{'fraction_non_alphanumeric': 0.03366382291906848, 'fraction_numerical': 0.010145261701637076, 'mean_word_length': 4.240695988400193, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Standard Model of Particle Physics

A concise introduction to the Standard Model of fundamental particle interactions is presented.


Introduction
The Standard Model (SM) is a consistent, finite and, within the limitations of our present technical ability, computable theory of fundamental microscopic interactions that successfully explains most of the known phenomena in elementary particle physics. The SM describes strong, electromagnetic and weak interactions. All so far observed microscopic phenomena can be attributed to one or the other of these interactions. For example, the forces that hold together the protons and the neutrons in the atomic nuclei are due to strong interactions, the binding of electrons to nuclei in atoms or of atoms in molecules is caused by electromagnetism and the energy production in the sun and the other stars occurs through nuclear reactions induced by weak interactions. In principle gravitational forces should also be included in the list of fundamental interactions but their impact on fundamental particle processes at accessible energies is totally negligible.
The structure of the SM is a generalisation of that of Quantum Electrodynamics (QED) in the sense that it is a renormalizable field theory based on a local symmetry (i.e. separately valid at each space-time point x) that extends the gauge invariance of electrodynamics to a larger set of conserved currents and charges. There are eight strong charges, called "color" charges and four electro-weak charges (which in particular include the electric charge). The commutators of these charges form the SU(3) SU(2) U(1) algebra. In QED the interaction between two matter particles with electric charges, for example two electrons, is mediated by the exchange of one (or more) photons emitted by one electron and reabsorbed by the second. In the SM the matter fields, all of spin 1/2, are the quarks, the constituents of protons, neutrons and all hadrons, endowed with both color and electro-weak charges, and the leptons (the electron e − , the muon µ − , the tauon τ − plus the three associated neutrinos ν e ,ν µ and ν τ ) with no color but with electro-weak charges. The matter fermions come in three generations or families with identical quantum numbers but different masses. The pattern is as follows: Each family contains a weakly charged doublet of quarks, in three color replicas, and a colorless weakly charged doublet with a neutrino and a charged lepton. At present there is no explanation for this triple repetition of fermion families. The force carriers, of spin 1, are the photon γ, the weak interaction gauge bosons W + , W − and Z 0 and the eight gluons g that mediate the strong interactions. The photon and the gluons have zero masses as a consequence of the exact conservation of the corresponding symmetry generators, the electric charge and the eight color charges. The weak bosons W + , W − and Z 0 have large masses (m W ∼ 80.4 GeV , m Z = 91.2 GeV ) signalling that the corresponding symmetries are badly broken. In the SM the spontaneous breaking of the electroweak gauge symmetry is induced by the Higgs mechanism which predicts the presence in the physical spectrum of one (or more) spin 0 particles, the Higgs boson(s), not yet experimentally observed. A tremendous experimental effort is underway or planned to reveal the Higgs sector as the last crucial missing link in the SM verification.

Quantum Chromodynamics
The statement that QCD is a renormalizable gauge theory based on the group SU(3) with color triplet quark matter fields fixes the QCD lagrangian density to be: Here: q j are the quark fields (of n f different flavors) with mass m j ; D / = D µ γ µ , where γ µ are the Dirac matrices and D µ is the covariant derivative: and e s is the gauge coupling; in analogy with QED: (in natural unitsh = c = 1, always used throughout this summary); g A µ , A = 1, 8, are the gluon fields and t A are the SU(3) group generators in the triplet representation of quarks (i.e. t A are 3x3 matrices acting on q); the generators obey the commutation relations [t A , t B ] = iC ABC t C where C ABC are the complete antisymmetric structure constants of SU(3) (the normalisation of C ABC and of e s is specified by T r[t A t B ] = 1/2δ AB ); The physical vertices in QCD include the gluon-quark-antiquark vertex, analogous to the QED photon-fermion-antifermion coupling, but also the 3-gluon and 4-gluon vertices, of order e s and e 2 s respectively, which have no analogue in an abelian theory like QED. In QED the photon is coupled to all electrically charged particles but itself is neutral. In QCD the gluons are colored hence self-coupled. This is reflected in the fact that in QED F µν is linear in the gauge field, so that the term F 2 µν in the lagrangian is a pure kinetic term, while in QCD F A µν is quadratic in the gauge field so that in F A2 µν we find cubic and quartic vertices beyond the kinetic term. The QCD lagrangian in eq.(2) has a simple structure but a very rich dynamical content, including the observed complex spectroscopy with a large number of hadrons. The most prominent properties of QCD are asymptotic freedom and confinement. In field theory the effective coupling of a given interaction vertex is modified by the interaction. As a result, the measured intensity of the force depends on the tranferred (four)momentum squared, Q 2 , among the participants. In QCD the relevant coupling parameter that appears in physical processes is α s (see eq.