Transition form factors: $\gamma^\ast + p \to \Delta(1232)$, $\Delta(1600)$

TL;DR

This paper calculates the transition form factors for gamma-star p to Delta(1232) and Delta(1600) using a relativistic diquark-quark model, providing predictions that align with existing data and extend to higher momentum transfers.

Contribution

It introduces a fully-dynamical relativistic model for computing transition form factors of Delta resonances, including the first predictions for the Delta(1600) transition form factors.

Findings

Calculated form factors agree with data for Delta(1232) outside meson-cloud domain.

Predicted form factors for Delta(1600) match empirical values at the real photon point.

The electric quadrupole form factor indicates deformation of Delta baryons.

Abstract

Electroproduction form factors describing the $\gamma^\ast p \to \Delta^+(1232), \Delta^+(1600)$ transitions are computed using a fully-dynamical diquark-quark approximation to the Poincar\'e-covariant three-body bound-state problem in relativistic quantum field theory. In this approach, the $\Delta(1600)$ is an analogue of the Roper resonance in the nucleon sector, appearing as the simplest radial excitation of the $\Delta(1232)$. Precise measurements of the $\gamma^\ast p \to \Delta^+(1232)$ transition already exist on $0 \leq Q^2 \lesssim 8\,$GeV$^2$ and the calculated results compare favourably with the data outside the meson-cloud domain. The predictions for the $\gamma^\ast p \to \Delta^+(1600)$ magnetic dipole and electric quadrupole transition form factors are consistent with the empirical values at the real photon point, and extend to $Q^2 \approx 6 m_p^2$, enabling a meaningful direct comparison with experiment once analysis of existing data is completed. In both cases, the electric quadrupole form factor is particularly sensitive to deformation of the $\Delta$-baryons. Interestingly, whilst the $\gamma^\ast p \to \Delta^+(1232)$ transition form factors are larger in magnitude than those for $\gamma^\ast p \to \Delta^+(1600)$ in some neighbourhood of the real photon point, this ordering is reversed on $Q^2 \gtrsim 2 m_p^2$, suggesting that the $\gamma^\ast p \to \Delta^+(1600)$ transition is more localised in configuration space.

Tables

Table 1. Table 1: Static properties computed from the Δ + ​ ( 1232 ) superscript Δ 1232 \Delta^{+}(1232) and Δ + ​ ( 1600 ) superscript Δ 1600 \Delta^{+}(1600) elastic form factors. An empirical value of G M ​ 1 ​ ( 0 ) subscript 𝐺 𝑀 1 0 G_{M1}(0) is available for the Δ + ​ ( 1232 ) superscript Δ 1232 \Delta^{+}(1232) Tanabashi:2018oca : 3.6 − 1.7 + 1.3 ± 2.0 ± 4 plus-or-minus subscript superscript 3.6 1.3 1.7 2.0 4 3.6^{+1.3}_{-1.7}\pm 2.0\pm 4 . Point-particle values for J = 3 / 2 𝐽 3 2 J=3/2 states are: G M ​ 1 ​ ( 0 ) = 3 subscript 𝐺 𝑀 1 0 3 G_{M1}(0)=3 , G E ​ 2 ​ ( 0 ) = − 3 subscript 𝐺 𝐸 2 0 3 G_{E2}(0)=-3 , G M ​ 3 ​ ( 0 ) = − 1 subscript 𝐺 𝑀 3 0 1 G_{M3}(0)=-1 . Lorce:2009bs . All radii listed in units of the quark-core proton charge radius, r p = 0.61 subscript 𝑟 𝑝 0.61 r_{p}=0.61\, fm.

baryon	$r_E$	$G_{M1}\left(0\right)$	$r_{M1}$	$G_{E2}\left(0\right)$	$r_{E2}$	$G_{M3}\left(0\right)$	$r_{M3}$
${\Delta }^{+}\left(1232\right)$	$1.23$	$2.86$	$1.10$	$-6.67$	$1.20$	$-3.00$	$0.48$
${\Delta }^{+}\left(1600\right)$	$1.68$	$1.50$	$1.05$	$-3.00$	$0.79$	$0.80$	$0.64$

Table 2. Table 2: Interpolation coefficients for each of our computed γ ∗ ​ p → Δ → superscript 𝛾 ∗ 𝑝 Δ \gamma^{\ast}p\to\Delta transition form factors, Eq. ( 14 ). Blank entries indicate “0”.

	$ℱ$	$a_0^{ℱ}$	$a_1^{ℱ}$	$b_1^{ℱ}$	$b_2^{ℱ}$	$c_0^{ℱ}$
$\Delta \left(1232\right)$	$M$	$1.93$	$4.15$	$3.92$	$3.85$	$0.55$
	$E$	$0.041$	$-0.010$	$4.62$	$0.68$	$0.55$
	$C$	$0.30$	$0.030$	$1.58$	$0.35$	$0.55$
$\Delta \left(1600\right)$	$M$	$0.32$			$0.22$	$0.08$
	$E$	$-0.022$		$-0.10$	$0.15$	$0.45$
	$C$	$0.14$			$0.23$	$0.07$

Equations

$$\begin{array}[]{ccc}m_{p}&m_{\Delta(1232)}&m_{\Delta(1600)}\\ 1.19&1.35&1.79\end{array}\,.$$

$${\psi }^{\Delta }(p_i,{\alpha }_i,{\sigma }_i)$$

$$\times {\Delta }_{\mu \nu }^{1^{+}d}(K)[{\phi }_{\nu \rho }^{\Delta d}(\ell ;P)u_{\rho }(P){]}_{{\sigma }_3}^{{\alpha }_3},$$

$${\phi }_{\nu \rho }^{\Delta d}(\ell ;P)$$

$${\text{mathpzc}D}_{\nu \rho }^k={\text{mathpzc}S}^k{\delta }_{\nu \rho },k=1,2,$$

$${\text{mathpzc}D}_{\nu \rho }^k=i{\gamma }_5{\text{mathpzc}A}_{\nu }^{k-2}{\ell }_{\rho }^{\perp },k=3,\dots ,8,$$

$${\text{mathpzc}S}^1$$

$${\text{mathpzc}A}_{\nu }^1$$

$${\text{mathpzc}A}_{\nu }^4$$

$$\widetilde{\text{mathpzc}a}({\ell }^2;P^2)=\frac{2}{\pi }{\int }_{-1}^{1}du\sqrt{1-u^2}\widetilde{\text{mathpzc}a}({\ell }^2,u;P^2),$$

