Express Letter

Empirical Determination of the Pion Mass Distribution

  • Corresponding author:

    C. D. Roberts, E-mail: cdroberts@nju.edu.cn

  • Received Date: March 01, 2023
  • Published Date: March 19, 2023
  • Existing pion+nucleus Drell-Yan and electron+pion scattering data are used to develop ensembles of model-independent representations of the pion generalized parton distribution (GPD). Therewith, one arrives at a data-driven prediction for the pion mass distribution form factor, θπ2. Compared with the pion elastic electromagnetic form factor, θπ2 is harder: the ratio of the radii derived from these two form factors is rθ2π/rπ=0.79(3). Our data-driven predictions for the pion GPD, related form factors and distributions should serve as valuable constraints on theories of pion structure.
  • Article Text

  • Science is operating, building, and planning high-luminosity, high-energy facilities in order to reveal the source of the mass of visible material in the Universe.[110] That matter is chiefly built from the nuclei that can be found on Earth, which are themselves composed of neutrons and protons (nucleons) bound together by the exchange of π-mesons (pions)[11] – and other contributors at shorter ranges. Partly, therefore, some of the mass is generated by Higgs boson couplings to matter fields in the Standard Model (SM).[1215]

    However, insofar as nucleons and pions are concerned, the Higgs-generated mass component is only a small part. Regarding the nucleon – a bound state with mass mN ≈ 940 MeV, yet built from three light valence quarks, ∑mq ≈ 9 MeV, Higgs boson couplings are directly responsible for ≲ 1% of its mass: the remainder has its origin in some other mechanism. The story for the pions, a triplet of three electric charge states, whose positive member is a bound state of a valence u quark and a valence ˉd quark, is very subtle because pions are the SM’s (would-be) Nambu-Goldstone bosons whose emergence can be traced to exactly the same source as 99% of the nucleon mass, whatever that may be. This second source is called emergent hadron mass (EHM).[1626]

    Modern theory indicates[1626] that EHM is a feature of SM strong interactions, i.e., quantum chromodynamics (QCD). Further, that owing to its Nambu-Goldstone boson character, expressed in QCD symmetry identities,[2729] pion properties provide the clearest window onto EHM.[1618,20,21] It is thus imperative that theory provide constraints on the distribution of mass within the pion in advance of new facility operation, so forthcoming experiments can truly test the EHM paradigm, avoiding a common pattern of models being developed to describe novel data after they have been collected. To this end, herein, we use existing data on pion valence quark distribution functions (DFs)[3033] and the pion elastic electromagnetic form factor[3440] to develop data-based predictions for the three-dimensional (3D) structure of the pion and, therefrom, the pion mass distribution.

    Pion Valence-Quark Distribution Function and the Hadron Scale. Consider the pion valence u quark DF, uπ(x;ζ), which is the probability density for finding a valence u quark with light-front momentum fraction x of the pion’s total momentum when the pion is probed at energy scale ζ.[41] In the G-parity symmetry limit, which is an accurate reflection of Nature, ˉdπ(x;ζ)=uπ(x;ζ).

    Extant data relevant to extraction of uπ(x;ζ) has been obtained using the Drell-Yan process[42,43]π + A+ + X, where is a lepton, A is a nuclear target, and X denotes the debris produced by the deeply inelastic reaction.[3033] New data will be obtained using Drell-Yan[1,2] and other processes.[37,21]

    The most malleable set of Drell-Yan data[33] was collected at a large resolving scale, viz. ζ5 = 5.2 GeV. In this case, QCD perturbation theory can be employed in the analysis and a reasonable interpretative basis is provided by the quark and gluon parton degrees-of-freedom used to define the QCD Lagrangian density.

    On the other hand, development of a theory prediction for uπ(x;ζ) is best begun at the hadron scale, ζζ5, whereat dressed-quark and dressed-antiquark quasiparticle degrees-of-freedom can be used to deliver symmetry-preserving, parameter-free predictions for pion properties.[4448] Of particular importance and utility is the fact that the bound-state’s quasiparticle degrees-of-freedom carry all measurable properties of the hadron at this scale, including the light-front momentum; hence:[46,47]
    uπ(x;ζ)=uπ(1x;ζ).

    (1)

    In QCD perturbation theory, the evolution of uπ(x;ζ) with changing scale, ζ, is described by the DGLAP equations.[4952] A nonperturbative extension is explained in Refs. [5355]. It is based on the following proposition:

    Such charges are discussed elsewhere.[5658] They need not be process-independent (PI); hence, not unique. Nevertheless, an efficacious PI charge is not excluded; and that discussed in Refs. [48,53,59] has proved suitable. Connections with experiment and other nonperturbative extensions of QCD’s running coupling are given in Refs. [6062].

    Experiment and theory can now be joined at ζ = ζ because Proposition 1 and Eq. (1) entail the following evolution equation:[59]ζζ,
    xnζuπ=xnζuπ(2xζuπ)γn0/γ10,

    (2)
    where the DF Mellin moments are
    xnζuπ:=xnuπ(x;ζ)=10dxxnuπ(x;ζ)

    (3)
    and, with nf = 4 flavors of active quarks,
    γn0=43(3+2(n+1)(n+2)4n+1j=11j).

    (4)
    Owing to Eq. (1), 2xζuπ=1.

    Using Eq. (2), any set of valence-quark DF Mellin moments at a scale ζ can be converted into an equivalent set of hadron-scale moments; hence, one has a direct mapping between the pion valence-quark DF at any two scales. Crucially, the leading moment, 2xζuπ, is the kernel of the mapping; so, although existence of α1(k2) is essential, its pointwise form is largely immaterial.

    If one analyzes the data in Ref. [33] using methods that ensure consistency with QCD endpoint (x ≃ 0, 1) constraints, then realistic connections with SM properties can be drawn.[54,55] Two such studies are available: Ref. [63] whose result we label uπA(x;ζ5); and the Mellin–Fourier analysis in Ref. [64] reappraised in Ref. [54] and labelled uπB(x;ζ5) herein.

    Following Steps 1 and 2, one arrives at the ζ = ζ representation ensembles of E615 data drawn in Fig. 1. Evidently, the data-driven results are a fair match with modern theory predictions, albeit those based on the analysis in Ref. [63] deliver better agreement.

    Fig. Fig. 1.  Ensembles of ζ5ζ pion valence-quark DF replicas: uπA(x;ζ)[63] – blue band; uπB(x;ζ)[54] – orange band. Comparison curves: dashed purple – pion valence-quark DF calculated using continuum Schwinger function methods (CSMs);[48] grey band – ensemble of valence-quark DFs developed in Ref. [55] from results obtained using lattice Schwinger function methods.[6567]

    Reconstructing the GPD. Generalized parton distributions (GPDs) are discussed in Refs.[6870]. They provide an extension of one-dimensional (1D – light-front longitudinal) DFs into 3D images of hadrons because they also express information about the distribution of partons in the plane perpendicular to the bound-state’s total momentum, i.e., within the light front. Data that may be interpreted in terms of GPDs can be obtained via deeply virtual Compton scattering on a target hadron, T, viz. γ*(q) T(p) → γ*(q′)T(p′), so long as at least one of the photons [γ*(q), γ*(q′)] possesses large virtuality, and in the analogous process of deeply virtual meson production: γ*(q) T(p) → M(q′)T(p′), where M is a meson. Moreover, GPDs connect DFs with hadron form factors because any DF may be recovered as a forward limit (p′ = p) of the relevant GPD and any elastic form factor can be expressed via a GPD-based sum rule.

    Exploiting this last feature, the valence-quark DFs obtained above can be used to develop a 3D image of the pion by working with the light-front wave function (LFWF) overlap representation of GPDs. Capitalizing on the property of LFWF factorization, which is an excellent approximation for the pion,[59,71,72] one may write the |x|≥|ξ| pion GPD as follows:
    Huπ(x,ξ,Δ2;ζH)=θ(x)uπ(x;ζH)uπ(x+;ζH)Φπ(z2;ζH),

    (9)
    where P = (p + p′)/2; Δ = p′−p; ξ = − [nΔ] / [2 nP], with n a light-like four-vector, n2 = 0; x± = (x ± ξ)/(1 ± ξ); and, with mπ being the pion mass,
    z2=Δ2(1x)2/(1ξ2)2,

    (10a)
    Δ2=Δ2(1ξ2)+4ξ2m2π.

    (10b)
    In principle, the shape of Φπ(z;ζH) is determined by the pion LFWF – see Ref. [59][Eq. (18b)]. However, in our data-driven approach, we exploit the following GPD sum rule, which relates the zeroth moment of the GPD to the pion elastic electromagnetic form factor:
    Fπ(Δ2)Fuπ(Δ2)=11dxHuπ(x,0,Δ2;ζH)

    (11a)
    =10dxuπ(x;ζH)Φπ(Δ2[1x]2;ζH),

    (11b)
    where we have used Eqs. (10). Charge and baryon number conservation entail Φπ(0;ζH) ≡ 1.

    At this point, one can use available precision data on the pion elastic form factor[3640] and a recent objective determination of the pion charge radius[73] to complete a reconstruction of Huπ(x,0,Δ2;ζH). To achieve this, we proceed as follows.

    In concert with this rπ distribution, the values of (β,γ) are now varied at random about the best-fit values obtained in Step 3; and one selects a data-driven GPD, H, using Eqs. (9)–(12), with probability PH=PH|uPu, where both the conditional probability of the GPD given a particular DF, PH|u, and that of the DF, Pu, are given by Eq. (7) evaluated with their respective χ2 functions, defined from DF and elastic form factor data.

    This final step is repeated for every member of each DF ensemble. We thereby arrive at data-driven GPDs specified by Eqs. (8), (9), (12) and the parameters in Table 1. A check on the Fπ(Δ2) part of the procedure is provided by Fig. 2, which displays the pion electromagnetic form factor obtained via Eqs. (11) using these pion GPD ensembles. Once again, the data-driven results are in accord with modern theory predictions.

      Table 1.  Data-constrained GPD characterization parameters, Eqs. (8) and (12), obtained using Step 4. Regarding the lQCD row, derived from the valence-quark DF ensemble in Ref. [55]: given the range of ρ variation in the ensemble, the random selection of replicas was made from log ρ ∈ (−3,3), with the corresponding standard deviation translated into an asymmetric uncertainty. (Each DF ensemble has K ≳ 100 elements: A derived from Ref. [63] and B from Ref. [54]. Values of rπ are displayed and confirm that rπ was generated with a Gaussian probability centered at 0.64 fm[73] with width 0.02 fm.)
    rπ (fm)ˉρ(δρ)ˉβ(δβ) (GeV)ˉγ(δγ) (GeV)
    A0.640(20)0.060(16)6.30(51)5.20(50)
    B0.638(18)0.025(7)6.14(46)6.15(60)
    lQCD0.639(19)0.041+0.0540.0236.34(52)6.10(91)
     | Show Table
    DownLoad: CSV
    Fig. Fig. 2.  Pion elastic electromagnetic form factor, Fπ(Δ2), obtained from Eqs. (11) using the GPD ensembles generated via Step 4. (a) DFs uπA(x;ζ)[63] (blue band). (b) DFs uπB(x;ζ)[54] (orange band). Comparison curves: dashed purple – Fπ(Δ2) calculated using CSMs;[21,45] grey band – Fπ(Δ2) ensemble obtained with valence-quark DFs developed in Ref. [55] from results obtained using lattice Schwinger function methods.[6567] The form factor data are taken from Refs.[3640].

