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Lecture Notes 31 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 31 Thursday, November 5, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Contents 1 Standard Model and Confrontation with Data 1 1.1 Some Comparisons with Data . .1 1.2 Cabibbo Angle . .4 1.2.1 GIM Mechanism . .7 1.3 CKM Matrix . .8 1.4 The Unitarity Triangle . .8 1 Standard Model and Confrontation with Data The Standard Model appears to be in complete agreement with all measurements. Apart form neutrino mass, there have been no confirmed deviations between data and predictions of the model. 1.1 Some Comparisons with Data Let us look at some of the candidate examples where an agreement is observed between expectations from QCD and collisions studied at high energies. Figs. 1 and 2 show such cases. In these figures we show the data and theoretical predictions, for the production of W and Z bosons, and for production of particle jets, in pp¯ and pp collisions. Consider the differential production cross section as a function of the variable say, the transverse momentum pT . It can be written schematically in QCD in terms of the elastic scattering of a parton a from hadron A and a parton b from hadron B, as dσ Z dσ^ = dxb fA(xa; µ)fB(xb; µ)dxa: (1) dpT dpT PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 The term dσ=dp^ T refers to the point cross section for elastic scattering of the two partons. This term can be calculated from fundamental principles of QFT. xi is the fraction of the momentum of hadron I carried by i. 2 2 fI (xi; µ) represents the momentum distribution of parton i within hadron I at the scale q = µ . Just as for the case of α in QED, and the strength of the color interaction in QCD, the parton distribution function f(xi; µ) also depends on the momentum scale in any collision. That is f(xi; µ) runs with the scale. Such dependence of parameters on q2 is usually referred to as the scaling violation of QCD. 2 Given some f(xi; µ0) at q = µ0, it is possible to calculate within QCD the resulting function 2 2 f(xi; µ) at some other q = µ , using what are known as the DGLAP evolution equations. The integrations in Eq. (1) have to be performed over all values of the xi. The primary uncertainty in the theory (displayed as the allowed regions between the two sets of smooth curves in Fig. 1) originates from the inability to predict the content and the momentum distributions of constituents that are bound within hadrons. Figure 1: The cross section for W and Z production in pp¯ and pp collisions compared to theoretical predictions based on the Standard Model. This is an issue related to confinement, and interactions of quarks and gluons at low momentum transfer, which cannot be calculated reliably in perturbation theory. 2 2 However, parton distribution functions f(xi; µ0) at some known scale q = µ0, can be extracted from other reactions (e.g., from electron scattering off protons), and then applied to predict results for collisions of any partons located within separate hadrons. Thus, for the case of W production, the main contribution to the yield arises from the interaction of u¯ (d¯) quarks in the p¯ that fuse with d (or u) quarks within p to produce a W − (or W +), and possible remnant jets of particles. The uncertainty on the gluon content of hadrons is larger than for quarks. The reason is photons, W and Z bosons can interact directly with quarks, but only indirectly (at higher order in perturbation theory) with gluons. This can be inferred from Fig. 3. 2 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Figure 2: Cross section and prediction from QCD (solid curve) for production of particle jets at large momentum transfers in pp¯ and pp collisions. Figure is taken from Review of Particle Properties, Particle Data Group, 2020. For the case of jet production in hadron-hadron collisions, any parton in one of the interacting hadrons can scatter elastically off any parton in the other hadron, and then both partons can evolve into jets. The scattered partons can appear at large angles relative to the collision axis, while the other (un-scattered) constituents tend to evolve into color-neutral states at small angles along the collision axis. Since momentum must be conserved in the direction transverse to the collision axis, we expect the scattered parton jets to be emitted back-to-back. An event of this kind are shown in Fig. 5. This type of display is referred to as a lego plot. The height of any entry is proportional to the energy observed (deposited in the calorimeter) in that region of coordinates. The axes correspond to the azimuth (φ) around the collision axis, and the polar angle θ relative to the collision axis. The calorimeter may be thought as a cylinder surrounding the beam axis around the point where the particles collide: the cylinder is cut along its height, unrolled, and the lego plot is drawn on top of its surface. Fig. 5 shows and exclusive dijet production candidate isolated by the DZERO experiment in 2010 There indeed is nothing in the event but two well-separated, back-to-back streams of hadrons. 