Matrix element for the P c + decay chain

We now turn to the discussion of Λ⁰_b → P_cjK⁻, P_cj → ψp decays, in which we allow more than one pentaquark state, j = 1, 2, . . . . Superscripts containing the P_cdecay chain name without curly brackets, e.g. φ^Pc, will denote quantities belonging to this decay chain and should not be confused with the superscript “{P_c}” denoting the P_c⁺ rest frame, e.g. φ^{Pc}. With only a few exceptions, we omit the Λ^∗ decay chain label.

The weak decay Λ⁰_b → PcjK⁻ is described by the term,

λ_Pc are resonance (i.e. j) dependent helicity couplings. The helicity of the pentaquark state, λ_Pc, can take values of ±¹₂ independently of its spin, J_Pcj = ¹₂,³₂, . . . . Therefore, there are two independent helicity couplings to be determined for each P_cj state.

K

Figure 8: Definition of the decay angles in the P_c⁺ decay chain.

The above mentioned φ_Pc, θ^P_Λ^c0 b

symbols refer to the azimuthal and polar angles of P_c in the Λ⁰_b rest frame (see Fig. 8).

Similar to Eq. (17), the Λ⁰_b helicity angle in the P_c decay chain can be calculated as, cos θ_Λ^Pc0

The strong decay P_cj → ψp is described by a term H^Pcj→ψp

λ^Pc_ψ ,λ^Pcp D^JPcj

λ_Pc, λ^Pc_ψ −λ^Pcp (φ_ψ, θ_Pc, 0)^∗ R_Pcj(M_ψp), (37) where φ^P_ψ^c, θ_Pc are the azimuthal and polar angles of the ψ in the P_c rest frame (see Fig. 8).

They are defined analogously to Eqs. (20)−(22). The ˆz₀^{Pc} direction is defined by the boost direction from the Λ⁰_b rest frame, which coincides with the −~p_K^{Pc} direction. This leads to

cos θ_Pc = −ˆp_K^{Pc}· ˆp_ψ^{Pc}. (38) The azimuthal angle of the ψ can now be determined in the P_crest frame (see Fig. 8) from

φ^P_ψ^c = atan2

−(ˆp_K^{Pc}× ˆx₀^{Pc}) · ˆp_ψ^{Pc}, ˆx₀^{Pc}· ˆp_ψ^{Pc}

. (39)

In Eq. (37), the ˆx₀^{Pc} direction is defined by the convention that we used in the Λ⁰_b rest frame. Thus, similar to Eq. (21) we have

~a_z^{Λ0b} clear that the spin quantization axes are different than in the Λ^∗ decay chain. Since the

Λ Λ

^*

K

Λ

−

*

z0{p} Λ∗ z⁰{p} Pc

p

rest frame P

c

ψ p

p

b

rest frame ψ

P

c

rest frame

ψ K

K K

p p

rest frame p

θ

− −

Figure 9: Definition of the θp angle.

ψ is an intermediate particle, this has no consequences after we sum (coherently) over λ^P_ψ^c = −1, 0, +1. The proton, however, is a final-state particle. Before the P_c terms in the matrix element can be added coherently to the Λ^∗ terms, the λ^P_p^c states must be rotated to λ_p states (defined in the Λ^∗ decay chain). The proton helicity axes are different, since the proton comes from a decay of different particles in the two decay sequences, the Λ^∗ and P_c. The quantization axes are along the proton direction in the Λ^∗ and the P_c rest frames, thus antiparallel to the particles recoiling against the proton: the K⁻ and ψ, respectively.

These directions are preserved when boosting to the proton rest frame (see Fig. 9). Thus, the polar angle between the two proton quantization axes (θ_p) can be determined from the opening angle between the K⁻ and ψ mesons in the p rest frame,

cos θ_p = ˆp_K^{p}· ˆp_ψ^{p}. (41) (A similar problem is discussed in Ref. [4], where the two different χ_c1 helicity frames in B⁰ → K⁺π⁻χ_c1 decays, in the interference of B⁰ → K^∗χ_c1, K^∗ → K⁺π⁻ and of B⁰ → Z⁻K⁺, Z⁻ → χ_c1π⁻ contributions, are realigned.) The dot product above must be calculated by operating on the ~p_K^{p} and ~p_ψ^{p} vectors in the proton rest frame obtained by the same sequence of boost transformations, either according to the Λ^∗ or P_c decay chains, or even by a direct boost transformation from the lab frame.⁶

No azimuthal rotation is needed to align the two proton helicity frames, since the decay planes of the Λ^∗ and the P_c are the same (see Fig. 9). Therefore, the relation between λ_p and λ^P_p^c states is

|λ_pi =X

λ^Pcp

D^Jp

λ^Pcp , λp(0, θ_p, 0)^∗|λ^P_p^ci =X

λ^Pcp

d^Jp

λ^Pcp , λp(θ_p)|λ^P_p^ci. (42)

6Numerical values of momentum vector components, (px, py, pz), depend on the boost sequence taken and are related between different boosts via the rotation matrix. However, the dot product between the two vectors remains independent of the boost sequences.

