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 byfour 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 twocomponent 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 "pathintegral" 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 antiparticles, 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 lowenergy/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: