Nucleon generalized parton distributions from full lattice QCD

We present a comprehensive study of the lowest moments of nucleon generalized parton distributions in $N_f=2+1$ lattice QCD using domain-wall valence quarks and improved staggered sea quarks. Our investigation includes helicity dependent and independent generalized parton distributions for pion masses as low as 350 MeV and volumes as large as $(3.5\mathrm{fm}{)}^3$, for a lattice spacing of 0.124 fm. We use perturbative renormalization at one-loop level with an improvement based on the nonperturbative renormalization factor for the axial vector current, and only connected diagrams are included in the isosinglet channel.

Figure 1

GPDs as part of a scattering amplitude.

Figure 2

Connected (a) and disconnected (b) diagrams in unquenched lattice QCD with an operator-insertion at $\tau$ and finite momentum transfer $\Delta$.

Figure 3

Residual quark mass as a function of $L_5$ for the two samples (heavy and light) of 25 configurations each.

Figure 4

Dependence of the pion mass on the extent of the fifth dimension $L_5$ for heavy quarks.

Figure 5

Dependence of the pion mass on the extent of the fifth dimension $L_5$ for light quarks.

Figure 6

Dependence of the nucleon mass on the extent of the fifth dimension $L_5$ for heavy quarks.

Figure 7

Dependence of the nucleon mass on the extent of the fifth dimension $L_5$ for light quarks.

Figure 8

Unpolarized (vector) generalized $n=2$ form factors for the flavor combinations $u-d$ (left) and $u+d$ (right). Disconnected contributions are not included.

Figure 9

Polarized (axial vector) generalized $n=2$ form factors for the flavor combinations $u-d$ (left) and $u+d$ (right). Disconnected contributions are not included.

Figure 10

Flattening of the slope of the $A_{n0}$ GFFs with increasing $n$ for flavor combinations $u-d$ (left) and $u+d$ (right). The solid curves and error bands correspond to dipole fits described in the text. Disconnected contributions are not included.

Figure 11

Two-dimensional rms radii of the vector GFFs versus $m_{\pi }^2$ for the flavor combination $u-d$. The results for $m_{\pi }=354\mathrm{MeV}$, $L^3={20}^3$ are displayed in gray.

Figure 12

Two-dimensional rms radii of the axial vector GFFs versus $m_{\pi }^2$ for the flavor combination $u-d$. The results for $m_{\pi }=354\mathrm{MeV}$, $L^3={20}^3$ are displayed in gray and slightly shifted to the right for clarity.

Figure 13

Ratio of generalized form factors $A_{30}(t)/A_{10}(t)$ for the flavor combinations $u-d$ (left) and $u+d$ (right) compared with the parametrization in Ref. [11]. Disconnected contributions are not included.

Figure 14

Ratio of polarized generalized form factors ${\tilde{A}}_{30}(t)/{\tilde{A}}_{10}(t)$ for the flavor combinations $u-d$ (left) and $u+d$ (right) compared with the parametrization in Ref. [11]. Disconnected contributions are not included.

Figure 15

Total quark spin and orbital angular momentum contributions to the spin of the nucleon. The filled and open stars represent values given in HERMES 2007 [52] and 1999 [51] respectively, and open symbols represent earlier LHPC/SESAM calculations. The error bands are explained in the text. Disconnected contributions are not included.

Figure 16

Quark spin and orbital angular momentum contributions to the spin of the nucleon for up- and down-quarks. Squares and triangles denote $\Delta {\Sigma }^u$ and $\Delta {\Sigma }^d$ respectively, and diamonds and circles denote $L^u$ and $L^d$ respectively. The filled and open stars represent values given in HERMES 2007 [52] and 1999 [51] respectively and open symbols represent earlier LHPC/SESAM calculations. The error bands are explained in the text. Disconnected contributions are not included.

Figure 17

Lattice results for $A_{20}^{u-d}$ at $t=0{\mathrm{GeV}}^2$ versus $m_{\pi }^2$. The error band is the result of a global simultaneous chiral fit using Eqs. (24, 25, 26). The phenomenological result from CTEQ6 is indicated by the star. The heavy-baryon-limit of the CBChPT fit is shown by the dotted line, and a HBChPT fit to the lattice data for $|t|<0.3{\mathrm{GeV}}^2$ and $m_{\pi }<0.5\mathrm{GeV}$ is shown by the dashed line.

Figure 18

Lattice results for $A_{20}^{u-d}$ at $t\approx -0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 19

Lattice results for $A_{20}^{u-d}(t)$ at $m_{\pi }\approx 0.35\mathrm{GeV}$ together with the result of a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 20

Lattice results for $A_{20}^{u-d}(t)$ at $m_{\pi }\approx 0.496\mathrm{GeV}$ together with the result of a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 21

Lattice results for $A_{20}^{u-d}$ at $t=0{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with chiral fits based on Eq. (24). The four different error bands represent chiral fits to lattice results including pion masses in the regions $m_{\pi }<500$, 600, 685 and 760 MeV.