    Pion Generalized Parton Distribution. Pion GPDs, reconstructed, as described above, from available analyses of relevant Drell-Yan and electron+pion scattering data, are drawn in Fig. 3. Although the different ensembles are only marginally compatible with each other, owing to differences between the analyses in Refs. [63,64] they both agree with the lQCD based ensembles, within mutual uncertainties, because the lQCD-constrained ensemble possesses a large uncertainty. The CSM prediction favors the uπA(x;ζ) ensemble. In all cases, one sees that the support of the valence-quark GPD becomes increasingly concentrated in the neighborhood x ≃ 1 with increasing Δ2. Namely, greater probe momentum focuses attention on the domain in which one valence-quark carries a large fraction of the pion’s light-front momentum.

    Fig. Fig. 3.  Pion GPDs. (a) Working with DFs uπA(x;ζ)[63] – blue band. (b) Using DFs uπB(x;ζ)[54] – orange band. Comparison curves, both panels: CSM prediction in Refs. [59,72] – dashed purple curve; GPD ensemble generated from valence-quark DFs developed in Ref. [55] obtained from results computed using lattice Schwinger function methods[6567] – grey band.
    Pion Mass Distribution. The first Mellin moment of the ξ = 0 GPD is the mass distribution form factor:
    θπ2(Δ2)=11dx2xHuπ(x,0,Δ2;ζ),

    (13)
    which is a principal, dynamical coefficient in the expectation value of the QCD energy-momentum tensor in the pion.[77] Noticeably, θπ2(Δ2) is ζ-independent but the manner by which its strength is shared amongst different parton species does evolve with ζ. Herein we exploit the fact that, at ζ = ζ, θπ2(Δ2) is completely determined by the contribution from dressed valence degrees-of-freedom.

    The pion mass distributions obtained from the data-driven GPDs are depicted in Fig. 4. Plainly, the mass distribution form factor is harder than the elastic electromagnetic form factor, i.e., the distribution of mass in the pion is more compact than the distribution of electric charge. Since every curve herein is built solely from available data, then this is an empirical fact. It is also readily understood theoretically, as we now explain.

    Fig. Fig. 4.  Pion mass distribution form factor, θπ2(Δ2). (a) Developed from the uπA(x;ζ) ensemble[63] – blue band. (b) Developed from the uπB(x;ζ) ensemble[54] – orange band. Comparison curves, both panels: CSM prediction for θπ2(Δ2) in Refs. [59,72] – solid purple; GPD ensemble generated from valence-quark DFs developed in Ref. [55] using lQCD results[6567] – grey band. In addition, each panel displays the CSM prediction for Fπ(Δ2)[21,45] – dashed purple curve. The data are those for Fπ(Δ2) from Refs. [3640] included so as to highlight the precision required to distinguish the mass and electromagnetic form factors.
    Beginning with Eqs. (11) and (13), consider the difference between mass and charge distributions,
    θπ2(Δ2)Fπ(Δ2)=10dˉx(12ˉx)uπ(ˉx;ζ)Φπ(Δ2ˉx2;ζH),

    (14)
    where ˉx=1x and we have used Eq. (1) to simplify the right-hand side (rhs). For Δ2 = 0, the rhs is zero as a consequence of charge and baryon-number conservation. Suppose now that Φπ(z;ζH) is a non-negative, monotonically decreasing function of its argument, as required to produce a realistic pion electromagnetic form factor, then it is straightforward to establish that
    Δ2>0,θπ2(Δ2)Fπ(Δ2)>0.

    (15)
    Consequently, the pion’s mass distribution is more compact than its charge distribution in any realistic description of pion properties; in particular, the mass radius is smaller than the charge radius: (rθ2π)2<r2π, as demonstrated previously in Ref. [59][Eq. (41)].
    Using Eq. (13) and the ensemble characterization parameters in Table 1, one finds (in fm):
    ABlQCDrθ2π0.518(16)0.498(14)0.512(21),

    (16)
    to be compared with rπ/fm = 0.64(2). These results yield the data-driven prediction rθ2π/rπ=0.79(3), which translates into a spacetime volume ratio of 0.39(6).

    Considering Ref. [59][Eq. (41)], it is plain that the radii and associated uncertainties in Eq. (16) are determined by knowledge of rπ and the DF ensembles. This simplicity is a feature of the factorized Ansatz for the LFWF. As evident from Ref. [59][Fig. 6A], on the domain Δ2r2π0, relevant to radius determination, there is practically no difference between pion GPDs obtained using factorized and nonfactorized LFWFs. Hence, in the context of Eq. (16), the factorized representation is a reliable approximation.

