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 dσˆ = dxb fA(xa, µ)fB(xb, µ)dxa. (1) dpT dpT PHY401 - Nuclear and Particle Physics Monsoon Semester 2020

The term dσ/dpˆ 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.

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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.

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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 |∆S| = 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

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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.

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+ 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. (3), namely, ! c , (12) s0 with 0 s = − sin θcd + cos θcs. (13)

This, indeed, leads to a resolution of all the leptonic decay modes of the strange mesons, and this proposal is commonly referred to as the GIM mechanism. The ideas of Cabibbo and GIM can be summarized by saying that for the two doublets ! ! u c and (14) d0 s0 the weak eigenstates are related to the eigenstates of the strong Hamiltonian through the following

7 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 unitary matrix ! ! ! d0 cos θ sin θ d = c c . (15) 0 s − sin θc cos θc s

1.3 CKM Matrix

With the subsequent discovery of the b (in 1977) and t (in 1995) quarks, the Standard Model is now characterized by three doublets of quarks of the form ! ! ! u c t , , and . (16) d0 s0 b0

The relation between the three states d0, s0 and b0 and the eigenstates d, s and b is somewhat more complicated and involves a 3 × 3 matrix, which is known as the Cabibbo-Kobayashi-Maskawa (or CKM) unitary matrix  0     d Vud Vus Vub d  0     s  = Vcd Vcs Vcb  s . (17) 0 b Vtd Vts Vtb b The elements of this matrix reflect the couplings of the W boson to all the possible quark pairs, e.g., W → t¯b, c¯b, cs¯ etc. Clearly, the dominant transitions in W decay are to members of the same doublet, and therefore, to first order, the diagonal elements of the matrix are expected to be large (close to unity) and the off-diagonal elements small. In writing down the original 3 × 3 matrix (prior to the discovery of the b and t quarks!), Makoto Kobayashi and Toshihide Maskawa noted that this kind of matrix representation introduces at least one complex phase into the Standard Model, which can then encompass the phenomenon of CP violation in transitions of neutral mesons. With the generalization of Eq. (17), in which all quarks of same electric charge can mix, and the quark eigenstates of the weak Hamiltonian b0, s0 and d0 correspond to specific superpositions of the d, s and b quarks, the Standard Model can accommodate all measured particle decays and transitions. For example, the CP violating decays of the K0 mesons can be calculated from the kind of processes indicated in Fig. 9 and Fig 10, for the indirect and direct terms in the CP violating processes, respectively.

1.4 The Unitarity Triangle

The CKM matrix is required to be unitary. This produces nine equations of the form

∗ ∗ ∗ VudVub + VcdVcb + VtdVtb = 0. (18)

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s¯ u, c, t d¯

K0 W+ W+ K¯ 0

d u, c, t s

s¯ W+ d¯

K0 u, c, t u, c, t K¯ 0

d W+ s

Figure 9: The ∆S = 2 transition “box diagrams” via two consecutive weak processes that are responsible for K0 − K¯ 0 mixing and indirect CP violation in K0 decay.

This equation is known as the unitarity triangle as it can be visualized as a triangle in the complex plane. If all components of the CKM matrix were real (or even if they simply had the same phase), the unitarity triangle would simply be a line segment. Its triangle-ness is a testament to the fundamental CP asymmetry of the system. Putting current best estimates in terms of the CP -violating term in the CKM matrix, we obtain

δ = 68.8 ± 4.6◦. (19)

This tells us that the CP violation in the quark family is maximal.

References

[1] Dave Goldberg, The Standard Model in a Nutshell, Princeton University Press (2017).

[2] A. Das and T. Ferbel, Introduction To Nuclear And Particle Physics, World Scientific (2003).

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s W+ d¯ ¯ π+ u, c, t u K0 g Z0 γ , , u¯ π− d d

Figure 10: The ∆S transition, or “penguin diagram”, that is responsible for direct CP violation in K0 decay.

Figure 11: A penguin diagram superimposed on an image of a Gentoo penguin.

Figure 12: Current constraints on the unitarity triangle from CP symmetry violations in B-meson decay and other reactions. Credit: CKMfitter Group (J. Charles et al.), Eur. Phys. J. C41, 1-131 (2005) [hep-ph/0406184], updated results and plots are available at http://ckmfitter.in2p3.fr.

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