-
The electromagnetic (EM) Dalitz decay of a hadron A, namely,
A→Bγ∗→Bl+l− , refers to the decay process where A decays into B by emitting a time-like photon that then converts to a lepton pairl+l− . The differential partial decay width with respect to the invariant massq2≡m2l+l− of the lepton pair can be expressed bydΓ(q2)/dq2=(dΓ(q2)/dq2)point−like|fAB(q2)|2 [1], where(dΓ(q2)/dq2)point−like can be calculated exactly in QED for point-like particles A and B;fAB(q2) is called the transition form factor (TFF) of the transitionA→B and is an important probe to the EM structure of theABγ vertex as well as the internal structure of the hadron A (and B if it is also structured). Experimentally, the TFFfAB(q2) can be derived by taking the ratio[dΓ(q2)/dq2]/Γ(A→Bγ)∝|fAB(q2)|2/|fAB(0)|2≡|FAB(q2)|2 with the normalizationFAB(0)=1 , where many systematic uncertainties cancel. The Dalitz decays of light hadrons, such asϕ→π0e+e− [2, 3],ϕ→ηe+e− [4],ω→π0e+e− [5], andω→π0μ+μ− [6], have been widely studied in experiments. It should be noted that experimental studies on Dalitz decays usually require large statistics, as they are rare decays for a hadron.The BESIII Collaboration (BESIII) has accumulated more than
1010 J/ψ events [7], based on which the Dalitz decays ofJ/ψ to light hadrons can be researched. Meanwhile, the light pseudoscalars (P), such asη,η′,η(1405/1475) andX(1835) , are observed to have large production rates on theJ/ψ radiative decays [8]. Thus, it is expected that the Daltiz decaysJ/ψ→Pl+l− can be investigated to a high precision. Actually, BESIII has performed experimental studies on the processesJ/ψ→e+e−(η,η′,π0) [9],J/ψ→e+e−η [10],J/ψ→e+e−η′ [11, 12],J/ψ→e+e−η(1405) [13], andJ/ψ→e+e−(X(1835),X(2120),X(2370) [14]. With the largeψ(3686) data ensemble, BESIII also studied the processψ(3686)→e+e−ηc . The TTFsFψp(q2) are extracted for the processesJ/ψ→η,η′,η(1405),X(1835) , andψ(3686)→ηc , and theq2 -dependence can be described by the single-pole modelFψP=11−q2/Λ2,
(1) based on the vector meson dominance (VMD) [1, 15, 16]. The pole parameter Λ varies in the range of 1.7 to 3.8 GeV.
Intuitively, the Dalitz and radiative decays of charmonium into light pseudoscalars happens through the annihilation of the charm quark and antiquark. According to the OZI rule, the dominant contribution comes from the initial state radiation of the virtual and real photons from the charm (anti)quark (strictly speaking, the initial state radiation can also take place through light sea quarks; however, although the intermediate gluons can be soft such that the OZI rule is not conceptually justified, the approximate flavor SU(3) symmetry leads to a cancellation among the contributions from
u,d,s quarks due to their electric charges). In this sense, the TFF ofJ/ψ to pseudoscalars should reflect the electromagnetic properties ofJ/ψ . Therefore, for the same initial vector charmonium, Dalitz decays are insensitive to the properties of the final state light hadrons. A theoretical derivation ofFψP(q2) from QCD is desirable but is still challenging becauseFψP(q2) is obviously in the non-perturbative regime of QCD. Phenomenological studies ofFψη(′)(q2) can be found in Ref. [17], where the analysis is carried out in the full kinematic region based on QCD models, and in Ref. [18], where theJ/ψ→γ∗P is discussed within the framework of the effective Lagrangian approach andηc−η−η′ mixing is considered.Lattice QCD may take the mission to give reliable predictions of TFF of
J/ψ to light hadrons. A recentNf=2 lattice QCD calculation confirms the large production rate of the flavor singlet pseudoscalar mesonη(2) in theJ/ψ radiative decay [19]. In that work, the EM form factor was obtained at numerous values of time-likeq2 , from which the on-shell form factor atq2=0 was obtained through a polynomial interpolation. By assuming theUA(1) anomaly dominance and using theη−η′ mixing angle, this on-shell form factor results in the branching fractions ofJ/ψ→γη andJ/ψ→γη′ that are close to the experimental values. Actually, theq2 dependence of this decay form factor is described better by the single-pole model in Eq. (1) below.Recently, we generated a large gauge ensemble with
Nf=1 strange sea quarks. TheNf=1 QCD is a well defined theory and simplified version of QCD. It has no chiral symmetry breaking, but theUA(1) anomaly has a close relation with the unique light pseudoscalar mesonη(1) . Thus, we will revisit the production rate ofη(1) in theJ/ψ radiative decays. We will test theUA(1) anomaly dominance in this process by looking at theNf dependence of the partial decay width, as theUA(1) anomaly is proportional toNf . In the meantime, we will explore theq2 -dependence of the related TFF and its sensitivity to the light pseudoscalar mass since our sea quark is much heavier than that in Ref. [19]. The related calculations necessarily involve the annihilation effect of strange quarks, which are dealt with using the distillation method [20].The remainder of this paper is organized as follows. The numerical procedures and results are presented in Section II. Section III is devoted to the discussion and physical implications of our results. Section IV gives the summary of this work.