    Exploiting these features further and adapting the arguments that lead to Ref. [55][Eq. (7)], one arrives at the following constraints on the radii ratio for pion-like bound-states:
    12rθ2π/rπ1,

    (17)
    where the lower bound is saturated by a point-particle DF, u(x;ζ)=θ(x)θ(1x), and the upper by the DF of a bound-state formed from infinitely massive valence degrees-of-freedom, u(x;ζ)=δ(x1/2). In the context of Eqs. (16) and (17), we note that an analysis of γ*γπ0π0 data[78] using a generalized distribution amplitude formalism, with subsequent analytic continuation to space-like momenta to enable extraction of a mass radius, yields:[79]rθ2π/rπ=0.49–0.59. Although the radii size ordering is qualitatively similar to our determination, the value of the ratio is smaller and inconsistent with the bounds in Eq. (17). Furthermore, regarding theory, CSMs produce the results drawn in Figs. 2 and 4, which yield rθ2π/rπ=0.81(3), viz. a value consistent with the data-driven result. On the other hand, analyses based on momentum-independent quark+antiquark interactions produce smaller values: rθ2π/rπ=0.71[80] and rθ2π/rπ=0.53.[81] Here, for reasons explained elsewhere,[82] the mutual inconsistency is attributable to the different treatments of the momentum-independent interaction used in those studies. Moreover, Ref. [80] result is consistent with Eq. (17) because it omits resonance pole contributions to both radii and that in Ref. [81] fails the test because it only includes such contributions to rπ.

    It is worth recording here that the pion LFWF, hence GPD, is independent of the probe: it is the same whether a photon or graviton is the probing object. However, the probe itself is sensitive to different features of the target constituents. A target dressed-quark carries the same charge, irrespective of its momentum. So, the pion wave function alone controls the distribution of charge. On the other hand, the gravitational interaction of a target dressed-quark depends on its momentum – see Eq. (13); after all, the current relates to the energy momentum tensor. The pion effective mass distribution therefore depends on interference between quark momentum growth and LFWF momentum suppression. This pushes support to a larger momentum domain in the pion, which corresponds to a smaller distance domain.

    GPD in Impact Parameter Space. An impact parameter space (IPS) GPD is defined via a Hankel transform:
    uπ(x,b2;ζH)=0dΔ2πΔJ0(|b|Δ)Huπ(x,0,Δ2;ζH),

    (18)
    where J0 is a cylindrical Bessel function. This is a density, which reveals the probability of finding a valence-quark within the light-front at a transverse distance |b| from the meson’s center of transverse momentum. Using Eq. (9), Eq. (18) simplifies:
    uπ(x,b2;ζH)=uπ(x;ζH)(1x)2Ψπ(|b|1x;ζH),

    (19a)
    Ψπ(u;ζH)=0ds2πsJ0(us)Φπ(s2;ζH).

    (19b)
    Considering the character of the entries in Eqs. (19), it becomes clear that the global maximum of the IPS GPD is given by the value of uπ(x=1,b2=0;ζH).

    Herein, motivated by the presence of scaling violations in Fπ(Δ2)[7476] and informed by Ref. [59][Eq. (38)] and Ref. [83][Eq. (14)], we ensure the Hankel transforms are finite by matching each member of a form factor ensemble to (y = Δ2) [(1 + a1y)/(1 + a2y + (a3y)2)][(1 + a0y)/(1 + a0y ln [1 + a0y])], with a0,1,2,3 being fitting parameters.

    In depicting the IPS GPD, it is usual to draw 2π|b|uπ(x,b2;ζH). The peak in this function is shifted to |b| > 0 by an amount that reflects aspects of bound-state structure. These features are evident in Fig. 5, which displays the IPS GPDs obtained from both A and B ensembles. The x > 0 locations of the global maxima and their intensities, given as a triplet {x,b/rπ, iπ}, are
    xb/rπiπCSM[59]0.880.133.29A0.89(2)0.10(2)3.21(30)B0.95(1)0.05(1)4.58(50)lQCD0.91(6)0.08(5)4.04(1.67).

    (20)
    The valence IPS GPDs become increasingly broad as x → 0 because, with diminishing x, the valence degree-of-freedom plays a progressively smaller role in defining the pion’s center of transverse momentum.
    Fig. Fig. 5.  Pion GPDs in impact parameter space, displayed as functions of |b| at x = 0.9, 0.8, 0.6, 0.4, 0.2, 0. (a) Working with DFs uπA(x;ζ)[63] – blue band. (b) Using DFs uπB(x;ζ)[54] – orange band. Comparison curves, both panels: CSM prediction in Refs. [59,72] – dashed purple curve; GPD ensemble generated from valence-quark DFs developed in Ref. [55] obtained from results computed using lattice Schwinger function methods[6567] – grey band.
    Density distributions in the light-front transverse plane are obtained by integrating a given IPS GPD and its x-weighted partner over the light-front momentum fraction:
    ρπ{F,θ2}(|b|)=11dx{1,2x}uπ(x,b2;ζH)=0dΔ2πΔJ0(|b|Δ){Fπ(Δ2),θ2(Δ2)}.

    (21)
    Here we have used isospin symmetry to identify the charge density associated with a given valence-quark flavor with the net pion density. Again, these quantities are ζ-independent.

    The light-front transverse densities in Eq. (21) are drawn in Fig. 6. Evidently, consistent with the analysis presented above, the pion’s light-front transverse mass distribution is more compact than the analogous charge distribution. Moreover, the data-built results are consistent with modern theory predictions.

    Fig. Fig. 6.  Light-front transverse density distributions, Eq. (21), built from: uπA(x;ζ) ensemble[63] – (a) (charge = light blue, mass = dark blue); and uπB(x;ζ) ensemble[54] – (b) (charge = light orange, mass = dark orange). Comparison curves, both panels: dashed purple curve – charge density calculated using CSM prediction for Fπ(Δ2)[21,45]; solid purple curve – mass density obtained using CSM prediction for θπ2(Δ2) in Refs. [59,72] silver band – lattice-based charge density ensemble; and grey band – lattice-based mass density ensemble.

