Particle Physics 2018 Final Exam (Answers with Words Only) - Inside Mines

Particle Physics 2018 Final Exam (Answers with Words Only)
This was a hard course that likely covered a lot of new and complex ideas. If you are feeling as if
you could not possibly recount all of the details of what we have done, please don't worry. The
truth is that no one usually does. Rather what happens is that we study something like this, learn
it a bit, then feel like we forgot it all. That is, until we go to learn or use it again later, in which case
we realize that it is all a bit easier the second time through. I do not expect that any of you will
leave this course knowing how to calculate a Feynman amplitude "off the couch", nor should you
be able to recite the intricate form of a Higgs'ed Lagrangian. But what I do want you to leave this
class with is an ability to talk about the big picture of what you saw and worked with. So to that
end, I want you to answer each of the questions below qualitatively. There is no need to pull out
fancy equations. Imagine you are trying to explain to an undergraduate physics major what types
of things they could learn about in particle physics. If you are not enjoying this exercise, then you
are missing the point.

1) (10 pts) What is a group? What is a representation of a group? Why are either of these
important for particle physics?

A group is set of objects together with a composition rule for combing two of the objects that
satisfies the following: Closure (the composition of any two elements of the group should return
an element of the group), Identity (the group should contain one element such that when it is
composed with any other element of the group it returns that same element), Inverse (for any
element of the group there is an element which composes with it to form the identity),
Associativity (for a specified order of three group elements, the composition of all three elements
is independent of the order in which the composition is performed).

For a given group we can associate each element to an object to form a representation. If each
group element is mapped to a distinct object then the representation is called "faithful". If more
than one group element is mapped to the same object, the representation is called "unfaithful".
For groups whose elements are transformations themselves, it is often useful to construct linear
(or matrix) representations of the groups for ease of calculation. In this case the objects are
represented by column matrices and the group elements (transformations) are represented by
square matrices.

In particle physics we employ the notion of symmetry to a) simplify calculations, b) enforce the
structure of special relativity and c) deduce the interaction terms in the Standard Model
Lagrangian. To formally describe symmetries, we first need a mathematical means of describing
transformations, and that is where groups become important, i.e. we consider sets of
transformations that form groups. However in physics we typically start with the objects that
these transformations act on, so we usually start with the representations themselves. Another

way to say this is that degrees of freedom of a physical theory must be in some representation of
the group of symmetries being used.

2) (10 pts) What is special relativity?

Special relativity is a replacement of Galilean relativity that is applicable at any relative speed (the
speed connecting two frames of reference), but reduces to Galilean relativity for relative speeds
small compared to the speed of light. It's mathematical structure can be described by starting
with a four‐dimensional space‐time where the distance between two space‐time events is
determined in terms of the rectangular space‐time coordinates of those events (where the time
coordinate is corrected by a constant factor to have spatial dimensions) and the space‐time
metric which takes the form of (almost) the identity with a relative sign difference between its
time and spatial terms. Then one simply posits that the forms of the laws of physics are invariant
under any coordinate change which preserves the form of this metric, e.g. the form of the
Lagrangian is unchanged. Teasing out the explicit form of these transformations reveals invariance
under space‐time translations, spatial rotations and space‐time boosts (altogether forming the
Poincaré group). From this starting point, one obtains all of the love of special relativity including
the constancy of the speed of light (or any massless degree of freedom) to all inertial observers,
the relativity of time, length contraction, the speed of light as an ultimate limit, relation of
energy/momentum/mass, a meaningful description of the behavior of massless degrees of
freedom, and a refinement of the notion of causality

3) (10 pts) What is a spinor?

When considering the case of spatial rotations in three dimensions, we typically start by looking at
how intuitive quantities with a (space‐related) magnitude and possibly direction are transformed.
These include scalars, vectors and other tensors, all of which are linear representations of the
rotation group. While the scalar is not very useful in determining the structure of the group (since
it is extremely degenerate), we also find that higher rank tensors can be understood in terms of
their vector indices. So in the end we can just focus on the vector representation itself. Then we
can deduce that the rotation group is most simply described by three‐by‐three real orthogonal
matrices of unit determinant which act on three real component column vectors. To generalize
this, we abstract the definition of the group in question (rotations in three dimensions) by taking
three "independent" matrices (corresponding to rotations in any three mutually orthogonal
planes) and considering their infinitesimal form. At this stage we are "undoing" the exponential
map used for Lie groups to build finite transformations from the generators of infinitesimal
transformations. This will give us matrix expressions for the generators, from which we can then
determine the Lie algebra by considering the commutators of all pairs of the generators.