-
We generate gauge configurations with
Nf=1 dynamical strange quarks on anL3×T=163×128 anisotropic lattice. We use the tadpole-improved Symanzik's gauge action for anisotropic lattices [21, 22] and the tadpole-improved anisotropic clover fermion action [23, 24]. The RHMC algorithm implemented in Chroma software [25] is used to generate theNf=1 gauge configurations. The parameters in the action are tuned to give the anisotropyξ=as/at≈5 , whereat andas are the temporal and spatial lattice spacings, respectively. The scale setting takes the following procedure. Experimentally, there is an interesting relation between pseudoscalar meson massesmPS and the vector meson massesmV of the quark configurationqlˉq :Δm2≡m2V−m2PS≈0.56−0.58GeV2,
(2) where
ql represents the u, d, s quarks, and q representsu,d,s,c quarks. The masses of these vector and pseudoscalar mesons from PDG [8] are listed in Table 1 along with their mass squared differences. Similar to the scale setting in Ref. [26], we assume the relation of Eq. (2) is somewhat general for light mesons and use it to set the scale parameterat . We make the least squares fitting to the mass squared differences over thenˉn ,nˉs ,nˉc , andsˉc systems, where n refers to theu,d quarks, and get the mean value¯Δm2=0.568(8) GeV2 , which serves as an input to give the lattice scale parametera−1t=6.66(5) GeV. As the HPQCD collaboration determined thesˉs pseudoscalar meson mass to bemηs=0.686(4) GeV from the connected quark diagram [27], we use the ratiomϕ/mηs= 1.487(9) to set the bare mass parameters of strange quarks. Althoughηs is not a physical state, the mass squared differencem2ϕ−m2ηs≈0.570GeV2 also satisfies the empirical relation of Eq. (2). Finally, we obtainmηs=693.1(3)(6.0) MeV,mϕ=1027.2(5)(7.7) MeV, andm2ϕ−m2ηs=0.570GeV2 on our gauge ensemble. This serves as a self-consistent check of our lattice setup. The details of the gauge ensemble are given in Table 2. For the valence charm quark, we use the same fermion action as the strange sea quarks, and the charm quark mass parameters are tuned to give(mηc+3mJ/ψ)/4=3069 MeV.qlˉq mV /GeVmPS /GeVm2V−m2PS/GeV2 nˉn 0.775 0.140 0.581 nˉs 0.896 0.494 0.559 sˉs 1.020 0.686 [27] 0.570 nˉc 2.010 1.870 0.543 sˉc 2.112 1.968 0.588 nˉb 5.325 5.279 0.481 sˉb 5.415 5.367 0.523 Table 1. Experimental values of the masses of the pseudoscalar (P) and vector mesons (V) of quark configurations
nˉn ,nˉs ,nˉc ,sˉc ,nˉb , andsˉb [8] , where n refers to theu,d quarks. The rightmost column lists them2V−m2PS(GeV2) values. In the row ofsˉs states, the mass of thesˉs pseudoscalarηs is determined by the HPQCD collaboration from lattice QCD calculations [27].L3×T β a−1t /GeVξ mηs /MeVmϕ /MeVNcfg 163×128 2.0 6.66(5) ∼5.0 693(5) 1027(8) 1547 Table 2. Parameters of the gauge ensemble.
