# Emergence of Hadron Mass and Structure

^{1}

^{2}

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^{4}

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^{†}

## Abstract

**:**

Contents | ||

1 | Introduction | 2 |

2 | Hadron Mass Budgets | 4 |

3 | Gluons and the Emergence of Mass | 6 |

4 | Process-Independent Effective Charge | 8 |

5 | Confinement | 11 |

6 | Spectroscopy | 16 |

7 | Baryon Wave Functions | 22 |

8 | Meson Form Factors | 26 |

9 | Baryon Form Factors | 29 |

10 | Transition Form Factors of Heavy+Light Mesons | 36 |

11 | Distribution Functions | 42 |

12 | Conclusions | 50 |

References | 51 |

## 1. Introduction

## 2. Hadron Mass Budgets

## 3. Gluons and the Emergence of Mass

## 4. Process-Independent Effective Charge

- (a)
- Does QCD possess a unique, nonperturbatively well-defined and calculable effective charge, viz. a veritable analogue of QED’s Gell-Mann–Low running coupling; and
- (b)
- Does Equation (8) express the large-${k}^{2}$ behaviour of that charge?

**Absence of a Landau pole**. Whereas the perturbative running coupling, e.g., Equation (8), diverges at ${k}^{2}={\Lambda}_{\mathrm{QCD}}^{2}$, revealing the Landau pole, the PI charge is a smooth function on ${k}^{2}\ge 0$: the Landau pole is eliminated owing to the appearance of a gluon mass scale, Equation (7).Implicit in the “screening function”, $\mathcal{K}\left({k}^{2}\right)$, is a screening mass:$${\zeta}_{\mathcal{H}}=\mathcal{K}({k}^{2}={\Lambda}_{\mathrm{QCD}}^{2})\approx 1.4\phantom{\rule{0.166667em}{0ex}}{\Lambda}_{\mathrm{QCD}}<{m}_{p},$$These features emphasise the role of EHM as expressed in Equation (7): the existence of ${m}_{0}\approx {m}_{p}/2$ guarantees that long-wavelength gluons are screened, so play no dynamical role. Consequently, ${\zeta}_{H}$ marks the boundary between soft/nonperturbative and hard/perturbative physics. It is therefore a natural choice for the “hadron scale”, viz. the renormalisation scale at which valence quasiparticle degrees-of-freedom should be used to formulate and solve hadron bound state problems [71]. Implementing that notion, then those quasiparticles carry all hadron properties at $\zeta ={\zeta}_{\mathcal{H}}$. This approach is today being used to good effect in the prediction of hadron parton distribution functions (DFs)—see Section 11 and References [124,125,126,127,128,129,130,131,132,133,134,135,136].**Match with the Bjorken process-dependent charge**. The theory of process-dependent (PD) charges was introduced in References [137,138]: “…to each physical quantity depending on a single scale variable is associated an effective charge, whose corresponding Stückelberg–Peterman–Gell-Mann–Low function is identified as the proper object on which perturbation theory applies.” PD charges have since been widely canvassed [94,139,140].One of the most fascinating things about the PI running coupling is highlighted by its comparison with the data in Figure 3, which express measurements of the PD effective charge, ${\alpha}_{{g}_{1}}\left({k}^{2}\right)$, defined via the Bjorken sum rule [141,142]. The charge calculated in Reference [71] is an essentially PI charge. There are no parameters; and, prima facie, no reason to expect that it should match ${\alpha}_{{g}_{1}}\left({k}^{2}\right)$. The almost precise agreement is a discovery, given more weight by new results on ${\alpha}_{{g}_{1}}\left({k}^{2}\right)$ [95], which now reach into the conformal window at infrared momenta.Mathematically, at least part of the explanation lies in the fact that the Bjorken sum rule is an isospin non-singlet relation, which eliminates many dynamical contributions that might distinguish between the two charges. It is known that the two charges are not identical; yet, equally, on any domain for which perturbation theory is valid, the charges are nevertheless very much alike:$$\frac{{\alpha}_{{g}_{1}}\left({k}^{2}\right)}{\widehat{\alpha}\left({k}^{2}\right)}\stackrel{{k}^{2}\gg {m}_{0}^{2}}{=}1+\frac{1}{20}{\alpha}_{\overline{\mathrm{MS}}}\left({k}^{2}\right)\phantom{\rule{0.166667em}{0ex}},$$$$\frac{{\alpha}_{{g}_{1}}\left({k}^{2}\right)}{\widehat{\alpha}\left({k}^{2}\right)}\stackrel{{k}^{2}\ll {m}_{0}^{2}}{=}1.03\left(4\right)\phantom{\rule{0.166667em}{0ex}}.$$**Infrared completion**. Being process-independent, $\widehat{\alpha}\left({k}^{2}\right)$ serves numerous purposes and unifies many observables. It is therefore a good candidate for that long-sought running coupling which describes QCD’s effective charge at all accessible momentum scales [139], from the deep infrared to the far ultraviolet, and at all scales in between, without any modification.Significantly, the properties of $\widehat{\alpha}\left({k}^{2}\right)$ support the conclusion that QCD is actually a theory, viz. a well-defined $D=4$ quantum gauge field theory. QCD therefore emerges as a viable tool for use in moving beyond the SM by giving substructure to particles that today seem elementary. A good example was suggested long ago; namely, perhaps all spin-$J=0$ bosons may be [57] “…secondary dynamical manifestations of strongly coupled primary fermion fields and vector gauge fields …”. Adopting this position, the SM’s Higgs boson might also be composite, in which case, inter alia, the quadratic divergence of Higgs boson mass corrections would be eliminated.

