Particle Physics

Prerequisites: Standard Model of Particle Physics. Full path (32 pages)

Best explanations

Concrete Particle physics basics lectures (Oxford)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Richard P. Feynman once said that doing elementary particle physics is a lot like banging two fine Swiss watches against each other and trying to figure out their workings by examining the debris. That was the challenge.

From My Life As A Quant by Emanuel Derman

Recommended Books:

A Zeptospace Odyssey: A Journey Into the Physics of the LHC by Gian Francesco Giudice
The Second Creation by Crease, Mann
Deep Down Things by Schumm
The Particle Odyssey by Close
Reminiscences: A Journey through Particle Physics by Adrian Melissinos

Concrete

Particle physics uses natural units $\hbar = c = 1$ , in which energy, momentum and mass are all measured in electronvolts and lengths in inverse energy ( $\hbar c \approx 197\,\text{MeV fm}$ ). The known particles and their interactions are described by the Standard Model, a gauge theory with gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ . Its content is: - quarks ( $u, d, c, s, t, b$ ), which carry color and feel the strong force; - leptons ( $e, \mu, \tau$ and three neutrinos), which do not; - the gauge bosons: eight gluons, the $W^\pm$ and $Z$ of the weak interaction, and the photon; - the Higgs boson, whose vacuum expectation value gives mass to the $W$ , $Z$ and the fermions.

What is computed. The two quantities that connect theory to experiment are decay rates and cross sections. Both follow from the invariant amplitude $\mathcal M$ , obtained from Feynman diagrams. For a particle of mass $M$ decaying at rest, $d\Gamma = \frac{1}{2M}\,|\mathcal M|^2\, d\Phi_n ,$ with $d\Phi_n$ the Lorentz-invariant phase space of the $n$ final particles. For a two-body decay this integrates to $\Gamma = \frac{|\mathbf p|}{8\pi M^2}\,\overline{|\mathcal M|^2}$ , where $|\mathbf p|$ is the momentum of either daughter and the bar denotes the average over initial and sum over final spins (with a factor $1/2$ for identical daughters). The lifetime is $\tau = 1/\Gamma_{\text{tot}}$ . For two massless colliding particles the cross section is $d\sigma = \frac{1}{2s}\,|\mathcal M|^2\,d\Phi_n$ , with $s$ the squared center-of-mass energy.

Worked example. At lowest order in QED and for $\sqrt s \gg m_\mu$ , the process $e^+e^- \to \mu^+\mu^-$ through a virtual photon has $\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4s}\left(1 + \cos^2\theta\right), \qquad \sigma = \frac{4\pi\alpha^2}{3s} ,$ with $\alpha \approx 1/137$ the fine-structure constant and $\theta$ the angle between the incoming electron and the outgoing muon. Replacing the muon by a quark $q$ of charge $Q_q e$ multiplies the result by $Q_q^2$ and by the number of colors $N_c$ , because the quark can be produced in any of the colors. Summing over quarks light enough to be produced gives the ratio $R = \frac{\sigma(e^+e^- \to \text{hadrons})}{\sigma(e^+e^- \to \mu^+\mu^-)} \approx N_c \sum_q Q_q^2 .$ Below the charm threshold ( $u, d, s$ ) this is $N_c \cdot \frac{6}{9} = 2$ for $N_c = 3$ , and measurements agree with $N_c = 3$ (up to QCD corrections of order $\alpha_s/\pi$ ). This is one of the classic pieces of evidence for color.