The quark propagators are calculated in the framework of the distillation method [20]. Let
{Vi,i=1,2,…,70} be the set of theNV=70 eigenvectors (with smallest eigenvalues) of the gauge covariant Laplacian operator on the lattice. We use these eigenvectors to calculate the perambulators of strange and charm quarks, which are encoded with the all-to-all quark propagators and facilitate the treatment of quark disconnected diagrams. In the meantime, these eigenvectors provide a smearing scheme for quark fields, namely,ψ(s)=V†Vψ , whereψ(s) is the smeared quark field ofψ , and V is a matrix with each column being an eigenvector. All the meson interpolation operators in this work are built from the smeared charm and strange quark fields. -
We use two interpolation operators for
η(1) , namely,Oγ5=ˉs(s)γ5s(s) andOγ4γ5=ˉs(s)γ4γ5s(s) , to calculate the correlation functionsCγ5γ5(t) andC(γ4γ5)(γ4γ5)(t) .Cγ5γ5(t) has a finite volume artifact that approaches a nonzero constant when t is large, as shown in Fig. 1. This artifact comes from the topology of QCD vacuum and can be approximately expressed asa5(χtop+Q2/V)/T , where a is the lattice spacing (in the isotropic case),χtop is the topological susceptibility, Q is the topological charge, V is the spatial volume, and T is the temporal extension of the lattice [28−30]. In contrast,C(γ4γ5)(γ4γ5)(t) damps to zero for large t, which is the normal large t behavior. The constant term ofCγ5γ5(t) can be subtracted by taking the differenceFigure 1. (color online) Lattice results of the energies of
η(1) . Left panel: effective mass of two-point correlation functions ofη(1) andηs , whereη(1) uses the operatorsOγ5 andOγ4γ5 . The green points are from the original two-point function ofOγ5 , the blue points are from the subtracted two-point function in Eq. (3) witht0=3at , and the red points are from the two-point function ofOγ4γ5 . The plateau regions of red and blue points merge together in the larget/at range. The purple points represent unphysicalηs , which only include connected diagram. Middle panel: effective energiesE(→q)η(1)(t) with momentum mode→n of→q up to|→n|2=9 , where the data points are from the correlation functions ofOγ4γ5(→q) . The fittedE(→q)η(1) values are given in Table 3. Right panel: dispersion relation ofη(1) . The grey band illustrates the dispersion relation in Eq. (9) with the fittedξ=4.88(1) andχ2 per degree of freedomχ2/d.o.f=0.32 .C′γ5γ5(t)=Cγ5γ5(t)−Cγ5γ5(t+t0),
(3) and we take
t0=3at in practice. The effective mass functionsmeff(t)=lnC(′)ΓΓ(t)C(′)ΓΓ(t+1) of the two correlations are shown in Fig. 1, where one can see that notable mass plateaus appear whent/at>15 and agree with each other. The effective masses of the connected parts of the two correlation functions are also shown for comparison. Their plateaus correspond to the massmηs ofηs . The data analysis gives the resultsmηs=693.1(3)MeV,mη(1)=783.0(5.5)MeV.
