Nucleon structure from mixed action calculations using $2+1$ flavors of asqtad sea and domain wall valence fermions

We present high statistics results for the structure of the nucleon from a mixed-action calculation using $2+1$ flavors of asqtad sea and domain-wall valence fermions. We perform extrapolations of our data based on different chiral effective field theory schemes and compare our results with available information from phenomenology. We discuss vector and axial form factors of the nucleon, moments of generalized parton distributions, including moments of forward parton distributions, and implications for the decomposition of the nucleon spin.

Figure 1

Layout of smeared nucleon sources (stars) and coherent, fixed momentum sinks (vertical lines) on our lattices: (left) even configurations; (right) odd configurations.

Figure 2

Comparison of coherent sink technique and two source-sink separations with our previous calculation. The observables are the axial charge, $g_A$ in the left panel [see Sec. 4a], and the isovector charge radius, $a^{-2}⟨r_1^2⟩$ in the right panel [see Sec. 4b].

Figure 3

Comparison of two separations on two volumes for the isovector axial charge, $g_A$. This plot shows the resulting plateau plot, i.e. $g_A(T)$ as a function of the location of the operator insertion at fixed source-sink separation.

Figure 4

Comparison of bootstrap (left panels) and superjackknife (right panels) resampling plans for chiral fits to the nucleon axial charge, $g_A$. The upper two plots show fits to the lowest three masses, the lower two show fits to the lowest four masses.

Figure 5

Mass plateau at $am=0.010$ on the ${20}^3$ and the ${28}^3$ lattices. The band corresponds to Eq. (26).

Figure 6

Isovector Dirac radius $⟨r_1^2⟩$ as a function of the pion mass for all ensembles.

Figure 7

Isovector Pauli radius, $⟨r_2^2⟩$, and anomalous magnetic moment, ${\kappa }_v^{\mathrm{norm}}\equiv \frac{m_N^{\mathrm{phys}}}{m_N(m_{\pi })}{F}_2^v(0)$, as a function of the pion mass for all ensembles.

Figure 8

Direct comparison of finite-volume effects for the isovector form factors, $F_1^v(Q^2)$ and $F_2^v(Q^2)$, at $m_{\pi }=356\mathrm{MeV}$ on the two volumes $\Omega ={28}^3\times 64$ and $\Omega ={20}^3\times 64$.

Figure 9

SSE fit to the axial charge. Left: three-parameter fit with $m_{\pi }<500\mathrm{MeV}$. Right: two-parameter fit with $m_{\pi }<360\mathrm{MeV}$.

Figure 10

Chiral fits to isovector Dirac radii, $⟨r_1^2⟩$, for $m_{\pi }<500\mathrm{MeV}$ (SSE) and $m_{\pi }<400\mathrm{MeV}$ (BChPT).

Figure 11

SSE best fit to isovector form factor $F_1^v(Q^2,m_{\pi }=293\mathrm{MeV})$ with lattice data for that ensemble.

Figure 12

Isovector Dirac radii $⟨r_1^2⟩$ with best fit using kinematic cuts $Q^2<0.5{\mathrm{GeV}}^2$ and $m_{\pi }<400\mathrm{MeV}$. The data points shown are dipole radii and experiment, which serve to compare this method with the previous one. They have not been included in the fit.

Figure 13

Comparison of dipole and tripole fits for isovector $F_2^v(Q^2)$ on the ${28}^3$ lattice at $m_{\pi }=356\mathrm{MeV}$.

Figure 14

Results of SSE and BChPT fits to the normalized anomalous magnetic moment ${\kappa }_v^{\mathrm{norm}}$ (left panel) and to the isovector mean squared radius $⟨r_2^2⟩$ (right panel).

Figure 15

Isovector form factor $F_2^v(Q^2,m_{\pi })$ lattice data with best fit SSE at $m_{\pi }=293\mathrm{MeV}$.

Figure 16

Isovector normalized anomalous magnetic moment ${\kappa }_v^{\mathrm{norm}}$ (left panel) and mean squared radius $⟨r_2^2⟩$ (right panel) as a function of the pion mass as obtained from an SSE fit to the chiral expansion of $F_2^v(Q^2,m_{\pi })$. The data points shown are tripole radii and experiment, which serve to compare this method to the previous one and to phenomenology. They have not been included in the fit.

Figure 17

Data for the ratio of $Q{F}_2^v(Q^2)/F_1^v(Q^2)$.

Figure 18

Comparison between dipole and tripole fit for isovector $G_M(Q^2)$.

Figure 19

Linear fits to $F_1^d/F_1^u(Q^2)$ ratio.

Figure 20

Chiral fit to the slope of the ratio $F_1^d/F_1^u(Q^2=0{\mathrm{GeV}}^2)$. The physical pion mass is marked by the dashed red line.

Figure 21

Linear slope of $F_1^d/F_1^u(Q^2)$ at the physical pion mass.

Figure 22

Isoscalar form factor $F_2^s(Q^2)$ as a function of $Q^2$ with SSE fit at $m_{\pi }=293\mathrm{MeV}$.

Figure 23

Comparison of dipole and chiral fit to $G_A(Q^2)$ for the ${28}^3$ lattice at $m_{\pi }=356\mathrm{MeV}$.

Figure 24

Comparison of pion-pole fits to isovector $G_P(Q^2)$ on the ${28}^3$ lattice with fixed pion pole and with the pion pole as a free parameter.

Figure 25

Dipole fit to the isoscalar axial form factor ${\tilde{A}}_{10}(Q^2)$ on the ${28}^3$ lattice at $m_{\pi }=356\mathrm{MeV}$.