## 5. Confinement

“Quantum Yang–Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang–Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the ‘mass gap’: the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang–Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.”

## 6. Spectroscopy

## 7. Baryon Wave Functions

- $\Delta \left(1232\right){{\textstyle \frac{3}{2}}}^{+}$ … $0{\phantom{\rule{0.166667em}{0ex}}}^{4}{S}_{{\textstyle \frac{3}{2}}}=\mathsf{S}$-wave ground state;
- $\Delta \left(1600\right){{\textstyle \frac{3}{2}}}^{+}$ … $1{\phantom{\rule{0.166667em}{0ex}}}^{4}{S}_{{\textstyle \frac{3}{2}}}=\mathsf{S}$-wave radial excitation of $\Delta \left(1232\right){{\textstyle \frac{3}{2}}}^{+}$;
- $\Delta \left(1700\right){{\textstyle \frac{3}{2}}}^{-}$ … $0{\phantom{\rule{0.166667em}{0ex}}}^{2}{P}_{{\textstyle \frac{3}{2}}}=\mathsf{P}$-wave orbital angular momentum excitation of $\Delta \left(1232\right){{\textstyle \frac{3}{2}}}^{+}$;
- $\Delta \left(1940\right){{\textstyle \frac{3}{2}}}^{-}$ … $1{\phantom{\rule{0.166667em}{0ex}}}^{4}{P}_{{\textstyle \frac{3}{2}}}=\mathsf{P}$-wave excitation of $\Delta \left(1600\right){{\textstyle \frac{3}{2}}}^{+}$.

- $N\left(1520\right){{\textstyle \frac{3}{2}}}^{-}$ … $0{\phantom{\rule{0.166667em}{0ex}}}^{2}{P}_{{\textstyle \frac{1}{2}}}=\mathsf{P}$-wave ground state in this channel and an angular momentum coupling partner of $N\left(1535\right){{\textstyle \frac{1}{2}}}^{-}$;
- $N\left(1700\right){{\textstyle \frac{3}{2}}}^{-}$ … $0{\phantom{\rule{0.166667em}{0ex}}}^{4}{P}_{{\textstyle \frac{3}{2}}}=\mathsf{P}$-wave angular momentum coupling partner of $N\left(1520\right){{\textstyle \frac{3}{2}}}^{-}$;
- $N\left(1720\right){{\textstyle \frac{3}{2}}}^{+}$ … $0{\phantom{\rule{0.166667em}{0ex}}}^{2}{D}_{{\textstyle \frac{3}{2}}}=\mathsf{D}$-wave orbital angular momentum excitation of $N\left(1520\right){{\textstyle \frac{3}{2}}}^{-}$;
- $N\left(1900\right){{\textstyle \frac{3}{2}}}^{+}$ … $0{\phantom{\rule{0.166667em}{0ex}}}^{4}{D}_{{\textstyle \frac{3}{2}}}=\mathsf{D}$-wave orbital angular momentum excitation of $N\left(1700\right){{\textstyle \frac{3}{2}}}^{-}$.

## 8. Meson Form Factors

## 9. Baryon Form Factors

^{2}[274,275,276]. Extensions to even larger ${Q}^{2}$ are feasible. Where data are available, the predictions confirm the measurements. More significantly, the results are serving as motivation for new experiments at high-luminosity facilities.

## 10. Transition Form Factors of Heavy+Light Mesons

^{2}—see Figure 21B. Today, lattice analyses typically employ such results to construct a least-squares fit to the form factor points, using some practitioner-favoured functional form. That fit is then employed to define the form factor on the whole kinematically accessible domain: $0\lesssim t\lesssim 25\phantom{\rule{0.166667em}{0ex}}$ GeV

^{2}in this case. It is worth noting that, at this time, given the small number of points and their limited precision, the SPM cannot gainfully be used to develop function–form unbiased interpolations and extrapolations of the lQCD output.

## 11. Distribution Functions

- P1
- There is an effective charge, ${\alpha}_{1\ell}\left({k}^{2}\right)$, which, when used to integrate the one-loop perturbative-QCD DGLAP equations, defines a DF evolution scheme that is all orders exact.

**(i)**- The power-laws express measurable effective exponents, obtained from separate linear fits to $ln\left[xp\right(x\left)\right]$ on the domains $0<x<0.005$, $0.85<x<1$. (Here, $p\left(x\right)$ denotes a generic DF.)
**(ii)**- Within mutual uncertainties, proton and pion DFs have the same power-law behaviour on $x\simeq 0$:$${\alpha}_{\mathrm{valence}}^{p,\pi}\approx -0.22\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\alpha}_{\mathrm{glue}}^{p,\pi}\approx -1.6\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\alpha}_{\mathrm{sea}}^{p,\pi}\approx -1.5\phantom{\rule{0.166667em}{0ex}}.$$
**(iii)**- On $x\simeq 1$, the following relationships exist for and between pion and proton DF exponents:$$\begin{array}{cc}\hfill {\beta}_{\mathrm{valence}}^{\pi}& \approx 2.5\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\beta}_{\mathrm{valence}}^{p}\approx {\beta}_{\mathrm{valence}}^{\pi}+1.6\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\beta}_{\mathrm{glue}}^{p,\pi}& \approx {\beta}_{\mathrm{valence}}^{p,\pi}+1.4\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}{\beta}_{\mathrm{sea}}^{p,\pi}\approx {\beta}_{\mathrm{valence}}^{p,\pi}+2.4\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$
**(iv)**- Given (ii) and (iii), then the CSM predictions are consistent with the QCD expectations discussed in connection with Equation (56).
**(v)**

**Observation****A**- … Applied to MARATHON data, the SPM yields ${\left.{F}_{2}^{n}/{F}_{2}^{p}\right|}_{x\to 1}=0.437\left(85\right)$⇒${\left.{d}^{p}/{u}^{p}\right|}_{x\to 1}=0.227\left(100\right)$.8 The possibility ${\left.{d}^{p}/{u}^{p}\right|}_{x\to 1}=0$ is thus excluded with a 98.7% level of confidence; hence, scalar-diquark-only models of proton structure are excluded with equal likelihood. On the other hand, with this same 98.7% level of confidence, the SPM analysis confirms the QCD parton model prediction [412,446]: ${d}^{p}\left(x\right)\propto {u}^{p}\left(x\right)$ on $x\simeq 1$.
**Observation****B**- … The value of ${\left.{F}_{2}^{n}/{F}_{2}^{p}\right|}_{x\to 1}$ inferred from nuclear DIS [456] agrees with the SPM prediction; hence, they may be averaged to yield$${\left.{F}_{2}^{n}/{F}_{2}^{p}\right|}_{x\to 1}^{\mathrm{SPM}\phantom{\rule{0.166667em}{0ex}}\&\phantom{\rule{0.166667em}{0ex}}\mathrm{DIS}-\mathrm{A}}=0.454\pm 0.047\phantom{\rule{0.166667em}{0ex}}.$$This result is drawn on Row 3 in Figure 27. It corresponds to$$\underset{x\to 1}{lim}\frac{{d}^{p}\left(x\right)}{{u}^{p}\left(x\right)}=0.230\pm 0.057$$
**Observation****C**- … Within uncertainties, the result in Equation (64) agrees with both: (i) the value obtained by assuming an SU$\left(4\right)$-symmetric spin–flavour wave function for the proton and helicity conservation in high-${Q}^{2}$ interactions [412,446]; and (ii) the prediction developed from proton Faddeev wave functions that contain both scalar and axial vector diquarks, with the axial vector contributing approximately 40% of the proton charge [271,450]. (Recall that this was the axial vector diquark fraction built into the analyses in References [134,135].)

