X Y Z. Are they cusp effects? Feng-Kun Guo. Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn

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1 X Y Z Are they cusp effects? Feng-Kun Guo Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn Hadrons and Hadron Interactions in QCD 2015 Feb. 15 Mar. 21, 2015, YITP Based on: FKG, C. Hanhart, Q. Wang, Q. Zhao, arxiv: [hep-ph] Feng-Kun Guo (UniBonn) XY Z / 28

2 Charmonia spectrum Feng-Kun Guo (UniBonn) XY Z / 28

3 Charmonia spectrum Feng-Kun Guo (UniBonn) XY Z / 28

4 Charmonia spectrum Feng-Kun Guo (UniBonn) XY Z / 28

5 Charmonia spectrum Feng-Kun Guo (UniBonn) XY Z / 28

6 Near-threshold prominent structures X(3872) X(3872) Belle, PRL91(2003) Discovered in B ± K ± J/ψππ, mass extremely close to the D 0 D 0 threshold M = ( ± 0.17) MeV M D 0 + M D 0 = ( ± 0.12) MeV J P C = 1 ++ LHCb (2013) S-wave coupling to D D Observed in the D 0 D 0 mode as well BaBar, PRD77(2008) Feng-Kun Guo (UniBonn) XY Z / 28

7 Near-threshold prominent structures X(3872) X(3872) Belle, PRL91(2003) Discovered in B ± K ± J/ψππ, mass extremely close to the D 0 D 0 threshold M = ( ± 0.17) MeV M D 0 + M D 0 = ( ± 0.12) MeV J P C = 1 ++ LHCb (2013) S-wave coupling to D D Observed in the D 0 D 0 mode as well BaBar, PRD77(2008) Feng-Kun Guo (UniBonn) XY Z / 28

8 Near-threshold prominent structures X(3900) Z c (3900) (2013) BESIII, Belle, Xiao et al Discovered by BESIII and Belle in e + e ππj/ψ at s = 4.26 GeV Recall: the vector state Y (4260) A structure in e + e π ± [D D ] as well Mass from Breit-Wigner fits is close to the D D threshold M = ( ± 3.4) MeV assuming the two structures have the same origin Feng-Kun Guo (UniBonn) XY Z / 28

9 Near-threshold prominent structures more states Z c (4020) Discovered in e + e π ± π h c A structure with similar mass observed in e + e π ± [D D ] Close to the D D threshold: Z b (10610) and Z b (10650) M = ( ± 2.4) MeV Discovered in 5 different channels: Υ(10860) ππυ(1s, 2S, 3S)/πh b (1P, 2P ) Observed in Υ(10860) π ± [B ( ) B ] well M Zb (10610) M B + M B, M Zb (10650) 2M B BESIII, PRL111(2013) BESIII, PRL112(2014) Belle, PRL108(2012) Belle, arxiv: [hep-ex] Feng-Kun Guo (UniBonn) XY Z / 28

10 Near-threshold prominent structures more states Z c (4020) Discovered in e + e π ± π h c A structure with similar mass observed in e + e π ± [D D ] Close to the D D threshold: Z b (10610) and Z b (10650) M = ( ± 2.4) MeV Discovered in 5 different channels: Υ(10860) ππυ(1s, 2S, 3S)/πh b (1P, 2P ) Observed in Υ(10860) π ± [B ( ) B ] well M Zb (10610) M B + M B, M Zb (10650) 2M B BESIII, PRL111(2013) BESIII, PRL112(2014) Belle, PRL108(2012) Belle, arxiv: [hep-ex] Feng-Kun Guo (UniBonn) XY Z / 28

11 Two types of interpretations Poles in the S-matrix: tetraquarks hadronic molecules Cusps due to kinematical effect there is always a cusp at an S-wave threshold if they couple see talk by J. Nieves Can we distinguish them? Feng-Kun Guo (UniBonn) XY Z / 28

12 Two types of interpretations Poles in the S-matrix: tetraquarks hadronic molecules Cusps due to kinematical effect there is always a cusp at an S-wave threshold if they couple see talk by J. Nieves q cm (s) θ(s-(m 1 +m 2 ) 2 ) = disc A(s) C (s) q cm(s) s B(s)θ(s (m 1 + m 2 ) 2 ) Can we distinguish them? Feng-Kun Guo (UniBonn) XY Z / 28

