Homework 3: Group Theory and the Quark Model Due February 16

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1 Homework 3: Group Theory and the Quark Model Due February Lorentz Group. From the defining requirement that a Lorentz transformation implemented by a matrix Λ leave the metric invariant: Λ µ ρη ρσ Λ ν σ = η µν, prove that the set of such matrices {Λ} form a group. Assume that the matrix Λ is non-singular.. Baker-Campbell-Hausdorff Formula. In class we discussed unitary representations of the rotation group SO(3 could be implemented by exponentiating matrices: M = e it = I + it T +, where T is a matrix and the exponential is defined by its Taylor expansion. What is the multiplication law of exponentiated matrices? That is, for two matrices T 1 and T, determine T 3 such that e it 1 e it = e it 3. Find T 3 up through quadratic order in matrices T 1 and T. Hint: Express T 3 as it 3 = log ( e it 1 e it, and Taylor expand to quadratic order. 3. Pauli Spin Matrices. The Pauli spin matrices are the generators for the fundamental (two-dimensional representation of SU(. Therefore they define the structure of isospin doublets, like the proton and neutron. The isospin matrices are σ 1 = 1 ( 1 1, σ = 1 ( i i, σ 3 = 1 ( 1 1 These isospin matrices are Hermitian and so correspond to possible physical measurements. In this problem, we will study their properties applied to the isospin doublet and triplet of the proton and the neutron. (a Construct the total isospin matrix: I = σ 1 + σ + σ 3. What is the value of the total isospin I when acting on either the proton or the neutron in the nucleon doublet? This is defined by the eigenvalue of I acting on a state: I N = I(I + 1 N.. 1

2 (b It is useful to characterize particles in the same isospin multiplet by their third component of isospin, denoted I 3. What the the value of I 3 for the proton and the neutron? This is defined as the eigenvalue of σ 3 ; for example, the third component of isospin for the proton is defined as σ 3 p = I 3 p. (c What are the I 3 values of the three states in the nucleon-nucleon triplet that we found in class? To determine these values, you can just sum I 3 of the individual nucleons in the particular state. 4. Decays of the Higgs Boson. The Higgs Boson has been observed by the ATLAS and CMS experiments at the Large Hadron Collider to decay into two photons, H γγ. In this problem, we will analyze the spin of the photons. (a A plot of the most recent analysis of H γγ decays is shown below. Here, the rate of observing two photons is plotted against their invariant mass. If the Higgs boson decays from rest, write the momentum four-vectors of the two photons in the decay. Assume the photons momenta is aligned along the ẑ axis and use the plot to estimate the energy of the photons. weights / GeV Data Background Signal + Background Signal ATLAS Preliminary -1 s = 13 TeV, 13.3 fb H γγ, m = 15.9 GeV H S/B weighted sum of event categories weights - bkg m γγ [GeV] Figure 1: Rate for production of two photons in pp collisions in the ATLAS experiment as a function of the photons invariant mass. From ATLAS-CONF

3 (b The photon is a spin-1 particle, and it is useful to define its spin with respect to its direction of momentum, this is called helicity. Helicity of a photon can take two values: +1 or right-handed (R, if the spin aligns with the momentum, and 1 or left-handed (L, when the spin is anti-aligned with the momentum. Argue that the helicity of a photon is Lorentz invariant. (c For a photon with momentum in the +ẑ direction, the two helicity states can be represented by the polarization vectors ɛ R = 1 (ˆx + iŷ, ɛ L = 1 (ˆx iŷ. Rotate these polarization vectors about the ẑ axis. Under a rotation of an angle φ about the ẑ axis, the polarization of a particle with helicity h will change by a phase factor of e ihφ. Show that these two polarization vectors correspond to helicity +1 and 1, respectively. (d The Higgs boson is a spin- particle. What are the allowed helicity configurations of the two photons in the decay? 5. π-p Scattering. We are able to estimate the relative rates of interactions between different isospin multiplets with the corresponding Clebsch-Gordon coefficients. We introduced in lecture Clebsch-Gordon coefficients for determining the decomposition of product isospin states into irreducible representations of isospin. In practice, the procedure for determining the Clebsch-Gordon coefficients is to simply look them up in a table. A nice table of Clebsch-Gordon coefficients is located in the PDG. Go to the PDG website, click on Reviews, Tables, Plots and then scroll down to Mathematical Tools. From there, you can click on the link to the Clebsch-Gordon coefficient tables. In the following questions, we will use these tables to estimate relative pion-proton scattering rates. To see how these tables work, look at the 1/ 1/ table. Horizontally, the rows correspond to the coefficients of combining an isospin 1/ state with an isospin 1/ state with the corresponding I 3 components at the left. For example, consider looking for the coefficient of the I = 1 and I 3 = combination of a proton (I 3 = 1/ and a neutron (I 3 = 1/. This is found from looking at the following row and columns: 1 1/ 1/ 1/ That is, the coefficient of this term is 1/, which is exactly what we found in class. (a The proton and neutron are in the isospin I = 1/ doublet with I 3 = 1/ and I 3 = 1/, respectively, while the pions are isospin I = 1 with I 3 = 1 for π +, I 3 = for π, and I 3 = 1 for π. Determine the Clebsch-Gordon coefficients for the combinations: 3

4 i. π n in the I = 3/, I 3 = 1/ state, ii. π p in the I = 1/, I 3 = 1/ state, iii. and π + p in the I = 3/, I 3 = 1/ state. (b The baryon is a particle that is similar to a proton or a neutron, but in excited state configuration of up and down quarks. It has isospin I = 3/. From the Clebsch-Gordon table, estimate the ratio of the probability that the scattering of a proton and a π + produce a baryon versus a proton and a π. (c Below is a plot of the scattering probability for pion-proton collisions over a range of pion-proton system masses. The horizontal axis of the plot is the invariant mass of the pion-proton system (in MeV and the vertical axis is a measure of the probability of interaction at that mass (in units of millibarns, or mb. π + p results are solid and π p results are dashed. What is the minimum value that the mass of the pion-proton system can be? Is this consistent with the plot? Figure : Rates for scattering of π + p and π p as a function of the pion-proton invariant mass. From S. Gasiorowicz, Elementary Particle Physics, (

5 (d On this plot, the baryon corresponds to the peak (resonance at 13 MeV. From this plot, estimate the ratio of the probability for π + p scattering to produce a versus π p scattering. How does this compare to your result in part (b? (e Consider the bump at 155 MeV in the π p scattering data. This is now called N(15. Using the PDG, look up the isospin of this resonance. Why doesn t this resonance appear in the π + p data? 5