## 12. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACM | anomalous chromomagnetic moment |

AdS/CFT (duality) | anti-de Sitter/conformal field theory (duality) |

$\overline{\mathrm{ard}}$ | mean absolute relative difference |

CKM | Cabibbo–Kobayashi–Maskawa (matrix) |

CSMs | continuum Schwinger function methods |

DCSB | dynamical chiral symmetry breaking |

DF | (parton) distribution function |

DIS | deep inelastic scattering |

DSE | Dyson–Schwinger equation |

EHM | emergent hadron mass |

FF | (parton) fragmentation function |

JLab | Thomas Jefferson National Accelerator Facility |

lQCD | lattice-regularised quantum chromodynamics |

NG (mode/boson) | Nambu–Goldstone (mode/boson) |

PD (charge) | process-dependent (charge) |

PDG | Particle Data Group and associated publications |

PI (charge) | process-independent (charge) |

QCD | quantum chromodynamics |

QED | quantum electrodynamics |

RGI | renormalisation-group-invariant |

RL | rainbow ladder (truncation) |

SCI | symmetry-preserving treatment of a vector×vector contact interaction |

SM | Standard Model of particle physics |

SPM | Schlessinger point method |

VMD | vector meson dominance |

1 | Here, the value “10” is arbitrary. More generally, the number should be small enough to ensure that predictive power is not lost through a need to fit too many renormalised observables to measured quantities. |

2 | When working with a Euclidean formulation, as we do, Poincaré covariance maps straightforwardly into Euclidean covariance, viz. valid Schwinger functions must transform covariantly under O$\left(4\right)$ rotations and linear translations in ${\mathbb{R}}^{4}$. Owing to the simplicity of this connection, we avoid transliteration and speak of Poincaré covariance and invariance throughout. |

3 | This is well known and explains why the truncated interaction in Equation (26) was not used for any of the calculations described in Section 7, Section 8, Section 9, Section 10 and Section 11 below. All those studies are based on interactions that at least preserve QCD’s ultraviolet power-law behaviour, where more has not yet been achieved—Section 7 and Section 9—and also the one-loop logarithmic improvement, when the necessary algorithms are already available—Section 8, Section 10, and Section 11. |

4 | Dressed-quark propagators form an important part of the kernels of all bound state equations. As on-shell meson masses increase, poles in those propagators enter the complex plane integration domain sampled by the Bethe–Salpeter equation [199]. For such cases—here, meson excited states—a direct on-shell solution cannot be obtained using simple algorithms. Therefore, to obtain the masses of those mesons, Reference [222] used an extrapolation procedure based on Padé approximants. This is the origin of the uncertainty bar on the CSM predictions. |

5 | The Chebyshev (or hyperspherical) expansion of Poincaré-invariant functions of two scalar variables is discussed, e.g., in Reference [199] (IV.B). |

6 | If one eliminates axial vector diquarks from the proton wave function, then ${g}_{A}^{d}/{g}_{A}^{u}=-0.054\left(13\right)$, a result disfavoured by experiments at the level of $5.1\sigma $, i.e., the probability that the scalar-diquark-only proton result could be consistent with data is $1/7,100,000$. |

7 | |

8 | Extrapolations based on $[1,1]$ Padé fits to MARATHON data, obtained using a one-point jackknife procedure, yield ${F}_{2}^{n}/{F}_{2}^{p}=0.395\left(3\right)$ on $x\simeq 1$⇒${d}^{p}/{u}^{d}=0.169\left(3\right)$ [182]. Another analysis [460], employing practitioner-chosen polynomials as the basis for extrapolation, obtains ${F}_{2}^{n}/{F}_{2}^{p}=0.37\left(7\right)$ on $x\simeq 1$⇒${d}^{p}/{u}^{d}=0.13\left(8\right)$. The latter is less precise, but both results are consistent with the function form unbiased SPM prediction. |

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