Figure 22

Chiral fits including the terms in Eq. (24) plus an additional $m_{\pi }^3$-term as described in the text. The narrow band is the original band omitting this term, the dashed line corresponds to ${\delta }_A^{(3),u-d}=1$, and the dotted line to ${\delta }_A^{(3),u-d}=-1$.

Figure 23

Lattice results for $B_{20}^{u-d}$ at $t\approx -0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 24

Lattice results for $B_{20}^{u-d}(t)$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $(-t)$ together with the result of a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 25

Lattice results for $C_{20}^{u-d}$ at $t\approx -0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 26

Lattice results for $C_{20}^{u-d}(t)$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $(-t)$ together with the result a global simultaneous chiral fit using Eqs. (24, 25, 26).

Figure 27

Lattice results for $A_{20}^{u+d}$ at $t=0{\mathrm{GeV}}^2$ versus $m_{\pi }^2$. The error band is the result of a global simultaneous chiral fit using Eqs. (27, 28, 29). The phenomenological value from CTEQ6 is denoted by a star. The heavy-baryon-limit of the CBChPT fit is shown by the dotted line, and a HBChPT fit to the lattice data for $|t|<0.3{\mathrm{GeV}}^2$ and $m_{\pi }<0.5\mathrm{GeV}$ is shown by the dashed line.

Figure 28

Lattice results for $A_{20}^{u+d}(t)$ at $t\approx -0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global simultaneous chiral fit using Eqs. (27, 28, 29).

Figure 29

Lattice results for $A_{20}^{u+d}(t)$ at $m_{\pi }\approx 0.35\mathrm{GeV}$ together with the result of a global simultaneous chiral fit using Eqs. (27, 28, 29).

Figure 30

Lattice results for $A_{20}^{u+d}(t)$ at $m_{\pi }\approx 0.496\mathrm{GeV}$ together with the result of a global simultaneous chiral fit using Eqs. (27, 28, 29).

Figure 31

Lattice results for $B_{20}^{u+d}$ at $t\approx -0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global simultaneous chiral fit based on Eqs. (27) and (29) and a variant of Eq. (28), as described in the text.

Figure 32

Lattice results for $B_{20}^{u+d}(t)$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $(-t)$ together with the result of a global simultaneous chiral fit based on Eqs. (27) and (29) and a variant of Eq. (28), as described in the text.

Figure 33

Lattice results for $C_{20}^{u+d}$ at $t\approx -0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global simultaneous chiral fit using Eqs. (27, 28, 29).

Figure 34

Lattice results for $C_{20}^{u+d}(t)$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $(-t)$ together with the result of a global simultaneous chiral fit using Eqs. (27, 28, 29).

Figure 35

Combined $(t,m_{\pi })$-dependence of $C_{20}^{u+d}$ from a global simultaneous chiral fit using Eqs. (27, 28, 29) compared to lattice data.

Figure 36

Lattice results for $E_{20}^{u+d}$ at $t=0$ versus $m_{\pi }^2$ together with the result of a global chiral fit using Eq. (30).

Figure 37

Lattice results for $E_{20}^{u+d}$ at $-t\approx 0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global chiral fit using Eq. (30).

Figure 38

Lattice results for $E_{20}^{u+d}$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $-t$ together with the result of a global chiral fit using Eq. (30).

Figure 39

Lattice results for $E_{20}^{u+d}$ at $m_{\pi }\approx 496\mathrm{MeV}$ versus $-t$ together with the result of a global chiral fit using Eq. (30).

Figure 40

Lattice results for $M_{20}^{u+d}$ at $|t|\approx 0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global chiral fit using Eq. (31).

Figure 41

Lattice results for $M_{20}^{u+d}$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $(-t)$ together with the result of a global chiral fit using Eq. (31).

Figure 42

Lattice results for $B_{20}^{u+d}$ at $|t|\approx 0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global chiral fit using a linear combination of Eqs. (30) and (31).

Figure 43

Lattice results for $B_{20}^{u+d}$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $(-t)$ together with the result of a global chiral fit using a linear combination of Eqs. (30) and (31).

Figure 44

Combined $(t,m_{\pi })$-dependence of the quark anomalous gravitomagnetic moment $B_{20}^{u+d}$ from a global chiral fit using a linear combination of Eqs. (30) and (31) compared to lattice data.

Figure 45

Lattice results for $C_{20}^{u+d}$ at $|t|\approx 0.24{\mathrm{GeV}}^2$ versus $m_{\pi }^2$ together with the result of a global chiral fit using Eq. (32).

Figure 46

Lattice results for $C_{20}^{u+d}$ at $m_{\pi }\approx 350\mathrm{MeV}$ versus $-t$ together with the result of a global chiral fit using Eq. (32).

Figure 47

Combined $(t,m_{\pi })$-dependence of $C_{20}^{u+d}$ from a global chiral fit using Eq. (32) compared to lattice data.

Figure 48

Chiral extrapolation of $J^{u+d}$ using HBChPT including the $\Delta$ resonance, Eq. (34). The fit and error band on the axis are explained in the text.

Figure 49

Comparison of a global fit to $A_{20}^{u+d}+B_{20}^{u+d}$ using Eq. (31) and the lattice results for $J^{u+d}$. The fit and error band on the axis are explained in the text.