Quantum Field Theory and the Electroweak Standard Model
Alexander Bednyakov



 


BLTP JINR, Dubna, Russia


 

These lecture notes cover the basics of Quantum Field Theory (QFT) and peculiarities in the construction of the Electroweak (EW) sector of the Standard Model (SM). 
In addition, the present status, issues, and prospects of the SM are discussed.


Quantum Field Theory and the Electroweak Standard Model[Lectures 
given at the European School of High-Energy Physics (ESHEP), 
June 2018, Maratea, Italy]

    A.V. Bednyakov
    December 30, 2023
===========================================================================================================================================================


 


§ INTRODUCTION



The Standard Model (SM) <cit.> was established in the mid-1970s. Its success is incredible: even after almost half a century, no significant deviations from the SM predictions have been found. 


But what is the SM?


After the discovery<cit.> of the Higgs boson at the LHC, it is fair to give the following short answer <cit.>:


	 The Absolutely Amazing Theory of Almost Everything.


There are many excellent lectures (<cit.>) and textbooks
(<cit.>) that can provide a lot of convincing arguments for such a fancy name. In this course we are not able to cover all the aspects of the SM, but just review some basic facts and underlying principles of the model emphasizing salient features of the latter.

Let us start with a brief overview of the SM particle content (see Fig. <ref>).  One usually distinguishes fermions (half-integer spin) from bosons (integer spin). 
Traditionally, fermions are associated with “matter”, while bosons take the role of “force carriers” that mediate interactions between spin-1/2 particles. 

In the SM, there are three generations involving two types of fermions - quarks and leptons. In total, we have

				
  * 6 quarks of different flavour (q=u,d,c,s,t,b),
				
  * 3 charged (l=e,μ,τ) and 3 neutral (ν_l = ν_e, ν_μ, ν_τ)
								leptons.

All of them participate in the weak interactions. 
Both quarks q and charged leptons l take part in the electromagnetic interactions.
In addition,  
quarks carry a colour charge 
and are influenced by the strong force. 
In the SM the above-mentioned interactions are mediated 
by the exchange of spin-1 (or vector) bosons:

				
  * 8 gluons are responsible for the strong force between quarks;
				
  * 4 electroweak bosons mediate the electromagnetic (photon - γ) 
								and weak (Z,W^±) interactions.

There is also a famous spin-0 Higgs boson h, which plays an important role in the construction of the SM. It turns out that only gluons and photons (γ) are assumed to be massless.[Initially neutrinos ν_l were assumed to be massless in the SM but experiments show that it is not the case.]
All other elementary particles are massive due to the Higgs  mechanism.

In the SM the properties of the particle interactions can be read off the SM Lagrangian _SM. One can find its compact version on the famous CERN T-shirt. However, there is a lot of structure behind the short expression and it is Quantum Field Theory or QFT (see, textbooks <cit.>) that allows us to derive the full Lagrangian and  understand why the T-shirt Lagrangian is unique in a sense.

The form of _SM is restricted by various kinds of (postulated) symmetries.
Moreover, the SM is a renormalizable model. 
The latter fact allows us to
use perturbation theory (PT)
to
provide high-precision predictions for thousands and thousands observables and verify the model experimentally.
All these peculiarities will be discussed during the lectures, which have the following structure.  

We begin by introducing quantum fields in Sec. <ref> as the key objects of the relativistic quantum theory of particles. 
Then we discuss (global) symmetries in Sec. <ref> 
and emphasize the relation between symmetries and particle properties.
We switch from free to interacting fields in Sec. <ref> and
give a brief overview of techniques used to perform calculations in QFT models.
We introduce gauge (or local) symmetries in Sec. <ref>
and discuss how they are realized in the SM (Sec. <ref>).
The experimental status of the SM can be found in Sec. <ref>.
Final remarks and conclusions are provided in Sec. <ref>.






§ FROM PARTICLES TO QUANTUM FIELDS


Before we begin our discussion of quantum fields let us set up our notation. We work in natural units with the speed of light c=1 and the (reduced) Planck constant ħ=1. In this way, all the quantities in particle physics are expressed in powers of electron-Volts (eV). To recover ordinary units, the following conversion constants can be used:

    [
    				-0.5
        < g r a p h i c s >
    ]   ħ≃  6.58 
    				· 10^-22 MeV· s,     ħ c ≃ 1.97 
    				· 10^-14 GeV·cm  [
    				-0.5
        < g r a p h i c s >
    ].

In High-Energy Physics (HEP) we routinely deal with particles traveling at speed v≲ c.
As a consequence, we require that our theory should respect Lorentz symmetry that leaves a scalar product[Summation over repeated indices is implied.] 

    px ≡  p_μ x_μ = g_μν p_μ  x_ν 
    		= p_0 x_0 - p·x,    g_μν = (1,-1,-1,-1)

of any four-vectors, space-time coordinates x_μ and energy-momenta p_μ
	
    x_μ   = {x_0,x},  t≡ x_0, 
    
    					p_μ    = {p_0,p},  E≡ p_0,

	invariant under rotations and boosts parametrized by Λ_μν:

    x_μ→ x'_μ = Λ_μν x_ν,
    			     x_μ x_μ = x'_μ x'_μ⇒Λ_μαΛ_μβ = g_αβ


			It is this requirement that forces us to use QFT as a theory of relativistic particles. Relativistic quantum mechanics (RQM) describing a fixed number of particles turns out to be inconsistent.  Indeed, from the energy-momentum relation for a free relativistic particle

    E^2 = p^2 + m^2    (instead of  E=p^2/2m in the non-relativistic case),

			and the correspondence principle 
				
    E  → i ∂/∂ t,
    					    p→ - i ∇

				one obtains a relativistic analog of the Shrödinger equation  - the Klein-Gordon (KG) equation 
				
    (∂_t^2 - ∇^2 + m^2)ϕ(t,x) = 0
    								   (instead of  i∂_t ψ  = -∇^2/2 mψ)

				for a wave-function ϕ(t, x)≡⟨x|ϕ(t)⟩.
			It has two plane-wave solutions for
	any three-dimensional p:	
	
    ϕ_p(t,x) = e^-i E t + px,   E= ±ω_p,   ω_p = + √(p^2 + m^2).

