1. a) What does one mean by running of the strong coupling, and by asymptotic freedom of QCD? (1p)

Transcription

1 FYSH300 Particle physics 2. half-course exam (2. välikoe) : 4 problems, 4 hours. Return the the question sheet and particle tables together with your answer sheets remember to write down your name in the problem sheet! 1. a) What does one mean by running of the strong coupling, and by asymptotic freedom of QCD? (1p) b) Explain, using Feynman diagrams, how one can probe the strong coupling constant α s by measuring the decay width Γ(τ ν τ hadrons). Indicate how the width Γ(τ ν τ hadrons) depends on α s. (1p) c) 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 ratio of the following 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) d) Using lowest order QED perturbation theory, estimate the value of the ratio R = σ(e e hadrons) σ(e e µ µ ) in the region s = GeV in the cases of N C = 3 and N C = 4 colors. Draw the Feynman diagrams for these processes and indicate the coupling strengths in each vertex. Above the mass thresholds of the final state particles, you can assume an ultrarelativistic case, i.e. massless particles. [Note: this is a short calculation but answer, however, in sufficient details!] (2p)

2 2. 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. (1p) b) Show, as briefly as possible, that φ φ and (D µ φ) (D µ φ), and F µν and thus also F µν F µν, are invariant in these gauge transformations. (1p) c) 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. You don t have to do a detailed calculation here but explain the principle, i.e. write down how the fields φ(x) and A µ (x), and the terms φ φ and (D µ φ) (D µ φ) and F µν F µν transform in the Higgs mechanism. (1p) d) After the Higgs mechanism, the Lagrangian of the broken-symmetry theory becomes L = 1 4 F µνf µν 1 2 µh µ h λv 2 h e2 v 2 A µ A µ λvh λh4 1 2 e2 A µ A µ h 2 ve 2 A µ A µ h where v 2 = µ 2 /λ and h = h(x) is the real scalar field. i) Identify the mass terms and the particle masses in this theory (the factors of 2 you do not need to specify). (1p) ii) Identify all the interaction terms of the broken-symmetry theory, and draw the vertices which each of these terms describes. In the figure, indicate the interaction strength in the vertex too. (1p) iii) Draw all the tree-level (=non-loop) Feynman diagrams for the scattering A A h h which this broken-symmetry theory predicts. Indicate the coupling strengths in the graphs. (1p)

3 3. Let s consider the Standard Model (SM) Higgs particle (H 0 ) production and decays here. a) One production channel through which the SM Higgs particle is searched for in the high-energy pp collisions at the LHC, is the heavy-vector-boson fusion with tagged jets (one in the forward direction and one in the backward direction). Draw a parton model example graph of such a SM Higgs production channel, and identify the colliding partons, beam jets, tagged jets and Higgs in the figure. (1p) b) Write down, schematically, an expression for such a Higgs production cross section, dσ(p p H 0 2 jets X), according to collinear factorization. (1p) Consider then the following possible SM Higgs decay channels, marked in the figure on the next page (these are simulations for different Higgs masses): 38 CMS Collaboration / Physics Letters B 716 (2012) c) Draw an example Feynman graph of the SM Higgs decay shown in each panel (draw these below each panel). Identify the particles and draw the arrows in your diagrams. (1p) s Letters B 716 (2012) Table 4 Observed number of events, background estimates, and signal predictions for m H = 125 GeV in each category of the WW analysis of the 8 TeV data set. All the selection requirements have been applied. The combined experimental and ected background yields andtheoretical, expected number systematic of signal and statistical uncertainties are shown. The Zγ process includes the dimuon, dielectron, and ZZ analysis. The estimates ττof the llz final X states. background ass range from 110 to 160 GeV. The total background for the three bins ( signal region ) Category: of Fig. 4 where an 0-jet eμ 0-jet ll 1-jet eμ 1-jet ll 2-jet eμ 2-jet ll WW 87.6 ± ± ± ± ± ± 0.1 4μ 2e2μWZ ZZ Zγ 4l 2.2 ± ± ± ± ± ± 1.8 Top 9.3 ± ± ± ± ± ± ± ± W 0.8 jets 15.6 ± ± ± ± ± ± ± Wγ ( ) ± ± ± ± ± ± ± ± ± All backgrounds ± ± ± ± ± ± 2.2 d) Into each panel, identify the Higgs signal particles and the following detector parts of the CMS experiment: Tracking chamber (TC), electromagnetic calorimeter (ECAL), hadron calorimeter (HCAL), muon detector (MD) (2p) Let s then look at the recent measurements, shown in the two figures below. Left: The invariant mass distribution of four leptons (l l l l ) 1.7 Right: The invariant mass distribution of e ± µ ± Signal ± 0.44(m H 7.54 = 125 ± 0.78 GeV) 23.9 ± ± ± ± ± ± 0.1 e) Explain why in the four-lepton channel there is a peak at m llll 125 GeV, while in the eµ channel such a peak is not seen at m ll 125 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 ɛ loose The/(1 points ɛ loose represent ). The thenormalisation data, filled of histograms the top-quark represent the background, is estimated and the openby histogram counting shows the the number signal expectation of top-tagged for a events Higgs boson andof applying the corresponding top-tagging efficiency. The nonresonant mass m H = 125 GeV, added to the background expectation. The inset shows the m 4l distribution after selection of events with K D > 0.5, as described in the text. WW contribution is normalised by using events with a dilepton mass larger than 100 GeV, where the Higgs boson signal contamination is negligible, extrapolated to the signal region using 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.

