Homework 4: Fermi s Golden Rule & Feynman Diagrams

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1 tot = Nel + N inel, (4) where N el and N inel represent respectively the elastic and inelastic rates integrated over a given data taking period. Using the above equation, the elastic and inelastic measurements performed (independently but simultaneously) in the same Due analyses February of reference 28 (note [9] different and [11] due respectively, date!) Homework 4: Fermi s Golden Rule & Feynman Diagrams σ el (green), σ inel (blue) and σtot (red) [mb] A measurement of the forward charged partic using the data taken trigger during special As a key aspect o track reconstruction, a (up to 10%), domina 1. Proton-Proton were combined Total Cross to obtain Section. tot The without TOTEM taking experiment into account in the analysis in ord (TOTal Elastic and diffractive the luminosity cross section[15]. Measurement) Furthermore, at the also Large el and Hadron inel Collider were measures ases, found the to be domi total cross derived section independently for proton collisions. from the Theluminosity TOTEM experiment using theisme- asured ratio N located far T2 down- stream from the collisions at CMS. The basic way that it measures the total cross quarters. The relat section of the proton is el /N much inel. The errors on this measurement within a T2 quarter in the same as how you would measure the cross section of any object. are dominated by the uncertainties on the extrapolation alignment) have been to t = 0 and on the correction for the contribution from ods (iterative and MIL A very efficient way to measure the cross section of an object is to shine light on it and measure lowthemass size of di raction. the shadow. The Thecross-sections TOTEM experiment measured sits far with in the related correctio from the collisions at CMS this so L-independent that it can measure method, the size reported of the effective in table shadow 1, are well µm. The relative align from the proton collisions. in agreement A proton occupies with the a finite previous size in TOTEM space, and measurements, quarters of an arm ha so particles produced in the collision which must be confirms deflectedthe around understanding the proton, of producing the systematic a shadow, uncertainties and of the corrections applied in the di erent or lack of structed particlesin the overlap observed. ment respect to its no The plot analysis below are strategies. results from These TOTEM measurements and various areother alsoexperiments reported onand the shifts), measurements of main i in figure of the total, 5, showing elastic (kinetic a very energy good agreement preserving), with and inelastic the ex-crospectations from the overall fit of previously measured data sections has beenof the proton, measured in millibarns, as a function of center-of-mass collision energy achieved by e s. the track parameters d Estimate the radius of the proton in meters as measured by the TOTEM experiment, at center-of-mass over a large collision rangeenergies of energies of 7 [10]. on each T2 plane of th and 8 TeV. shaped zone character 140 due to primary particl pp (PDG) 130 pp (PDG) pipe cone in front of 120 Auger + Glauber ALICE methods resulted in co 110 ATLAS precision of 1 mm a 100 CMS 90 TOTEM (L indep.) precision of 0.4 mra best COMPETE σ tot fits ln s ln 2 s σ tot Secondary track re 70 this analysis as about 60 is due to secondaries. 50 σ inel cedure for the primar based on a proper trac 20 σ el in detailed MC studies 10 and the most stable ag track e ciency (found s [GeV] corrections have been Figure 5. Compilation of the tot, inel and el measurements. The results of the m The black points show the L-independent measurements performed by TOTEM at p and total Figure 1: Plots of the elastic, inelastic, s = 7 and p cross sections for proton scattering withmea- sured by various experiments, as a function of the center-of-mass collision energy, at least one char s = 8 TeV. The continuous black lines (lower for pp, upper for p p) represent the best fit to and s. From > s go AG. Latino [TOTEM Collaboration], Summary of Physics Results from the TOTEM tot data by the COMPETE Collaboration [10], while the dashed reported Experiment, EPJ Web Conf. 49, (2013). in figure 6. H line is related to a fit to the el data. The dash-dotted lines, corresponding to inel, are derived as the di erence between the two with the four T2 qua mental values in terms previous fits. 1 ting the combination o

2 2. Proton Collision Beam at ATLAS. In class, we derived the expression for the number of proton collision events per second: events/second = n A n B Vol v A v B σ. Here, the beams are denoted by A and B, with n A the number density of protons in bunch A and v A the velocity of bunch A. Vol is the volume of a bunch of protons, and σ is the cross section for proton collisions. Everything on the right of this equation, except for σ is called the luminosity L, with L = n A n B Vol v A v B. This has units of area 1 time 1. The integrated luminosity is the time integral of the luminosity: Integrated Luminosity = t1 t 0 dt L, and has units of area 1. The integrated luminosity in collider physics experiments is often measured in units of inverse barns (b 1 ). Below is a plot from the ATLAS experiment of the integrated luminosity of proton collisions during data taking in 2016, as a function of date. Note that the date is written in the European standard Day/Month. The integrated luminosity is measured in inverse femtobarns (fb 1 ). ] -1 Total Integrated Luminosity [fb 50 ATLAS Online Luminosity s = 13 TeV LHC Delivered ATLAS Recorded -1 Total Delivered: 38.9 fb -1 Total Recorded: 36.0 fb 0 18/04 16/05 13/06 11/07 08/08 05/09 03/10 31/10 Day in /16 calibration Figure 2: Total integrated luminosity of the ATLAS experiment versus the date in 2016 when data was taken. From the ATLAS Experiment Luminosity Public Results Run 2. (a) From this plot, estimate the average luminosity over the data taking period. Express the result in inverse femtobarns per second and inverse centimeters squared per second. 2

3 (b) Using the TOTEM plot from the previous problem, estimate the number of proton collision events that took place in the ATLAS experiment during data taking in On average, approximately how many proton collision events occurred, per second? (c) Protons at the LHC are accelerated in bunches in an RF electromagnetic field that has a frequency of 400 MHz. At the collision points, in experiments like ATLAS and CMS, the radius of a bunch of protons is about 10 micrometers. Using the plot of the integrated luminosity above, estimate the number of protons in each bunch at the LHC. (d) The plot below is nicknamed the Stairway to Heaven plot, and shows the cross section in picobarns for numerous final states in proton-proton collisions as measured in the ATLAS experiment at the LHC. For example, the cross section for W boson production pp W at a center-of-mass collision energy of 13 TeV is about σ pp W pb = b. For the following processes estimate the number of collision events recorded in 2016 (13 TeV collisions): i. pp Z ii. pp H iii. pp t t Standard Model Total Production Cross Section Measurements Status: August 2016 σ [pb] 500 µb µb ATLAS Preliminary 1 Theory 10 6 Run 1,2 s = 7, 8, 13 TeV LHC pp s = 7 TeV Data fb pb 1 35 pb 1 81 pb 1 35 pb 1 LHC pp s = 8 TeV Data 20.3 fb 1 LHC pp s = 13 TeV Data fb total 2.0 fb 1 VBF 1 VH t th 10 1 pp W Z t t t t-chan WW H Wt WZ ZZ t s-chan t tw t tz Figure 3: Measured cross sections for the production of numerous Standard Model particles. From the ATLAS Experiment Standard Model Public Results. 3

4 3. e + e e + e Scattering. In class, we discussed the details of the (lowest-order) Feynman diagram for the process e + e µ + µ. In this problem, we will study the process e + e e + e. We discussed some of the rules for drawing Feynman diagrams with electrons, muons, and photons in class. To summarize, they are: Identify all external fermions. Draw a line for each external fermion and an arrow along the line that points with time (for particles) or against time (for anti-particles). Connect pairs of external particle lines that correspond to the same type of fermion (electron or muon) and that have arrows that point in the same direction. Draw photons to connect pairs of these external particle lines. Repeat for all possible unique configurations/diagrams. Using these rules draw all of the Feynman diagrams for e + e e + e scattering. Each diagram should contain only one internal photon, and be sure to clearly label the momentum of each external particle. Hint: Unlike e + e µ + µ scattering, there is more than one Feynman diagram for e + e e + e scattering. 4. Non-relativistic limit of Feynman diagrams. Feynman diagrams may seem somewhat magical and disconnected from familiar non-relativistic or classical physics. In this problem, we will demonstrate how the contents of a Feynman diagram connect to the non-relativistic limit. Consider the scattering of electrons and muons e µ e µ which is represented by the following Feynman diagram: e p p 0 e q µ k k 0 time µ (a) Express the four-momentum q flowing through photon in terms of the four momenta of the external particles. (b) Describe the physical configuration of the electron and muon if the internal photon goes on-shell. 4