				One can see that the spectrum (<ref>) is not bounded from below. Another manifestation of this problem is the fact that for a general wave-packet solution
		
    ϕ(t,x) = 1/(2π)^3/2∫d  p/√(2 ω_p)[ a(p) 
                e^-i ω_p t + i px 
    			+ b(p) 
        e^+i ω_p t - i px]

we are not able to introduce a positive-definite probability density  ρ 
			
    ρ   ≡  j_0 = i ( ϕ^* ∂_t ϕ - ϕ∂_t ϕ^*) ⇒ 2 E ϕ∝ e^- i E t,

			required to interpret ϕ as a wave-function of a single particle.  
			Of course, one can try to 
			impose the positive-energy condition [b(p)≡ 0] but it is not stable under interactions. A single-particle interpretation fails to account for the appearance of negative-energy modes and we need a new formalism to deal with such situations. Moreover, in RQM space coordinates play a role of dynamical variables and are represented by operators, while time is an evolution parameter. Obviously, a consistent relativistic theory should treat space and time on equal footing.

			In order to circumvent these difficulties,  one can re-interpret ϕ(x, t) satisfying (<ref>) as a quantum field, an operator[We use the Heisenberg picture, in which operators 𝒪_H(t) depend on time, while
			in the Schrödinger picture it is the states that evolve:
			⟨ψ(t)|𝒪_S |ψ(t)⟩ = ⟨ψ|𝒪_H(t) |ψ⟩ with 𝒪_S = 𝒪_H(t=0), |ψ⟩ = |ψ(t=0)⟩.]
			ϕ̂(x,t). 
			The space coordinates x can be treated as a label for infinitely many dynamical variables and we are free to choose a system of reference, in which we evolve these variables. As a consequence, a single field can account for an infinite number of particles, which are treated as field excitations. 
			In the QFT notation the solution of the KG equation
			(p_0 = ω_p) can be rewritten 
		
    ϕ(x)     = 1/(2π)^3/2∫d p/√(2 ω_p)[ a^-_p
    						e^-i p x 
    			+ b^+_p 
        		e^+i p x ]

		as a linear combination of operators
		a^±_p and b^±_p obeying

    a^-_p a^+_p' - a^+_p' a^-_p≡[ a^-_p, a^+_p'] = δ^3(p - p'),   [ b^-_p, b^+_p']  =  δ^3(p - p').

		All other commutators are zero, [ a^±_p, a^±_p']  = 0. 
	The operators satisfy a^±_p=(a^∓_p)^† and b^±_p = (b^∓_p)^†, and for a^±_p≡ b^±_p the field is hermitian ϕ^†(x) = ϕ(x).

	The operator (<ref>) needs some space to act on and in QFT we consider the Fock space. It consists of a vacuum |0⟩, which is annihilated by a^-_p (and b^-_p)	
		for every p 

    ⟨ 0 | 0 ⟩    = 1, 
    						   a^-_p |0⟩ =0, 
    						   ⟨ 0|a^+_p = (a^-_p |0⟩)^† = 0,

		and field excitations. The latter are created from the vacuum by acting with a^+_k (and/or b^+_k) , 
    |f_1 ⟩    = ∫ d k · f_1(k) a_k^+|0⟩  
    								,	   1-particle state; 
    |f_2⟩    = 
    		∫ d k_1 d k_2 ·  f_2(k_1,k_2) a^+_k_1 a^+_k_2|0⟩   2-particle state, 
    …

where f_i(k, …) are supposed to be square-integrable, so that, ⟨f_1|f_1⟩ = ∫ |f_1(k)|^2 dk < ∞. In spite of the fact that it is more appropriate to  deal with such normalizable states, in QFT we usually consider (basis) states that have definite momentum p, we assume that f_1(k) = δ(k - p). 

The two set of operators a^± and b^± correspond to particles and antiparticles. From the commutation relations we deduce that a^+_p a^+_k = a^+_k a^+_p, so particles are not distinguishable by construction. 
			
		The commutation relations (<ref>)
		should remind us about a bunch of independent quantum harmonic oscillators. Indeed, the corresponding Hamiltonian 

    _osc    = 
    		∑_j1/2 (p_j^2 + ω_j^2 x_j^2)
    			= ∑_j ω_j/2( a^+_j a^-_j + 
    										a^-_j a^+_j) 
    						 = ∑_jω_j( n_j  + 1/2)

can be expressed in terms of ladder operators √(2 ω_j) a_j^± = (ω_j x_j ∓ i p_j) (no summation), which  satisfy 
		[ a_j^-, a_k^+]  = δ_jk similar to (<ref>). 
	For convenience we re-order operators entering into _osc and introduce 
	n_j = a_j^+ a_j^-
	that counts energy quanta  n_j|n_j⟩ = n_j |n_j⟩. 
	A direct consequence of the re-ordering is the fact that the lowest possible state (vacuum |0⟩) has non-zero energy, which is equal to the sum of  zero-point energies ∑_jω_j/2 of all oscillators. 

	We can make the analogy between a (free) field and harmonic oscillators more pronounced if we put our field in a box of size L. In this case,  the energy ω_p  and momentum p are quantized

    p   →p_j = (2π/L) j,  ω_p  →ω_j = √( (2π/L)^2 j^2 + m^2),	
    				  j = (j_1,j_2,j_3),  j_i ∈ℤ.

		The corresponding _osc (<ref>) can be used to deduce the (QFT) Hamiltonian (by taking the limit L→∞):
		
    ℋ_part = lim_L→∞[(2π/L)^3 ∑_j]_∫ d pω_j[ .
    										.
    										(L/2π)^3/2 a^+_j_a^+_p(L/2π)^3/2 a^-_j_a^-_p + 
    						1/2(L/2π)^3_δ(0).
    . ].

