EPJ manuscript No.(will be inserted by the editor)

Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays

Wei-Hong Liang1, Hua-Xing Chen2 a, Eulogio Oset1,3 b, and En Wang4 c

1 Department of Physics, Guangxi Normal University, Guilin 541004, China2 School of Physics, Beihang University, Beijing 100191, China3 Departamento de Fısica Teorica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigacion de Paterna,

Aptdo. 22085, 46071 Valencia, Spain4 School of Physics and Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China

Received: date / Revised version: date

Abstract. We study the J/ψ → K+K−f0(980)(a0(980)) reaction and find that the mechanism to producethis decay develops a triangle singularity around Minv(K−f0/K

−a0) ≈ 1515 MeV. The differential widthdΓ/dMinv(K−f0/K

−a0) shows a rapid growth around the invariant mass being 1515 MeV as a consequenceof the triangle singularity of this mechanism, which is directly tied to the nature of the f0(980) and a0(980)as dynamically generated resonances from the interaction of pseudoscalar mesons. The branching ratiosobtained for the J/ψ → K+K−f0(980)(a0(980)) decays are of the order of 10−5, accessible in presentfacilities, and we argue that their observation should provide relevant information concerning the natureof the low-lying scalar mesons.

PACS. XX.XX.XX No PACS code given

1 Introduction

Discussed already in Ref. [1] and systematized by Landauin Ref. [2], the triangle singularities (TS) were fashionablein the sixties [3,4,5,6,7,8] and efforts were done to under-stand some reactions through TS mechanisms [9,10]. Atriangle mechanism stems from the decay of a particle Ainto 1+2, followed by the decay of 1 into 3+B, and poste-rior fusion of 2 + 3 to give a new particle C (see Fig. 1(a))or 2 + 3 (see Fig. 1(b), rescattering), or a different pair ofparticles. It was shown in Ref. [2] that when all these par-ticles, 1, 2, 3, can be placed on shell in the correspondingFeynman diagram, a singularity can develop in the corre-sponding amplitude. The conditions for the singularity aremade more specific by Coleman-Norton [7] showing thatparticles 1 and B have to be parallel in the A rest frameand the process has to be possible at the classical level.Analytical expressions of these conditions can be see inRef. [11] and in a simpler form in Ref. [12]. The formal-ism of Ref. [12] allows one to see the explicit effect of thewidth of particle 1 in the shape of the singularity, andthis is explicitly shown in Ref. [13] where some considera-tions are made concerning the Schmid theorem [8], whichstates that in the case of the mechanism with 2+3→ 2+3(rescattering) the triangle singularity can be absorbed by

Send offprint requests to:a hxchen@buaa.edu.cnb eulogio.oset@ific.uv.esc wangen@zzu.edu.cn

the tree level diagram A→ 1 + 2 (1→ 3 +B). In Ref. [13]it is shown that this only occurs in the limit of zero widthfor particle 1, where the triangle mechanism is negligiblewith respect to the tree level one.

With the advent of a large amount of experimental in-formation, examples of triangle singularities have becomeavailable lately, and the field has experienced a revival.The spark was raised by the puzzle of the anomalouslylarge isospin breaking in the η(1405) → π0f0(980) reac-tion [14], which was explained in Refs. [15,16] in termsof a triangle singularity (see also following references [17,18]). It is interesting to recall the mechanism of Refs. [15,16], which has served to disentangle related reactions andto make predictions for new reactions that should see TSeffects. The mechanism studied in Refs. [15,16] consists ofη(1405) → K∗K, followed by K∗ → πK and later fusionof KK to give a0(980) or f0(980). There is no problemin having KK → a0(980), since πa0(980) can match tozero isospin of the η(1405), but KK → f0(980) is isospinforbidden, since πf0(980) has isospin one. The differenceof masses between K+K− and K0K0 in the triangle loopprevents the exact cancellation of two diagrams that oc-curs in the limit of equal masses where isospin is con-served. This also leads to a peculiar very narrow shape ofthe f0(980) that is not tied to the natural width of thef0(980) but to the difference of masses between K+K−

and K0K0.

The above reaction is also very enlightening from thepoint of view of the nature of the f0(980) and a0(980),since these resonances are not directly produced but come

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2 Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays

A 1

C

2 3

B

3

2

(a)

A 1

C

2 3

B

3

2

(b)

Fig. 1. Mechanism for a triangle singularity.

from the rescattering of the KK components, i.e., themechanism for the formation of these resonances in thechiral unitary approach [19,20,21,22]. This same TS mech-anism appears in the τ− → ντπ

−f0(a0) decay [23], theB0s → J/ψπ0f0(a0) decay [24], the D+

s → π+π0f0(a0)decay [25], and the B− → D∗0π−f0(a0) decay [26]. Thesame triangle singularity was shown in Refs. [27,28] toprovide a natural explanation for the peak around 1420MeV observed in the πf0(980) final state in diffractiveπp collisions by the COMPASS Collaboration, which wasoriginally branded as a new “a1(1420)” resonance [29].It is easy to envisage many reactions of this type, onealso from the J/ψ decay, as J/ψ → π 1©K

∗K followed byK∗ → Kπ 2© and KK → f0(980). Once again one can an-ticipate a peak at Minv(π 2©f0) ≈ 1420 MeV like in theother reactions. However, the possibility to have differentpairs from the π+π−π+π− final state forming the f0(980)makes the experimental analysis and the theoretical workmore involved.

