Set #1

Theoretical Physics
Homework Problems #1 in
SI2400 Theoretical Particle Physics, 7.5 credits
Spring 2015
Deadline:
Teachers:
Examiner:
GOOD LUCK!
April 17, 2015 @ 17:00
Dr. Sushant Raut (raut@kth.se)
Dr. Juan Herrero (juhg@kth.se)
Stella Riad (sriad@kth.se)
Prof. Tommy Ohlsson (tommy@theophys.kth.se)
1. Which of the following reactions are possible in the Standard Model? If they are
not possible, indicate which conservation laws are violated.
a)
c)
e)
g)
i)
p → e+ + π 0
µ− → e − + γ
76
Ge → 76 Se + 2e−
Υ → Ψ + τ + + e−
p → n + e + + νe
b)
d)
f)
h)
j)
µ+ + µ− → γ
p¯ + p → 2π + + π −
76
Ge → 76 Se + 2e− + 2¯
νe
n → p + e− + ν¯e
p + e − → n + νe
2. Noether’s theorem states that every continuous symmetry of the Lagrangian leads
to
current j µ (x), i.e., ∂µ j µ (x) = 0, implying a conserved charge Q =
R a3 conserved
d xj 0 (x). Consider the simplest theory of a real scalar field
1
1
L(x) = ∂ µ φ(x)∂µ φ(x) − m2 φ2 (x) ,
2
2
and show that the total energy and momentum of field configuration
Z
Z
h
i
1
3
2
2
2 2
i
˙
˙ iφ
E=
d x φ + (∇φ) + m φ ,
p = d3 xφ∂
2
are the conserved charges corresponding to the space-time translational invariance.
3. Particle X produced in a detector decays electromagnetically into two photons:
X → γγ. The two photons have energy 75 MeV and 81 MeV, with an opening
angle of 120◦ . Using a table of particle masses, identify X. Calculate its relativistic
γ-factor. What can you conclude about its spin and parity from the decay process?
4. A Lorentz boost along the z-direction can be represented by a matrix of the form


γ 0 0 βγ
 0 1 0 0 


 0 0 1 0  ,
βγ 0 0 γ
p
where γ = 1/ 1 − β 2 . β is interpreted as the boost velocity. Show that Lorentz
boosts along the z-direction form a group. Can you use this to ‘derive’ the relativistic velocity addition formula?
5. (a) Show that if the neutron decays as n → p e, the emitted electron must be
mono-energetic.
(b) The experimentally measured electron spectrum is not mono-energetic, but is a
continuous spectrum. As Pauli proposed in 1930, this must be due to the emission
of an additional particle (the neutrino, ν). Assume that ν has a small, but non-zero,
mass mν . In a beta decay n → p e ν, what is the maximal energy of the electron
in the rest frame of the neutron?
6. The decays π + → `+ + ν` , where ` = e, µ, are mediated by a W + boson, which only
couples to left-handed particles and right-handed anti-particles. The decay into µ+
dominates and the decay to e+ only occurs in 0.012 % of the cases. Why is the
decay to positrons so suppressed?
Hint: What would be the decay rate if e+ was massless?
7. Using the usual isospin assignments, find the ratio of the two decay modes of the
∆0 : ∆0 → n π 0 , ∆0 → p π − .
√
8. (a) In the CM frame, show that the energy of a collision is s, where s is the
Mandelstam variable.
(b) The Tevatron collided protons and antiprotons with energy of around 1 TeV to
find heavy particles. At these hadron colliders, the interacting constituents of the
hadrons carry only a fraction x of the total hadron momentum. Assuming x ≈ 0.3
for both the interacting constituents, what is the maximum mass of particles that
the Tevatron could have produced?
Rules and guidelines for homework problems in SI2400,
Theoretical Particle Physics
When solving the homework problems, you are allowed to use books or other sources of
information, as well as to discuss the problems with one another. However, the solutions
that you hand in have to reflect your own knowledge. Therefore, make sure that you
motivate the steps in your solutions. If we receive nearly identical solutions, we might
ask you to orally describe what you have done.
Also, you should follow the following simple guidelines:
• Your solutions should be handwritten
• Motivate your computations, it should be clear that you understand
what you are doing and why!
• Start each problem on a separate sheet of paper
• Hand in the problems stapled together, in the correct order
• Write clearly, so that your solutions are readable
• Do not forget to put your name on your solutions
• Hand in the solutions in time—if, for some reason, you cannot do this, please talk
to one of the teachers rather than just handing in the solutions too late.
Failure to follow these rules might result in a deduction of points.