(4) Here,
mηs is the mass parameter from the connected diagram and is consistent with the valuemηs=686(4) obtained by HPQCD at the physical strange quark mass [27]. This indicates that our sea quark mass parameter is tuned to be almost at the strange quark mass.mη(1) is determined from the correlation function that includes the connected diagram and quark annihilation diagram and is therefore the mass of the well-defined pseudoscalar mesonη(1) . -
The transition matrix element
M for the processJ/ψ→γ∗η(1) can be expressed in terms of one form factorM(q2) , namely,Mμψη(1)γ∗≡⟨η(1)(pη)|jμem(0)|ψ(pψ,λ)⟩=M(q2)ϵμνρσpψ,νpη,ρϵσ(pψ,λ),
(5) where
q2=(pψ−pη)2 is the virtuality of the photon,ϵσ(pJ/ψ,λ) is the polarization vector ofJ/ψ , andjμem=ˉcγμc is the electromagnetic current of charm quark (we only consider the initial state radiation and ignore photon emissions from sea quarks and the final state). The matrix elementM is encoded in the following three-point functions:Cμi(3)(→q;t,t′)=∑→yei→q⋅→y⟨Oη(→p′,t)jμem(→y,t′)Oi,†ψ(→p,0)⟩
(6) with
→q=→p′−→p , whereOη(→p′,t) andOiψ(→p,t) are the interpolating field operators forη(1) andJ/ψ with spatial momenta→p′ and→p , respectively. Fort≫t′ ,t′≫0 , and in the rest frame ofJ/ψ (→p=0 ), the explicit spectral expression ofCμi(3)(→q;t,t′) readsCμi(3)(→q;t,t′)≈Zη(→q)Z∗ψ4V3Eη(→q)mψe−Eη(→q)(t−t′)e−mψt′×∑λ⟨η(1)(→q)|jμem|J/ψ(→0,λ)⟩ϵ∗,i(→0,λ),
(7) where
V3 is the spatial volume,Zη(→q)=⟨Ω|Oη(1)(→q)|η(1)(→q)⟩ andZψϵi(→0,λ)=⟨Ω|Oiψ(→0)|J/ψ(→0,λ)⟩ . Note thatZη has a→q dependence due to the smeared operatorOη [31]. The parametersmψ ,Eη(→q) ,Zψ , andZη(→q) can be derived from the two-point correlation functionsC(2),η(→q,t)≈12Eη(→q)V3|Zη(→q)|2e−Eη(→q)t,Cii(2),ψ(t)≈12mψV3|ZJ/ψ|2e−mψt.
(8) Thus, we can extract the matrix element
⟨η(1)|jμem|J/ψ⟩ through Eqs. (7) and (8).Therefore, the major numerical task is the calculation of
Cμi(3)(→q;t,t′) . The local EM currentjμem(x)=[ˉcγμc](x) mentioned above (the charm quark field c andˉc are the original field, which are not smeared) is not conserved anymore on the finite lattice and should be renormalized. We determine the renormalization factorsZtV=1.147(1) andZsV=1.191(2) for the temporal and spatial components ofjμem(x) , respectively, by calculating the relevant electromagnetic form factors ofηc [32, 33]. In practice, onlyZsV is involved and is incorporated implicitly injμem(x) . We use the operatorOγ4γ5=ˉs(s)γ4γ5s(s) forOη , andOiψ takes the formˉc(s)γic(s) in Eq. (6). The three-point functionCμi(3)(→q;t,t′) is calculated in the rest frame ofJ/ψ (→p=0 ) such thatη(1) moves with spatial momentum→p′=→q . The right panel of Fig. 1 shows the dispersion relation ofη(1) a2tE2η(→q)=a2tm2η+1ξ2(2πL)2|→n|2,
(9) where
→n represents the momentum mode of→q=2πLas→n . It can be seen thatE2η(→q) exhibits a perfect linear behavior in|→q|2 up to|→n|2=9 and the fitted slope givesξ=4.88(1) , which deviates from the renormalized anisotropyξ≈5.0 by less than 3%.After the Wick's contractions, the three-point function
Cμi(3) is expressed in terms of quark propagators, and the schematic quark diagram is illustrated in Fig. 2. There are two separated quark loops connected by gluons. The strange quark loop on the right-hand side can be calculated in the framework of the distillation method. The left partGμi comes from the contraction ofOiψ and the currentjμem , namely,Gμi(→p,→q;t′+τ,τ)=∑→yei→q⋅→yjμem(→y,t′+τ)Oi†ψ(→p,τ),
(10) and is dealt with by the distillation method [34]. Considering
Oiψ(→p,t)=∑→ye−i→p⋅→y[ˉc(s)γic(s)](→y,t) , the explicit expression ofGμi at the source time sliceτ=0 isGμi(→p,→q;t,0)=∑→xei→q⋅→xTr{γ5[ScV(0)]†(→x,t)γ5γμ×[ScV(0)](→x,t)[V†(0)D(→p)γiV(0)]},
