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Quantum Field Theory and the Electroweak Standard Model

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Quantum Field Theory and the Electroweak Standard Model Published by CERN in the Proceedings of the 2013 European School of High-Energy Physics, Parádfürdo,˝ Hungary, 5 – 18 June 2013, edited by M. Mulders and G. Perez, CERN–2015–004 (CERN, Geneva, 2015) Quantum Field Theory and the Electroweak Standard Model E. Boos M. V. Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics (SINP MSU), Moscow 119991, Russia Abstract The Standard Model is one of the main intellectual achievements for about the last 50 years, a result of many theoretical and experimental studies. In this lecture a brief introduction to the electroweak part of the Standard Model is given. Since the Standard Model is a quantum field theory, some aspects for understanding of quantization of abelian and non-abelian gauge theories are also briefly discussed. It is demonstrated how well the electroweak Standard Model works in describing a large variety of precise experimental measure- ments at lepton and hadron colliders. 1 Introduction The Standard Model (SM) of strong and electroweak (EW) interactions is the basis for understanding of nature at extremely small distances. In high-energy physics usually the relativistic system of units is used in which the Planck constant ~ and the speed of light c are equal to unity, ~ = c = 1. Taking into account well-known values for ~ = 1.055 1027 erg s, c = 3 1010 cm/s and the positron electric · · charge e = 1.6 10 19 C and using the relation between the electronvolt and erg (1 eV = e 1 V = 1 V · − · 1.6 10 19 C = 1.6 10 12 erg), one easily gets the following very useful relation between length and · · − · − energy units: 1/GeV = 2 10 14 cm. Due to the Heisenberg uncertainty principle, ∆x∆p 1/2, the · − ≥ above relation allows us to understand which energies (momentum transfers) are needed approximately to probe certain distances: 16 100 GeV 10− cm, → 17 1 TeV 10− cm, → 18 10 TeV 10− cm. → 18 17 Therefore, at the LHC one can study the structure of matter at distances of 10− –10− cm. For small 16 distances of the order of 10− cm or correspondingly 100 GeV energies the SM works very well, as follows from many studies and measurements. The SM is a quantum field theory; it is based on a few principles and requirements: – gauge invariance with lowest dimension (dimension four) operators; SM gauge group: SU(3) C × SU(2) U(1) ; L × Y – correct electromagnetic neutral currents and correct charge currents with (V–A) structure as fol- lows from four fermion Fermi interations (1) GF [¯νµ γα(1 γ5) µ ] [¯e γα(1 γ5) νe ] + h.c.; (1) √2 · − · · − · – three generations without chiral anomalies; – Higgs mechanism of spontaneous symmetry breaking. Fermions are combined into three generations forming left doublets and right singlets with respect to the weak isospin (see Fig.1). 1 f = (1 γ )f, L,R 2 ∓ 5 0531-4283 – c CERN, 2015. Published under the Creative Common Attribution CC BY 4.0 Licence. 1 http://dx.doi.org/10.5170/CERN-2015-004.1 E. BOOS Fig. 1: Fermion generation ν u L = e , e = e ,Q = , u = u , d = d , 1 e R1 R− 1 d R1 R R1 R −L L 3L,3R 1 νµ c If = , 0 : L2 = , eR2 = µR−,Q2 = , uR1 = cR, dR1 = sR, ±2 µ− s L L ν t L = τ , e = τ ,Q = , u = t , d = b . 3 τ R3 R− 3 b R1 R R1 R −L L The SM Lagrangian written in accord with the mentioned requirements looks very simple: L = 1 W i (W µν)i 1 B Bµν 1 Ga (Gµν)a − 4 µν − 4 µν − 4 µν ¯ f L µ ¯ f R µ + f=`,q ΨL(iDµγ )ΨL† + f=`,q ΨR(iDµ γ )ΨR† + LH, P LH = LΦ + LPYukawa, µ 2 4 L = D Φ†D Φ µ Φ†Φ λ(Φ†Φ) , Φ µ − − ij i j ij i C j ij i j L = Γ Q¯ Φd0 + h.c. Γ Q¯ Φ u0 + h.c. Γ L¯ Φe0 + h.c. Yukawa − d L0 R − u L0 R − e L0 R The field strength tensors and covariant derivatives have very familiar forms: W i = ∂ W i ∂ W i + g εijkW jW k, µν µ ν − ν µ 2 µ ν B = ∂ B ∂ B , µν µ ν − ν µ Ga = ∂ Aa ∂ Aa + g f abcAb Ac , µν µ ν − ν µ S µ ν Y f DL = ∂ ig W i τ i ig B L ig Aa ta, µ µ − 2 µ − 1 µ 2 − S µ Y f DR = ∂ ig B R ig Aa ta, µ µ − 1 µ 2 − S µ i where i = 1, 2, 3, a = 1,..., 8; Wµ are gauge fields for the weak isospin group, Bµ are gauge fields for the weak hypercharge group and Aµ are gluon gauge fields for the strong SUC (3) colour group. 