Triangle Singularities in the Λ b J/ψK p Reaction

Triangle Singularities in the Λ b J/ψK p Reaction M. Bayar Kocaeli University Hadron 27 Salamanca, 25 September 27 Collaborators: F. Aceti, F. K. Guo and E. Oset Based On: M. Bayar, F. Aceti, F.-K. Guo and E. Oset, Phys. Rev.D 94, 7439 (26)

Outline Introduction Detailed analysis of the triangle singularity Results for the Λ b K J/ψ p Conclusion

Introduction Triangle singularities in physical processes were introduced by Landau (L. D. Landau, Nucl. Phys. 3, 8 (959)) Some examples of effects of the triangle singularity : J. J. Wu, X. H. Liu, Q. Zhao and B. S. Zou, PRL8(22)883 X. G. Wu, J. J. Wu, Q. Zhao and B. S. Zou, PRD87(23)423(23) F. Aceti, W. H. Liang, E. Oset, J. J. Wu and B. S. Zou, Phys. Rev. D 86, 47 (22) M. Mikhasenko, B. Ketzer and A. Sarantsev, Phys. Rev. D 9, 945 (25) F. Aceti, L. R. Dai and E. Oset, Phys.Rev. D94 (26) no.9, 965 LHCb pentaquark-like structures: in the Λ b J/ψK p reaction in the J/ψp spectrum a narrow peak at 445 MeV R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 5, 72 (25), R. Aaij et al. [LHCb Collaboration], Chin. Phys. C 4, (26) The possibility that this peak is due to triangle singularity is discussed: F.-K. Guo, U.-G. Meißner, W. Wang and Z. Yang, Phys. Rev. D 92, 752 (25), X. H. Liu, Q. Wang and Q. Zhao, Phys. Lett. B 757, 23 (26), F.-K. Guo, U.-G. Meißer, J. Nieves and Z. Yang Eur.Phys.J. A52 (26) no., 38

Detailed analysis of the triangle singularity

The Λ b J/ψK p Reaction Λ b (P) (P q) K (k) Λ (q) c c p (P q k) J/Ψ p Figure: Triangle diagram for Λ b J/ψK p decay, where Λ stands for the different Λ and c c stands for different charmonium states. The scalar three-point loop integral I = i d 4 q (2π) 4 )[ ]. () (q 2 m c c 2 + i ǫ (P q) 2 m Λ 2 + i ǫ ][(P q k) 2 mp 2 + i ǫ

Detailed analysis of the triangle singularity m p3 P m3 m2, q p23 Figure: A triangle diagram. The two dashed vertical lines correspond to the two relevant cuts. The scalar three-point loop integral I = i d 4 q (2π) 4 (q 2 m 2 2 + i ǫ )[(P q) 2 m 2 + i ǫ ][(p 23 q) 2 m 2 3 + i ǫ ]. (2) Rewriting a propagator into two poles ) = (q 2 m 2 2 ( + i ǫ q ω 2 + i ǫ )( q + ω 2 + i ǫ ), ω 2 = m 2 2 + q 2 (3) focus on the positive-energy poles i I 8m m 2 m 3 dq d 3 q (2π) 4 ( q ω 2 + i ǫ )[ P q ω + i ǫ ][ p 23 q ω 3 + i ǫ]. (4)

m p 3 P m 3 m 2, q p 23 I(m 23 ) = d 3 q [ P ω ( q ) ω 2 ( q ) + i ǫ ][ E 23 ω 2 ( q ) ω 3 ( k + q ) + i ǫ] = 2π dq q 2 P f(q) (5) ω (q) ω 2 (q) + i ǫ f(q) = dz E 23 ω 2 (q) m 3 2 + q2 + k 2 + 2 q k z + i ǫ, (6) = singularity of integrand does not necessarily give a singularity of integral

Relation between singularities of integrand and integral Im q Im q Im q analytic end point singularity pinch singularity Re q Re q Re q = 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: end point singularity pinch singularity

m p3 P m3 m2, q p23 I dq q 2 P ω (q) ω 2 (q) + i ǫ f(q), f(q) dz E 23 ω 2 (q) m 3 2 + q2 + p 23 2 + 2 q p 23 z + i ǫ ( Singularities of the integrand of I in the rest frame of initial particle: cut : P ω (q) ω 2 (q)+i ǫ = = q on+ = q on + i ǫ with q on = λ(m 2M 2, m 2, m2 2 ). cut 2: E 23 ω 2 (q) m3 2 + q2 + p23 2 + 2 q p 23 z + i ǫ =, end point singularities z = : q a+ = γ ( v E2 + 2) p + i ǫ qa = γ ( v E2 2) p i ǫ z = +: q b+ = γ ( v E2 + 2) p + i ǫ qb = γ ( v E2 + 2) p i ǫ v = k, γ = E = E 23 23 v 2 m 23 E2 (p 2 ) are the energy (momentum) of particle-2 in the center-of-mass frame of the (2,3) system