    Summary and Perspective. Supposing that there exists an effective charge which defines an evolution scheme for parton distribution functions (DFs) that is all-orders exact[5355] and working solely with existing pion+nucleus Drell-Yan and electron+pion scattering data,[3340] we used a χ2-based selection procedure to develop ensembles of model-independent representations of the three-dimensional pointwise behavior of the pion generalized parton distribution (GPD). These ensembles yield a data-driven prediction for the pion mass distribution form factor, θ2. In comparison with the pion elastic electromagnetic form factor, obtained analogously, θ2 is more compact; and based on extant data, the ratio of the radii derived from these form factors is rθ2π/rπ=0.79(3).

    Our results are reported at the hadron scale, ζ, whereat, by definition, all hadron properties are carried by dressed valence quasiparticle degrees-of-freedom. Nevertheless, using the all-orders evolution scheme, one can readily determine the manner in which these distributions are shared amongst the various parton species at any scale ζ>ζ – see, e.g., Ref. [59].

    Improvement of our data-driven predictions must await new data relevant to pion DFs and improvements in associated analysis methods; and pion form factor data that extends to larger momentum transfers than currently available. Meanwhile, our results for the pion GPD, related form factors and distributions should serve as valuable constraints on modern pion structure theory.

    No similar analysis for the kaon will be possible before analogous empirical information becomes available. In this case, today, the kaon charge radius cannot be considered known[73] because kaon elastic form factor data are sketchy[84,85] and only eight data relevant to kaon valence-quark DFs are available.[86]

    Given the importance of contrasting pion and proton mass distributions in the search for an understanding of emergent hadron mass, completing a kindred analysis that leads to data-driven ensembles of proton GPDs should be given high priority. Precise elastic form factor data are available;[8789] however, despite a wealth of data relevant to proton DFs, improved analyses are required, including, e.g., effects of next-to-leading-logarithm resummation. Meanwhile, one might begin with existing theory predictions that provide a unified description of pion and proton DFs.[90]

    Acknowledgments: We are grateful for constructive comments from G. M. Huber. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12135007 and 12233002), the Natural Science Foundation of Jiangsu Province (Grant No. BK20220122), Spanish Ministry of Science and Innovation (MICINN Grant No. PID2019-107844GB-C22); and Junta de Andalucí a (Grant No. P18-FR-5057).
  • [1]
    Adams B, et al. 2018 arXiv:1808.00848 [hep-ex] https://arxiv.org/abs/1808.00848

    Google Scholar

    [2]
    Quintans C 2022 Few Body Syst. 63 72 doi: 10.1007/s00601-022-01769-7

    CrossRef Google Scholar

    [3]
    Aguilar A C, et al. 2019 Eur. Phys. J. A 55 190 doi: 10.1140/epja/i2019-12885-0

    CrossRef Google Scholar

    [4]
    Brodsky S J, et al. 2020 Int. J. Mod. Phys. E 29 2030006 doi: 10.1142/S0218301320300064

    CrossRef Google Scholar

    [5]
    Chen X R, Guo F K, Roberts C D, Wang R 2020 Few Body Syst. 61 43 doi: 10.1007/s00601-020-01574-0

    CrossRef Google Scholar

    [6]
    Anderle D P, et al. 2021 Front. Phys. Beijing 16 64701 doi: 10.1007/s11467-021-1062-0

    CrossRef Google Scholar

    [7]
    Arrington J, et al. 2021 J. Phys. G 48 075106 doi: 10.1088/1361-6471/abf5c3

    CrossRef Google Scholar

    [8]
    Khalek R A, et al. 2022 Nucl. Phys. A 1026 122447 doi: 10.1016/j.nuclphysa.2022.122447

    CrossRef Google Scholar

    [9]
    Wang R, Chen X R 2022 Few Body Syst. 63 48 doi: 10.1007/s00601-022-01751-3

    CrossRef Google Scholar

    [10]
    Carman D S, Gothe R W, Mokeev V I, Roberts C D 2023 Particles 6 416 doi: 10.3390/particles6010023

    CrossRef Google Scholar

    [11]
    Machleidt R, Entem D R 2011 Phys. Rept. 503 1 doi: 10.1016/j.physrep.2011.02.001

    CrossRef Google Scholar

    [12]
    Aad G, et al. 2012 Phys. Lett. B 716 1 doi: 10.1016/j.physletb.2012.08.020

    CrossRef Google Scholar

    [13]
    Chatrchyan S, et al. 2012 Phys. Lett. B 716 30 doi: 10.1016/j.physletb.2012.08.021

    CrossRef Google Scholar

    [14]
    Englert F 2014 Rev. Mod. Phys. 86 843 doi: 10.1103/RevModPhys.86.843

    CrossRef Google Scholar

    [15]
    Higgs P W 2014 Rev. Mod. Phys. 86 851 doi: 10.1103/RevModPhys.86.851

    CrossRef Google Scholar

    [16]
    Roberts C D 2017 Few Body Syst. 58 5 doi: 10.1007/s00601-016-1168-z

    CrossRef Google Scholar

    [17]
    Roberts C D, Schmidt S M 2020 Eur. Phys. J. ST 229 3319 doi: 10.1140/epjst/e2020-000064-6

    CrossRef Google Scholar

    [18]
    Roberts C D 2020 Symmetry 12 1468 doi: 10.3390/sym12091468

    CrossRef Google Scholar

    [19]
    Krein G, Peixoto T C 2020 Few Body Syst. 61 49 doi: 10.1007/s00601-020-01581-1

