Phys 624 - Quantum Field Theory
University of Maryland, College Park
Fall 2015, Professor: Ted Jacobson
Homework
HW0
- due at beginning of
class Thursday, Sept. 17.
HW12
-
due
Tuesday, Dec.
15, 5pm.
1. (Renormalization
of
pseudoscalar
Yukawa theory)
(Problem
borrowed from
"An Introduction
to Quantum
Field Theory",
by Peskin
and Schroeder.)
Add part (c)
Compute the
functions
and
to leading
order in
coupling
constants,
assuming that
and
are of the
same order.
Sketch the
coupling
constant flows
in the
-
plane.
2. (The Gross-Neveu model)(Problem
borrowed from
"An Introduction
to Quantum
Field Theory",
by Peskin
and
Schroeder.)
Add part (c)
Compute
.
You should
find that the
model is
asymptotically
free.
Guidance:
This homework is based on the material covered by Anton. You
should use dimensional regularization and minimal subtraction in
your renormalization computations.
Here's what Anton had to say about the material:
"I ended up covering sections
19.1-19.3 as well as 23.2 and 23.5.1. Basically, I went through
the mechanics of renormalized perturbation theory at 1-loop for
both phi^4 theory and QED. I drew all the relevant loop
diagrams, applied the Feynman rules to them, and then just wrote
down the final answers. I then fixed all the counterterms to
cancel the 1/epsilon poles (dim reg with minimal subtraction).
From the counterterms, it's just a quick calculation to get the
beta functions and anomalous dimensions by demanding \mu
invariance of the bare couplings and masses (secs 23.2 and
23.5.1). This lecture was more algorithmic and less conceptual,
but I think that's actually the whole point of renormalized
perturbation theory."
"I forgot to mention in class that choosing the counterterms to
cancel just the 1/epsilon poles is called "minimal subtraction."
That's the way it should be done in the homework as well. Also,
in deriving the beta functions, all you need is the
counterterms. So keep that in mind when evaluating the diagrams.
It's often much easier to extract just the 1/epsilon piece than
it is to calculate the whole diagram exactly."
HW11
-
due
at beginning
of class
Tuesday, Dec.
1.
1. P16.4 (Landau
pole with
standard model
fermions)
(Note: To keep
this simple,
define the
renormalized
charges at p2
= mtop2,
and run up the
energy scale
starting from
mtop,
using the |p2|
>> mtop2
approximation
in the vacuum
polarization
graphs. The
fine structure
constant at mtop
is eR2(mtop)/4π
~ 1/127.
2.
Read Chapter
17 (The
anomalous
magnetic
moment)
3. (Exotic contributions to g-2)(This problem is borrowed from "An
Introduction to Quantum Field
Theory", by Peskin and
Schroeder. You may assume the
axion mass is much smaller than
the electron mass, as that is
anyway implied by other bounds.)
HW10
-
due
at beginning
of class
Thursday, Nov.
19.
1. P24.1 (cutting
rules)
(Note: You
will need to
read section
24.1 to do
this problem.
You will see
that you have
already
considered
some of this
material in
the process of
doing problem
7.9 on HW4.
The solution
to that
problem,
posted at our
piazza site
may be helpful
in doing this
problem as
well.)
HW9
-
due
at beginning
of class
Thursday, Nov.
12.
1. (e+e-
-> µ+µ-
through scalar
channels)
(a) Consider
the analysis
of Sec. 13.3
with the
photon
replaced by a
massless
scalar
particle, and
the
interaction
Lagrangian .
2. P13.6
(Parity
violation)
HW8
-
due
at beginning
of class
Thursday, Nov.
5.
1. (Fermi
oscillator) Suppose
b and b*
satisfy the
algebraic
relations
{b,b}=0={b*,b*},
and {b,b*}=1,
where b* is
the Hermitian
conjugate of b
and { , }
represents the
anticommutator.
Show that the
spectrum of
b*b consists
of just two
values, 0 and
1,
corresponding
to eigenstates
|0> and
|1>, and
that
b|0>=0,
b*|0>=|1>,
b|1>=|0>.
2.
P13.1
(Møller
scattering,
e−e− → e−e−)
3.
P13.3
(particle
decays to
e+e−)
HW7
- due
at beginning of
class Thursday,
Oct. 29.
