Nucleon form factors and moments of generalized parton distributions using ${N}_{f}\mathbf{=}2\mathbf{+}1\mathbf{+}1$ twisted mass fermions

Nucleon form factors and moments of generalized parton distributions using $N_{f} = 2 + 1 + 1$ twisted mass fermions

C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, and G. Koutsou

Phys. Rev. D 88, 014509 – Published 16 July 2013

Abstract

We present results on the axial and the electromagnetic form factors of the nucleon, as well as on the first moments of the nucleon generalized parton distributions using maximally twisted mass fermions. We analyze two $N_{f} = 2 + 1 + 1$ ensembles having pion masses of 213 MeV and 373 MeV each at a different value of the lattice spacing. The lattice scale is determined using the nucleon mass computed on a total of 17 $N_{f} = 2 + 1 + 1$ ensembles generated at three values of the lattice spacing, $a$ . The renormalization constants are evaluated nonperturbatively with a perturbative subtraction of $O (a^{2})$ terms. The moments of the generalized parton distributions are given in the $\bar{MS}$ scheme at a scale of $μ = 2 GeV$ . We compare with recent results obtained using different discretization schemes. The implications on the spin content of the nucleon are also discussed.

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Received 27 March 2013

DOI:https://doi.org/10.1103/PhysRevD.88.014509

Authors & Affiliations

C. Alexandrou^1,2, M. Constantinou¹, S. Dinter³, V. Drach³, K. Jansen^1,3, C. Kallidonis¹, and G. Koutsou²

¹Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
²Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus
³NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany

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Vol. 88, Iss. 1 — 1 July 2013

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Figure 1
Connected nucleon three-point function.Reuse & Permissions
Figure 2
Ratios for the matrix elements of the local axial-vector operator (upper) and one-derivative vector operator (lower) for a few exemplary choices of the momentum. The solid lines with the bands indicate the fitted plateau values with their jackknife errors. From top to bottom the momentum takes values $\vec{p} = (0, 0, 0)$ , (1, 0, 0), $(0, - 1, 0)$ and (1, 0, 1).Reuse & Permissions
Figure 3
Nucleon mass at three lattice spacings. The solid lines are fits to $O (p^{3})$ (upper panel) and $O (p^{4})$ (lower panel) $HB χ PT$ with explicit $Δ$ degrees of freedom in the so-called small scale expansion (SSE). The dotted lines denote the error band. The physical point is shown with the asterisk.Reuse & Permissions
Figure 4
Upper panel: One-derivative renormalization functions for $β = 2.10$ , $a μ = 0.0015$ using $N_{f} = 4$ gauge configurations, where $Z_{D V 1} (Z_{D A 1}) \equiv Z_{D V}^{μ μ} (Z_{D A}^{μ μ})$ and $Z_{D V 2} (Z_{D A 2}) \equiv Z_{D V}^{μ \neq ν} (Z_{D A}^{μ \neq ν})$ . Black circles are the unsubtracted data, and the magenta diamonds the data after subtracting the perturbative $O (a^{2})$ terms. For comparison, we show the subtracted data using $N_{f} = 2 + 1 + 1$ gauge configurations at the same value of the quark mass and $β$ (blue crosses). Lower panel: One-derivative renormalization functions for $β = 1.95$ using $N_{f} = 4$ gauge configurations as a function of the twisted quark mass.Reuse & Permissions
Figure 5
Upper panel: $Z_{A}$ , $Z_{V}$ for $β = 1.95$ , and $a μ = 0.0055$ . Lower panel: Renormalization constants for one-derivative operators for $β = 1.95$ and $a μ = 0.0055$ , where $Z_{D V 1} (Z_{D A 1}) \equiv Z_{D V}^{μ μ} (Z_{D A}^{μ μ})$ and $Z_{D V 2} (Z_{D A 2}) \equiv Z_{D V}^{μ \neq ν} (Z_{D A}^{μ \neq ν})$ . The lattice data are shown as black circles, and the data after the $O (a^{2})$ terms have been subtracted are shown as magenta diamonds. The solid diamond at $(a p)^{2} = 0$ is the value obtained after performing a linear extrapolation of the subtracted data.Reuse & Permissions
Figure 6
Results for the nucleon axial charge with (i) $N_{f} = 2$ twisted mass fermions with $a = 0.089 fm$ (filled red circles for $L = 2.1 fm$ and filled blue squares for $L = 2.8 fm$ ), $a = 0.070 fm$ (filled green triangles), and $a = 0.056 fm$ (open star for $L = 2.7 fm$ and open square for $L = 1.8 fm$ ) [3] and (ii) $N_{f} = 2 + 1 + 1$ twisted mass fermions with $a = 0.082 fm$ (open circle) and $a = 0.064 fm$ (square with a cross). The asterisk is the physical value as given by the PDG [21].Reuse & Permissions
Figure 7
The nucleon axial charge for twisted mass fermions, $N_{f} = 2$ (filled red circles) and $N_{f} = 2 + 1 + 1$ (filled blue squares), as well as results using other lattice actions: Filled (green) triangles correspond to a mixed action with $2 + 1$ flavors of staggered sea and domain wall valence fermions [51], crosses to $N_{f} = 2 + 1$ domain wall fermions [5], open triangles to $N_{f} = 2$ clover fermions [50] and open (cyan) circles to $N_{f} = 2 + 1$ of tree-level clover-improved Wilson fermions coupled to double HEX-smeared gauge fields [49].Reuse & Permissions
Figure 8
The nucleon axial charge for twisted mass fermions ( $N_{f} = 2$ and $N_{f} = 2 + 1 + 1$ ), as well as results using other lattice actions versus $L m_{π}$ . Black symbols denote results at almost physical pion mass obtained using $N_{f} = 2$ [50] and $N_{f} = 2 + 1$ [49] clover fermions. The rest of the notation is the same as that in Fig. 7.Reuse & Permissions
Figure 9
Comparison of the $N_{f} = 2 + 1 + 1$ twisted mass data on $G_{A} (Q^{2})$ (upper) and $G_{p} (Q^{2})$ (lower) for the two different pion masses considered. Filled blue squares correspond to $β = 2.10$ and $m_{π} = 213 MeV$ , while filled red circles correspond to $β = 1.95$ and $m_{π} = 373 MeV$ . The dashed lines are the dipole fits on the lattice data, while the solid green line is the dipole fit of experimental data for $G_{A} (Q^{2})$ [54] in combination with pion-pole dominance for $G_{p} (Q^{2})$ .Reuse & Permissions
Figure 10
The $Q^{2}$ dependence of the form factors $G_{A}$ and $G_{p}$ for (i) $N_{f} = 2$ at $m_{π} = 377 MeV$ , $a = 0.089 fm$ (filled red circles) and (ii) $N_{f} = 2 + 1 + 1$ at $m_{π} = 373 MeV$ , $a = 0.082 fm$ . The solid line in the upper plot shows the resulting dipole fit to the experimental data on $G_{A} (Q^{2})$ [54]. Assuming a pion-pole dependence for $G_{p} (Q^{2})$ and using the fit on $G_{A} (Q^{2})$ shown in the upper panel produces the solid line shown in the lower panel for $G_{p} (Q^{2})$ .Reuse & Permissions
Figure 11
The $Q^{2}$ dependence of the form factors $G_{A}$ (upper) and $G_{p}$ (lower) for $N_{f} = 2 + 1 + 1$ twisted mass fermions at $m_{π} = 213 MeV$ , $a = 0.064 fm$ (filled blue squares) and $N_{f} = 2$ twisted mass fermions at $m_{π} = 262 MeV$ and $a = 0.056 fm$ (filled red circles). The rest of the notation is the same as that in Fig. 10.Reuse & Permissions
Figure 12
$Q^{2}$ dependence of $G_{A} (Q^{2})$ for $N_{f} = 2 + 1 + 1$ at $m_{π} = 373 MeV$ (filled blue squares) and the $N_{f} = 2$ [3] at $m_{π} = 298 MeV$ (filled red circles) twisted mass data on a lattice with spatial length $L = 2.8 fm$ and similar lattice spacing. We also show results with $N_{f} = 2 + 1$ DWF at $m_{π} = 329 MeV$ , $L = 2.7 fm$ (crosses) [5] and with a hybrid action with $N_{f} = 2 + 1$ staggered sea and DWF at $m_{π} = 356 MeV$ and $L = 3.5 fm$ (open orange circles) [51].Reuse & Permissions
Figure 13
The $Q^{2}$ dependence of $G_{p} (Q^{2})$ . The notation is the same as that in Fig. 12.Reuse & Permissions
Figure 14
Comparison of the $N_{f} = 2 + 1 + 1$ twisted mass data on $G_{E} (Q^{2})$ (upper) and $G_{M} (Q^{2})$ (lower) for the two different pion masses considered. The solid lines are Kelly’s parametrization of the experimental data [55], whereas the dashed lines are dipole fits to the lattice QCD data.Reuse & Permissions
Figure 15
The $Q^{2}$ dependence of $G_{E} (Q^{2})$ (upper) and $G_{M} (Q^{2})$ (lower) for $N_{f} = 2 + 1 + 1$ TMF at $m_{π} = 213 MeV$ (filled blue squares) and $N_{f} = 2$ TMF at $m_{π} = 262 MeV$ (filled red circles).Reuse & Permissions
Figure 16
The $Q^{2}$ dependence of $G_{E} (Q^{2})$ . We show results for $N_{f} = 2 + 1 + 1$ at $m_{π} = 373 MeV$ (filled blue squares) and $N_{f} = 2$ [1] at $m_{π} = 298 MeV$ (filled red circles) TMF data on a lattice with spatial length $L = 2.8 fm$ and similar lattice spacing. We also show results with $N_{f} = 2 + 1$ DWF at $m_{π} = 297 MeV$ , $L = 2.7 fm$ (crosses) [4], with a hybrid action with $N_{f} = 2 + 1$ staggered sea and DWF at $m_{π} = 293 MeV$ and $L = 2.5 fm$ (open orange circles) [51], and $N_{f} = 2$ clover at $m_{π} = 290 MeV$ and $L = 3.4 fm$ (asterisks) [52]. The solid line is Kelly’s parametrization of the experimental data [55] from a number of experiments as given in Ref. 55.Reuse & Permissions
Figure 17
The $Q^{2}$ dependence of $G_{M} (Q^{2})$ . The notation is the same as that in Fig. 16.Reuse & Permissions
Figure 18
Twisted mass fermion results with $N_{f} = 2$ [3] and with $N_{f} = 2 + 1 + 1$ , for the isovector anomalous magnetic moment, $κ^{p - n}$ in Bohr magnetons (upper), Dirac r.m.s. radius (middle) and Pauli r.m.s. radius (lower). The notation is the same as that in Fig. 6.Reuse & Permissions
Figure 19
Results for $⟨ x ⟩_{u - d}$ (upper) and $⟨ x ⟩_{Δ u - Δ d}$ (lower) using $N_{f} = 2$ and $N_{f} = 2 + 1 + 1$ twisted mass fermions as a function of the pion mass. We show results for (i) $N_{f} = 2$ twisted mass fermions with $a = 0.089 fm$ (filled red circles for $L = 2.1 fm$ and filled blue squares for $L = 2.8 fm$ ), $a = 0.070 fm$ (filled green triangles), and $a = 0.056 fm$ (open stars for $L = 2.7 fm$ and open square for $L = 1.8 fm$ ) and (ii) $N_{f} = 2 + 1 + 1$ twisted mass fermions with $a = 0.0820 fm$ (open circle) and $a = 0.0657 fm$ (square with a cross). The physical point, shown by the asterisk, is from Ref. 59 for the unpolarized and from Refs. 62, 63 for the polarized first moment.Reuse & Permissions
Figure 20
Results for $⟨ x ⟩_{u - d}$ (upper) and $⟨ x ⟩_{Δ u - Δ d}$ (lower) obtained in this work are shown with the red filled circles for $N_{f} = 2$ and with the blue filled squares for $N_{f} = 2 + 1 + 1$ . We compare with (i) $N_{f} = 2 + 1$ DWF for $a = 0.114 fm$ [72], (ii) $N_{f} = 2 + 1$ using DWF for the valence quarks on staggered sea [51] with $a = 0.124 fm$ , and (iii) $N_{f} = 2$ clover with $a = 0.075 fm$ [73]. For $⟨ x ⟩_{u - d}$ we also show recent results using $N_{f} = 2$ clover with $a = 0.071 fm$ [6] and $N_{f} = 2 + 1$ of tree-level clover-improved Wilson fermions coupled to double HEX-smeared gauge fields with $a = 0.116 fm$ [7].Reuse & Permissions
Figure 21
The $Q^{2}$ dependence of $A_{20} (Q^{2})$ (upper) and ${\tilde{A}}_{20} (Q^{2})$ (lower) for $N_{f} = 2$ with $a = 0.056 fm$ and $m_{π} = 262 MeV$ (filled green diamonds), and $N_{f} = 2 + 1 + 1$ with (i) $a = 0.064 fm$ and $m_{π} = 213 MeV$ (filled blue squares) and (ii) $a = 0.082 fm$ and $m_{π} = 373 MeV$ (filled red circles).Reuse & Permissions
Figure 22
The $Q^{2}$ dependence of $A_{20} (Q^{2})$ (upper) and ${\tilde{A}}_{20} (Q^{2})$ (lower) shown for (i) $N_{f} = 2$ twisted mass fermions for $a = 0.089 fm$ , $m_{π} = 377 MeV$ (filled red circles) [1], (ii) $N_{f} = 2 + 1 + 1$ twisted mass fermions (this work) for $a = 0.082 fm$ and $m_{π} = 373 MeV$ (iii) $N_{f} = 2$ clover fermions for $a \sim 0.08 fm$ and $m_{π} \sim 350 MeV$ (open cyan diamonds) [74], and (iv) $N_{f} = 2 + 1$ with DWF valence on a staggered sea for $a = 0.124 fm$ and $m_{π} = 356 MeV$ (open orange circles) [51].Reuse & Permissions
Figure 23
The $Q^{2}$ dependence of $B_{20} (Q^{2})$ , $C_{20} (Q^{2})$ and ${\tilde{B}}_{20} (Q^{2})$ for $N_{f} = 2 + 1 + 1$ computed at $β = 1.95$ ( $m_{π} = 373 MeV$ ) and $β = 2.10$ ( $m_{π} = 213 MeV$ ). The dashed lines show the linear fits to $B_{20} (Q^{2})$ , $C_{20} (Q^{2})$ and ${\tilde{B}}_{20} (Q^{2})$ to extract the value at $Q^{2} = 0$ shown here.Reuse & Permissions
Figure 24
The $Q^{2}$ dependence of the isoscalar $G_{A} (Q^{2})$ , $A_{20} (Q^{2})$ and $B_{20} (Q^{2})$ for $N_{f} = 2 + 1 + 1$ computed at $β = 1.95$ ( $m_{π} = 373 MeV$ ) and $β = 2.10$ ( $m_{π} = 213 MeV$ ).Reuse & Permissions
Figure 25
Comparison of TMF results (filled symbols) to those using a hybrid action [51] (open symbols). The upper panel shows the angular momenta $J^{u}$ and $J^{d}$ for $u$ and $d$ quarks, respectively (blue filled squares for $N_{f} = 2 + 1 + 1$ and filled red circles for $N_{f} = 2$ ). The lower panel shows the quark spin (same symbols as for $J^{q}$ ) and the orbital angular momentum (filled green triangles for $N_{f} = 2$ and filled magenta diamonds for $N_{f} = 2 + 1 + 1$ ). The errors are determined by carrying out a superjackknife analysis described in Ref. 51. The experimental values of $Δ Σ^{u, d}$ are shown by the asterisks and are taken from the HERMES 2007 analysis [71].Reuse & Permissions
Figure 26
Chiral extrapolation using $CB χ PT$ (upper) and $HB χ PT$ (lower) for the angular momentum carried by the $u$ and $d$ quarks. The red band is the chiral fit using the data for $B_{20} (Q^{2} = 0)$ obtained by a linear extrapolation of $B_{20} (Q^{2})$ using $Q^{2}$ values up to $Q^{2} = 4 {GeV}^{2}$ , whereas the green band is the fit using values of $B_{20} (0)$ extracted from a linear extrapolation of $B_{20} (Q^{2})$ using $Q^{2}$ values up to $\sim 0.25 {GeV}^{2}$ . The data shown in the plot are obtained from the extended linear $Q^{2}$ extrapolation. Filled red circles are data for $N_{f} = 2$ at $β = 3.9$ , filled green triangles for $N_{f} = 2$ at $β = 4.05$ , filled magenta diamonds for $N_{f} = 2$ at $β = 4.2$ , filled light blue inverted triangle for $N_{f} = 2 + 1 + 1$ at $β = 1.95$ and filled blue square for $N_{f} = 2 + 1 + 1$ at $β = 2.10$ .Reuse & Permissions
Figure 27
Chiral extrapolation using $HB χ PT$ . The upper graph shows the spin and orbital angular momenta carried by $u$ and $d$ quarks, whereas the middle and lower graphs show the spin and orbital angular momenta carried separately by the $u$ and $d$ quarks. The errors are determined through a superjackknife analysis. The physical points, shown by the asterisks, are from the HERMES 2007 analysis [71]. The notation is the same as that in Fig. 26.Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology

Nucleon form factors and moments of generalized parton distributions using $N_{f} = 2 + 1 + 1$ twisted mass fermions

C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, and G. Koutsou

Phys. Rev. D 88, 014509 – Published 16 July 2013

Abstract

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Physical Review D

covering particles, fields, gravitation, and cosmology

Nucleon form factors and moments of generalized parton distributions using Nf=2+1+1 twisted mass fermions

C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, K. Jansen, C. Kallidonis, and G. Koutsou

Phys. Rev. D 88, 014509 – Published 16 July 2013

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Nucleon form factors and moments of generalized parton distributions using $N_{f} = 2 + 1 + 1$ twisted mass fermions