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 Incoming links: Bjorken scaling variable

Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 305 / 4 / c / Solution Created 2026-10-03 Updated 2026-10-06

In the parton model, the incoming parton has

k = ξ P_{H}

and the outgoing massless parton has

k^{'} = ξ P_{H} + q

. Neglecting

P_{H}^{2}

and imposing its on-shell condition gives

0 = (ξ P_{H} + q)^{2} = 2 ξ P_{H} \cdot q + q^{2}, ξ = \frac{- q ^{2}}{2 P _{H} \cdot q} = x .

Thus the Bjorken scaling variable measures the struck parton's longitudinal momentum fraction in this approximation. The inelasticity is unchanged when the hadron momentum is replaced by its parton's collinear momentum:

y_{q} = \frac{ξ P _{H} \cdot q}{ξ P _{H} \cdot p} = \frac{P _{H} \cdot q}{P _{H} \cdot p} = y .

Let

S_{H} = (p + P_{H})^{2} ≃ 2 p \cdot P_{H}

. The partonic invariant in part (b) is

s (ξ) = ξ S_{H}

, and

Q^{2} = - q^{2} = x y S_{H}

. Insert its two parton cross sections into the parton distribution function convolution:

\frac{d σ _{H}}{d y} = \frac{G _{F}^{2} S _{H}}{π} \int_{0}^{1} ξ [q_{d} (ξ) + (1 - y)^{2} q_{\overset{u}{ˉ}} (ξ)] d ξ .

Since the event's measured

x

fixes

ξ = x

, the integrand is the cross-section density in

x

. No additional Jacobian from

y_{q}

is needed because

y_{q} = y

. Therefore

\frac{d ^{2} σ _{H}}{d y d x} = \frac{G _{F}^{2} S _{H}}{π} x [q_{d} (x) + (1 - y)^{2} q_{\overset{u}{ˉ}} (x)] .

Equivalently

S_{H} = 2 E_{ν} E_{H} (1 - cos ϑ_{ν H})

in the chosen massless-hadron frame. The explicit factor

x

comes from the partonic energy

s = x S_{H}

, not from redefining the parton distribution functions. This expression uses the stated two-flavor and low-energy approximations, with masses and generation mixing omitted.

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