Thus, the term given by Eq. (37) must be preceded by X

λ^Pcp =±¹₂

d^Jp

λ^Pcp , λp(θ_p). (43)

Parity conservation in Pcj → ψp decays leads to the following relation H^Pcj→ψp

−λ^Pc_ψ ,−λ^Pcp = P_ψP_pP_Pcj(−1)^{Jψ+Jp−JPcj} H^Pcj→ψp

λ^Pc_ψ , λ^Pcp

= PP_cj(−1)^12−JPcj H^Pcj→ψp

λ^Pc_ψ , λ^Pcp , (44)

where P_Pcj is the parity of the P_cj state. This relation reduces the number of inde-pendent helicity couplings to be determined from the data to 2 for J_Pcj = ¹₂ and 3 for J_Pcj ≥ ³₂. Since the helicity couplings enter the matrix element formula as a product, H_λ^Λ0b→PcjK

Pc H^Pcj→ψp

λ^Pc_ψ , λ^Pcp

, the relative magnitude and phase of these two sets must be fixed by a convention. For example, H^Λ

0 b→P_cjK

λ_Pc=−¹₂ can be set to (1, 0) for every Pcj resonance, in which case H^Λ0b→PcjK

λ_Pc=+¹₂ develops a meaning of the complex ratio of H^Λ0b→PcjK

λ_Pc=+¹₂ /H^Λ0b→PcjK

λ_Pc=−¹₂ , while all H^Pcj→ψp

λ^Pc_ψ , λ^Pcp

couplings should have both real and imaginary parts free in the fit.

The term R_Pcj(M_ψp) describes the ψp invariant mass distribution of the P_cj resonance and is given by Eq. (23) after appropriate substitutions. Angular momentum conservation limits L^P_Λ^cj0

b

in Λ⁰_b → P_cjK⁻ decays to J_Pcj ±¹₂. The angular momentum conservation also imposes max(J_Pcj − ³₂ , 0) ≤ L_Pcj ≤ J_Pcj + ³₂, which is further restricted by the parity conservation in the P_cj decays, P_Pcj = (−1)^LPcj+1. We assume the minimal values of L^P_Λ^cj0

b

and of L_Pcj in R_Pcj(m_ψp).

The electromagnetic decay ψ → µ⁺µ⁻ in the P_c decay chain contributes a term D_λ¹Pc

ψ , ∆λ^Pcµ (φ^P_µ^c, θ^P_ψ^c, 0)^∗, (45)

which is the same as Eq. (27), except that since the ψ meson comes from the decay of different particles in the two decay chains, the azimuthal and polar angle of the muon in the ψ rest frame, φ^P_µ^c, θ_ψ^Pc, are different from φ_µ, θ_ψ introduced in the Λ^∗ decay chain. The ψ helicity axis is along the boost direction from the P_c to the ψ rest frames, which is given by

ˆ

z₀^{{ψ} Pc} = − ˆp_p^{ψ}, (46)

and so

cos θ_ψ^Pc = −ˆp_p^{ψ}· ˆp_µ^{ψ}. (47) The x axis is inherited from the P_c rest frame (Eq. (11)),

~a_z^{Pc}

0⊥ψ = −~p_K^{Pc}+ (~p_K^{Pc}· ˆp_ψ^{Pc}) ˆp_ψ^{Pc} ˆ

x₀^{{ψ} Pc} = ˆx₃^{Pc} = − ~a_z^{Pc}

0⊥ψ

| ~a_z^{Pc}

0⊥ψ|, (48)

which leads to

φ^P_µ^c = atan2

−(ˆp_p^{ψ}× ˆx₀^{{ψ} Pc}) · ˆp_µ^{ψ}, ˆx₀^{{ψ} Pc} · ˆp_µ^{ψ}

. (49)

Since the muons are final-state particles, their helicity states in the P_c decay chain,

|λ^P_µ^ci, need to be rotated to the muon helicity states in the Λ^∗ decay chain, |λ_µi, before the Pc matrix element terms can be coherently added to the Λ^∗ matrix element terms.