Related reactions, τ− → ντπA (A for axial-vector mesons) [30]and concretely the τ− → ντπ

−f1(1285) [31], rely upon theK∗K∗K intermediate states with a good description of theτ− → ντπ

−f1(1285) data [33] (see alternative approach inRef. [32] based on the Nambu-Jona-Lasinio model). Forthese latter reactions, the K∗K in the triangle loop fuseto give the axial vector mesons, which are also dynami-cally generated from the pseudoscalar-vector mesons in-teractions according to the chiral unitary approach [34,35,36].

A related mechanism is also used in Ref. [37] to de-scribe the Bc → Bsππ reaction with the 1, 2, 3 particlesbeing K∗BK, the B− → K−π−D∗+s0 (2317) reaction [38]withK∗DK in the intermediate states, the Λc → ππΛ(1405)reaction [39] with K∗pK in the intermediate states, andthe Λc → ππΣ∗ reaction [40] with K∗pK in the intermedi-ate states. Other reactions rely on very different interme-diate states, like in the Ψ → KKJ/ψ reaction [41] withDs1DsD

∗ in the intermediate states, or the Λc → πφpreaction [42] with Σ∗K∗Σ in the intermediate states. Arecent review of reactions explained in terms of TS mech-anisms and predictions made for many other physical pro-cesses can be seen in Ref. [13].

The reaction proposed here relies upon a different in-termediate state, not discussed previously, which involves

the KKφ intermediate states, as depicted in Fig. 2. Thereaction is J/ψ → K+K−f0(980)(a0(980)). It involvesthe strong J/ψ decay and the mechanism proceeds viaJ/ψ → K+K−φ, followed by φ→ K−K+ and the poste-rior fusion of K−K+ to give either the f0(980) or a0(980).Both reactions are possible in the present case but therates of production are tied to the way the K+K− gen-erates the f0(980) or a0(980) resonance through its inter-action, hence, providing new information on the natureof these resonances. The intermediate states in the loop,1, 2, 3, are now φK−K+. In addition, the φ is very narrow,Γφ = 4.25 MeV [33], and in particular the TS structureshould be narrow around the TS point given by the equa-tion [12]

qon − qa− = 0 , (1)

where qon is the on-shell momentum of the φ and qa− theon-shell K− momentum in the loop, antiparallel to the φ.

It should be noted that the TS condition of Eq. (1) isnow fulfilled only in a very narrow window of K+K− en-ergies between 987 MeV and 993 MeV, where the f0(980)and a0(980) peak. Away but close to the point wherethe TS appears, the amplitudes are no longer singularin the Γφ → 0 limit, but a peak structure still remainsby inertia. Yet, this feature confers the amplitude a spe-cial signature that makes the shapes different to ordi-nary cases of f0(980) or a0(980) production, which is tiedagain to the dynamical nature of these resonances formedfrom the pseudoscalar-pseudoscalar interaction in coupledchannels. The reaction, hence, contains relevant informa-tion concerning the nature of the f0(980) and a0(980) res-onances.

The condition of Eq. (1) is very useful to know whenone has a TS, and helps rule out related triangle mech-anisms which however do not develop a singularity, andhence do not compete with the singular mechanisms. Inthis sense we can envisage a primary J/ψ → KK∗π de-cay with K∗ → Kπ and the two pions merging into thef0(980). We can apply the condition of Eq. (1) to thismechanism containing K∗ππ in the loop, but with thesame final KKf0(980) state as in the mechanism of Fig. 2.However, in this case one can see that qon and qa− are veryfar from each other, leading to a mechanism that cannotcompete with the singular one, and which in any case onlyprovides a smooth background in the region where the

Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays 3

J/ψ

K

f(P-q)

f /a 0

+

K (q)-

0

K (P-q-k)+

K (k)-

(a)

J/ψ

K

f(P-q)

f /a 0

-

K (q)+

0

K (P-q-k)-

K (k)+

(b)

Fig. 2. Triangle diagrams for the J/ψ → K+K−f0(980)(a0(980)) decays. The parentheses give the four-momenta of the particleswith (a) P = pJ/ψ − pK+ and (b) P = pJ/ψ − pK− .

mechanism of Fig. 2 produces a peak, which is what wewant to investigate. The smaller coupling of f0(980) to ππthan to KK also helps to make this background smaller.

2 Formalism

Our mechanism for the J/ψ → K+K−f0(980)(a0(980))reaction is depicted in Fig. 2. There is a primary decayof J/ψ → K+φK−, a second decay of φ → K−K+, andthe posterior fusion of K−K+ to produce the f0(980) ora0(980) resonance. Given the complicated dynamics of thewhole process, our strategy to provide absolute numbersfor the decay width and mass distributions consists of tak-ing the information for the primary step J/ψ → K+φK−

from experiment and the rest can be calculated reliably.In view of this, we study in a first step the reaction J/ψ →K+φK−.

