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Theoretical Physics
Homework Problems #3 in
SI2400 Theoretical Particle Physics, 7.5 credits
Spring 2014
Deadline:
Teachers:
May 26, 2015 @ 17:00
Dr. Sushant Raut ([email protected])
Dr. Juan Herrero ([email protected])
Stella Riad ([email protected])
Examiner: Prof. Tommy Ohlsson ([email protected])
GOOD LUCK!
1. Higgs boson interactions with fermions.
The SM Higgs boson interactions with the fermions f are proportional to their
masses:
X mf
Hff
(1)
LH = −
v
f
a) Using dimensional analysis, estimate Γ(H → f f ). Which are the dominant
decay modes in the SM?
b) Compute the decay width into fermions for the dominant channels using mH =
125 GeV.
c) A future muon collider may be tuned to have the exact energy to be on the
Higgs peak. Using dimensional analysis, estimate the s-channel σ(µ+ µ− → bb)
assuming just Higgs exchange for centre of mass energy s mh .
d) Compute σ(µ+ µ− → bb) by using the complete scalar propagator with the
Higgs width:
1
(2)
2
s − mH + iΓH mH
e) What happens at s = m2H ? Express the result in terms of the branching ratios
(BRf = Γf /ΓH ) to muons and b quarks using the decay widths computed in
b).
2. Higgs boson interactions with gauge bosons.
The Feynman rules for the SM Higgs boson interactions with the gauge bosons W
and Z are:
m2W
m2Z
+
−
HWµ Wν → 2i
gµν
HZµ Zν → 2i
gµν
(3)
v
v
a) Using dimensional analysis, estimate the decay rate into gauge bosons Γ(H →
W + W − ) and Γ(H → ZZ) (of course at least one of gauge bosons is virtual as
mH < 2mW,Z , but leave everything in terms of mH ).
b) Compute the decay rate Γ(H → W + W − ). Use the result to get the one
Γ(H → ZZ). Do they fulfill the estimations of a)? Why?
c) Goldstone Boson Equivalence Theorem states that in the high energy limit the
decay of the Higgs into gauge bosons is the same as the decay into Goldstone
bosons. In the SM, the third polarization of the massive gauge field comes
from the Higgs doublet Φ = (φ+ , φ0 + iφ3 ), with φ3 eaten by the Z and φ+
eaten by the W). Using that the vertex of the triple Higgs potential is given
by
m2
(4)
Hφ+ φ− → −i H ,
v
compute Γ(H → φ− φ+ ). Do they agree with the computation of b) for Γ(H →
W + W − )? In which limit?
d) Similarly, use these results to relate Γ(H → φ3 φ3 ) with Γ(H → ZZ), using
that the vertex is:
m2
Hφ3 φ3 → −i H ,
(5)
v
and discuss the result.
3. Charged currents.
In the SM, the W boson interacts with the leptons via the Lagrangian:
o
g n † X
a
√
LW =
Wa
[ν l γ (1 − γ5 ) l] + H.c. .
2 2
l=e,µ,τ
(6)
and similarly for quarks.
a) What are the possible decay channels (in the weak basis)? Estimate the
expected branching ratios without using the experimental value for the decay
width.
b) Using dimensional analysis, estimate Γ(W → eνe ).
c) Compute the width Γ(W → eνe ) neglecting the mass of the neutrino and the
electron.
d) Use the previous result to compute the branching ratios into all channels using
that ΓW
tot = 2.085 GeV.
e) Compare and discuss qualitatively the results with the experimental measurements in the PDG.
4. Neutral currents.
At tree level, the Z boson can decay only into fermion-antifermion pairs, i.e.,
Z → f f . Dynamically, Z couples to all fermions in the Standard Model, but kinematically, it cannot decay into a top-antitop pair, since mt ' 174 GeV > mZ /2.
The relevant vertex for the decays is
f
Z
=−
ig
γ µ (v f − af γ 5 )
2 cos θw
f
where θw is the Weinberg angle and the constants v f and af are given in the table
below.
vf
af
f
1
1
νe , νµ , ντ
2
2
2
1
1
e− , µ− , τ − − 2 + 2 sin θw − 2
1
1
u, c, t
− 43 sin2 θw
2
2
d, s, b
− 12 + 23 sin2 θw − 12
a) Calculate the partial decay rate of the Z boson into one generic fermionantifermion pair, where the fermion could in general have color degrees of
freedom. Since mf mZ for all the Standard Model fermions except for
the top quark, you may neglect the fermion masses. Do not forget to sum and
average over spin and color degrees of freedom (a massive vector boson has
three spin degrees of freedom)! You will need to use the result
X
p µ pν
µ (p)∗ν (p) = −gµν + 2 ,
mZ
spin
which holds for the polarization vectors of the Z boson.
your result
√ Express
g2
2
f
f
in terms of mZ , v , a , and the Fermi constant, GF ≡ 8 cos2 θw m2 .
Z
b) Calculate the total decay rate of the Z boson in the Standard Model. Provide
a numerical answer, using values for mZ , sin2 θw , and GF from pdg.lbl.gov,
and compare your result to the experimentally measured value. What is the
mean lifetime of the Z boson, in SI units?
c) In the Large Electron-Positron collider at CERN, the decays of the Z boson
could be studied in the process e+ e− → Z → f f . Naively, due to the Z
1
boson propagator, the cross section would seem to be proportional to (s−m
2 )2 ,
Z
√
and hence, it would diverge at the “resonance energy” s = mZ . However, a
proper treatment taking the finite lifetime of the Z boson into account shows
that, close to the resonance energy, the cross section is actually proportional
to the so-called Breit–Wigner peak, which is given by
fBW =
(s −
m2Z )2
1
.
