FYSH300 Particle physics 2. half-course exam (2. välikoe) : 4 problems, 4 hours.

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1 ics Letters B 716 (2012) FYSH300 Particle physics 2. half-course exam (2. välikoe) : 4 problems, 4 hours. 1. a) What is meant by deep inelastic electron-proton scattering, and what is the main purpose of such collision measurements? (1p) b) Draw a detailed parton-model graph of the Standard Model (SM) Higgs production&decay process, where the Higgs is produced in p + p collisions through heavyvector-boson 4 fusion with tagged jets (one in the forward direction and one in the Table Observed backward numberdirection), of events, background and where estimates, the and signal Higgs predictions then decays for m H = 125 into GeVainmuon-antimuon each category of the pair WW analysis of the 8 TeV data set. All the selection requirements have been applied. The combined experimental and theoretical, and 2 jets. systematic Alland the statistical vertices uncertainties appearing are shown. inthe your Zγ process graphincludes mustthebe dimuon, basic dielectron, SM vertices and ττconsult ll finalthe states. attachments if needed. Identify the colliding protons and partons, beam Category: jets, tagged jets, decay-jets 0-jet eμ and 0-jet all ll other 1-jet particles eμ 1-jet in your ll graph. 2-jet eμ(1p) 2-jet ll 38 CMS Collaboration / Physics Letters B 716 (2012) ected background yields and expected number of signal ZZ analysis. The estimates of the Z + X background mass range from 110 to 160 GeV. The total background n for the three bins ( signal region ) of Fig. 4 where an WW 87.6 ± ± ± ± ± ± 0.1 4μ 2e2μ 4l ± ± ± WZ + ZZ + Zγ 2.2 ± ± ± ± ± ± 1.8 c) Let s look at the SM Higgs measurements shown in the figures below. Top 9.3 ± ± ± ± Left: The invariant mass distribution of four leptons (l l l l ± ± 1.2 W + jets ) ± ± ± ± ± ± 0.0 Wγ 1.7 Right: ( ) 6.0 ± ± The invariant mass distribution of e ± ± µ ± ± ± 0.0. All Explain backgrounds why in the124.2 four-lepton ± channel ± 15.0 there 61.7 ± 7.0 is a 33.1 peak ± 5.7 at m4.1 ± 1.9 llll ± GeV, 2.2 while Signal in the (m H eµ = 125 channel GeV) such 23.9 ± a5.2 peak is 14.9 not ± 3.3seen10.3 at ± m3.0 ll ± GeV (1p) ± ± ± ± ± ± ± ± ± ± Data ± The background from low-mass resonances is rejected by requiring a dilepton invariant mass greater than 12 GeV. To suppress the top-quark background, a top tagging technique based on soft-muon and b-jet tagging is applied. The first method is designed to veto events containing muons in b jets coming from decays of top quarks. The second method uses a b-jet tagging algorithm, which looks within jets for tracks with large impact parameters. The algorithm is applied also in the case of zero-jet events, which may contain low-p T jets below the selection threshold. To reduce the background from WZ production, events with a third lepton passing the identification and isolation requirements are rejected. Yields for the dominant backgrounds are estimated using control regions in the data. The W + jets contribution is derived from data using a tight loose sample in which one lepton passes the standard criteria and the other does not, but instead satisfies a loose set of requirements. The efficiency ɛ loose for a jet that satisfies the loose selection to pass the tight selection is determined using data from an independent loose lepton-trigger sample dominated by jets. The background contamination is then estimated Fig. 4. using Distribution the events of the four-lepton of the tight loose invariant masssample for the weighted ZZ 4l analysis. by The points represent the data, the filled histograms represent the background, ɛ loose /(1 ɛ loose ). The normalisation of the top-quark background and the open histogram shows the signal expectation for a Higgs boson of mass is estimated m H = by counting the number of top-tagged events and applyintribution theafter corresponding selection of events top-tagging with K D > 0.5, efficiency. as described Thein nonresonant the text. 125 GeV, added to the background expectation. The inset shows the m 4l dis- WW contribution is normalised by using events with a dilepton mass larger than 100 GeV, where the Higgs boson signal contamination is tabulated is negligible, using a simulation extrapolated of the to the qq signal ZZ/Zγregion process. using The simulated statistical samples. analysisthe only same-flavour includes events Drell Yan with mbackground 4l > 100 GeV. is normalised Fig. using 5 (upper) the number shows of the events distribution observed with of K D a dilepton versus mmass 4l for within events 7.5selected GeV of in the the Z boson 4l subchannels. mass, after The subtracting colour-coded the regions non- Drell Yan show thecontribution. expected background. Other minor Fig. 5 backgrounds (lower) shows from the WZ, samezz, two- Wγ are estimated distribution from of simulation. events, but this time superimposed anddimensional on Thethe 7 TeV expected data are event analysed density by from training a SM a BDT Higgs for boson each Higgs (m H = boson 125 mass GeV). Ahypothesis clusteringin of the events zero-jet is observed and one-jet around event 125 GeV categories, a large while value a simple of K D, where selection thestrategy background is employed expectation in is thelowvbf and with category the signal [26]. expectation In the BDT analysis, high, corresponding the Higgs boson to signal the excess is separated in the fromone-dimensional the backgroundmass by using distribution. a binned The maximum-likelihood m 4l distribution of seen fit events to thesatisfying classifier Kdistribution. D > 0.5 shown The 8inTeV the analysis inset in Fig. is based 4. on a simple There selection are three strategy final states optimized and two for each data sets mass(7hypothesis, and 8 TeV), where and additional thus the statistical kinematictreatment and topological requiresrequirements six simultaneous are appliedimensional to improve maximum-likelihood the signal-to-background fits forratio. each value One of ofthe m H,inthe two- most Fig. 7. Distribution of m ll for the zero-jet eμ category in the H WW search at 8 TeV. The signal expected from a Higgs boson with a mass m H = 125 GeV is shown added to the background. in the zero-jet eμ category are expected to be produced by the gluon gluon fusion process, whereas 83% of the signal in the twojet eμ category is expected to be produced by the VBF process. The 95% CL expected and observed limits for the combination of the 7 and 8 TeV analyses are shown in Fig. 8. Abroadexcessisobserved that is consistent with a SM Higgs boson of mass 125 GeV. This is illustrated by the dotted curve in Fig. 8 showing the median expected limit in the presence of a SM Higgs boson with m H = 125 GeV. The expected significance for a SM Higgs of mass 125 GeV is 2.4σ and the observed significance is 1.6σ H ττ The decay mode H ττ is searched for in four exclusive subchannels, corresponding to different decays of the τ pair: eμ, μμ, eτ h, and μτ h, where electrons and muons arise from leptonic τ decays, and τ h denotes hadronic τ decays. The latter are recon-

