Fernando Garcia
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Particle Physics
Formulas
One morning I woke up thinking "I have been studying BSM models for a while, but do I really understand the backbone of the SM?" and thus this page was created.
Below you will find a collection of important values, tables, rules, and formulas used in (SM particle physics. These are not for QFT. I follow the structure of Griffiths's "Introduction to Elementary Particles."
Contents
Constants.
Special Relativity.
C, P, and T.
Decays and Scattering.
The ABC Model.
QED.
Electro and colordynamics of quarks.
Constants.
Metric convention:
\[g_{\mu \nu }=\text{diag} (+1,-1,-1,-1) \]Scales of the interactions:
Fine structure constant
\[\alpha = \frac{e^2 }{\hbar c}=\frac{1}{137} \]Barns
\[1\;\text{b} =10^{-24}\;\text{cm} ^2 \]Special Relativity.
Lorentz factor
\[\gamma =\frac{1}{\sqrt{1-\frac{v^2 }{c^2 }}} \]Lorentz transformation (for primed frame moving in the positive unprimed $x $-direction):
\begin{align*} x' &= \gamma (x-vt) \\ t' &= \gamma \left( t-\frac{v}{c^2 }x \right) \end{align*}With
\[u^2 =(u^0 )^2 -\mathbf{u}^2 \]We call
- $u^{\mu } $ timelike if $a^2 >0 $.
- $u^{\mu } $ spacelike if $a^2 <0 $.
- $u^{\mu } $ lightlike if $a^2 =0 $.
Proper time
\begin{align*} \eta ^{\mu } &= \frac{dx^{\mu }}{d\tau } \\ &= \gamma (c,v_x,v_y,v_z) \end{align*}Momentum
\[p^{\mu }=m\eta ^{\mu } \]Relativistic energy:
\[E=\gamma mc^2 \]Energy-momentum four-vector:
\[p^{\mu }= \left( \frac{E}{c},p_x,p_y,p_z \right) \]Rest energy:
\[R=mc ^2 \]Relativistic kinetic energy:
\[T=mc ^2 (\gamma -1) \]For massless particles:
\begin{align*} v &= c \\ E &= |\mathbf{p}|c \end{align*}Frequency of a photon:
\[E=h\nu \]In a relativistic collision:
- Energy is conserved.
- Momentum is conserved.
- Kinetic energy may or may not be conserved.
C, P, and T.
Parity
- Scalar under parity: $P(s)=s $.
- Pseudoscalar under parity: $P(p)=-p $.
- Vector under parity: $P(\mathbf{v})=-\mathbf{v} $.
- Pseudovector under parity: $P(\mathbf{a})=\mathbf{a} $.
- Fermions have opposite intrinsic parity with respect to their antiparticle counterparts.
- Bosons have the same intrinsic parity with respect to their antiparticle counterparts.
- Parity is a multiplicative quantum number.
- Conserved in electromagnetic and strong interactions, not in weak interactions.
- Helicity: \[\frac{m_s}{s} \]
Charge (conjugation)
- Charge conjugation turns a particle into its antiparticle.
- Charge conjugation is a multiplicative quantum number.
- Conserved in electromagnetic and strong interactions, not in weak interactions.
Time
- Harder to measure.
- Violation of CP implies violation of T due to the CPT theorem.
Decays and Scattering.
Number of particles as a function of time:
\[N(t)=N(0)e^{-\Gamma t} \]Where $\Gamma $ is the decay rate:
\[\Gamma (=)\text{probability per unit time} \]Mean lifetime:
\[\tau =\frac{1}{\Gamma } \]If a particle can decay in various ways, the total decay rate is given by:
\[\Gamma _{\text{tot} }=\sum_{i=1}^{n} \Gamma _i \]With lifetime
\[\tau =\frac{1}{\Gamma _{\text{tot} }} \]Branching ratios:
\[\text{Branching ratio for the $i $th decay mode} =\frac{\Gamma _i }{\Gamma _{\text{tot} }} \]Total cross section:
\[\sigma _{\text{tot} }=\sum_{i=1}^{n} \sigma _i \]With a scattering solid angle $d\Omega $, we define the differential scattering cross section $D $:
\[D(\theta )=\frac{d\sigma }{d\Omega } \]Giving a differential cross section:
\[d\sigma =D(\theta )d\Omega \]Total cross section:
\begin{align*} \sigma &= \int d\sigma \\ &= \int D(\theta )d\Omega \end{align*}Luminosity $\mathcal{L} $:
\[\mathcal{L}(=)\frac{\text{number of particles passing down} }{\text{time} \cdot \text{area} } \]The Event rate $dN $ is the number of particles per unit time passing through area $d\sigma $:
\begin{align*} dN &= \mathcal{L}d\sigma \\ &= \mathcal{L}D(\theta )d\Omega \end{align*}Standard terminology:
\begin{align*} \text{Amplitude} &\Leftrightarrow \text{matrix element} \\ \text{Phase space} &\Leftrightarrow \text{density of final states} \end{align*}Fermi's golden rule for multiparticle ($1\rightarrow 2+3+4+\cdots +n $) decay:
\[\Gamma =\frac{S}{2\hbar m_1 }\int |\mathcal{M}|^2 (2\pi )^4 \delta ^4 (p_1 -p_2 -p_3 -\cdots -p_n )\times \prod ^{n}_{j=2}\frac{1}{2\sqrt{\mathbf{p}_j ^2 +m_j ^2 c^2 }}\frac{d^3 \mathbf{p}_j }{(2\pi )^3 } \]Where $S $ is a statistical counting factor.
Fermi's golden rule for two-particle ($1\rightarrow 2+3 $) decay (with outgoing momentum $\mathbf{p} $):
\[\Gamma =\frac{S|\mathbf{p}|}{8\pi \hbar m_1 ^2 c}|\mathcal{M}|^2 \]Fermi's golden rule for Scattering $1+2\rightarrow 3+4+\cdots +n $:
\[\sigma =\frac{S\hbar ^2 }{4\sqrt{(p_1 \cdot p_2 )^2 -(m_1 m_2 c^2 )^2 }}\int |\mathcal{M}|^2 (2\pi )^4 \delta ^4 (p_1 +p_2 -p_3 -\cdots -p_n )\times \prod _{j=3}^n \frac{1}{2\sqrt{\mathbf{p}_j ^2 +m_j ^2 c^2 }}\frac{d^3 \mathbf{p}_j }{(2\pi )^3 } \]Fermi's golden rule for two-body Scattering $1+2\rightarrow 3+4 $:
\[\frac{d\sigma }{d\Omega }= \left( \frac{\hbar c}{8\pi } \right) ^2 \frac{S|\mathcal{M}|^2 }{(E_1 +E_2 )^2 }\frac{|\mathbf{p}_f|}{|\mathbf{p}_i |} \]The dimensions of $\mathcal{M} $ depend on $n $:
\[\text{dim} (\mathcal{M})=(mc)^{4-n} \]