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Spatial and Momentum Imaging of the Pion and Kaon
High Energy Physics - Phenomenology · Submitted 6 Mar 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: $\langle \vec{b}_T^2\rangle_{u}^\pi=0.11$fm$^2$, $\langle \vec{b}_T^2\rangle_{s}^K=0.08$fm$^2$, and $\langle \vec{b}_T^2\rangle_{u}^K=0.13$fm$^2$, and therefore the $s$ quark in the kaon is much closer to the center of transverse momentum than the $u$ quark.
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The abstract, sentence by sentence
We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. Using these LFWFs we study the multi-dimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: $\langle \vec{b}_T^2\rangle_{u}^\pi=0.11$fm$^2$, $\langle \vec{b}_T^2\rangle_{s}^K=0.08$fm$^2$, and $\langle \vec{b}_T^2\rangle_{u}^K=0.13$fm$^2$, and therefore the $s$ quark in the kaon is much closer to the center of transverse momentum than the $u$ quark. From the electromagnetic and gravitational form factors we find that the light-cone energy radii are about 60\% smaller than the light-cone charge radii for each quark sector in the pion and kaon. A quantitative measure of the importance of the leading Fock state is obtained via comparison with a full DSE calculation (containing an infinite tower of Fock states) for the pion form factor.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. | method 0.79 | 0.96 | 0.09 | asserted 1.00 |
| s1 | Using these LFWFs we study the multi-dimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). | contribution 0.65 | 0.91 | 0.05 | asserted 0.98 |
| s2 | Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. | method 0.82 | 0.91 | 0.07 | asserted 0.99 |
| s3 | Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: $\langle \vec{b}_T^2\rangle_{u}^\pi=0.11$fm$^2$, $\langle \vec{b}_T^2\rangle_{s}^K=0.08$fm$^2$, and $\langle \vec{b}_T^2\rangle_{u}^K=0.13$fm$^2$, and therefore the $s$ quark in the kaon is much closer to the center of transverse momentum than the $u$ quark. | result 0.99 | 0.94 | 0.32 | asserted 0.94 |
| s4 | From the electromagnetic and gravitational form factors we find that the light-cone energy radii are about 60\% smaller than the light-cone charge radii for each quark sector in the pion and kaon. | result 1.00 | 0.96 | 0.17 | asserted 1.00 |
| s5 | A quantitative measure of the importance of the leading Fock state is obtained via comparison with a full DSE calculation (containing an infinite tower of Fock states) for the pion form factor. | method 0.41 | 0.88 | 0.11 | asserted 1.00 |
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The paper
Original abstract
We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. Using these LFWFs we study the multi-dimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: $\langle \vec{b}_T^2\rangle_{u}^\pi=0.11$fm$^2$, $\langle \vec{b}_T^2\rangle_{s}^K=0.08$fm$^2$, and $\langle \vec{b}_T^2\rangle_{u}^K=0.13$fm$^2$, and therefore the $s$ quark in the kaon is much closer to the center of transverse momentum than the $u$ quark. From the electromagnetic and gravitational form factors we find that the light-cone energy radii are about 60\% smaller than the light-cone charge radii for each quark sector in the pion and kaon. A quantitative measure of the importance of the leading Fock state is obtained via comparison with a full DSE calculation (containing an infinite tower of Fock states) for the pion form factor.
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References
How the literature uses it
14 arXiv papers cite this in 28 sentences
Papers citing it · 11 sentences without a confident label
Builds on it 1 paper
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In this work, based on the study of the pion at physical mass in [ref] , we will predict LF-LFWFs of fictitious pion at masses of 310 MeV and 690 MeV which are directly accessible by lattice QCD, as well as $η_c$ and $η_b$ from the heavy sector.
Ground state pseudoscalar mesons on the light front: from the light to heavy sector · INTRODUCTION · 0.97
Uses its method or tool 3 papers
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To compute the $J/ψ$ LFWFs, we adopt the technique used in [ref] : by fitting the meson BS amplitude with Nakanishi-like representation [ref] and the quark propagator with pairs of complex conjugate poles form which is particularly accurate in heavy sector [ref] , we are able to compute point-wisely accurate LFWFs.
