Nucleon form factors, generalized parton distributions and quark angular momentum

Abstract

We extract the individual contributions from u and d quarks to the Dirac and Pauli form factors of the proton, after a critical examination of the available measurements of electromagnetic nucleon form factors. From this data we determine generalized parton distributions for valence quarks, assuming a particular form for their functional dependence. The result allows us to study various aspects of nucleon structure in the valence region. In particular, we evaluate Ji’s sum rule and estimate the total angular momentum carried by valence quarks at the scale μ=2 GeV to be $J_{v}^{u} = 0.230^{+ 0.009}_{- 0.024}$ and $J_{v}^{d} = -0.004^{+ 0.010}_{- 0.016}$.

Appendices

Appendix A: Tables of form factors

In Table 20 we give the values and errors of R ⁿ for the data set we have selected. Except for the entries from Plaster 05, Geis 08 and Riordan 10, which directly quote results on R ⁿ, we have computed this ratio from $G_{E}^{n}$ and the assumed value of $G_{M}^{n}$ for the reasons explained in Sect. 3.2.

Table 20 Values of R ⁿ measured in polarization experiments on deuterium or ³He. References for the data sets are given in Table 4

In Table 21 we list the results for the flavor form factors we have extracted from our default data set as explained in Sect. 5.1.

Table 21 The flavor form factors we obtain by interpolation of the data, as explained in Sect. 5.1

Appendix B: Matrices for computing fit errors

In this appendix we give the information that is needed to compute parametric errors for our default fit ABM 1 and for the power-law fit of Sect. 3.4. A convenient procedure to propagate errors is the so-called Hessian method used in modern PDF determinations, see e.g. [64, 128]. We briefly describe this method and then list the relevant matrices.

Let us introduce the column vector p of the n original fit parameters, as well as the vector of transformed parameters z defined by

$$ \boldsymbol{p} - \boldsymbol{p}_0 = E \boldsymbol{z} , $$

(B.1)

where p ₀ is the set of parameters that minimizes χ ². The matrix E satisfies

$$ E E^{T} = V $$

(B.2)

with the standard covariance matrix V for the parameters p. The deviation of χ ² from its minimum value is then given by

$$ \varDelta \chi^2 = (\boldsymbol{p} - \boldsymbol{p}_0)^T V^{-1} ( \boldsymbol{p} - \boldsymbol{p}_0) = \boldsymbol{z}^T \boldsymbol{z} . $$

(B.3)

The error on a function f of the parameters, as given by linear error propagation, can be written as

(B.4)

where the parameter set $\boldsymbol{p}_{i}^{\pm}$ is specified by the condition

$$ \bigl( \boldsymbol{p}_i^{\pm} - \boldsymbol{p}_0 \bigr)_j = \pm E_{ji} . $$

(B.5)

In the second step of (B.4) we have approximated the derivative by a difference quotient, which is consistent in the region where linear error propagation is adequate. The vector given by the ith column of the matrix E thus gives the amount by which the central values of the parameters need to be shifted to obtain a set of parameters on the Δχ ²=1 contour.

In Table 22 we give the matrix E _D for the default GPD fit (ABM 1) and the matrix E _P for the power law fit of Sect. 3.4. The order of entries in the matrices corresponds to the following vectors of parameters:

(B.6)

The central values of the fit parameters are given in (50) and Table 11 for the GPD fit, and those of the power-law fit are given in Table 5.

Table 22 The matrix E _D for the default GPD fit ABM 1 and the matrix E _P for the power-law fit of Sect. 3.4. Entries are given in units of GeV⁻² for rows 1 to 9 in E _D and for rows 1 to 8 in E _P