Processing math: 100%

Form factor for Dalitz decays from J/ψ to light pseudoscalars

  • We calculate the form factor M(q2) for the Dalitz decay J/ψγ(q2)η(Nf=1) with η(Nf) being the SU(Nf) flavor singlet pseudoscalar meson. The difference among the partial widths Γ(J/ψγη(Nf)) at different Nf can be attributed in part to the Nf and quark mass dependences induced by the UA(1) anomaly dominance. M(q2) in both Nf=1,2 is well described by the single pole model M(q2)=M(0)/(1q2/Λ2). Combined with the known experimental results of the Dalitz decays J/ψPe+e, the pseudoscalar mass mP dependence of the pole parameter Λ is approximated by Λ(m2P)=Λ1(1m2P/Λ22) with Λ1=2.65(5) GeV and Λ2=2.90(35) GeV. These results provide inputs for future theoretical and experimental studies on the Dalitz decays J/ψPe+e.
  • The electromagnetic (EM) Dalitz decay of a hadron A, namely, ABγBl+l, refers to the decay process where A decays into B by emitting a time-like photon that then converts to a lepton pair l+l. The differential partial decay width with respect to the invariant mass q2m2l+l of the lepton pair can be expressed by dΓ(q2)/dq2=(dΓ(q2)/dq2)pointlike|fAB(q2)|2 [1], where (dΓ(q2)/dq2)pointlike can be calculated exactly in QED for point-like particles A and B; fAB(q2) is called the transition form factor (TFF) of the transition AB and is an important probe to the EM structure of the ABγ vertex as well as the internal structure of the hadron A (and B if it is also structured). Experimentally, the TFF fAB(q2) can be derived by taking the ratio [dΓ(q2)/dq2]/Γ(ABγ)|fAB(q2)|2/|fAB(0)|2|FAB(q2)|2 with the normalization FAB(0)=1, where many systematic uncertainties cancel. The Dalitz decays of light hadrons, such as ϕπ0e+e [2, 3], ϕηe+e [4], ωπ0e+e [5], and ωπ0μ+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 of J/ψ to light hadrons can be researched. Meanwhile, the light pseudoscalars (P), such as η,η,η(1405/1475) and X(1835), are observed to have large production rates on the J/ψ radiative decays [8]. Thus, it is expected that the Daltiz decays J/ψPl+l can be investigated to a high precision. Actually, BESIII has performed experimental studies on the processes J/ψe+e(η,η,π0) [9], J/ψe+eη [10], J/ψe+eη [11, 12], J/ψe+eη(1405) [13], and J/ψe+e(X(1835),X(2120),X(2370) [14]. With the large ψ(3686) data ensemble, BESIII also studied the process ψ(3686)e+eηc. The TTFs Fψp(q2) are extracted for the processes J/ψη,η,η(1405),X(1835), and ψ(3686)ηc, and the q2-dependence can be described by the single-pole model

    FψP=11q2/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 of J/ψ to pseudoscalars should reflect the electromagnetic properties of J/ψ. Therefore, for the same initial vector charmonium, Dalitz decays are insensitive to the properties of the final state light hadrons. A theoretical derivation of FψP(q2) from QCD is desirable but is still challenging because FψP(q2) is obviously in the non-perturbative regime of QCD. Phenomenological studies of Fψη()(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 the J/ψγP is discussed within the framework of the effective Lagrangian approach and ηcηη mixing is considered.

    Lattice QCD may take the mission to give reliable predictions of TFF of J/ψ to light hadrons. A recent Nf=2 lattice QCD calculation confirms the large production rate of the flavor singlet pseudoscalar meson η(2) in the J/ψ radiative decay [19]. In that work, the EM form factor was obtained at numerous values of time-like q2, from which the on-shell form factor at q2=0 was obtained through a polynomial interpolation. By assuming the UA(1) anomaly dominance and using the ηη mixing angle, this on-shell form factor results in the branching fractions of J/ψγη and J/ψγη that are close to the experimental values. Actually, the q2 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. The Nf=1 QCD is a well defined theory and simplified version of QCD. It has no chiral symmetry breaking, but the UA(1) anomaly has a close relation with the unique light pseudoscalar meson η(1). Thus, we will revisit the production rate of η(1) in the J/ψ radiative decays. We will test the UA(1) anomaly dominance in this process by looking at the Nf dependence of the partial decay width, as the UA(1) anomaly is proportional to Nf. In the meantime, we will explore the q2-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 an L3×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 the Nf=1 gauge configurations. The parameters in the action are tuned to give the anisotropy ξ=as/at5, where at and as are the temporal and spatial lattice spacings, respectively. The scale setting takes the following procedure. Experimentally, there is an interesting relation between pseudoscalar meson masses mPS and the vector meson masses mV of the quark configuration qlˉq:

    Δm2m2Vm2PS0.560.58GeV2,

    (2)

    where ql represents the u, d, s quarks, and q represents u,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 parameter at. We make the least squares fitting to the mass squared differences over the nˉn, nˉs, nˉc, and sˉc systems, where n refers to the u,d quarks, and get the mean value ¯Δm2=0.568(8) GeV2, which serves as an input to give the lattice scale parameter a1t=6.66(5) GeV. As the HPQCD collaboration determined the sˉs pseudoscalar meson mass to be mηs=0.686(4) GeV from the connected quark diagram [27], we use the ratio mϕ/mηs= 1.487(9) to set the bare mass parameters of strange quarks. Although ηs is not a physical state, the mass squared difference m2ϕm2ηs0.570GeV2 also satisfies the empirical relation of Eq. (2). Finally, we obtain mηs=693.1(3)(6.0) MeV, mϕ=1027.2(5)(7.7) MeV, and m2ϕm2ηs=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.

    Table 1

    Table 1.  Experimental values of the masses of the pseudoscalar (P) and vector mesons (V) of quark configurations nˉn, nˉs, nˉc, sˉc, nˉb, and sˉb [8] , where n refers to the u,d quarks. The rightmost column lists the m2Vm2PS(GeV2) values. In the row of sˉs states, the mass of the sˉs pseudoscalar ηs is determined by the HPQCD collaboration from lattice QCD calculations [27].
    qlˉq mV/GeV mPS/GeV m2Vm2PS/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
    DownLoad: CSV
    Show Table

    Table 2

    Table 2.  Parameters of the gauge ensemble.
    L3×T β a1t/GeV ξ mηs/MeV mϕ /MeV Ncfg
    163×128 2.0 6.66(5) 5.0 693(5) 1027(8) 1547
    DownLoad: CSV
    Show Table

    The quark propagators are calculated in the framework of the distillation method [20]. Let {Vi,i=1,2,,70} be the set of the NV=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)=VVψ, where ψ(s) is the smeared quark field of ψ, 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) and Oγ4γ5=ˉs(s)γ4γ5s(s), to calculate the correlation functions Cγ5γ5(t) and C(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 as a5(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 [2830]. In contrast, C(γ4γ5)(γ4γ5)(t) damps to zero for large t, which is the normal large t behavior. The constant term of Cγ5γ5(t) can be subtracted by taking the difference

    Figure 1

    Figure 1.  (color online) Lattice results of the energies of η(1). Left panel: effective mass of two-point correlation functions of η(1) and ηs, where η(1) uses the operators Oγ5 and Oγ4γ5. The green points are from the original two-point function of Oγ5, the blue points are from the subtracted two-point function in Eq. (3) with t0=3at, and the red points are from the two-point function of Oγ4γ5. The plateau regions of red and blue points merge together in the large t/at range. The purple points represent unphysical ηs, which only include connected diagram. Middle panel: effective energies E(q)η(1)(t) with momentum mode n of q up to |n|2=9, where the data points are from the correlation functions of Oγ4γ5(q). The fitted E(q)η(1) values are given in Table 3. Right panel: dispersion relation of η(1). The grey band illustrates the dispersion relation in Eq. (9) with the fitted ξ=4.88(1) and χ2 per degree of freedom χ2/d.o.f=0.32.

    Cγ5γ5(t)=Cγ5γ5(t)Cγ5γ5(t+t0),

    (3)

    and we take t0=3at in practice. The effective mass functions meff(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 when t/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 mass mηs of ηs. The data analysis gives the results

    mηs=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 value mηs=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 meson η(1).