All singularities of the integrand: Imq Imq Imq qon+ qa+ qon+ qa+ qon+ qa+ qa Req Req qa qa Req (a) Imq (b) (c) qon+ qb+ qa+ m p3 Req P m3 m2, q p23 q on+, q a+ = γ ( v E2 + 2) p + i ǫ qa = γ ( v E2 2) p i ǫ threshold singularity = q a+ = q a triangle singularity = q on+ = q a q a <, q b+ = q a

The Coleman Norton Theorem The singularity is in the physical region only when the process can happen classically: all the intermediate states are on shell, and the particle 3 emitted from the decay of the particle moves along the same direction as the particle 2 with a large speed than the particle 2 and can catch up with it to rescatter. m p3 P m3 m2, q p23 For q on and q a (ǫ = ), physical regions m M m 2 and m 23 m 2 + m 3 using q on+ = q a [ M m 2 2 m 3 + m3 2 m ] 2 m 2 m 3, (M m 2 ) 2, (8) m 2 + m 3 m 2 23 [ (m 2 + m 3 ) 2, Mm3 2 m2 3 m ] 2 + Mm 2. (9) M m 2

Results for the Λ b K J/ψ p Λb (P) (P q) K (k) Λ (q) c c p (P q k) J/Ψ p Λ = Λ(45), Λ(52), Λ(6), Λ(67), Λ(69), Λ(8), Λ(8), Λ(82), Λ(83), Λ(89), Λ(2), Λ(2), Λ(235), Λ(2585) c c Most relevant range of M Λ (MeV) Range of triangle singularity (MeV) η c [2226, 2639] [399, 4283] J/ψ [25, 2523] [435, 4366] χ c [949, 225] [4353, 4588] χ c [887, 29] [4449, 4654] χ c2 [858, 263] [4494, 4686] h c [878, 294] [4464, 4664] η c(2s) [86, 983] [4575, 474] ψ(2s) [774, 933] [4624, 4775]

The χ c intermediate charmonium state Assume the χ c p S wave in the χ c p J/ψ p.2 =67 MeV =8 MeV =89 MeV =2 MeV I 2 (x -6 MeV -2 ).8.6.4.2 42 43 44 45 46 47 48 49 5 m J/ p (MeV) Figure: The value of I for Λ χ c with a width Γ = MeV for the hyperon. peak around 445 MeV!! The largest strength Λ(89); the threshold and the triangle singularities merge Λ(67) and Λ(8): the cusp structure comes from the threshold singularity Λ(2): inside the range of the triangle singularities

Detailed analysis of the S and P-wave amplitudes for χ c and ψ(2s) and Λ(89) Λb (P) (P q) K (k) Λ (q) c c p (P q k) J/Ψ p In the c c p J/ψ p we have several situations: Λ(89) [I(J P ) = ( 3 2+ )] The quantum numbers of the J/ψ p J P = 3/2 ( P c (445)) : a) c c = χ c : requires a P-wave in the χ c p system b) c c = ψ(2s): requires an S-wave in both the ψ(2s) p and J/ψ p 2 The quantum numbers of the J/ψ p are J P = /2 + or 3/2 + : a) c c = χ c : the χ c p system (S-wave)and the J/ψp ( P-wave) b) c c = ψ(2s): requires a P-wave in both the ψ(2s)p and J/ψp systems, and will not play a role in the discussion.

.7..6.9 I 2 (x -6 MeV -2 ).5.4.3 I 2 (x -6 MeV -2 ).8.7.6.5.4.3.2.2.. 42 43 44 45 46 47 48 49 5 5 m J/ p (MeV) 42 43 44 45 46 47 48 49 5 5 m J/ p (MeV) Figure: (.a) The value of I 2 for P-wave Λ(89)χ c Γ Λ(89) = MeV (.b) The value of I 2 for S-wave Λ(89)ψ(2S). the amplitude (.a) is very much suppressed than the S-wave case (.b) This is natural!! the singularity appears the χ c p on shell and at threshold the P-wave factor vanishes. (.a) the P-wave has a background" below the peak the shape is too broad!! (.b) the S-wave structure is very much peaked and narrow a narrow peak around 4624 MeV the experimental data the J/ψp invariant mass distribution in this region is flat. Conclusion: if the narrow J/ψ p has 3/2 the triangle singularities due to Λ c c p intermediate states cannot play an important role in the decay Λ b K J/ψ p

Conclusion Analyzed the singularities of a triangle loop integral in detail Derived a formula for an easy evaluation of the triangle singularity on the physical boundary applied to the Λ b J/ψK p process via Λ -charmonium-proton intermediate states most relevant states the χ c and the ψ(2s) among all possible charmonia up to the ψ(2s) In many of the other cases not develop a triangle singularity, but the threshold cusp The Λ(89)χ c p loop is very special normal threshold and triangle singularities merge at about 445 MeV in the case of J P = 3 2, 5+ 2 for the narrow Pc, the χ c p J/ψ p requires P- and D-waves in χ c p drastically reduce the strength of the contribution In the case of /2 +, 3/2 + : the χ c p in the χ c p J/ψ p amplitude be in an S-wave the Λ(89)χ c p triangle diagram could play an important role