13 Two types of interpretations Poles in the S-matrix: tetraquarks hadronic molecules Cusps due to kinematical effect there is always a cusp at an S-wave threshold if they couple see talk by J. Nieves q cm (s) θ(s-(m 1 +m 2 ) 2 ) = disc A(s) C (s) q cm(s) s B(s)θ(s (m 1 + m 2 ) 2 ) Can we distinguish them? Feng-Kun Guo (UniBonn) XY Z / 28

14 Cusp models Bugg Bugg speculated that the X(3872) could be a cusp effect but no calculation PLB598(2004)8 Then he realized in 2008 that such a cusp model could not reproduce the data for X(3872) in the J/ψρ and D D channels, a pole is needed JPG35(2008) For the Z b (10610, 10650) EPL96(2011)11002 could produce a narrow peak by using Re T (s) = 1 π P s th ds g2 ρ(s ) s s, with R = 1.41 fm F F (s) = exp ( q 2 cm(s)r 2 /3 ) (1/R = 0.14 GeV) ρ(s) = 2q cm(s) F F (s) s Feng-Kun Guo (UniBonn) XY Z / 28

15 Cusp models Bugg Bugg speculated that the X(3872) could be a cusp effect but no calculation PLB598(2004)8 Then he realized in 2008 that such a cusp model could not reproduce the data for X(3872) in the J/ψρ and D D channels, a pole is needed JPG35(2008) For the Z b (10610, 10650) EPL96(2011)11002 could produce a narrow peak by using Re T (s) = 1 π P s th ds g2 ρ(s ) s s, with R = 1.41 fm F F (s) = exp ( q 2 cm(s)r 2 /3 ) (1/R = 0.14 GeV) ρ(s) = 2q cm(s) F F (s) s Feng-Kun Guo (UniBonn) XY Z / 28

16 Cusp models Bugg Bugg speculated that the X(3872) could be a cusp effect but no calculation PLB598(2004)8 Then he realized in 2008 that such a cusp model could not reproduce the data for X(3872) in the J/ψρ and D D channels, a pole is needed JPG35(2008) For the Z b (10610, 10650) EPL96(2011)11002 could produce a narrow peak by using Re T (s) = 1 π P s th ds g2 ρ(s ) s s, with R = 1.41 fm F F (s) = exp ( q 2 cm(s)r 2 /3 ) (1/R = 0.14 GeV) ρ(s) = 2q cm(s) F F (s) s Feng-Kun Guo (UniBonn) XY Z / 28

17 Cusp models Swanson Swanson s model for both Z c and Z b states PRD91(2015) Π αβ (s) = 1 π s th ds Im Π αβ(s ) s s iɛ, Im Π αβ (s) = ( q cm,i (s)f αi (s)f βi (s), F αi (s) = g αi exp s ) 2β 2 i αi Impressive agreement with data by adjusting 2 parameters for each channel (g αi, β αi ) Feng-Kun Guo (UniBonn) XY Z / 28

18 Cusp models Chen, Liu, Matsuki D.-Y.Chen, X.Liu, PRD84(2011)094003; PRD84(2011)034032; Chen, Liu, Matsuki, PRD84(2011)074032; PRD88(2013)036008; PRL110(2013)232001;... Form factor ( Λ 2 m 2 B ( ) Λ 2 q 2 )2 B B + c.c. loops B B loops Belle data B B loops Feng-Kun Guo (UniBonn) XY Z / 28

19 Well-known cusp effect Opening of an S-wave threshold can produce a structure. What can be learned? Cusp effect has been well-known for a long time: example of the cusp in K ± π ± π 0 π 0 the strength of the cusp is determined by the interaction strength! Meißner, Müller, Steininger (1997); Cabibbo (2004); Colangelo, Gasser, Kubis, Rusetsky (2006);... Feng-Kun Guo (UniBonn) XY Z / 28