$$J_{\mu \lambda }(K,Q)={\Lambda }_{+}(P_f)R_{\lambda \alpha }(P_f)i{\gamma }_5{\Gamma }_{\alpha \mu }(K,Q){\Lambda }_{+}(P_i),$$

$${\check{K}}_{\mu }^{\perp }={\mathcal{T}}_{\mu \nu }^Q{\check{K}}_{\nu }=({\delta }_{\mu \nu }-{\check{Q}}_{\mu }{\check{Q}}_{\nu }){\check{K}}_{\nu },$$

$${\Gamma }_{\alpha \mu }(K,Q)=\text{mathpzc}k[\frac{{\lambda }_m}{2{\lambda }_{+}}(G_M^{\ast }-G_E^{\ast }){\gamma }_5{\epsilon }_{\alpha \mu \gamma \delta }{\check{K}}_{\gamma }{\check{Q}}_{\delta }$$

$$-G_E^{\ast }{\mathcal{T}}_{\alpha \gamma }^Q{\mathcal{T}}_{\gamma \mu }^K-\frac{i\varsigma }{{\lambda }_m}{G}_C^{\ast }{\check{Q}}_{\alpha }{\check{K}}_{\mu }^{\perp }],$$

$$\text{mathpzc}{t}_1$$

$$\text{mathpzc}{t}_2$$

$$\text{mathpzc}{t}_3$$

$$G_M^{\ast }=3\left[\text{mathpzc}{t}_2+\text{mathpzc}{t}_1\right],G_E^{\ast }=\text{mathpzc}{t}_2-\text{mathpzc}{t}_1,G_C^{\ast }=\text{mathpzc}{t}_3.$$

$$R_{\mathrm{EM}}=-\frac{G_E^{\ast }}{G_M^{\ast }},R_{\mathrm{SM}}=-\frac{|\vec{Q}|}{2m_{\Delta }}\frac{G_C^{\ast }}{G_M^{\ast }}=-\frac{{\lambda }_m}{m_{\Delta }}\frac{G_C^{\ast }}{G_M^{\ast }}.$$

$$J_{\mu ,\alpha }(P_f,$$

$$\times {\bar{\Psi }}_{\alpha }(-p;P_f)J_{\mu }^n(p,P_f,k,P_i)\Psi (k;P_i),$$

$$r_E^{\Delta (1600)}\approx 1.4r_E^{\Delta (1232)},r_{M1}^{\Delta (1600)}\approx 0.95r_{M1}^{\Delta (1232)},$$

$$G_{\text{mathpzc}F}^{\ast }(x)=\frac{a_0^{\text{mathpzc}F}+a_1^{\text{mathpzc}F}x}{1+b_1^{\text{mathpzc}F}x+b_2^{\text{mathpzc}F}{x}^2}{\mathrm{e}}^{-c_1^{\text{mathpzc}F}x},$$

$$R_{EM}\stackrel{Q^2\to \infty }{=}1,R_{SM}\stackrel{Q^2\to \infty }{=}\text{mbox}\mathrm{constant},$$

Full text

I Introduction

The $\Delta(1232)$ family of baryons were the first resonances discovered in $\pi N$ reactions Fermi:1952zz ; Anderson:1952nw ; Nagle:1984sg . With positive parity, isospin $I=\frac{3}{2}$, total-spin $J=\frac{3}{2}$ and no net strangeness Tanabashi:2018oca , the $\Delta^{+,0}$ members of this quadruplet have conventionally been viewed as the lightest isospin- and spin-flip excitations of the proton and neutron, respectively. Hence, since protons and neutrons (nucleons, $N$) are the basic elements of all nuclei, developing a detailed understanding of the $\Delta$-baryons is of fundamental importance. Without this, hadron physics remains at a level akin to atomic physics based only on knowledge of the hydrogen atom’s ground state.

Given that pions are a complex probe, there are advantages in exploiting the relative simplicity of virtual photons in order to chart $\Delta$-resonance structure. Elastic form factors are empirically inaccessible because the $\Delta(1232)$-baryon lifetime is too small: $\tau_{\Delta}\sim 10^{-26}\,\tau_{n}$, where $\tau_{n}$ is the lifetime of a free neutron Tanabashi:2018oca . On the other hand, by exploiting intense, energetic electron-beams at the Thomas Jefferson National Accelerator Facility, $\gamma^{\ast}p\to\Delta^{+}$ data are now available for $0\leq Q^{2}\lesssim 8\,$GeV2 Aznauryan:2011ub ; Aznauryan:2011qj ; Aznauryan:2012ba . These data have stimulated much theoretical analysis and speculation about, inter alia: the relevance of perturbative QCD (pQCD) to processes involving moderate momentum transfers Carlson:1985mm ; Pascalutsa:2006up ; Aznauryan:2011qj ; Aznauryan:2012ba ; Eichmann:2011aa ; Segovia:2013rca ; Segovia:2013uga ; Segovia:2014aza ; Segovia:2016zyc ; hadron shape deformation Eichmann:2011aa ; Alexandrou:2012da ; Santopinto:2012nq ; Segovia:2013rca ; Segovia:2013uga ; Segovia:2014aza ; Segovia:2016zyc ; Sanchis-Alepuz:2017mir ; Buchmann:2018nmu ; and the role that resonance electroproduction experiments can play in exposing nonperturbative aspects of QCD, such as the nature of confinement and dynamical chiral symmetry breaking (DCSB) Aznauryan:2012ba ; Roberts:2015dea ; Burkert:2018oyl ; Roberts:2018hpf .

Just above the $\Delta$-baryon level lies the nucleon’s first positive-parity excitation, i.e. the Roper resonance, labelled $N(1440)\,1/2^{+}$. Discovered in 1963 Roper:1964zza ; BAREYRE1964137 ; AUVIL196476 ; PhysRevLett.13.555 ; PhysRev.138.B190 , its characteristics were long the source of puzzlement because, e.g. constituent-quark potential models typically (and erroneously) produce a spectrum in which this excitation lies above the first negative-parity state $N(1535)\,1/2^{-}$ Capstick:2000qj ; Crede:2013sze ; Giannini:2015zia . This has now changed following: acquisition and analysis of high-precision proton-target exclusive electroproduction data with single- and double-pion final states, on a large energy domain and with momentum-transfers out to $Q^{2}\approx 5\,$GeV2; development of a dynamical reaction theory capable of simultaneously describing all partial waves extracted from available, reliable data; and formulation and application of a Poincaré covariant approach to the continuum bound-state problem in relativistic quantum field theory. Today, it is widely accepted that the Roper is, at heart, the first radial excitation of the nucleon, consisting of a well-defined dressed-quark core that is augmented by a meson cloud, which both reduces the Roper’s core mass by approximately 20% and contributes materially to the electroproduction form factors at low-$Q^{2}$ Golli:2017nid ; Burkert:2017djo .

