This collection of problems contain short exercises and discussion questions on QCD.
a) With respect to what scale(s) are the \( c,b,t \) quarks called heavy?
b) Have you heard about the \( s \) quark before? If yes, in what context?
c) A possible way to see quarks and gluons is in jets. What happens in these events?
d) Using the Particle Data Group website, discuss which properties of the neutron and proton are similar and what are differences? What about for the three pions?
e) Which is more important in making a neutron more massive than a proton: the light quark mass difference or the electromagnetic contribution? Or do you think such considerations are too simplistic?
f) What is the evidence for spontaneous chiral symmetry breaking in
h) If you and your friend each do a QCD calculation with the same diagrams but use \( \alpha_s \) at different scales, will you get the same answer? If not, how could that happen?
i) Does the running coupling in QCD mean that the QCD Hamiltonian is not unique? Would you say that if you used \( \alpha_s \) at two different scales that you were using two different Hamiltonians?
j) If the neutron lifetime is so short, why are there any stable nuclei?
k) One observes a marked resonance when a \( \pi^+ \) pion is scattering off a proton. Which baryon does this correspond to and at which energy of the \( \pi^+ \) does this occur (the proton is at rest)?
l) At sufficient energy in proton-proton collisions it is possible to create a pion, \( p + p \rightarrow p + n + \pi^+ \). At which energy in the center-of-mass frame does pion production start?
a) We typically use units in which \( \hbar=c=1 \) and express quantities as powers of MeV or fm or both, using \( \hbar c\approx 197.33 \) MeVfm to convert between them. If we take for the nucleon mass \( M_N=939 \) MeV/$c^2$, what is \( \hbar^2/M_N \) numerically in terms of MeV and fm?
Hint: This should be almost immediate if you insert the right factors of \( c \).
b) For the scattering of equal mass (nonrelativistic) particles, if the laboratory energy \( E_{\rm lab} \) is related to the magnitude of the relative momentum \( k_{\rm rel} \) (i.e., the momentum each particle has in the center-of-mass frame) by \( E_{\rm lab} = C k_{\rm rel}^2 \), what is \( C \)? If the mass is \( M_N=939 \) MeV, what is the value of \( C \) in MeVfm$^2$?
c) We write the partial-wave momentum space Schroedinger equation (see Lecture notes) as $$ \frac{k^2}{2\mu}\langle klm | \psi \rangle + \frac{2}{\pi}\sum_{l'm'} \int_0^\infty\! dk'\,k'{}^2\, \langle klm | V | k'l'm' \rangle \langle k'l'm' | \psi \rangle = E_k \langle klm | \psi \rangle \;, $$ what are the units of \( V_{ll'}(k,k') \equiv \langle klm | V | k'l'm \rangle \)? In coordinate space the potential is local, \( V(r) \), with units of MeV, and \( k \) is given in inverse fm. If you see a plot in a journal article of \( V_{ll'}(k,k') \) with units of fm, how would you convert it to the units you just found?
Hint: use the results from the first exercise here.
d) In Figure 18 of the review by Scott Bogner et al., Prog. Nucl. Part. Phys. 65, 94 (2010) the momentum-space matrix elements of different chiral effective field theory potentials are given in units of fm. Consider the value at zero relative momenta. \( \tilde{C}_{^1S_0} \), see Eq. (2.5) and the article by Epelbaum et al. in GeV$^{-2}$. How do you convert to fm units? Do the values for the matrix elements then match?
e) What do on-shell and off-shell mean in the context of scattering?
f) Under what conditions is a partial-wave expansion of the potential useful?
g) Derive the standard result: $$ \frac{e^{i\delta_l(k)}\sin\delta_l(k)}{k} = \frac{1}{k\cot\delta_l(k) - i k} $$
h) Given a potential that is not identically zero as \( r\rightarrow\infty \) (e.g., a Yukawa), how would you know in practice where the asymptotic (large \( r \)) region starts?
i) What is the physical interpretation of the relation between the (partial-wave) \( S \)-matrix and the scattering amplitude? (Note that \( S_l(k) = 1 + 2 i k f_l(k) \).)
a) Using the Schr\"odinger equation for the scattering of two particles with mass \( m \), $$ (H_0 + V)|\psi_E\rangle = E |\psi_E\rangle \;, $$ where \( H_0 \) is the free Hamiltonian, show that the Lippmann-Schwinger equation for the wave function, $$ |\psi_E^{\pm}\rangle = |\phi_k\rangle + \frac{1}{E-H_0\pm i\epsilon}V |\psi_E^{\pm}\rangle \;, $$ is satisfied. Here \( E = k^2/m \) and the plane wave state satisfies \( H_0 |\phi_k\rangle = E |\phi_k\rangle \). Why do you need the \( \pm i\epsilon \)?