    The transition matrix element M for the process J/ψγη(1) can be expressed in terms of one form factor M(q2), namely,

    Mμψη(1)γη(1)(pη)|jμem(0)|ψ(pψ,λ)=M(q2)ϵμνρσpψ,νpη,ρϵσ(pψ,λ),

    (5)

    where q2=(pψpη)2 is the virtuality of the photon, ϵσ(pJ/ψ,λ) is the polarization vector of J/ψ, and jμem=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 element M is encoded in the following three-point functions:

    Cμi(3)(q;t,t)=yeiqyOη(p,t)jμem(y,t)Oi,ψ(p,0)

    (6)

    with q=pp, where Oη(p,t) and Oiψ(p,t) are the interpolating field operators for η(1) and J/ψ with spatial momenta p and p, respectively. For tt, t0, and in the rest frame of J/ψ (p=0), the explicit spectral expression of Cμi(3)(q;t,t) reads

    Cμi(3)(q;t,t)Zη(q)Zψ4V3Eη(q)mψeEη(q)(tt)emψt×λη(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)and Zψϵi(0,λ)=Ω|Oiψ(0)|J/ψ(0,λ). Note that Zη has a q dependence due to the smeared operator Oη [31]. The parameters mψ, Eη(q), Zψ, and Zη(q) can be derived from the two-point correlation functions

    C(2),η(q,t)12Eη(q)V3|Zη(q)|2eEη(q)t,Cii(2),ψ(t)12mψV3|ZJ/ψ|2emψt.

    (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 current jμem(x)=[ˉcγμc](x) mentioned above (the charm quark field c and ˉc are the original field, which are not smeared) is not conserved anymore on the finite lattice and should be renormalized. We determine the renormalization factors ZtV=1.147(1) and ZsV=1.191(2) for the temporal and spatial components of jμem(x), respectively, by calculating the relevant electromagnetic form factors of ηc [32, 33]. In practice, only ZsV is involved and is incorporated implicitly in jμem(x). We use the operator Oγ4γ5=ˉs(s)γ4γ5s(s) for Oη, and Oiψ takes the form ˉc(s)γic(s) in Eq. (6). The three-point function Cμi(3)(q;t,t) is calculated in the rest frame of J/ψ (p=0) such that η(1) moves with spatial momentum p=q. The right panel of Fig. 1 shows the dispersion relation of η(1)

    a2tE2η(q)=a2tm2η+1ξ2(2πL)2|n|2,

    (9)

    where n represents the momentum mode of q=2πLasn. It can be seen that E2η(q) exhibits a perfect linear behavior in |q|2 up to |n|2=9 and the fitted slope gives ξ=4.88(1), which deviates from the renormalized anisotropy ξ5.0 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 part Gμi comes from the contraction of Oiψ and the current jμem, namely,

    Figure 2

    Figure 2.  (color online) Schematic diagram for the process J/ψγη(1).

    Gμi(p,q;t+τ,τ)=yeiqyjμem(y,t+τ)Oiψ(p,τ),

    (10)

    and is dealt with by the distillation method [34]. Considering Oiψ(p,t)=yeipy[ˉc(s)γic(s)](y,t), the explicit expression of Gμi at the source time slice τ=0 is

    Gμi(p,q;t,0)=xeiqxTr{γ5[ScV(0)](x,t)γ5γμ×[ScV(0)](x,t)[V(0)D(p)γiV(0)]},

    (11)

    where Sc=cˉcU is the all-to-all propagator of charm quark for a given gauge configuration U , and D(p) is a 3L3×3L3 diagonal matrix with diagonal elements δabeipy (y labels the column or row indices, and a,b=1,2,3 refer to the color indices). The γ5-hermiticity Sc=γ5Scγ5 implies [V(0)Sc](x,t)=γ5[ScV(0)](x,t)γ5, such that only ScV(0) is required, while ScV(0) can be obtained by solving the system of linear equations

    M[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 by M[U;mc]x=b is applied 4NV times for Dirac indices α=1,2,3,4 and all columns of V(0)). To increase the statistics, the above procedure runs over the entire time range, say, τ[0,T1]. Averaging over τ[0,T1] improves the precision of the calculated Cμi(3) drastically.

    It is observed that theJ/ψ contribution dominates Cμi(3)(q;t,t) when t>40. Combining Eqs. (5), (7), and (8), we have the expression

    Rμi(q;t,t)ZψZη(q)Cμi(3)(q;t,t)V3C(2),η(q,tt)C(2),ψ(t)M(q2;tt)ϵμijqj

    (13)

    for fixed t/at=40, from which we obtain M(q2,tt) for each q2. Figure 3 shows the tt dependence of M(q2,tt) at several q2 close to q2=0. It can be seen that a plateau region appears beyond tt>10 for each q2, where M(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 form M(q2,tt)=M(q2)+c(q2)eδE(tt), with the exponential term being introduced to account for the higher state contamination. The fitted values of M(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 of M(q2). The derived M(q2) up to q2=4.3GeV2 data points are listed in Table 3.

    Figure 3

    Figure 3.  (color online) Fit of form factor M(q2) for J/ψγη(1). The lattice data are plotted as data points, and the grey bands show the fit by constants to the plateau regions. The fitted M(q2) values are given in Table 3.

    Table 3

    Table 3.  Fit values of η(1) form factor M(q2). The momentum modes n represent the relation q=2πLn. The two-point function CΓΓ(t) and three-point ratio function Rμi(q;t,t) corresponding to the same momentum mode n have been averaged for increasing signal of the energy E(q)η(1) and form factor M(q2), respectively.
    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)/GeV1 0.00380(80) 0.00447(97) 0.00645(82) 0.0071(10) 0.00828(89) 0.0137(11) 0.0174(18)
    DownLoad: CSV
    Show Table

    Instead of a polynomial function form used by Ref. [19], we use the single-pole model to describe the q2 dependence of M(q2)

    M(q2)=M(0)1q2/Λ2M(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 parameters

    Figure 4

    Figure 4.  (color online) The form factor M(q2) for J/ψγη(1,2). The data points are the lattice QCD results, and the shaded bands illustrate the fit model M(q2)=M(0)1q2/Λ2 with the best fit parameters M(0)=0.01066(36)GeV1 for Nf=1 and Λ=2.442(36)GeV for Nf=2. The M(q2) data of J/ψγη(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 fit M(q2)=M(0)+aq2+bq4.

    M(0)=0.00498(36)GeV1,Λ=2.44(60)GeV.

    (15)

    The partial decay width Γ(J/ψγη(1)) is dictated by the on-shell form factor M(q2=0) through the relation

    Γ(J/ψγη(1))=4α27|M(0)|2|pγ|3,

    (16)

    where the electric charge of charm quark Q=+2e/3 has been incorporated, αe2/(4π)=1/134 is the fine structure constant at the charm quark mass scale, and |pγ|=(m2ψm2η(1))/2mψ is the on-shell momentum of the photon. Using the value of M(0) in Eq. (15), the partial decay width and corresponding branching fraction are predicted as

    Γ(J/ψγη(1))=0.087(13)keVBr(J/ψγη(1))=0.93(14)×103,

    (17)

    where the experimental value ΓJ/ψ=92.6keV is used. Obviously, Br(J/ψγη(1)) is four or five times smaller than Br(J/ψγη(2))=4.16(49)×103 in the Nf=2 case of mπ350MeV [19] and the experimental value Br(J/ψγη)=5.25(7)×103 [8].

    This large difference can be attributed to the dependence of quark masses and the flavor number Nf. The decay process J/ψγη(Nf) takes place in the procedure that the cˉc pair annihilates into gluons (after a photon radiation), which then convert into η(Nf). There are two mechanisms for gluons to couple to η(Nf). The first is the UA(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=1NfNfk=1ˉqkγ5γμqk is the flavor singlet axial vector current for Nf flavor quarks, and q(x) is the topological charge density. The UA(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 [3542]. 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)=1Nffη(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, where χtop 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 the UA(1) anomaly dominates the production of η(Nf) in the process J/ψγη(Nf), then one expects the Nf dependence for the partial decay width

    Γ(J/ψγη(Nf))|0|q(0)|η(Nf)|2Nff2η(Nf).

    (20)

    This kind of Nf dependence can be tested in a Nf=3 lattice QCD calculation. Note that fη(Nf) has quark mass dependence for massive quarks and becomes larger when quark mass increases. Thus, Eq. (20) partially explains the small value of Γ(J/ψγη(1)) we obtain at the strange quark mass in Nf=1 QCD.

