May 7, Physics Beyond the Standard Model. Francesco Fucito. Introduction. Standard. Model- Boson Sector. Standard. Model- Fermion Sector

Transcription

1 - Boson - May 7, 2017

2 - Boson - The standard model of particle physics is the state of the art in quantum field theory All the knowledge we have developed so far in this field enters in its definition: it incorporates all the principles of quantum mechanics and relativity Except gravity all other fundamental forces of nature are described in its framework using gauge theories with gauge group SU(3) c SU(2) w U(1) em The gauge bosons mediate the forces: gluons, W ± and Z, photons

3 - Boson - Gauge theories can exist in different phases: the Coulomb (massless gauge bosons electrodynamics), the Higgs (massive W ± and Z with spontaneous breaking of the gauge symmetry), confined (the gauge bosons do not appear in the spectrum quantum chromodynamics). All of these phases are realized in nature. Symmetries play a central role: global and local symmetries Global symmetries are usually only approximate. Exact symmetries are locally realized and require the existence of a gauge field. Many symmetries are broken. The simplest form of symmetry breaking is explicit breaking which is due to non-invariant symmetry breaking terms in the classical Lagrangian of the theory

4 - Boson - The quantization of the theory may also lead to explicit symmetry breaking, even if the classical Lagrangian is invariant: the anomaly is an explicit symmetry breaking in the measure of the Feynman path integral Theories with explicitly broken gauge symmetries are inconsistent (perturbatively and even non-perturbatively non- renormalizable) In the standard model all gauge anomalies are canceled due to the properly arranged fermion content of each generation

5 - Boson - A more interesting form of symmetry breaking is spontaneous symmetry breaking which is a dynamical effect When a continuous global symmetry breaks spontaneously, massless Goldstone bosons appear in the spectrum. If there is, in addition, a weak explicit symmetry breaking, the Goldstone bosons pick up a small mass. This is the case for the pions, which arise as a consequence of the spontaneous breaking of the approximate global chiral symmetry in QCD

6 - Boson - When a gauge symmetry is spontaneously broken one has the so-called Higgs mechanism which gives mass to the gauge bosons. This gives rise to an additional helicity state. This state has the quantum numbers of a Goldstone boson that would arise if the symmetry were global. One says that the gauge boson eats the Goldstone boson and thus becomes massive. The fermions in the standard model are either leptons or quarks. Leptons participate only in the electromagnetic and weak gauge interactions, while quarks also participate in the strong interactions Quarks and leptons also pick up their masses through the Higgs mechanism

7 - Boson - The weak interaction eigenstates mix to form the mass eigenstates. The quark mixing Cabbibo-Kobayashi-Maskawa (CKM) matrix contains several more free parameters of the standard model The values of these masses are free parameters of the standard model that are presently not understood on the basis of a more fundamental theory There is experimental evidence for non-zero neutrino masses. This implies that there are not only additional free mass parameters for the electron-, muon-, and tau-neutrinos, but an entire lepton mixing matrix

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9 - Boson - There is a very interesting parameter in the standard model, the CP violating QCD θ-vacuum angle, which seems to be zero in the real world. The strong CP problem is to understand why this is the case The θ-angle is related to the topology of the gluon field which manifests itself e.g. in so-called instanton The standard model can be extended by the introduction of a second Higgs field which gives rise to an additional U(1) PQ symmetry as first suggested by Peccei and Quinn, and it naturally leads to θ = 0. The spontaneous breaking of the Peccei-Quinn symmetry leads to an almost massless Goldstone boson, the axion. If this particle would be found in experimental searches, it could be a first concrete hint to the physics beyond the standard model.

10 - Boson - Non-trivial topology also arises for the electroweak gauge field. This leads to an anomaly in the fermion number, or more precisely in the U (1) B+L global symmetry of baryon plus lepton number In particular, baryon number itself is not strictly conserved in the standard model. This has been discussed as a possible explanation of the baryon asymmetry in the universe It is more likely that baryon number violating processes beyond the standard model are responsible for the baryon asymmetry. For example, in the SU (5) grand unified theory (GUT) baryon number violating processes appear naturally at extremely high energies close to the GUT scale

11 - Boson - Although the U (1) B+L symmetry is explicitly broken by an anomaly, the global U(1) B-L symmetry remains intact both in the standard model and in the SU (5) GUT, at least if the neutrinos were massless. This would, in fact, be quite strange (an exact symmetry should be local, not global) and we now know that neutrinos are indeed massive. A grand unified theory that naturally incorporates massive neutrinos is based on the symmetry group SO(10). In this model B-L is also violated and all exact symmetries are locally realized. In addition, the so-called see-saw mechanism gives a natural explanation for very small neutrino masses.

