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2 - Collider Physics

2 - Collider Physics. 2.1 Phase space and rapidity - the “plateau” 2.2 Source Functions - protons to partons 2.3 Pointlike scattering of partons 2.4 2-->2 formation kinematics 2.5 2--1 Drell-Yan processes 2.6 2-->2 decay kinematics - “back to back” 2.7 Jet Fragmentation.

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2 - Collider Physics

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  1. 2 - Collider Physics • 2.1 Phase space and rapidity - the “plateau” • 2.2 Source Functions - protons to partons • 2.3 Pointlike scattering of partons • 2.4 2-->2 formation kinematics • 2.5 2--1 Drell-Yan processes • 2.6 2-->2 decay kinematics - “back to back” • 2.7 Jet Fragmentation

  2. Kinematics - Rapidity • One Body Phase Space • NR Rapidity Relativistic Kinematically allowed range in y of a proton with PT=0 If transverse momentum is limited by dynamics, expect a uniform distribution in y

  3. Rapidity “Plateau” Monte Carlo results are homebuilt or COMPHEP - running under Windows or Linux Region around y=0 (90 degrees) has a “plateau” with width y ~ 6 for LHC LHC

  4. Rapidity Plateau - Jets For ET small w.r.t sqrt(s) there is a rapidity plateau at the Tevatron with y ~ 2 at ET < 100 GeV.

  5. Parton and Hadron Dynamics For large ET, or short distances, the impulse approximation means that quantum effects can be ignored. The proton can be treated as containing partons defined by distribution functions. f(x) is the probability distribution to find a parton with momentum fraction x. Proceed left to right

  6. The “Underlying Event” The residual fragments of the pp resolve into soft - PT ~ 0.5 GeV pions with a density ~ 5 per unit of rapidity (Tevatron) and equal numbers of +o-. At higher PT, “minijets” become a prominent feature s dependence for PT < 5 GeV is small

  7. COMPHEP - Minijets p-p at 14 TeV, subprocess g+g->g+g, cut on Ptg> 5 GeV. Note scale is mb/GeV

  8. Minijets - Power Law? pp(g+g) -> g + g The very low PT fragments change to “minijets” - jets at “low” PT which have mb cross sections at ~ 10 GeV. The boundary between “soft, log(s)” physics and “hard scattering” is not very definite. Note log-log, which is not available in COMPHEP – must export the histogram

  9. The Distribution Functions • Suppose there was very weak binding of the u+u+d “valence” quarks in the proton. • But quarks are bound, . • Since the quark masses are small the system is relativistic - “valence” quarks can radiate gluons ==> xg(x) ~ constant. Gluons can “decay” into pairs ==> xs(x) ~ constant. The distribution is, in principle, calcuable but not perturbatively. In practice measure in lepton-proton scattering. x ~ 1/3, f(x) is a delta function

  10. Radiation - Soft and Collinear ,k The amplitude for radiation of a gluon of momentum fraction z goes as ~ 1/z. The radiated gluon will be ~ collinear -  ~ k ==>  ~ 0. Thus, radiated objects are soft and collinear. P (1-z)P Cherenkov relation

  11. COMPHEP, e+t->e+t+A Use heavy quark as a source of photons – needed to balance E,P. See strong forward (electron-photon) peak.

  12. Parton Distribution Functions In the proton, u and d quarks have largest probability at large x. Gluons and “sea” anti-quarks have large probability at low x. Gluons carry ~ 1/2 the proton momentum. Distributions depend on distance scale (ignore). “valence” “sea” gluons

  13. Proton – Parton Density Functions g dominates for x < 0.2 At large x, x > 0.2, u dominates over d and g. “sea” dominates for x < 0.03 over valence. Points are simple xg(x) parametrization.

  14. 2-->2 Formation Kinematics x1 x2 E.g. for top quark pairs at the Tevatron, M ~ 2Mt ~ 350 GeV. <x> ~ ~350/1800 ~ 0.2 Top pairs produced by quarks.

  15. Linux COMPHEP • g + g->g + g with Pt of final state gluons > 50 GeV at 14 TeV p-p • n.b. To delete diagrams use d, o to turn them back on one at a time • Cross section is 0.013 mb (very large) • Write out full events – but no fragmentation. COMPHEP does not know about hadrons.

  16. gg -> gg in Linux COMPHEP Note the kinematic boundary, where <x> ~ 0.007 is the y=0 value for x1=x2 for M = 100, C.M. = 14000.

  17. CDF Data – DY Electron Pairs DY Plateau x1,x2 at Z mass ~ 0.045

  18. The Fundamental Scattering Amplitude

  19. Pointlike Parton Cross Sections Pointlike partons have Rutherford like behavior  ~ (12)|A|2/s s,t,u are Mandelstam variables. |A|2 ~ 1 at y=0.