A similar pattern of energy levels is found in the spectrum of $\Delta$-baryons. Namely, contradicting quark-model predictions Capstick:2000qj ; Crede:2013sze ; Giannini:2015zia , the first positive-parity excitation, $\Delta(1600)\,3/2^{+}$, lies below the negative parity $\Delta(1700)\,3/2^{-}$, with the splitting being approximately the same as that in the nucleon sector. This being the case and given the Roper-resonance example, it is likely that elucidating the nature of the $\Delta(1600)\,3/2^{+}$-baryon will require both (i) data on its electroproduction form factors which extends well beyond the meson-cloud domain and (ii) predictions for these form factors to compare with that data. The data exist Trivedi:2018rgo ; Burkert:2019opk ; and can be analysed with this aim understood. Herein, therefore, we provide the theoretical predictions.

Our treatment of the nucleon, $\Delta(1232)$- and $\Delta(1600)$-baryons, and the associated $\gamma N\to\Delta$ transitions is based on Refs. Segovia:2014aza ; Chen:2019fzn . Capitalising on this tight connection, herein we only sketch the elements of our calculation. (Isospin symmetry is assumed throughout.) Moreover, with nothing changed, our study delivers a unification of the $N\to\Delta(1232)$ and $N\to\Delta(1600)$ transitions.

Section II explains the quark-diquark approximation to the baryon problem in the context of a Poincaré-covariant Faddeev equation and discusses the solutions obtained for the $\Delta(1232)$-baryon and its first positive-parity excitation. The $\gamma^{\ast}p\to\Delta$ transition current and associated form factors are described in Sec. III. Section IV reports results for the $\gamma^{\ast}p\to\Delta(1232)$ transition, providing comparisons with data and other analyses. The $\gamma^{\ast}p\to\Delta(1600)$ transition form factors are discussed in Sec. V; Sec. VI describes their diquark and scatterer dissections; and Sec. VII provides a summary and offers perspectives.

II Baryon Wave Functions

In relativistic quantum field theory, baryon structure is described by a Faddeev amplitude, obtained from a Poincaré-covariant Faddeev equation, which sums all possible quantum field theoretical exchanges and interactions that can take place between the three dressed-quarks that characterise its valence-quark content. A dynamical prediction of Faddeev equation studies that employ realistic quark-quark interactions Binosi:2014aea ; Binosi:2016wcx ; Binosi:2016nme ; Rodriguez-Quintero:2018wma is the appearance of nonpointlike quark$+$quark (diquark) correlations within baryons, whose characteristics are determined by DCSB Cahill:1987qr ; Maris:2002yu ; Maris:2004bp . Consequently, the baryon bound-state problem is transformed into solving the linear, homogeneous matrix equation in Fig. 1 Cahill:1988dx ; Burden:1988dt ; Cahill:1988zi ; Reinhardt:1989rw ; Efimov:1990uz . Its key elements are the dressed-quark and -diquark propagators, and the diquark Bethe-Salpeter amplitudes.

Evidence supporting the presence of diquark correlations in baryons is accumulating, e.g. Refs. Eichmann:2009qa ; Cates:2011pz ; Segovia:2013rca ; Roberts:2013mja ; Segovia:2014aza ; Segovia:2015ufa ; Eichmann:2016jqx ; Segovia:2016zyc ; Eichmann:2016hgl ; Eichmann:2016nsu ; Lu:2017cln ; Chen:2017pse ; Mezrag:2017znp ; Roberts:2018hpf ; Chen:2018nsg ; Chen:2019fzn . It should be emphasised that these correlations are fully dynamical and appear in a Faddeev kernel which requires their continual breakup and reformation. Consequently, they are vastly different from the static, pointlike diquarks introduced originally Anselmino:1992vg in an attempt to solve the so-called “missing resonance” problem Aznauryan:2011ub . In fact, consistent with numerical simulations of lattice-regularised QCD Edwards:2011jj , the spectrum of states produced by the Faddeev equation in Fig. 1 possesses a richness that cannot be explained by a two-body model.

To define the Faddeev equation in Fig. 1, we employ the elements specified in Refs. Segovia:2014aza ; Chen:2019fzn , which provide a successful description of the spectrum and structure of octet and decuplet baryons and their positive-parity excitations, and are part of a body of work that unifies a large array of hadron properties Roberts:2015lja ; Horn:2016rip ; Eichmann:2016yit ; Burkert:2017djo . A key to these successes is DCSB, which produces a dressed-quark mass-scale Binosi:2016wcx : $M_{D}\simeq 0.4\,$GeV, whose value underlies the natural size for mass-dimensioned quantities in the light-quark sector of the Standard Model.

With the inputs drawn from Refs. Segovia:2014aza ; Chen:2019fzn (including light-quark scalar and axial-vector diquark masses $m_{0^{+}}=0.79\,$GeV, $m_{1^{+}}=0.89\,$GeV, respectively) one can readily construct the relevant Faddeev equation kernels and use ARPACK software Arpack to obtain the mass and Faddeev amplitude of the $(I,J^{P})=(1/2,1/2^{+})$ ground-state (proton) and the two lightest $(I,J^{P})=(3/2,3/2^{+})$ states, which we identify with the $\Delta(1232)$- and $\Delta(1600)$-baryons. The masses are (in GeV):

[TABLE]

These values correspond to the locations of the lowest-magnitude poles in the three-quark scattering problems in the given channels.