    CrossRef Google Scholar

    [20]
    Roberts C D 2021 AAPPS Bull. 31 6 doi: 10.1007/s43673-021-00005-4

    CrossRef Google Scholar

    [21]
    Roberts C D, Richards D G, Horn T, Chang L 2021 Prog. Part. Nucl. Phys. 120 103883 doi: 10.1016/j.ppnp.2021.103883

    CrossRef Google Scholar

    [22]
    Binosi D 2022 Few Body Syst. 63 42 doi: 10.1007/s00601-022-01740-6

    CrossRef Google Scholar

    [23]
    Papavassiliou J 2022 Chin. Phys. C 46 112001 doi: 10.1088/1674-1137/ac84ca

    CrossRef Google Scholar

    [24]
    Ding M H, Roberts C D, Schmidt S M 2023 Particles 6 57 doi: 10.3390/particles6010004

    CrossRef Google Scholar

    [25]
    Roberts C D 2022 arXiv:2211.09905 [hep-ph]

    Google Scholar

    [26]
    Ferreira M N, Papavassiliou J 2023 Particles 6 312 doi: 10.3390/particles6010017

    CrossRef Google Scholar

    [27]
    Bhagwat M S, Chang L, Liu Y X, Roberts C D, Tandy P C 2007 Phys. Rev. C 76 045203 doi: 10.1103/PhysRevC.76.045203

    CrossRef Google Scholar

    [28]
    Brodsky S J, Roberts C D, Shrock R, Tandy P C 2010 Phys. Rev. C 82 022201R doi: 10.1103/PhysRevC.82.022201

    CrossRef Google Scholar

    [29]
    Qin S X, Roberts C D, Schmidt S M 2014 Phys. Lett. B 733 202 doi: 10.1016/j.physletb.2014.04.041

    CrossRef Google Scholar

    [30]
    Corden M, et al. 1980 Phys. Lett. B 96 417 doi: 10.1016/0370-26938090800-X

    CrossRef Google Scholar

    [31]
    Badier J, et al. 1983 Z. Phys. C 18 281 doi: 10.1007/BF01573728

    CrossRef Google Scholar

    [32]
    Betev B, et al. 1985 Z. Phys. C 28 15 doi: 10.1007/BF01550244

    CrossRef Google Scholar

    [33]
    Conway J S, et al. 1989 Phys. Rev. D 39 92 doi: 10.1103/PhysRevD.39.92

    CrossRef Google Scholar

    [34]
    Amendolia S R, et al. 1984 Phys. Lett. B 146 116 doi: 10.1016/0370-26938490655-5

    CrossRef Google Scholar

    [35]
    Amendolia S R, et al. 1986 Nucl. Phys. B 277 168 doi: 10.1016/0550-32138690437-2

    CrossRef Google Scholar

    [36]
    Volmer J, et al. 2001 Phys. Rev. Lett. 86 1713 doi: 10.1103/PhysRevLett.86.1713

    CrossRef Google Scholar

    [37]
    Horn T, et al. 2006 Phys. Rev. Lett. 97 192001 doi: 10.1103/PhysRevLett.97.192001

    CrossRef Google Scholar

    [38]
    Tadevosyan V, et al. 2007 Phys. Rev. C 75 055205 doi: 10.1103/PhysRevC.75.055205

    CrossRef Google Scholar

    [39]
    Blok H P, et al. 2008 Phys. Rev. C 78 045202 doi: 10.1103/PhysRevC.78.045202

    CrossRef Google Scholar

    [40]
    Huber G, et al. 2008 Phys. Rev. C 78 045203 doi: 10.1103/PhysRevC.78.045203

    CrossRef Google Scholar

    [41]
    Holt R J, Roberts C D 2010 Rev. Mod. Phys. 82 2991 doi: 10.1103/RevModPhys.82.2991

    CrossRef Google Scholar

    [42]
    Peng J C, Qiu J W 2016 The Universe 43 34

    Google Scholar

    [43]
    Dove J, et al. 2021 Nature 590 561 doi: 10.1038/s41586-021-03282-z

    CrossRef Google Scholar

    [44]
    Gao F, Chang L, Liu Y X, Roberts C D, Tandy P C 2017 Phys. Rev. D 96 034024 doi: 10.1103/PhysRevD.96.034024

    CrossRef Google Scholar

    [45]
    Chen M Y, Ding M H, Chang L, Roberts C D 2018 Phys. Rev. D 98 091505R doi: 10.1103/PhysRevD.98.091505

    CrossRef Google Scholar

    [46]
    Ding M, Raya K, Binosi D, Chang L, Roberts C D, Schmidt S M 2020 Chin. Phys. C 44 031002 doi: 10.1088/1674-1137/44/3/031002

    CrossRef Google Scholar

    [47]
    Ding M, Raya K, Binosi D, Chang L, Roberts C D, Schmidt S M 2020 Phys. Rev. D 101 054014 doi: 10.1103/PhysRevD.101.054014

    CrossRef Google Scholar

    [48]
    Cui Z F, Ding M, Gao F, Raya K, Binosi D, Chang L, Roberts C D, Rodríguez-Quintero J, Schmidt S M 2020 Eur. Phys. J. C 80 1064 doi: 10.1140/epjc/s10052-020-08578-4

    CrossRef Google Scholar

    [49]
    Dokshitzer Y L 1977 Sov. Phys. JETP 46 641

    Google Scholar

    [50]
    Gribov V N, Lipatov L N 1971 Phys. Lett. B 37 78 doi: 10.1016/0370-26937190576-4