(11) where
Sc=⟨cˉc⟩U is the all-to-all propagator of charm quark for a given gauge configuration U , andD(→p) is a3L3×3L3 diagonal matrix with diagonal elementsδabei→p⋅→y (→y labels the column or row indices, anda,b=1,2,3 refer to the color indices). Theγ5 -hermiticitySc=γ5S†cγ5 implies[V†(0)Sc](→x,t)=γ5[ScV(0)]†(→x,t)γ5 , such that onlyScV(0) is required, whileScV(0) can be obtained by solving the system of linear equationsM[U;mc][ScV(0)]=V(0),
(12) where
M[U;mc] is the fermion matrix in the charm quark action (the linear system solver defined byM[U;mc]x=b is applied4NV times for Dirac indicesα=1,2,3,4 and all columns ofV(0) ). To increase the statistics, the above procedure runs over the entire time range, say,τ∈[0,T−1] . Averaging overτ∈[0,T−1] improves the precision of the calculatedCμi(3) drastically.It is observed that the
J/ψ contribution dominatesCμi(3)(→q;t,t′) whent′>40 . Combining Eqs. (5), (7), and (8), we have the expressionRμi(→q;t,t′)≡ZψZη(→q)Cμi(3)(→q;t,t′)V3C(2),η(−→q,t−t′)C(2),ψ(t′)≈M(q2;t−t′)ϵμijqj
(13) for fixed
t′/at=40 , from which we obtainM(q2,t−t′) for eachq2 . Figure 3 shows thet−t′ dependence ofM(q2,t−t′) at severalq2 close toq2=0 . It can be seen that a plateau region appears beyondt−t′>10 for eachq2 , whereM(q2) is obtained through a constant fit. The grey bands illustrate the fitted values and fitting time ranges, along with the jackknife errors. We also test the fit function formM(q2,t−t′)=M(q2)+c(q2)e−δE(t−t′) , with the exponential term being introduced to account for the higher state contamination. The fitted values ofM(q2) in this way are consistent with those in the constant fit but have much larger errors. Therefore, we use the results from the constant fit for the values ofM(q2) . The derivedM(q2) up toq2=4.3GeV2 data points are listed in Table 3.Figure 3. (color online) Fit of form factor
M(q2) forJ/ψ→ γ∗η(1) . The lattice data are plotted as data points, and the grey bands show the fit by constants to the plateau regions. The fittedM(q2) values are given in Table 3.mode →n of→q (1,2,2) (0,2,2) (1,1,2) (0,1,2) (1,1,1) (0,1,1) (0,0,1) q2/GeV2 − 0.6800(66)− 0.1869(73)0.8777(91) 1.459(10) 2.756(14) 3.499(16) 4.337(20) E(→q)η(1)/GeV 1.803(14) 1.710(17) 1.5279(29) 1.4291(24) 1.2119(29) 1.0886(20) 0.9466(18) M(q2)/GeV−1 0.00380(80) 0.00447(97) 0.00645(82) 0.0071(10) 0.00828(89) 0.0137(11) 0.0174(18) Table 3. Fit values of
η(1) form factorM(q2) . The momentum modes→n represent the relation→q=2πL→n . The two-point functionCΓΓ(t) and three-point ratio functionRμi(→q;t,t′) corresponding to the same momentum mode→n have been averaged for increasing signal of the energyE(→q)η(1) and form factorM(q2) , respectively.Instead of a polynomial function form used by Ref. [19], we use the single-pole model to describe the
q2 dependence ofM(q2) M(q2)=M(0)1−q2/Λ2≡M(0)Fψη(q2).
(14) As indicated by the red band in Fig. 4, the model fits the overall behaviors of
M(q2) very well with the parametersFigure 4. (color online) The form factor
M(q2) forJ/ψ→ γ∗η(1,2) . The data points are the lattice QCD results, and the shaded bands illustrate the fit modelM(q2)=M(0)1−q2/Λ2 with the best fit parametersM(0)=0.01066(36)GeV−1 forNf=1 andΛ=2.442(36)GeV forNf=2 . TheM(q2) data ofJ/ψ→γ∗η(2) are the same as those in Table II of Ref. [19], and the fit is performed using the jackknife method on the original data sample. The light-shaded bands are the comparison with the polynomial fitM(q2)=M(0)+aq2+bq4 .M(0)=0.00498(36)GeV−1,Λ=2.44(60)GeV.
