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Symmetries and Their Breaking in the Fundamental Laws of Physics

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S S symmetry Review Symmetries and Their Breaking in the Fundamental Laws of Physics Jose Bernabeu Department of Theoretical Physics, University of Valencia and IFIC, Joint Centre Univ. Valencia-CSIC, E-46100 Burjassot, Spain; [email protected] Received: 22 June 2020; Accepted: 29 July 2020; Published: 6 August 2020 Abstract: Symmetries in the Physical Laws of Nature lead to observable effects. Beyond the regularities and conserved magnitudes, the last few decades in particle physics have seen the identification of symmetries, and their well-defined breaking, as the guiding principle for the elementary constituents of matter and their interactions. Flavour SU(3) symmetry of hadrons led to the Quark Model and the antisymmetric requirement under exchange of identical fermions led to the colour degree of freedom. Colour became the generating charge for flavour-independent strong interactions of quarks and gluons in the exact colour SU(3) local gauge symmetry. Parity Violation in weak interactions led us to consider the chiral fields of fermions as the objects with definite transformation properties under the weak isospin SU(2) gauge group of the Unifying Electro-Weak SU(2) U(1) symmetry, which predicted × novel weak neutral current interactions. CP-Violation led to three families of quarks opening the field of Flavour Physics. Time-reversal violation has recently been observed with entangled neutral mesons, compatible with CPT-invariance. The cancellation of gauge anomalies, which would invalidate the gauge symmetry of the quantum field theory, led to Quark–Lepton Symmetry. Neutrinos were postulated in order to save the conservation laws of energy and angular momentum in nuclear beta decay. After the ups and downs of their mass, neutrino oscillations were discovered in 1998, opening a new era about their origin of mass, mixing, discrete symmetries and the possibility of global lepton-number violation through Majorana mass terms and Leptogenesis as the source of the matter–antimatter asymmetry in the universe. The experimental discovery of quarks and leptons and the mediators of their interactions, with physical observables in spectacular agreement with this Standard Theory, is the triumph of Symmetries. The gauge symmetry is exact only when the particles are massless. One needs a subtle breaking of the symmetry, providing the origin of mass without affecting the excellent description of the interactions. This is the Brout–Englert–Higgs Mechanism, which produces the Higgs Boson as a remnant, discovered at CERN in 2012. Open present problems are addressed with by searching the New Physics Beyond-the-Standard-Model. Keywords: flavour families; colour charges; gauge symmetries; chirality; discrete symmetries; neutrinos; spontaneous breaking 1. Symmetry as Guiding Principle for Particles and Interactions In ordinary life, we observe symmetry of objects, like characteristic features of geometrical forms, material objects or biological bodies. The concept is related to the invariance of the object under definite transformations: One object is symmetric if, after a transformation is applied, the result remains the same, i.e., it remains “invariant”. However, we also observe Symmetry Breaking, which is particularly of interest when it is not a random effect but follows a definite pattern. In Figure1, we show the three-span arch of the FermiLab entrance, near Chicago, which appears perfectly symmetric when viewed from below, but has a calculated asymmetry from its other views. Symmetry and Symmetry Symmetry 2020, 12, 1316; doi:10.3390/sym12081316 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1316 2 of 27 Symmetry 2020, 12, x FOR PEER REVIEW 2 of 26 SymmetryBreaking are and very Symmetry important Breaking concepts are in thevery field important of elementary concepts particle in the physics, field of however elementary notreferring particle physics,to objects, however but to the not Fundamental referring to objects, Laws of but Physics. to the Fundamental Laws of Physics. FigureFigure 1. 