+ (mZ ΓZ )2
√
Plot fBW (s) in the energy range mZ /2 ≤ s ≤ 2mZ , for the three cases that
the number of light (mν < mZ /2) neutrinos is equal to 2, 3, and 4 (and the
charged fermions are the same as in the Standard Model).
Remark: The decay rate Γ is related to the width of the Breit–Wigner peak, and
for this reason Γ is sometimes also referred to as the decay width. Actually, ΓZ is
experimentally determined by studying this peak, since the Z boson is too shortlived to leave observable tracks in any detector.
5. Astrophysical neutrinos.
A common assumption for astrophysical neutrinos is that they are produced through
the following decay chain
π + → µ+ + νµ ,
µ+ → e+ + ν¯µ + νe ,
and its corresponding CP conjugate. The astrophysical distances travelled by these
neutrinos are so large that any phase information or coherence between mass eigenstates is lost (this is equivalent to actually being able to tell which mass eigenstate
was produced). Assuming that there are as many initial π − as π + :
a) What is the initial neutrino flavour composition at the source? What will be
the neutrino flavour composition when the neutrinos arrive at the Earth?
b) Estimate neutrino energy threshold for the Glashow resonance ν e + e− →
W − → anything. This reaction can be used to probe ultrahigh-energy antineutrinos in neutrino telescopes, such as IceCube at the South Pole.
Note: Flavour composition is typically given on the form 1 : nµ : nτ , where nα is
the number of neutrinos of flavor α divided by the number of electron neutrinos νe .
6. The survival probability of νe → νe oscillations is given by (i, j = 1, 2, 3)
P (νe → νe ) = 1 − 4
X
|Uei |2 |Uej |2 sin2 Fji ,
i<j
where Fji ≡ ∆m2ji L/(4E) with |∆m221 | |∆m231 | ≈ |∆m232 |. The two-flavor approximation leads to
∆m232 L
P (νe → νe ) ≈ 1 − sin 2θ sin
4E
2
2
for the reactor neutrino experiment.
a) Prove that sin2 2θ = 4|Ue3 |2 (1 − |Ue3 |2 ).
b) Given ∆m232 ∼ 2.4 × 10−3 eV2 and E ∼ 4 MeV, the optimal baseline length
L of the reactor experiment is determined by the requirement P (νe → νe ) ≈
1−sin2 2θ. What is the minimal value of L for which this condition is fulfilled?
7. Neutrino oscillation probability P (νe → νµ ) can be derived by solving the time
evolution of neutrino flavor eigenstates. The relevant Hamiltonian for two-flavor
neutrino oscillations in vacuum is
2
1
m1 0
cos θ sin θ
†
U
U
with
U=
Hv =
0 m22
− sin θ cos θ
2E
where mi (for i = 1, 2) stand for masses of neutrino mass eigenstates νi , E is neutrino energy, and θ is the mixing angle in vacuum. When neutrinos are propagating
in matter, however, the coherent forward scattering of neutrinos off matter particles
will contribute an effective potential energy term to the total Hamiltonian
2
1
m1 0
A 0
†
U
U +
,
Hm =
0 m22
0 0
2E
√
where A ≡ 2 2GF Ene with ne being the electron number density in matter.
a) Suppose that the total Hamiltonian Hm can be diagonalized by the following
unitary matrix
cos θm sin θm
,
Um =
− sin θm cos θm
where θm is the effective neutrino mixing angle in matter. Given neutrino
mass-squared difference ∆m2 = m22 − m21 and matter potential term A, derive
the relation between sin2 2θm and sin2 2θ.
b) Given ∆m2 = 2.4 × 10−3 eV2 and θ = 9◦ , calculate the matter density ρ
in units of g/cm3 where the resonance condition ∆m2 cos 2θ = A is satisfied
for supernova neutrinos of energy 15 MeV. Note that the electron fraction
Ye = ne /nB , i.e, the electron to baryon ratio, in ordinary matter is about 0.5.
8. Consider a very heavy Majorana neutrino (SM singlet or right-handed neutrino νR )
which has Yukawa couplings only with the first generation SU (2)L lepton doublet,
`e = (νe , e− )T , and the SM Higgs doublet, φe = (φ+ , φ0 )T and φ˜ = iσ2 φ∗ :
L ⊂ Yν `νR φ˜ + H.c.
(7)
a) Demonstrate that the Yukawa interaction is SU (2)L invariant, i.e., that φ˜
transforms as a doublet.
b) It is a Majorana fermion, so the mass eigenstate is N = νR + νRc . Obtain the
other interaction vertex involving νRc . [Hint: take the transpose and use the
charge conjugation matrix].
c) Compute (before spontaneous symmetry breaking) the branching ratio BR(N →
φ+ e− ).
d) What is the value (after symmetry breaking, SSB) of BR(N → W+ e− ), in the
limit mW mN ? [Hint: think about the Goldstone Boson Theorem (as in
problem 2).]
e) Write the Majorana mass term for νR (use mR as notation). Diagonalize the
complete mass matrix after SSB and obtain the spectrum of
√ active νL and
sterile νR neutrinos on the limit mD mR , with mD = Yν v/ 2.
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
• Write only on one side of each 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.