(15) -
The partial decay width
Γ(J/ψ→γη(1)) is dictated by the on-shell form factorM(q2=0) through the relationΓ(J/ψ→γη(1))=4α27|M(0)|2|→pγ|3,
(16) where the electric charge of charm quark
Q=+2e/3 has been incorporated,α≡e2/(4π)=1/134 is the fine structure constant at the charm quark mass scale, and|→pγ|=(m2ψ−m2η(1))/2mψ is the on-shell momentum of the photon. Using the value ofM(0) in Eq. (15), the partial decay width and corresponding branching fraction are predicted asΓ(J/ψ→γη(1))=0.087(13)keVBr(J/ψ→γη(1))=0.93(14)×10−3,
(17) where the experimental value
ΓJ/ψ=92.6keV is used. Obviously,Br(J/ψ→γη(1)) is four or five times smaller thanBr(J/ψ→γη(2))=4.16(49)×10−3 in theNf=2 case ofmπ≈350MeV [19] and the experimental valueBr(J/ψ→γη′)=5.25(7)×10−3 [8].This large difference can be attributed to the dependence of quark masses and the flavor number
Nf . The decay processJ/ψ→γη(Nf) takes place in the procedure that thecˉc pair annihilates into gluons (after a photon radiation), which then convert intoη(Nf) . There are two mechanisms for gluons to couple toη(Nf) . The first is theUA(1) anomaly manifested by the anomalous axial vector current relation (in the chiral limit)∂μjμ5(x)=√Nfg232πGaμν(x)˜Ga,μν(x)≡√Nfq(x),
(18) where
jμ5=1√NfNf∑k=1ˉqkγ5γμqk is the flavor singlet axial vector current forNf flavor quarks, andq(x) is the topological charge density. TheUA(1) anomaly induces the anomalous gluon-η coupling with the strength described by the matrix element⟨0|q(0)|η(Nf)⟩ , which has been discussed extensively in theoretical studies [35−42]. With the matrix element⟨0|∂μjμ5(0)|η(Nf)⟩=fη(Nf)m2η(Nf) , from Eq. (18), one has the relation⟨0|q(0)|η(Nf)⟩=1√Nffη(Nf)m2η(Nf)
(19) in the chiral limit. According to the Witten and Veneziano mechanism [43, 44] for the mass of
η(Nf) ,m2η(Nf)=4Nff2πχtop , whereχtop is the topological susceptibility of the SU(3) pure Yang-Mills theory, one has⟨0|q(0)|η(Nf)⟩∝√Nf/fη(Nf) in the chiral limit. Consequently, if theUA(1) anomaly dominates the production ofη(Nf) in the processJ/ψ→γη(Nf) , then one expects theNf dependence for the partial decay widthΓ(J/ψ→γη(Nf))∝|⟨0|q(0)|η(Nf)⟩|2∝Nff2η(Nf).