3 1 4 2 Yf = 2Qf 2I YL = 1,Ye = 2,YQ = ,Yu = ,Yd = . − f ⇒ i − Ri − i 3 Ri 3 Ri −3 The Lagrangian is so compact that its main part can be presented on the CERN T-shirt (see Fig. 2). It is hard to imagine that such a simple Lagrangian allows one to describe basically all the phe- nomena of the microworld. But the SM Lagrangian, being expressed in terms of physics components, is not that simple, leading after quantization to many interaction vertices between particles or quanta of corresponding quantum fields. This lecture is organized as follows. In the next section some aspects of quantum field theory are briefly discussed. After a motivation as to why do we need a quantum field theory, we consider scalar 2 2 QUANTUM FIELD THEORY AND THE ELECTROWEAK STANDARD MODEL Fig. 2: CERN T-shirt with SM Lagrangian fields and introduce the Feynman propagator and functional integral approach as a quantization method. The functional integral given in the holomorphic representation allows us to clarify boundary conditions and show the connection between the Green functions and S-matrix elements. Feynman diagrams are introduced. The formalism is extended to the fermion and gauge fields stressing peculiarities in the quantization procedure and Feynman rule derivation. In the next section a construction of the EW SM Lagrangian is presented. We discuss experimental facts and theory principles based on which the EW part of the SM Lagrangian for fermion and gauge fields is constructed. We show explicitly which conditions on weak hypercharges allow us to get correctly electromagnetic and charge current (CC) interactions and predict additional neutral currents (NCs). We demonstrate how potentially dangerous chiral anoma- lies cancelled out. Then spontaneous symmetry breaking, the Goldstone theorem and the appearance of Nambu–Goldstone bosons are briefly discussed. The Brout–Englert–Higgs–Hagen–Guralnik–Kibble mechanism of spontaneous symmetry breaking is introduced leading to non-zero masses of the gauge fields and appearance of the Higgs boson. Very briefly we discuss in addition to the unitary gauge the covariant gauge, propagators of Goldstone bosons and ghosts. At the end of the section it is shown how the spontaneous symmetry breaking mechanism leads to non-zero masses for the fermions in the SM and how very naturally the Cabibbo–Kobayashi–Maskawa mixing matrix appears. In the next section we concentrate on some phenomenological aspects of the EW SM such as connections between the Fermi constant GF, the Higgs vacuum expectation value v, consistency of low-energy measurements and W, Z mass measurements, W-, Z-boson decay widths and branching ratios, number of light neutrinos, two- + fermion processes in e e− collisions, tests of the gauge boson self-interactions, top-quark decays and the EW top production (single top). Briefly we discuss the EW SM beyond the leading order, renormal- ization and running coupling in quantum electrodynamics (QED), as a simplest example, running masses and running parameters in the SM, precision EW data and global parameter fits. Concluding remarks are given in the next section. The quantum chromodynamics (QCD) part of the SM and the phenomenol- ogy of the Higgs boson are not discussed in these lectures as they are addressed in other lectures of the School. For a deeper understanding of the topics discussed, one can recommend a number of very good textbooks and reviews [1–9] and lectures given at previous schools and specialized reviews [10, 12–15], which have been used in preparation of this lecture. 2 Introductory words to quantum theory In classical mechanics a system evolution follows from the principle of least action: tf tf ∂L ∂L d δS = δ dtL(q(t), q˙(t)) = 0; δq + δ(q ˙) = 0; δ(q ˙) = δq. ∂q ∂q˙ dt tZi tZi 3 3 E. BOOS For an arbitrarily small variation δq, one gets the well-known Lagrange equation of motion ∂L d ∂L = . (2) ∂q dt ∂q˙ 2 For a non-relativistic system described by the Lagrangian L = mq˙ V (q), the second Newton law 2 − follows from Eq. (2), ∂V mq¨ = = F. − ∂q The Hamiltonian of the system is related to the Lagrangian in the following well-known way: H(p, q) = pq˙ L(q, q˙), − ∂L where q˙ is a solution of the equation p = ∂q˙ . In quantum mechanics the coordinate and momentum are replaced by corresponding operators p, q p,ˆ qˆ with postulated commutator relation [ˆp(0), qˆ(0)] = i~. In the Heisenberg picture the → − system evolution is described by the Heisenberg equation with time-dependent operators; for example, the equation for the coordinate operator has the following form: ∂qˆ i = [H,ˆ qˆ] (3) ∂t ~ with a formal solution i Htˆ i Htˆ qˆ(t) = e ~ qˆ(0)e− ~ .
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