2.1 The J/ψ → K+K−φ decay

The J/ψ → K+K−φ decay can proceed via S-wave. Wetake its amplitude, suited to the production of two vectors,as in Refs. [26,38]:

tJ/ψ→K+K−φ = A ε(J/ψ) · ε(φ) , (2)

where ε(J/ψ) and ε(φ) are the polarization vectors of theJ/ψ and φ, respectively. Then we write the K−φ invariantmass distribution of this decay as

dΓJ/ψ→K+K−φ

dMinv(K−φ)(3)

=1

(2π)31

4M2J/ψ

pK+ pK−∑∑∣∣tJ/ψ→K+K−φ

∣∣2 ,where pK+ is the momentum of the K+ in the J/ψ restframe, and pK− is the momentum of the K− in the K−φ

rest frame:

pK+ =λ1/2(M2

J/ψ,m2K+ ,M2

inv(K−φ))

2MJ/ψ,

pK− =λ1/2(M2

inv(K−φ),m2K− ,m

2φ)

2Minv(K−φ). (4)

Here λ(x, y, z) is the Kallen function λ(x, y, z) = x2+y2+z2 − 2xy − 2yz − 2xz.

After summing over polarizations of Eq. (2),∑∑∣∣tJ/ψ→K+K−φ

∣∣2=∑ 1

3A2 ε(J/ψ) · ε(φ) ε(J/ψ) · ε(φ)

= A2 , (5)

we can simplify Eq. (3) to be

dΓJ/ψ→K+K−φ

dMinv(K−φ)=

1

(2π)31

4M2J/ψ

pK+ pK− A2 . (6)

Thus, the branching ratio of the J/ψ → K+K−φ decaycan be calculated through

Br(J/ψ → K+K−φ)

=1

ΓJ/ψ

∫dΓJ/ψ→K+K−φ

dMinv(K−φ)dMinv(K−φ)

=A2

ΓJ/ψ

∫1

(2π)31

4M2J/ψ

pK+ pK− dMinv(K−φ) , (7)

where the integration is performed from Minv(K−φ)min =mK− +mφ to Minv(K−φ)max = mJ/ψ −mK+ .

The branching ratio of the J/ψ → K+K−φ decay hasbeen experimentally measured to be [33]

Br(J/ψ → K+K−φ) = (8.3± 1.2)× 10−4 . (8)

Using this as the input, we can determine the value of theconstant A, satisfying:

A2

ΓJ/ψ=

Br(J/ψ → K+K−φ)∫1

(2π)31

4M2J/ψ

pK+ pK−dMinv(K−φ)

= 0.018± 0.003 MeV−1 , (9)

4 Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays

where the error is taken from Eq. (8).

2.2 Triangle diagram mechanism for theJ/ψ → K+K−f0(980)(a0(980)) decays

In the former subsection we studied the J/ψ → K+K−φdecay. In this subsection we show how theJ/ψ → K+K−f0(980)(a0(980)) decays can be producedusing this input. To do this we look into the triangle dia-grams depicted in Fig. 2. The two mechanisms of Fig. 2(a)and Fig. 2(b) are clearly distinguishable. The kinematicsof the reaction around the TS point provide for the mech-anism of Fig. 2(a) a momentum pK+ ≈ 1114 MeV/c andpK− ≈ 170 MeV/c, and opposite for the mechanism ofFig. 2(b). In addition, as we shall see later (see Eq. (16)),in one diagram we have ε(J/ψ) · k(K−) and in the otherone ε(J/ψ) · k(K+). Upon squaring the sum of the twoamplitudes and summing over the J/ψ polarizations, weget the crossed term proportional to k(K−)·k(K+), linearin the cosine of the K− and K+ angle, which will cancelupon angle integrations in the phase space. Because ofthis, there is also no interference between these mecha-nisms. The two mechanisms give the same width and wecan study just one of them.

We take the first diagram Fig. 2(a) and the J/ψ →K+K−f0(980) decay as an example, and write down itsamplitude as:

−it = −iA∫

d4q

(2π)4i

(P − q)2 −m2φ + iε

i

q2 −m2K− + iε

× i

(P − q − k)2 −m2K+ + iε

× ε(J/ψ) · ε(φ) ε(φ) · (2k + q − P )

× (−igV ) (−igf0,K+K−)

= A∫

d4q

(2π)41

(P − q)2 −m2φ + iε

1

q2 −m2K− + iε

× 1

(P − q − k)2 −m2K+ + iε

× ε(J/ψ) · (2k + q − P ) gV gf0,K+K− , (10)

where in the last equation we have summed over the φpolarization. The φ → K+K− vertex is obtained fromthe P -wave Lagrangian [46,47],

LV PP = −igV 〈[P, ∂µP ] V µ〉 , gV =mV

2fπ, (11)

where P and V are the ordinary pseudoscalar and vec-tor meson SU(3) matrices, mV is the vector mass (mV ∼800 MeV), and fπ is the pion decay constant (fπ = 93 MeV).This Lagrangian produces a vertex

tφ→K+K− = gV (pµK+ − pµK−)εµ(φ) , (12)

and the ε0(φ) component can be neglected as shown in Ap-pendix A of Ref. [38], since the three-momentum of the φis very small compared to its mass for φ on-shell in that

diagram. The f0(980) → K+K− and a0(980) → K+K−

vertices are obtained from the chiral unitary approach ofRef. [19] with gf0,K+K− = 2567 MeV and ga0,K+K− =3875 MeV. Note that these two vertices will be more care-fully examined in the next subsection.