		Since our field (<ref>) involves two kinds of ladder operators, we have 
		
    = _part + _antipart
    						= ∫ dp ω_p 
    										[n_p + n̅_p] + 
    						∫ dp ω_p δ(0)

		with n̅_p≡ b^+_p b^-_p and
		n_p≡ a^+_p a^-_p.
		The interpretation of the first term is straightforward: (n̅_p) n_p  counts (anti-)particles with definite momentum p 
		and there is a sum over the corresponding energies. The second term in (<ref>) looks disturbing. It is associated with infinite vacuum (no particles) energy:

    E_0 = ⟨0||0⟩ =  ∫ dp ω_p δ(0).

Actually, there are two kinds of infinities in E_0:
		
						
  * [1.8cm][l]InfraRed  (large distances, L→∞) due to L^3 → (2π)^3 δ(0);
						
  * [1.8cm][l]UltraViolet (small distances, p,ω_p →∞).
		
		One usually “solves” this problem by 
	introducing normal-ordered Hamiltonians, 
    _osc 
    		=  ω_j/2(a_j^+ a_j^- + a_j^- a_j^+) 
    		= ω_j a_j^+ a_j^- = ω_j a_j^+ a_j^-.

		With  we measure all energies with respect to the vacuum 
		ℋ→ =  - ⟨0|ℋ|0⟩ and ignore (non-trivial) dynamics of the latter.  
In what follows we assume that operators are normal-ordered by default.

		It is easy to check that 
		[, a^±_p] = ±ω_p a^±_p and 
	  [, b^±_p] = ±ω_p b^±_p. As a consequence,   	single-particle states with definite momentum p
		
    |p⟩    = a^+_p|0⟩,   ℋ|p⟩ = ω_p |p⟩,      |p̅⟩ = b^+_p|0⟩,   ℋ|p̅⟩  = ω_p |p̅⟩

		are eigenvectors of the Hamiltonian with positive energies and we avoid introduction of negative energies in our formalism from the very beginning. 
		One can generalize Eq. (<ref>) and  “construct” the momentum P 		
		and charge Q operators[It is worth pointing here that by construction both Q and P do not depend on time and commute. In the next section, we look at this fact from a different perspective and connect it to various symmetries.]: 
		
    P    = ∫ dp p[ n_p + n̅_p],   P|0⟩ = 0 |0⟩,    P|p⟩ = p|p⟩    P|p⟩ = 
    		p|p̅⟩,
    		
    Q    = 
    						p∫ dp[ n_p - n̅_p],   Q|0⟩  = 0 |0⟩,
    						    Q|p⟩ = + |p⟩    Q|p̅⟩ 
    						= - |p̅⟩.

		The charge operator Q distinguishes particles from anti-particles. One can
		show that the field ϕ^† (ϕ) increases (decreases) the charge of a  state 
				
    Q,ϕ^† (x) = + ϕ^† (x),
    								  Q,ϕ (x) = - ϕ (x)

		and consider the following amplitudes: 		
	

		
						t_2 > t_1:   ⟨0|ϕ(x_2)_a^-ϕ^† (x_1)_a^+|0⟩ 	
						        
						t_1 > t_2:   ⟨0|ϕ^† (x_1)_b^-ϕ(x_2)_b^+|0⟩ 		
						

		Particle (charge +1) 
		        
		Antiparticle (charge -1) 
 
	  propagates from x_1 to x_2	
		        
	  propagates from x_2 to x_1
	
	

	Both possibilities can be taken into account in one function: 
	
    ⟨0| T [ϕ(x_2) ϕ^†(x_1) ] |0⟩_-i D_c(x-y)   ≡θ(t_2-t_1) ⟨0|ϕ(x_2) ϕ^†(x_1)  |0⟩
        + θ(t_1-t_2) ⟨0|ϕ^†(x_1) ϕ(x_2)  |0⟩,

	with T being the time-ordering operation (θ(t)=1 for t≥0 and zero otherwise). 

	Equation (<ref>) is nothing else but the famous Feynman propagator, which has the following momentum representation:
	
    D_c(x-y) = -1/(2π)^4∫d^4 pe^-i p(x - y)/p^2 - m^2 + i ϵ.

	r0.5
				
    < g r a p h i c s >

Integration contours in p_0 plane.


	The iϵ-prescription (ϵ→0)  picks up certain poles in the complex p_0 plane (see Fig. <ref>) and gives rise to the time-ordered expression (<ref>). 
	The propagator plays a key role in the construction of perturbation theory for interacting fields (see Sec. <ref>). 

	For the moment, let us mention a couple of facts about D_c(x).
	 It is a Green-function for the KG equation, 
    (∂^2_x + m^2) D_c(x-y)  
    					= δ(x-y).

	This gives us an alternative way to find the expression (<ref>). One can also see that D_c(x-y) is a Lorentz and translational invariant function.

The propagator of particles can be connected to the force between two classical  static sources J_i(x)=δ(x-x_i) 
				located at x_i=(x_1,x_2). The sources disturb the vacuum |0⟩→|Ω⟩, since  the Hamiltonian of the system is modified →_0 + J ·ϕ.  Assuming for simplicity that
				ϕ = ϕ^†, 	we can find the energy of the disturbed vacuum from 
				
    ⟨Ω| e^-i  T|Ω⟩   ≡ e^-i E_0(J) T⇒in the limit T→∞
         = e^i^2/2!∫ dx dy J(x) ⟨0| T(ϕ(x) ϕ(y)) |0⟩ J(y)  = 
    								e^ + i/2∫ dx dy J(x) D_c(x-y)J(y)

				Evaluating the integral for J(x) = J_1(x) + J_2(x) and neglecting “self-interactions“,  
				we get the contribution δ E_0  to E_0(J) due to interactions between two sources
				
    lim_T→∞δ E_0 T 
    								    = - ∫ dx dy J_1(x) D_c(x-y) J_2(y) 
    								
    δ E_0     = 
    								- ∫dp/(2π)^3e^+i p (x_1 - x_2)/p^2 + m^2  = -1/4π r e^- m r,      r = |x_1 - x_2|

				This is nothing else but the Yukawa potential. 
				It is attractive and falls off exponentially over the distance scale 1/m.  Obviously, for m=0 we get a Coulomb-like potential. 



§ SYMMETRIES AND FIELDS


Let us switch to the discussion of symmetries and their role in QFT. 
A convenient way to deal with quantum fields and the symmetries of the corresponding physical systems is to consider the following Action functional[Contrary to ordinary functions that produce numbers from numbers, a functional takes a function and produces a number.]
	
    [ϕ(x)] = ∫ d^4 x  (ϕ(x),∂_μϕ)_Lagrangian (density)    = ∫ d^4 x ( ∂_μϕ^†∂_μϕ - m^2 ϕ^†ϕ)_ϕ^†·  K ·ϕ.
	
	To have an analogy with a mechanical system, one can rewrite [ϕ] as 

    [ϕ(x)]      = ∫ dt  L(t),      L = T - U,    H=T + U 
    
    					T     = ∫ dx |∂_t ϕ|^2,   
    					U  = ∫ dx (|∂_xϕ|^2 + m^2 |ϕ|^2)
	
	with T and U being kinetic and potential energy of a system of coupled oscillators (a “mattress”).

	Given a Lagrangian , one can derive the equations of motions (EOM) via the Action Principle. For this we consider variation of the action 
	
    [ϕ'(x)] - [ϕ(x)]_δ[ϕ(x)]= 0 =  
    				∫ d^4 x 
    								[
    								(
    												∂_μ∂ℒ/∂∂_μϕ
    								- ∂ℒ/∂ϕ)
    _(∂_μ^2 + m^2) ϕ = 0δϕ
    				+ ∂_μ( 
    						∂ℒ/∂∂_μϕδϕ)_surface term=0].
	
	due to tiny (infinitesimal) shifts in the field ϕ'(x) = ϕ(x) + δϕ(x). 
	If we require that δ[ϕ(x)]=0 for any variation δϕ(x) of some ϕ(x), we will immediately deduce that this can be achieved only for specific ϕ(x) that satisfy EOM. 
	These particular fields are usually called “on-mass-shell”.    
	From the Lagrangian for our free scalar field (<ref>) we derive the KG equation. It is related in a straightforward way to the quadratic form K in (<ref>). Having in mind (<ref>), one can see that the (Feynman) propagator 
	can also be obtained by inverting K. This statement is easily generalized to the case of other fields. 



 The Action functional for a physical system allows one to study Symmetries. The latter are intimately connected with
												transformations, which leave something invariant.
												The transformations can be discrete, such as   		
    Parity   :         ϕ'(x,t) = P ϕ(x,t) = ϕ(-x,t), 
    Time-reversal   :  ϕ'(x,t) = T ϕ(x,t) = ϕ(x,-t), 
    Charge-conjugation   : ϕ'(x,t) = C ϕ(x,t) = ϕ^†(x,t),

										or depend on continuous parameters. 
										One distinguishes space-time from internal transformations.
										Lorentz boosts, rotations, and translations are typical examples of the former, while phase transformations belong to the latter (see Fig. <ref>).  
										At the moment, we only consider global symmetries with parameters independent of space-time coordinates and postpone the discussion of x-dependent or  local (gauge) transformations to Sec. <ref>. 

Given [ϕ], one can find its symmetries, which can be defined as  
particular infinitesimal variations δϕ(x) that for any ϕ(x) leave [ϕ] invariant up to a surface term (  the Action Principle)
          
    [ϕ'(x)] - [ϕ(x)] = ∫ d^4 x  ∂_μ𝒦_μ,   ϕ'(x) ≡ϕ(x) + δϕ(x).

          If we compare this with the general expression
  
    [ϕ'(x)] - [ϕ(x)] =  
            
            ∫ d^4 x 
                    [
                            (
                            ∂_μ∂ℒ/∂∂_μϕ
                    - ∂ℒ/∂ϕ)
            δϕ            
            + ∂_μ( 
                ∂ℒ/∂∂_μϕδϕ)
    ].

	and require in addition that ϕ satisfy EOM[This requirement is crucial.], we get a local conservation law

    ∂_μ J_μ = 0,   
    J_μ≡𝒦_μ  - ∂/∂∂_μϕδϕ.

The integration of Eq. (<ref>) over space leads to conserved charge:

    d/d t Q = 0,      Q = ∫ d x J_0.

If δϕ = ρ_i δ_i ϕ depends on parameters ρ_i, we have a conservation law for every ρ_i. This is the essence of the first Noether theorem <cit.>. 


	

										A careful reader might notice that we somehow forgot about the quantum nature of our fields and in our discussion of symmetries treat them as classical objects. Let us comment on this fact. In Classical Physics symmetries allow one to find 
										 
														
  * new solutions to EOM from the given one, keeping some features of the solutions
																		(invariants) intact;

														
  * how a solution in one coordinate system (as seen by one observer) looks in other coordinates (as seen by another observer).  
										
										In a quantum world a symmetry 𝒮 guarantees that transition probabilities 𝒫 
										between states do not change upon transformation: 

    |A_i⟩𝒮→|A'_i⟩,     𝒫(A_i → A_j) = 𝒫(A'_i → A'_j) ,      
    										|⟨A_i|A_j⟩|^2 = |⟨A'_i|A'_j⟩|^2.

										One can see that symmetries 
										can be represented by unitary[or anti-unitary (in the case of time reversal).] operators U: 

    |A'_i⟩ = U |A_j⟩,   ⟨A'_i|A'_j⟩ = ⟨A_i|U^† U_1|A_j⟩.
 