    There may be also other sources for the quark mass dependence. First, an additional term 1NfNfk=12imkˉqkγ5qk appears on the right-hand side of Eq. (18) when the quark masses mk0. Secondly, the coupling of perturbative gluons to η(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 decay J/ψη(1)l+l when q2>4m2l, which is seen to be well described by the single pole model with Λ=2.439(60)GeV. In Ref. [19], the Dalitz TFF M(q2) is also obtained in the Nf=2 lattice QCD at mπ350MeV, and the value M(0) is interpolated using a polynomial function form, namely, M(q2)=M(0)+aq2+bq4. We refit the q2-dependence of M(q2) in Ref. [19] using the same single-pole model and present the result in Fig. 4, where the polynomial fits to both the Nf=1 data in this work and the Nf=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 for Nf=1 and 0.32 for Nf=2) of χ2/d.o.f than the polynomial fit (1.00 for Nf=1 and 0.86 for Nf=2), even though the latter has one more parameter. Especially for the Nf=2 case, the pole model with Λ=2.442(36)GeV describes the whole q2 range very well. This indicates that the single-pole model is suitable for describing the Dalitz decay TFFs of J/ψ to light pseudoscalar mesons P.

    In experiments, the TFF FψP is extracted from the ratio

    dΓ(ψPl+l)/dq2Γ(ψPγ)=A(q2)|FψP(q2)|2,

    (21)

    where A(q2) is a known kinematic factor [1, 15, 16]

    A(q2)=α3π1q2(14m2lq2)1/2(1+2m2lq2)×[(1+q2m2ψm2P)24m2ψq2(m2ψm2P)2]3/2

    (22)

    derived from the QED calculation. BESIII has measured many Dalitz decay processes of J/ψPe+e with P=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 the J/ψη,η Dalitz decays are compatible with the lattice values, the values of Λ for J/ψη(1405),X(1835) are substantially smaller. Thus, it is possible that Λ depends on the mass of the final state pseudoscalar meson.

    Table 4

    Table 4.  Values of the pole parameter Λ of the TFF for different Dalitz decays J/ψPe+e. The Nf=1,2 lattice QCD results of Λ are also shown in the bottom two rows for comparison.
    Ve+eP Λ/GeV Ref.
    J/ψe+eη 2.56±0.04±0.03 [10]
    J/ψe+eη 3.1±1.0 [9]
    J/ψe+eη(1405) 1.96±0.24±0.06 [13]
    J/ψe+eX(1835) 1.75±0.29±0.05 [14]
    J/ψγη(2)(718) 2.44±0.04 [19]
    J/ψγη(1)(783) 2.44±0.06 this work
    DownLoad: CSV
    Show Table

    In principle, the production of each light pseudoscalar P in the J/ψ radiative decay or Dalitz decay undergoes the same procedure that the cˉc pair emits a photon of the virtuality q2 and then annihilates into gluons, whose invariant mass squared is labelled as s. As the single-pole model describes M(q2) very well while the q2 and s in the J/ψγ(q2)(gg)(s) vertex are correlated, one expects the s-dependence of Λ. We assume an empirical linear function for Λ(s), namely,

    Λ(s)=Λ1(1sΛ22),

    (23)

    and then use the experimental values of Λ at different s=m2P in Table 4 to determine the parameters Λ1 and Λ2. Finally, we get

    Λ1=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 at Nf=1,2 reside almost entirely on the fitting curve. Actually, the function in Eq. (23) with fitted parameters Λ1,2 gives the interpolated values Λ(s=m2η(1))=2.465(40) and Λ(s=m2η(2))=2.495(42), which are in good agreement with the lattice QCD results 2.44(4) GeV and 2.44(6)GeV .

    Figure 5

    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), where mP is the mass of the psuedoscalar meson in the process J/ψPγ. The shaded blue band shows the fitting of the experimental data using the model Λ(s)=Λ1(1sΛ22) with fitted parameters Λ1=2.65(5)GeV and Λ2=2.90(35)GeV. The χ2/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 process J/ψηe+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 prediction

    Λ(s=m2η)=2.37(5)GeV.

    (25)

    Then, according to Eq. (21) and using the experimental result of Br(J/ψγη)=5.25(7)×103, the branching fraction of J/ψηe+e is estimated to be 6.04(4)(8)×105, which is compatible with the BESIII result 6.59(7)(17)×105 [12]. When the ρ resonance contribution is included, as it was by BESIII for J/ψηe+e in Ref. [10], |Fψη(q2)|2 reads

    |Fψη(q2)|2=|Aρ|2(m4ρ(q2m2ρ)2+m2ρΓ2ρ)+|AΛ|2(11q2/Λ2)2,

    (26)

    where Aρ is the coupling constant of the ρ meson, and AΛ is the coupling constant of the non-resonant contribution. For J/ψηe+e, BESIII determines Aρ=0.23(4) and AΛ=1.05(3) [10], which give |Fψη(q20)|2=1.11±0.07±0.07. If we take the same value for Aρ=0.23(4) and assume AΛ=1 for the case of η (the |Fψη(q2)|2 at q20 in Ref. [9] is consistent with one within errors), then using the PDG values of mρ and Γρ [47], we get

    Br(J/ψηe+e)=6.57+2017(4)(9)×105,

    (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 of Br(J/ψγη). 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 anisotropy as/at5.0. The pseudoscalar mass is measured to be mηs=693.1(3)MeV without considering the quark annihilation effect, and mη(1)=783.0(5.5)MeV with the inclusion of quark annihilation diagrams. We calculate the EM form factor M(q2) for the decay process J/ψγ(q2)η(1) with q2 being the virtuality of the photon. By interpolating M(q2) to the value at q2=0 through the VMD-inspired single-pole model in Eq. (14), the decay width and branching fraction of J/ψγη(1) are predicted to be Γ(J/ψγη(1))=0.087(13)keV and Br(J/ψγη(1))=0.93(14)×103, respectively, which are much smaller than those from previous Nf=2 lattice QCD results [19] and the experimental results for J/ψγη. This large difference can be attributed in part to the Nf dependence owing to the UA(1) anomaly and quark mass dependence.

    It is interesting to see that M(q2) in both Nf=1,2 is well described by the single-pole model M(q2)=M(0)/(1q2/Λ2). By assuming an empirical s=m2P dependence Λ(m2P)=Λ1(1m2P/Λ22), the parameters Λ1,2 are determined to be Λ1=2.65(5)GeV and Λ2=2.90(35)GeV by using the experimental results of the Dalitz decays J/ψPe+e with P being light pseudoscalar mesons. This parameterization of Λ(m2P) gives a consistent result with lattice QCD data at the same mP and may provide meaningful inputs for future theoretical and experimental studies on Dalitz decays J/ψPe+e. As a direct application, this mP dependence expects a pole parameter Λ(s=m2η)=2.37(5)GeV, which is more precise than the value 3.1(1.0)GeV measured by BESIII [9] and provides a more precise prediction for Br(J/ψηe+e).