The next step is to use the Lie algebra itself as a means of "defining" rotations in three dimensions.
With this in mind, we are now free to seek any other set of generators that satisfy this same Lie
algebra, and infer that they must then somehow be connected to objects which can be "rotated"
in three dimensions. Among the possible solutions (most are three‐by‐three or higher), there turns
out to be a set of generators described by complex two‐by‐two matrices. While at this point it is
often pointed out that the set of generators is given by the Pauli matrices (which also satisfy a
Clifford algebra related to the spatial metric), it is more appropriate (and important later) to
identify that they are expressible as commutators of Pauli matrices. If we take this set of
generators and then apply the exponential map to build up the set of finite transformations, we
find that these take the form of complex two‐by‐two unitary matrices with unit determinant which
then act on complex two component objects (represented by a column matrix) which we call
spinors. Note that I never said anything about "vectors" here. They are just column matrices to be
acted on by the "spinor" version of rotation operators. Altogether, this forms the spinor
representation of rotations in three dimensions. Interestingly, we find that even though the vector
and spinor representations share the same Lie algebra, they are not the same (isomorphic) as
groups. Recall that the Lie algebra of a group describes the behavior of transformations which are
infinitesimal. If we take a large enough transformation, e.g. rotation through 2 , we find that
vectors return to their original state while spinors pick up a minus sign. Another way to say this is
that for any finite rotation by an angle , there is only one result for vectors, while for spinors
there are two possible outcomes: one if we rotate by , and another (with a relative minus sign) if
we rotate by 2         . So we can see that as representations of rotations in three dimensions, the
spinor representation is more faithful than the vector.

All of this can be extended to the case of Lorentz transformations in four‐dimensional space‐time.
In this case we find that for vectors, the corresponding transformations are enacted by four‐by‐
four real matrices which have unit determinant and are orthogonal with respect to the metric of
space‐time (which means that you have a metric factor on each side of the usual orthogonality
condition). Again, we can take the infinitesimal form of an independent set of six of these
matrices (one for each mutually orthogonal plane in four dimensions) and deduce the Lie algebra.
In this case the resulting Lie algebra is complicated because it takes different forms for different
pairings of generators. We can simplify it by using certain linear combinations of the generators to
arrive at a Lie algebra that "factorizes" into two independent algebras, each of which takes the
form of a three dimensional rotation algebra. Then we can immediately carry over the earlier
results and form two‐component spinor representations of each, which must be tensored
together to form the spinor representation of the entire Lorentz group. Tensoring together two‐
component representations results in a 2 dimensional representation, which in this case means
our spinor has four components. But one should note that these are not in any simple way related
to the four components of the vector representations. The generators in the spinor representation
are given by commutators of the Dirac matrices which themselves satisfy a Clifford algebra which

is related to the metric of space‐time. From these generators we can then use the exponential
map to build up the finite form of Lorentz transformations as they act on spinors.

4) (10 pts) How are the Dirac, Proca and Klein‐Gordon equations related?

The Klein‐Gordon equation is a relativistic field equation that is appropriate for scalar fields where
the only degree of freedom is the energy‐momentum of the field configuration, and the KG
equation simply imposes that these degrees of freedom satisfy the relativistic mass‐energy
relation. The Dirac equation on the other hand is relevant for spinor fields where we have
additional degrees of freedom coming from the intrinsic angular momentum (or spin) of the field.
Since Dirac fields also carry energy‐momentum, they must also satisfy the mass‐energy relation
and in fact we find that each component of the Dirac spinor field separately satisfies the KG
equation. So in a sense, the Dirac equation contains the information in the KG equation, but also
provides additional constraints between the various components of the spinor field. Similarly, the
Proca equation, valid for vector fields, contains the information in the KG equation, but also
imposes constraints between the components (or polarizations states) of the vector field.

5) (10 pts) What is a gauge theory?