1. SymmetrySymmetry Breaking.Breaking. We show here how Symmetry has acted as a guiding principle for both the existence of new particles and and the the formulation formulation of of interactions. interactions. One One can can claim claim that that “Symmetry “Symmetry dictates dictates Interaction”, Interaction”, as statedas stated by byYang. Yang. In InQuantum Quantum Mechanics, Mechanics, the the Symmetry Symmetry isis implemented implemented by by aa Unitary Unitary Transformation Transformation Uˆ acting on states and observables. If If the Dynamics, described described by by the the Hamiltonian Hamiltonian Hˆ,, are Invariant under the transformation, one can obtain h,i = 0 Hˆ , Uˆ = 0 (1) (1) = − / For continuous groups characterized by a parameter “a”, the transformation n o is Forgenerated continuous by groups = characterizedand one obtains by immediately a parameter “a”, the transformation Uˆ = exp i Gaˆ /} − ˆ ˆ is generated by G = Gy and one obtains immediately 〈〉 = 〈,〉 = 0 (2) d h i Gˆ = i Hˆ , Gˆ = 0 (2) As is Hermitian, it correspondsdt toh ian Observableh i that satisfies a Conservation Law if is symmetryAs Gˆ isof Hermitian,. Well known it corresponds examples toare an momentum Observable for that translations, satisfies a Conservation angular momentum Law if U ˆforis rotationssymmetry or of chargeHˆ . Well for knowngauge symmetry. examples For are momentumlocal gauge symmetry, for translations, the requirement angular momentum of Invariance for leadsrotations to a orCovariant charge for Derivative gauge symmetry. with a Mediator For local Field gauge responsible symmetry, for the interactions. requirement This of is Invariance valid for eitherleads toQED a Covariant with the DerivativeAbelian U(1) with gauge a Mediator group or Field non-Abelian responsible gauge for interactions. groups with This the isinteraction valid for fieldeither transforming QED with the as Abelian the adjoint U(1) representation. gauge group or non-Abelian gauge groups with the interaction field transformingIn Section as 2, the we adjoint develop representation. the ideas leading from hadrons to quarks and the symmetries of strong interactions.In Section In2 ,Section we develop 3, a parallel the ideas discussion leading from is made hadrons for toelectroweak quarks and interactions the symmetries starting of strong from Parityinteractions. Violation In Sectionand leading3, a parallel to the Standard discussion Mo isdel made with for neutral electroweak currents interactions and the need starting for charm, from plusParity the Violation third family and leading of bottom to the and Standard top quarks, Model including with neutral Quark–Lepton currents and Symmetry. the need forSection charm, 4 discussesplus the third Discrete family Symmetries, of bottom and with top the quarks, observat includingion and Quark–Lepton implications Symmetry.of the independent Section4 discussesbreaking ofDiscrete CP and Symmetries, T, as well withas tests the observationof CPT. In andSection implications 5, the fascinating of the independent history of breaking neutrino of physics CP and is T, summarizedas well as tests and of the CPT. present In Section open5, thequestions fascinating of CP-Vio historylation of neutrino in the lepton physics sector is summarized and Global and Lepton the Numberpresent open Violation questions are discussed. of CP-Violation Section in the6 pres leptonents sector the Brout–Englert–Higgs and Global Lepton Number Mechanism Violation for the are Origindiscussed. of Mass, Section breaking6 presents the the ElectroWeak Brout–Englert–Higgs Gauge Symmetry. Mechanism Some for Conclusions the Origin of and Mass, Outlook breaking in thethe fieldElectroWeak of Symmetries Gauge Symmetry.are provided Some in Section Conclusions 7. and Outlook in the field of Symmetries are provided in Section7. 2. Quarks and Strong Interactions 2. QuarksThe proliferation and Strong of Interactions non-strange and strange Hadrons in the 1960’s of the 20th century led to the EightfoldThe proliferationWay of Gell, of Mann non-strange and Ne’eman and strange with Hadronsthe use of in the the Flavour 1960’s of SU(3) the 20th symmetry. century The led fundamentalto the Eightfold representations Way of Gell, 3, Mann 3 are andthe Ne’emanelementary with building the use blocks of the for Flavour all higher-dimensional SU(3) symmetry. representations.The fundamental In representations terms of their3, basis3 are states, the elementary Mesons are building q − q blocksstates for3 3 all higher-dimensional= 1+8, Baryons are q-q-qrepresentations. states 3 × 3 In × terms3 = 1 + of 8 theirs + 8a basis+ 10, states,with three Mesons quark are qq = u,q statesd, s states.3 3 In= 1Figure+ 8, Baryons 2, the octet are q-q-q and − × decupletstates 3 3representations3 = 1 + 8s + 8 aof+ Baryons10, with threeare given quark qin= termsu, d, s of states. third In component Figure2, the octetof Isospin and decuplet I3 and × × hypercharge Y axes. According to the Gell–Mann–Nishijima rule, the electric charge is Q = I3 + Y/2, with Y = B + S, B the baryonic number and S strangeness. Symmetry 2020, 12, 1316 3 of 27 representations of Baryons are given in terms of third component of Isospin I3 and hypercharge Y axes. According to the Gell–Mann–Nishijima rule, the electric charge is Q = I3 + Y/2, with Y = B + S, B the baryonicSymmetry 2020 number, 12, x FOR and PEER S strangeness. REVIEW 3 of 26 FigureFigure 2.2. Octet and Decuplet of Baryons. AtAt thethe timetime ofof thisthis formulation,formulation, thethe
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