5 (c) We can test the mass of the photon in processes like this. Call the four-momentum of the internal photon q. For a massive photon, the denominator of the propagator is changed from q 2 to q 2 m 2 γ, where m γ is the purported mass of the photon. Multiplying by the electric charges e of the electron and muon, we can define a momentum electric potential, Ṽ (q): Ṽ (q) = e2 q 2 m 2 γ. Take the non-relativistic limit of this expression; that is, using the result from part (a), take the limit where the masses of the electron and muon are much larger than their magnitude of (three-)momentum. (d) Call the non-relativistic limit of the potential from the previous question Ṽ (q); that is, it depends on the three-momentum of the photon. We aren t used to dealing with potentials defined by particles momenta; typically, we deal with potentials that are functions of (relative) position. To determine the electric potential V (r), Fourier transform the potential Ṽ (q) from a function of momentum to position. Hint: Because momentum is a vector, you will need to integrate over all three components of the momentum. An efficient way to do the Fourier integral is to change to spherical coordinates. A simple way to evaluate the final integral is as a contour integral in the complex plane and using Cauchy s theorem. (e) For m γ 0, what is the effective range of the potential V (r)? What is the potential if m γ 0? 5. Three-body Phase Space. In class, we defined n-body phase space as the integral (including momentum conservation) n ( ) d 4 p i n dπ n = (2π) 4 2πδ(p2 i m 2 i ) (2π) 4 δ (4) Q p i, i=1 where Q is the total four-vector of the initial state. The textbook simplifies two-body phase space (n = 2); in this problem, we will simplify three-body phase space (n = 3), which will be useful when we discuss processes in QCD. In this problem, we will denote the energies of the three final-state particles as E i, for i = 1, 2, 3, and they have masses m i, for i = 1, 2, 3. We will work in the center-ofmass frame where the total momentum is 0 and the total energy is E CM. The total four-vector of the process will be denoted by Q = (E CM, 0, 0, 0). (a) Very useful quantities for defining three-body phase space are the x i variables, for x = 1, 2, 3. Define x i = 2Q p i Q 2. 5 i=1

6 Show that x 1 + x 2 + x 3 = 2. (b) In the center of mass frame, determine expressions for the energy of particle i, E i and the magnitude of the three-momentum p i in terms of the x i and the masses of the particles. (c) In the textbook, the on-shell δ-functions are integrated over, producing the integral for three-body phase space: d 3 p 1 d 3 p 2 d 3 p dπ 3 = (2π) 4 δ (4) (Q p (2π) 3 (2π) 3 (2π) 3 1 p 2 p 3 ), 2E 1 2E 2 2E 3 We can eliminate the integral over p 3 by enforcing the three-momentum conserving δ-functions. Integrate over p 3 and express the result in spherical coordinates as an integral over the four remaining angles and the magnitudes of momentum p 1 and p 2, with one δ-function constraint remaining. Work in the center-of-mass frame. (d) By integrating over p 3, the energy E 3 is a function of p 1 + p 2 and therefore a function of the angle between p 1 and p 2, cos θ 12. Do the integral over cos θ 12, eliminating the last δ-function. (e) The remaining three angles simply rotate the three-particle system. over these angles. Integrate (f) Now, there just remain two integrals over the magnitude of momenta p 1 and p 2. Using part (a), express the remaining integrals in terms of x 1 and x 2. Show that dπ 3 = E2 CM dx 128π 3 1 dx 2. 6