In QFT one usually reformulates a symmetry transformation of states as a change of operators 𝒪_k via

    ⟨A_i|𝒪_k(x) |A_j⟩𝒮→⟨A'_i|𝒪_k (x) |A'_j⟩ =
    														⟨A_i|𝒪'_k(x)|A_j⟩, 
    												
    										
    														  𝒪_k'(x) ≡   U^†𝒪_k(x) U.

For example,  translational invariance leads to a relation between matrix elements of quantum fields, 
    ⟨A_i|ϕ(x) |A_j⟩ = 
            ⟨A_i|ϕ'(x+a) |A_j⟩ =  
            ⟨A_i| U^† (a) ϕ(x+a) U(a) |A_j⟩

for any states. As a consequence, any quantum field in a translational invariant theory should satisfy 

    ϕ(x+a)  = U(a) ϕ(x) U^†(a).

One can also find similar constraints on quantum operators due to  other symmetries. 

By means of the Noether theorem we can get almost at no cost the expressions for energy-momentum P_μ = (, P) and charge Q, which we used in Sec. <ref>. For example, 
P_μ is nothing else  but the conserved “charges”, which correspond to space-time translations. 
Indeed, the Noether current in this case is just the energy-momentum tensor T_μν

    ϕ'(x + a)     = ϕ(x),     expand in a⇒δϕ(x)
                 = - a_ν∂_νϕ(x),  
    δ(ϕ(x),∂_μϕ(x))     = ∂_ν
                (- a_ν) 
    						⇒ 
    J_μ  =  - a_μ + a_ν∂/∂∂_μϕ∂_νϕ 
                = a_ν T_μν.
 	
		According to  (<ref>), for every a_μ we have P_ν = ∫ d x T_0ν, conserved total energy-momentum. In the same way, we can apply the Noether theorem to phase transformations of our complex field and get
		
    ϕ'(x)     = e^i αϕ(x),  δϕ(x)  =    i αϕ(x),    
    													J_μ  = i (ϕ^†∂_μϕ - ϕ∂_μϕ^†),   Q = ∫ d x J_0.
	

		 The corresponding quantum operators,  (<ref>) or Q (<ref>), are obtained[We leave this as an exercise.] (modulo ordering issues) from these (classical) expressions by plugging in quantum field 
		 ϕ̂ from (<ref>).   
     
				It turns out that the charges act as generators of symmetries, for space-time translations the unitary operator from (<ref>) is given by
         
    U(a) = exp(i P_μ a_μ),     ϕ(x+a) = U(a) ϕ̂(x) U^†(a).

		 In addition, conserved quantities 
		 can be used to define a convenient basis of states, we characterize our particle states by eigenvalues of P_μ, and Q:
				
    |p⟩≡|p,+1⟩, |p̅⟩≡|p,-1⟩⇒Q|p,q⟩ = q|p,q⟩, P|p,q⟩ = p|p,q⟩.


It is worth mentioning that some symmetries can mix fields, 
    ϕ'_i(x') = S_ij(a) ϕ_j(x)  ⇒ϕ_i(x') = S_ij(a) U(a) ϕ_j(x) U^†(a),    x' = x'(x,a).

Typical examples are fields with non-zero spin: they have several components, which also change under coordinate rotations (more generally, under Lorentz transformations). Moreover, it is the Lorentz symmetry that allows us to classify fields as different representations of the corresponding group. 

Let us discuss this in more detail.
We can describe fields involving several degrees of freedom (per space point) 
				by adding more (and more) indices ϕ(x) →Φ_α^i(x). 
				One can split the indices into two groups: 	
				space-time (α) and internal (i). The former are associated with space-time transformations, while the latter with transformations in the “internal” space:

    Lorentz transform Λ: 
    				Φ^'i_α(Λ x)     = 
    				S_αβ(Λ) Φ^i_β(x), 
    				
    Internal transform a: 
    				ΛΦ^'i_α(x)     =    
    				U^ij(a)Φ^j_α(x).

A quantum field in this case can be represented as

    Φ_α^i(x)  = 1/(2π)^3/2∑_s
    		∫d p/√(2 ω_p)[ 
    						u_α^s(p)  (a_p^-)_s^i    
    						e^-i p x 
    						+
    						v_α^*s(p)(b^+_p)_s^i   e^+i p x ].

Here the factors e^± i p x with p_0 = ω_p (plane waves) guarantee that every component of Φ^i_α satisfies the KG equation. The sum in (<ref>) is over all polarization states, which are characterized by polarization “vectors” for particles 
u_α^s( p) annihilated by 
(a_p^-)_s^i, and anti-particles v_α^*s ( p) created by (b^+_p)_s^i 
. The conjugated field (Φ^i_α)^† involves (conjugated) polarization vectors for (anti) particles that are (annihilated) created.  
Let us give a couple of examples:


				
  * Quarks are coloured fermions Ψ_α^i and, (a^-_p)^b_s annihilates the “blue” quark in a spin state s. The latter is characterized by a spinor u_α^s(p);
				
  * There are eight vector gluons G^a_μ. So (a^-_p)^a_s annihilates a gluon a in spin state s having polarization 
								u_α^s(p) →ϵ^s_μ(p).

Since Lorentz symmetry plays a key role in QFT, we elaborate on some of its non-trivial representations and consider vector and fermion fields in more detail.


 §.§ Massive vector fields

A charged Vector Field (e.g., a W-boson) can be written as 

    W_μ(x)  = 1/(2π)^3/2∑_λ=1^3∫d p/√(2 ω_p)[ 
    						( ϵ^λ_μ(p) a^-_λ(p)	
    						e^-i p x 
    						+
    						ϵ^*λ_μ(p)
    						b^+_λ(p)  e^+i p x )
        ].

				A massive spin-1 particle has 
				3 independent polarization vectors, which satisfy

    p_μϵ^λ_μ(p) 
    				= 0,   ϵ^λ_μ(p) ϵ_μ^*λ'(p) = - δ^λλ', 
    				  ∑_λ=1^3ϵ^λ_μϵ^*λ_ν =
    				-  ( g_μν - p_μ p_ν/m^2)  [ p_0 = ω_p ].

