Triangle singularity in light meson spectroscopy

Transcription

1 Triangle singularity in light meson spectroscopy M. Mikhasenko, B. Ketzer, A. Sarantsev HISKP, University of Bonn April 16, 2015 M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

2 New a f 0 (980) P wave COMPASS [arxiv: ] COMPASS [PoS Hadron2013, 087 (2013)] oldves K + K [PoS Hadron2013, 088 (2014)] [Berdnikov et al., Nuovo Cim. 107, 1941 (1994)] M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

3 Is a 1 (1420) a resonance? Might be, because narrow bump, sharp phase motion Issues to be clarified: a 1 (1420) doesn t fit to radial excitation trajectory as well as Regge-trajectory. Closeness of a 1 (1260). Width is too small compared to a 1 (1260). Strange coincidence of the mass to K (892) K threshold, GeV/c 2. Phase (deg) t' GeV 2 /c Phase [(1 1 0 f 0 (980) p + p (COMPASS 2008) ++ + P) - (1 1 0 ρ(770) S)] Preliminary Mass of + System (GeV/c 2 ) [arxiv: ] M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

4 The interpretations of a 1 (1420) 4-quark state candidate [Hua-Xing Chen et al., arxiv: ], [Zhi-Gang Wang, arxiv: ]. K K molecule (similar to XYZ interpretation) Dynamic effect of interference with Deck [Basdevant et al., arxiv: ]. Triangle singularity [Mikhasenko et al., arxiv: ] : K (892) K (892) a 1 (1260) K K a 1 (1260) K K f 0 (980) final state rescattering of almost real particle. Logarithmic singularity in the amplitude of the processes: M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

5 Contents 1 Triangle singularity Essence Anomalous threshold Decay kinematics 2 Nature of a 1 (1420) Peculiar kinematics Issues of the calculation Magnitude of the effect 3 Conclusion About calculation M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

6 Triangle singularity Essence What is the triangle singularity? Feynman approach d A(s 0, s 1, s 2 ) = g 3 4 k 1 1 (2) 4 = g 3 1 i dx dy dz δ(1 x y z), D i = m 2 i k 2 i, D = x m y m z m 2 3 x y s 0 z x s 1 y z s 2. Positions of singularities are given by equations: [Landau, Nucl. Phys. 13, 181 (1959)] p 1, s 1 k 1, m 2 1 p 0, s 0 k 2, m 2 2 k 3, m 2 3 p 2, s 2 k 2 i = m 2 i, i = , x k 1µ y k 2µ + z k 3µ = 0, x, y, z [0, 1], x + y + z = 1, leading singularity: log. branching point, A log(s s sing ), square root branching points, A s s sing (normal thresholds). M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

7 Triangle singularity Essence The different kinematical regions, [Eden et al., Cambridge Univ.Pr.(2002)] Introduce normalized invariants y 0 = (s 0 m1 2 m2 2 )/(2m 1m 2 ), y 1 = (s 1 m1 2 m2 3 )/(2m 1m 3 ), y 2 = (s 2 m2 2 m2 3 )/(2m 2m 3 ) s 0 m 2 1 m 2 2 s 1 = M 2 1 To simplify representation, we fix internal masses and s 1. Depending on value of s 1 (y 1 ), we have: y 1 < 1 No decays : M 1 < m 1 + m 3, m 1 < M 1 + m 3, m 3 < M 1 + m 1. The triangle singularity is known as anomalous threshold. It sits below the normal threshold for s 0, s 2. Appears in form-factors. y 1 < 1 Internal decay : m 1 > M 1 + m 3 or m 3 > M 1 + m 1. Position is given by decay kinematics. Appears in decays. y 1 > 1 External decay : M 1 > m 1 + m 3. Is covered above, just if we choose another variable to fix. M. Mikhasenko (HISKP) Triangle singularity April 16, / 21 m 2 3 s 2

8 Triangle singularity Anomalous threshold No decays region: position of singularities 2 Invariant s 1 is fixed, y 1 = (s 1 m 2 1 m2 3 )/(2m 1m 3 ) < 1. 1 s 0 m 2 1 s 1 m 2 3 y 2 0 m 2 2 s y 0 ReA ImA y 0 the position of leading log singularity is arc of ellipse, solid y 0 = +1, y 2 = +1 lines are normal thresholds: (m 1 + m 2 ) 2 for s 0 and (m 2 + m 3 ) 2 for s 2 (lower order singularities). Green area: ImA 0 (above threshold). M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

