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Phenomenology Phenomenology Lectures given at the 2012 RAL Summer School P. Richardson, IPPP, Durham University. 1 Introduction Historically the lecture notes for the phenomenology course have consisted of the slides presented in the lectures. These notes are intended to provide additional information, and more mathematical detail, on the more theoretical aspects of the course which don’t change from year to year. The recent experimental results, which as the LHC experiments take more and more data change from day-to-day, will continue to be presented solely on the slides used in the lectures. The course will focus primarily, and unapologetically, on hadron collider, and specifi- cally LHC, phenomenology. In order to study hadron collisions we need to understand the basics of cross section calculations, Quantum Chromodynamics (QCD) and jets which we + will first consider in the simpler environment of e e− and lepton-hadron collisions before we go on to study hadron–hadron collisions. This occupies the first three lectures of the course. The next two lectures consider the electroweak and Higgs sector of the Standard Model. Lectures six and seven consider possible physics beyond the Standard Model. The final lecture looks at the physics of Monte Carlo event generators which are now an essential tool in all aspects of modern collider physics. Unfortunately there is no single good book on modern phenomenology. Two old classics but now a bit dated are: Quarks and Leptons Halzen and Martin [1]; • Collider Physics Barger and Phillips [2]. • Two good books, although mainly focused on QCD and probably at a bit too high a level for this course, are: QCD and Collider Physics Ellis, Stirling and Webber [3]; • Quantum Chromodynamics Dissertori, Knowles and Schmelling [4]; • and of course the classic on Higgs physics The Higgs Hunter’s Guide Gunion, Haber, Kane and Dawson [5]. • In addition the recent reviews: Towards Jetography [6] which provides a good primer on jet physics; • General-purpose event generators for LHC physics [7] which gives a detailed descrip- • tion of the physics of Monte Carlo event generators; are good sources of additional information. 2 e+e− Annihilation + Historically this course would spend a significant amount of time looking at e e− anni- hilation. It is less relevant for current experiments but the results from LEP are still important and it is worth discussing as the simplest type of collisions. If we consider what happens when electrons and positrons collide, then the most likely thing is that some hadrons are produced. However, none of the Lagrangians or Feynman rules you’ve learnt involve hadrons. This is the key issue in most collider physics, we can calculate things for quarks and gluons but we observe hadrons. 2.1 Leading Order + We will start by studying one of the simplest possible processes, e e− annihilation via the exchange of a photon or Z0 boson, as shown in Fig. 1. This process can produce either e+ ℓ+, ν¯ e+ q γ/Z0 γ/Z0 e− ℓ−, ν e− q¯ + Figure 1: Feynman diagrams for e e− annihilation into leptons and quarks. quarks or leptons. Unfortunately due to quark confinement we cannot observe free quarks directly, instead quarks and antiquarks will produce hadrons with unit probability. Much of what we will study in this course will be concerned with the question, given we observe hadrons how do we infer what was going on in the fundamental process involving quarks? We will start with the simplest example, given quarks and antiquarks produce quarks + with unit probability we can measure the cross section for the process e e− qq¯, which + → we can calculate perturbatively, by measuring the cross section for e e− hadrons. This → is the basis of most collider phenomenology, we want to measure things using hadrons + that we can calculate using quarks. The total cross section for e e− annihilation into hadrons is the simplest such observable. Using the techniques you have learnt in the other courses you can now calculate the + total cross section for e e− annihilation. In reality it is more common to study the ratio + σ(e e− hadrons) R → , (1) ≡ σ(e+e µ+µ ) − → − as this reduces experimental uncertainties. At