Question 01#

Generate and plot shapes for $n=1$ to $5$ (one plot for each value of $n$). For example, for $n=1$ you would have $f(x)=1+{\epsilon }_1\cos \varphi$. Take each value of ${\epsilon }_n=0.1$ for plotting purposes (you can do this with with python, demos, or any other tool…).

Read about the Quark Gluon Plasma (QGP) here. Write a short paragraph explaining the connection between the geometry, the measured Fourier coefficients and the QGP properties.

Question 02#

It’s an open question as to whether the angles various angles ${\mathrm{\Psi }}_n$ are correlated with each other (e.g. is ${\mathrm{\Psi }}_2$ correlated with ${\mathrm{\Psi }}_3$?).

Generate a bunch of particles ($\varphi$ values) (say 50 particle per event and 1000 events) according to:

in two cases:

• Case 1: where ${\mathrm{\Psi }}_2={\mathrm{\Psi }}_3$ but ${\mathrm{\Psi }}_2$ is different in every event

• Case 2: where ${\mathrm{\Psi }}_2$ and ${\mathrm{\Psi }}_3$ are independent of each other and different for every event

(let $v_2=0.2$ and $v_3=0.1$ that are the same for every event).

Write code to do a least-squares fit of the two-particle correlations for Case 1 and Case 2. Use the same principle as in the Week 11: $$\frac{d{N}^{ij}}{d\mathrm{\Delta }\varphi }\propto 1+\sum _{n=1}^{\mathrm{\infty }}2v_n^i{v}_n^j\cos (n(\mathrm{\Delta }\varphi ))$$onlyinsteadofusingonlythe$n=2$term,alsoaddthe$n=3$ term.

Question 03#

Do you get consistent answers for the fit parameters $v_2$ and $v_3$ between Case 1 and Case 2 when you fit ? Explain what you see.

Question 04#

The $v_n$ values of the particles are proportional to the corresponding ${\epsilon }_n$ values; for each $n$, $v_n\propto {\epsilon }_n$. Write a Monte Carlo that generates values for $v_2$ and $v_3$ according to a Bivariate Gaussian distribution with five parameters $⟨v_2⟩$ and ${\sigma }_{v2}$, $⟨v_3⟩$ and ${\sigma }_{v3}$ and $\rho$, the correlation between $v_2$ and $v_3$.

Question 05#

Verify the Monte Carlo using the parameters:

• $⟨v_2⟩=0.25$

• $⟨v_3⟩=0.18$

• ${\sigma }_{v2}=0.05$

• ${\sigma }_{v3}=0.05$

• for $\rho$ of -1.0, 0.0, 0.5 and 1.0 and making a scatter plot of $v_2$ vs $v_3$.

One way to look for correlations between the flow harmonics is a technique called Symmetric Cumulants (SC). These quantities measure the correlation between the coefficients of two separate orders $n$ and $m$. The SC between orders $m$ and $n$ can be written as $$SC(n,m)=⟨⟨\cos (m{\varphi }_i+n{\varphi }_j-m{\varphi }_k-n{\varphi }_l)⟩⟩-⟨⟨\cos m({\varphi }_i-{\varphi }_j)⟩⟩⟨⟨\cos n({\varphi }_i-{\varphi }_j)⟩⟩$$$$=⟨v_n^2{v}_m^2⟩-⟨v_n^2⟩⟨v_m^2⟩$$

Question 06#

Symmetric cumulants are time consuming to calculate from particle angles due to the four-particle correlation $⟨⟨\cos (m{\varphi }_i+n{\varphi }_j-m{\varphi }_k-n{\varphi }_l)⟩⟩$ (in practice people use a different technique that avoids directly running through these combinations).

However, with the Monte Carlo above, it is possible to calculate the SC(3,2) using the $v_2$ and $v_3$ values you calculated. Do thta for each value of $\rho$ and use a bootstrap method to get the uncertainties. Plot the value of the SC(n,m) as a function of $\rho$.