    CrossRef Google Scholar

    [51]
    Lipatov L N 1975 Sov. J. Nucl. Phys. 20 94

    Google Scholar

    [52]
    Altarelli G, Parisi G 1977 Nucl. Phys. B 126 298 doi: 10.1016/0550-32137790384-4

    CrossRef Google Scholar

    [53]
    Cui Z F, Zhang J L, Binosi D, de Soto F, Mezrag C, Papavassiliou J, Roberts C D, Rodríguez-Quintero J, Segovia J, Zafeiropoulos S 2020 Chin. Phys. C 44 083102 doi: 10.1088/1674-1137/44/8/083102

    CrossRef Google Scholar

    [54]
    Cui Z F, Ding M, Morgado J M, Raya K, Binosi D, Chang L, Papavassiliou J, Roberts C D, Rodríguez-Quintero J, Schmidt S M 2022 Eur. Phys. J. A 58 10 doi: 10.1140/epja/s10050-021-00658-7

    CrossRef Google Scholar

    [55]
    Cui Z F, Ding M, Morgado J M, Raya K, Binosi D, Chang L, De Soto F, Roberts C D, Rodríguez-Quintero J, Schmidt S M 2022 Phys. Rev. D 105 L091502 doi: 10.1103/PhysRevD.105.L091502

    CrossRef Google Scholar

    [56]
    Grunberg G 1980 Phys. Lett. B 114 271 doi: 10.1016/0370-2693(82)90494-4

    1982 Erratum: Phys. Lett. B 110 501 doi: 10.1016/0370-26938290494-4

    CrossRef Google Scholar

    [57]
    Grunberg G 1984 Phys. Rev. D 29 2315 doi: 10.1103/PhysRevD.29.2315

    CrossRef Google Scholar

    [58]
    Dokshitzer Y L 1998 arXiv:hep-ph/9812252 https://arxiv.org/abs/hep-ph/9812252

    Google Scholar

    [59]
    Raya K, Cui Z F, Chang L, Morgado J M, Roberts C D, Rodríguez-Quintero J 2022 Chin. Phys. C 46 013105 doi: 10.1088/1674-1137/ac3071

    CrossRef Google Scholar

    [60]
    Deur A, Brodsky S J, de Teramond G F 2016 Prog. Part. Nucl. Phys. 90 1 doi: 10.1016/j.ppnp.2016.04.003

    CrossRef Google Scholar

    [61]
    Deur A, Burkert V, Chen J P, Korsch W 2022 Particles 5 171 doi: 10.3390/particles5020015

    CrossRef Google Scholar

    [62]
    Deur A, Brodsky S J, Roberts C D 2023 arXiv:2303.00723 [hep-ph] https://arxiv.org/abs/2303.00723

    Google Scholar

    [63]
    Aicher M, Schäfer A, Vogelsang W 2010 Phys. Rev. Lett. 105 252003 doi: 10.1103/PhysRevLett.105.252003

    CrossRef Google Scholar

    [64]
    Barry P C, Ji C R, Sato N, Melnitchouk W 2021 Phys. Rev. Lett. 127 232001 doi: 10.1103/PhysRevLett.127.232001

    CrossRef Google Scholar

    [65]
    Joó B, Karpie J, Orginos K, Radyushkin A V, Richards D G, Sufian R S, Zafeiropoulos S 2019 Phys. Rev. D 100 114512 doi: 10.1103/PhysRevD.100.114512

    CrossRef Google Scholar

    [66]
    Sufian R S, Karpie J, Egerer C, Orginos K, Qiu J W, Richards D G 2019 Phys. Rev. D 99 074507 doi: 10.1103/PhysRevD.99.074507

    CrossRef Google Scholar

    [67]
    Alexandrou C, Bacchio S, Cloet I, Constantinou M, Hadjiyiannakou K, Koutsou G, Lauer C 2021 Phys. Rev. D 104 054504 doi: 10.1103/PhysRevD.104.054504

    CrossRef Google Scholar

    [68]
    Belitsky A, Radyushkin A 2005 Phys. Rept. 418 1 doi: 10.1016/j.physrep.2005.06.002

    CrossRef Google Scholar

    [69]
    Mezrag C 2022 Few Body Syst. 63 62 doi: 10.1007/s00601-022-01765-x

    CrossRef Google Scholar

    [70]
    Mezrag C 2023 Particles 6 262 doi: 10.3390/particles6010015

    CrossRef Google Scholar

    [71]
    Xu S S, Chang L, Roberts C D, Zong H S 2018 Phys. Rev. D 97 094014 doi: 10.1103/PhysRevD.97.094014

    CrossRef Google Scholar

    [72]
    Zhang J L, Raya K, Chang L, Cui Z F, Morgado J M, Roberts C D, Rodríguez-Quintero J 2021 Phys. Lett. B 815 136158 doi: 10.1016/j.physletb.2021.136158

    CrossRef Google Scholar

    [73]
    Cui Z F, Binosi D, Roberts C D, Schmidt S M 2021 Phys. Lett. B 822 136631 doi: 10.1016/j.physletb.2021.136631

    CrossRef Google Scholar

    [74]
    Lepage G, Brodsky S J 1979 Phys. Rev. Lett. 43 545 doi: 10.1103/PhysRevLett.43.1625.2

    1979 Erratum: Phys. Rev. Lett. 43 1625 doi: 10.1103/PhysRevLett.43.1625.2

    CrossRef Google Scholar

    [75]
    Efremov A V, Radyushkin A V 1980 Phys. Lett. B 94 245 doi: 10.1016/0370-26938090869-2

    CrossRef Google Scholar

    [76]
    Lepage G P, Brodsky S J 1980 Phys. Rev. D 22 2157 doi: 10.1103/PhysRevD.22.2157