A gauge theory is a physical theory where the form of the interaction terms is determined by a
principle of local (or gauge) invariance under some continuous symmetry group. To build a gauge
theory, one usually starts with a non‐interacting theory, i.e. only (Dirac or Klein‐Gordon) kinetic
terms for the matter fields, that possesses a global symmetry (where the same transformation is
enacted at each point in space‐time) and then promotes this to a local symmetry (where the
transformation can differ at different points in space‐time). To maintain invariance, one modifies
the derivative (space‐time gradient) in the matter kinetic terms to a new covariant form with the
addition of (dual) vector gauge fields. One can determine the required transformation rule for the
new gauge fields such that the new matter field kinetic term is invariant. Expanding out this new
matter kinetic term, one observes terms that involve products of the original matter field and the
new gauge fields which implies that there is an interaction between them. The final step is to
allow the gauge field to propagate by providing it with its own (gauge invariant) kinetic term. Since
the gauge field is a (dual) vector one uses the Proca Lagrangian. The Proca term itself will be gauge
invariant only if the gauge field is massless. The field strength that appears in the Proca term can
be determined by considering the action of the commutator of the covariant derivative with itself
(with two different space‐time indices). In the case of a non‐abelian gauge group, owing to the
non‐vanishing structure constants, the gauge field strength (and hence the kinetic term) will
involve products between the various gauge fields and thus leads to interactions of the gauge
fields with each other. Such gauge‐gauge interactions are absent in the abelian case, e.g. QED.

6) (10 pts) Explain the Higgs mechanism. Why do we need it? What does it do?

The unified electroweak symmetry group acts differently on the left and right handed chiral states
of matter in the Standard Model. However, to form matter mass terms we must take bilinear
combinations of left and right chiral states. So we find that the matter mass terms are not
invariant under the combined electroweak symmetry group. This despite the fact that we see the
fundamental fields as having mass. Additionally, we observe that the weak interaction bosons
have nonzero mass, which (as stated in the answer to the previous question) spoils invariance of
the gauge kinetic term under the gauge symmetry group.

To address both of these issues, the idea is that the masses we see in the low‐energy form of the
Standard Model Lagrangian are not true fundamental mass terms, but rather are the residual
manifestation of an interaction between the matter and gauge fields with an additional scalar field
called the Higgs. The way this works is that at some higher energy (early time) state of the
universe, the Higgs potential, which itself had a (hard to draw) invariance under the electroweak
symmetry group, afforded a stable solution for a symmetric field configuration. Studying the
fluctuations of all fields around this solution results in massless matter and gauge fields interacting
with the Higgs, e.g. matter‐matter‐Higgs terms, and manifest invariance (linearly realized) under
the electroweak symmetry. At some lower energy (later time) the potential deformed and the
symmetric solution became unstable, so the Higgs field transitioned to a new but non‐symmetric
stable solution. Studying the fluctuations of all fields around this new solution results in nonzero
mass terms for the matter and (some) gauge fields due to their coupling to the Higgs which now
has a nonzero constant term as part of its background configuration, i.e. leading to matter‐matter‐
(constant+Higgs) which includes matter‐matter‐constant mass terms. Parts of the originally
manifest electroweak symmetry group are now obscured (realized nonlinearly) and we call the
gauge symmetry spontaneously broken. Fluctuations of the Higgs field along the original
symmetry directions are referred to as Goldstone modes and are eaten by the gauge fields which
acquire mass in order to provide the extra polarization state they need compared to their massless
version.

7) (10 pts) Explain what you are doing when you draw and evaluate a Feynman diagram.

A single Feynman diagram represents one realization of how to connect the incoming and
outgoing states in a given physical process. The initial incoming and final outgoing states are "free"
and the diagrams connecting them represent their interactions. The collection of all such diagrams
for a given process is used to calculate the amplitude which encodes the quantum description
of the dynamics (interactions and propagation of virtual states) involved. As per the "path‐
integral" approach to quantum mechanics, we must sum over all possible ways of connecting the
initial and final states. For coupling values less than one, we can get away with considering less
complicated diagrams (with a small number of vertices) and calculate to some order (including

all diagrams up to and at that order) which can a provide an estimate for . The amplitude        can
then be used in Fermi's golden rule to evaluate decay rates and scattering cross‐sections.

When evaluating a given diagram, we are essentially using a set of rules to assign a (complex)
number to the diagram. The rules include writing down factors which describe the spin states of
incoming and outgoing particles, vertex factors and momentum conserving delta‐functions for
each interaction vertex, and propagators for internal virtual particles. The order of elements is
written down in a manner such that overall expression is a Lorentz and gauge scalar, e.g.
constructing spin and color sandwiches. Finally one performs integrals over all internal momenta.
We can use all but one of the delta functions to evaluate some of the internal momentum
integrals. The final delta function will involve only external momenta and should be eliminated to
avoid redundancy with the delta function that appears in the golden rule.

