Past exam of the particle physics course of the University of Oxford / 2018 / 2

The valence quark distributions describe net quark content: $\int u_v(x)dx=2$ and $\int d_v(x)dx=1$ for a proton. Their momentum densities $x{u}_v,x{d}_v$ are small at very small $x$, peak at moderate momentum fractions, and vanish toward $x=1$, where one parton carrying almost all proton momentum is unlikely. The larger $u$ distribution reflects two valence up quarks and its comparatively harder shape; its pointwise value need not be exactly twice the down distribution.

The sea quark distribution is generated largely by gluon radiation and $g\to q\bar{q}$ splitting. It grows strongly toward small $x$ and falls at large $x$. The plotted sea curve is scaled by $0.05$, so it must be multiplied by twenty before numerical use. The total up-quark PDF isusing the valence-up distribution and the up-sea distribution. The summed sea curve alone does not fix $\bar{u}$ without a flavour assumption. Under the later stated four-flavour symmetry, $\bar{u}=\Sigma /8$.

At leading order, neglect the transverse momentum of the incoming partons and of the produced on-shell boson. In the hadron centre-of-mass frame,Then $m_W^2=(p_{+}+p_{-}{)}^2=x_{+}{x}_{-}s$, andSolving these product and ratio equations givesOver $-2<y<2$, each momentum fraction ranges betweenThey are anticorrelated: increasing $y$ increases $x_{+}$ while decreasing $x_{-}$. Higher-order recoil changes the simple two-to-one kinematics.

The leading subprocess is $u\bar{d}\to W^{+}$. There are two beam assignments, and charge conjugation relates antiproton PDFs to proton PDFs. ThereforeIn a narrow-width leading-order normalization the prefactor is $\pi g^2/(12s)$ when $|V_{ud}|=1$; only the PDF dependence is needed for the fractions below.

Write the common sea PDF as $S=\bar{u}=\bar{d}=\bar{s}=\bar{c}=\Sigma /8$. The Valence and sea decomposition of proton-antiproton W production gives the three parton luminositiesThe factor two in the sea-sea term includes both beam assignments. The antiproton's valence antiquarks make valence-valence Annihilation possible.

At $y=0$, both fractions equal $x=0.0402$. Reading the original plot near this value gives representative momentum densitiesThese are approximate graphical readings, not a precision PDF fit. Undo the sea-curve scale and divide by eight flavours/antiflavours:All products have the same $1/x^2$ factor, which cancels in the fractions. The three relative weights areTheir sum is $0.3128$, givingReading the central curves more finely changes these percentages by about a percentage point, well within what the plot supports. Omitting its factor $0.05$ would substantially underestimate the sea contribution.

Use leptonic decays: an isolated high transverse momentum positive electron or muon, together with substantial missing transverse momentum from the neutrino, provides a clean $W^{+}$ signature. The lepton transverse momentum has a Jacobian scale near $m_W/2$, and the lepton–missing-momentum transverse mass has an edge near $m_W$, broadened by width, recoil and resolution. Lepton identification, isolation and a prompt primary-vertex origin suppress Jets faking leptons and heavy-flavour decays. Missing-momentum quality requirements suppress mismeasured multijet events; a veto on a second oppositely charged lepton suppresses $Z\to {\ell }^{+}{\ell }^{-}$ when both are reconstructed. Hadronic $W$ decays have a much larger ordinary-QCD background.