2 d) Sketch a cross-section figure of the LHC s CMS detector, where you put the beam pipe (BP), muon detector (MD), electromagnetic calorimeter (ECAL), tracking chamber (TC), and hadron calorimeter (HCAL), at their correct places. (1p) e) The absolute values of the elements of the CKM-matrix have been measured to be as follows (ignoring the experimental error bars here): V ud V us V ub V cd V cs V cb = V td V ts V tb Using this, estimate the following ratio of the B + meson decay widths: Γ(B + (u b) K + (u s) + π 0 (uū)) Γ(B + (u b) π + (u d) + D 0 (u c)) where the quark content of the hadrons is shown in the parentheses, and where you don t have to think about any phase-space effects. Draw also the Feynman graphs for these processes. (2p) 2. Consider a theory whose Lagrange density is L = 1 2 µφ µ φ µ2 2 φ2 η 3! φ3 λ 4! φ4 + i ψγ µ µ ψ m ψψ + ig ψγ 5 ψφ where φ is a real field for a spin-0 particle which we call here φ, and ψ is the Dirac spinor for a spin- 1 particle which we call here F. The γ 2 µ are the Dirac gamma matrices and γ 5 is given in the collection of the formulae. As usual, ψ ψ γ 0. The constants µ 2, η, λ, m and g are positive. Let s also assume that the potential of the theory has the minimum at φ = 0, so that the particle content, masses and interactions of the theory can be directly read off from the above Lagrangian. a) Identify the kinetic terms, mass terms and interaction terms of this theory. (1p) b) For each interaction term, draw the basic vertex, indicate the particles participating in the vertex, and indicate also the interaction strength of each vertex. (2p) c) Using the basic vertices of this theory [be extra careful in the b)-item above!], draw all the tree-level (=non-loop) Feynman diagrams which this theory predicts for the following scatterings, and indicate for each graph what is its dependence on the interaction strengths (3p): i) F + F F + F ii) F + F φ + φ iii) φ + φ φ + φ

