Triangle singularity and the LHCb P c (4450) structure

Transcription

1 Triangle singularity and the LHCb P c (4450) structure Feng-Kun Guo Institute of Theoretical Physics, Chinese Academy of Sciences INPAC, Shanghai Jiao Tong University, March 10, 2017 Based on: FKG, U.-G. Meißner, W. Wang, Z. Yang, Phys. Rev. D 92, (R) (2015) [arxiv: [he-h]] FKG, U.-G. Meißner, J. Nieves, Z. Yang, Eur. Phys. J. A 52, 318 (2016) [arxiv: [he-h]] M. Bayar, A. Aceti, FKG, E. Oset, Phys. Rev. D 94, (2016) [arxiv: ] Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

2 LHCb entaquark-like structures: Big news in 2015! PRL115(2015) [arxiv: ] aeared on arxiv on , and acceted by PRL on 24.07! 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) Triangle singularity and Pc(4450) / 25

3 LHCb entaquark-like structures (II) Quantum numbers not fully determined, for ( P c (4380), P c (4450) ): (3/2, 5/2 + ), (3/2 +, 5/2 ), (5/2 +, 3/2 ) In J/ψ invariant mass distribution, with hidden charm entaquarks if they are really hadron states Narrow entaquark-like structures with hidden-charm had been redicted 5 years ago ( ): Prediction of narrow N and Λ resonances with hidden charm above 4 GeV, J. J. Wu, R. Molina, E. Oset, B. S. Zou, Phys. Rev. Lett. 105 (2010) Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

4 A flood of short aers The LHCb aer aeared on line, arxiv: R. Chen, X. Liu, X. Q. Li and S. L. Zhu, arxiv: [he-h]. H. X. Chen, W. Chen, X. Liu, T. G. Steele and S. L. Zhu, arxiv: [he-h] L. Roca, J. Nieves and E. Oset, arxiv: [he-h] A. Mironov and A. Morozov, arxiv: [he-h] weekend, but everybody was working hard (NOT including me) F.-K. Guo, U.-G. Meißner, W. Wang and Z. Yang, arxiv: [he-h]. L. Maiani, A. D. Polosa and V. Riquer, arxiv: [he-h] J. He, arxiv: [he-h]; X. H. Liu, Q. Wang, Q. Zhao, arxiv: [he-h] R. F. Lebed, arxiv: [he-h] Exotic! why no new aers? M. Mikhasenko, arxiv: [he-h] U.-G. Meißner and J. A. Oller, arxiv: [he-h] V. V. Anisovich et al., arxiv: [he-h] Guan-Nan Li, Min He, Xiao-Gang He, arxiv: [he-h] Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

5 Two kinds of singularities of the S matrix Poles in the S-matrix: dynamics bound states (real axis, 1st Riemann sheet (RS) of the comlex energy lane) virtual states (real axis, 2nd RS) resonances (2nd RS) Landau singularities: kinematics (a): two-body threshold cus (b): triangle singularity... Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

6 Two kinds of singularities of the S matrix Poles in the S-matrix: dynamics bound states (real axis, 1st Riemann sheet (RS) of the comlex energy lane) virtual states (real axis, 2nd RS) resonances (2nd RS) Landau singularities: kinematics (a): two-body threshold cus (b): triangle singularity... Λ 0 b K χ c1 (a) J/ψ Λ 0 b Λ χ c1 (b) K J/ψ Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

7 Triangle singularity literature Some recent work using triangle singularity to exlain (art of) eak structures [η(1405/1475), a 1 (1420),... ]: J. J. Wu, X. H. Liu, Q. Zhao and B. S. Zou, PRL108(2012)081803; X. G. Wu, J. J. Wu, Q. Zhao and B. S. Zou, PRD87(2013)014023(2013); Q. Wang, C. Hanhart and Q. Zhao, PLB725(2013)106; M. Mikhasenko, B. Ketzer and A. Sarantsev, PRD91(2015)094015; N. N. Achasov, A. A. Kozhevnikov and G. N. Shestakov, PRD92(2015)036003; X. H. Liu, M. Oka and Q. Zhao, PLB753(2016)297; A. P. Szczeaniak, PLB747(2015)410; PLB757(2016)61; F. Aceti, L. R. Dai and E. Oset, arxiv: [he-h]; Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