The light front wave functions and diffractive electroproduction of vector mesons · From vector meson Bethe-Salpeter wave functions to LF-LFWFs. · 1.00
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Ref. [ref] has showed the process in their appendix, the inhomogeneous BSE in Fig. [ref], on the right side, the first diagram is the bare vertex corresponds to the bare vector GPD $H^a(x,ξ,t)$, the second term corresponds to the dressed GPD term $H^c(x,ξ,t)$ which is plotted in Fig. [ref], after some calculation, we get the final form of $H^c$
Regularization dependence of pion generalised parton distributions · The definition and calculation of pion GPDs · 0.89
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Using Eqs.([ref]-[ref]) and Eqs.([ref],[ref]), one can reproduce the pointwise behavior of $ψ(x,k_T^2)$ with very high precision, using same method explained in section II of [ref] .
Ground state pseudoscalar mesons on the light front: from the light to heavy sector · From Bethe-Salpeter wave functions to LF-LFWFs · 0.86
Compares with it 4 papers
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A more detailed view of the fall-off can be gained from the right panel of Fig. [ref], where $f^S⊥(γ,ξ=0.5)$ is compared with the results obtained by using i) the LF constituent quark model of Ref. [ref] (cf. the second line in Eq. [ref], without the gluonic term). ii) the LF wave function from DSE calculations [ref] and iii) the NJL model [ref] .
Unpolarized transverse-momentum dependent distribution functions of a quark in a pion with Minkowskian dynamics · Transverse degree of freedom · 1.00
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Defining their mean squared impact parameter $⟨b_T^2 ⟩^q= ∫d b_T^2 b_T^2 ρ^q,(0)(b_T^2)$ as a measure for their spatial extension, we find $⟨b_T^2 ⟩^c_D=(0.13 fm)^2$, $⟨b_T^2 ⟩^u̅_D=(0.40 fm)^2$, $⟨b_T^2 ⟩^b_B=(0.056 fm)^2$, $⟨b_T^2 ⟩^u̅_B=(0.47 fm)^2$ and $⟨b_T^2 ⟩^b_B_c=(0.077 fm)^2$, $⟨b_T^2 ⟩^c̅_B_c=(0.22 fm)^2$, in comparison with our results in pion and heavy quarkonium $⟨b_T^2 ⟩^u_π=(0.33 fm)^2$ [ref] , $⟨b_T^2 ⟩^c_η_c=(0.16 fm)^2$ and $⟨b_T^2 ⟩^b_η_b=(0.092 fm)^2$ [ref] .
Heavy flavor-asymmetric pseudoscalar mesons on the light front · Three-dimensional Parton distribution functions of the heavy-light pseudoscalar mesons. · 0.95
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Comparing with our results on pion and kaon $⟨b_T^2 ⟩^u_π=(0.332 fm)^2$, $⟨b_T^2 ⟩^u_K=(0.361 fm)^2$ and $⟨b_T^2 ⟩^s_K=(0.283 fm)^2$ from [ref] , one finds the heavy mesons are considerably more compact in the transverse plane on the light front.
Ground state pseudoscalar mesons on the light front: from the light to heavy sector · Spatial and Momentum Tomography of the heavy pseudoscalar mesons. · 0.93
Cites it as background 3 papers
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In view of our study, one has to separately mention the approaches developed within the so-called continuum-QCD, that are based on solutions (actually in Euclidean space) of dynamical equations like the homogeneous 4D Bethe-Salpeter equation (BSE) [ref] in combination or not with the quark gap-equation (see, e.g. Refs. [ref] ).
Unpolarized transverse-momentum dependent distribution functions of a quark in a pion with Minkowskian dynamics · Introduction · 0.82
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Meanwhile, promising theoretical frameworks for accessing TMDs also include the discretized space-time Euclidean lattice [ref] and the Dyson–Schwinger equations approach [ref] .
Transverse momentum structure of proton within the basis light-front quantization framework · Introduction · 0.93
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Meanwhile, promising theoretical frameworks for illuminating TMDs also include the Dyson–Schwinger equations approach [ref] and the discretized space-time euclidean lattice [ref] .
Transverse structure of electron in momentum space in basis light-front quantization · Introduction · 0.91
What it cites · 61 sentences without a confident label
Uses the method or tool of 3 papers
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To aid the calculation of the moments $< x^m >_l_z$ we use an accurate parametrization of numerical solutions to the gap and BSEs in the DCSB-improved truncation to the DSEs [ref] .
Dressed-quark anomalous magnetic moments · Parameterization of $S(k)$ and $Γ(k;P)$ · 0.88
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To aid the calculation of the moments $< x^m >_l_z$ we use an accurate parametrization of numerical solutions to the gap and BSEs in the DCSB-improved truncation to the DSEs [ref] .