9 Triangle singularity Anomalous threshold An example: deuteron size [Gribov lecture, Cambridge Univ.Pr.(2009)] Non-relativistic quantum mechanics Proton and neutron bound state, binding energy: ε = M D m p m n, ψ is the wave function, ψ e r εm. Electron scattering amplitude: f (q) = e2 q 2 F (q), F (q) is a FF. F (q) is given by charge density inside of the deuteron F (q) = d 3 r ψ 2 (r) e iqr 1 q2 R O(q 4 ) R is electromagnetic radius, R 2 (εm) 1 Relativistic-theory framework Radius is determined by t-channel singularities. d n p e p d γ, Q 2 The singularities in Q 2 : Normal threshold Q 2 0 = (2m p) 2 Anomalous threshold Q εm M. Mikhasenko (HISKP) Triangle singularity April 16, / 21 e

10 Triangle singularity Anomalous threshold Further examples Selected examples: Some Experimental Consequences of Triangle Singularities in Production Processes [Landshoff et al, Phys. Rev. 127, 649 (1962) ] The Deuteron as a composite two nucleon system [Anisovich et al, Nucl.Phys. A544 (1992) ] Form-factors of meson decays [Melikhov, Phys.Rev. D53 (1996) ] M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

11 Triangle singularity Decay kinematics Internal decay region: position of singularities 2 Invariant s 1 is fixed, y 1 = (s 1 m 2 1 m2 3 )/(2m 1m 3 ) < 1. y s 0 m 2 1 m 2 2 s 1 m 2 3 s ReA ImA y y 0 position of leading log singularity is solid arc of hyperbola, solid y 0 = +1, y 2 = +1 lines are normal thresholds: (m 1 + m 2 ) 2 for s 0 and (m 2 + m 3 ) 2 for s 2. Green area: ImA 0 (above threshold). M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

12 Triangle singularity Decay kinematics Examples Selected examples: The nature of C-meson. [Achasov et al., Phys.Lett. B207 (1988) 199] ρ(1450) (K K K K) φ η(1405/1475) puzzle. η (K K K K) a 0 (980), f 0 (980), f 1 (1420) (K K K K) a0 (980), f 0 (980). [Jia-Jun Wu et al., Phys.Rev.Lett. 108 (2012) ], [Xiao-Gang Wu et al., Phys.Rev. D87 (2013) 1, ] Some of XYZ states. [Szczepaniak, arxiv: ] Z c (3900) (D D D D) J/Ψ Υ(5s) (B B B B) Υ(1s) The nature of a 1 (1420). [Mikhasenko et al., arxiv: ] a 1 (1260) (K K K K) f0 (980) M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

13 Nature of a 1 (1420) Peculiar kinematics waves in COMPASS: f 0 P-wave and ρ S-wave Low t f 0 (ρ) beam P(S)-wave p target p recoil M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

14 Nature of a 1 (1420) Peculiar kinematics The triangle diagram condition of internal decay is satisfied: m K > m K + m a 1 (1260) is rather wide, so s 0 can go above K K threshold. f 0 sits at the K K threshold. K (892) a 1 (1260) K K f 0 (980) The triangle singularity can appear in the physical region. Whether it does is given by Landau equations. Two isospin combinations contribute: a 1 (1260) K 0 K K + K f 0 (980) a 1 (1260) K K 0 K 0 K 0 f 0 (980) M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

15 Nature of a 1 (1420) Peculiar kinematics Kinematical formulation of Landau equations K * K (892) a 1 (1260) K K K a 1(1260) K Imagine cascade reaction a 1 (1260) K (892) K, then K K, and calculate invariant mass of K and K for the case when K is parallel to K M(K K ) M(f 0 ) p K >p K p K <p K Particular form of Landau conditions: All particles in loop are on mass shell. The alignment of moments p K p K. K is faster then K M(a 1) M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

