Particle Physics -2-

This week we’ve been mostly talking about natural units and relativity, with some mentions of basic ROOT usage during sections.

Natural Units#

As we all know, physics rely on the concept of *dimensions *quite often. For example, numbers with different units cannot be added, it makes no sense to add 1 s with 1 m. But SI units are human constructs, they are useful and convenient for describing scales close to our daily life – 1 meter, 1 second etc. But in the world of high energy physics, they’re not what the nature meant us to use. As far as we know, nature offers at least two *constants *that we should try to use:

speed of light c, with dimension Length / Time
(reduced) Plank constant $\hbar$ , with dimension Energy * Time

In fact, now that kilogram has been redefined by fixing $\hbar$ as a constant, humanity has officially got rid of ‘human artifact’ for defining SI units :)

The trick is to set $c=1, \hbar=1$ when conducting calculation, and put them back as needed in the end. For example, a 4-momentum has:

$p = (\frac{E}{c}, p _ x, p _ y, p _ z)$

But $E = \gamma m c ^2$ , so if we let $c=1$ , at a rest frame, we can use energy unit (usually GeV or so) to represent mass, with an implicit $\frac{1}{c ^2}$ omitted. Which is much easier to use. To put back the correct units, just give each m a ${c ^2}$ and then use $[c \hbar] = \text{Length * Energy}$ to fix things.

Relativistic Kinematics#

There are many ways to ‘derive’ special relativity, after that, we call all the physical vectors that transform according to Lorentz Transformation a 4-vector, and we have to re-formulate physical quantities (such as velocity, momentum) with 4-vectors – because now they will be used in relativistic conservation laws.

The canonical 4-vector is the space-time 4-vector, which is what we used to derive special relativity in the first place, and they mark the coordinates of events in space-time. We call 4-vectors transform according to the normal LT (get $t^\prime = ...$ ) a contra-variant vector: $\{X^\mu\}$ . We call 4-vectors transform according to the inverse LT (get non-prime from prime) a covariant vector: $\{X_\mu\}$

A big realization comes when we find that velocity needs to be replaced by 4-velocity because frame-dependent time $dt$ is not good (because (dt, dx, dy, dz) is already a 4-vector so we can only divide by invariant quantity), instead, we use proper time $d\tau$ to construct 4-velocity:

\{v^\mu\} = \frac{dx^\mu}{d\tau} = \frac{dx^\mu}{dv} \frac{dv}{d\tau} = \frac{dx^\mu}{dv} \gamma

Following through some algebra, we find that:

$\frac{\partial}{\partial x^\mu}$ means taking derivative respect to contra-variant component but itself transforms as a co-variant component
$\frac{\partial}{\partial x_\mu}$ itself transforms as a contra-variant component

We found this by taking derivative of a Lorentz invariant function $\Psi{x}$ :

\frac{\partial \Psi}{\partial X^\mu} = \frac{\partial \Psi}{\partial X^{\prime\nu}} \frac{\partial X^{\prime\nu}}{\partial X ^\mu}

Let’s just do one case of $\mu = 0$ , assuming boost is in x-axis:

\begin{cases} \frac{\partial \Psi}{\partial X^0} &= \frac{\partial \Psi}{\partial t'} \frac{\partial t'}{\partial t} + \frac{\partial \Psi}{\partial x'} \frac{\partial x'}{\partial t} + \frac{\partial \Psi}{\partial y'} \frac{\partial y'}{\partial t} + \frac{\partial \Psi}{\partial z'} \frac{\partial z'}{\partial t}\\ &= \frac{\partial \Psi}{\partial t'} \gamma + \frac{\partial \Psi}{\partial x'} (-\beta\gamma) + \frac{\partial \Psi}{\partial y'} 0 + \frac{\partial \Psi}{\partial z'} 0\\ &= \gamma \frac{\partial \Psi}{\partial t'} -\beta\gamma\frac{\partial \Psi}{\partial x'} \end{cases}

We can see this is a close resemble of $t' = \gamma( t - \beta x)$ , but the prime is switched, so $\frac{\partial }{\partial X^\mu}$ transforms like $\{X _ \mu\}$ !

ROOT basics#

This week we’re just learning how to use TF1 and TH1D, I will just paste some snippet:

Double_t BlackBodyFcn(Double_t* E, Double_t* T)
{
// E is the photon energy in eV
// T is the equilibrium temperature in K
// The function returns the Black Body spectrum (1/V)dE_tot/dE_gam
    Double_t hbarc = 1973; // units: eV Angstroms
    Double_t k = 8.617e-5; // units: eV K^{-1}
//
    return 8.0*TMath::Pi()*pow( E[0]/(2.0*TMath::Pi()*hbarc),
    3)/(exp(E[0]/k*T[0]) - 1.0);
}

void myFunc(){
    TF1* myFunc = new TF1("myFunc", "pow(x, 3)/(exp(x/[0]) - 1.0)", 0, 200);
    myFunc->SetParameter(0, 20.0);
    myFunc->Draw();
    TF1 *fbb = new TF1("fbb", BlackBodyFcn, 0, 0.0001, 1);
    fbb->SetParameter(0, 50);
    fbb->Draw();
}

\\



TF1 f1("f1", "-10.0*pow(x, 2) + 0.5*pow(x,4)", -5.0, 5.0);
TCanvas c1("c1","My Canvas 1",400,800);
c1.cd();
f1.Draw();