    [1] L. G. Landsberg, Phys. Rept. 128, 301 (1985) doi: 10.1016/0370-1573(85)90129-2
    [2] M. N. Achasov et al., JETP Lett. 75, 449 (2002) doi: 10.1134/1.1494039
    [3] A. Anastasi et al. (KLOE-2), Phys. Lett. B 757, 362 (2016), arXiv: 1601.06565[hep-ex] doi: 10.1016/j.physletb.2016.04.015
    [4] D. Babusci et al. (KLOE-2), Phys. Lett. B 742, 1 (2015), arXiv: 1409.4582[hep-ex] doi: 10.1016/j.physletb.2015.01.011
    [5] R. R. Akhmetshin et al. (CMD-2), Phys. Lett. B 613, 29 (2005), arXiv: hep-ex/0502024 doi: 10.1016/j.physletb.2005.03.019
    [6] R. I. Dzhelyadin et al., Phys. Lett. B 84, 143 (1979) doi: 10.1016/0370-2693(79)90669-5
    [7] M. Ablikim et al. (BESIII), Chin. Phys. C 46, 074001 (2022), arXiv: 2111.07571[hep-ex] doi: 10.1088/1674-1137/ac5c2e
    [8] R. L. Workman et al., (Particle Data Group), PTEP 2022, 083C01 (2022) doi: 10.1093/ptep/ptaa104
    [9] M. Ablikim et al. (BESIII), Phys. Rev. D 89, 092008 (2014), arXiv: 1403.7042[hep-ex] doi: 10.1103/PhysRevD.89.092008
    [10] M. Ablikim et al. (BESIII), Phys. Rev. D 99 , 012006 (2019) [Erratum: Phys. Rev. D 104 , 099901 (2021)], arXiv: 1810.03091[hep-ex]
    [11] M. Ablikim et al. (BESIII), Phys. Lett. B 783, 452 (2018), arXiv: 1803.09714[hep-ex] doi: 10.1016/j.physletb.2018.05.038
    [12] M. Ablikim et al. (BESIII), Phys. Rev. D 99, 012013 (2019), arXiv: 1809.00635[hep-ex] doi: 10.1103/PhysRevD.99.012013
    [13] M. Ablikim et al. (BESIII), (2023), arXiv: 2307.14633[hep-ex]
    [14] M. Ablikim et al., Phys. Rev. Lett. 129, 022002 (2022), arXiv: 2112.14369[hep-ex] doi: 10.1103/PhysRevLett.129.022002
    [15] J. Fu, H.-B. Li, X. Qin et al., Mod. Phys. Lett. A 27, 1250223 (2012), arXiv: 1111.4055[hep-ph] doi: 10.1142/S0217732312502239
    [16] L.-M. Gu, H.-B. Li, X.-X. Ma et al., Phys. Rev. D 100, 016018 (2019), arXiv: 1904.06085[hep-ph] doi: 10.1103/PhysRevD.100.016018
    [17] J.-K. He and C.-J. Fan, Phys. Rev. D 105, 094034 (2022), arXiv: 2005.13568[hep-ph] doi: 10.1103/PhysRevD.105.094034
    [18] Y.-H. Chen, Z.-H. Guo, and B.-S. Zou, Phys. Rev. D 91, 014010 (2015), arXiv: 1411.1159[hep-ph] doi: 10.1103/PhysRevD.91.014010
    [19] X. Jiang, F. Chen, Y. Chen et al., Phys. Rev. Lett. 130, 061901 (2023), arXiv: 2206.02724[hep-lat] doi: 10.1103/PhysRevLett.130.061901
    [20] M. Peardon, J. Bulava, J. Foley et al. (Hadron Spectrum), Phys. Rev. D 80, 054506 (2009), arXiv: 0905.2160[hep-lat] doi: 10.1103/PhysRevD.80.054506
    [21] C. J. Morningstar and M. J. Peardon, Phys. Rev. D 56, 4043 (1997), arXiv: hep-lat/9704011 doi: 10.1103/PhysRevD.56.4043
    [22] Y. Chen et al., Phys. Rev. D 73, 014516 (2006), arXiv: hep-lat/0510074 doi: 10.1103/PhysRevD.73.014516
    [23] J.-h. Zhang and C. Liu, Mod. Phys. Lett. A 16, 1841 (2001), arXiv: hep-lat/0107005 doi: 10.1142/S0217732301005096
    [24] S.-q. Su, L.-m. Liu, X. Li et al., Int. J. Mod. Phys. A 21, 1015 (2006), arXiv: hep-lat/0412034 doi: 10.1142/S0217751X06024967
    [25] R. G. Edwards and B. Joo (SciDAC, LH PC, UKQCD), Nucl. Phys. B Proc. Suppl. 140, 832 (2005), arXiv: heplat/0409003 doi: 10.1016/j.nuclphysbps.2004.11.254
    [26] X. Jiang, W. Sun, F. Chen et al., Phys. Rev. D 107, 094510 (2023), arXiv: 2205.12541[hep-lat] doi: 10.1103/PhysRevD.107.094510
    [27] C. T. H. Davies, E. Follana, I. D. Kendall et al. (HPQCD), Phys. Rev. D 81, 034506 (2010), arXiv: 0910.1229[hep-lat] doi: 10.1103/PhysRevD.81.034506
    [28] S. Aoki, H. Fukaya, S. Hashimoto et al., Phys. Rev. D 76, 054508 (2007), arXiv: 0707.0396[hep-lat] doi: 10.1103/PhysRevD.76.054508
    [29] G. S. Bali, S. Collins, S. Dürr et al., Phys. Rev. D 91, 014503 (2015), arXiv: 1406.5449[hep-lat] doi: 10.1103/PhysRevD.91.014503
    [30] P. Dimopoulos et al., Phys. Rev. D 99, 034511 (2019), arXiv: 1812.08787[hep-lat] doi: 10.1103/PhysRevD.99.034511
    [31] G. S. Bali, B. Lang, B. U. Musch et al., Phys. Rev. D 93, 094515 (2016), arXiv: 1602.05525[hep-lat] doi: 10.1103/PhysRevD.93.094515
    [32] J. J. Dudek, R. G. Edwards, and D. G. Richards, Phys. Rev. D 73, 074507 (2006), arXiv: hep-ph/0601137 doi: 10.1103/PhysRevD.73.074507
    [33] Y.-B. Yang, Y. Chen, L.-C. Gui et al. (CLQCD), Phys. Rev. D 87, 014501 (2013), arXiv: 1206.2086[hep-lat] doi: 10.1103/PhysRevD.87.014501
    [34] F. Chen, X. Jiang, Y. Chen et al., Phys. Rev. D 107, 054511 (2023), arXiv: 2207.04694[hep-lat] doi: 10.1103/PhysRevD.107.054511
    [35] V. A. Novikov, M. A. Shifman, A. I. Vainshtein et al., Nucl. Phys. B 165, 55 (1980) doi: 10.1016/0550-3213(80)90305-3
    [36] T. Feldmann, Int. J. Mod. Phys. A 15, 159 (2000), arXiv: hep-ph/9907491 doi: 10.1142/S0217751X00000082
    [37] M. Beneke and M. Neubert, Nucl. Phys. B 651, 225 (2003), arXiv: hep-ph/0210085 doi: 10.1016/S0550-3213(02)01091-X
    [38] H.-Y. Cheng, H.-n. Li, and K.-F. Liu, Phys. Rev. D 79, 014024 (2009), arXiv: 0811.2577[hep-ph] doi: 10.1103/PhysRevD.79.014024
    [39] J. P. Singh, Phys. Rev. D 88, 096005 (2013), arXiv: 1307.3311[hep-ph] doi: 10.1103/PhysRevD.88.096005
    [40] W. Qin, Q. Zhao, and X.-H. Zhong, Phys. Rev. D 97, 096002 (2018), arXiv: 1712.02550[hep-ph] doi: 10.1103/PhysRevD.97.096002
    [41] M. Ding, K. Raya, A. Bashir et al., Phys. Rev. D 99, 014014 (2019), arXiv: 1810.12313[nucl-th] doi: 10.1103/PhysRevD.99.014014
    [42] G. S. Bali, V. Braun, S. Collins et al. (RQCD), JHEP 08, 137 (2021), arXiv: 2106.05398[hep-lat] doi: 10.1007/JHEP08(2021)137
    [43] E. Witten, Nucl. Phys. B 156, 269 (1979) doi: 10.1016/0550-3213(79)90031-2
    [44] G. Veneziano, Nucl. Phys. B 159, 213 (1979) doi: 10.1016/0550-3213(79)90332-8
    [45] M. Chanowitz, Phys. Rev. Lett. 95, 172001 (2005), arXiv: hep-ph/0506125 doi: 10.1103/PhysRevLett.95.172001
    [46] K.-T. Chao, X.-G. He, and J.-P. Ma, Phys. Rev. Lett. 98, 149103 (2007), arXiv: 0704.1061[hep-ph] doi: 10.1103/PhysRevLett.98.149103
    [47] B. Colquhoun, L. J. Cooper, C. T. H. Davies et al. (Particle Data Group, HPQCD, (HPQCD Collaboration)?), Phys. Rev. D 108, 014513 (2023), arXiv: 2305.06231[hep-lat] doi: 10.1103/PhysRevD.108.014513
    [48] M. A. Clark, R. Babich, K. Barros et al., Comput. Phys. Commun. 181, 1517 (2010), arXiv: 0911.3191[hep-lat] doi: 10.1016/j.cpc.2010.05.002
    [49] R. Babich, M. A. Clark, B. Joo et al., in SC11 International Conference for High Performance Computing, Networking, Storage and Analysis (2011), arXiv: 1109.2935 [hep-lat]