8 Triangle singularity literature Very old knowledge from 1960s: Classical books: R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The Analytic S-Matrix Cambridge University Press, T.-S. Chang, Introduction to Disersion Relation (2 volumes, in Chinese, written in 1965), Science Press, Beijing 1980, Recent lecture notes by one of the key layers: I. J. R. Aitchison, arxiv: [he-h]. Unitarity, Analyticity and Crossing Symmetry in Twoand Three-hadron Final State Interactions. Tsung-Sui Chang ( ) Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

9 P c (4450) is at the χ c1 threshold Mass: M = ( ± 1.7 ± 2.5) MeV The LHCb aer says: the closest threshold is at ( ± 0.3) MeV [Λ c (2595) D 0 ] difficult to exlain with threshold effect It could be more comlicated It is located exactly at the χ c1 threshold: M Pc(4450) M χc1 M = (0.9 ± 3.1) MeV and at a triangle singularity at the same time! Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

10 P c (4450) is at the χ c1 threshold Mass: M = ( ± 1.7 ± 2.5) MeV The LHCb aer says: the closest threshold is at ( ± 0.3) MeV [Λ c (2595) D 0 ] difficult to exlain with threshold effect It could be more comlicated It is located exactly at the χ c1 threshold: M Pc(4450) M χc1 M = (0.9 ± 3.1) MeV and at a triangle singularity at the same time! Λ 0 b K χ c1 (a) J/ψ Λ 0 b Λ χ c1 (b) K J/ψ Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

11 Landau equation 13 m 1 12 m 3 m 2 23 Triangle singularity: leading Landau singularity for a triangle diagram, anomalous threshold studied extensively in 1960s Solutions of Landau equation: Landau (1959) y 12 y 23 y 13 = y y y 2 13, y ij m2 i + m2 j 2 ij 2 m i m j quadratic equation of y ij, always two solutions Do they affect the hysical amlitude? Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

12 Some details (I) Consider the scalar three-oint loo integral d 4 q 1 I = i (2π) 4 [(P q) 2 m iɛ] (q2 m iɛ) [( 23 q) 2 m iɛ] Rewriting a roagator into two oles: 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 ositive-energy oles 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) ( 0 23 q0 ω 3 + iɛ) Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

13 Some details (II) Contour integral over q 0 cut-1 cut-2 d 3 q 1 I (2π) 3 [P 0 ω 1 (q) ω 2 (q) + i ɛ][ 0 23 ω 2(q) ω 3 ( 23 q ) + i ɛ] q 2 dq 0 P 0 ω 1 (q) ω 2 (q) + i ɛ f(q) The second cut: f(q) = dz 0 23 ω 2(q) m q qz + i ɛ Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

14 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: endoint singularity inch singularity Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

15 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: endoint singularity inch singularity Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

16 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 0 23 ω 2(q) m q qz + i ɛ Singularities of the integrand in the rest frame of initial article: 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 endoint singularities of f(q) z = +1 : q a+ = γ (β E 2 + 2) + i ɛ, q a = γ (β E 2 2) i ɛ, z = 1 : q b+ = γ ( β E 2 + 2) + i ɛ, q b = γ (β E 2 + 2) i ɛ β = 23 /E 23, γ = 1/ 1 β 2 = E 23 /m 23 E 2( 2): energy (momentum) of article-2 in the cmf of the (2,3) system Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

17 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 0 23 ω 2(q) m q qz + i ɛ Singularities of the integrand in the rest frame of initial article: 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 endoint singularities of f(q) z = +1 : q a+ = γ (β E 2 + 2) + i ɛ, q a = γ (β E 2 2) i ɛ, z = 1 : q b+ = γ ( β E 2 + 2) + i ɛ, q b = γ (β E 2 + 2) i ɛ β = 23 /E 23, γ = 1/ 1 β 2 = E 23 /m 23 E 2( 2): energy (momentum) of article-2 in the cmf of the (2,3) system Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