20 Z c (3900) as an example (I) Logic: first, fit to data with the one-loop expression which produces a cusp; then, try to understand the implications of the resulting parameter values Example: Y (4260) D D π: A 1-loop = g Y [1 C G Λ (E)] regularize the loop with a Gaussian form factor with a cutoff Λ d 3 ) q f Λ (q) G Λ (E) = (2π) 3 E m 1 m 2 q 2 /(2µ), f Λ(q) = exp ( 2q2 Λ 2 three parameters: g Y, C, Λ Feng-Kun Guo (UniBonn) XY Z / 28

21 Z c (3900) as an example (I) Logic: first, fit to data with the one-loop expression which produces a cusp; then, try to understand the implications of the resulting parameter values Example: Y (4260) D D π: A 1-loop = g Y [1 C G Λ (E)] regularize the loop with a Gaussian form factor with a cutoff Λ d 3 ) q f Λ (q) G Λ (E) = (2π) 3 E m 1 m 2 q 2 /(2µ), f Λ(q) = exp ( 2q2 Λ 2 three parameters: g Y, C, Λ Feng-Kun Guo (UniBonn) XY Z / 28

22 Z c (3900) as an example (II) Events / 4MeV m D 0 D *- [GeV] Implicit assumption of using g Y [1 C G Λ (E)] as the decay amplitude: the D D interaction is perturbative The two-loop contribution is large nonperturbative Resumming all the bubbles by 1 + C G Λ (E) g Y C G Λ (E th ) = 1.3! with the parameters determined from the 1-loop fit gives a bound state pole very close to the threshold (binding energy: 0.6 MeV) Feng-Kun Guo (UniBonn) XY Z / 28

23 Z c (3900) as an example (II) Events / 4MeV m D 0 D *- [GeV] Implicit assumption of using g Y [1 C G Λ (E)] as the decay amplitude: the D D interaction is perturbative The two-loop contribution is large nonperturbative Resumming all the bubbles by 1 + C G Λ (E) g Y C G Λ (E th ) = 1.3! with the parameters determined from the 1-loop fit gives a bound state pole very close to the threshold (binding energy: 0.6 MeV) Feng-Kun Guo (UniBonn) XY Z / 28

24 Z c (3900) as an example (II) Events / 4MeV m D 0 D *- [GeV] Implicit assumption of using g Y [1 C G Λ (E)] as the decay amplitude: the D D interaction is perturbative The two-loop contribution is large nonperturbative Resumming all the bubbles by 1 + C G Λ (E) g Y C G Λ (E th ) = 1.3! with the parameters determined from the 1-loop fit gives a bound state pole very close to the threshold (binding energy: 0.6 MeV) Feng-Kun Guo (UniBonn) XY Z / 28

25 Z c (3900) as an example (III) For perturbative interaction, we need C G Λ (E th ) Events / 4 MeV m D 0 D *- [GeV] Black curve: up to 1 loop with C G Λ (E th ) = 1/2, no narrow peak any more! Conclusion: A pronounced near-threshold peak in the elastic channel cannot be explained by kinematical cusp effect Feng-Kun Guo (UniBonn) XY Z / 28

26 Z c (3900) as an example (III) For perturbative interaction, we need C G Λ (E th ) Events / 4 MeV m D 0 D *- [GeV] Black curve: up to 1 loop with C G Λ (E th ) = 1/2, no narrow peak any more! Conclusion: A pronounced near-threshold peak in the elastic channel cannot be explained by kinematical cusp effect Feng-Kun Guo (UniBonn) XY Z / 28

27 Z c (3900) as an example (IV) For Y (4260) J/ψπ + π Inelastic channel: g Y ψ g Y G Λ (E)g ψ g ψ cannot be determined separately π π π D π Compare with the elastic case: g Y [1 CG Λ (E)] Y Y π (a) J/ψ D D * Y (c) C D * D D * (b) J/ψ π J/ψ Fit to the data with Λ fixed above Events / 0.02 GeV M J/ψ π - [GeV] A near-threshold peak in inelastic channel cannot distinguish a cusp effect from a pole Feng-Kun Guo (UniBonn) XY Z / 28