  20. Hadronic Cross Sections To form the system need x1 from A and x2 from B picked out of probability distributions with the joint probability PAPB to form a system of mass M moving with momentum fraction x. C is a color factor (later). The cross section is  ~ (d/dy)y=0y. The value of y varies only slowly with mass ~ ln(1/M)

  21. 2-->2 and 2-->1 Cross Sections “scaling” behavior – depends only on  and not M and s separately

  22. DY Formation: 2 --> 1 At a fixed resonant mass, expect rapid rise from “threshold” -  ~ (1-M/s)2a - then slow “saturation”. W ~ 30 nb at the LHC

  23. DY Z Production – F/B Asymmetry CDF – Run I The Z couples to L and R quarks differently -> parity violating asymmetry in the photon-Z interference.

  24. F/B Asymmetry Coupling of leptons and quarks to Z specified in SM by gauge principle. Coupling to L and R fermions differs => P violation ~ R-L coupling. Predict asymmetry , A ~ I3/Q. Thus, A for muons = 1, that for u quarks is 3/2, while for d quarks it is 3.

  25. COMPHEP At 500 GeV the asymmetry is large and positive – here not p-p but u-U

  26. COMPHEP - Assym Option in “Simpson” to get F/B asymmetry in COMPHEP

  27. DY Formation of Charmonium Cross section =  ~ 2(2J+1)/M3 for W, width ~ 2 GeV,  = 47 nb. For charmonium, width is 0.000087 GeV, and estimate cross section in gg formation as 34 nb. The PT arises from ISR and intrinsic parton transverse momentum and is only a few GeV, on average. Use for lepton momentum scale and resolution. g  g

  28. Charmonium Calibration Cross section in |y|<1.5 is ~ 800 nb at the LHC. Lepton calibration – mass scale, width?

  29. Upsilon Calibration Cross section * BR about 2 nb at the LHC. Resolve the spectral peaks? Mass scale correct?

  30. ZZ Production vs CM Energy VV production also has a steep rise near threshold. There is a 20 fold rise from the Tevatron to the LHC. Measure VVV coupling. ZZ has ~ 2 pb cross section at LHC. Not much gain in using anti-protons once the energy is high enough that the gluons or “sea” quarks dominate.

  31. WWZ – Quartic Coupling Not accessible at Tevatron. Test quartic couplings at the LHC.

  32. Jet-Jet Mass, 2 --> 2 Expect 1/M3 behavior at low mass. When M/s becomes substantial, the source effects will be large. E.g. for M = 400 GeV, at the Tevatron, M/s=0.2, and (1-M/s)12 is ~ 0.07.

  33. Jets - 2 TeV- |y|<2 1/M3[1-M/s]12 behavior ET ~ M/2 for large scattering angles.

  34. COMPHEP Linux

  35. Scaling ? Tevatron runs at 630 and 1800 GeV in Run I. Test of scaling in inclusive jet production. Expect a function of only in lowest order.

  36. Direct Photon Production Expect a similar spectrum with a rate down by ratio of coupling constants and differences in u and g source functions. /s~14 u/g~6 at x~0.

  37. D0 Single Photon Process dominated by q + g – a la Compton scattering. COMPHEP – 2 TeV p-p

  38. x1 x2 x,y,M y3, y4 y*, * 2--> 2 Kinematics - “Decays” Formation System Decay CM Decay The measured values of y3, y4 and ET allow one to solve for the initial state x1 and x2 and the c.m. decay angle.

  39. COMPHEP - Linux g+g-> g+ g, in pp at 14 TeV with cut of Pt of jets of 50 GeV. See a plateau for jets and the t channel peaking. Want to establish jet cross section, angular distributions and to look at jet “balance” – missing Et distribution in dijet events. MET angle ~ jet azimuthal angle and no non-Gaussian tails.

  40. Parton-->Hadron Fragmentation For light hadrons (pions) as hadronization products, assume kT is limited (scale ~. The fragmentation function, D(z) has a radiative form, leading to a jet multiplicity which is logarithmic in ET Plateau widens with s, <n>~ln(s)

  41. CDF Analysis – Jet Multiplicity Different Cone radii Jet cluster multiplicity within a cone increases with dijet mass as ~ ln(M).

  42. Jet Transverse Shape There is a “leading fragment” core localized at small R w.r.t. the jet axis - 40% of the energy for R< 0.1. 80% is contained in R < 0.4 cone

  43. Jet Shape - Monte Carlo Simple model with zD(z) ~ (1-z)5 and <kt> ~ 0.72 GeV. “Leading fragment” with <zmax> ~ 0.24. On average the leading fragment takes ~ 1/4 of the jet momentum. Fragmentation is soft and non-perturbative.

  44. Low Mass LHC Rates For small x and strong production, the cross section is a large fraction of the inelastic cross section. Therefore, the probability to find a “small Pt “minijet” in an LHC crossing is not small.

  45. V V Production - W +  The angular distribution at the parton level has a zero. The SM prediction could be confirmed with a large enough event sample. – pp at 2 TeV with Pt > 10 GeV, 0.6 pb Asymmetry somewhat washed out by the contribution of sea anti-quarks in the p and sea quarks in the anti-proton.

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