Introduction to the Standard Model

1. Introduction to the Standard Model 1. Costituents of Matter 2. Fundamental Forces 3. Particle Detection 4. Symmetries and Conservation Laws 5. Relativistic Kinematics 6. The Quark Model 7. The Weak Interaction 8. Introduction to the Standard Model

2. What is the Standard Model? A quantum relativistic theory of fundamental constituents It is based on concepts of gauge symmetry (and fields) describing three out of four fundamental interactions. Actually… two out of three. The Modello Standard is the best candidate we have to A complete theory of fundamental interactions Fundamental constituents and interactions between them

3. Construction of a QED Lagrangian The Dirac Lagrangian: (Eulero-Lagrange) (Free Dirac equation) The Lagrangian is invariant for global gauge transformations: (phase transformation) We require that this global property also holds true locally. Gauge invariance becomes a dynamical principle. Now the gauge transformation depends on the spacetime point. Let us see how L behaves

4. Using This lagrangian is not gauge-invariant If we want to have a gauge-invariant lagrangian, we need to introduce a compensating field with a suitable transformation law: This new lagrangian is locally gauge-invariant. It was necessary to introduce a new field (the Electromagnetic Field).

5. Local gauge transformation The gauge field A must also have a free-field term. This term will be the free Electromagnetic Field.

6. Lagrangian of the free Electromagnetic Field : (Eulero-Lagrange) (Free field Maxwell equation) This new term is gauge-invariant by itself : It is interesting to note that the presence of a mass for the photon would spoil the gauge invariance.

7. A vector massive field (…if the photon had a mass!). The Proca lagrangian Recall: the e.m. field interacting with a current would be : The mass term violates gauge invariance. Gauge invariance and masslessness of the photon are connected

8. The Dirac Lagrangian with the interaction with the e.m. field (gauge-invariant): Free massive ½ fermion field Interaction Free e.m. field In order for the gauge-invariance to be preserved, the gauge field must be massless. This is the U(1) gauge symmetry

9. A little to-do list for the construction of the Standard Model Lagrangian: Build an electroweak lagrangian based on the symmetry group SU(2)xU(1). Add QCD. Solve the mass problem for the W, Z. Mass must be generated in a gauge-invariant way Solve the mass problem for the fermion constituents. Mass must be introduced in a gauge-invariant way

10. The SU(2) symmetry of the Yang-Mills Field Let us consider two non-interacting Dirac fields : It can be written as : Rewriting the Lagrangian : (mass matrix) If the two masses are equal, M = m (scalar) We can apply the SU(2) symmetry among the two fields (Yang-Mills)

11. U a unitary matrix to be written as : τ: the Pauli matrices U(1) SU(2) Let us concentrate on SU(2): The initial lagrangian has an SU(2) global invariance : Let us now impose the local (x-dependent) invariance :

12. In order to keept this lagrangian (locally) invariant, it is necessary to introduce a triplet of compensating fields, such that: The transformation law of the compensating fields The new interaction term is introduced One should introduce ther free-field terms of the three compensating fields The free fields need to preserve the gauge-invariance. Therefore they should transform : (a consequence of the transformation law of A)

13. This is the Yang-Mills, SU(2) invariat, lagrangian describing two Dirac fields, iteracting by means of three massless gauge fields: The structure of the SU(2) fields and of their transformation law depends on the fact that the symmetry group is not abelian. This structure bears a strong analogy with the Standard Model lagrangian, in particular as long as the symmetry group is concerned. If we did not worry about the mass, we could in principle write the Electroweak lagrangian just by using the SU(2) and U(1) symmetries

14. The mass problem What is the relation of a mass term to the chirality of states? Let us study a mass term An explicit mass term in the Lagrangian will involve both left and right states. But left handed fermions are in doublets and right-handed fermions in singlets! An explicit mass term is forbidded ! This is solved in the frame of the Higgs Mechanism!

15. Quantum Chromodynamics as a Gauge Theory QCD is the Theory of Strong Interactions The quarks are structureless spin ½ elementary particles A quark is described by a 4-component spinor obeying (in a free theory) the Dirac Equation The general form of the free wavefunction for a fixed flavor f and a fixed color i Spin-dependent momentum-space wavefunction Four-momentum and polarization

16. The free-quark Lagrangian Is invariant under a SU(Nc=3) global transformation U is an SU(3) matrix acting on the color part of the wavefunction A generic SU(3) matrix requires 8 real parameters : The group SUC(3) depends on 32 – 1 = 8 real parameters. The matrices are unitary, hermitian, traceless. They are the generators of the fundamental representation of SU(3), to which the field ψif belongs. With the SU(3) generators (hermitian 3x3 matrices) : Gell-Mann matrices Antisymmetric structure constants