12 - Boson - Until the scale of the TeV the SM has been experimentally tested. Still it suffers from the hierarchy problem (how to stabilize the electro-weak scale with respect to the Planck mass) and from the presence of neutrino masses It is important to stress the implications of the SM for the evolution of the Universe immediately after the big bang expecially for the QCD confinement transition and electroweak transitions (spontaneoous gauge symmetry breaking)

13 - Boson - Now (15 billion years) Stars form (1 billion years) Atoms form (300,000 years) Nuclei form (180 seconds) Nucleons form (10-10 seconds) Quarks differentiate (10-34 seconds?) 4x10-12 seconds

14 Boson - Boson - The boson sector of the SM is given by the scalar Higgs and the gauge bosons In the electroweak sector particles get masses through spontaneous symmetry breaking which produces a massless photon and massive W ±, Z 0 ϕ e iαa τ a e iβ 2 A gauge transformation ( ) with α 1,2 = 0, α 3 = β leaves the 0 vacuum ϕ = invariant so that a gauge boson stays v massless

15 Boson - Boson - Now 1 2 (D µϕ) 2 = 1 2 (0 v)(gaa µτ a g B µ )(ga b µτ b + 1 ( ) 2 g 0 B µ ) v = 1 8 v 2 [g 2 ((A 1 µ) 2 + (A 2 µ) 2 ) + ( ga 3 µ + g B µ ) 2 ) particle mass W ± = 1 (A 1 µ ia 2 µ) m W = gv 2 2 Zµ 0 1 (ga 3 µ g B µ ) = m g 2 + g 2 Z = v g g 2 1 A µ = g 2 + g 2 (ga3 µ + g B µ ) m A = 0

16 - Boson - Given that a Dirac spinor is ψ = ( ) ψl ψ R, since W ±, Z only couple to the L part, ψ L, ψ R must have different quantum numbers L fermions are assigned to the fundamental of SU(2) with T 3 = ± 1 2 while R fermions have T 3 = 0 The values of Y come from the experimentally measured charges ( ) νe The fields in the fundamental of SU(2) are E L = e ) and Q L = ( ul d L

17 - Boson - Given Q = T 3 + Y Q T 3 Y ν el el u L 3 2 d L u R d R er 1 0 1

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19 -Masses - Boson - masses do not come easily in the SM ē L e R + ē R e L We introduce Yukawas λ l Ē l ϕe R λ d Q L ϕd R λ u ɛ ab Q La ϕ b u R + h.c. Now the v.e.v. of ϕ gives the mass

20 -Masses - Boson - It s time to see the behaviour under C, P and CP L = 1 4 i F a iµνf a iµν + J ψ J (i /D)ψ J the fermion part is ψ L σµ µ ψ L + ψ R σµ µ ψ R while under we get C : { ψl σ 2 ψr = { ψ L ψl ψ R ψ R σ 2 ψl = P : ψ R ψ R ψ L CP : ψ L σ 2 ψ L

21 - Boson - We find that ψ L σµ µ ψ L + ψ R σµ µ ψ R is invariant under P and C while ψ L σµ µ ψ L is invariant under CP. For the Yukawa s we find yψ t Lσ 2 χ L ϕ y ψ L σ 2χ Lϕ CP y ψ t Lσ 2 χ L ϕ yψ L σ 2χ Lϕ which implies yϕ = y ϕ. If we do not require specific properties for y, CP is violated If we have three families λ l,u,d λ ij l,u,d

22 - Boson - The quarks come in three families the up s and down s quarks can mix. Then the mass matrix is d (d L s L b L)M D R s R + (u b R L c L t L)M U u R c R t R The v.e.v. is real the Yukawa s complex. In general λλ = U L Λ diag U L and λ λ = U R Λ diag U R so that λ = U L Λ diag U R i.e. d L s L = U D d L d b L L s R L s R = U D d R b L b R R s R b R u L c L = U U u L u R L c L c R = U U u R R c R t L t L t R t R

23 - Boson - Let us now express the weak interaction currents in the basis of mass eigenstates. The quark neutral current contains terms such as i ū il,r γµ u il,r When one rotates these terms into the basis of mass eigenstates one can simply drop the primes, because = 1. The neutral current do not lead to changes among different quark flavors. The SM is free of flavor-changing neutral currents For the charged currents U U,D L,R UU,D L,R ū ilγ µ d il = (u L c L t L)γ µ U U L UD L i = (u L c L t L)γ µ d L V s L b L d L s L b L

24 - Boson - The quarks can also be rotated so that U U,D L = U U,D D U,D and V = U U L U L D = DU VD D. The total number of parameters is N 2 2N + 1 = (N 1) 2 for three families is 4, 3 euler type and 1 complex CP violating phase. Plus 6 quark masses For massless neutrino the lepton sector is no problem otherwise we have another CKM type matrix

25 The θ Term - Boson - In the gauge sector we can always add a term θɛ µνρτ F µν F ρτ which is apparently P, T violating We still have two important problems: the U(1) problem and the strong CP problem QCD has global U(N f ) L U(N f ) R while in the spectrum only SU(N f ) L+R U(1) L+R = U(N f ) L+R is manifest. We expect N 2 f + N 2 f N 2 f = N 2 f d.o.f. but we find N 2 f 1. The missing one is the η with M η =.958GeV. Too heavy for a Goldstone. It can be also formulated as why the axial U(1) is not spontaneously broken. The solution is that it is not a Goldstone and the axial U(1) is not a QCD symmetry In the Euclidean action the vacuum angle manifests itself as an additional term iθq. For This term explicitly breaks the CP symmetry.