The Feynman propagator can be found by considering  time-ordered product of two fields

    ⟨0| T(W_μ(x) W^†_ν(y)) |0⟩ =     
    				1/(2π)^4∫ d^4 p e^-i p(x - y)[
    								- i(g_μν - p_μ p_ν/m^2)/p^2 - m^2 + i ϵ]    [p_0 - arbitrary]

or by inverting the quadratic form of the (free) Lagrangian

    = -1/2 W^†_μν W_μν + m^2 W^†_μ W_μ,
    				   W_μν≡∂_μ W_ν - ∂_ν W_μ.

One can show that one of the polarization vectors ϵ_μ^L≃ p_μ/m + 𝒪(m) and diverges in the limit p_μ→∞ (m→0). This indicates that one should be careful when constructing models with massive vector fields. We will return to this issue later.



 §.§ Massless vector fields



				Massless (say photon) vectors are usually represented by 
				
    A_μ(x)  = 1/(2π)^3/2∑_λ=0^3∫d p/√(2 ω_p)[ 
    						ϵ^λ_μ(p) a^-_λ(p)	
    						e^-i p x 
    						+ ].

				with 

    ϵ^λ_μ(p) ϵ_μ^*λ'(p) = g^λλ', 
    				  ϵ^λ_μ(p) ϵ_ν^*λ(p) = g_μν, 
    			
    				  
    				[a^-_λ(p), a^+_λ'(p')] = - g_λλ'δ_p,p'.

The corresponding Feynman propagator can be given by 

    ⟨0| T(A_μ(x) A_ν(y)) |0⟩ =     
    				1/(2π)^4∫ d^4 p e^-i p(x - y)[
    								- i g_μν/p^2  + i ϵ.
    ]

In spite of the fact that we sum over four polarizations in (<ref>) only two of them are physical! 
This reflects the fact that the vector-field Lagrangian in the massless case m=0

    = -1/4 F_μν F_μν,    
    				F_μν≡∂_μ A_ν - ∂_ν A_μ

is invariant under A_μ→ A_μ + ∂_μα(x) for arbitrary α(x) (gauge symmetry).  Additional conditions (gauge-fixing) are needed to get rid of unphysical states.



 §.§ Fermion fields


Spin-1/2 fermion  fields (leptons) are given by[There exists a charge-conjugation matrix C=iγ_2, which relates spinors for particles u and antiparticles v, v=C u^*.] 


    ψ^α(x)     
    = 1/(2π)^3/2∫d p/√(2 ω_p)∑_s=1,2[ 
    u^α_s(p) a_s^-(p)
    						e^-i p x 
    			+
    			v^α_s(p) b_s^+(p)
        		e^+i p x ],

where we explicitly write the spinor (Dirac) index α for u_s, v_s and the quantum operator ψ.  The former satisfy the 4× 4 matrix (Dirac) equations 

    (p̂ - m) u_s(p) = 0,    (p̂ + m) v_s(p) = 0,   p̂≡γ_μ p_μ,   p_0 ≡ω_p

and correspond to particles (u_s) or antiparticles (v_s). 
In (<ref>) we use gamma-matrices

    γ_μγ_ν + γ_νγ_μ≡[ γ_μ, γ_ν]_+ = 2 g_μν1  ⇒γ_0^2 = 1,   γ_1^2 =γ_2^2=γ_3^2 = -1

to account for two spin states (s=1,2) of particles and antiparticles. 
Fermion fields transform under the Lorentz group x'=Λ x as ( (<ref>))

    ψ'(x') = 𝒮_Λψ(x),     ψ'(x')^† = ψ(x) 𝒮_Λ^†.

		It turns out that the 4 × 4 matrix 𝒮_Λ^†≠𝒮_Λ^-1  but 𝒮^-1 = γ_0 𝒮^†γ_0. Due to this, it is convenient to  introduce a Dirac-conjugated spinor ψ̅(x)≡ψ^†γ_0. The latter enters into

    ψ̅'(x')ψ'(x')     = ψ̅(x) ψ(x), Lorentz scalar; 
    ψ̅'(x')γ_μψ'(x')     = Λ_μνψ̅(x)γ_νψ(x),   Lorentz vector.

This allows us to convince ourselves that the Dirac Lagrangian 

    = ψ̅( i ∂̂- m ) ψ

is also a Lorentz scalar, respects Lorentz symmetry.
Dirac-conjugated spinors can be used to impose Lorentz-invariant normalization on u and v:

    u̅_s(p) u_r(p) = 2 m δ_rs, 
    				    v̅_s(p) v_r(p) = - 2 m δ_rs,


An important fact about quantum fermion fields it that, contrary to the case of scalar or vector (boson) fields, the creation/annihilation operators for fermions  a^±_s,p and antifermions b^±_s,p anticommute: 

    [a^-_r,p,a^+_s,p']_+    = 
    				[b^-_r,p,b^+_s,p']_+ = δ_srδ(p - p') 
    				
    [a^±_r,p,a^±_s,p']_+    = 
    				[b^±_r,p,b^±_s,p']_+ = 
    				[a^±_r,p,b^±_s,p']_+ = 0.

Due to this, fermions obey the Pauli principle, a^+_r,p a^+_r,p=0.
Moreover, one can explicitly show that quantization of bosons (integer spin) with anticommutators or fermions (half-integer spin) with commutators leads to inconsistencies (violates the Spin-Statistics theorem).  


Let us continue our discussion of free fermions by emphasizing the difference between the notions of Chirality and Helicity.  Two independent solutions for massive fermions (u_1,2) can be chosen to correspond to two different helicities — projections of spin vector s onto direction of p: 

    ℋ = s·n,   n = p/|p|.
    				       
