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Introduction to

Swedish Teacher program 2019 Lecture II

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 1 EP-UAI Lecture II Introduction to the SM

– The

– The Higgs mechanism in the SM (hints)

– The detection of the Higgs

The most part of figures and formulas are from A. Bettini, Introduction to elementary . II edition Cambridge U.P. 2016

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 2 EP-UAI SM mathematical features

• SM is based on:

– Gauge (QFT) that incorporates:

• The Special Relativity (already introduced); • The Quantum Mechanics (already introduced); • Gauge invariance of the interactions (to be introduced!)

• Rather complex mathematical framework mediated from the Quantum Electro Dynamics (QED) that successfully describe the electromagnetic interaction as gauge QFT !

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 3 EP-UAI Quantum Field Theory

• Lamb and Retherford experiment in 1947 showed that the non-relativistic QM was not sufficient to describe the Nature!! • The result showed that the motion of the atomic could not be described simply by the Dirac equation in an external classical given field (that of the in this case);

Virtual particle! : the of the interaction

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 4 EP-UAI QED Quantum Field Theory

• In QFT , it is not only the that are quanta of a field but also the charged , like and are quanta of a field

• Dirac equation is still valid but its interpretation changes: the 휓 becomes a mass field, the quantum field of electrons. 푒+푎푛푑 푒− are the source of the mass fields!

• The fields are quantum operators to create and annihilate their quanta.(Feynman diagrams). Fundaments of the QED.

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 5 EP-UAI Speaking of gauge invariance

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 6 EP-UAI Gauge invariance in the EM field

1. There are (Coulomb) between stationary charges and additional forces (magnetic) between moving charges;

2. Electromagnetic radiation is emitted by accelerating charges and can be absorbed or scattered by bodies with electrical structure

3. Classical EM defines the fields E and B which obey Maxwell’s equations (1830, Unification of electrostatic and magnetism)  DivE   0 DivB  0 B CurlE   (2.1) t E CurlB   0(j  0 ) t • But this formulation is not invariant under Lorentz transform (L.I.) of space- time coordinates. 7/3/2018 Dep. of Physics, Giacinto de Cataldo INFN, Ba-It and 7 Malta CERN-CH [email protected] Gauge invariance in the EM field

1st step: 1st and 4th Maxwell equations (E,B) implies the conservation of the charge: 휕휌 (2.2) 휵 ∙ 푱 − = 0 휕푡 To express the Maxwell equations in a LI form, it is convenient to define the vector potential A and the (scalar) scalar potential 휑, such: B=퐶푢푟푙 푨 휕푨 E= - grad휑 - (2.3) 휕푡 The properties of A are such that the four-vector (A, 휑/c) transforms exactly as (r,ct) or the (p,E/c). So, the reformulation of the Maxwell equations using A and 휑 become L.I.!

7/3/2018 Dep. of Physics, Malta Giacinto de Cataldo INFN, Ba-It and CERN- 8 CH [email protected] Gauge invariance in the EM. field

2nd step In addition, the EM field shows a very important property, that of the gauge invariance. If A and 휑 are simultaneously changed in the following way: 휕χ 푨 → 푨′ = 푨 + 𝑔푟푎푑χ; 휑 → 휑′ = 휑 - 휕푡 (2.4)

where 휒 is a scalar function of space and time 휒 = 휒(푥, 푦, 푧, 푡) (called gauge function), the fields E and B are unchanged! In fact replacing (2.4) in (2.3) , there is no change in E and B.

B′ =푐푢푟푙푨′ = 푐푢푟푙(푨 + 𝑔푟푎푑휒) curlA = B) 휕푨′ 휕휒 휕(푨+품풓풂풅휒) 휕푨 E’= - grad휑’ - = −𝑔푟푎푑 휑 − − =− grad휑 − =E 휕푡 휕푡 휕푡 휕푡 The (2.4 ) is called gauge transformation. The EM theory is said gauge invariant or the EM has the gauge symmetry property.

7/3/2018 Dep. of Physics, Malta Giacinto de Cataldo INFN, Ba-It and CERN- 9 CH [email protected] Gauge invariance in QED

• The theoretical motivations for charge conservation are extremely strong , since they are consequence of the gauge invariance of the theory;

• Up to now we were speaking of the Relativistic Electrodynamics, NOT yet of gauge QFT. Now let’s start to describe it!

• The scattering amplitude in QED is given by the matrix element of the interaction Hamiltonian:

푀푓푖 = 휓푓 퐻푄퐸퐷 휓푖 where 퐻푄퐸퐷= 퐻푄퐸퐷(푨푄푀 푓푖푒푙푑, 휑푄푀 푓푖푒푙푑)

7/3/2018 Dep. of Physics, Malta Giacinto de Cataldo INFN, Ba-It and CERN- 10 CH [email protected] Gauge invariance in QED

• The gauge invariance of the QED Lagrangian (replacing the Hamiltonian) is ensured if the wave function of the charged particles is transformed at the same time as the quantum potential field 푨푄푀 푓푖푒푙푑, 휑푄푀 푓푖푒푙푑(V. Fock 1929) .

• For an electron: 휓 ⟹ 휓′ = 푒푖휒(풓,푡) 휓

• 휓 is itself an operator of the field electrons that create and destroy the electron. The local gauge invariance of the Lagrangian implies the electric charge conservation,

• the invariance held if the , the photon in this case, is massless, as it is !

7/3/2018 Dep. of Physics, Malta Giacinto de Cataldo INFN, Ba-It and CERN- 11 CH [email protected] Feynman diagrams for the QED

constant is the EM

훼 is NOT constant! depends on the E because of Scattering amplitude: the electric charge depends on E! Running constant

Probability amplitude for the mediator

The probability of this process, cross section, is proportional to the 2 2, 29/Oct/2019|1/m -t| the coupling constants, G de Cataldo INFN Ba, It/CERN CH 12 and to the phase space. EP-UAI • WI are responsible of the decays in the radioactivity;

• E. Fermi 1933 first theoretical description of B decay;

• and anti-mater behave differently w.r.t WI!! Reason of matter/anti-matter asymmetry in the ??

• WI would be a gauge QFT if the vector mediating the interaction are assumed massless! WARNING:W±, Z0 are massive!! • We need a mechanism providing mass to the W±, Z0 without destroying the gauge invariance! Higgs mechanism! 29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 13 EP-UAI Some Feynman diagrams of the WI

Similarity with EM scattering! Z0 couples matter and antimatter partners partners

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 14 EP-UAI Quantum Chromo Dynamics (QCD)

• Deep inelastic scattering with e on p and n demonstrate the existence of partons : finally the ;

• Quarks interact strongly. They strong charges are named colors, scattering with scattering with gluon exchange gluon exchange • Their values are: Red, Green Blue (RGB). Anti-colors exists.

• 8 are the interaction mediators. They are massless. Differently from the photon, they are charged as well!

• QCD is a “gauge QFT”. The strong charge is conserved! 29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 15 EP-UAI Origin of masses and Quantum

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 16 EP-UAI Unification in the Standard Model • Space-time and local gauge symmetry are of fundamental importance in building the basic laws of the nature, the Standard Model;

• EW unification started in 1960. From Fermi to gauge QFT:

At that time, with a raw approximation, neglecting the quadri-

transfer, the MW could already be estimated: 2 2 4 2 퐺퐹 𝑔 푀푊푐 퐺퐹 𝑔 2 ≈ 2 4 푠표 푡ℎ푎푡 3 ≈ ℏ푐 푀푊푐 4휋 ℏ푐 4휋ℏ푐

2 2 푔 푒 1 2 If 𝑖푠 푡표 ℎ푎푣푒 푡ℎ푒 푠푎푚푒 푣푎푙푢푒 = 훼 = then 푀푊푐 ≈ 88 퐺푒푉! 4휋ℏ푐 4휋휀0ℏ푐 137 (83 the measured value!) 29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 17 EP-UAI Unification in the Standard Model

• More formally: • The EW predict three gauge bosons and the relationship of their

masses with two parameters: GF and 푠푒푛휃푊 with 휃푊 called “weak mixing angle. • The latter parameter imposes the photon to not couple with neutral current and the Z0 to be neutral.

• 푠푒푛휃푊 can be measured only indirectly and it was done by UA1 and UA2 measuring the 푀푊 푎푛푑 푀푍0 (1983).

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 18 EP-UAI 푈퐴1: 푀푀푤= 83 ± 3 퐺푒푉

0 − − 푍 → 푒 +푒 푈퐴1: 푀푍0= 93 ± 3 퐺푒푉

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 19 EP-UAI And the interaction coupling constant constants?

• The EM and Weak coupling constant are not independent but correlated by the EW theory. On the contrary , EW and DCD , both being gauge theories, are unified by the theoretical framework while their coupling constant are independent.

• EW and QCD together form the SM of fundamental interactions . What about the evolution of their coupling constant?

Comments in Beyond the SM! (lecture III)

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 20 EP-UAI The Higgs mechanism

• The Higgs mechanism was proposed (1960) to provide mass to the weak gauge bosons W± and Z°, the interaction carriers, without destroying the gauge invariance of the EW Lagrangian !

• this was to unify the Weak and EM interactions in a gauge QFT (Electroweak interaction);

• After about 50 years, The has been observed at the LHC and presented on the 4th July 2012 at CERN;

• The Higgs field fills the Universe with a non zero value and provides mass to the fundamental particles;

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 21 EP-UAI Observation of the Higgs boson

Two interacting in the LHC excite the Higgs field that manifests itself with the emission of the boson with a mass of 126 GeV.

14/03/2018 Giacinto de Cataldo INFN, Ba-It and 22 CERN-CH [email protected] The “giant” of LHC: ATLAS

14/03/2018 Giacinto de Cataldo INFN, Ba-It and CERN-CH 23 [email protected] 23 HZ0Z04

14/03/2018 Giacinto de Cataldo INFN, Ba-It and CERN-CH 24 [email protected] What an Invariant mass plot is

14/03/2018 Giacinto de Cataldo INFN, Ba-It and CERN-CH 25 [email protected] Invariant mass plot HZZ4 leptons

14/03/2018 Giacinto de Cataldo INFN, Ba-It and CERN-CH 26 [email protected] Hgg

14/03/2018 Giacinto de Cataldo INFN, Ba-It and CERN-CH 27 [email protected] Hgg

14/03/2018 Giacinto de Cataldo INFN, Ba-It and CERN-CH 28 [email protected] the SM Lagrangian

Gauge bosons: W,Z, photon and gluons Quarks and leptons And their gauge interactions Fermion interactions With H boson Higgs field

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 29 EP-UAI Summary lecture II

• The SM is a paradigm of the gauge QFT. The SM incorporates the Quantum Mechanics and the Special Relativity;

• The requirement of the gauge invariance for the EW and Storng interactions imposes the existence of 4 electo-weak gauge bosons W± , Z°, g and 8 strong gauge bosons, the gluons;

• The Higgs mechanism provides mass to three electroweak bosons W± Z° without breaking the local gauge invariance;

• The detection of the H boson confirmed the effectiveness of SM but still some holes remain (dark matter, dark , many ad hoc parameters, no …)

29/Oct/2019 G de Cataldo INFN Ba, It/CERN CH 30 EP-UAI