17. Since SUC(3) is an exact symmetry, we can «gauge» it (which means: makes it local) by requiring invariance : If we require local gauge invariance Where A linear combination of 8 gluon fields A possible realization of the Gell-Mann matrices

18. The locally SU(3) invariant Quark lagrangian 8 gauge potentials (gluons) and their transformation laws This lagrangian contains a free quark term and an interaction term: To have the full QCD Lagrangian, one needs the gluon part: Where the field strength tensor: Contains self-interaction terms of the field with itself f: SU(3) structure constants The full QCD Lagrangian:

19. The costituents in the Standard Model Lagrangian A set of elementary constituents: quarks and leptons (pointlike, no internal structure down to 10-18 m) • A set of forcesgenerated by gaugesymmetries : • Electroweak (quarks & leptons) • Strong (quarks) Leptonic sector characterized by the SU(2)xU(1) scheme classification : (assume Massless Neutrinos)

20. The hadronic sector consists of the left-handed quarks: The primes on the lower components of the quark doublets signal that the weak eigenstates are mixtures of the mass eigenstates (CKM matrix):

21. The non-interacting Standard Model Lagrangian Massles non interacting Leptons Massles non interacting Quarks Let us require local gauge invariance of the type SU(2)xU(1)xSU(3)

22. The free gauge fields The gauge potentials: Hypercharge phase Weak Isospin rotation Color rotation 22

23. Now the masses ! We introduce a doublet of scalar field: With a Lagrangian: A two parameters potential: This potential gives mass to W,Z in a gauge invariant way (more on this later) This potential can give mass to the constituent fermions by adding to the Lagrangian terms like The Full Lagrangian of the Standard Model (prior to 1996,massless neutrinos) : 23

24. The very many interactions of vector bosons in the Standard Model

25. Introduction to the Higgs Mechanism On the mass terms in a Lagrangian Suppose we consider the L To understand where is the mass term, we compare with the KG: By expanding in power series : So, this lagrangian describes a massive field with

26. On the fundamental state of the Lagrangian In general, in order to find the ground state, it is necessary to write L in the form L = T-U and then minimize U with respecto to the fields. For instance, in the Klein-Gordon: The minimum of U with respect to the fields Let us now start with our prototype lagrangian : The minima of Are the two points :

27. We can then reformulate the theory in terms of deviations from the ground state : In terms of this new variable, the lagrangian becomes And this brings into evidence a mass term : To correctly identify a mass term in a lagrangian, it is necessary first to expand around the fundamental state.

28. Spontaneous Symmetry Breaking The starting lagrangian apparently has not mass term. It has, however, a symmetry In doing the expansion around the fundamental state one identifies the mass term In this second form, the lagrangian has lost its symmetry. This is because the vacuum (fundamental state) does not have the symmetry of the lagrangian itself. The symmetry is spontaneously broken. In this case we have a discrete symmetry, made by two values. We can have a continuous symmetry by using two fields.

29. This lagrangian involves two fields It is invariant by rotations in The “potential energy” part is featuring a ring of minima along the circumference : We make an expansion around a point of minimum that we can arbitrarily choose :

30. L gets written as a function of fields that are fluctuations around the ground state: Couplings Free KG with Free KG with There is a massless field, which is typical of the case when a continuous global symmetry gets broken

31. The Higgs Mechanism Spontaneous symmetry breaking applied to the case of local gauge invariance. Equivalence between a complex field and two real fields like this : The prototype lagrangian that we have used : Features the “usual” trivial U(1) global symmetry (phase) If we now make the symmetry local : It is necessary to introduce a compensating field (and a free field term):

32. Let us now expand around the minimum of U Scalar particle with mass Massless Goldstone Boson Compensating field term that now has aquired a mass.

33. Origin of the mass term: the original term in the “gauged” lagrangian : The shift of the Φ field on this term generates a mass : This is a “Proca” term. The value of the mass depends on the Vacuum Expectation Value (VEV) of the Higgs field.

34. Epilogue What we have not described in these lectures (among many things) Neutrino mass and oscillation The discovery of the Higgs Boson The Standard Model : a great success A Model that accounts for all constituents and 2 out of 3 fundamental forces of nature in a quantum-relativistic way, explaining an enormous amount of data Drawbacks of the Standard Model • The Model has very many free parameters (constants of Nature) • It has few internal consistency problems (source of CP and radiative correction of masses, understood up to the TeV scale, strong interaction problems out of the perturbative regime) • Gravity not included • Dark Matter not included • Dark Energy not included Dark Matter and Dark Energy notincluded

35. Neutrino Physics: extension of the Standard Model (M. Mezzetto)

36. Thank you for your attention 36

37. Backup slides 37

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