16 Nature of a 1 (1420) Issues of the calculation Calculation of the rescattering: a 1 (1260) K K f0 For the realistic decay, we have expression in the numerator K (892) p 1 M (vpp) = a 1 f 0 p 0 K a 1 (1260) K p 2 f 0 (980) M (vpp) a 1 f 0 = g 3 dk 4 1 (2) 4 i Spin-Parity of particles. Width of K ) (p 1 k 3 ) ν ε 0µ (g µν kµ 1 kν 1 k 1 2 (m1 2 k2 1 iɛ)(m2 2 (p 0 k 1 ) 2 iɛ)(m3 2 (k 1 p 1 ) 2 iɛ) M (sc) a 1 f 0 p. If one fixes mass of f 0, i.e. pf 2 0 = mf 2 0, then only p0 2 = s is variable. VPP and SSS + finite width of K M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

17 Nature of a 1 (1420) Issues of the calculation Further corrections to the amplitude K (892) K, P-wave decay gives tail to the amplitude. A left-hand singularity is introduced to correct the amplitude g K K g K K F (k 1 ). k 1 is K four-momentum. F (k 1 ) = M2 mk 2 M 2 k1 2, M 2 = (m + m K ) 2 4 R 2. M is position of the left singularity, it corresponds to the size of K : F (1 + R 2 p 0 2 )/(1 + R 2 p 2 ). p is K K break up momentum. M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

18 Nature of a 1 (1420) Magnitude of the effect Magnitude of the effect Couplings: a 1 (1260) K K, ρ, K K, Following the experiment, we compare main channel a 1 (1260) ρ(770) S-wave decay, channel a 1 (1260) f 0 (980) only via kaon rescattering f 0 K K. General analysis of,r K K/ τ decay K decay [Coan et al., Phys.Rev.Lett. 92, (2004)] [Garcia-Martin et al., Phys.Rev.Lett. 107, (2011)] a 1 (1260) ρ(770) K (892) a 1 (1260) K K f 0 (980) M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

19 Nature of a 1 (1420) Magnitude of the effect Result a 1 f 0 A narrow peak and sharp phase motion are obtained Peak-to-peak ratio σ(a 1 f 0 )/σ(a 1 ρ, S wave) 1%, close to experimental value. M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

20 Conclusion Conclusions Huge data samples and advanced analysis tools make possible to see tiny features of strong interactions. Rescattering in the triangle diagram is an important mechanism in hadron interactions. Simplest diagram to produce dynamic singularity which gives infinity. Triangle singularity appears not only as anomalous threshold, but also as additional threshold above normal one in decay kinematics. The position of singularity in the decay kinematics is given by simple condition of rescattering of real particles. The full rescattering amplitude in presence of many inelastic channels can t be calculated, but difference to tree level can be noticed. We have now several hints that the triangle singularity plays important role in meson spectroscopy. X K K Y (K K) is ideal kinematics to produce narrow bump and sharp phase motion, which can be recognised in data. M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

21 Backup slides Backup M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

22 Backup slides About calculation The pinch singularity K (892) p 1 p 0 K a 1 (1260) K a 1 (1260) f 0 (980) f 0 (980) p 2 g a1 f 0 (s) d g a1 f 0 (s s=p 2) = g 3 4 k (2) 4 = g 3 i dy 1 y 0 dz 1 D, i = m 2 i k 2 i iɛ, D = (1 y z)m ym2 2 + zm2 3 y(1 y z)p2 0 z(1 z y)p2 1 yzp2 2. If Landau conditions [Nucl. Phys. 13, 181 (1959)] are satisfied, g a1 f 0 log(s s 0 ). For a box loop, A (s s0) 1/2, for 5-leg loop A (s s 0 ) 1 (pole). M. Mikhasenko (HISKP) Triangle singularity April 16, / 21

23 Backup slides About calculation The imaginary part based on Cutkosky cutting rules Im, Disc K _ K /2, Disc K * K / ImA = Disc K K A + Disc K K A (vpp) A a1 f 0 p 0 K 0 (k 1) a 1 (1260) Disc K K A: K (k 2) K + (k 3) f 0(980) p 1 p (sc) M a1 f s (GeV) K 0 (k 1) Disc K K A: p 1 K + (k 3) f 0(980) p 2 a 1 p (1260) 0 K (k 2) M. Mikhasenko (HISKP) Triangle singularity April 16, / 21