3 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 νμ μ νμ μ W W Jet q Jet q g q¯ e e e e γ Z0 Jet γ g Z0 q¯ q Jet Figure 3: Production of jets in lepton-nuleon scattering. 1.2 Cabibbo Angle We have learned that the W and Z bosons can produce transitions between members of the same weak isospin doublet. However, if W and Z bosons could not also provide transitions among particles belonging to different multiplets, it would clearly present a great puzzle concerning the origin of the j∆Sj = 1 strangeness-changing weak decays. The solution to this issue comes from our previous observation that strangeness is a quantum number that is not conserved in weak interactions. Consequently, the eigenstates of the weak Hamiltonian are different from those of the strong Hamiltonian. In particular, these weak eigenstates do not have unique strangeness. In analogy with our analysis of the K0 − K¯ 0 system, we can try to redefine the quark doublet eigenstates of the weak Hamiltonian as mixed states of the doublets of ! ! ! u c t ; and : (2) d s b Before the discovery of the charm quark (it was discovered in 1974), and based on the experi- mental results available at that time, Nicola Cabibbo showed (in 1963) that all data were consistent 4 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 p¯ p¯ g g Dijet g p p Figure 4: An exclusive process is one where the two protons, in fact, do not fragment at all: they exchange at least two gluons, and thus are capable of retaining their null color charge, and remain unbroken. One of the two gluons may then radiate the two partons which produce the observed two jets. Figure 5: Energy flowing in the direction transverse to the collision axis for production of particle jets in pp¯ collisions. This event was observed by the DZERO collaboration in 2010 during pp¯ collisions at the Tevatron 2-TeV accelerator at Fermilab. with altering the doublet corresponding to the first family of quarks in the following way ! ! u u ! : (3) d d0 The newly defined state d0 is a mixture of d and s quarks 0 d = cos θcd + sin θcs: (4) This kind of state clearly does not have a unique strangeness quantum number, and, if the weak gauge bosons can give rise to transitions within the u, d0 multiplet, then they can, in fact, induce strangeness-changing processes. The angle θc parameterizing the mixing between the d and s quarks in Eq. (4) is commonly 5 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 called the Cabibbo angle. Its value determines the relative rates for processes such as W + ! us;¯ (5) W + ! ud;¯ (6) Z0 ! uu;¯ (7) Z0 ! ds:¯ (8) The Cabibbo angle can be determined experimentally through a comparison of ∆S = 0 and ◦ ∆S = 1 transitions, and has the value sin θc = 0:23. (That is, θc ' 13:1 .) Figs. 6, 7 and 8 show some of the strangeness changing transitions in the Standard Model. Fig. 7 shows how the decay of a K0 into a π+ and π− can now be described in the Standard Model. u u K+ π0 s¯ u¯ W+ μ+ νμ Figure 6: Strangeness-changing transition in the Standard Model. s¯ u¯ π− d K0 W+ d¯ π+ d u¯ Figure 7: Strangeness-changing transition in the Standard Model. 6 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 + u μ W+ K+ s¯ νμ Figure 8: Strangeness-changing transition in the Standard Model. 1.2.1 GIM Mechanism While the Cabibbo hypothesis accommodated most of the W ± induced decays, certain strangeness- changing processes, particularly involving leptonic decay modes of the K0 remained puzzling. For example 1 + + 8 −1 Γ(K ! µ νµ) ≈ 0:5 × 10 sec ; (9) ~ 1 0 + − −1 Γ(KL ! µ µ ) ≈ 0:14 sec ; (10) ~ leading to 0 + − Γ(KL ! µ µ ) −9 + + ≈ 3 × 10 : (11) Γ(K ! µ νµ) The smallness of this ratio could not be explained within the framework of the Cabibbo analysis. In fact, further investigation of this problem led Sheldon Glashow, John Illiopoulos and Luciano Maiani (in 1970) to propose the existence of a fourth quark (charm quark) in a doublet structure of the type given in Eq.
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