Mathematical formulation of the Standard Model
The Standard Model is renormalizable and mathematically self-consistent;[1] however, despite having huge and continued successes in providing experimental predictions, it does leave some unexplained phenomena.At first the basic fields given above may not seem to correspond well with the "fundamental particles" in the chart above, but there are several alternative presentations that, in particular contexts, may be more appropriate than those that are given above.This is very important in the Standard Model because left and right chirality components are treated differently by the gauge interactions.As an aside, if a complex phase term exists within either of these matrices, it will give rise to direct CP violation, which could explain the dominance of matter over antimatter in our current universe.Finally, the quantum fields are sometimes decomposed into "positive" and "negative" energy parts: ψ = ψ+ + ψ−.To retain gauge invariance, the underlying fields must be massless, but the observable states can gain masses in the process.In the alternative Heisenberg picture, state vectors are kept constant, at the price of having the operators (in particular the observables) be time-dependent.The interaction picture constitutes an intermediate between the two, where some time dependence is placed in the operators (the quantum fields) and some in the state vector.and ar(p) as creation and annihilation operators comes from comparing conserved quantities for a state before and after one of these have acted upon it.The Lagrangian can also be derived without using creation and annihilation operators (the "canonical" formalism) by using a path integral formulation, pioneered by Feynman building on the earlier work of Dirac.We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density.Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an observer.The three factors of the gauge symmetry together give rise to the three fundamental interactions, after some appropriate relations have been defined, as we shall see.The electroweak sector interacts with the symmetry group U(1) × SU(2)L, where the subscript L indicates coupling only to left-handed fermions.where Bμ is the U(1) gauge field; YW is the weak hypercharge (the generator of the U(1) group); Wμ is the three-component SU(2) gauge field; and the components of τ are the Pauli matrices (infinitesimal generators of the SU(2) group) whose eigenvalues give the weak isospin.As explained above, these currents mix to create the physically observed bosons, which also leads to testable relations between the coupling constants.This is an interaction corresponding to a "rotation in weak isospin space" or in other words, a transformation between eL and νeL via emission of a W− boson.The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, with SU(3) symmetry, generated by Ta.This can be seen by writing ψ in terms of left and right-handed components (skipping the actual calculation):The spin-half particles have no right/left chirality pair with the same SU(2) representations and equal and opposite weak hypercharges, so assuming these gauge charges are conserved in the vacuum, none of the spin-half particles could ever swap chirality, and must remain massless.The solution to both these problems comes from the Higgs mechanism, which involves scalar fields (the number of which depend on the exact form of Higgs mechanism) which (to give the briefest possible description) are "absorbed" by the massive bosons as degrees of freedom, and which couple to the fermions via Yukawa coupling to create what looks like mass terms.An obvious solution[4] is to simply add a right-handed neutrino νR, which requires the addition of a new Dirac mass term in the Yukawa sector:This field however must be a sterile neutrino, since being right-handed it experimentally belongs to an isospin singlet (T3 = 0) and also has charge Q = 0, implying YW = 0 (see above) i.e. it does not even participate in the weak interaction.terms are consistently all left (or all right) chirality (note that a left-chirality projection of an antiparticle is a right-handed field; care must be taken here due to different notations sometimes used).It is possible to include both Dirac and Majorana mass terms in the same theory, which (in contrast to the Dirac-mass-only approach) can provide a “natural” explanation for the smallness of the observed neutrino masses, by linking the right-handed neutrinos to yet-unknown physics around the GUT scale[6] (see seesaw mechanism).Since in any case new fields must be postulated to explain the experimental results, neutrinos are an obvious gateway to searching physics beyond the Standard Model.[9] Upon writing the most general Lagrangian with massless neutrinos, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment.The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously.Experimentally, neutrino oscillations imply that individual electron, muon and tau numbers are not conserved.[15] Another problem lies within the mathematical framework of the Standard Model itself: the Standard Model is inconsistent with that of general relativity, and one or both theories break down under certain conditions, such as spacetime singularities like the Big Bang and black hole event horizons.
Standard Modelelementary particlesHiggs bosongenerationsquarksleptonsgauge bosonsstrongelectromagneticelectroweak symmetry breakingparticle physicsQuantum field theoryGauge theorySpontaneous symmetry breakingHiggs mechanismElectroweak interactionQuantum chromodynamicsCKM matrixStrong CP problemHierarchy problemNeutrino oscillationsPhysics beyond the Standard ModelRutherfordThomsonChadwickSudarshanDavis JrAndersonFeynmanRubbiaGell-MannKendallTaylorFriedmanPowellGlashowIliopoulosLedermanMaianiChamberlainCabibboSchwartzMajoranaWeinbergKobayashiMaskawaYukawa't HooftVeltmanPolitzerReinesSchwingerWilczekCroninEnglertGuralnikKibblede MayoloLattesFeynman diagramHistoryField theoryElectromagnetismWeak forceStrong forceQuantum mechanicsSpecial relativityGeneral relativityYang–Mills theorySymmetriesSymmetry in quantum mechanicsC-symmetryP-symmetryT-symmetryLorentz symmetryPoincaré symmetryGauge symmetryExplicit symmetry breakingNoether chargeTopological chargeAnomalyBackground field methodBRST quantizationCorrelation functionCrossingEffective actionEffective field theoryExpectation valueLattice field theoryLSZ reduction formulaPartition functionPath Integral FormulationPropagatorQuantizationRegularizationRenormalizationVacuum stateWick's theoremWightman axiomsDirac equationKlein–Gordon equationProca equationsWheeler–DeWitt equationBargmann–Wigner equationsSchwinger-Dyson equationRenormalization group equationQuantum electrodynamicsString theorySupersymmetryTechnicolorTheory of everythingQuantum gravityAnselmBargmannBecchiBelavinBerezinBjorkenBleuerBogoliubovBrodskyBuchholzCachazoCallanColemanConnes