The residues associated with these poles are the Poincaré-covariant wave functions, $\chi(\ell^{2},\ell\cdot P;P^{2})$, where $\ell$ is the quark-diquark relative momentum. For every baryon considered herein, eight scalar functions are required to completely describe the system, each associated with a particular Dirac-matrix structure. For instance, the (amputated) Faddeev amplitude of any $(I,J^{P})=(3/2,3/2^{+})$ baryon can be written in the following form:

[TABLE]

where: $(p_{i},\sigma_{i},\alpha_{i})$ are the momentum, spin and isospin labels of the quarks constituting the bound state; $P=p_{1}+p_{2}+p_{3}=p_{d}+p_{q}$ is the total momentum of the baryon; $k=(p_{1}-p_{2})/2$, $K=p_{1}+p_{2}=p_{d}$, $\ell=(-K+2p_{3})/3$; $d$ counts the diquarks participating in the baryon111In $\Delta^{+}$-baryons, the sum ranges over isovector-pseudovector $\{uu\}$, $\{ud\}$ correlations; and in $\Delta^{0}$-baryons, $\{ud\}$ and $\{dd\}$. Assuming isospin-symmetry, as we do throughout, the correlation amplitudes and propagators are identical for all these diquarks. and $\Gamma^{d}_{1^{+}\mu}$, $\Delta^{1^{+}d}_{\mu\nu}$ are, respectively, the associated correlation amplitude and propagator; $u_{\rho}(P)$ is a Rarita-Schwinger spinor (Ref. Segovia:2014aza , Appendix B); and

[TABLE]

with

[TABLE]

$\hat{\ell}^{2}=1$, $\hat{P}^{2}=-1$, $\ell^{\perp}=\hat{\ell}_{\nu}+\hat{\ell}\cdot\hat{P}\hat{P}_{\nu}$, $\gamma^{\perp}=\gamma_{\nu}+\gamma\cdot\hat{P}\hat{P}_{\nu}$. The (unamputated) Faddeev wave function, $\chi(\ell^{2},\ell\cdot P;P^{2})$, can be computed from the amplitude specified by Eqs. (2) – (3) simply by attaching the appropriate dressed-quark and diquark propagators. It may also be decomposed in the form of Eqs. (3). Naturally, the scalar functions are different, and we label them $\tilde{\mathpzc a}_{\Delta k}^{d}$.

In order to visualise the wave function solutions of the Faddeev equations, it is convenient to work with univariate Chebyshev projections of the scalar functions used to express them. Furthermore, as with mesons Maris:1997tm ; Maris:1999nt , one usually focuses on the zeroth functional moment of the given function ($u=\ell\cdot P/\sqrt{\ell^{2}P^{2}}$):

[TABLE]

because it is typically dominant in realistic solutions and hence expresses the largest amount of information.

The order-zero Chebyshev projections of the Faddeev wave function for the proton are plotted in Ref. Chen:2017pse , Fig. 4; and our calculation reproduces those results. Herein, therefore, in Fig. 2 we depict the projections for the ground-state $\Delta$-baryon and its first positive-parity excitation. In all cases, we plot that combination of functions which has a well-defined value of quark-diquark orbital angular momentum in the baryon’s rest frame. A key observation here is that, for the ground-state, each projection is of a unique sign (positive or negative). On the other hand, with the exception of two $D$-wave components $(-\tilde{\mathpzc a}_{6}^{0,1}+\tilde{\mathpzc a}_{8}^{0,1})$, all excited-state projections possess a single zero. As noted elsewhere Segovia:2015hra ; Chen:2017pse ; Mezrag:2017znp ; Qin:2018dqp , this pattern of behaviour indicates that the positive-parity excitation may be interpreted as the simplest radial excitation of its ground-state partner.

Figure 2 also shows that the $\Delta$-baryon ground-state and positive-parity excitation are primarily $S$-wave in character: the magnitudes of the curves in the top row are greater than those in the other rows. Naturally, we replicate the results of Ref. Chen:2019fzn , viz. the ground-state mass is almost insensitive to non-$S$-wave components; and in the first positive-parity excitation, $P$-wave components generate a little repulsion, some attraction is provided by $D$-waves, and $F$-waves have no measurable impact. Evidently, too, some $S$-wave strength is shifted into $P$- and $D$-wave contributions within the positive-parity excitation Eichmann:2016nsu ; Qin:2018dqp ; Chen:2019fzn . Notwithstanding their smaller magnitudes, we will see that the higher partial-waves have noticeable effects on electroproduction form factors.

Let us return to the masses in Eq. (1). Empirical values of these pole locations are Tanabashi:2018oca (in GeV): $0.939$, $1.21-i0.05$, $1.51-i0.14$. (The physical $\Delta$-baryons are unstable and hence the associated pole has an imaginary part.) At first glance, these values appear unrelated to those in Eq. (1). However, deeper consideration reveals Eichmann:2008ae ; Eichmann:2008ef that the kernel in Fig. 1 has an intrinsic weakness: resonant contributions , viz. meson-baryon final-state-interactions (MB FSIs), are omitted. It is such effects which are resummed in dynamical coupled channels models, generating the widths and thereby transforming bare-baryons into the observed states Kamano:2018sfb ; Doring:2018kue . Our Faddeev equation should therefore be understood as producing the dressed-quark core of the bound-state, not the completely-dressed object.

Clothing the nucleon’s dressed-quark core by including resonant contributions to the kernel produces a physical nucleon whose mass is $\approx 0.2$ GeV lower than that of the core Ishii:1998tw ; Hecht:2002ej . Similarly, MB FSIs reduce the $\Delta(1232)$-baryon’s core mass by $\approx 0.16\,$GeV JuliaDiaz:2007kz ; JuliaDiaz:2007fa ; Suzuki:2009nj and the Roper resonance’s core-mass by $0.3\,$GeV Suzuki:2009nj . Evidently, such reductions shift the mass of a given baryon’s dressed-quark core into alignment with the measured Breit-Wigner mass of the associated physical states. Moreover, this pattern is seen to prevail broadly, extending to baryons in the multiplets of flavour-$SU(5)$ Qin:2019hgk ; Yin:2019bxe .

Our approach thus delivers the dressed-quark-core contribution to a given observable and that this should subsequently be corrected by incorporating MB FSIs. These features have long been appreciated and exploited in developing a successful body of work on the baryon spectrum and elastic and transition form factors, e.g. Refs. Segovia:2014aza ; Chen:2019fzn ; and we capitalise on such experience herein.

III Transition Current

Electromagnetic $N\to\Delta$ transitions are described by three form factors Jones:1972ky : magnetic-dipole, $G_{M}^{\ast}$; electric quadrupole, $G_{E}^{\ast}$; and Coulomb (longitudinal) quadrupole, $G_{C}^{\ast}$. They arise through consideration of the transition current:

[TABLE]

where: $P_{i}$, $P_{f}$ are, respectively, the incoming nucleon and outgoing $\Delta$ momenta, $P_{i}^{2}=-m_{N}^{2}$, $P_{f}^{2}=-m_{\Delta}^{2}$; $Q_{\mu}=(P_{f}-P_{i})_{\mu}$ is the incoming photon momentum, $K=(P_{i}+P_{f})/2$; and $\Lambda_{+}(P_{i})$, $\Lambda_{+}(P_{f})$ are, respectively, positive-energy projection operators for the nucleon and $\Delta$, with the Rarita-Schwinger tensor projector $R_{\lambda\alpha}(P_{f})$ arising in the latter connection. (See Ref. Segovia:2014aza , Appendix B.)