low energies this process is dominated by photon exchange so we can neglect the Z0 boson. In this limit 2 + + 4πα σ(e e− µ µ−)= , (2) → 3s -2 10 2 10 φ u, d, s ω φ ω -3 J/ψ 3 loop pQCD 10 ψ(2S) 10 Naive quark model ρ′ Υ ρ ′ -4 ρ ρ 10 Z 1 -5 10 Sum of exclusive Inclusive [mb] measurements measurements σ -6 -1 10 10 0.5 1 1.5 2 2.5 3 7 ψ(2S) -7 J/ψ ψ4160 c 10 6 Mark-I Mark-I + LGW ψ4415 -8 Mark-II ψ4040 5 ψ3770 10 PLUTO 2 R DASP 1 10 10 4 Crystal Ball BES 3 3 Υ 10 J/ψ ψ(2S) 2 Z 3 3.5 4 4.5 5 10 2 8 φ Υ(1S) Υ(3S) b ω 7 Υ(2S) R Υ(4S) 10 6 ρ′ 5 4 1 3 ρ ARGUS CLEO CUSB DHHM MD-1 -1 2 CrystalBall CLEOII DASP LENA 10 2 9.5 10 10.5 11 1 10 10 √s [GeV] √s [GeV] σ(e+e− hadrons) → Figure 2: The ratio R σ(e+e− µ+µ−) as a function of energy taken from Ref. [8]. ≡ → where s is the centre-of-mass energy of the collision squared. The cross section for the production of quarks is 2 + 4πα 2 σ(e e− hadrons) = e N , (3) → 3s q C q X where eq is the charge of the quark in units of the positron charge and the sum runs over all quarks for which the centre-of-mass energy √s > 2mq, where mq is the mass of the quark. Remember we must sum over all the quantum numbers of the quarks so the cross section is multiplied by number of colours, Nc. Therefore for centre-of-mass energies much less than the mass of the Z0 boson, √s M , ≪ z 2 R = eqNC . (4) q X The experimental measurement of this ratio is shown in Fig. 2 as a function of energy showing the thresholds for the production of the charm and bottom quarks. Below 1 2 the charm threshold there are three active quarks down (ed = 3 ), up (eu = 3 ) and 1 2 − 10 strange (es = 3 ) giving R = 2. Above the charm (ec = 3 ) threshold R = 3 while above the bottom (e−= 1 ) threshold R = 11 . b − 3 3 2.2 Higher Order Corrections When we draw Feynman diagrams we are performing a perturbative expansion in the (hope- fully) small coupling constant. Unfortunately the strong coupling often isn’t very small, 0 at the Z mass, αS(MZ )=0.118. We therefore need to consider higher orders in the perturbative expansion. There are always two types of correction: real gluon emission; • virtual gluon loops. • 2.2.1 Real Emission There are two possible diagrams for gluon emission, see Fig. 3. The matrix element, only e+ q e+ q γ/Z0 γ/Z0 g g e− q¯ e− q¯ + Figure 3: Feynman diagrams for e e− qqg¯ . → considering photon exchange for simplicity, is gµν = e2e g ta v¯(p )γ u(p )− (5) M q s ij b µ a q2 p +p p +p u¯ (p ) γ 6 1 6 3 γ γ 6 2 6 3 γ v (p )ǫσ(p ), i 1 σ (p + p )2 ν − ν (p + p )2 σ j 2 a 3 1 3 2 3 where pa,b are the 4-momenta of the incoming electron and positron, respectively. The out- going quark, antiquark and gluon have 4-momenta p1,2,3, respectively. The total momen- tum of the system q = p +p = p +p +p . The gluon has colour index a =1,...,N 2 1 a b 1 2 3 C − whereas the quark/antiquark have colour indices i, j =1,...,NC . Summing/averaging over spins and colours 2 2 2 2 2 2 2 4e e g Nc (p p ) +(p p ) +(p p ) +(p p ) 2 = q s C 1 · a 1 · b 2 · a 2 · b . (6) |M| s F p p p p 1 · 3 2 · 3 The colour algebra gives a colour factor N 2 1 C − 1 1 ta ta ∗ = ta ta = δaa = (N 2 1) = N C , (7) ij ij ij ji 2 2 C − C F a X where the colour charges in the fundamental (quarks and antiquarks) and adjoint (gluons) representations are 1 2 CF (NC 1) and CA NC , (8) ≡ 2NC − ≡ respectively. The three-body phase space is dΦn(pa + pb; p1,p2,p3) 3 3 3 4 d p1 d p2 d p3 = δ (pa + pb p1 p2 p3) 3 3 3 − − − (2π) 2E1 (2π) 2E2 (2π) 2E3 1 1 √ = 9 p1dp1d cos θdφp2dp2d cos βdα δ( s p1 p2 p3), 8(2π) p3 − − − where θ and φ are the polar and azimuthal angles, respectively, of the outgoing quark with respect to the beam direction. The polar and azimuthal angles of the antiquark with respect to the quark direction are β and α, respectively. We have integrated over p3 using the δ-function and assumed that the outgoing particles are massless. Using momentum conservation ~p = ~p + ~p = p2 + p2 +2p p cos β. (9) | 3| | 1 2| 1 2 1 2 q Therefore the integral over the remaining δ-function is p d cos βδ(√s p p p )= 3 , (10) − 1 − 2 − 3 p p Z 1 2 so 1 dΦ (p + p ; p ,p ,p ) = dp d cos θdφdp dα (11) n a b 1 2 3 8(2π)9 1 2 s d cos θdφdα = dx dx , 16(2π)7 1 2 2(2π)2 where x 2p /√s.
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