26 The θ Term - Boson - As a consequence, the neutron would have an electric dipole moment proportional to θ, while without CP violation the dipole moment vanishes. Indeed, the observed electric dipole moment of the neutron is indistinguishable from zero. This puts a stringent bound on the vacuum angle θ < The question arises why in Nature θ = 0 to such a high accuracy. This is the strong CP-problem In QCD we could require CP invariance, but in the standard model we already have CP violations A possible solution could be that CP is spontaneously broken: then θ = 0 at classical level but to achieve 10 9 one needs also 1-loop to be very close to zero. Too complex also because experimental data agree with the explicit CKM breaking

27 The θ Term - Boson - Until now the best solution has been proposed by Peccei and Quinn who proposed to enlarge the gauge simmetry of the theory by an extra U(1) PQ. This introduces a new field, the axion, which is the Goldstone boson, of the spontaneously broken U(1) PQ L = L SM + g 2 32π 2 θf µν F µν 1 2 µa(x) µ a(x)+l int +ξ a f a F µν F µν Instanton must be summed in the dilute gas approximation (even number) with F µν F µν an integer then < 0 e ET 0 > 1 ξa n! ein(θ+ n=0 fa ) e cos(θ+ ξa fa ) Minimizing the energy brings to < a >= faθ ξ and a phys = a < a > sets to zero the θ term

28 - Boson - Why quarks are not visible? The best answer comes from t Hooft and it is the dual Meissner effect. The force lines of an electric field get repelled by a superconductor. At the same time the magnetic field (if any) stays constant since Ḃ = E = 0. If the external magnetic field grows stronger the field lines get into the superconductor (thus making a trnsition to a non superconducting state) but get squeezed into thin flux tubes and get quantized. If there existed magnetic monopoles in the superconductor they woud be at the end points of the flux tubes and permanently confined into it with a potential linearly rising with the distance between two monopoles This scenario needs: monopoles as elementary particles, an Higgs mechanism to go to the superconducting phase a dual electric charge (color) to be at the end of the flux

29 - Boson - The idea is simple: embed SU(3) c SU(2) w U(1) em inside a larger group which breaks spontaneously at a certain scale. In the new gauge groups couplings must be equal. k k Aµ Aν µ q + k We compute Tr(T a T b ) = N R δ ab q ν

30 - Boson - SU(3) : Tr( 1 2 g 3λ 3 ) 2 = 4 n g ( g 3 2 )2 + n g ( g 3 }{{} 2 )2 = 2n g g3 2 }{{} eigenvalue+1 eigenvalue 1 SU(2) : Tr(t 3 g 2 ) 2 = 3 n g (( g 2 2 )2 + ( g 2 2 )2 ) }{{} quarks + n g (( g 2 2 )2 + ( g 2 2 )2 ) = 2n g g2 2 }{{} leptons U(1) : Tr(g y) 2 = 2n g ( g 2 )2 + 2n g ( g ) 2 + 6n g ( g }{{}}{{} 6 )2 }{{} R leptons L leptons L quarks + 3n g ( 2g 3 ) n g ( g }{{} 3 ) }{{} 3 n gg 2

31 - Boson - Equating these results we get g 2 3 = g 2 2 = 5 3 g 2 leading to sin 2 θ = g 2 /(g g 2 ) =.375 vs (sin 2 θ) exp =.231 Georgi, Quinn and Weinberg proposed that the unification is meant to be at the scale of grand unification, M, which is reached from the scale of the measurement E with µ g µ = g t = b 1g 3 16π 2 Repeating the computation we get M GeV and sin 2 θ.21 In supersymmetry the same computation yields M GeV (depleting by a 10 4 the proton lifetime) and sin 2 θ.231

32 of the Coupling Constants in the SM and the minimal MSSM - Boson - 1/α i log Q 1/α i 60 1/α 1 MSSM /α /α log Q

33 SM - Boson - We have seen a long list of problems and questions. What generates the patters of masses in the SM? How to deal with neutrinos and their masses? How do we confine? How do we generate the matter-antimatter asymmetry? Are neutrinos Maioranas or Dirac? How to protect the SM from the hierarchy problem? And what about gravity? The various approaches are:, Technicolor (and its various declinations), Supersymmetry, Supergravities, String Theory Some scenarios are more theoretical: show that the pattern of confinement can be realized, how to we generate the Maiorana mass at the Planck scale for the see-saw, is an inflationary potential realized in some model, etc. In the past we have used models to show some of these mechanisms Other could be already tested already at LHC: extra