Then we follow Refs. [12,43] and perform analyticallythe q0 integration in Eq. (10) in the K−f0(980) rest frame,with the result

t = A∫

d3q

(2π)31

8ωK+ωK−ωφ

1

k0 − ωK+ − ωφ + iΓφ/2

× 1

Minv(K−f0) + ωK− + ωK+ − k0

× 1

Minv(K−f0)− ωK− − ωK+ − k0 + iε

× 1

Minv(K−f0)− ωφ − ωK− + iΓφ/2

×

(2Minv(K−f0)ωK− + 2k0ωK+

− 2(ωK− + ωK+)(ωK− + ωK+ + ωφ)

)× gV gf0,K+K− ε(J/ψ) · (2k + q) , (13)

where

ωK− =√m2K− + q2 ,

ωK+ =√m2K+ + (q + k)2 ,

ωφ =√m2φ + q2 ,

k0 =M2

inv(K−f0) +m2K− −m

2f0

2Minv(K−f0),

k = |k| =λ1/2(M2

inv(K−f0),m2K− ,m

2f0

)

2Minv(K−f0). (14)

In addition, since k is the only momentum not integratedin Eq. (13), we can replace

∫d3q q −→ k

∫d3q

q · kk2

. (15)

Then the amplitude t of Eq. (13) can be rewritten as

t = A gV gf0,K+K− ε(J/ψ) · k tT , (16)

Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays 5

where

tT =

∫d3q

(2π)31

8ωK+ωK−ωφ

1

k0 − ωK+ − ωφ + iΓφ/2

× 1

Minv(K−f0) + ωK− + ωK+ − k0

× 1

Minv(K−f0)− ωK− − ωK+ − k0 + iε

× 1

Minv(K−f0)− ωφ − ωK− + iΓφ/2

×

(2Minv(K−f0)ωK− + 2k0ωK+

− 2(ωK− + ωK+)(ωK− + ωK+ + ωφ)

)

×(

2 +q · kk2

). (17)

As in Ref. [12], the above integration is regularized withthe factor θ(qmax−|q ∗|), where q ∗ is the momentum of theK− in the rest frame of f0(980), with qmax = 600 MeV asit is needed in the chiral unitary approach that reproducesthe f0(980) [44,45]. After summing over polarizations inEq. (16), we obtain∑∑

|t|2

=∑A2 g2V g2f0,K+K− |tT |

2 1

3ε(J/ψ) · k ε(J/ψ) · k

= A2 g2V g2f0,K+K− |tT |2 1

3|k|2 . (18)

Now we can write the K−f0(980) invariant mass dis-tribution of the J/ψ → K+K−f0(980) decay as

dΓJ/ψ→K+K−f0(980)

dMinv(K−f0)=

1

(2π)31

4M2J/ψ

p′K+ p ′K−∑∑

|t|2 .

(19)where p′K+ is the momentum of the K+ in the J/ψ restframe, and p ′K− = k is the momentum of the K− in theK−f0(980) rest frame:

p′K+ =λ1/2(M2

J/ψ,m2K+ ,M2

inv(K−f0))

2MJ/ψ,

p ′K− = k =λ1/2(M2

inv(K−f0),m2K− ,m

2f0

)

2Minv(K−f0). (20)

Recalling Eq. (9) and Eq. (18), we obtain the differentialbranching ratio of the J/ψ → K+K−f0(980) decay to be

1

ΓJ/ψ

dΓJ/ψ→K+K−f0(980)

dMinv(K−f0)(21)

=A2

ΓJ/ψ

1

(2π)31

4M2J/ψ

1

3p′K+ p ′3K− g

2V g

2f0,K+K− |tT |

2 .

The case for a0(980) production is identical replacing gf0,K+K−

by ga0,K+K− .

2.3 The J/ψ → K+K−f0(980)→ K+K−π+π− andJ/ψ → K+K−a0(980)→ K+K−π0η reactions

In the former subsection we studied the triangle diagrammechanism for the J/ψ → K+K−f0(980)(a0(980)) de-cays. In this subsection we further consider that the f0(980)and a0(980) will be seen in the π+π− and π0η mass dis-tribution, respectively, as depicted in Fig. 3. Take the firstdiagram Fig. 3(a) as an example, the J/ψ first decays intoK+K−φ, next the φ decays into K+K−, then the K− andK+ merge to give the f0(980), and finally the f0(980) de-cays into π+π−. We can write down its amplitude as:

−it′ = −iA∫

d4q

(2π)4i

(P − q)2 −m2φ + iε

i

q2 −m2K− + iε

× i

(P − q − k)2 −m2K+ + iε

× ε(J/ψ) · ε(φ) ε(φ) · (2k + q − P )

× (−igV ) (−itK+K−,π+π−)

= A∫

d4q

(2π)41

(P − q)2 −m2φ + iε

1

q2 −m2K− + iε

× 1

(P − q − k)2 −m2K+ + iε

× ε(J/ψ) · (2k + q − P ) gV tK+K−,π+π− .

(22)

This amplitude t′ is very similar to t given in Eq. (10)in the former subsection, just with gf0,K+K− replaced bythe transition amplitude tK+K−,π+π− . We can follow thesame procedure to simplify it to be

t′ = A gV tK+K−,π+π− ε(J/ψ) · k′ t′T , (23)

where t′T is also very similar to tT given in Eq. (17), justwith the following replacements:

k0 → k′0 =M2

inv(K−f0) +m2K− −M

2inv(π+π−)

2Minv(K−f0), (24)

k → k′ =λ1/2(M2

inv(K−f0),m2K− ,M

2inv(π+π−))

2Minv(K−f0).(25)

The K+K− → π+π− and K+K− → π0η scattering hasbeen studied in detail in Refs. [44,45] within the chiral uni-tary approach, where altogether six channels were takeninto account, including π+π−, π0π0, K+K−, K0K0, ηη,and π0η. In the present study we use this as input, andwe shall see simultaneously both the f0(980) (with I = 0)and a0(980) (with I = 1) productions.

Now we can write down the double differential massdistribution for the J/ψ → K+K−f0(980)→ K+K−π+π−

reaction, as a function ofMinv(K−f0) andMinv(π+π−) [26]:

d2ΓJ/ψ→K+K−f0(980)→K+K−π+π−

dMinv(K−f0)dMinv(π+π−)

=1

(2π)51

4M2J/ψ

p′′K+ p ′′K− p′′π+

∑∑|t′|2 , (26)

6 Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays

J/ψ

K

f(P-q)

f 0

+

K (q)-

K (P-q-k)+

K (k)-

π+

π-

(a)

J/ψ

K

f(P-q)

f 0

-

K (q)+ K (P-q-k)-

K (k)+

π+

π-

(b)

J/ψ

K

f(P-q)

a 0

+

K (q)-

K (P-q-k)+

K (k)-

π0

η

(c)

J/ψ

K

f(P-q)

a 0

-

K (q)+ K (P-q-k)-

K (k)+

π0

η

(d)

Fig. 3. Triangle diagrams for the (a, b) J/ψ → K+K−f0(980)→ K+K−π+π− and (c, d) J/ψ → K+K−a0(980)→ K+K−π0ηreactions.

where p′′K+ is the momentum of the K+ in the J/ψ restframe, p ′′K− = k′ is the momentum of the K− in theK−f0(980) rest frame, and p ′′π+ is the momentum of theπ+ in the π+π− rest frame:

p′′K+ =λ1/2(M2

J/ψ,m2K+ ,M2

inv(K−f0))

2MJ/ψ,

p ′′K− = k′ =λ1/2(M2

inv(K−f0),m2K− ,M

2inv(π+π−))

2Minv(K−f0),

p ′′π+ =λ1/2(M2

inv(π+π−),m2π+ ,m2

π−)

2Minv(π+π−). (27)

Recalling Eq. (9) and Eq. (23), we obtain the double dif-ferential branching ratio of the J/ψ → K+K−f0(980) →K+K−π+π− reaction to be

1

ΓJ/ψ

d2ΓJ/ψ→K+K−f0(980)→K+K−π+π−

dMinv(K−f0)dMinv(π+π−)

=A2

ΓJ/ψ

1

(2π)51

4M2J/ψ

1

3p′′K+ p ′′3K− p

′′π+

× g2V |tK+K−,π+π− |2 |t′T |2 . (28)

With trivial changes, replacing π+π− by π0η, we get thecorresponding expressions for π0η production.

One should note that a different picture of these res-onances, like compact qq or tetraquarks, which happened

to provide the same couplings to KK, ππ, πη, would pro-duce the same results that we have reported here. Thedifference of the two pictures stems from the fact thatfor the dynamically generated resonances that come fromthe interaction of the KK and coupled channels, it is theinteraction of the KK primarily produced that producesthe f0 and a0 in the final state. A compact quark state,Q, necessarily has overlap with J/ψ and K+K− in a di-rect J/ψ → K+K−Q reaction. The J/ψ → K+K−f0(a0)production contains then two mechanisms, the direct one,that one can envisage important for a compact object,and the triangle mechanism. For a dynamically generatedresonance the triangle mechanism is the only productionmode. Hence, there would be different predictions for theproduction rate of f0(a0) in these pictures, but lacking thepredictions from the compact picture, one cannot go anyfurther. There are not many of such reactions where eval-uations are done with both pictures, but some exist, likethe Bs → J/ψf0 reaction [48], where a compact pictureis proposed in Ref. [49] and a more detailed descriptionof the data in line with the molecular picture is done inRef. [44] (see also related method in Ref. [50]). Other casesare discussed in Ref. [51].

Ultimately, it is the consistent and systematic descrip-tion of experimental facts what gives weight to a particularpicture, and the molecular picture has passed a great dealof these tests [51,52]. The agreement with experiment of

Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays 7

the present study should be considered within this per-spective. In any case the main aim of the present workis to point out the presence of the TS in this reaction,with the double purpose of finding experimental cases,where TS are manifested, and anticipating a warning fornot confusing the peaks predicted, when observed, withnew resonances.

3 Results

Firstly, we show our results for theJ/ψ → K+K−f0(980)(a0(980)) decays, which were previ-ously studied in Sec. 2.2. Let us begin by showing in Fig. 4the contribution of the triangle loop defined in Eq. (17).The TS condition of Eq. (1) requires all K+K−φ inter-mediate particles to be on shell. This forces mf0(a0) >mK+ + mK− . On the other hand, if we go to energies abit bigger than that, Eq. (1) is no longer fulfilled. There ishence a very narrow window of f0(a0) masses where theTS condition is exactly fulfilled, i.e., from 987 MeV to 993MeV. In view of this we plot in Fig. 4 the real, imaginaryparts and modulus of tT of Eq. (17) for different masses off0(a0). The magnitude depends on the f0(a0) mass, inde-pendent on whether we have f0 or a0, since the differentcouplings to K+K− have been factorized out of the in-tegral of tT . We show the results for six different masses.The first two are inside the window of energies where theTS appears, the other four are outside. We observe a neatpeak in the first two cases, which gets broader gradually aswe depart from the TS window. Note that the peak of theimaginary part is related to the triangle singularity, whilethe one of the real part is related to the K−φ threshold,as discussed in Refs. [38,39].

Then we show 1ΓJ/ψ

dΓJ/ψ→K+K−f0(a0)

dMinv(K−f0/K−a0), the differential

branching ratio of the J/ψ → K+K−f0(980)(a0(980)) de-cays defined in Eq. (21), as a function ofMinv(K−f0/K

−a0)in Fig. 5. We plot the results for three selected masses off0(a0), 989 MeV, 987 MeV, and 981 MeV. The resultsfor f0 or a0 production differ only in a factor because ofthe different couplings gf0,K+K− or ga0,K+K− . We observe

a peak indΓJ/ψ→K+K−f0(a0)

dMinv(K−f0/K−a0)around Minv(K−f0/K

−a0) =

1515 MeV. The peak is clear for the first mass of 989 MeV,but gradually the upper part of the spectrum falls downmore slowly. This is due to the factor p ′3K− in Eq. (21),which raises fast as Minv(K−f0/K

−a0) increases. If weremove this factor the peaks are sharper. Next we in-

tegratedΓJ/ψ→K+K−f0(a0)

dMinv(K−f0/K−a0)over Minv(K−f0/K

−a0) from

Minv(K−f0/K−a0)min = mK− +mf0(a0) to

Minv(K−f0/K−a0)max = mJ/ψ −mK+ to obtain

mf0(MeV) Br(J/ψ → K+K−f0(980))

989 1.56× 10−5

987 1.66× 10−5

981 1.37× 10−5

989 3.55× 10−5

987 3.79× 10−5

981 3.12× 10−5

(29)

We can see that the results for the integrated branching ra-tios depend on the mass assumed for the f0(a0) resonance.For the same mass, the f0 or a0 production rates differ bythe ratio of the square of their couplings to K+K−.

In view of the changing shape and strength of the re-sults on the mass assumed for the f0(a0) resonance, weapply next the method discussed in Sec. 2.3, taking intoaccount the mass distribution of the f0(980) and a0(980)reflected by the tK+K−,π+π− and tK+K−,π0η amplitudes.In Fig. 6 we plot |t′T |, the triangle loop defined in Eq. (23)for the J/ψ → K+K−f0(980) → K+K−π+π− reaction,as a function of Minv(π+π−) by fixing Minv(K−f0) = 1496MeV, 1516 MeV, and 1536 MeV. The distribution getsits largest strength when Minv(K−f0) is near 1516 MeV.The triangle loop |t′T | for the J/ψ → K+K−a0(980) →K+K−π0η reaction is the same as this one. The |t′T | func-tion is stopped at the Minv(π+π−) which makes k′ ofEq. (25) zero. Note that t′ of Eq. (23), proportional tok′, vanishes in this point, where one has the frontier ofthe phase space.

We also show 1ΓJ/ψ

d2ΓJ/ψ→K+K−f0(980)→K+K−π+π−

dMinv(K−f0)dMinv(π+π−) in the

left panel of Fig. 7, that is the double differential branch-ing ratio of the J/ψ → K+K−f0(980) → K+K−π+π−

reaction defined in Eq. (28). We show this as a func-tion of Minv(π+π−) by fixing Minv(K−f0) = 1496 MeV,1516 MeV, and 1536 MeV. A strong peak can be foundwhen Minv(π+π−) is around 980 MeV, corresponding tothe f0(980). Consequently, most of the contribution comesfrom Minv(π+π−) ∈ [900, 1050] MeV, and we can restrictthe integral in Minv(π+π−) to this region when calculating

the mass distribution 1ΓJ/ψ

dΓJ/ψ→K+K−f0(980)→K+K−π+π−

dMinv(K−f0).

Unlike in Fig. 6, the strength forMinv(K−f0) = 1536 MeVis a bit bigger than that for 1516 MeV. This is because of

the factor(p ′′K−

)3in Eq. (28). Similarly, we show

1ΓJ/ψ

d2ΓJ/ψ→K+K−a0(980)→K+K−π0η

dMinv(K−a0)dMinv(π0η) in the right panel of Fig. 7.

Again, we can restrictMinv(π0η) to the regionMinv(π0η) ∈

8 Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays

ÈtT ÈReHtT LImHtT L

1490 1500 1510 1520 1530 1540 1550-1

0

1

2

3

4

-1

00

1

2

3

4

MinvHK- f0L @MeVD

t T´

10-

7@M

eV-

2 D

(a)

ÈtT ÈReHtT LImHtT L

1490 1500 1510 1520 1530 1540 1550-1

0

1

2

3

4

-1

00

1

2

3

4

MinvHK- f0L @MeVD

t T´

10-

7@M

eV-

2 D

(b)

ÈtT ÈReHtT LImHtT L

1490 1500 1510 1520 1530 1540 1550-1

0

1

2

3

4

-1

00

1

2

3

4

MinvHK- f0L @MeVD

t T´

10-

7@M

eV-

2 D

(c)

ÈtT ÈReHtT LImHtT L

1490 1500 1510 1520 1530 1540 1550-1

0

1

2

3

4

-1

00

1

2

3

4

MinvHK- f0L @MeVD

t T´

10-

7@M

eV-

2 D

(d)