    CrossRef Google Scholar

    [77]
    Polyakov M V, Schweitzer P 2018 Int. J. Mod. Phys. A 33 1830025 doi: 10.1142/S0217751X18300259

    CrossRef Google Scholar

    [78]
    Masuda M, et al. 2016 Phys. Rev. D 93 032003 doi: 10.1103/PhysRevD.93.032003

    CrossRef Google Scholar

    [79]
    Kumano S, Song Q T, Teryaev O V 2018 Phys. Rev. D 97 014020 doi: 10.1103/PhysRevD.97.014020

    CrossRef Google Scholar

    [80]
    Shastry V, Broniowski W, Arriola E R 2022 Phys. Rev. D 106 114035 doi: 10.1103/PhysRevD.106.114035

    CrossRef Google Scholar

    [81]
    Xing Z, Ding M, Chang L 2023 Phys. Rev. D 107 3 L031502 doi: 10.1103/PhysRevD.107.L031502

    CrossRef Google Scholar

    [82]
    Chen C, Chang L, Roberts C D, Wan S L, Schmidt S M, Wilson D J 2013 Phys. Rev. C 87 045207 doi: 10.1103/PhysRevC.87.045207

    CrossRef Google Scholar

    [83]
    Qin S X, Chen C, Mezrag C, Roberts C D 2018 Phys. Rev. C 97 015203 doi: 10.1103/PhysRevC.97.015203

    CrossRef Google Scholar

    [84]
    Dally E, Hauptman J, Kubic J, et al. 1980 Phys. Rev. Lett. 45 232 doi: 10.1103/PhysRevLett.45.232

    CrossRef Google Scholar

    [85]
    Amendolia S, et al. 1986 Phys. Lett. B 178 435 doi: 10.1016/0370-26938691407-3

    CrossRef Google Scholar

    [86]
    Badier J, et al. 1980 Phys. Lett. B 93 354 doi: 10.1016/0370-26938090530-4

    CrossRef Google Scholar

    [87]
    Punjabi V, Perdrisat C F, Jones M K, Brash E J, Carlson C E 2015 Eur. Phys. J. A 51 79 doi: 10.1140/epja/i2015-15079-x

    CrossRef Google Scholar

    [88]
    Gao H, Vanderhaeghen M 2022 Rev. Mod. Phys. 94 015002 doi: 10.1103/RevModPhys.94.015002

    CrossRef Google Scholar

    [89]
    Cui Z F, Binosi D, Roberts C D, Schmidt S M 2022 Chin. Phys. C 46 122001 doi: 10.1088/1674-1137/ac89d0

    CrossRef Google Scholar

    [90]
    Lu Y, Chang L, Raya K, Roberts C D, Rodríguez-Quintero J 2022 Phys. Lett. B 830 137130 doi: 10.1016/j.physletb.2022.137130

    CrossRef Google Scholar

  • Related Articles

    [1]Yunjing Gao, Jianda Wu. Quantum magnetism from low-dimensional quantum Ising models with quantum integrability [J]. Chin. Phys. Lett., 2025, 42(4): 047501. doi: 10.1088/0256-307X/42/4/047501
    [2]LI Zhi-Ming, HAO Yue, ZHANG Jin-Cheng, CHEN Chi, CHANG Yong-Ming, XU Sheng-Rui, BI Zhi-Wei. Optimization and Finite Element Analysis of the Temperature Field in a Nitride MOCVD Reactor by Induction Heating [J]. Chin. Phys. Lett., 2010, 27(7): 070701. doi: 10.1088/0256-307X/27/7/070701
    [3]ZHU Kun-Yan, TAN Lei, GAO Xiang, WANG Daw-Wei. Quantum Fluids of Self-Assembled Chains of Polar Molecules at Finite Temperature [J]. Chin. Phys. Lett., 2008, 25(1): 48-51.
    [4]LI Hong-Qi, XU Xing-Lei, WANG Ji-Suo. Quantum Fluctuations of Current and Voltage for Mesoscopic Quartz Piezoelectric Crystal at Finite Temperature [J]. Chin. Phys. Lett., 2006, 23(11): 2892-2895.
    [5]PAN Rongshi, SU Rukeng, SU Chenggang. Scalar Wormhole at Finite Temperature [J]. Chin. Phys. Lett., 1995, 12(5): 261-264.
    [6]ZHENG Guotong, SU Rukeng. NNδ Interaction at Finite Temperature [J]. Chin. Phys. Lett., 1994, 11(8): 469-471.
    [7]DING Haogang, HUANG Chaoguang, ZHANG Yuan Zhong. On Representation Transformation in Einstein Quantum Cosmology [J]. Chin. Phys. Lett., 1993, 10(4): 201-204.
    [8]XIANG Yingming, LIU Liao. Third Quantization of a Solvable Model in Quantum Cosmology in Brans-Dicke Theory [J]. Chin. Phys. Lett., 1991, 8(1): 52-55.
    [9]ZHU zonghong, HUANG Chaoguang, LIU Liao. A SOLVABLE MODEL IN QUANTUM COSMOLOGY IN THE BRANS-DICKES THEORY [J]. Chin. Phys. Lett., 1990, 7(10): 477-480.
    [10]HUANG Fengyi. POSSIBLE EXPLANATION OF THE PLATEAU WIDTH IN THE QUANTUM HALL EFFECT AT FINITE TEMPERATURE [J]. Chin. Phys. Lett., 1989, 6(12): 541-545.
  • Other Related Supplements

Catalog

    Figures(6)  /  Tables(1)

    Article views (263) PDF downloads (408) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return