📖 Definition:
In this course we will be using natural units ie $\hbar=c=1$
Quantities in natural units: they are all described by $(\text{energy})^n$
$$ \begin{aligned} n=1:& \quad \text{energy,momentum,mass} \\ n=-1:& \quad \text{length,time}
\end{aligned} $$
The unit we will use is $1\,\rm eV = 1.6\times 10 ^{-19} \,\rm J$
Useful quantities to 🧠 Remember:
$$ m_e= 0.511\,{\rm Mev} \quad m_p\approx 170\,{\rm GeV} \quad m_t\sim 170\,{\rm GeV} $$
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Lorentz transformation of a boosted frame $S’$ moving at speed $v$ with respect to $S$ are
$$ \begin{aligned} x'&=\gamma(x-\beta t) \qquad \beta = \frac{v}{c}\\ y'&=y \\ z'&=z\\ t'&=\gamma(t-\beta x) \end{aligned} $$
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Four vector:
🗒️ Note: the first 4 components of the $\Lambda^{\mu}_{\nu}=(\genfrac{}{}{0pt}{}{\gamma}{-\beta\gamma}\genfrac{}{}{0pt}{}{-\beta\gamma}{\gamma})$
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Invariant quantities: For a general vector $V^\mu =(A,\vec B)$ we have $|V^\mu|^2=V_\mu V^\mu =A^2-\vec B \cdot \vec B$ which is invariant
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💃 Example: $\text ds^2 = |X^\mu|^2 = t^2-\vec x \cdot \vec x=t^2 -x^2-y^2-z^2$
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Invariant mass:
Definition from the energy-momentum four vector
$$ M^2=|P^\mu|^2 = E^2 -\vec p \cdot \vec p=E^2 -p_x^2 -p_y^2-p_x^2 $$
For $N$ particles we have
$$ {P^\mu_\text{tot}=\begin{pmatrix} \sum^N_{i=1} E_{i} & \sum_{i=1} ^N\vec p_i\end{pmatrix} \quad M^2=|P^\mu_\text{tot}|^2=\begin{pmatrix} \sum^N_{i=1} E_{i} \end{pmatrix}^2- \begin{pmatrix} \sum_{i=1} ^N\vec p_i\end{pmatrix}}^2
$$
Centre of mass frame is defined as $\sum_i p_i=0$ thus
$$ M^2=\begin{pmatrix} \sum^N_{i=1} E_{i} \end{pmatrix}^2=E_\text{CM}^2 $$
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💃 Example: find the minimum invariant mass needed for the protons in the interaction $pp\to ppp\overline p$
We work in the centre of mass frame $\sum_ip_i=0$ and in 1 dimensions. We need the total energy to be enough to produce $4$ protons so
$$ P^\mu_\text{tot}=(4m_p, \; 0) \qquad \Rightarrow \qquad M^2=(4m_p)^2 $$
🗒️ Note: the result $M^2=(4m_p)^2$ is valid for all reference frames (invariant)
If we now work in the lab frame we have the incoming proton and the stationary target proton
$$ \begin{aligned} \text{Beam:}& \; & \; P_b^\mu&=(E_b,\; p_b) \\ \text{Target:}& \; & \; P_t^\mu&=(m_p,\; 0) \end{aligned} $$
Using our general result above we know that
$$ \begin{aligned} M^2&=|P^\mu_\text{tot}|^2=|P_b^\mu + P^\mu_t|^2=|(E_b+m_p ,\; p_b )|^2 \\ &=(E_b+m_p)^2-p_b^2=(4m_p)^2
\end{aligned} $$
💎 Conclusion: solving the last equation for $E_b=7m_p$ which is the minimum energy required
💃 Example: Consider the decay of $X\to AB$ lets find the mass of $X$ here we dont assume $\rm 1D$
Using conservation of momentum we can write
$$ \begin{aligned} P_X^\mu &= P_A^\mu + P_B^\mu = (E_A+E_B,\; \vec p_A+\vec p_B) \\ M^2 &= |P_X^\mu|^2 \end{aligned} $$
💎 Conclusion: from $E_A,E_b,\vec p_A,\vec p_B$ we can infer $M_X$
Heavy particles are unstable and decay in finite time, this has consequences to the measured mass
We start by considering a set $N_0$ of heavy particle at time $t_0$ which are subject to decay
$$ N(t)=N_0 \exp\left ({-\frac{t}{\tau}} \right ) $$
where $\tau$ is the particle lifetime
Lets now try to find how this impacts the mass using $\Delta E \Delta t \sim 1$ and setting $\Delta t \approx \tau$
$$ \Delta E= \Delta m \sim \frac{1}{\tau} \equiv \Gamma $$
💎 Conclusion: $\Gamma$ is the decay width which represents the uncertainty in the mass
The relativistic version of Schrödinger's equation is the Klein-Gordon equation