    
    										Left-Handed     Right-Handed 
    
        < g r a p h i c s >
       
        < g r a p h i c s >

	 In free motion it is conserved 
	 and serves as a good quantum number.  However, it is not a Lorentz-invariant quantity. 
	 Indeed, we can flip the sign of particle momentum by moving with speed faster than v=|p|/p_0. As a consequence, n→ -n and ℋ→ - ℋ.
	 However, helicity for a massless particle is the same for all inertial 
	 observers and coincides with chirality, which is a Lorentz-invariant concept. 

	By definition Left (ψ_L) and Right (ψ_R) chiral spinors are eigenvectors of  
	
    γ_5  = i γ_0 γ_1 γ_2 γ_3  ⇒ [γ_μ,γ_5]_+ = 0,   γ_5^2 = 1,   γ_5^† = γ_5,

	where 
	
    γ_5 ψ_L  = - ψ_L,     γ_5 ψ_R = + ψ_R.

	Any spinor ψ can be decomposed as 

    ψ = ψ_L + ψ_R,     ψ_L/R = P_L/Rψ,    P_L/R = 
    				1 ∓γ_5/2.

			Rewriting the Dirac	Lagrangian it terms of chiral components

    = i (ψ̅_L ∂̂ψ_L
    				+   ψ̅_R ∂̂ψ_R_conserve chirality) 
    				-  m (ψ̅_L ψ_R +  ψ̅_R ψ_L_break chirality),

	we see that, indeed, it is the mass term that mixes two chiralities. Due to this, 
	it violates chiral symmetry corresponding to the independent rotation of left and right components
	
    ψ→ e^i γ_5 αψ.

	Consequently, if we drop the mass term, the symmetry of the Lagrangian is enhanced. 
	
	Up to now we were discussing the so-called Dirac mass term. For neutral fermions (neutrino) 
there is another possibility — a Majorana mass. 
Since charge-conjugation applied to fermion fields,  ψ→ψ^c,  flips chirality, we can use ψ_L^c in place of ψ_R to write 

    = 1/2 (i ψ̅_L ∂̂ψ_L - m ψ̅_L ψ_L^c).

As a consequence, to describe Majorana particles, we need only two components instead of four  since antiparticles coincide with particles in this case. At the moment, the nature of neutrinos is unclear and we refer to  <cit.> for more details. 



		




	

§ FROM FREE TO INTERACTING FIELDS



		 Let us summarize  what we have learned so far. If we have a Lagrangian  at hand, we can 
          
                  
  * Derive EOM (via the Action Principle);
									
  * Find the Symmetries of the Action = ∫ d^4 x;
                  
  * Find Conserved quantities (via the Noether Theorem).
           
 However, we usually 
                start building our models by postulating symmetries. 
								Indeed, we assume that a general QFT Lagrangian  is 
          
                  
  * a Lorentz (Poincare) invariant (a sum of Lorentz scalars),
                  
  * Local (involves a finite number of partial derivatives),
                  
  * Real (hermitian) (respects unitarity=conservation of probability)
          
					In addition, one can impose other symmetries and get further restrictions on the model. Having all this in mind, we can proceed further and discuss particle interactions.	



				In HEP, a typical collision/scattering experiment deals with “free”  initial and final states and considers transitions between these states.  To account for this in a quantum theory, one introduces the S-matrix with matrix elements 
			
    ℳ = ⟨β| S |α⟩,     ℳ = δ_αβ + (2π)^4 δ^4(p_α - p_β) i M_αβ

			giving amplitudes
				for possible transitions between in |α⟩ and out |β⟩ states: 

    |α⟩ = 
    				ã^+_p_1... ã^+_p_r|0⟩, 
    				  |β⟩ = 
    				ã^+_k_1... ã^+_k_s|0⟩, 
    				  ã^+_p = (2π)^3/2√(2 ω_p) a^+_p,

where for convenience[The states created by ã^+ are normalized in the relativistic-invariant way.] 
 (see also (<ref>)) we rescale our creation/annihilation operators.
Given the matrix element M_αβ, one can calculate the differential probability (per unit volume per unit time) to evolve from |α⟩ to |β⟩:

    d w  =   n_1...n_r/(2 ω_p_1)...
    				(2 ω_p_r)
    				|M_αβ|^2 dΦ_s,

where n_i correspond to initial-state particle densities, and an element of phase space is given by 

    d Φ_s  = 
    				(2π)^4 δ^4(p_in - k_out)
    				d k_1/(2π)^3 (2ω_k_1)
    				...	
    				d k_i/(2π)^3 (2ω_k_i)

with p_in = ∑ p_i and k_out = ∑ k_i.  Since we are usually interested in processes involving one or two particles in the initial state, it is more convenient 
	to consider the differential decay width dΓ in the rest frame of a particle with mass m, or cross-section dσ of a process 2→ s: 

    d Γ    =  Φ_Γ	|M_1→ s|^2 dΦ_s,     Φ_Γ = 1/2 m,  
    
    				d σ    =  Φ_σ	|M|^2 dΦ_s,     Φ_σ = 1/4 √((p_1 p_2)^2 - p_1^2 p_2^2).

In (<ref>) the factor Φ_σ is Lorentz-invariant and is expressed in terms of four-momenta of initial particles p_1 and p_2.  The total width Γ  and total cross-section σ can be obtained by integration over the
momenta of final particles restricted by energy-momentum conservation due to the four-dimensional δ-function in (<ref>).
				
In QFT, the S-matrix is given by the time-ordered exponent
				
    S = T e^-i∫ d^4x _I(x) = T e^i ∫ d^4x _I(x).

				involving the interaction Hamiltonian _I (Lagrangian _I).  

				The interaction Lagrangian _I = _full - _0 is a sum of Lorentz-invariant terms having more than two fields and more ∂_μ than in the quadratic part _0, which corresponds to free particles.
				It is worth noting that in (<ref>) we treat _I (_I)  as an operator built from free[More precisely, operators in the interaction picture.] quantum fields (certain combinations of a^± and b^±). 