ÈtT ÈReHtT LImHtT L

1490 1500 1510 1520 1530 1540 1550-1

0

1

2

3

4

-1

00

1

2

3

4

MinvHK- f0L @MeVD

t T´

10-

7@M

eV-

2 D

(e)

ÈtT ÈReHtT LImHtT L

1490 1500 1510 1520 1530 1540 1550-1

0

1

2

3

4

-1

00

1

2

3

4

MinvHK- f0L @MeVD

t T´

10-

7@M

eV-

2 D

(f)

Fig. 4. Triangle amplitude tT , defined in Eq. (17), as a function of Minv(K−f0/K−a0) for (a) mf0(a0) = 989 MeV, (b)

mf0(a0) = 988 MeV, (c) mf0(a0) = 987 MeV, (d) mf0(a0) = 985 MeV, (e) mf0(a0) = 983 MeV, and (f) mf0(a0) = 981 MeV. |tT |,Re(tT ), and Im(tT ) are plotted using the black, red, and green curves, respectively.

[900, 1050] MeV, and perform the integration

1

ΓJ/ψ

dΓJ/ψ→K+K−f0(980)→K+K−π+π−

dMinv(K−f0)

=1

ΓJ/ψ

∫ 1050 MeV

900 MeV

dMinv(π+π−)

×d2ΓJ/ψ→K+K−f0(980)→K+K−π+π−

dMinv(K−f0)dMinv(π+π−), (30)

1

ΓJ/ψ

dΓJ/ψ→K+K−a0(980)→K+K−π0η

dMinv(K−a0)

=1

ΓJ/ψ

∫ 1050 MeV

900 MeV

dMinv(π0η)

×d2ΓJ/ψ→K+K−a0(980)→K+K−π0η

dMinv(K−a0)dMinv(π0η). (31)

The (single) differential branching ratios obtained are shownin Fig. 8. We see a clear structure around 1515 MeV com-ing from the peak of the triangle loop t′T , but we also ob-serve strong contribution from the larger K−f0/K

−a0 in-

variant masses produced by the(p ′′K−

)3factor of Eq. (28).

Finally, we integrate fromMinv(K−f0/K−a0)min = mK−+

mf0(a0) to Minv(K−f0/K−a0)max = mJ/ψ −mK+ to ob-

tain

Br(J/ψ → K+K−f0(980)→ K+K−π+π−) = 7.6× 10−6 ,

Br(J/ψ → K+K−a0(980)→ K+K−π0η) = 5.2× 10−6 .

(32)

These rates should be multiplied by two to account forthe mechanisms of Fig 3(b) and Fig 3(d), if one looks for

J/ψ → K+K−f0(980)(a0(980)) independently of whichof the K’s is the fast one.

As we can see, the explicit consideration of the tK+K−,π+π−

and tK+K−,π0η changes the final shape of the differentialwidth and integrated branching ratio. We should note thatin the case of the f0 production the method of Sec. 2.2 ac-counts for all the decay modes of the f0, π+π− and π0π0,the latter with a strength 1/2 compared to the one ofπ+π−. To compare the result of Eq. (32) with those ofEq. (29) we must multiply the result of Eq. (32) by 3/2and then the results are more similar. The discrepanciesare bigger in the case of the a0 production. This shouldnot be a surprise since the a0(980) is a border line statebetween a bound KK state and a threshold cusp, as a con-sequence of which the coupling of a0 to K+K− has largeuncertainties, in which case, the method used in Sec. 2.3is more reliable.

4 Conclusion

We have studied the J/ψ → K+K−f0(980)(a0(980)) de-cays and have seen that they are driven by a triangle sin-gularity, peaking at Minv(K−f0/K

−a0) ≈ 1515 MeV. Theprocess proceeds as follows: In a first step the J/ψ decaysto K+K−φ. The K+ and K− momenta are very distinctin the process and we select for our study the mode withK+ with large momentum and K− with small momen-tum. The opposite case provides the same contribution. Ina second step the φ decays into K+K− and the primaryK− together with the K+ from the φ decay merge to give

Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays 9

1490 1500 1510 1520 1530 1540 15500

1

2

3

4

5

0

1

2

3

4

5

MHK- f0L @MeVD

1 G

dG

dM

invHK-

f 0L@1

0-8

MeV-

1 D

(a)

1490 1500 1510 1520 1530 1540 15500

1

2

3

4

5

0

1

2

3

4

5

MHK- f0L @MeVD

1 G

dG

dM

invHK-

f 0L@1

0-8

MeV-

1 D

(b)

1490 1500 1510 1520 1530 1540 15500

1

2

3

4

5

0

1

2

3

4

5

MHK- f0L @MeVD

1 G

dG

dM

invHK-

f 0L@1

0-8

MeV-

1 D

(c)

1490 1500 1510 1520 1530 1540 15500

2

4

6

8

10

0

2

4

6

8

10

MHK-a0L @MeVD

1 G

dG

dM

invHK-

a 0L@1

0-8

MeV-

1 D

(d)

1490 1500 1510 1520 1530 1540 15500

2

4

6

8

10

0

2

4

6

8

10

MHK-a0L @MeVD

1 G

dG

dM

invHK-

a 0L@1

0-8

MeV-

1 D

(e)

1490 1500 1510 1520 1530 1540 15500

2

4

6

8

10

0

2

4

6

8

10

MHK-a0L @MeVD

1 G

dG

dM

invHK-

a 0L@1

0-8

MeV-

1 D

(f)

Fig. 5. The differential branching ratio 1ΓJ/ψ

dΓJ/ψ→K+K−f0(a0)

dMinv(K−f0/K−a0)

, defined in Eq. (21), as a function of Minv(K−f0/K−a0), for (a)

mf0 = 989 MeV, (b) mf0 = 987 MeV, (c) mf0 = 981 MeV, (d) ma0 = 989 MeV, (e) ma0 = 987 MeV, and (f) ma0 = 981 MeV.Note that Eq. (21) is for f0 production, and for a0 production one multiplies it by (ga0,K+K−/gf0,K+K−)2.