In order to succinctly express $\Gamma_{\alpha\mu}(K,Q)$, we define

[TABLE]

with $\check{K}^{2}=1=\check{Q}^{2}$, in which case

[TABLE]

where $\mathpzc{k}=\sqrt{(3/2)}(1+m_{\Delta}/m_{N})$, $\varsigma=Q^{2}/[2\Sigma_{\Delta N}]$, $\lambda_{\pm}=\varsigma+t_{\pm}/[2\Sigma_{\Delta N}]$ with $t_{\pm}=(m_{\Delta}\pm m_{N})^{2}$, $\lambda_{m}=\sqrt{\lambda_{+}\lambda_{-}}$, $\Sigma_{\Delta N}=m_{\Delta}^{2}+m_{N}^{2}$, $\Delta_{\Delta N}=m_{\Delta}^{2}-m_{N}^{2}$.

With a concrete expression for the current in hand, one may obtain the form factors using any three sensibly chosen projection operations, e.g. with Eichmann:2011aa

[TABLE]

where $\mathpzc{d}=\Delta_{\Delta N}/[2\Sigma_{\Delta N}]$, $\mathpzc{n}=\sqrt{1-4\mathpzc{d}^{2}}/[4i\mathpzc{k}\lambda_{m}]$), then

[TABLE]

The following ratios are often considered in connection with $\gamma^{\ast}N\to\Delta$ transitions:

[TABLE]

Since they are identically zero in $SU(6)$-symmetric constituent-quark models, they can be read as measures of deformation in one or both of the hadrons involved.

Following Refs. Oettel:1999gc ; Segovia:2014aza , the transition current in Eq. (6) can be explicated as follows:

[TABLE]

where $\Psi_{\alpha}$, $\Psi$ are, respectively, the $\Delta$ and nucleon Faddeev amplitudes described in Sec. II; and the sum ranges over the six diagrams depicted and detailed in Ref. Segovia:2014aza , Appendix C. Each term in Eq. (12) can be evaluated using standard algebraic and numerical techniques, and results for the form factors obtained subsequently via the projections in Eqs. (9) and combinations in Eqs. (10).

In these calculations, the proton and $\Delta^{+}$-baryon Faddeev amplitudes must be canonically normalised. This is achieved by computing the elastic electric form factor in each case and rescaling the amplitude such that the associated $Q^{2}=0$ value (electric charge) is unity Segovia:2014aza . Given this necessity, we computed the low-$Q^{2}$ behaviour of all elastic form factors for each baryon and report the associated static properties of their dressed-quark cores in Table 1. These results lead to the following observations:

[TABLE]

which may sensibly be compared with $r_{E}^{\rm Roper}\approx 1.8\,r_{p}$, $r_{M}^{\rm Roper}\approx 1.6\,r_{M}^{p}$ Segovia:2015hra ; and the octupole moments of the $\Delta(1232)$ and $\Delta(1600)$ have opposite signs, an outcome that signals the impact of differences in the distribution and strength of higher partial-waves in the respective wave functions (see Fig. 2 herein and the discussion of Fig. 7 in Ref. Segovia:2014aza ). In addition, we find that the $\Delta(1600)$ elastic electric form factor possesses a zero, at $Q^{2}\approx 1.8\,m_{p}^{2}$. For the $\Delta(1232)$, this zero lies at $Q^{2}\approx 2.7\,m_{p}^{2}$ Segovia:2014aza . Notably, the ordering and locations are consistent with the electric radii reported in Table 1.

IV Calculated Form Factors: $\mathbf{\Delta(1232)}$

Our computed $\gamma^{\ast}p\to\Delta^{+}(1232)$ transition form factors are depicted in Fig. 3. They are accurately interpolated using a simple functional form Sato:2000jf :

[TABLE]

with the coefficients given in Table 2. (These forms should not be used for large-$x$ extrapolation.)

Considering Fig. 3, it is evident both that $G_{M}^{\ast}$, the magnetic dipole form factor, dominates this transition and our result agrees with modern data on $Q^{2}\gtrsim 0.5\,m_{p}^{2}$. As explained elsewhere Sato:2000jf ; JuliaDiaz:2007kz , incorporation of MB FSIs is crucial to ensuring agreement on $Q^{2}\lesssim 0.5\,m_{p}^{2}$, e.g. such effects increase the result by a factor of $\approx 1.5$ at $Q^{2}=0$. This “meson-cloud domain” is indicated by shading in the top panel of Fig. 3. Its size typically depends on the baryon(s) being considered, e.g. extending to $Q^{2}\approx 2m_{p}^{2}$ for nucleon elastic form factors Eichmann:2008ef ; Eichmann:2011vu and Roper electroproduction Burkert:2017djo .

The $\gamma^{\ast}p\to\Delta^{+}(1232)$ electric and Coulomb quadrupole form factors are small but nonzero, highlighting that the dressed-quark cores of the baryons involved are deformed, viz. not purely $S$-wave in their rest frames. Although this is obvious from inspection of their Poincaré-covariant wave functions (Ref. Chen:2017pse , Fig. 4, and Fig. 2 above), $G_{E,C}^{\ast}$ are a measurable manifestation of the distortion’s magnitude. (As will subsequently become apparent, it is deformation of the $\Delta$-baryons which is most important.)

Each panel in Fig. 3 contains three curves: the solid (black) curve is our complete prediction; the long-dashed (blue) curve is obtained when only those components of the $\Delta(1232)$ wave function are retained which correspond to $S$-waves in the rest frame; and the dashed (blue) curve is obtained when both the proton and $\Delta(1232)$ are reduced to $S$-wave states. Notably: the role played by higher partial waves in the wave functions increases with momentum transfer (something also observed in meson form factors Maris:1998hc ), here generating destructive interference; agreement with data on $G_{M}^{\ast}$ is impossible without the higher partial waves; and the effect of these components is very large in $G_{E}^{\ast}$, unsurprisingly, because it is a difference of two positive-definite functions. (The complete result for $G_{E}^{\ast}$ exhibits a zero at $x\approx 4$, which is absent in the S-wave-only result(s).)