28 Naive fit with resummed amplitude fix Λ = 0.8 GeV and fit C to the data Events / 4MeV bound state m D 0 D *- [GeV] has a bound state pole with a binding energy of 1.6 MeV [C = 0.92 fm 2 ] Caution: more complications in a realistic fit, e.g., possible effect of triangle singularity, data for angular distribution... Feng-Kun Guo (UniBonn) XY Z / 28

29 Naive fit with resummed amplitude fix Λ = 0.8 GeV and fit C to the data Events / 4MeV virtual state bound state m D 0 D *- [GeV] or a virtual state pole with the same mass [C = 0.68 fm 2 ] Caution: more complications in a realistic fit, e.g., possible effect of triangle singularity, data for angular distribution... Feng-Kun Guo (UniBonn) XY Z / 28

30 Naive fit with resummed amplitude fix Λ = 0.8 GeV and fit C to the data Events / 4MeV virtual state bound state m D 0 D *- [GeV] or a virtual state pole with the same mass [C = 0.68 fm 2 ] Caution: more complications in a realistic fit, e.g., possible effect of triangle singularity, data for angular distribution... Feng-Kun Guo (UniBonn) XY Z / 28

31 X(3872), Y (4260) and Z c (3900) (I) Suppose that the X(3872), Y (4260) and Z c (3900) are hadronic molecules: X(3872): J P C = 1 ++, D D Y (4260): J P C = 1, D 1 (2420) D [two D 1 s, should be the narrow one] Z(3900): J P C = 1 +, D D Features of hadronic molecules: spin partners with similar fine splitting of their components M Ds1(2460) M D s0 (2317) M D M D M Zc(4020) M Zc(3900) talk by J. Nieves couple strongly to their components. If the binding energy is small, g 2 16π(1 Z)(m 1 + m 2 ) 2 2E B µ 16π(m 1 + m 2 ) 2 2E B µ = g2 h.m. 1 Z: compositeness talks by T. Sekihara and E. Oset last week can decay through the decays of their components seemingly unrelated processes may be related;... Feng-Kun Guo (UniBonn) XY Z / 28

32 X(3872), Y (4260) and Z c (3900) (I) Suppose that the X(3872), Y (4260) and Z c (3900) are hadronic molecules: X(3872): J P C = 1 ++, D D Y (4260): J P C = 1, D 1 (2420) D [two D 1 s, should be the narrow one] Z(3900): J P C = 1 +, D D Features of hadronic molecules: spin partners with similar fine splitting of their components M Ds1(2460) M D s0 (2317) M D M D M Zc(4020) M Zc(3900) talk by J. Nieves couple strongly to their components. If the binding energy is small, g 2 16π(1 Z)(m 1 + m 2 ) 2 2E B µ 16π(m 1 + m 2 ) 2 2E B µ = g2 h.m. 1 Z: compositeness talks by T. Sekihara and E. Oset last week can decay through the decays of their components seemingly unrelated processes may be related;... Feng-Kun Guo (UniBonn) XY Z / 28

33 X(3872), Y (4260) and Z c (3900) (II) Wang et al, PRL111(2013)132003; Cleven et al, PRD90(2014)074039; FKG et al, PLB725(2013)127 Production of X(3872) and Z c (3900) in Y (4260) decays Loops are enhanced when the binding energies are small: ( ) ( ) v 5 1 A O (v 2 ) 3 V (q D1D π/γ) = O V D1D v (q π/γ) S-wave vertices for couplings to Y (4260), X(3872) and Z c (3900) intermediate mesons are nonrelativistic. v: velocity power counting: three-momentum O (v), energy O ( v 2) loop measure : v 5, propagator : Feng-Kun Guo (UniBonn) XY Z / 28 1 v 2

34 X(3872), Y (4260) and Z c (3900) (II) Wang et al, PRL111(2013)132003; Cleven et al, PRD90(2014)074039; FKG et al, PLB725(2013)127 Production of X(3872) and Z c (3900) in Y (4260) decays Loops are enhanced when the binding energies are small: ( ) ( ) v 5 1 A O (v 2 ) 3 V (q D1D π/γ) = O V D1D v (q π/γ) S-wave vertices for couplings to Y (4260), X(3872) and Z c (3900) intermediate mesons are nonrelativistic. v: velocity power counting: three-momentum O (v), energy O ( v 2) loop measure : v 5, propagator : Feng-Kun Guo (UniBonn) XY Z / 28 1 v 2