(20) This kind of
Nf dependence can be tested in aNf=3 lattice QCD calculation. Note thatfη(Nf) has quark mass dependence for massive quarks and becomes larger when quark mass increases. Thus, Eq. (20) partially explains the small value ofΓ(J/ψ→γη(1)) we obtain at the strange quark mass inNf=1 QCD.There may be also other sources for the quark mass dependence. First, an additional term
1√Nf∑Nfk=12imkˉqkγ5qk appears on the right-hand side of Eq. (18) when the quark massesmk≠0 . Secondly, the coupling of perturbative gluons toη(Nf) is proportional to quark mass [45, 46]. However, it is nontrivial to theoretically deduce a precise quark mass dependence for the decay process we are considering. This issue can be explored by the lattice QCD calculations at different light quark masses in the future. -
The form factor
M(q2) in Eq. (14) is actually the TFF for the Dalitz decayJ/ψ→η(1)l+l− whenq2>4m2l , which is seen to be well described by the single pole model withΛ=2.439(60)GeV . In Ref. [19], the Dalitz TFFM(q2) is also obtained in theNf=2 lattice QCD atmπ≈350MeV , and the valueM(0) is interpolated using a polynomial function form, namely,M(q2)=M(0)+aq2+bq4 . We refit theq2 -dependence ofM(q2) in Ref. [19] using the same single-pole model and present the result in Fig. 4, where the polynomial fits to both theNf=1 data in this work and theNf=2 data in Ref. [19] are also shown for comparison. In both cases, the pole model (only two parameters) fit gives smaller values (0.74 forNf=1 and0.32 forNf=2 ) ofχ2/d.o.f than the polynomial fit (1.00 forNf=1 and 0.86 forNf=2 ), even though the latter has one more parameter. Especially for theNf=2 case, the pole model withΛ=2.442(36)GeV describes the wholeq2 range very well. This indicates that the single-pole model is suitable for describing the Dalitz decay TFFs ofJ/ψ to light pseudoscalar mesons P.In experiments, the TFF
FψP is extracted from the ratiodΓ(ψ→Pl+l−)/dq2Γ(ψ→Pγ)=A(q2)|FψP(q2)|2,
(21) where
A(q2) is a known kinematic factor [1, 15, 16]A(q2)=α3π1q2(1−4m2lq2)1/2(1+2m2lq2)×[(1+q2m2ψ−m2P)2−4m2ψq2(m2ψ−m2P)2]3/2
(22) derived from the QED calculation. BESIII has measured many Dalitz decay processes of
J/ψ→Pe+e− withP=η [9, 10],η′ [9, 11, 12],η(1405) [13], and(X(1835), X(2120),X(2370)) [14]. For some of these processes, the TFF are obtained and fitted through the single-pole model (along with resonance terms if experimental data are precise enough [10]) in Eq. (1), and the fitted values of Λ are listed in Table 4, where the values of Λ derived from lattice QCD are also presented in the last two rows for comparison. Although the values of Λ for theJ/ψ→η,η′ Dalitz decays are compatible with the lattice values, the values of Λ forJ/ψ→η(1405),X(1835) are substantially smaller. Thus, it is possible that Λ depends on the mass of the final state pseudoscalar meson.Table 4. Values of the pole parameter Λ of the TFF for different Dalitz decays
J/ψ→Pe+e− . TheNf=1,2 lattice QCD results of Λ are also shown in the bottom two rows for comparison.In principle, the production of each light pseudoscalar P in the
J/ψ radiative decay or Dalitz decay undergoes the same procedure that thecˉc pair emits a photon of the virtualityq2 and then annihilates into gluons, whose invariant mass squared is labelled as s. As the single-pole model describesM(q2) very well while theq2 and s in theJ/ψ−γ∗(q2)−(gg⋯)∗(s) vertex are correlated, one expects the s-dependence of Λ. We assume an empirical linear function forΛ(s) , namely,Λ(s)=Λ1(1−sΛ22),
(23) and then use the experimental values of Λ at different
s=m2P in Table 4 to determine the parametersΛ1 andΛ2 . Finally, we getΛ1=2.65(5)GeV,Λ2=2.90(35)GeV,
(24) with
χ2/d.o.f=0.26 . The fit result is illustrated in Fig. 5 by a shaded blue band. It can be seen that the lattice QCD data atNf=1,2 reside almost entirely on the fitting curve. Actually, the function in Eq. (23) with fitted parametersΛ1,2 gives the interpolated valuesΛ(s=m2η(1))=2.465(40) andΛ(s=m2η(2))=2.495(42) , which are in good agreement with the lattice QCD results2.44(4) GeV and2.44(6)GeV .Figure 5. (color online) The s dependence of the pole parameter Λ. The data points indicate the experimental (blue points) and lattice result (red points) of Λ at different values of
s=m2P (listed in Table 4), wheremP is the mass of the psuedoscalar meson in the processJ/ψ→Pγ∗ . The shaded blue band shows the fitting of the experimental data using the modelΛ(s)=Λ1(1−sΛ22) with fitted parametersΛ1=2.65(5)GeV andΛ2=2.90(35)GeV . Theχ2/d.o.f of the fit is 0.26.The values of
Λ1,2 in Eq. (24) can give inputs for theoretical and experimental studies. Taking the processJ/ψ→η′e+e− for instance, the experimental value of Λ has huge uncertainties, but the model in Eq. (23) with the parameters in Eq. (24) gives a more precise predictionΛ(s=m2η′)=2.37(5)GeV.