In Fig. 4, to further elucidate the observable impacts of higher partial-waves in the Poincaré-covariant wave functions, we depict the ratios $R_{EM}$, $R_{SM}$ defined in Eq. (11). The long-dashed and dashed curves in the upper panel are each multiplied by $0.5$ so that they fit comfortably within the frame. The need for such multiplication highlights the substantial impact of higher partial-waves on $R_{EM}$. Such marked sensitivity of $R_{EM}$ has been observed elsewhere Eichmann:2011aa ; Segovia:2014aza ; Segovia:2016zyc ; but the difference between our prediction for the response and that in Ref. Eichmann:2011aa shows $R_{EM}$ to be particularly susceptible to model-details.

It is here worth reiterating a conclusion from Ref. Segovia:2016zyc , viz. in the $\gamma^{\ast}p\to\Delta^{+}(1232)$ transition, $G_{E}^{\ast}$ is dominated by terms involving a scalar diquark in the proton and a pseudovector diquark in the $\Delta^{+}(1232)$, with photon-diquark interactions controlling the transition away from $x=0$. It follows that, within the dressed-quark core, the electric quadrupole transition proceeds primarily by a photon transforming the $0^{+}$-diquark in the proton into a $1^{+}$-diquark ($\delta J=1$) in the $\Delta^{+}(1232)$, with the overlap of quark-diquark components in the rest-frame Faddeev wave functions of the proton and $\Delta^{+}(1232)$ that differ by one unit of angular momentum. This explains why the shift induced by adding $P$- and $D$-waves in the $\Delta(1232)$ is especially large.

Given, too, that axial-vector diquark contributions interfere constructively with MB FSIs Ishii:1998tw ; Hecht:2002ej , then these features also indicate that $G_{E}^{\ast}$ should be most sensitive to meson cloud contributions Segovia:2014aza .

V Calculated Form Factors: $\mathbf{\Delta(1600)}$

Predictions for the $\gamma^{\ast}p\to\Delta^{+}(1600)$ transition form factors are displayed in Fig. 5. Interpolations are provided by the simple functional form in Eq. (14), with the coefficients given in Table 2. (Again, these forms should not be used for large-$x$ extrapolation.) Empirical results are here only available at the real-photon point: $G_{M}^{\ast}(Q^{2}=0)$, $G_{E}^{\ast}(Q^{2}=0)$. Evidently, the quark model results – (shaded grey band) Capstick:1994ne , dot-dashed (brown) curve Aznauryan:2015zta and dot-dot-dashed (orange) curve Aznauryan:2016wwm ) – are very sensitive to the wave functions employed for the initial and final states. Furthermore, inclusion of relativistic effects has a sizeable impact on transitions to positive-parity excited states Capstick:1994ne .

Our prediction is the solid (black) curve in each panel of Fig. 5. In this instance, every transition form factor is of unique sign on the domain displayed. Notably, the mismatches with the empirical results for $G_{M}^{\ast}(Q^{2}=0)$, $G_{E}^{\ast}(Q^{2}=0)$ are commensurate in relative sizes with those in the $\Delta(1232)$ case, suggesting that MB FSIs are of similar importance in both channels.

As remarked above, axial-vector diquark contributions interfere constructively with MB FSIs; hence, regarding form factors, one can mimic some effects of a meson cloud by modifying the axial-vector diquark content of the participating hadrons. Accordingly, to illustrate the potential impact of MB FSIs, we computed the transition form factors using an enhanced axial-vector diquark content in the proton. This was achieved by setting $m_{1^{+}}=m_{0^{+}}=0.85\,$GeV, values with which the proton’s mass is practically unchanged. The procedure produced the dotted (green) curves in Fig. 5; better aligning the $x\simeq 0$ results with experiment and suggesting thereby that MB FSIs will improve our predictions.

The short-dashed (blue) curve in Fig. 5 is the result obtained when only rest-frame $S$-wave components are retained in the wave functions of the proton and $\Delta(1600)$-baryon; and the long-dashed (blue) curve is that computed with a complete proton wave function and a $S$-wave-projected $\Delta(1600)$. Once again, the higher partial-waves have a visible impact on all form factors, with $G_{E}^{\ast}$ being most affected: the higher waves produce a change in sign. This reemphasises one of the conclusions from the quark model studies, viz. data on the $\gamma^{\ast}p\to\Delta^{+}(1600)$ transition form factors will be sensitive to the structure of the $\Delta^{+}(1600)$.

A direct comparison between the $\gamma^{\ast}p\to\Delta^{+}(1232)$ and $\gamma^{\ast}p\to\Delta^{+}(1600)$ transition form factors is presented in Fig. 6. In all cases, the $\Delta(1232)$ form factors are larger in magnitude at small $x$. However, with increasing $x$, there is always a point at which the ordering is reversed: $x\approx 2$ for $G_{M}^{\ast}$; $x\approx 0.5$ for $G_{E}^{\ast}$; and $x\approx 1$ for $G_{C}^{\ast}$. These observations indicate that the dressed-quark-core component of the $\gamma^{\ast}p\to\Delta^{+}(1600)$ transition is more localised in configuration space, i.e. more pointlike, than that of the $\gamma^{\ast}p\to\Delta^{+}(1232)$ transition. In fact, using the dominant transition form factor, $G_{M}^{\ast}$, as a guide, the $\Delta^{+}(1600)$ transition radius is $\approx 1/3$ that of the $\Delta^{+}(1232)$.

Considering $\gamma^{\ast}p\to\Delta^{+}(1232)$, helicity conservation arguments within pQCD have been used to make the follow predictions for the ratios in Eq. (11) Carlson:1985mm :

[TABLE]

up to $\ln^{2}Q^{2}$ corrections Idilbi:2003wj . These predictions disagree markedly with the outcomes produced by $SU(6)$-based quark models: $R_{EM}\equiv 0\equiv R_{SM}$; and they are inconsistent with available data Aznauryan:2011ub ; Aznauryan:2011qj . Notwithstanding such contradictions, Eqs. (15) are indubitably correct, but evidence for approach to these limits will probably not become apparent until $x\gtrsim 20$ Segovia:2013uga .

Our predictions for the ratios in Eqs. (11) associated with the $\gamma^{\ast}p\to\Delta^{+}(1600)$ transition are depicted in Fig. 7. The reasoning in Ref. Carlson:1985mm should equally apply to this case; hence, Eqs. (15) will become evident at some (very) large value of $x$. At accessible scales, however, as we have repeatedly highlighted, dynamical features of the bound-state wave functions control the $x$-dependence of these ratios. Examining Fig. 7, one sees that $R_{EM}$ for the $\Delta(1600)$ transition is far larger in magnitude than the analogous result for the $\Delta(1232)$ final state (and opposite in sign). This is an observable manifestation of the enhanced $D$-wave strength in the $\Delta(1600)$ relative to that in the $\Delta(1232)$, which is apparent in Fig. 2.

VI Form Factor Dissections: $\mathbf{\Delta(1600)}$

In connection with Eq. (12), we noted that the vertex sufficient to express the interaction of a photon with a baryon generated by the Faddeev equation in Fig. 1 is a sum of six terms, with the photon separately probing the quarks and diquarks in various ways. Hence, diverse features of quark dressing and the quark-quark correlations all play a role in determining the form factors. To elaborate, electroproduction form factors involving the nucleon and its excitations may be dissected in two separate ways, each of which can be considered as a sum of three distinct terms Segovia:2016zyc .

DD = diquark dissection

DD1

scalar diquark, $[ud]$, in both the initial- and final-state baryon,

DD2

pseudovector diquark in both the initial- and final-state ($\Delta^{+}$: $\{uu\}$ or $\{ud\}$), and

DD3

a different diquark in the initial- and final-state.

DS = scatterer dissection

DS1

photon strikes a bystander dressed-quark, with the accompanying diquark untouched (Diagram 1 in Ref. Segovia:2014aza , Fig.C.1);

DS2

photon interacts with a diquark, elastically or causing a transition scalar $\leftrightarrow$ pseudovector whilst the accompanying bystander quark is unaffected (Diagrams 2 and 4 in Ref. Segovia:2014aza , Fig.C.1); and

DS3

photon strikes a dressed-quark in-flight, as one diquark breaks up and another is formed (Diagram  3 in Ref. Segovia:2014aza , Fig.C.1), or appears in one of the two associated “seagull” terms (Diagrams 5 and 6).

The anatomy of a given transition is revealed by merging the information provided by DD and DS. With a $\Delta$-baryon in the final state, DD1 does not contribute because the $I=0$ diquark plays no role in an $I=3/2$ baryon.

The structure of $G_{M}^{\ast}$ in the $\gamma^{\ast}p\to\Delta(1600)$ transition is revealed in the upper row of Fig. 8. The left panel shows that DD2 is far stronger than DD3 and the right panel reveals that DS1 is overwhelmingly dominant, viz. the largest contribution to $G_{M}^{\ast}$ is provided by diagrams in which a photon scatters from the bystander quark, flipping its spin, in the presence of an idle pseudovector diquark. This is similar to the nature of $G_{M}^{\ast}$ in the $\gamma^{\ast}p\to\Delta(1232)$ transition Segovia:2016zyc , although the $0^{+}$-to-$1^{+}$ diquark transition component is a much smaller fraction for the $\Delta(1600)$ final state. One should also recall Fig. 6, which depicts the $x$-dependence of the relative magnitudes of $G_{M}^{\ast}$ for the two final states.

The electric quadrupole transition form factor, $G_{E}^{\ast}$, for the $\Delta(1600)$ final state is dissected in the middle row of Fig. 8. The left panel shows that DD2 and DD3 are of comparable size; and the right panel, that DS1 is dominant, with DS2 and DS3 approximately cancelling. Hence, the transition is dominated by diagrams in which the photon scatters from the bystander quark, leaving its spin unchanged, with the strength of the transition resulting from the overlap between what may be said to be quark-diquark components in the rest-frame Faddeev wave functions of the proton and $\Delta^{+}(1600)$ that differ by two units of angular momentum. This is markedly different from $G_{E}^{\ast}$ in the $\Delta(1232)$ transition, described in connection with Fig. 5.

The anatomy of $G_{C}^{\ast}$ is revealed in the bottom row of Fig. 8. Evidently, the behaviour is largely determined by DD2 and DS1 processes, i.e. the transition strength and $x$-dependence measure the overlap between $S$- and $D$-wave quark-diquark angular momentum components in the rest-frame proton and $\Delta^{+}(1600)$ Faddeev wave functions. Much the same is true in the $\gamma^{\ast}p\to\Delta(1232)$ transition Segovia:2016zyc .

VII Summary and Perspectives

We computed $\gamma^{\ast}p\to\Delta^{+}(1232),\Delta^{+}(1600)$ transition form factors using a quark-diquark approximation to the Poincaré-covariant three-body bound-state problem in relativistic quantum field theory, unifying their treatment with that of nucleon elastic form factors Segovia:2014aza and $\gamma^{\ast}N\to R$ transitions Segovia:2015hra ; Chen:2018nsg . Crucially, the diquark correlations are nonpointlike and fully-dynamical, and the Faddeev kernel ensures that every valence-quark participates actively in all diquark correlations to the fullest extent allowed by kinematics and symmetries. Moreover, each dressed-quark is characterised by a nonperturbatively generated running mass function, expressing a signature consequence of dynamical chiral symmetry breaking in the Standard Model NAP13438 ; Brodsky:2015aia . The $\Delta(1600)$-baryon generated by this approach is the simplest radial excitation of the $\Delta(1232)$ (Sec. II, Fig. 2), viz. it is analogous to the Roper resonance in the nucleon sector Burkert:2017djo .

Regarding the $\gamma^{\ast}p\to\Delta^{+}(1232)$ transition, precise measurements already exist on $0\leq Q^{2}\lesssim 8\,$GeV2 Aznauryan:2011ub ; Aznauryan:2011qj ; and on $Q^{2}\gtrsim 0.5m_{p}^{2}$, i.e. outside the meson cloud domain for this process, our calculated magnetic dipole and Coulomb quadrupole form factors agree well with this data (Sec. IV, Figs. 3, 4). Consistent with the data, too, we find that the electric quadrupole form factor is very small in magnitude; hence, it is particularly sensitive to the diquark content and quark-diquark angular-momentum structure of the baryons involved, and also to meson-baryon final-state-interactions (MB FSIs) on a larger domain than the other form factors.