35 Explicit check of the power counting Check the 1/v with explicit calculation of the three-point loop function (normalized to 1/v at an arbitrary point) for Y (4260) Z c (3900)π: Enhancement /v loop function power counting is well satisfied a large enhancement to the reaction rate M Y [GeV] Feng-Kun Guo (UniBonn) XY Z / 28

36 X(3872), Y (4260) and Z c (3900) (III) If these three states are hadronic molecules, then the Z c (3900) can be easily produced in the Y (4260) decays, in line with the BESIII and Belle observations Prediction: the X(3872) can be easily produced in Y (4260) γx(3872) FKG et al, PLB725(2013)127 BESIII observation of e + e γx(3872) γj/ψπ + π at s = 4.26 GeV: BESIII, PRL112(2014) Feng-Kun Guo (UniBonn) XY Z / 28

37 Summary Near-threshold peaks (cusps) provide information on interaction strength A pronounced narrow peak in line shape of the elastic channel cannot be explained by the kinematical cusp. It requires a pole. The data so far are in line with the hadronic molecular interpretation for the X(3872), Y (4260) and Z c (3900) Feng-Kun Guo (UniBonn) XY Z / 28

38 Z c (3900) as an example (V) Check the dependence on form factors: use a Lorentzian form factor f Λ (q) = Λ2 Λ 2 + q 2 80 Events / 4 MeV m D 0 D *- [GeV] C G Λ (E th ) 10 Feng-Kun Guo (UniBonn) XY Z / 28

39 Bound state and virtual state (I) Suppose the scattering length is very large, the S- wave scattering amplitude f 0 (k) = 1 k cot δ 0 (k) ik 1 1/a ik bound state pole: 1/a = κ virtual state pole: 1/a = κ If the same binding energy, cannot be distinguished above threshold (k is real): f 0 (k) 2 1 κ 2 + k 2 Feng-Kun Guo (UniBonn) XY Z / 28

40 Bound state and virtual state (I) Suppose the scattering length is very large, the S- wave scattering amplitude f 0 (k) = 1 k cot δ 0 (k) ik 1 1/a ik bound state pole: 1/a = κ virtual state pole: 1/a = κ If the same binding energy, cannot be distinguished above threshold (k is real): f 0 (k) 2 1 κ 2 + k 2 Events / 4MeV virtual state bound state m D 0 D *- [GeV] Feng-Kun Guo (UniBonn) XY Z / 28

41 Bound state and virtual state (II) Bound state and virtual state with a small binding energy should be distinguished in inelastic channel GZ E GeV E MeV A bound state and virtual state with a 5 MeV binding energy, a small residual width to the inelastic channel is allowed. Cleven et al, EPJA47(2011)120 Feng-Kun Guo (UniBonn) XY Z / 28

42 S-wave loosely bound hadronic molecules (I) Suppose the physical state ψ contains a two-hadron continuum state h 1 h 2 = q and something else ψ 0 The time-independent Schrödinger Equation (Ĥ0 + ˆV ) ψ = E B ψ here H 0 is the free Hamiltonian, Ĥ 0 q = q 2 /(2µ), and E B > 0 is the binding energy. Multiplying by q, we get q ψ = q ˆV ψ E B + q 2 /(2µ) The probability of finding the physical state in the continuum state is d λ 2 3 q d 3 q q = (2π) 3 q ψ 2 = ˆV ψ 2 (2π) 3 [E B + q 2 /(2µ)] 2 Feng-Kun Guo (UniBonn) XY Z / 28

43 S-wave loosely bound hadronic molecules (I) Suppose the physical state ψ contains a two-hadron continuum state h 1 h 2 = q and something else ψ 0 The time-independent Schrödinger Equation (Ĥ0 + ˆV ) ψ = E B ψ here H 0 is the free Hamiltonian, Ĥ 0 q = q 2 /(2µ), and E B > 0 is the binding energy. Multiplying by q, we get q ψ = q ˆV ψ E B + q 2 /(2µ) The probability of finding the physical state in the continuum state is d λ 2 3 q d 3 q q = (2π) 3 q ψ 2 = ˆV ψ 2 (2π) 3 [E B + q 2 /(2µ)] 2 Feng-Kun Guo (UniBonn) XY Z / 28