Conservation laws. Many reactions can be checked quickly with conservation laws: energy-momentum, charge, baryon number and lepton number hold in all observed processes, while the strong and electromagnetic interactions also conserve quark flavor, parity and charge conjugation, and the weak interaction violates them.

http://www-pnp.physics.ox.ac.uk/~mjohn/notes/BasicsLectures.pdf

Recommended Textbooks:

Introduction to Elementary Particles by D. Griffith
An Introductory Course of Particle Physics by P. Pal
Quarks & Leptons by Halzen, Martin

Abstract

In relativistic quantum theory a one-particle state space carries a unitary representation of the symmetry group of spacetime. Following Wigner, an elementary particle is identified with an irreducible unitary representation of the universal cover of the Poincaré group, $\mathbb R^{1,3} \rtimes SL(2,\mathbb C)$ . The Casimir $P^\mu P_\mu = -m^2$ (signature $(-,+,+,+)$ ) fixes the mass, and the remaining label is a representation of the stabilizer of a reference momentum, the little group. For $m > 0$ the little group is $SU(2)$ and the label is the spin $j \in \{0, \tfrac12, 1, \dots\}$ . For $m = 0$ it is the double cover of $ISO(2)$ ; the physically realized representations are those where its translations act trivially, labeled by a helicity $h \in \tfrac12\mathbb Z$ . Internal quantum numbers enter as representations of internal symmetry groups that commute with the Poincaré group.

Interactions are described by quantum field theory. The data of a perturbative QFT of the Standard Model type are a compact gauge group $G$ , a set of Weyl fermions and complex scalars in unitary representations of $G \times$ Lorentz, and the renormalizable couplings allowed by the symmetries. For the Standard Model $G$ is $SU(3)\times SU(2)\times U(1)$ , possibly divided by a discrete subgroup of $\mathbb Z_6$ that acts trivially on all fields. Writing all fermions as left-handed Weyl fields, one generation is $Q:(3,2)_{1/6},\quad u^c:(\bar 3,1)_{-2/3},\quad d^c:(\bar 3,1)_{1/3},\quad L:(1,2)_{-1/2},\quad e^c:(1,1)_{1} ,$ with hypercharge normalized so that $Q_{\text{em}} = T_3 + Y$ , and the Higgs is $H:(1,2)_{1/2}$ . The representation is chiral, so gauge anomalies must cancel. For example the $U(1)_Y^3$ and gravitational $U(1)_Y$ conditions are $\sum Y^3 = 6\left(\tfrac16\right)^3 + 3\left(-\tfrac23\right)^3 + 3\left(\tfrac13\right)^3 + 2\left(-\tfrac12\right)^3 + 1^3 = 0, \qquad \sum Y = 0 ,$ where the sum runs over all Weyl components of one generation, and the remaining conditions hold as well.

Observables are encoded in the S-matrix, whose elements are obtained from time-ordered correlation functions through the LSZ reduction formula and are invariant under the Poincaré group. In the modern view the Standard Model is the leading part of an effective field theory: the most general local Lagrangian with this field content and gauge symmetry, organized by operator dimension. The first correction is the dimension-five operator $(LH)(LH)/\Lambda$ (unique up to flavor indices), which gives Majorana neutrino masses $m_\nu \sim v^2/\Lambda$ after electroweak symmetry breaking.

Why is it interesting?

“Often, this kind of physics is referred to as particle physics, which I don’t like,” Arkani-Hamed told me. “People get the mistaken impression that what we care about is the particles. The science is characterized like: What are things made of? What are the ultimate building blocks of matter? I hate that. That sounds a lot like chemistry, and it’s not like that at all. There are many, many more exciting things in nature than some random elementary particles.

“The reason we go to short distances isn’t to probe the building blocks of matter,” he went on. “It’s because for four hundred years fundamental physics has been on this trajectory of unifying seemingly disparate things. We’ve found that, as we understand more, apparently incredibly disparate phenomena turn out to be different aspects of a more surprising, more beautiful answer than we could have anticipated—and often even hoped for. This started with Newton, who realized that the force dragging the apple down was the same force holding the moon around the earth. It continued with the realization that electricity and magnetism are different aspects of the same thing. Relativity told us that space and time are different aspects of the same thing. There’s more and more unity in our understanding of nature. And we’ve seen, especially over the last hundred years, that the essential unity and the essential simplicity best reveal themselves at short-distance scales. So it’s not that we care about the particles. We care about the laws.”Crash Course by Elizabeth Kolbert published in the New Yorker

See also https://profmattstrassler.com/articles-and-posts/particle-physics-basics/particle-physics-why-do-it-and-why-do-it-that-way/

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