1.
(boost to
the
eigenspinors)
In class
we showed that
the spinors
u(p) and v(p)
in (11.25)
satisfy the
Dirac equation
with positive
and negative
energy
respectively.
These can also
be obtained
from (11.17)
by application
of a Lorentz
boost. (a)
What
hyperbolic
boost angle
produces
energy E and
momentum p?
(b) Apply the
appropriate
boosts to the
L & R
chirality
components of
(11.17) and
show that you
recover
(11.25). [Hint:
exp[w·]=cosh(w)
+ sinh(w) (w/w)·,
where
w = √w·w,
as
noted in
class.]
2.
P11.4
(Gordon
identity)
HW6 - due at beginning of
class Thursday, Oct. 29.
1. P10.1 (relativistic
effects in hydrogen atom) : Revision: This
problem is confusing and
confused. Hence change it as
follows. a) Delete all parts
except (h), and note that "m"
should be
/
in that part, and
"time-dependent" should be
time-independent".) Replace the rest of the
problem by this.
(A proper derivation of the
relativistic corrections is
tricky. I will write it up
and post it later.)
2. P11.1
(Dirac matrix identities)
3. P11.6 (physics
of spin & helicity)
The textbook
website has this correction: "
This problem has generated some
confusion. A cleaner version would
be: (a) Use the left and right
chirality projection operators to
show that the QED vertex vanishes
unless ψ and its conjugate are both
left-handed or both right-handed.
And (d) Suppose we take a spin-up
electron going in the +z direction
and turn it around carefully with
electric fields so that now it goes
in the -z direction but is still
spin up. Has its helicity or
chirality flipped (or both)? How is
your answer consistent with part
(a)?" For part (f), note that the
weak interaction, which is
responsible for beta decay, couples
only to the left handed chirality of
the electron.
4. P11.9 (right handed neutrino,
Majorana mass & see-saw mechanism) [Change
part (d) to: "If m=100GeV, what value of M would yield a
light neutrino mass 0.1 eV, and what would be the heavy
neutrino mass?] For
11.9(c), by "satisfies
the Klein-Gordon
equation" he means with
a mass-matrix that can
be diagonalized. By
taking linear
combinations of the
fields you can decouple
the system into
components, each of
which satisfies the
Klein-Gordon equation
with a different mass.
HW5 - due at beginning of class Thursday,
Oct. 15.
1. (gauge invariance and
the little group for massless spin-1
particles)
2.
(spontaneous symmetry breaking
and Higgs mechanism) Read Sections 1-3 of http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism
and verify equations (9) and (12).
3. (Linear sigma
model) This multi-part
problem is borrowed from "An Introduction
to Quantum Field Theory", by Peskin and
Schroeder.
HW4 - due
at beginning of class Tuesday, Oct. 6.
1. Problem 7.3 (non-relativistic
Møller scattering)
2.
Problem 7.4 (mass as an interaction)
3. Problem 7.8 (mu
and tau decay)
4. Problem 7.9 (unstable
particles and widths)
HW3 - due
at beginning of class Thursday, Sept. 24.
1. Read section 3.5: Green's functions
2. Problem 3.4 (second order classical Feynman
diagrams)
3. Problem 3.6 (Yukawa
potential)
4. Read section 7.1: Lagrangian derivation of position
space Feynman rules
5. Carry out the check suggested after Eq. (7.11)
HW2 - due
at beginning of class Thursday, Sept. 17.
1. The LHC is currently producing head-on collisions of
protons with energy 6.5 TeV each. If instead one of the
protons were at rest, what energy would the other proton need
to have in order to produce a collision with the same energy
in the zero-momentum frame?
2. For a given source distance, there is an effective maximum
energy above which the universe is opaque to extragalactic
gamma rays due to the pair creation process
,
where the second photon comes from the "background light". To
get a sense for where this cutoff energy lies, compute the
threshold energy for a gamma ray to collide with a 1 micron
photon to make an electron-positron pair. Compare your result
with the graph in Fig. 3 of http://arxiv.org/abs/0807.2822.
3. Problem 5.3 (muon decay phase space)
4. Problem 5.5 (Rutherford scattering)
HW1
- due at beginning of class Thursday, Sept.
10.