Imaging dynamical chiral symmetry breaking: pion wave function on the light front · Parameterization of $S(k)$ and $Γ(k;P)$ · 0.88
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Using a dressed operator that defines the GPD from the NJL model [ref] , we find in the impulse approximation that our leading Fock state DSE result is modified such that
Impact of DCSB and dynamical diquark correlations on proton GPDs · ELECTROMAGNETIC AND GRAVITATIONAL FORM FACTORS · 0.80
Uses data or values from 4 papers
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This approximation to the full valence quark distribution function is best at a low hadronic scale $μ_0$, which in Ref. [ref] was determined to be $μ_0 = 520$MeV.
Intrinsic Transverse Motion of the Pion's Valence Quarks · PION AND KAON LIGHT FRONT WAVE FUNCTIONS · 0.95
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The initial scale $μ_0$ is determined as follows (see Ref. [ref] ): At the scale of $Q^2 = 4$GeV$^2$, the $πN$ Drell-Yan analysis gives averaged momentum fraction of valence quark distribution in pion as $2 ⟨x ⟩_v = 0.47(2)$ [ref] and the lattice QCD gives $2 ⟨x ⟩_v = 0.48(4)$ [ref] .
Parton distribution functions in the pion from lattice QCD · GPDS AT ZERO SKEWNESS · 0.97
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The initial scale $μ_0$ is determined as follows (see Ref. [ref] ): At the scale of $Q^2 = 4$GeV$^2$, the $πN$ Drell-Yan analysis gives averaged momentum fraction of valence quark distribution in pion as $2 ⟨x ⟩_v = 0.47(2)$ [ref] and the lattice QCD gives $2 ⟨x ⟩_v = 0.48(4)$ [ref] .
Pionic Parton Distributions Revisited · GPDS AT ZERO SKEWNESS · 0.97
Compares with 3 papers
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In general we find that our result overshoots the data for all $Q^2$, and also the full DSE calculation that uses the Bethe-Salpeter wave function and a dressed quark-photon vertex to directly calculate the pion's form factor [ref] .
The $\pi$, $K^+$, and $K^0$ electromagnetic form factors · ELECTROMAGNETIC AND GRAVITATIONAL FORM FACTORS · 0.99
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A similar result was also found using a relativistic constituent quark model based on an effective $qq̅$ Hamiltonian [ref] , where a pion charge radius of $r_c = 0.45$ fm was found [ref] .
Hard Constituent Quarks and Electroweak Properties of Pseudoscalar Mesons · ELECTROMAGNETIC AND GRAVITATIONAL FORM FACTORS · 0.87
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It is closer to the NJL model result (blue dashed) [ref] than to the spectral quark model [ref] .
Gravitational and higher-order form factors of the pion in chiral quark models · ELECTROMAGNETIC AND GRAVITATIONAL FORM FACTORS · 1.00
Cites as support 1 paper
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The strong support of the LFWFs at infrared $k_T^2$ originates from the strength of the covariant Bethe-Salpeter wave functions at low $|k_T|$, which is closely connected to DCSB, as illustrated model-independently in Ref. [ref] .
Pion mass and decay constant · PION AND KAON LIGHT FRONT WAVE FUNCTIONS · 0.99
Cites as background 11 papers
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The IPDs have the interpretation of parton distributions in the transverse plane [ref] , with $x$ the light-cone momentum fraction and $b_T$ the transverse separation between the active parton and the origin of transverse center of momentum $R_T$.
Impact Parameter Dependent Parton Distributions and Off-Forward Parton Distributions for $\zeta\to 0$ · GPDS AT ZERO SKEWNESS · 0.91
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The IPDs have the interpretation of parton distributions in the transverse plane [ref] , with $x$ the light-cone momentum fraction and $b_T$ the transverse separation between the active parton and the origin of transverse center of momentum $R_T$.
Impact Parameter Space Interpretation for Generalized Parton Distributions · GPDS AT ZERO SKEWNESS · 0.91
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where the gauge link is unity [ref] , $x$ denotes the parton's averaged light-cone momentum fraction, the skewness parameter is $ξ=-Δ^+/2 P^+$, and the momentum transfer $t = Δ^2 = -4ξ^2 m_M^2+Δ_T^2/1-ξ^2$.
Generalized Parton Distributions · GPDS AT ZERO SKEWNESS · 0.85
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