Minv=1496Minv=1516Minv=1536

900 950 1000 10500

1

2

3

4

0

1

2

3

4

MinvHΠ+Π-L @MeVD

Èt TÈ´

10-

7@M

eV-

2 D

Fig. 6. The triangle loop |t′T | for the J/ψ → K+K−f0(980) → K+K−π+π− reaction, defined in Eq. (23), as a function ofMinv(π+π−). The red, black, and blue curves are obtained by setting Minv(K−f0) = 1496 MeV, 1516 MeV, and 1536 MeV,respectively. The triangle loop |t′T | for the J/ψ → K+K−a0(980)→ K+K−π0η reaction is the same as the this one. Curves arestopped at the end of phase space for the production of mK and Minv(π+π−) with the invariant mass Minv(K−f0), where t′ ofEq. (23) vanishes.

the f0(980) or a0(980) resonance. The mechanism implic-itly assumes that the f0(980) and a0(980) resonances arenot produced directly but are a consequence of the finalstate interaction of the K+K−. This is the basic findingof the chiral unitary approach where these two resonancesare the consequence of the pseudoscalar-pseudoscalar in-teraction in coupled channels and not qq objects.

Using as input empirical information from the J/ψ →K+K−φ decay, we are able to determine the double dif-ferential decay width in terms of the K−f0/K

−a0 in-variant mass and the π+π−/π0η from the decay of thef0(980)/a0(980), respectively. We find very distinct shapes

of the double differential distributions, and the single one,dΓ/dMinv(K−f0/K

−a0), with a sharp raise of this magni-tude around Minv(K−f0/K

−a0) ≈ 1515 MeV, where thetriangle singularity appears. All these features are tied tothe nature of the f0(980) and a0(980) resonances as dy-namically generated from the interaction of pseudoscalarmesons, and its experimental observation should bringvaluable information on the important issue of the natureof the low-lying scalar mesons. We find the branching ra-tios of the order of 10−5, which are accessible in presentfacilities, where many J/ψ branching ratios of the orderof 10−6 ∼ 10−7 have already been measured.

10 Wei-Hong Liang et al.: Triangle singularity in the J/ψ → K+K−f0(980)(a0(980)) decays

Minv=1496

Minv=1516

Minv=1536

900 950 1000 10500

2

4

6

0

2

4

6

MHΠ+Π-L @MeVD

1 G

d2G

dM

inv

dM

' inv

@10-

10M

eV-

1 D

(a)

Minv=1496

Minv=1516

Minv=1536

900 950 1000 10500

2

4

6

0

2

4

6

MHΠ0ΗL @MeVD

1 G

d2G

dM

inv

dM

' inv

@10-

10M

eV-

1 D

(b)

Fig. 7. (a) 1ΓJ/ψ

d2ΓJ/ψ→K+K−f0→K+K−π+π−

dMinv(K−f0)dMinv(π

+π−)as a function of Minv(π+π−). (b) 1

ΓJ/ψ

d2ΓJ/ψ→K+K−a0→K+K−π0η

dMinv(K−f0)dMinv(π

0η)as a function

of Minv(π0η). The red, black, and blue curves are obtained by setting Minv(K−f0/K−a0) = 1496 MeV, 1516 MeV, and 1536

MeV, respectively.

1490 1500 1510 1520 1530 1540 15500

0.5

1.0

1.5

0

0.5

1.0

1.5

MHK- f0L @MeVD

1 G

dG

dM

invHK-

f 0L@1

0-8

MeV-

1 D

(a)

1480 1490 1500 1510 1520 1530 1540 15500

0.5

1.0

1.5

0

0.5

1.0

1.5

MHK-a0L @MeVD

1 G

dG

dM

invHK-

a 0L@1

0-8

MeV-

1 D

(b)

Fig. 8. (a) the differential branching ratio 1ΓJ/ψ

dΓJ/ψ→K+K−f0→K+K−π+π−

dMinv(K−f0)

, defined in Eq. (30), as a function of Minv(K−f0).

(b) the differential branching ratio 1ΓJ/ψ

dΓJ/ψ→K+K−a0→K+K−π0η

dMinv(K−a0)

, defined in Eq. (31), as a function of Minv(K−a0).

Acknowledgments

This work is partly supported by the National NaturalScience Foundation of China under Grants Nos. 11505158,11565007, 11722540, and 11847317, the Fundamental Re-search Funds for the Central Universities, and the Aca-demic Improvement Project of Zhengzhou University. Thiswork is also partly supported by the Spanish Ministeriode Economia y Competitividad and European FEDERfunds under the contract number FIS2011-28853-C02-01,FIS2011-28853-C02-02, FIS2014-57026-REDT, FIS2014-51948-C2-1-P, and FIS2014-51948-C2-2-P.

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