  • [1] L. G. Landsberg, Phys. Rept. 128, 301 (1985) doi: 10.1016/0370-1573(85)90129-2
    [2] M. N. Achasov et al., JETP Lett. 75, 449 (2002) doi: 10.1134/1.1494039
    [3] A. Anastasi et al. (KLOE-2), Phys. Lett. B 757, 362 (2016), arXiv: 1601.06565[hep-ex] doi: 10.1016/j.physletb.2016.04.015
    [4] D. Babusci et al. (KLOE-2), Phys. Lett. B 742, 1 (2015), arXiv: 1409.4582[hep-ex] doi: 10.1016/j.physletb.2015.01.011
    [5] R. R. Akhmetshin et al. (CMD-2), Phys. Lett. B 613, 29 (2005), arXiv: hep-ex/0502024 doi: 10.1016/j.physletb.2005.03.019
    [6] R. I. Dzhelyadin et al., Phys. Lett. B 84, 143 (1979) doi: 10.1016/0370-2693(79)90669-5
    [7] M. Ablikim et al. (BESIII), Chin. Phys. C 46, 074001 (2022), arXiv: 2111.07571[hep-ex] doi: 10.1088/1674-1137/ac5c2e
    [8] R. L. Workman et al., (Particle Data Group), PTEP 2022, 083C01 (2022) doi: 10.1093/ptep/ptaa104
    [9] M. Ablikim et al. (BESIII), Phys. Rev. D 89, 092008 (2014), arXiv: 1403.7042[hep-ex] doi: 10.1103/PhysRevD.89.092008
    [10] M. Ablikim et al. (BESIII), Phys. Rev. D 99 , 012006 (2019) [Erratum: Phys. Rev. D 104 , 099901 (2021)], arXiv: 1810.03091[hep-ex]
    [11] M. Ablikim et al. (BESIII), Phys. Lett. B 783, 452 (2018), arXiv: 1803.09714[hep-ex] doi: 10.1016/j.physletb.2018.05.038
    [12] M. Ablikim et al. (BESIII), Phys. Rev. D 99, 012013 (2019), arXiv: 1809.00635[hep-ex] doi: 10.1103/PhysRevD.99.012013
    [13] M. Ablikim et al. (BESIII), (2023), arXiv: 2307.14633[hep-ex]
    [14] M. Ablikim et al., Phys. Rev. Lett. 129, 022002 (2022), arXiv: 2112.14369[hep-ex] doi: 10.1103/PhysRevLett.129.022002
    [15] J. Fu, H.-B. Li, X. Qin et al., Mod. Phys. Lett. A 27, 1250223 (2012), arXiv: 1111.4055[hep-ph] doi: 10.1142/S0217732312502239
    [16] L.-M. Gu, H.-B. Li, X.-X. Ma et al., Phys. Rev. D 100, 016018 (2019), arXiv: 1904.06085[hep-ph] doi: 10.1103/PhysRevD.100.016018
    [17] J.-K. He and C.-J. Fan, Phys. Rev. D 105, 094034 (2022), arXiv: 2005.13568[hep-ph] doi: 10.1103/PhysRevD.105.094034
    [18] Y.-H. Chen, Z.-H. Guo, and B.-S. Zou, Phys. Rev. D 91, 014010 (2015), arXiv: 1411.1159[hep-ph] doi: 10.1103/PhysRevD.91.014010
    [19] X. Jiang, F. Chen, Y. Chen et al., Phys. Rev. Lett. 130, 061901 (2023), arXiv: 2206.02724[hep-lat] doi: 10.1103/PhysRevLett.130.061901
    [20] M. Peardon, J. Bulava, J. Foley et al. (Hadron Spectrum), Phys. Rev. D 80, 054506 (2009), arXiv: 0905.2160[hep-lat] doi: 10.1103/PhysRevD.80.054506
    [21] C. J. Morningstar and M. J. Peardon, Phys. Rev. D 56, 4043 (1997), arXiv: hep-lat/9704011 doi: 10.1103/PhysRevD.56.4043
    [22] Y. Chen et al., Phys. Rev. D 73, 014516 (2006), arXiv: hep-lat/0510074 doi: 10.1103/PhysRevD.73.014516
    [23] J.-h. Zhang and C. Liu, Mod. Phys. Lett. A 16, 1841 (2001), arXiv: hep-lat/0107005 doi: 10.1142/S0217732301005096
    [24] S.-q. Su, L.-m. Liu, X. Li et al., Int. J. Mod. Phys. A 21, 1015 (2006), arXiv: hep-lat/0412034 doi: 10.1142/S0217751X06024967
    [25] R. G. Edwards and B. Joo (SciDAC, LH PC, UKQCD), Nucl. Phys. B Proc. Suppl. 140, 832 (2005), arXiv: heplat/0409003 doi: 10.1016/j.nuclphysbps.2004.11.254
    [26] X. Jiang, W. Sun, F. Chen et al., Phys. Rev. D 107, 094510 (2023), arXiv: 2205.12541[hep-lat] doi: 10.1103/PhysRevD.107.094510
    [27] C. T. H. Davies, E. Follana, I. D. Kendall et al. (HPQCD), Phys. Rev. D 81, 034506 (2010), arXiv: 0910.1229[hep-lat] doi: 10.1103/PhysRevD.81.034506
    [28] S. Aoki, H. Fukaya, S. Hashimoto et al., Phys. Rev. D 76, 054508 (2007), arXiv: 0707.0396[hep-lat] doi: 10.1103/PhysRevD.76.054508
    [29] G. S. Bali, S. Collins, S. Dürr et al., Phys. Rev. D 91, 014503 (2015), arXiv: 1406.5449[hep-lat] doi: 10.1103/PhysRevD.91.014503
    [30] P. Dimopoulos et al., Phys. Rev. D 99, 034511 (2019), arXiv: 1812.08787[hep-lat] doi: 10.1103/PhysRevD.99.034511
    [31] G. S. Bali, B. Lang, B. U. Musch et al., Phys. Rev. D 93, 094515 (2016), arXiv: 1602.05525[hep-lat] doi: 10.1103/PhysRevD.93.094515
    [32] J. J. Dudek, R. G. Edwards, and D. G. Richards, Phys. Rev. D 73, 074507 (2006), arXiv: hep-ph/0601137 doi: 10.1103/PhysRevD.73.074507
    [33] Y.-B. Yang, Y. Chen, L.-C. Gui et al. (CLQCD), Phys. Rev. D 87, 014501 (2013), arXiv: 1206.2086[hep-lat] doi: 10.1103/PhysRevD.87.014501
    [34] F. Chen, X. Jiang, Y. Chen et al., Phys. Rev. D 107, 054511 (2023), arXiv: 2207.04694[hep-lat] doi: 10.1103/PhysRevD.107.054511
    [35] V. A. Novikov, M. A. Shifman, A. I. Vainshtein et al., Nucl. Phys. B 165, 55 (1980) doi: 10.1016/0550-3213(80)90305-3
    [36] T. Feldmann, Int. J. Mod. Phys. A 15, 159 (2000), arXiv: hep-ph/9907491 doi: 10.1142/S0217751X00000082
    [37] M. Beneke and M. Neubert, Nucl. Phys. B 651, 225 (2003), arXiv: hep-ph/0210085 doi: 10.1016/S0550-3213(02)01091-X
    [38] H.-Y. Cheng, H.-n. Li, and K.-F. Liu, Phys. Rev. D 79, 014024 (2009), arXiv: 0811.2577[hep-ph] doi: 10.1103/PhysRevD.79.014024
    [39] J. P. Singh, Phys. Rev. D 88, 096005 (2013), arXiv: 1307.3311[hep-ph] doi: 10.1103/PhysRevD.88.096005
    [40] W. Qin, Q. Zhao, and X.-H. Zhong, Phys. Rev. D 97, 096002 (2018), arXiv: 1712.02550[hep-ph] doi: 10.1103/PhysRevD.97.096002
    [41] M. Ding, K. Raya, A. Bashir et al., Phys. Rev. D 99, 014014 (2019), arXiv: 1810.12313[nucl-th] doi: 10.1103/PhysRevD.99.014014
    [42] G. S. Bali, V. Braun, S. Collins et al. (RQCD), JHEP 08, 137 (2021), arXiv: 2106.05398[hep-lat] doi: 10.1007/JHEP08(2021)137
    [43] E. Witten, Nucl. Phys. B 156, 269 (1979) doi: 10.1016/0550-3213(79)90031-2
    [44] G. Veneziano, Nucl. Phys. B 159, 213 (1979) doi: 10.1016/0550-3213(79)90332-8
    [45] M. Chanowitz, Phys. Rev. Lett. 95, 172001 (2005), arXiv: hep-ph/0506125 doi: 10.1103/PhysRevLett.95.172001
    [46] K.-T. Chao, X.-G. He, and J.-P. Ma, Phys. Rev. Lett. 98, 149103 (2007), arXiv: 0704.1061[hep-ph] doi: 10.1103/PhysRevLett.98.149103
    [47] B. Colquhoun, L. J. Cooper, C. T. H. Davies et al. (Particle Data Group, HPQCD, (HPQCD Collaboration)?), Phys. Rev. D 108, 014513 (2023), arXiv: 2305.06231[hep-lat] doi: 10.1103/PhysRevD.108.014513
    [48] M. A. Clark, R. Babich, K. Barros et al., Comput. Phys. Commun. 181, 1517 (2010), arXiv: 0911.3191[hep-lat] doi: 10.1016/j.cpc.2010.05.002
    [49] R. Babich, M. A. Clark, B. Joo et al., in SC11 International Conference for High Performance Computing, Networking, Storage and Analysis (2011), arXiv: 1109.2935 [hep-lat]
  • 加载中

