10 MeV Xc E [MeV] B B W b2(2 ++ ) E Wb2 Γ Wb2 } Wb2 23 ± 2 MeV 6 ± 1 MeV Source of uncertainty: UV regulator (c-sector) and input (b-sector) Impact of HQSS violation: Stronger in c-sector than in b-sector Stronger than for perturbative pions (cf. Albaladejo et al. (2015)) Role of three-body effects Sizable in c-sector (reduce E Xc2 and Γ Xc2 by 20%) Marginal in b-sector 10 / 13

16 W b2 (2 ++ ): Dependence on input and parameters of interaction Dynamical OPE enhahces HQ symmetry breaking Dynamical OPE cannot be absorbed into LEC s Physical limit = Physical limit = E B [W b2 ] survives large input variations Γ[W b2 ] is robust against input variations Blue dashed line: contact+(static S-wave OPE) Black Solid line: contact+(full dynamic OPE) 11 / 13

17 Conclusions The proposed systematic approach to hadronic molecules respects all relevant symmetries (chiral symmetry, HQSS, unitarity, analyticity) and allows to build a practical parametrisation for line shapes extract parameters of resonances directly from data build chiral extrapolations investigate various molecular candidates in c- and b-sectors HQSS breaking and nonperturbative pions have significant impact on near-threshold states OPE is more operative in the c-sector than in the b-sector 12 / 13

18 Conclusions X c2 is broad and shifted away from the D D threshold (observable?) W b2 is narrow and attracted to the B B threshold = Must be resolvable agaist the B B threshold = May be produced in Υ(5S) γw b2 (Voloshin (2011)) = May be seen in B B, B B, πχ b1, πχ b2, ρυ(1s) modes Predictions for the Z b s spin partners are rather accurate due to Weak HQSS breaking in the b-sector (Λ QCD /m b 1) No bb quarkonium admixture in w.f. s (isovectors) Small uncertainty caused by UV regulator (damped by large m b ) Robust with respect to input variations (at least for the 2 ++ state) 13 / 13

19 Backup 1 / 5

20 Spin partners of Z b s: Results Z b1 (1 + ) Z b1 (1+ ) W b0 (0 ++ ) W b0 (0++ ) W b1 (1 ++ ) W b2 (2 ++ ) E B 5 (input) 1 (input) 5.3 ± ± ±2.2 Γ B 4.6 ± 1.0 E B 1 (input) 1 (input) 0.7 ± ± ±1.8 Γ B 6.2 ± 1.1 Energies and widths are given in MeV Full calculation (contact+ope+oee) Uncertainty comes from UV regulator varied from 800 to 1500 MeV The 0 ++ partner at the B B threshold does not exist as a bound state 2 / 5

21 Spin partners of Z b s: Dependence on input Contact only theory Full theory In the contact theory, all bound states turn to virtual states in unison In the full calculation, OPE preserves 2 ++ and (possibly) 1 ++ partners as bound states 3 / 5

22 Spin partners of Z b s: HQSS breaking and SU(3) effects Red dotted line: contact only Blue dashed line: contact+central (S-wave) OPE Blue dashed-dotted line: contact+full OPE Black solid line: contact+full OPE+full OEE) 4 / 5

23 Spin partners: HQSS breaking effects (contact theory) Relation for 2 ++ and 1 ++ partners (γ = 2µE B ) γ X2 = ( 1 δ ) γ X + δ ( γ 2 ) 2 m π m Λ + O X Λ, δ2 Λ m 2 δ = m m m = 1 4 (3m + m) Relation for 1 + partners ( γ X 1 = 1 δ ) γ X1 + Λδ 2 m π m (γ X 1 γ X ) 2 + i (γ X 1 γ X ) mδ mδ UV regulator-related uncertainties in c- and b-sectors ( ) ( ) Λδ Λδ γ m γ m c b 5 / 5