18 Some details (V) All singularities of the integrand: q on+, q a+ = γ (β E 2 + 2) + i ɛ, q a = γ (β E 2 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) Triangle singularity and Pc(4450) / 25

19 Some details (VI) Rewrite q a = 2 i ɛ, 2 γ (β E 2 2) Kinematics for 2 > 0, which is relevant to triangle singularity: 3 = γ (β E 3 + 2) > 0 articles 2 and 3 move in the same direction in the rest frame of initial article velocities in the rest frame of the initial article: v 3 > β > v 2 v 2 = β E 2 2/β E 2 β 2 < β, v 3 = β E 3 + 2/β E3 + β > β 2 article 3 moves faster than article 2 in the rest frame of initial article Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

20 Some details (VI) Rewrite q a = 2 i ɛ, 2 γ (β E 2 2) Kinematics for 2 > 0, which is relevant to triangle singularity: 3 = γ (β E 3 + 2) > 0 articles 2 and 3 move in the same direction in the rest frame of initial article velocities in the rest frame of the initial article: v 3 > β > v 2 v 2 = β E 2 2/β E 2 β 2 < β, v 3 = β E 3 + 2/β E3 + β > β 2 article 3 moves faster than article 2 in the rest frame of initial article Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

21 Coleman-Norton theorem Coleman Norton theorem: S. Coleman and R. E. Norton, Nuovo Cim. 38 (1965) 438 The singularity is on the hysical boundary if and only if the diagram can be interreted as a classical rocess in sace-time. hysical boundary: uer edge (lower edge) of the unitary cut in the first (second) Riemann sheet Translation: all three intermediate states can go on shell χc1, the roton can catch u with the χ c1 to rescatter Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

22 Coleman-Norton theorem Coleman Norton theorem: S. Coleman and R. E. Norton, Nuovo Cim. 38 (1965) 438 The singularity is on the hysical boundary if and only if the diagram can be interreted as a classical rocess in sace-time. hysical boundary: uer edge (lower edge) of the unitary cut in the first (second) Riemann sheet Translation: all three intermediate states can go on shell χc1, the roton can catch u with the χ c1 to rescatter Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

23 Analysis of the kinematics K Dalitz lot for Λ b χ c1 Λ χ c1 K: Starting from a large Λ mass, in Λ b rest frame Λ 0 b Λ χ c1 J/ψ when M Λ > M Λb M χc1, cannot go on-shell at oint A, M Λ = M Λb M χc1, χ c1 is at rest at oint B, roton and χ c1 has the same velocity between A and B, χc1 and roton moves faster than χ c1 Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

24 Analysis of the kinematics K Dalitz lot for Λ b χ c1 Λ χ c1 K: Starting from a large Λ mass, in Λ b rest frame Λ 0 b Λ χ c1 J/ψ when M Λ > M Λb M χc1, cannot go on-shell at oint A, M Λ = M Λb M χc1, χ c1 is at rest at oint B, roton and χ c1 has the same velocity between A and B, χc1 and roton moves faster than χ c1 Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

25 Analysis of the kinematics K Dalitz lot for Λ b χ c1 Λ χ c1 K: Starting from a large Λ mass, in Λ b rest frame Λ 0 b Λ χ c1 J/ψ when M Λ > M Λb M χc1, cannot go on-shell at oint A, M Λ = M Λb M χc1, χ c1 is at rest at oint B, roton and χ c1 has the same velocity between A and B, χc1 and roton moves faster than χ c1 Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

26 Analysis of the kinematics K Dalitz lot for Λ b χ c1 Λ χ c1 K: Starting from a large Λ mass, in Λ b rest frame Λ 0 b Λ χ c1 J/ψ when M Λ > M Λb M χc1, cannot go on-shell at oint A, M Λ = M Λb M χc1, χ c1 is at rest at oint B, roton and χ c1 has the same velocity between A and B, χc1 and roton moves faster than χ c1 Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

27 Trajectories of triangle singularities in comlex energy lane numbers: assumed masses for Λ blue: roton and χ c1 are arallel, in the 2nd Riemann sheet green: roton and χ c1 are anti-arallel M Λb = 5.62 GeV, M χc1 = 3.51 GeV, s M(χc1 ) MΛ M K,A = M Λb M χc1, M K,B = 2 M b +MK 2 Mχ c1 M χc1 +M M χc1 M Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