Figures(5) / Tables(4)

Get Citation
Chunjiang Shi, Ying Chen, Xiangyu Jiang, Ming Gong, Zhaofeng Liu and Wei Sun. Form factor for Dalitz decays from J/ψ to light pseudoscalars[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad641b
Chunjiang Shi, Ying Chen, Xiangyu Jiang, Ming Gong, Zhaofeng Liu and Wei Sun. Form factor for Dalitz decays from J/ψ to light pseudoscalars[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad641b shu
Milestone
Received: 2024-06-14
Article Metric

Article Views(2074)
PDF Downloads(57)
Cited by(0)
Policy on re-use
To reuse of Open Access content published by CPC, for content published under the terms of the Creative Commons Attribution 3.0 license (“CC CY”), the users don’t need to request permission to copy, distribute and display the final published version of the article and to create derivative works, subject to appropriate attribution.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Form factor for Dalitz decays from J/ψ to light pseudoscalars

  • 1. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
  • 2. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 4. Center for High Energy Physics, Peking University, Beijing 100871, China

Abstract: We calculate the form factor M(q2) for the Dalitz decay J/ψγ(q2)η(Nf=1) with η(Nf) being the SU(Nf) flavor singlet pseudoscalar meson. The difference among the partial widths Γ(J/ψγη(Nf)) at different Nf can be attributed in part to the Nf and quark mass dependences induced by the UA(1) anomaly dominance. M(q2) in both Nf=1,2 is well described by the single pole model M(q2)=M(0)/(1q2/Λ2). Combined with the known experimental results of the Dalitz decays J/ψPe+e, the pseudoscalar mass mP dependence of the pole parameter Λ is approximated by Λ(m2P)=Λ1(1m2P/Λ22) with Λ1=2.65(5) GeV and Λ2=2.90(35) GeV. These results provide inputs for future theoretical and experimental studies on the Dalitz decays J/ψPe+e.

    HTML

    I.   INTRODUCTION
    • The electromagnetic (EM) Dalitz decay of a hadron A, namely, ABγBl+l, refers to the decay process where A decays into B by emitting a time-like photon that then converts to a lepton pair l+l. The differential partial decay width with respect to the invariant mass q2m2l+l of the lepton pair can be expressed by dΓ(q2)/dq2=(dΓ(q2)/dq2)pointlike|fAB(q2)|2 [1], where (dΓ(q2)/dq2)pointlike can be calculated exactly in QED for point-like particles A and B; fAB(q2) is called the transition form factor (TFF) of the transition AB and is an important probe to the EM structure of the ABγ vertex as well as the internal structure of the hadron A (and B if it is also structured). Experimentally, the TFF fAB(q2) can be derived by taking the ratio [dΓ(q2)/dq2]/Γ(ABγ)|fAB(q2)|2/|fAB(0)|2|FAB(q2)|2 with the normalization FAB(0)=1, where many systematic uncertainties cancel. The Dalitz decays of light hadrons, such as ϕπ0e+e [2, 3], ϕηe+e [4], ωπ0e+e [5], and ωπ0μ+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 of J/ψ to light hadrons can be researched. Meanwhile, the light pseudoscalars (P), such as η,η,η(1405/1475) and X(1835), are observed to have large production rates on the J/ψ radiative decays [8]. Thus, it is expected that the Daltiz decays J/ψPl+l can be investigated to a high precision. Actually, BESIII has performed experimental studies on the processes J/ψe+e(η,η,π0) [9], J/ψe+eη [10], J/ψe+eη [11, 12], J/ψe+eη(1405) [13], and J/ψe+e(X(1835),X(2120),X(2370) [14]. With the large ψ(3686) data ensemble, BESIII also studied the process ψ(3686)e+eηc. The TTFs Fψp(q2) are extracted for the processes J/ψη,η,η(1405),X(1835), and ψ(3686)ηc, and the q2-dependence can be described by the single-pole model

      FψP=11q2/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 of J/ψ to pseudoscalars should reflect the electromagnetic properties of J/ψ. Therefore, for the same initial vector charmonium, Dalitz decays are insensitive to the properties of the final state light hadrons. A theoretical derivation of FψP(q2) from QCD is desirable but is still challenging because FψP(q2) is obviously in the non-perturbative regime of QCD. Phenomenological studies of Fψη()(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 the J/ψγP is discussed within the framework of the effective Lagrangian approach and ηcηη mixing is considered.

      Lattice QCD may take the mission to give reliable predictions of TFF of J/ψ to light hadrons. A recent Nf=2 lattice QCD calculation confirms the large production rate of the flavor singlet pseudoscalar meson η(2) in the J/ψ radiative decay [19]. In that work, the EM form factor was obtained at numerous values of time-like q2, from which the on-shell form factor at q2=0 was obtained through a polynomial interpolation. By assuming the UA(1) anomaly dominance and using the ηη mixing angle, this on-shell form factor results in the branching fractions of J/ψγη and J/ψγη that are close to the experimental values. Actually, the q2 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. The Nf=1 QCD is a well defined theory and simplified version of QCD. It has no chiral symmetry breaking, but the UA(1) anomaly has a close relation with the unique light pseudoscalar meson η(1). Thus, we will revisit the production rate of η(1) in the J/ψ radiative decays. We will test the UA(1) anomaly dominance in this process by looking at the Nf dependence of the partial decay width, as the UA(1) anomaly is proportional to Nf. In the meantime, we will explore the q2-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.

    II.   NUMERICAL DETAILS

      A.   Nf = 1 gauge ensemble

    • We generate gauge configurations with Nf=1 dynamical strange quarks on an L3×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 the Nf=1 gauge configurations. The parameters in the action are tuned to give the anisotropy ξ=as/at5, where at and as are the temporal and spatial lattice spacings, respectively. The scale setting takes the following procedure. Experimentally, there is an interesting relation between pseudoscalar meson masses mPS and the vector meson masses mV of the quark configuration qlˉq:

      Δm2m2Vm2PS0.560.58GeV2,

      (2)

      where ql represents the u, d, s quarks, and q represents u,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 parameter at. We make the least squares fitting to the mass squared differences over the nˉn, nˉs, nˉc, and sˉc systems, where n refers to the u,d quarks, and get the mean value ¯Δm2=0.568(8) GeV2, which serves as an input to give the lattice scale parameter a1t=6.66(5) GeV. As the HPQCD collaboration determined the sˉs pseudoscalar meson mass to be mηs=0.686(4) GeV from the connected quark diagram [27], we use the ratio mϕ/mηs= 1.487(9) to set the bare mass parameters of strange quarks. Although ηs is not a physical state, the mass squared difference m2ϕm2ηs0.570GeV2 also satisfies the empirical relation of Eq. (2). Finally, we obtain mηs=693.1(3)(6.0) MeV, mϕ=1027.2(5)(7.7) MeV, and m2ϕm2ηs=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/GeV mPS/GeV m2Vm2PS/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, and sˉb [8] , where n refers to the u,d quarks. The rightmost column lists the m2Vm2PS(GeV2) values. In the row of sˉs states, the mass of the sˉs pseudoscalar ηs is determined by the HPQCD collaboration from lattice QCD calculations [27].

      L3×T β a1t/GeV ξ mηs/MeV mϕ /MeV Ncfg
      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 the NV=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)=VVψ, where ψ(s) is the smeared quark field of ψ, 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.