Figure 26

Dipole fit to the isoscalar axial form factor ${\tilde{B}}_{10}(Q^2)$ on the ${28}^3$ lattice at $m_{\pi }=356\mathrm{MeV}$.

Figure 27

The unpolarized isovector and isosinglet GFFs $A_{20}$, $B_{20}$, $C_{20}$.

Figure 28

The polarized isovector and isosinglet GFFs ${\tilde{A}}_{20}$ and ${\tilde{B}}_{20}$.

Figure 29

Generalized form factors of twist-two operators of dimensions 3, 4, and 5, in the isovector and isosinglet channels. Notice, in particular, the flattening of the slope of the $A_{n0}$ GFF with increasing $n$.

Figure 30

Transverse isovector radii as extracted from a dipole fit with momentum cut $|t|<0.5{\mathrm{GeV}}^2$. The left panel corresponds to the unpolarized case, and the right panel to the polarized case. At $m_{\pi }=356\mathrm{MeV}$, the radius on the ${28}^3$ lattice is in all cases larger than on the ${20}^3$ lattice.

Figure 31

Ratio of the $A_{30}$ to the $A_{10}$ unpolarized GFFs, isovector and isosinglet. Disconnected contributions have been omitted.

Figure 32

Ratio of the ${\tilde{A}}_{30}$ to the ${\tilde{A}}_{10}$ polarized GFFs, isovector and isosinglet. Disconnected contributions have been omitted.

Figure 33

Ratios of $n=2$ generalized form factors.

Figure 34

Ratios of $n=3$ generalized form factors.

Figure 35

The extrapolation of the polarized momentum fraction using HBChPT. The value obtained in the chiral limit is 0.165(8), and the experimental value is from HERMES [49].

Figure 36

Simultaneous BChPT fit to isovector (left) and isosinglet (right) lattice data for $m_{\pi }<580\mathrm{MeV}$. The dotted line is the heavy-baryon limit of the BChPT fit. A HBChPT fit to the lattice data for $|t|<0.3{\mathrm{GeV}}^2$ and $m_{\pi }<500\mathrm{MeV}$ is shown by the dashed line. The phenomenological value of $⟨x{⟩}_{u\pm d}$ (CTEQ6) is indicated by the cross. Disconnected contributions are omitted in the isosinglet case. Notice that $B_{20}$ and $C_{20}$ are displayed for $t\approx -0.24{\mathrm{GeV}}^2$. At $m_{\pi }^2\approx 0.12{\mathrm{GeV}}^2$, the larger volume (${28}^3$) is the filled symbol, the smaller volume (${20}^3$) the open symbol; for all other pion masses, the volume is ${20}^3$. The ${\chi }^2/\mathrm{dof}$ is 1.5 and 2.8, respectively, in the isovector and isosinglet cases.

Figure 37

A study of the stability of the BChPT extrapolation of $A_{20}^{u-d}$ at $t=0{\mathrm{GeV}}^2$ based on Eqs. (61, 62, 63). The three different error bands represent chiral fits to lattice results for pion masses in the regions $m_{\pi }<400$, 500, and 600 MeV, respectively.

Figure 38

The $t$ and $m_{\pi }^2$ dependence of the isosinglet generalized form factor $C_{20}$. The two sheets indicate the statistical uncertainty of the BChPT fit.

Figure 39

Pion-mass and momentum transfer dependence of $E_{20}$. At $m_{\pi }=356\mathrm{MeV}$, the open symbol corresponds to ${20}^3$, the filled symbol to ${28}^3$. At all other pion masses, the volume is ${20}^3$. The extrapolation in $m_{\pi }$ at $t=0$ can be compared to the extrapolation of $A_{20}^{u+d}$ in Fig. 36.

Figure 40

The $t$ and $m_{\pi }^2$ dependence of the isosinglet generalized form factors $\frac{m_N^{\mathrm{phys}}}{m_N(m_{\pi })}{B}_{20}$ and $\frac{m_N^{\mathrm{phys}}}{m_N(m_{\pi })}{C}_{20}$, and a HBChPT fit. Its uncertainty is indicated by the two surfaces.

Figure 41

Pion and momentum transfer dependence of $M_{20}$. At $m_{\pi }=356\mathrm{MeV}$, the open symbol corresponds to ${20}^3$, the filled symbol to ${28}^3$. At all other pion masses, the volume is ${20}^3$.

Figure 42

Projections of $C_{20}^{u+d}$—as depicted in Fig. 40—on a constant $-t$ plane and on a constant $m_{\pi }^2$ plane. The HBChPT fit appears as a shaded band.

Figure 43

Chiral extrapolations of $J^{u,d}$ using BChPT. Note that the displayed lattice data points were not directly employed in the chiral fits. Details are given in the text.

Figure 44

Chiral extrapolation of $J^{u+d}$ using HBChPT including the $\Delta$ resonance, Eq. (74). The fit and error bands are explained in the text.

Figure 45

Total quark spin and orbital angular momentum contributions to the spin of the proton. The cross represents the value from the HERMES 2007 measurement [49]. The error bands are explained in the text. Disconnected contributions are not included.

Figure 46

Quark spin and orbital angular momentum contributions to the spin of the proton for up and down quarks. Filled and open squares denote $\Delta {\Sigma }^u/2$ and $\Delta {\Sigma }^d/2$, and filled and open diamonds denote $L^u$ and $L^d$, respectively. The crosses represent the values from the HERMES 2007 measurement [49]. The error bands are explained in the text. Disconnected contributions are not included.

Figure 47

Leading order evolution of lattice results for $L^{u-d}$ (solid line) and $\Delta {\Sigma }^{u-d}$ (dashed line) from ${\mu }_0^2=4{\mathrm{GeV}}^2$ down to low hadronic scales.