28 Triangle singularity for P c (4450) When M Λ = 1.89 GeV, the effective triangle singularity is located exactly at the χ c1 threshold, GeV! Coincidentally, four-star baryon Λ(1890): J P = 3/2 +, Γ : MeV triangle loo with S-wave χ c1 : 0.08 ΓΛ *=60 MeV [a.u.] ΓΛ *=100 MeV Events/(15 MeV) s [GeV] m J/ [GeV] imossible to roduce a narrow eak for χ c1 in other artial waves Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

29 More comments Strength of the triangle singularity is determined by coulings: Λ b Λ χ c1 is from b c cs, not measured, but should have a sizeable branching fraction: Br(B + J/ψK + ) , Br(B + χ c1 K + ) Λ (1890) N K: largest branching fraction, Br= 20 35% χ c1 J/ψ: OZI suressed, O (1/N c ) [recall: OZI suressed meson-meson scattering: O ( ) 1/Nc 2 ] b u d c c s lattice QCD redicts ossible c c-nucleus bound states at M π = 805 MeV NPLQCD, PRD91(2015) Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

30 More triangle singularities? χ c0,c1,c2 J/ψ are related through heavy quark sin symmetry Weak decay b c cs, V A Fierz cγ µ (1 γ 5 )c Λ b Λ J/ψ and Λ b Λ χ c1 are easy for χ c2 : strongly suressed, χ c0 : also suressed; for B + χ cj K + Br > Br Br no obvious eak around the χ c0 or χ c2 threshold Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

31 More triangle singularities? Considering other ossible c c Λ combinations: h c, η c (1S, 2S): sin-singlet h c [η c (1S, 2S)] J/ψ breaks heavy quark sin symmetry, suressed relative to χ c1 J/ψ J/ψ: J/ψ J/ψ: elastic, no eak will show u (due to Schmid theorem) ψ(2s): radial excitation different from J/ψ in comarison with χ c1 J/ψ left: strongly suressed; right: might be slightly suressed, not very clear For ossible triangle singularities for Λ b J/ψK, the χ c1 Λ (1890) seems the most rominent one among all c c Λ combinations Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

32 How to distinguish triangle-singularity from genuine resonance? Schmid theorem: C. Schmid, Phys. Rev. 154 (1967) 1363 see also, A. V. Anisovich, V. V. Anisovich, Phys. Lett. B 345 (1995) 321 Triangle singularity cannot roduce an additional eak in the invariant mass distribution of the elastic channel when neglecting inelasticity Λ 0 b Λ K Λ 0 b Λ K χc1 χc1 χc1 (a) (b) Nearby the effective singularity: A (a)+(b) (s) e 2i δχ c1 (s) A (a) (s) here δ χc1 is the elastic χ c1 scattering hase shift corrections from couled channels A. Szczeaniak, PLB757(2016)61 Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

33 How to distinguish triangle-singularity from genuine resonance? Method-1: measuring the rocess Λ 0 b χ c1 K if a narrow near-threshold eak in χ c1 a real exotic resonance otherwise, cannot conclude P c (4450) to be an exotic hadron Method-2: rocesses (such as hotoroduction) with a different kinematics Q. Wang, X.-H. Liu, Q. Zhao, PRD92(2015)034022; V. Kubarovsky, M. Voloshin, PRD92(2015)031502; M. Karliner, J. L. Rosner, PLB752(2015)329;... Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25

34 Summary Two coincidences for the LHCb P c (4450) structure: located exactly at the χ c1 threshold four-star Λ(1890) makes a triangle singularity exactly at the same osition To control the strength, we need: Br(Λ b Λ (1890)χ c1 ) LHCb χ c1 J/ψ, might get information from lattice QCD More measurements are necessary to reveal the nature of the P c (4450) J P unambiguously Λ b χ c1 K searching for P c (4450) in rocesses with a different kinematics THANK YOU FOR YOUR ATTENTION! Feng-Kun Guo (ITP) Triangle singularity and Pc(4450) / 25