    • B.   Pseudoscalar meson η(1)

    • We use two interpolation operators for η(1), namely, Oγ5=ˉs(s)γ5s(s) and Oγ4γ5=ˉs(s)γ4γ5s(s), to calculate the correlation functions Cγ5γ5(t) and C(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 as a5(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 [2830]. In contrast, C(γ4γ5)(γ4γ5)(t) damps to zero for large t, which is the normal large t behavior. The constant term of Cγ5γ5(t) can be subtracted by taking the difference

      Figure 1.  (color online) Lattice results of the energies of η(1). Left panel: effective mass of two-point correlation functions of η(1) and ηs, where η(1) uses the operators Oγ5 and Oγ4γ5. The green points are from the original two-point function of Oγ5, the blue points are from the subtracted two-point function in Eq. (3) with t0=3at, and the red points are from the two-point function of Oγ4γ5. The plateau regions of red and blue points merge together in the large t/at range. The purple points represent unphysical ηs, which only include connected diagram. Middle panel: effective energies E(q)η(1)(t) with momentum mode n of q up to |n|2=9, where the data points are from the correlation functions of Oγ4γ5(q). The fitted E(q)η(1) values are given in Table 3. Right panel: dispersion relation of η(1). The grey band illustrates the dispersion relation in Eq. (9) with the fitted ξ=4.88(1) and χ2 per degree of freedom χ2/d.o.f=0.32.

      Cγ5γ5(t)=Cγ5γ5(t)Cγ5γ5(t+t0),

      (3)

      and we take t0=3at in practice. The effective mass functions meff(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 when t/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 mass mηs of ηs. The data analysis gives the results

      mηs=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 value mηs=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 meson η(1).

    • C.   Form factor for J/ψγη(1)

    • The transition matrix element M for the process J/ψγη(1) can be expressed in terms of one form factor M(q2), namely,

      Mμψη(1)γη(1)(pη)|jμem(0)|ψ(pψ,λ)=M(q2)ϵμνρσpψ,νpη,ρϵσ(pψ,λ),

      (5)

      where q2=(pψpη)2 is the virtuality of the photon, ϵσ(pJ/ψ,λ) is the polarization vector of J/ψ, and jμem=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 element M is encoded in the following three-point functions:

      Cμi(3)(q;t,t)=yeiqyOη(p,t)jμem(y,t)Oi,ψ(p,0)

      (6)

      with q=pp, where Oη(p,t) and Oiψ(p,t) are the interpolating field operators for η(1) and J/ψ with spatial momenta p and p, respectively. For tt, t0, and in the rest frame of J/ψ (p=0), the explicit spectral expression of Cμi(3)(q;t,t) reads

      Cμi(3)(q;t,t)Zη(q)Zψ4V3Eη(q)mψeEη(q)(tt)emψt×λη(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)and Zψϵi(0,λ)=Ω|Oiψ(0)|J/ψ(0,λ). Note that Zη has a q dependence due to the smeared operator Oη [31]. The parameters mψ, Eη(q), Zψ, and Zη(q) can be derived from the two-point correlation functions

      C(2),η(q,t)12Eη(q)V3|Zη(q)|2eEη(q)t,Cii(2),ψ(t)12mψV3|ZJ/ψ|2emψt.

      (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 current jμem(x)=[ˉcγμc](x) mentioned above (the charm quark field c and ˉc are the original field, which are not smeared) is not conserved anymore on the finite lattice and should be renormalized. We determine the renormalization factors ZtV=1.147(1) and ZsV=1.191(2) for the temporal and spatial components of jμem(x), respectively, by calculating the relevant electromagnetic form factors of ηc [32, 33]. In practice, only ZsV is involved and is incorporated implicitly in jμem(x). We use the operator Oγ4γ5=ˉs(s)γ4γ5s(s) for Oη, and Oiψ takes the form ˉc(s)γic(s) in Eq. (6). The three-point function Cμi(3)(q;t,t) is calculated in the rest frame of J/ψ (p=0) such that η(1) moves with spatial momentum p=q. The right panel of Fig. 1 shows the dispersion relation of η(1)

      a2tE2η(q)=a2tm2η+1ξ2(2πL)2|n|2,

      (9)

      where n represents the momentum mode of q=2πLasn. It can be seen that E2η(q) exhibits a perfect linear behavior in |q|2 up to |n|2=9 and the fitted slope gives ξ=4.88(1), which deviates from the renormalized anisotropy ξ5.0 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 part Gμi comes from the contraction of Oiψ and the current jμem, namely,

      Figure 2.  (color online) Schematic diagram for the process J/ψγη(1).

      Gμi(p,q;t+τ,τ)=yeiqyjμem(y,t+τ)Oiψ(p,τ),

      (10)

      and is dealt with by the distillation method [34]. Considering Oiψ(p,t)=yeipy[ˉc(s)γic(s)](y,t), the explicit expression of Gμi at the source time slice τ=0 is

      Gμi(p,q;t,0)=xeiqxTr{γ5[ScV(0)](x,t)γ5γμ×[ScV(0)](x,t)[V(0)D(p)γiV(0)]},

      (11)

      where Sc=cˉcU is the all-to-all propagator of charm quark for a given gauge configuration U , and D(p) is a 3L3×3L3 diagonal matrix with diagonal elements δabeipy (y labels the column or row indices, and a,b=1,2,3 refer to the color indices). The γ5-hermiticity Sc=γ5Scγ5 implies [V(0)Sc](x,t)=γ5[ScV(0)](x,t)γ5, such that only ScV(0) is required, while ScV(0) can be obtained by solving the system of linear equations

      M[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 by M[U;mc]x=b is applied 4NV times for Dirac indices α=1,2,3,4 and all columns of V(0)). To increase the statistics, the above procedure runs over the entire time range, say, τ[0,T1]. Averaging over τ[0,T1] improves the precision of the calculated Cμi(3) drastically.

      It is observed that theJ/ψ contribution dominates Cμi(3)(q;t,t) when t>40. Combining Eqs. (5), (7), and (8), we have the expression

      Rμi(q;t,t)ZψZη(q)Cμi(3)(q;t,t)V3C(2),η(q,tt)C(2),ψ(t)M(q2;tt)ϵμijqj

      (13)

      for fixed t/at=40, from which we obtain M(q2,tt) for each q2. Figure 3 shows the tt dependence of M(q2,tt) at several q2 close to q2=0. It can be seen that a plateau region appears beyond tt>10 for each q2, where M(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 form M(q2,tt)=M(q2)+c(q2)eδE(tt), with the exponential term being introduced to account for the higher state contamination. The fitted values of M(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 of M(q2). The derived M(q2) up to q2=4.3GeV2 data points are listed in Table 3.

      Figure 3.  (color online) Fit of form factor M(q2) for J/ψγη(1). The lattice data are plotted as data points, and the grey bands show the fit by constants to the plateau regions. The fitted M(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)/GeV1 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 factor M(q2). The momentum modes n represent the relation q=2πLn. The two-point function CΓΓ(t) and three-point ratio function Rμi(q;t,t) corresponding to the same momentum mode n have been averaged for increasing signal of the energy E(q)η(1) and form factor M(q2), respectively.

      Instead of a polynomial function form used by Ref. [19], we use the single-pole model to describe the q2 dependence of M(q2)

      M(q2)=M(0)1q2/Λ2M(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 parameters

      Figure 4.  (color online) The form factor M(q2) for J/ψγη(1,2). The data points are the lattice QCD results, and the shaded bands illustrate the fit model M(q2)=M(0)1q2/Λ2 with the best fit parameters M(0)=0.01066(36)GeV1 for Nf=1 and Λ=2.442(36)GeV for Nf=2. The M(q2) data of J/ψγη(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 fit M(q2)=M(0)+aq2+bq4.

      M(0)=0.00498(36)GeV1,Λ=2.44(60)GeV.

      (15)
    III.   DISCUSSION

      A.   The partial decay width of J/ψγη(1)

    • The partial decay width Γ(J/ψγη(1)) is dictated by the on-shell form factor M(q2=0) through the relation

      Γ(J/ψγη(1))=4α27|M(0)|2|pγ|3,

      (16)

      where the electric charge of charm quark Q=+2e/3 has been incorporated, αe2/(4π)=1/134 is the fine structure constant at the charm quark mass scale, and |pγ|=(m2ψm2η(1))/2mψ is the on-shell momentum of the photon. Using the value of M(0) in Eq. (15), the partial decay width and corresponding branching fraction are predicted as

      Γ(J/ψγη(1))=0.087(13)keVBr(J/ψγη(1))=0.93(14)×103,

      (17)

      where the experimental value ΓJ/ψ=92.6keV is used. Obviously, Br(J/ψγη(1)) is four or five times smaller than Br(J/ψγη(2))=4.16(49)×103 in the Nf=2 case of mπ350MeV [19] and the experimental value Br(J/ψγη)=5.25(7)×103 [8].

