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Spontaneous breaking of continuous symmetries based on S-32 It is also instructive to use a different parameterization: real scalar fields Consider the theory of a complex scalar field: the U(1) transformation takes a simple form: the lagrangian is invariant under the U(1) transformation: for negative, the minimum of the potential parameterizes the flat direction now we get: is achieved for is massless! spectrum and scattering amplitudes are independent of field redefinitions! The Goldstone boson remains massless even after including loop corrections! we have a continuous family of minima (ground states): 327 329 To see it, note that if the GB remains massless, the exact propagator: is achieved for has to have a pole at we have a continuous family of minima (ground states): for we can write: and we get: b is massless! real scalar fields all the minima are equivalent, there is a flat direction in the field space (we can move along without changing the energy) the physical consequence of a flat direction is the existence of a massless particle - Goldstone boson 328 . we can calculate it by summing 1PI diagrams with two external lines with . The vertices are proportional to momenta of (because of derivatives) and so they vanish. Note: the theory expanded around the minimum of the potential has interactions that were not present in the original theory. However it turns out that divergent parts of the counterterms for the theory with (where the symmetry is unbroken) will also serve to cancel the divergencies in the theory with where the symmetry is spontaneously broken. This is a general rule for theories with spontaneous symmetry breaking. 330 Let’s now generalize our results for a nonabelian case: Let’s work it out in detail; we can write our lagrangian as: where summed from 1 to N the lagrangian is invariant under the SO(N) transformation: Let’s shift the field by the VEV and define: set of for infinitesimal parameters set of antisymmetric generator matrices with a single nonzero entry -i above the main diagonal , the minimum of the potential is achieved for: we get: where: with summed from 1 to N-1 N-1 massless particles - Goldstone bosons the vacuum expectation value, N-component vector with arbitrary direction we can choose the coordinate system (SO(N) rotation) so that SO(N-1) - manifest symmetry of the lagrangian! 331 333 Spontaneous breaking of gauge symmetries Let’s make an infinitesimal SO(N) transformation; this changes the VEV: based on S-84,85 Consider scalar electrodynamics: zero for all the generators with no non-zero entry in the last column UNBROKEN GENERATORS where non-zero for N-1 generators with a non-zero entry in the last column BROKEN GENERATORS for these do not change the VEV of the field, and correspond to the unbroken symmetry For SO(N) we have unbroken generators these change the VEV of the field but not the energy; each of them corresponds to a flat direction in field space; each flat direction implies the existence of a massless particle - Goldstone boson. , the minimum of the potential is achieved for: we write: and the scalar potential becomes: SO(N-1) - obvious symmetry of the lagrangian! 332 334 the general solution is: The gauge symmetry allows us to shift the phase of spacetime function; we can set: by an arbitrary Unitary gauge with this choice the kinetic term becomes: mass term for the gauge field! in the rest frame we can choose the polarization vector to correspond to definite spin along the z-axis: polarization vectors together with the timelike unit vector orthonormal and complete set: form an Higgs mechanism: the Goldstone boson disappears and the gauge field acquires a mass. The Goldstone boson has become the longitudinal polarization of the massive gauge field. The scalar field that is used to break the gauge symmetry is called the Higgs field. 335 337 Part of the lagrangian quadratic in gauge fields: the equation of motion: acting on both sides with In analogy with the scalar field theory or QED the propagator for the “massive” gauge field is: we find: this is not a gauge-fixing condition! thus each component obeys the Klein-Gordon equation: and interactions can be read off of: the general solution is: and with 336 338 There are complications with calculating loop diagrams! taking the unitary gauge, , corresponds to inserting a functional delta function: cross term between the vector field and the b-field to For abelian gauge theory in the absence of spontaneous symmetry breaking we fixed the gauge by adding the gauge-fixing and ghost terms: ghost-ghost-scalar vertex: arbitrary constant ghost propagator: no ghost-gauge field interactions the ghost propagator doesn’t contain momentum and so