(4)). Asymptotic freedom means that the effective coupling becomes a function of Q 2 : α s (Q 2 ) decreases for increasing Q 2 and vanishes asymptotically. Thus, the QCD interaction becomes very weak in processes with large Q 2 , called hard processes or deep inelastic processes (i.e. with a final state distribution of momenta and a particle content very different than those in the initial state). One can prove that in 4 space-time dimensions all gauge theories based on a non commuting group of symmetry are asymptotically free and conversely. The effective coupling decreases very slowly at large momenta with the inverse logarithm of Q 2 : α s (Q 2 ) = 1/b log Q 2 /Λ 2 where b is a known constant and Λ is an energy of order a few hundred MeV. Since in quantum mechanics large momenta imply short wavelenghts, the result is that at short distances the potential between two color charges is similar to the Coulomb potential, i.e. proportional to α s (r)/r, with an effective color charge which is small at short distances. On the contrary the interaction strenght becomes large at large distances or small transferred momenta, of order Q < ∼ Λ. In fact the observed hadrons are tightly bound composite states of quarks, with compensating color charges so that they are overall neutral in color. The property of confinement is the impossibility of separating color charges, like individual quarks and gluons. This is because in QCD the interaction potential between color charges increases at long distances linearly in r. When we try to separate the quark and the antiquark that form a color neutral meson the interaction energy grows until pairs of quarks and antiquarks are created from the vacuum and new neutral mesons are coalesced instead of free quarks. For example, consider the process e + e − → qq at large center of mass energies. The final state quark and antiquark have large energies, so they separate in opposite directions very fast. But the color confinement forces create new pairs in between them. What is observed is two back-to-back jets of colorless hadrons with a number of slow pions that make the exact separation of the two jets impossible. In some cases a third well separated jet of hadrons is also observed: these events correspond to the radiation of an energetic gluon from the parent quark-antiquark pair.

Electroweak Interactions
We split the electroweak Lagrangian into two parts by separating the Higgs boson couplings: We start by specifying L symm , which involves only gauge bosons and fermions (a sum over all flavors of quark and leptons, generically indicated by ψ, is understood): This is the Yang-Mills Lagrangian for the gauge group SU(2)⊗U(1) with fermion matter fields. Here are the gauge antisymmetric tensors constructed out of the gauge field B µ associated with U(1), and W A µ corresponding to the three SU(2) generators; ǫ ABC are the group structure constants, see eq. (11), which, for SU(2), coincide with the totally antisymmetric Levi-Civita tensor (recall the familiar angular momentum commutators).
The standard EW theory is a chiral theory, in the sense that ψ L and ψ R behave differently under the gauge group. In particular, all ψ R are singlets and all ψ L are doublets in the minimal Standard Model (SM). Thus, mass terms for fermions (of the formψ L ψ R + h.c.) are forbidden in the symmetric limit. Fermion masses will be introduced, together with W ± and Z masses, by the mechanism of symmetry breaking. The covariant derivatives D µ ψ L,R are explicitly given by where t A L,R and 1/2Y L,R are the SU(2) and U(1) generators, respectively, in the reducible representations ψ L,R . The commutation relations of the SU(2) generators are given by We use the normalization T r[t A t B ] = 1/2δ AB in the fundamental representation of SU(2). The electric charge generator Q (in units of e, the positron charge) is given by All fermion couplings to the gauge bosons can be derived directly from Eqs. (7) and (10). The charged-current (CC) couplings are the simplest. From where t ± = t 1 ± it 2 and W ± = (W 1 ± iW 2 )/ √ 2, we obtain the vertex In the neutral-current (NC) sector, the photon A µ and the mediator Z µ of the weak NC are orthogonal and normalized linear combinations of B µ and W 3 µ : Equations (15) define the weak mixing angle θ W . The photon is characterized by equal couplings to left and right fermions with a strength equal to the electric charge. Recalling Eq. (12) for the charge matrix Q, we immediately obtain or equivalently, tg θ W = g ′ /g Once θ W has been fixed by the photon couplings, it is a simple matter of algebra to derive the Z couplings, with the result where Γψ ψZ is a notation for the vertex. In the MSM, t 3 R = 0 and t 3 L = ±1/2. In order to derive the effective four-fermion interactions that are equivalent, at low energies, to the CC and NC couplings given in Eqs. (14) and (18), we anticipate that large masses, as experimentally observed, are provided for W ± and Z by L Higgs . For left-left CC couplings, when the momentum transfer squared can be neglected with respect to m 2 W in the propagator of Born diagrams with single W exchange, from Eq. (14) we can write By specializing further in the case of doublet fields such as ν e − e − or ν µ − µ − , we obtain the tree-level relation of g with the Fermi coupling constant G F measured from µ decay (G F = 1.16639(2) 10 −5 GeV −2 ): By recalling that g sin θ W = e, we can also cast this relation in the form with where α is the fine-structure constant of QED (α ≡ e 2 /4π = 1/137.036).