(25) Then, according to Eq. (21) and using the experimental result of
Br(J/ψ→γη′)=5.25(7)×10−3 , the branching fraction ofJ/ψ→η′e+e− is estimated to be6.04(4)(8)×10−5 , which is compatible with the BESIII result6.59(7)(17)×10−5 [12]. When theρ resonance contribution is included, as it was by BESIII forJ/ψ→ηe+e− in Ref. [10],|Fψη′(q2)|2 reads|Fψη′(q2)|2=|Aρ|2(m4ρ(q2−m2ρ)2+m2ρΓ2ρ)+|AΛ|2(11−q2/Λ2)2,
(26) where
Aρ is the coupling constant of theρ meson, andAΛ is the coupling constant of the non-resonant contribution. ForJ/ψ→ηe+e− , BESIII determinesAρ=0.23(4) andAΛ=1.05(3) [10], which give|Fψη(q2≈0)|2=1.11±0.07±0.07 . If we take the same value forAρ=0.23(4) and assumeAΛ=1 for the case ofη′ (the|Fψη′(q2)|2 atq2≈0 in Ref. [9] is consistent with one within errors), then using the PDG values ofmρ andΓρ [47], we getBr(J/ψ→η′e+e−)=6.57+20−17(4)(9)×10−5,
(27) where the first error is due to the uncertainty of
Aρ , the second is from that of Λ, and the third is from that of the experimental value ofBr(J/ψ→γη′) . This value agrees with the experimental value better. -
We generate a large gauge ensemble with
Nf=1 dynamical strange quarks on an anisotropic lattice with the anisotropyas/at≈5.0 . The pseudoscalar mass is measured to bemηs=693.1(3)MeV without considering the quark annihilation effect, andmη(1)=783.0(5.5)MeV with the inclusion of quark annihilation diagrams. We calculate the EM form factorM(q2) for the decay processJ/ψ→γ∗(q2)η(1) withq2 being the virtuality of the photon. By interpolatingM(q2) to the value atq2=0 through the VMD-inspired single-pole model in Eq. (14), the decay width and branching fraction ofJ/ψ→γη(1) are predicted to beΓ(J/ψ→γη(1))=0.087(13)keV andBr(J/ψ→γη(1))=0.93(14)×10−3 , respectively, which are much smaller than those from previousNf=2 lattice QCD results [19] and the experimental results forJ/ψ→γη′ . This large difference can be attributed in part to theNf dependence owing to theUA(1) anomaly and quark mass dependence.It is interesting to see that
M(q2) in bothNf=1,2 is well described by the single-pole modelM(q2)=M(0)/(1−q2/Λ2) . By assuming an empiricals=m2P dependenceΛ(m2P)=Λ1(1−m2P/Λ22) , the parametersΛ1,2 are determined to beΛ1=2.65(5)GeV andΛ2=2.90(35)GeV by using the experimental results of the Dalitz decaysJ/ψ→Pe+e− with P being light pseudoscalar mesons. This parameterization ofΛ(m2P) gives a consistent result with lattice QCD data at the samemP and may provide meaningful inputs for future theoretical and experimental studies on Dalitz decaysJ/ψ→Pe+e− . As a direct application, thismP dependence expects a pole parameterΛ(s=m2η′)=2.37(5)GeV , which is more precise than the value3.1(1.0)GeV measured by BESIII [9] and provides a more precise prediction forBr(J/ψ→η′e+e−) .
![]() | ![]() | ![]() | ![]() |
![]() | 0.775 | 0.140 | 0.581 |
![]() | 0.896 | 0.494 | 0.559 |
![]() | 1.020 | 0.686 [27] | 0.570 |
![]() | 2.010 | 1.870 | 0.543 |
![]() | 2.112 | 1.968 | 0.588 |
![]() | 5.325 | 5.279 | 0.481 |
![]() | 5.415 | 5.367 | 0.523 |