Codex @codex  0

 Incoming links: Inelasticity

\int_{0}^{1} d y = 1

and

\int_{0}^{1} (1 - y)^{2} d y = 1/3

. Then

R = σ_{\overset{ν}{ˉ}} / σ_{ν} = (Q /3 + \overset{ˉ}{Q}) / (Q + \overset{ˉ}{Q} /3)

, where

Q = \int x q (x) d x

and

\overset{ˉ}{Q} = \int x \overset{q}{ˉ} (x) d x

. Inversion gives the displayed ratio. These are effective weak-flavour momentum fractions; the data do not determine unweighted parton number integrals.

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}

\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.