Our predictions for the $\gamma^{\ast}p\to\Delta^{+}(1600)$ magnetic dipole and electric quadrupole transition form factors are consistent with the empirical values at the real photon point, but we expect inclusion of meson-baryon final-state-interactions to improve the agreement on $Q^{2}\simeq 0$ (Sec. V, Fig. 5). On the other hand, the predictions extend to $Q^{2}\approx 6m_{p}^{2}$, i.e. beyond the meson-cloud domain; hence, a meaningful direct comparison with existing data Trivedi:2018rgo ; Burkert:2019opk will be possible once the analysis is completed.

It is interesting to observe that whilst all $\gamma^{\ast}p\to\Delta^{+}(1232)$ transition form factors are larger in magnitude than those for $\gamma^{\ast}p\to\Delta^{+}(1600)$ in some neighbourhood of $Q^{2}=0$, this ordering is reversed on $Q^{2}\gtrsim 2m_{p}^{2}$ (Sec. V, Fig. 6). One can thus argue that the $\gamma^{\ast}p\to\Delta^{+}(1600)$ transition is more localised in configuration space.

It is also notable that $R_{SM}$ is qualitatively similar for both transitions considered herein; but $R_{EM}$ is markedly different, being of opposite sign on $Q^{2}\lesssim 4m_{p}^{2}$ and uniformly larger in magnitude for the $\Delta(1600)$ (Sec. V, Fig. 7). These observations again highlight the sensitivity of the electric quadrupole form factor to the degree of deformation of the $\Delta$-baryons. Diquark and scatterer dissections of the transition form factors were useful in developing an understanding of the key reaction mechanisms for each electroproduction form factor (Sec. VI).

There are numerous worthwhile extensions of our analysis, e.g. calculation of the $\gamma^{\ast}p\to N(1535)\,1/2^{-}$ transition form factors is already underway. Here the final-state is the nucleon’s parity-partner, which holds a special place in QCD owing to the manifest role of dynamical chiral symmetry breaking (DCSB) in generating the mass-splitting between this state and the nucleon. The $N(1535)\,1/2^{-}$ wave function is qualitatively different to that of the near-lying Roper resonance Chen:2017pse : the pointwise behaviour of each component is simpler, but there are more components because pseudoscalar and vector diquark correlations are also present in this negative-parity bound-state. Consequently, analyses of $\gamma^{\ast}p\to N(1535)\,1/2^{-}$ explore novel aspects of baryon structure. For instance, as $\gamma^{\ast}p\to\Delta$ transitions are sensitive to the relative strength of scalar and axial-vector diquarks within the proton (Sec. VI), then one should expect $\gamma^{\ast}p\to N(1535)\,1/2^{-}$ to reveal the relative strength of positive and negative parity diquarks in the $N(1535)\,1/2^{-}$ because negative-parity diquarks are negligible in the proton. Any predictions one makes can immediately be tested because data exists on $0\leq Q^{2}\lesssim 6m_{p}^{2}$ Aznauryan:2011ub ; Aznauryan:2011qj . Moreover, given the relative ease of separating low-lying states of opposite parity, lattice-regularised QCD may also be able to contribute Braun:2009jy .

An analogue of $\gamma^{\ast}p\to N(1535)\,1/2^{-}$ is $\gamma^{\ast}p\to\Delta(1700)3/2^{-}$, in which the final state is the $\Delta$-baryon’s parity partner. Comparison between the electroproduction form factors for this process and those calculated herein would provide additional insights into the role played by DCSB in hadron structure.

Computation of $\gamma^{\ast}p\to N(1710)\,1/2^{+}$ electroproduction form factors is also valuable because the structure of the $N(1710)\,1/2^{+}$ is unclear. In quark models, the profile of its wave function is sensitive to the formulation. For instance, it can be Roper-like, with two peaks skewed relative to those in the kindred Roper wave function Capstick:1992xn ; Melde:2008yr , in which case it may be a candidate for the system which is predominantly quark-plus-radially-excited-diquark; or it can have three peaks, located on the same trajectory as the two in the related Roper wave function Santopinto:2012nq ; deTeramond:2014asa , viz. the second radial excitation of the quark-plus-diquark system. A third possibility, realised in some dynamical coupled channels (DCC) calculations Suzuki:2009nj , sees the Roper and $N(1710)\,1/2^{+}$ as both derived from the same quark core state. Given that $N(1710)\,1/2^{+}$ electroproduction data exist on $Q^{2}\lesssim 4\,m_{p}^{2}$ Park:2014yea and that each helicity amplitude appears to be of unique sign, unlike those for the Roper Aznauryan:2009mx ; Mokeev:2012vsa ; Mokeev:2015lda , it is worth testing these possibilities by exploring the solution space of the Poincaré-covariant Faddeev equation and using the results to compute the transition form factors.

As a final class of examples, we note that a complement to the analyses highlighted above is offered by studies of electroproduction form factors for low-lying baryons with “mixed” spin-isospin structure, viz. $(I,J)=(1/2,3/2^{\pm}),(3/2,1/2^{\pm})$. For such systems, the normal level-ordering has negative-parity states lighter than positive-parity states: DCSB must still generate the (large) splitting from the ground state baryon, but the connection with parity is reversed. Data on the $N(1520)3/2^{-}$ electrocouplings are available to $Q^{2}\lesssim 4m_{p}^{2}$ Dugger:2009pn ; Aznauryan:2009mx ; Mokeev:2012vsa ; Mokeev:2015lda . Based upon this, some coupled-channels studies indicate that MB FSIs are (almost) negligible for the $A_{1/2}$ helicity amplitude JuliaDiaz:2007fa ; Aznauryan:2012ba , the calculation of which might therefore serve as a good test of the dressed-quark-core approach exploited herein.

Acknowledgements.

We are grateful for constructive comments from L. Chang, R. Gothe, V. Mokeev, K. Raya, F. Wang and S.-S. Xu; for the hospitality and support of RWTH Aachen University, III. Physikalisches Institut B, Aachen - Germany; and likewise for the hospitality and support of the University of Huelva, Huelva - Spain, and the University of Pablo de Olavide, Seville - Spain, during the “4th Workshop on Nonperturbative QCD” (University of Pablo de Olavide, 6-9 November 2018). Work supported by: National Natural Science Foundation of China, grant nos. 11535005, 11690030, 11805097; Jiangsu Province Natural Science Foundation grant no. BK20180323; Jiangsu Province Hundred Talents Plan for Professionals; Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP grant no. 2015/21550-4; U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract no. DE-AC02-06CH11357; Forschungszentrum Jülich GmbH; and Ministerio de Economia Industria y Competitividad (MINECO), under grant no. FPA2017-86380-P.