In the same way, for neutral currents we obtain in Born approximation from Eq. (18) the effective four-fermion interaction given by and All couplings given in this section are obtained at tree level and are modified in higher orders of perturbation theory. In particular, the relations between m W and sin θ W [Eqs. (21) and (22)] and the observed values of ρ (ρ = ρ 0 at tree level) in different NC processes, are altered by computable small EW radiative corrections.
The gauge-boson self-interactions can be derived from the F µν term in L symm , by using Eq. (15) and W ± = (W 1 ± iW 2 )/ √ 2. For the three-gauge-boson vertex with This form of the triple gauge vertex is very special: in general, there could be departures from the above SM expression, even restricting us to SU(2) ⊗ U(1) gauge symmetric and C and P invariant couplings. In fact some small corrections are already induced by the radiative corrections. The SM form of the triple gauge vertex has been experimentally confirmed by measuring the cross-section e + e − → W + W − at LEP.
We now turn to the Higgs sector of the EW Lagrangian. The Higgs Lagrangian is specified by the gauge principle and the requirement of renormalizability to be where φ is a column vector including all Higgs scalar fields; it transforms as a reducible representation of the gauge group. The quantities Γ (which include all coupling constants) are matrices that make the Yukawa couplings invariant under the Lorentz and gauge groups. The potential V (φ † φ), symmetric under SU(2) ⊗ U(1), contains, at most, quartic terms in φ so that the theory is renormalizable: Spontaneous symmetry breaking is induced if the minimum of V which is the classical analogue of the quantum mechanical vacuum state (both are the states of minimum energy) is obtained for non-vanishing φ values. This occurs because we have taken µ 2 and λ positive in V (note the "wrong" sign of the mass term). Precisely, we denote the vacuum expectation value (VEV) of φ, i.e. the position of the minimum, by v: The fermion mass matrix is obtained from the Yukawa couplings by replacing φ(x) by v: with In the SM, where all left fermions ψ L are doublets and all right fermions ψ R are singlets, only Higgs doublets can contribute to fermion masses. There are enough free couplings in Γ, so that one single complex Higgs doublet is indeed sufficient to generate the most general fermion mass matrix. It is important to observe that by a suitable change of basis we can always make the matrix M hermitian, γ 5 -free and diagonal. In fact, we can make separate unitary transformations on ψ L and ψ R according to and consequently This transformation does not alter the general structure of the fermion couplings in L symm .
If only one Higgs doublet is present, the change of basis that makes M diagonal will at the same time diagonalize also the fermion-Higgs Yukawa couplings. Thus, in this case, no flavor-changing neutral Higgs exchanges are present. This is not true, in general, when there are several Higgs doublets. But one Higgs doublet for each electric charge sector i.e. one doublet coupled only to u-type quarks, one doublet to d-type quarks, one doublet to charged leptons would also be all right, because the mass matrices of fermions with different charges are diagonalized separately. In fact, at the moment, the simplest model with only one Higgs doublet seems adequate for describing all observed phenomena.
Weak charged currents are the only tree level interactions in the SM that change flavor: by emission of a W an up-type quark is turned into a down-type quark, or a ν l neutrino is turned into a l − charged lepton (all fermions are letf-handed). If we start from an up quark that is a mass eigenstate, emission of a W turns it into a down-type quark state d' (the weak isospin partner of u) that in general is not a mass eigenstate. In general, the mass eigenstates and the weak eigenstates do not coincide and a unitary transformation connects the two sets: or, in shorthand, D'=VD, where V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Thus in terms of mass eigenstates the charged weak current of quarks is of the form: Since V is unitary (i.e. V V † = V † V = 1) and commutes with T 2 , T 3 and Q (because all d-type quarks have the same isospin and charge) the neutral current couplings are diagonal both in the primed and unprimed basis (if the Z down-type quark current is abbreviated asD ′ ΓD ′ then by changing basis we getDV † ΓV D and V and Γ commute because, as seen from eq.(24), Γ is made of Dirac matrices and T 3 and Q generator matrices). It follows thatD ′ ΓD ′ =DΓD. This is the Glashow-Iliopoulos-Maiani (GIM) mechanism that ensures natural flavor conservation of the neutral current couplings at the tree level. For three generations of quarks the CKM matrix depend on four physical parameters: three mixing angles and one phase. This phase is the unique source of CP violation in the SM.