b) We can define the \( T \)-matrix on-shell as the transition matrix that acting on the plane wave state yields the same result as the potential acting on the full scattering state. That is, \( T^{(\pm)}(E = k^2/m)|\phi_k\rangle = V |\psi_E^{\pm}\rangle \). What does it mean that the \( T \)-matrix is on-shell? (This is a really quick question!)
c) Show that matrix elements of the \( T \)-matrix satisfy the Lippmann-Schwinger equation $$ \langle {\bf k}'|T^{(\pm)}(E)|{\bf k}\rangle = \langle {\bf k}'|V|{\bf k}\rangle + \int\! d^3p\, \frac{\langle {\bf k}'|V|{\bf p}\rangle \langle {\bf p}|T^{(\pm)}(E)|{\bf k}\rangle}{E-\frac{p^2}{m}\pm i\epsilon}. $$ What normalization is used for the momentum states? Are the matrix elements of the \( T \)-matrix on the right side on-shell?
d) Write the Lippmann-Schwinger equation for the wave function in coordinate space for a local potential \( V = V({\bf r}) \). To this end, show first that the free Green's function $$ G^{\pm}({\bf r}',{\bf r}; E = k^2/m) = \langle {\bf r} | \frac{1}{E-H_0\pm i\epsilon} | {\bf r}'\rangle, $$ is given by $$ G^{\pm}({\bf r}',{\bf r}; E = k^2/m) = -\frac{m}{4\pi}\frac{e^{\pm ik|{\bf r}-{\bf r}'|}}{|{\bf r}-{\bf r}'|}. $$
e) Show that when the \( T \)-matrix is evaluated on-shell, it is proportional to the scattering amplitude, \( T^+(E =k^2/m) = -\frac{1}{4\pi^2 m}f(k,\theta) \), by analyzing the asymptotic form of the Lippmann-Schwinger equation and comparing to $$ \langle {\bf r} | \psi_E^+ \rangle \stackrel{r\rightarrow\infty}{\longrightarrow} (2\pi)^{-3/2} \left( e^{i\boldsymbol{k\cdot r}} + f(k,\theta) \frac{e^{ikr}}{r} \right). $$
f) Start from the momentum-space partial wave expansion of the potential, $$ \langle {\bf k'} | V | {\bf k} \rangle = \frac{2}{\pi}\sum_{l,m} V_l(k',k)Y^{\ast}_{lm}(\Omega_{k'})Y_{lm}(\Omega_k), $$ and a similar expansion of the \( T \)-matrix to derive the partial wave version of the Lippmann-Schwinger equation (with the correct factor for the integral): $$ T_l(k',k;E) = V_l(k',k) + \frac{2}{\pi} \int_0^\infty \! dp\, p^2 \frac{V_l(k',q)T_l(q,k;E)}{E - p^2/m + i\epsilon}. $$
g) Scattering phase shifts for a square well potential. Calculate the S-wave scattering phase shifts for an attractive square-well potential \( V(r) = -V_0 \theta(R-r) \) and show that $$ \delta_0(E) = \arctan\left[ \sqrt{\frac{E}{E+V_0}}\tan\bigl(R\sqrt{2\mu(E+V_0)\bigr)} \right] - R\sqrt{2\mu E}. $$
h) Let's consider the analytic structure of the corresponding partial-wave S matrix, which is given by $$ S_0(k) = e^{-2 i k R} \frac{k_0 \cot k_0 R + ik}{k_0 \cot k_0 R - ik}, $$ where \( E = k^2/2\mu \) and \( k_0^2 = k^2 + 2\mu V_0 \). Show that \( S_l(k) = e^{2i\delta_l(k)} \) for \( l=0 \) is satisfied. Treat \( S_0(k) \) as a function of the complex variable \( k \) and find its singularities.
Hint: write \( e^{2i\delta} = e^{i\delta}/e^{-i\delta} \).
i) Bound states are associated with poles on the imaginary axis in the upper half plane. Show that the condition for such a pole here gives the same eigenvalue condition (a transcendental equation) that you would get from a conventional solution to the square well by matching logarithmic derivatives.
Hint: Define \( k= i\kappa \) with \( \kappa>0 \) when analyzing such a pole.