44 S-wave loosely bound hadronic molecules (II) Denoting gnr 2 (q) = q ˆV ψ 2, we have λ 2 = 4µ 2 dωq (2π) 3 0 dqq 2 g 2 NR (q) (q 2 + 2µE B ) 2 If the binding energy is very small, so that the binding momentum 2µE B 1/r with r the range of forces, we have an expansion here L is the orbital angular momentum. The integral is only convergent for S-wave. Therefore, gnr(q) 2 = q 2L gnr(0) 2 + O (r ) 2µE B the probability of finding the physical state in an S-wave two-hadron state with a small binding energy is related to the coupling constant g NR (0) λ 2 Landau (1960), Weinberg (1963,1965), Baru et al (2004),... µ 2 2π 2µE B g 2 NR(0) Feng-Kun Guo (UniBonn) XY Z / 28

45 S-wave loosely bound hadronic molecules (II) Denoting gnr 2 (q) = q ˆV ψ 2, we have λ 2 = 4µ 2 dωq (2π) 3 0 dqq 2 g 2 NR (q) (q 2 + 2µE B ) 2 If the binding energy is very small, so that the binding momentum 2µE B 1/r with r the range of forces, we have an expansion here L is the orbital angular momentum. The integral is only convergent for S-wave. Therefore, gnr(q) 2 = q 2L gnr(0) 2 + O (r ) 2µE B the probability of finding the physical state in an S-wave two-hadron state with a small binding energy is related to the coupling constant g NR (0) λ 2 Landau (1960), Weinberg (1963,1965), Baru et al (2004),... µ 2 2π 2µE B g 2 NR(0) Feng-Kun Guo (UniBonn) XY Z / 28

46 S-wave loosely bound hadronic molecules (II) Denoting gnr 2 (q) = q ˆV ψ 2, we have λ 2 = 4µ 2 dωq (2π) 3 0 dqq 2 g 2 NR (q) (q 2 + 2µE B ) 2 If the binding energy is very small, so that the binding momentum 2µE B 1/r with r the range of forces, we have an expansion here L is the orbital angular momentum. The integral is only convergent for S-wave. Therefore, gnr(q) 2 = q 2L gnr(0) 2 + O (r ) 2µE B the probability of finding the physical state in an S-wave two-hadron state with a small binding energy is related to the coupling constant g NR (0) λ 2 Landau (1960), Weinberg (1963,1965), Baru et al (2004),... µ 2 2π 2µE B g 2 NR(0) Feng-Kun Guo (UniBonn) XY Z / 28

47 S-wave loosely bound hadronic molecules (III) From nonrelativistic quantum mechanics to relativistic QFT: g = 2m 1 2m2 2(m1 + m 2 )g NR (0) here g is the coupling constant in the relativistic Lagrangian L = gψ h 1 h 2 + h.c. Therefore, the coupling constant contains the structure information g 2 16πλ 2 (m 1 + m 2 ) 2 2E B µ 16π(m 1 + m 2 ) 2 2E B µ It is bounded from above! Feng-Kun Guo (UniBonn) XY Z / 28

48 e + e D D π Y (4260) D D π with an explicit Z c pole: e + π e + π Y D 1 Y D 1 D D e (a) D D e D (b) Z c D Feng-Kun Guo (UniBonn) XY Z / 28

49 Radiative decays of the X(3872) FKG, Hanhart, Kalashnikova, Meißner, Nefediev, PLB742(2015)394 B(X(3872) ψ γ) The ratio = 2.46 ± 0.64 ± 0.29 B(X(3872) J/ψγ) is insensitive to the molecular component of the X(3872): LHCb, NPB886(2014)665 loops are sensitive to unknown couplings g ψdd /g ψ DD loops are divergent, needs a counterterm (short-distance physics) Feng-Kun Guo (UniBonn) XY Z / 28