loop diagrams with arbitrarily many external lines diverge more and more (also the gauge field propagator scales like for large momenta); difficult to establish renormalizability. For spontaneously broken theory we choose: can ghost fields have no interactions and can be ignored. l and ce to the path integral. Changing integration variables from and we must include the functional determinant: rearranging the kinetic term: and we get: mass term for b integration by parts 339 The gauge doesn’t suffer from this problem: 341 Let’s choose: real scalar fields W IE gauge-fixing function with this parameterization, the potential is: then V E -1 for and the covariant derivative is: and so the kinetic term can be written as: or 340 R we will get the gauge acting as an anticommuting object 342 Collecting terms quadratic in the vector field: W IE now we can easily perform the path integral over B: in the momentum space we have: it is equivalent to solving the classical equation of motion, V E and substituting the result back to the formula: we obtained the gauge fixing lagrangian and the ghost lagrangian R which can be easily inverted: propagator for the massive vector field in the gauge 343 345 propagator for the massive vector field in the gauge: For spontaneously broken theory we choose: transverse components propagate with mass M longitudinal component propagate with mass (the same as ghost and b) since scattering amplitudes do not depend on they all are unphysical particles and for the ghost term we find: Propagator highly simplifies for : ghost has the same mass as the b-field! typically used for loop calculations 344 346 Summary of Spontaneously broken nonabelian gauge theory gauge: gauge field (3 polarizations) with mass: based on S-84,86 Generalization to a nonabelian gauge theory is straightforward: ghost and the b-field propagators: physical Higgs field with mass: interactions are collected here: the potential is minimized for: plugging the vev to the kinetic term we get a mass term for gauge fields: where 347 It is instructive to consider 349 plugging the vev to the kinetic term we get a mass term for gauge fields: limit: gauge field propagator: where this is what we found in the unitary gauge ghost and the b-fields become infinitely heavy but we cannot just drop the ghost field because its interaction becomes infinite as well; instead we can define: and consider the ghost propagator to be and the vertex this is what we found in the unitary gauge The in the limit gauge fields corresponding to unbroken generators, , remain massless gauge fields corresponding to broken generators, , get a mass (these are the gauge fields of unbroken gauge symmetry) (and they form representations of the unbroken group) we can repeat everything that we did for the abelian case, see S-86, instead of working out details for the general case let’s focus on a specific example... is equivalent to unitary gauge! 348 350 The Standard Model: gauge and Higgs sector the potential is (the usual one): based on S-87 The Standard Model of elementary particles describes almost everything we know about our world. It is based on SU(3)xSU(2)xU(1) gauge symmetry spontaneously broken to SU(3)xU(1). Glashow-Weinberg-Salam model All particles acquire mass from interactions with the Higgs field in the process of electroweak symmetry breaking the scalar field get a vev, and we can use a global SU(2) transformation to bring it to the form: plugging the vev to the kinetic term: we get the mass term for gauge fields: 351 Let’s start with the electroweak part which is described by SU(2)xU(1) gauge symmetry and one complex scalar field - the Higgs field in the representation: 353 we have to diagonalize the mass matrix for gauge fields, we define: the weak mixing angle the covariant derivative is: then we have can be conveniently written as remains massless, it is the gauge field of 352 354 let’s work out the complete lagrangian: we will use unitary gauge (it is simple and sufficient for tree-level calculations) a complex scalar field (which is a doublet under SU(2)) can be written in terms of 4 real scalar fields; three of them become the longitudinal components of W and Z, thus we can write the Higgs doublet as: real scalar field the Higgs field has the usual kinetic term and the potential is now: we also have: where: we obtained the mass terms for gauge fields by inserting just the vev; to get the whole lagrangian that contains also interactions of Higgs and gauge fields we simply multiply the mass term for gauge fields by and we get another part of the lagrangian: 355 357 Putting all the terms together: finally, let’s look at the kinetic terms for the gauge fields: where we can combine where charge of is , e is positive (opposite sign that we used in QED) The lagrangian exhibits manifest and we get a part of the lagrangian: 356 gauge invariance! 