We now consider the gauge-boson masses and their couplings to the Higgs. These effects are induced by the (D µ φ) † (D µ φ) term in L Higgs [Eq. (28)], where Here t A and 1/2Y are the SU(2) ⊗ U(1) generators in the reducible representation spanned by φ. Not only doublets but all non-singlet Higgs representations can contribute to gauge-boson masses. The condition that the photon remains massless is equivalent to the condition that the vacuum is electrically neutral: The charged W mass is given by the quadratic terms in the W field arising from L Higgs , when φ(x) is replaced by v. We obtain whilst for the Z mass we get [recalling Eq. (15)] where the factor of 1/2 on the left-hand side is the correct normalization for the definition of the mass of a neutral field. For Higgs doublets we obtain Note that by using Eq. (20) we obtain It is also evident that for Higgs doublets This relation is typical of one or more Higgs doublets and would be spoiled by the existence of, for example, Higgs triplets. These result is valid at the tree level and is modified by calculable small EW radiative corrections. The ρ 0 parameter has been measured from the intensity of NC interactions (recall eq. (25)) and confirmed to be close to unity at a few per mille level.
In the minimal version of the SM only one Higgs doublet is present. Then the fermion-Higgs couplings are in proportion to the fermion masses. In fact, from the Yukawa couplings g φf f (f L φf R + h.c.), the mass m f is obtained by replacing φ by v, so that m f = g φf f v. In the minimal SM three out of the four Hermitian fields are removed from the physical spectrum by the Higgs mechanism and become the longitudinal modes of W + , W − , and Z which acquire a mass. The fourth neutral Higgs is physical and should be found. If more doublets are present, two more charged and two more neutral Higgs scalars should be around for each additional doublet.
The couplings of the physical Higgs H to the gauge bosons can be simply obtained from L Higgs , by the replacement In the minimal SM the Higgs mass m 2 H ∼ λv 2 is of order of the weak scale v but cannot be predicted because the value of λ is not fixed. The dominant decay mode of the Higgs is in the bb channel below the WW threshold, while the W + W − channel is dominant for sufficiently large m H . The width is small below the WW threshold, not exceeding a few MeV, but increases steeply beyond the threshold, reaching the asymptotic value of Γ ∼ 1/2m 3 H at large m H , where all energies and masses are in TeV.
A central role in the experimental verification of the standard electroweak theory has been played by CERN, the European Laboratory for Particle Physics, located near Geneva, between France and Switzerland. The indirect effects of the Z 0 , i.e. the occurrence of weak processes induced by the neutral current, were first observed in 1974 at CERN by the Collaboration Gargamelle (the name of the bubble chamber used in the experiment). Later, in 1982, always at CERN, the W ± and the Z 0 were for the first time directly produced and observed in protonantiproton collisions by the UA1 and UA2 collaborations and then further studied with the same technique both at CERN and subsequently at the Tevatron of Fermilab near Chicago. Starting from 1989 LEP, the large e + e − collider was functioning at CERN till 2000. In the LEP circular ring of about 27Km of circumference electrons and positrons were accelerated in opposite directions to an equal energy in the range between 45 and 103 GeV. The beams were made to cross and collide in correspondence of four experimental areas where the ALEPH, DELPHI, L3 and OPAL detectors were located to study the final states produced in the collisions. In its first phase, called LEP1, from 1989 to 1995 the LEP operation has been completely dedicated at a precise study of the Z 0 properties, mass, lifetime, decay modes in order to accurately test the predictions of the SM. The main lesson of the precision tests of the standard electroweak theory can be summarised as follows. It has been checked that the couplings of quark and leptons to the weak gauge bosons W ± and Z are indeed precisely those prescribed by the gauge symmetry. The accuracy of a few 0.1% for these tests implies that, not only the tree level, but also the structure of quantum corrections has been verified. Then, since the end of 1995, the energy of LEP was increased and the phase of LEP2 was started. The total energy was gradually increased up to 206 GeV. The main physics goals of LEP2 were the search for the Higgs and for possible new particles, the precise measurement of m W and the experimental study of the triple gauge vertices W W γ and W W Z 0 . The Higgs particle of the SM could in principle be produced at LEP2 in the reaction e + e− → Z 0 H which proceeds by Z 0 exchange. The non observation of the Higgs particle at LEP2 has allowed to establish a lower limit on its mass: m H > ∼ 114 GeV . Indirect indications on the Higgs mass were also obtained from the precision tests of the SM, as the radiative corrections effects depend logarithmically on m H . The indication is that the Higgs mass cannot be too heavy if the SM is valid: m H < ∼ 219 GeV at 95% c.l. In 2001 LEP was dismantled and in its tunnel a new double ring of superconducting magnets is being installed. The new accelerator, the LHC (Large Hadron Collider), will be a proton-proton collider of 14 T eV of total center of mass energy. Two large experiments ATLAS and CMS will continue the Higgs hunting starting in the year 2007. The sensitivity of LHC experiments to the SM Higgs will go up to masses m H of about 1 TeV.