Systematic uncertainties in parton distribution functions from lattice QCD simulations at the physical point

We present a detailed study of the helicity-dependent and helicity-independent collinear parton distribution functions (PDFs) of the nucleon, using the quasi-PDF approach. The lattice QCD computation is performed employing twisted mass fermions with a physical value of the light quark mass. We give a systematic and in-depth account of the salient features entering in the evaluation of quasi-PDFs and their relation to the light-cone PDFs. In particular, we give details for the computation of the matrix elements, including the study of the various sources of systematic uncertainties, such as excited-states contamination. In addition, we discuss the nonperturbative renormalization scheme used here and its systematics, effects of truncating the Fourier transform and different matching prescriptions. Finally, we show improved results for the PDFs and discuss future directions, challenges and prospects for evaluating precisely PDFs from lattice QCD with fully quantified uncertainties.

Figure 1

Schematic representation of the nucleon two- (left) and three-point (right) functions. The creation point of the nucleon is at $(\mathbf{0},0)$, while the annihilation is at $(\mathbf{x},t_s)$ and $(\mathbf{x},t_s)$ for the two- and three-point functions, respectively. The solid lines represent the quark propagators and the curly line denotes the Wilson line of length $z$.

Figure 2

Left: Relative error of the nucleon two-point correlator as a function of the Euclidean time $t/a$, for $P_3=8\pi /L$ and different values of the momentum smearing parameter $\xi$. The value $\xi =0$ (black points) corresponds to the standard Gaussian smearing. Right: Effective energy of the nucleon with boost $P_3=8\pi /L$ for different values of $\xi$.

Figure 3

Scaling of the statistical errors of the matrix elements, for the operator with $z=0$, varying the number of source positions on each configuration (left) and averaging over the directions of the nucleon boost (right). The absolute errors for the unpolarized, helicity and transversity matrix elements (red circles, blue squares, and yellow triangles, respectively) are normalized to the one obtained from only one point source or one boost direction. The results are compared with the ideal scaling (dashed line in both figures) expected for uncorrelated measurements.

Figure 4

Nucleon energy as a function of the momentum in lattice units. From the fit of the continuum dispersion relation, $E^2=m_N^2{c}^4+P_3^2{c}^2$, to the lattice data (red squares), we find a value of the squared speed of light equal to $c^2=1.00(3)$, and the nucleon mass in lattice units $a^2{m}_N^2{c}^4=0.207(4)$, which is also in agreement with the estimate given in Ref. [79].

Figure 5

Real and imaginary parts of the bare matrix elements for 0, 5, 10, and 15 steps of stout smearing at a nucleon momentum of $P_3=8\pi /L$. From top to bottom: The bare nucleon matrix element for the unpolarized, helicity and transversity operators.

Figure 6

Comparison of the spatial ${\gamma }^3$ (cyan circles) and temporal ${\gamma }^0$ (purple diamonds) unpolarized bare matrix elements for 5 stout steps and momentum $8\pi /L$.

Figure 7

Bare matrix elements for momenta $6\pi /L$ (green circles), $8\pi /L$ (red diamonds), and $10\pi /L$ (blue stars), extracted at a source-sink separation of $t_s=12a\simeq 1.13\mathrm{fm}$. From top to bottom: Matrix elements of the unpolarized, helicity and transversity operators.

Figure 8

Effective mass together with the extracted values of the ground-state energy from one-state (blue band) and two-state fits (orange band). The length of the bands indicates the fit range. The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 9

Real (left) and imaginary (right) parts of the matrix element for the unpolarized PDF, at a source-sink separation of $8a$ (red stars), $9a$ (blue circles), $10a$ (green diamonds) and $12a$ (yellow squares). The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 10

Real (left) and imaginary (right) parts of the matrix element for the unpolarized PDF from the summation method, using source-sink separations $8a–12a$ (red circles) and $9a–12a$ (blue squares). The nucleon is boosted with momentum $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 11

Summation fits for the real part of the matrix element for the unpolarized PDF, for $z/a=0$ (left) and $z/a=12$ (right). In both cases, the (a) fits to all source-sink separations and the (b) fits to the three largest separations are shown. The nucleon boost is $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 12

Real (left) and imaginary (right) parts of the matrix element for the unpolarized PDF from two-state fits, using source-sink separations $8a–12a$ (circles) and $8a–10a$ (squares). The sequential fit (filled symbols) is also compared to the simultaneous fit (open symbols). The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 13

Real (left) and imaginary (right) parts of the matrix element for the unpolarized PDF from the plateau method (points labeled with appropriate $t_s$), the summation method (using $t_s\ge 9a$) and sequential two-state fits (to $t_s=8a$, $9a$, $10a$ and to all $t_s$). The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 14

Real (left) and imaginary (right) parts of the matrix element for the unpolarized PDF at fixed $z/a=5$ (upper) and $z/a=10$ (lower). We compare results from the plateau method, the summation method (using $t_s\ge 9a$) and sequential two-state fits (to $t_s=8a$, $9a$, $10a$ and to all $t_s$). The final value that we use for extracting PDFs is indicated with an open symbol. The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 15

Real (left) and imaginary (right) parts of the matrix element for the helicity PDF from the plateau method (points labeled with appropriate $t_s$), the summation method (using all $t_s$) and sequential two-state fits (to $t_s=8a$, $9a$, $10a$ and to all $t_s$). The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 16

Real (left) and imaginary (right) parts of the matrix element for the transversity PDF from the plateau method (points labeled with appropriate $t_s$), the summation method (using all $t_s$) and sequential two-state fits (to $t_s=8a$, $9a$, $10a$ and to all $t_s$). The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 17

Real (left) and imaginary (right) parts of the matrix element for the helicity and transversity PDFs at fixed $z/a=5$ (upper) and $z/a=10$ (lower). We compare results from the plateau method, the summation method (using all $t_s$) and sequential two-state fits (to $t_s=8a$, $9a$, $10a$ and to all $t_s$). The final value that we use for extracting PDFs is indicated with an open symbol. The nucleon is boosted with $10\pi /L\simeq 1.38\mathrm{GeV}$.

Figure 18

Real (left) and imaginary (right) parts of the $Z$ factor $Z_{\mathcal{O}}^{{\mathrm{RI}}^{\prime }}(z,{\mu }_0,m_{\pi })$ for the unpolarized (blue circles), helicity (red squares) and transversity (orange diamonds) PDFs as a function of $m_{\pi }^2$ of the ensemble used. The scale is $(a{\mu }_0{)}^2=2.5$. The upper and lower panels show the results for $z/a=1$ and $z/a=5$, respectively. The dashed line corresponds to the chiral fit of Eq. (22), and the open symbols are the extrapolated values, $Z_{\mathcal{O},0}^{{\mathrm{RI}}^{\prime }}(z,{\mu }_0)$.

Figure 19

Plots of ${\mathcal{R}}_{\mathcal{O}}(z)$ and ${\mathcal{I}}_{\mathcal{O}}(z)$ to study volume effects between the ensembles $48^3\times 96$ and $64^3\times 128$, for the unpolarized (circles), helicity (diamonds) and transversity (squares) operators as a function of the length of the Wilson line, $z/a$. The ${\mathrm{RI}}^{\prime }$ scale is $(a{\mu }_0{)}^2=2.5$ and the pion mass is 130 MeV.

Figure 20

Real (left) and imaginary (right) parts of $Z_{\mathcal{O}}^{\mathrm{M}\overline{\mathrm{MS}}}(z,\overline{\mu },{\mu }_0)$ for the unpolarized (squares), helicity (circles) and transversity (diamonds) operators as a function of the initial ${\mathrm{RI}}^{\prime }$ scale. The upper (lower) panel corresponds to $z/a=1$ ($z/a=5$). The dashed line corresponds to the fit of Eq. (24), and the open symbols are the extrapolated values ${\mathcal{Z}}_{\mathcal{O},0}^{\mathrm{M}\overline{\mathrm{MS}}}$. We show results for all $Z$ factors with $Z_{V_0}$ and $Z_T$ shifted vertically for clarity of the presentation. The darker colored filled symbols correspond to the purely nonperturbative estimates, while the corresponding lighter colored open symbols have been partly improved using the perturbative results of the finite lattice spacing effects to $\mathcal{O}(g^2{a}^{\infty })$ computed in Ref. [104] on the same ensembles.

Figure 21

Final values of the $Z$ factors in the $\mathrm{M}\overline{\mathrm{MS}}$ scheme for the vector operator upon chiral and $(a{\mu }_0{)}^2$ extrapolation. The real and imaginary parts are shown in the left and right panels, respectively.

Figure 22

Real (left) and imaginary (right) parts of renormalized matrix elements for the axial operator (helicity PDF) for a momentum of $\frac{6\pi }{L}$, as a function of the length of the Wilson line. Green stars/blue circles/red diamonds/orange stars correspond to $0/5/10/15$ iterations of stout smearing.

Figure 23

One-loop diagrams entering the perturbative calculation for the matching between quasi-PDFs and light-cone PDFs.

Figure 24

Comparison of matched PDFs from quasi-PDFs renormalized in the $\overline{\mathrm{MS}}$ (red band) scheme and in the $\mathrm{M}\overline{\mathrm{MS}}$ (cyan band) scheme, for the unpolarized (top left panel), helicity (top right panel) and transversity (bottom panel) cases and for a nucleon momentum of $P_3=10\pi /L$.

Figure 25

Comparison of matched PDFs from quasi-PDFs renormalized in the $\mathrm{M}\overline{\mathrm{MS}}$ scheme (green band) and in the “ratio” scheme (cyan band) for the unpolarized case. The nucleon momentum is $P_3=10\pi /L$.

Figure 26

Left panel: Comparison of renormalized quasi-PDFs in the RI scheme with $\cancel{p}$ projection (red band) and the $\mathrm{M}\overline{\mathrm{MS}}$ scheme (green band). Right panel: Matched PDFs obtained from the same bare matrix elements, but with different scheme conversion, evolution and matching procedures: either one-step ($\mathrm{RI}\to \overline{\mathrm{MS}}$) or two-step (${\mathrm{RI}}^{\prime }\to \mathrm{M}\overline{\mathrm{MS}}\to \overline{\mathrm{MS}}$) (see text for details). This is for the unpolarized case, with a nucleon momentum of $P_3=10\pi /L$.

Figure 27

Real (left) and imaginary (right) parts of the bare (filled symbols) and $\mathrm{M}\overline{\mathrm{MS}}$-renormalized (open symbols) matrix elements for the unpolarized (upper row), helicity (middle row) and transversity PDFs (lower row) (for a nucleon momentum of $P_3=10\pi /L$).

Figure 28

Matched unpolarized PDF (upper right), helicity PDF (lower left) and transversity PDF (lower right) for different values of the cutoff $z_{\max }/a=10$ (red), 12 (blue), 14 (green). Our final choice is always the middle value of $z_{\max }/a$ from the ones shown. The nucleon momentum is $P_3=10\pi /L$ for all plots.

Figure 29

The unpolarized quasi-PDFs for $P_3=10\pi /L$. We show the bare matrix element (purple), ${\mathrm{RI}}^{\prime }$ (yellow) and $\overline{\mathrm{MS}}$ (cyan) quasi-PDFs for $z_{\max }=10a$.

Figure 30

Comparison of matched PDFs obtained from the same renormalized matrix elements, but with different Fourier transform definitions: the standard one (green band) or the “derivative” one (red band). The matching procedure is the same for both and proceeds via the $\mathrm{M}\overline{\mathrm{MS}}$ scheme. We show the unpolarized (upper left), helicity (upper right) and transversity cases (lower). The nucleon momentum is $P_3=10\pi /L$.

Figure 31

Effect of matching and nucleon mass corrections in the unpolarized PDF (upper left), helicity PDF (upper right) and transversity PDF (lower). The nucleon momentum is $P_3=10\pi /L$ for all plots.

Figure 32

Nucleon boost dependence for the unpolarized PDF (upper left), helicity PDF (upper right) and transversity PDF (lower), using $P_3=6\pi /L$ (green curve), $P_3=8\pi /L$ (red curve) and $P_3=10\pi /L$ (blue curve).

Figure 33

Dependence of the unpolarized PDF (upper), helicity PDF (middle) and transversity PDF (lower) on the nucleon momentum for six selected values of $x$.

Figure 34

Final results for the unpolarized PDF (upper left), helicity PDF (upper right) and transversity PDF (lower), using the largest momentum $P_3=10\pi /L$ (blue curve). The global fits of Refs. [114, 115, 116] (unpolarized), Refs. [117, 118, 119] (helicity), and Refs. [120] (transversity) are shown for qualitative comparison.