358 The Standard Model: Lepton sector Lagrangians for spinor fields based on S-88 as far as we know there are three generations of leptons: W IE based on S-36 we want to find a suitable lagrangian for left- and right-handed spinor fields. it should be: Lorentz invariant and hermitian quadratic in V E and equations of motion will be linear with plane wave solutions (suitable for describing free particles) R terms with no derivative: and they are described by introducing left-handed Weyl fields: + h.c. terms with derivatives: just a name would lead to a hamiltonian unbounded from below 359 361 to get a bounded hamiltonian the kinetic term has to contain both and , a candidate is: the covariant derivatives are: W IE is hermitian up to a total divergence and the kinetic term is the usual one for Weyl fields: there is no gauge group singlet contained in any of the products: R are hermitian it is not possible to write any mass term! 360 V E does not contribute to the action 362 Our complete lagrangian is: W IE it is convenient to label the components of the lepton doublet as: the phase of m can be absorbed into the definition of fields V E and so without loss of generality we can take m to be real and positive. Equation of motion: R Taking hermitian conjugate: are hermitian then we have: we define a Dirac field for the electron and we see that the electron has acquired a mass: and the neutrino remained massless! 363 365 consider a theory of two left-handed spinor fields: however we can write a Yukawa term of the form: W IE i = 1,2 the lagrangian is invariant under the SO(2) transformation: V E it can be written in the form that is manifestly U(1) symmetric: there are no other terms that have mass dimension four or less! in the unitary gauge we have: R and the Yukawa term becomes: 364 366 Thus the lagrangian can be written as: Equations of motion for this theory: W IE we can define a four-component Dirac field: V E Dirac equation we want to write the lagrangian in terms of the Dirac field: R Let’s define: W IE The U(1) symmetry is obvious: R V E The Nether current associated with this symmetry is: numerically but different spinor index structure later we will see that this is the electromagnetic current 367 Then we find: R V E Thus we have: 369 If we want to go back from 4-component Dirac or Majorana fields to the two-component Weyl fields, it is useful to define a projection matrix: W IE W IE just a name We can define left and right projection matrices: R V E And for a Dirac field we find: 368 370 We can describe the neutrino with a Majorana field: since we have: and it is also convenient to define: the electric charge assignments are as expected: a “Dirac field” for the neutrino covariant derivatives in terms of the four-component fields: because then the kinetic term: can be written as: 371 Now we have to figure out lepton-lepton-gauge boson interaction terms: (we want to write covariant derivatives in terms of , and ) 373 Putting pieces together, we have we find the interaction lagrangian: and where we identify it with electric charge: 372 374 The connection with Fermi theory: We can calculate e.g. muon decay: if we want to calculate some scattering amplitude (or a decay rate) of leptons with momenta well below masses of W and Z, the propagators effectively become: the relevant part of the charged current: and we get 4-fermion-interaction terms: we get the effective lagrangian: and the rest is straightforward... where the Fermi constant is: 375 377 The Standard Model: Quark sector Generalization to three lepton generations: based on S-89 as far as we know there are three generations of quarks: the most general Yukawa term: currents remain diagonal, just add the generation index a complex 3x3 matrix; it can be diagonalized (with positive or zero entries on the diagonal) by a bi-unitary transformation and they are described by introducing left-handed Weyl fields: fermion masses are then given by: just a name diagonal entries 376 378 the covariant derivatives are: it is convenient to label the components of the quark doublet as: then we have: and the kinetic terms are the usual ones for Weyl fields: we define Dirac fields for the down and up quarks there is no gauge group singlet contained in any of the products; it is not possible to write any mass term! and we see that the up and down quarks have acquired masses: ...work out interactions with gauge fields, and generalization to three families... 379 however we can write a Yukawa terms of the form: there are no other terms that have mass dimension four or less! in the unitary gauge we have: and the Yukawa terms become: 380 381