How Bright is the Proton? A Precise Determination of the Photon Parton Distribution Function

It has become apparent in recent years that it is important, notably for a range of physics studies at the Large Hadron Collider, to have accurate knowledge on the distribution of photons in the proton. We show how the photon parton distribution function (PDF) can be determined in a model-independent manner, using electron-proton ($ep$) scattering data, in effect viewing the $ep\to e+X$ process as an electron scattering off the photon field of the proton. To this end, we consider an imaginary, beyond the Standard Model process with a flavor changing photon-lepton vertex. We write its cross section in two ways: one in terms of proton structure functions, the other in terms of a photon distribution. Requiring their equivalence yields the photon distribution as an integral over proton structure functions. As a result of the good precision of $ep$ data, we constrain the photon PDF at the level of 1%–2% over a wide range of momentum fractions.

Figure 1

Our breakup of the $(x,Q^2)$ plane and the data for $F_2(x,Q^2)$ and $F_L(x,Q^2)$ we use in each region. The white region is inaccessible at leading order in QED.

Figure 2

Contributions to the photon PDF at $\mu =100\mathrm{GeV}$, multiplied by $10^3{x}^{0.4}/(1-x{)}^{4.5}$, from the various components discussed in the text. For the inelastic part, the area below the white line is the contribution from $Q^2\le 1{(\mathrm{GeV})}^2$ in Eq. (6) including the elastic part. The PDF would be the dashed blue line without the $\overline{\mathrm{MS}}$ conversion term.

Figure 3

Linearly stacked relative uncertainties on the photon PDF, from all sources we have considered, and their total sum in quadrature shown as a black line, which is our final uncertainty.

Figure 4

The ratio of common PDF sets to our luxqed result, along with the luxqed uncertainty band (light red). The ct14 and mrst bands correspond to the range from the PDF members shown in brackets (68% cl. in ct14’s case). The nnpdf bands span from $\max ({\mu }_r-{\sigma }_r,r_{16})$ to ${\mu }_r+{\sigma }_r$, where ${\mu }_r$ is the average (represented by the blue line), ${\sigma }_r$ is the standard deviation over replicas, and $r_{16}$ denotes the ${16}^{\mathrm{t}\mathrm{h}}$ percentile among replicas. Note the different $y$ axes for the panels.

Figure 5

$\gamma \gamma$ luminosity in $pp$ collisions as a function of the $\gamma \gamma$ invariant mass $M$, at four collider center-of-mass energies. The nnpdf30 results are shown only for 8 and 100 TeV. The uncertainty of our luxqed results is smaller than the width of the lines.