4 YOUR NAME: 4. For the highest MH, in the range TeV, the promising channels for 10 5 pb -1 are H 0 ZZ νν, H 0 ZZ jj and H 0 W W ± νjj. Detection relies on leptons, jets and missing transverse energy (E t miss ), for which the hadronic calorimeter( HCAL ) performance is very important 2-3. In the M H range GeV the most promising channel is H 0 ZZ* 2 2 or H 0 ZZ 2 2. The detection relies on the excellent performance from the muon chambers, the tracker and the electromagnetic calorimeter. For MH 170 GeV a mass resolution of ~1 GeV should be achieved with the 4 Tesla magnetic field and the high resolution of the crystal calorimeter 1. H 0 γγ is the most promising channel if M H is in the range GeV. The high performance PbWO 4 crystal electromagnetic calorimeter in CMS has been optimized for this search. The γγ mass resolution at Mγγ ~ 100 GeV is better than 1%, resulting in a S/B of 1/20. With larger data samples ( 10 5 pb -1 ) the "associated" modes (pp WH 0 and pp tth 0 ) should give higher S/B ratios for the same H 0 decay channel M (GeV) Higgs signal Events / 500 MeV for 10 5 pb -1 H γγ M H 130 GeV M (4 ± ) GeV Events / 2 GeV for L int = 105 pb -1 tt Zbb ZZ* E t > 20,15,10,10 GeV; p t > 20,10,5,5 GeV; μ η e, < 2.5, 2.4; μ e H ZZ* 4 ± 130 M H 170 GeV H ZZ jj M H = 800 GeV M( jj), GeV N ev / 200GeV / 3*10 4 pb -1 Signal Bkgd CENTRAL CUTS (Iη < 2.4): P l > 50 GeV/c P z > 150 GeV/c TAGGING JETS(IηI > 2.4): 1 cluster, No addit. jets with P t > 40 GeV I M cl - M Z I < 15 GeV I M ll - M Z I < 10 GeV t t I 2 jets with E > 400 GeV and P t > 10GeV/c acquire mass through their interaction with the Higgs field. This implies the existence of a new particle: the Higgs boson H 0. The theory does not predict the mass of the H 0, but it does predict its production rate and decay modes for each possible mass. CMS has been optimized to discover the Higgs in the full expected mass range 0.08 TeV < MH < 1 TeV ~ ~ The decay signature of the Higgs depends on its mass: H(800 GeV) e e - jet jet ZZ ZZ* H(130 GeV) e e - e e - 3 H(150 GeV) μ μ - μ μ - ZZ* H(100 GeV) γ γ

5 4. Consider the scattering e (p a ) µ (p b ) e (p c ) µ (p d ) at the ultrarelativistic limit, where the particle masses can be neglected, and in the leading order of the electromagnetic coupling. The 4-momenta of the particles are shown in the parentheses. a) Using the Feynman rules of QED (see the attachment), compute the unpolarized differential cross section dσ dω c = M 2 64π 2 s of this scattering and express the final result in terms of the Mandelstam variables s = (p a p b ) 2, t = (p a p c ) 2 and u = (p a p d ) 2. [Hint: Formulate the calculation in terms of the leptonic tensors L e µν and L µν muon. See the collection of formulae in the end of the paper for help in doing the spin summations.] (6p) Bonus problem (extra 2p available) do if you still have time and energy left! b) Let s then suppose that instead of a spin- 1 particle the muon is a spin-0 particle, keeping however the electron as a spin- 1 particle. Compute the unpola- 2 2 rized differential cross section dσ again. In this case the Feynman rule for the dω c muon photon vertex is not ieγ ν but ie(p b p d ) ν, and the Feynman rule for the external muon legs is just 1. Express again the result in terms of the Mandelstam variables. [Hint: Before squaring the amplitude M, use momentum conservation and the Dirac equations p/u(p) 0, ū(p)p/ 0 to simplify the expression. Make use of the leptonic tensor you derived above.]

6 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 2n1 ) = 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

7

8