The situation is simpler than for the rotation of the proton helicities discussed above, as the muons come from the ψ decay in both decay chains. This makes the polar angle θ_µ (analogous to θ_p in Eq. (43)) equal to zero, which leads to d

1 2

λ^Pcµ , λµ(0) = δ_λPc

µ , λµ, where δ_i,j is the Kronecker symbol. However, the muon helicity states are not identical since the x axes are offset by the azimuthal angle αµ. Since the boost to the µ rest frame is the same for both decay chains (i.e. always from the ψ rest frame), we can determine α_µ in the ψ rest frame

αµ= atan2



(ˆz₃^{ψ}× ˆx₃^{{ψ} Pc}) · ˆx₃^{{ψ} Λ∗}, ˆx₃^{{ψ} Pc}· ˆx₃^{{ψ} Λ∗}



, (50)

where ˆz₃^{ψ}= ˆpµ^{ψ}, and from Eq. (11) ˆ

x₃^{{ψ} Pc} = − ˆa_z^{ψ}

0⊥µ Pc, (51)

~a_z^{ψ}

0⊥µ

Pc = − ˆp_p^{ψ}+ (ˆp_p^{ψ}· ˆp_µ^{ψ}) ˆp_µ^{ψ}, (52) as well as

ˆ

x₃^{{ψ} Λ∗} = − ˆa_z^{ψ}

0⊥µ Λ^∗, (53)

~a_z^{ψ}

0⊥µ

Λ^∗ = − ˆp_Λ^{ψ}∗ + (ˆp_Λ^{ψ}∗ · ˆp_µ^{ψ}) ˆp_µ^{ψ}. (54) The term aligning the muon helicity states between the two reference frames is given by

X

λ^Pcµ

D^Jµ

λ^Pcµ λµ

(α_µ, 0, 0)^∗ =X

λ^Pcµ

e^{i λPcµ αµ}δ_λPc

µ , λµ = e^{i λµαµ}. (55)

The transformation of µ⁻ states will be similar to that of the µ⁺ states, except that since ˆ

z_ψ will have the opposite direction, α_µ⁺ = −α_µ⁻. The transformation of |λ^P_µ+^ci|λ^P_µ^c−i to

|λ_µ⁺i|λ_µ⁻i states will require multiplying the terms for the P_c decay chain by

e^{i λµαµ}e^{i λµ¯α¯µ} = e^{i (λµ−λµ¯)αµ} = e^{i ∆λµαµ}. (56) An alternative derivation of Eq. (56) is discussed in Ref. [5] (Eqs. (20)−(22) therein) for the interference of B⁰ → K^∗ψ, K^∗ → Kπ and of B⁰ → ZK⁻, Z → ψπ (ψ → `<sup>+</sup>`⁻) terms, which are analogous to the two decay chains discussed here with the substitution B⁰ → Λ⁰_b, K^∗ → Λ^∗, Z → P_c and π → p. The rotation by α_µ about the `⁺ direction in the ψ rest frame in the Z decay chain is incorporated by setting γ = α_µ, instead of γ = 0 in Eq. (45).

This leads to the same formulae since

D_λ¹Pc, ∆λµ(φ^P_µ^c, θ_ψ^Pc, α_µ)^∗ = D_λ¹Pc, ∆λµ(φ^P_µ^c, θ^P_ψ^c, 0)^∗e^{i ∆λµαµ}. (57)

We use the more generic derivation here to demonstrate that the methods of transforming the muon and proton helicity states between the two decay chains are the same.

Collecting terms from the three subsequent decays in the P_c chain together, M^Pc and adding them coherently to the Λ^∗ matrix element, via appropriate relation of

|λ_pi|λ_µ⁺i|λ_µ⁻i to |λ^P_p^ci|λ^P_µ+^ci|λ^P_µ^c−i states as discussed above, leads to the final matrix

Assuming approximate CP symmetry, the helicity couplings for Λ⁰_b and Λ⁰_b can be made equal, but the calculation of the angles requires some care, since parity (P ) conservation does not change polar (i.e. helicity) angles, but does change azimuthal angles. Thus, not only must ~p_µ⁺ be used instead of ~p_µ⁻ for Λ⁰_b candidates (with K⁺ and ¯p in the final-state) in Eqs. (28), (29), (47), (49) and (50), but also all azimuthal angles must be reflected before entering the matrix element formula: φ_K → −φ_K, φ_µ→ −φ_µ, φ_Pc → −φ_Pc, φ^P_ψ^c → −φ^P_ψ^c, φ^P_µ^c → −φ^P_µ^c and α_µ → −α_µ [5].

It is clear from Eq. (59) that various Λ_n^∗ and P_c resonances interfere in the differential distributions. By integrating the matrix element squared over the entire phase space the interferences cancel in the integrated rates unless the resonances belong to the same decay chain and have the same quantum numbers.⁷

In document Observation of J/ψ p resonances consistent with pentaquark states in. Supplementary material (Page 16-21)