      This large difference can be attributed to the dependence of quark masses and the flavor number Nf. The decay process J/ψγη(Nf) takes place in the procedure that the cˉc pair annihilates into gluons (after a photon radiation), which then convert into η(Nf). There are two mechanisms for gluons to couple to η(Nf). The first is the UA(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=1NfNfk=1ˉqkγ5γμqk is the flavor singlet axial vector current for Nf flavor quarks, and q(x) is the topological charge density. The UA(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 [3542]. 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)=1Nffη(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, where χtop 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 the UA(1) anomaly dominates the production of η(Nf) in the process J/ψγη(Nf), then one expects the Nf dependence for the partial decay width

      Γ(J/ψγη(Nf))|0|q(0)|η(Nf)|2Nff2η(Nf).

      (20)

      This kind of Nf dependence can be tested in a Nf=3 lattice QCD calculation. Note that fη(Nf) has quark mass dependence for massive quarks and becomes larger when quark mass increases. Thus, Eq. (20) partially explains the small value of Γ(J/ψγη(1)) we obtain at the strange quark mass in Nf=1 QCD.

      There may be also other sources for the quark mass dependence. First, an additional term 1NfNfk=12imkˉqkγ5qk appears on the right-hand side of Eq. (18) when the quark masses mk0. Secondly, the coupling of perturbative gluons to η(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.

    • B.   Dalitz decay form factors J/ψPl+l

    • The form factor M(q2) in Eq. (14) is actually the TFF for the Dalitz decay J/ψη(1)l+l when q2>4m2l, which is seen to be well described by the single pole model with Λ=2.439(60)GeV. In Ref. [19], the Dalitz TFF M(q2) is also obtained in the Nf=2 lattice QCD at mπ350MeV, and the value M(0) is interpolated using a polynomial function form, namely, M(q2)=M(0)+aq2+bq4. We refit the q2-dependence of M(q2) in Ref. [19] using the same single-pole model and present the result in Fig. 4, where the polynomial fits to both the Nf=1 data in this work and the Nf=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 for Nf=1 and 0.32 for Nf=2) of χ2/d.o.f than the polynomial fit (1.00 for Nf=1 and 0.86 for Nf=2), even though the latter has one more parameter. Especially for the Nf=2 case, the pole model with Λ=2.442(36)GeV describes the whole q2 range very well. This indicates that the single-pole model is suitable for describing the Dalitz decay TFFs of J/ψ to light pseudoscalar mesons P.

      In experiments, the TFF FψP is extracted from the ratio

      dΓ(ψPl+l)/dq2Γ(ψPγ)=A(q2)|FψP(q2)|2,

      (21)

      where A(q2) is a known kinematic factor [1, 15, 16]

      A(q2)=α3π1q2(14m2lq2)1/2(1+2m2lq2)×[(1+q2m2ψm2P)24m2ψq2(m2ψm2P)2]3/2

      (22)

      derived from the QED calculation. BESIII has measured many Dalitz decay processes of J/ψPe+e with P=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 the J/ψη,η Dalitz decays are compatible with the lattice values, the values of Λ for J/ψη(1405),X(1835) are substantially smaller. Thus, it is possible that Λ depends on the mass of the final state pseudoscalar meson.

      Ve+eP Λ/GeV Ref.
      J/ψe+eη 2.56±0.04±0.03 [10]
      J/ψe+eη 3.1±1.0 [9]
      J/ψe+eη(1405) 1.96±0.24±0.06 [13]
      J/ψe+eX(1835) 1.75±0.29±0.05 [14]
      J/ψγη(2)(718) 2.44±0.04 [19]
      J/ψγη(1)(783) 2.44±0.06 this work

      Table 4.  Values of the pole parameter Λ of the TFF for different Dalitz decays J/ψPe+e. The Nf=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 the cˉc pair emits a photon of the virtuality q2 and then annihilates into gluons, whose invariant mass squared is labelled as s. As the single-pole model describes M(q2) very well while the q2 and s in the J/ψγ(q2)(gg)(s) vertex are correlated, one expects the s-dependence of Λ. We assume an empirical linear function for Λ(s), namely,

      Λ(s)=Λ1(1sΛ22),

      (23)

      and then use the experimental values of Λ at different s=m2P in Table 4 to determine the parameters Λ1 and Λ2. Finally, we get

      Λ1=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 at Nf=1,2 reside almost entirely on the fitting curve. Actually, the function in Eq. (23) with fitted parameters Λ1,2 gives the interpolated values Λ(s=m2η(1))=2.465(40) and Λ(s=m2η(2))=2.495(42), which are in good agreement with the lattice QCD results 2.44(4) GeV and 2.44(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), where mP is the mass of the psuedoscalar meson in the process J/ψPγ. The shaded blue band shows the fitting of the experimental data using the model Λ(s)=Λ1(1sΛ22) with fitted parameters Λ1=2.65(5)GeV and Λ2=2.90(35)GeV. The χ2/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 process J/ψηe+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 prediction

      Λ(s=m2η)=2.37(5)GeV.

      (25)

      Then, according to Eq. (21) and using the experimental result of Br(J/ψγη)=5.25(7)×103, the branching fraction of J/ψηe+e is estimated to be 6.04(4)(8)×105, which is compatible with the BESIII result 6.59(7)(17)×105 [12]. When the ρ resonance contribution is included, as it was by BESIII for J/ψηe+e in Ref. [10], |Fψη(q2)|2 reads

      |Fψη(q2)|2=|Aρ|2(m4ρ(q2m2ρ)2+m2ρΓ2ρ)+|AΛ|2(11q2/Λ2)2,

      (26)

      where Aρ is the coupling constant of the ρ meson, and AΛ is the coupling constant of the non-resonant contribution. For J/ψηe+e, BESIII determines Aρ=0.23(4) and AΛ=1.05(3) [10], which give |Fψη(q20)|2=1.11±0.07±0.07. If we take the same value for Aρ=0.23(4) and assume AΛ=1 for the case of η (the |Fψη(q2)|2 at q20 in Ref. [9] is consistent with one within errors), then using the PDG values of mρ and Γρ [47], we get

      Br(J/ψηe+e)=6.57+2017(4)(9)×105,

      (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 of Br(J/ψγη). This value agrees with the experimental value better.

    IV.   SUMMARY
    • We generate a large gauge ensemble with Nf=1 dynamical strange quarks on an anisotropic lattice with the anisotropy as/at5.0. The pseudoscalar mass is measured to be mηs=693.1(3)MeV without considering the quark annihilation effect, and mη(1)=783.0(5.5)MeV with the inclusion of quark annihilation diagrams. We calculate the EM form factor M(q2) for the decay process J/ψγ(q2)η(1) with q2 being the virtuality of the photon. By interpolating M(q2) to the value at q2=0 through the VMD-inspired single-pole model in Eq. (14), the decay width and branching fraction of J/ψγη(1) are predicted to be Γ(J/ψγη(1))=0.087(13)keV and Br(J/ψγη(1))=0.93(14)×103, respectively, which are much smaller than those from previous Nf=2 lattice QCD results [19] and the experimental results for J/ψγη. This large difference can be attributed in part to the Nf dependence owing to the UA(1) anomaly and quark mass dependence.

      It is interesting to see that M(q2) in both Nf=1,2 is well described by the single-pole model M(q2)=M(0)/(1q2/Λ2). By assuming an empirical s=m2P dependence Λ(m2P)=Λ1(1m2P/Λ22), the parameters Λ1,2 are determined to be Λ1=2.65(5)GeV and Λ2=2.90(35)GeV by using the experimental results of the Dalitz decays J/ψPe+e with P being light pseudoscalar mesons. This parameterization of Λ(m2P) gives a consistent result with lattice QCD data at the same mP and may provide meaningful inputs for future theoretical and experimental studies on Dalitz decays J/ψPe+e. As a direct application, this mP dependence expects a pole parameter Λ(s=m2η)=2.37(5)GeV, which is more precise than the value 3.1(1.0)GeV measured by BESIII [9] and provides a more precise prediction for Br(J/ψηe+e).

Reference (49)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return