When adding up contributions from different diagrams, we must anti‐symmetrize the sum of any
diagrams that are related by the exchange of incoming (outgoing) identical particles or anti‐
particles, or incoming (outgoing) particles with their outgoing (incoming) anti‐particle partners.

8) (10 pts) What are the similarities/differences between QED, QCD and the weak interactions.

Common to all three: The forms of all three interactions are (more or less) based on principles of
local gauge invariance (see the answer to 5 above). Thus all three interactions have a basic
interaction vertex that includes two matter fields and one gauge field.

Common to QED and QCD and different for the weak interactions: Both QCD and QED are based
on manifest symmetries observed at low energies and hence have massless gauge bosons. The
weak interactions are the broken part of the electroweak interaction (see the answer to 6 above)
and hence have massive gauge bosons. For the basic QED and QCD vertices, the flavor of the two
matter fields is the same, while for the weak interactions mediate by the        the flavor of the two
matter fields is different. The latter is why the charged weak interactions are responsible for
particle decay. One could be more technical and argue that two quarks of the same flavor but
different color are in some sense different "flavors" and then conclude that some of the QCD
vertices change "flavor". But this misses the important point that fields of different flavor tend to
have different masses, while two quarks that only differ by color would have the same mass.

Common to QCD and the weak interactions and different for QED: Both the weak interactions and
QCD are based on non‐abelian symmetry groups and hence include interactions between the
gauge bosons themselves. QED on the other hand is based on an abelian group which is why
photons do not directly interact with each other. Both QCD and the weak interactions are also
effectively short range interactions compared to QED, but for different reasons. The weak
interactions are short ranged due to the large mass of the mediating gauge bosons. QCD on the
other hand is short ranged due to confinement which dictates that quarks only appear in color

singlet bound states, and hence the interactions we see between these hadrons are indirect at
best.

Common to QED and the weak interactions and different for QCD: At low energies the couplings of
QED and the weak interactions are small enough to make perturbation theory (see the answer to
7 above) a useful approach, while the QCD has a coupling greater than one and hence must be
treated non‐perturbatively. All of these couplings change with energy scale (see answer to 9
below) and for QCD in particular, the coupling decreases with increasing energy leading to
asymptotic freedom and the ability to use perturbation theory effectively for QCD at high
energies.

Differences between all three: QED only involves the electrically charged fields of the Standard
Model, i.e. all quarks and the charged leptons, QCD only involves the quarks, and last but not least
the weak interactions involve every single matter field. Oh, and they all have different names.

9) (10 pts) WTF is renormalization, and why does it matter?

In a nutshell, the program of renormalization involves taking a divergent Feynman amplitude,
regularizing it in some way to split it into a finite piece and a piece which diverges when the
regulator is taken away, then explaining away the divergent piece in terms of shifts to inaccessible
"bare" parameters.

At the heart of renormalization is the idea that we often define quantum field theories in terms of
fundamental parameters, e.g. masses and couplings of fields, whose true values we do not
actually know and which must be "corrected" in order to obtain effective field theories whose
content can be evaluated by leading order diagrams in the Feynman expansion. Often these
corrections can come in the way of infinities cancelling infinities which was bothersome at first but
has since become widely accepted.

There ends up being three values of any parameter that are of interest:

      The bare (or fundamental) value is what we could probe if we could do experiments at
       arbitrarily high energy, which of course we can't do and so we do not have experimental
       access to these values. In some situations it seems that these values may need to be
       infinite to provide appropriate cancellations.
      The zero momentum transfer value is what we experimentally measure with low‐
       energy/large distance experiments. These are the values which are cataloged in reference
       tables and are always finite.
      The parameter value at a particular momentum transfer scale is what we use to simplify
       calculations.

When doing calculations we use these in different ways. If we are doing a calculation for a process
where a finite nonzero momentum is exchanged then:

       Working in terms of the bare parameters often leads to divergences from diagrams with
        internal loops which must be cancelled by redefinitions of the parameters which involve
        cancellations between infinities.
       Working in terms of the zero momentum transfer values will require that we include higher
        order diagrams (including loop contributions), but now these will be finite since we are
        working between two experimentally accessible regimes.
       Working in terms of the parameters at the momentum scale of the process at hand means
        that we can evaluate only the leading order Feynman diagram in order to get the correct
        (not approximate) answer.

10) (10 pts) For those who begged….draw a turtle.

There were a lot of great turtles, all of which will make it on to my Particle Physics '18 turtle collage. But the
winner is from Madi Clark: