Advanced Topics in Particle Physics

11.1 Deep Inelastic Scattering and Parton Model

Deep inelastic scattering (DIS) experiments (SLAC, 1968) scattered high-energy electrons off protons. The key observation: at large momentum transfer $Q^2$The proton behaves as if composed of nearly free point-like constituents --- the partons (identified with quarks and gluons by Feynman and Bjorken).

Structure functions. The inclusive cross section for $ep \to eX$ is parameterised by structure functions $F_1(x, Q^2)$ and $F_2(x, Q^2)$Where $x = Q^2/(2p \cdot q)$ is the Bjorken scaling variable.

The Callan—Gross relation (for spin-1/2 partons):

$F_2(x) = 2xF_1(x)$

This relation was experimentally verified, confirming that the partons are fermions (quarks).

Bjorken scaling: At large $Q^2$The structure functions depend only on $x$Not on $Q^2$ separately. This is a consequence of the parton model (impulse approximation). In reality, QCD predicts logarithmic scaling violations from gluon radiation and quark-antiquark pair production.

PDFs. The parton distribution functions $f_i(x, Q^2)$ give the probability of finding parton $i$ with momentum fraction $x$ at resolution scale $Q^2$. The structure function is:

$F_2(x, Q^2) = \sum_i e_i^2\, x\, f_i(x, Q^2)$

Where $e_i$ is the electric charge of parton $i$.

The PDFs evolve with $Q^2$ according to the DGLAP equations:

$\frac{\partial f_i(x, Q^2)}{\partial \ln Q^2} = \frac{\alpha_s(Q^2)}{2\pi}\sum_j \int_x^1 \frac{dz}{z}\, P_{ij}(z)\, f_j\!\left(\frac{x}{z}, Q^2\right)$

Where $P_{ij}(z)$ are the splitting functions: $P_{qq}$ (quark emitting a gluon), $P_{qg}$ (gluon splitting into $q\bar{q}$), $P_{gq}$ (quark emitting a gluon), $P_{gg}$ (gluon splitting into two gluons).

11.2 The Quark-Gluon Plasma

At temperatures above $T_c \approx 170$ MeV ($\sim 2 \times 10^{12}$ K), hadrons “melt” into a quark-gluon plasma (QGP) --- a deconfined state of quarks and gluons.

Phase diagram of QCD:

• Low $T$Low $\mu_B$: Hadronic phase (confined)
• High $T$Low $\mu_B$: QGP (deconfined, crossover transition)
• High $\mu_B$Low $T$: Colour superconductor (predicted)
• Very high $\mu_B$: Colour-flavour locked phase (predicted)

Experimental evidence. The QGP is produced in heavy-ion collisions at RHIC and the LHC. Key signatures:

1. Jet quenching: High-$p_T$ partons lose energy traversing the QGP, reducing the jet yield (observed at RHIC and LHC).
2. Elliptic flow: The azimuthal anisotropy of particle emission ($v_2$) indicates strong collective behaviour, consistent with a nearly ideal fluid ($\eta/s \approx 0.12$Close to the KSS bound $1/(4\pi)$).
3. J/$\psi$ suppression: In a deconfined medium, the $c\bar{c}$ potential is screened (Debye screening), suppressing quarkonium production.
4. Strangeness enhancement: Increased production of strange hadrons ($s\bar{s}$ pairs) relative to $pp$ collisions.

The QGP at LHC reaches temperatures of $T \sim 300$—$600$ MeV ($\sim 5$ $\times$ the transition temperature) and behaves as the most perfect fluid known.

11.3 Anomalies and the Axion

Chiral anomaly. In the Standard Model, the classically conserved axial current $J_5^\mu = \bar{\psi}\gamma^\mu\gamma^5\psi$ is not conserved at the quantum level:

$\partial_\mu J_5^\mu = \frac{g^2}{16\pi^2}F_{\mu\nu}\tilde{F}^{\mu\nu}$

Where $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ is the dual field strength tensor.

Consequences:

• The $\pi^0 \to \gamma\gamma$ decay rate is accurately predicted by the anomaly
• The anomaly cancels between quark generations in the SM (gauge anomaly cancellation, a constraint on fermion representations)

Strong CP problem. QCD allows a term $\mathcal{L}_\theta = \theta\frac{g_s^2}{32\pi^2}F_{\mu\nu}^a\tilde{F}^{a\mu\nu}$ in the Lagrangian. This gives the neutron an electric dipole moment $d_n \propto \theta$But experiments find $d_n < 1.8 \times 10^{-26}\,e\cdot\text{cm}$Implying $|\theta| < 10^{-10}$. Why is $\theta$ so small?

Axion solution. The Peccei—Quinn mechanism (1977) promotes $\theta$ to a dynamical field --- the axion $a(x)$. The axion potential has a minimum at $\theta_{\text{eff} = 0}$Dynamically solving the strong CP problem. The axion acquires a small mass:

m_a \approx \frac{m_\pi f_\pi}{f_a} \approx 6\ \text{meV}\times\left(\frac{10^{12}\ \text{GeV}{f_a}\right)}

Where $f_a$ is the axion decay constant. Axions in the “window” $10^9$—$10^{12}$ GeV are viable cold dark matter candidates and are searched for by ADMX, CASPEr, and other experiments.

11.4 CP Violation in Detail

CP violation in the SM arises from a single irreducible complex phase in the CKM matrix. The unitarity triangle provides a convenient parameterisation:

$V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0$

This can be rescaled to form a triangle in the complex plane with sides and angles $(\alpha, \beta, \gamma)$.

Experimental status: All three angles have been measured:

• $\sin 2\beta = 0.691 \pm 0.017$ (B factories, $B^0 \to J/\psi\, K_S$)
• $\alpha = (84.7^{+2.6}_{-2.9})^\circ$ (LHCb, $B \to \pi\pi$, $\rho\rho$)
• $\gamma = (65.4 \pm 3.4)^\circ$ (LHCb, $B \to DK$ tree-level)

The triangle closes, confirming the CKM mechanism as the source of CP violation in quark decays. However, the amount of CP violation in the CKM matrix is far too small to explain the matter-antimatter asymmetry of the universe (Sakharov conditions require additional CP-violating sources, e.g., in the neutrino sector or from BSM physics).