3 3. Consider a theory whose Lagrange density is L = 1 4 F µνf µν + (D µ φ) (D µ φ) µ 2 φ φ λ(φ φ) 2, where φ is a complex scalar field which depends on the coordinate 4-vector x and which describes a charged spin-0 particle. The gauge field is A µ and the field strength tensor is F µν = µ A ν ν A µ. This theory is invariant in local U(1) gauge (phase) transformations, φ U(x) U(x)φ, where U(x) = e iα(x) where α(x) is real. The covariant derivative, D µ = µ iea µ, is required to transform in these gauge transformations as D µ φ U(x) U(x)D µ φ. a) Derive the transformation law for the gauge field A µ in these gauge transformations. (2p) b) To test your understanding of the 2013 Physics Nobel prize(!): Let s assume that λ > 0 but µ 2 < 0, so that the Higgs mechanism is needed to find out the physical fields and masses of the theory. Perform the Higgs mechanism for this theory in detail. In the end, identify the physical particle content of the theory, the Higgs particle and the masses of the particles obtaining the masses 2 up to factors 2 is just fine, you don t need to consider Euler-Lagrange equations here. (4p) Hint: Start by writing φ(x) = U(x) 1 φ(x) where φ(x) is a real field which accounts for the oscillations around the potential s minimum at φ 2 min = v2 µ2. 2 2λ

4 4. The figure below describes the measurement of the angular distribution (in the CMS frame) of muons produced in the unpolarized scattering e + + e µ + + µ Starting from the Feynman rules, compute the differential cross section dσ/d cos θ of this process, in the lowest order in the electromagnetic interaction. Express the final result in terms of the scattering angle θ in the CMS-frame, fine-structure constant α = e2 and the CMS energy s, assuming that s is in the range π GeV. After this, form the quantity dσ d cos θ, dσ d cos θ cos θ =0 which is shown in the figure and compare your result with the figure. In the end, explain briefly what one can learn from measuring such an angular distribution. Instructions: Let s consider here only the high-energy limit, i.e. you can set the particle masses in the initial and final states to zero. Please use the following notation: p a for the 4-momentum of the positron, p b for the 4-momentum of the electron, p c for the antimuon and p d for the muon. Mark all the intermediate steps you perform in your answer sheet and not on a scrap paper. Make use of the attached table (Field C.1) of the Feynman rules and the collection of the formulae.

5 Collection of formulae g µν = g µν ˆ= diag(1, 1, 1, 1) A µ B µ = A 0 B 0 A B {γ µ, γ ν } γ µ γ ν + γ ν γ µ = 2g µν 1 4 γ µ = γ 0 γ µ γ 0 γ µ γ µ = 41 4 γ µ a/ γ µ = 2a/, where a/ γ µ a µ γ µ a/ b/ γ µ = 4a b γ µ a/ b/ c/ γ µ = 2c/ b/ a/ γ 5 = γ 5, where γ 5 = iγ 0 γ 1 γ 2 γ 3 (γ 5 ) 2 = 1 4 {γ 5, γ µ } = 0 Tr(γ µ γ ν ) = 4g µν Tr(γ µ γ ν γ ρ γ σ ) = 4(g µν g ρσ g µρ g νσ + g µσ g νρ ) Tr(γ 5 ) = 0 Tr(γ µ 1 γ µ 2... γ µ 2n+1 ) = 0 Projection operators for Dirac spinors: u (s) (p)u (s) (p) = p/ + m s=1,2 v (s) (p)v (s) (p) = p/ m, s=1,2 where u u γ 0 and v v γ 0 Dirac equations: (p/ m)u(p) = 0 (p/ + m)v(p) = 0 Cross section ab cd (when m a,b = m c,d ): dσ dω = M 2 64π 2 s Spherical coordinates: dω = 2π 0 1 dφ d(cos θ) 1

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