QCD at Lepton Colliders

In QED/EW theory, $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ is a fundamental process. The outgoing ${\mu }^{+}{\mu }^{-}$ or their decay products can be observed in the detector.

For QCD, we instead consider the total inclusive cross-section for $e^{+}{e}^{-}\to \text{hadrons}$. The $e^{+}{e}^{-}$ collide to form an off-shell photon, ${\gamma }^{\ast }$ or $Z$-boson with virtual mass $Q^2=s$. The ${\gamma }^{\ast }$ or $Z$ then fluctuates into a $q\bar{q},q\bar{q}g$ state. These partons may radiate additional partons and gradually lose energy, but will eventually combine to form hadrons, which are then observed in the detector.

When calculating this process, we factorise the short- and long-distance physics present:

• We rely on the asymptotic freedom of QCD; at large Q the coupling is small and we can calculate the fluctuations perturbatively (use Feynman diagrams).
• At some later time, the produced quarks/gluons hadronise, forming bound-state hadrons.

When $Q^2<M_Z^2$, we can neglect the $Z$-boson exchange Feynman diagram.

For collisions at $e^{+}{e}^{-}$ colliders, we factorise the amplitude into the product of a Leptonic tensor $L_{\mu \nu }$ and a Hadronic tensor $W^{\mu \nu }$. This can be understood by considering the squared amplitude (diagram)

Neglecting the mass of the electron,

$L_{\mu \nu }=\text{Tr}[/p_a(-ie{Q}_e{\gamma }_{\mu })/p_b(ie{Q}_e{\gamma }_{\nu })]=e^2{Q}_e^2\text{Tr}[/p_a{\gamma }_{\mu }/p_b{\gamma }_{\nu }]$

$=e^2{Q}_e^{2}4[p_{a,\mu }{p}_{b,\nu }-g_{\mu \nu }(p_a.p_b)+p_{a,\nu }{p}_{b,\mu }]$.

Similarly, neglecting the quark masses,

$W_{q\bar{q}}^{\mu \nu }=e^2{Q}_q^{2}4[p_1^{\mu }{p}_2^{\nu }-g^{\mu \nu }(p_1.p_2)+p_1^{\nu }{p}_2^{\mu }].\text{underbracket}{N_c}_{\text{colour}}$.

Then, for $D=4-2ϵ$, we get $⟨|M_{e^{+}{e}^{-}\to q\bar{q}}|{}^2⟩=e^4{Q}_e^2{Q}_q^{2}8\left[\frac{t^2+u^2}{s^2}-ϵ\right].N_c$.

Then, for a general $2\to 2$ scattering process, the cross-section is

$\sigma (e^{+}{e}^{-}\to q\bar{q})=\frac{4\pi {\alpha }^2}{3s}{N}_c{Q}_q^2$, where the QED coupling is given by ${\alpha }^2=e^2/4\pi$, $s,t$ and $u$ are the Mandelstam variables for a $2\to 2$ process.

R-ratio