				The time-ordering operation, which was used to define particle propagators, is generalized in (<ref>) to account for more than two fields originating from _I 
				
    TΦ_1 (x_1) ... Φ_n(x_n) = (-1)^kΦ_i_1(x_i_1)
    								...
    								Φ_i_n(x_i_n),      x^0_i_1 > ...> x^0_i_n.

				Here the factor (-1)^k appears due to k possible permutations of fermion fields.

				As it was mentioned earlier, (interaction) Lagrangians should be hermitian. Any scalar combination of quantum fields can, in principle, be included in _I, 
    _I :      gϕ^3(x),     λϕ^4(x),     yψ̅(x) ψ(x) ϕ(x) 
         ψ̅(x) γ_μψ(x)  A_μ(x),     G[ (ψ̅_1 γ_μψ_2)   (ψ̅_3 γ_μψ_4) + ]

The parameters (couplings) g, λ, ,  y, and G set the strength of the interactions. 
An important characteristic of any coupling in the QFT model is its dimension, which can be deduced from the fact that Lagrangian
 has dimension [] = 4.
 One can notice that all the couplings (hidden) in the T-shirt Lagrangian are dimensionless.  This fact has crucial consequences for the self-consistency of the SM model. 
				





 §.§ Perturbation theory



In an interacting theory it is very hard, if not impossible, to calculate the S-matrix (<ref>) exactly. Usually, we assume that the couplings in _I are small allowing us to treat the terms in _I as perturbations to _0. 
As a consequence, we expand the T-exponent and restrict ourselves to a finite number of terms. In the simplest case of _I = -λϕ^4/4! we have at the nth order
				
    i^n/n![λ/4!]^n
    								⟨0|ã^-_k_1...ã^-_k_s∫ d x_1 ... d x_n
    								T [
    									ϕ(x_1)^4
    									...
    									ϕ(x_n)^4
    								]
    								ã^+_p_1...ã^+_p_r|0⟩.


				To proceed, one uses the Wick theorem: 
				
    T Φ_1...Φ_n  = ∑ (-1)^σ⟨0| T (Φ_i_1Φ_i_2) |0⟩
    								...
    								⟨0| T (Φ_i_m-1Φ_i_m) |0⟩Φ_i_m+1...Φ_i_n,

				where the sum goes over all possible ways to pair the fields. The Wick theorem (<ref>) expresses time-ordered products of fields in terms of normal-ordered
				ones and propagators.  				The normal-ordered operation puts all annihilation operators originating from different Φs to the right. It also cares about fermions, 
    a^-_1 a^+_2 a^-_3 a^-_4 a^+_5 a^-_6 = (-1)^σ
    				a^+_2 a^+_5 a^-_1 a^-_3 a^-_4 a^-_6,

where σ  correspond to the number of fermion permutations ( (<ref>)).  In Fig. <ref> a cartoon, which illustrates (<ref>) for one of the contributions to T[_I(x) _I(y)], is provided. 

	After application of the Wick theorem we have to calculate  
				
    ⟨0|ã^-_k_1...ã^-_k_sΦ_i_m+1...Φ_i_nã^+_p_1...ã^+_p_r|0⟩.

				To get a non-zero matrix element, all a^-(a^+) in the normal product of fields from the Lagrangian have to be “killed” by (commuted with) a^+(a^-) from the initial (final) states. 


				
				For our generalized field (<ref>) we have

    [Φ^i_α(x),(a^+_p)^i_s]     = 
    				e^-i p x/(2π)^3/2√(2 ω_p)_common to all fieldsu^s_α(p),     initial state polarization (particle); 
    [(b^-_p)^i_s,Φ^i_α(x)]     = 
    				
    								e^+i p x/(2π)^3/2√(2 ω_p)
    								v^*s_α(p),    final state polarization (antiparticle).

and one clearly sees that the factors in the denominators (<ref>) are avoided  when the rescaled ã^± (or b̃^±) operators (<ref>) are used. 

All this machinery can be implemented in a set of Feynman rules, which 
are used to draw (and evaluate) Feynman diagrams. Every Feynman diagram involves interaction vertices, external and internal lines. Internal lines connect two vertices and correspond to propagators. The expression for propagators can be derived from _0, 
    .
    																⟨0| T(ϕ(x) ϕ^†(y) |0⟩
    ⟨0| T(ψ(x) ψ̅(y) |0⟩
    ⟨0| T(W_μ(x) W^†_ν(y) |0⟩}    = 
    																∫d^4 p/(2π)^4i e^-i p (x-y)/p^2 - m^2 + i ϵ{
    																				1     
        < g r a p h i c s >
       ϕ;
    																							
    p̂ + m
    																				   
        < g r a p h i c s >
       ψ;
    																				
     
    																				- g_μν + p_μ p_ν / m^2 
    																				   
        < g r a p h i c s >
        W_μ.
    																.

												One can notice that all the dependence on x_i of the integrand in (<ref>) comes from 
												either (<ref>) or (<ref>). As a consequence, it is possible to carry out the integration for every x_i

    ∫ d^4 x_i e^-i x_i (p_1 + ... + p_n) = (2π)^4 δ^4(p_1 + ... + p_n)

											and obtain a δ-function reflecting energy-momentum conservation at the corresponding vertex.
		

												
											Depending on the direction of momenta, the external lines represent incoming or outgoing particles (see Table <ref>). Again, the corresponding factors (=polarization vectors) are derived from _0. 
	Notice that we explicitly write the Lorentz indices for vector particles and suppress the Dirac indices for fermions. 
	To keep track of the index contractions in the latter case, one uses arrows on the fermion lines.[There are subtleties when interactions involve Majorana fermions.] 

	Let us turn to interaction vertices. The corresponding Feynman rules can be derived from _I = ∫ d^4 _I. 
	It is convenient to do this by carrying out a Fourier transform to “convert” coordinate derivatives to momenta and considering variations of the action. In the case of   _I = - λϕ^4/4! we have (all momenta are assumed to be incoming)

    .	iδ^4 _I[ϕ]/δϕ(p_1) 
    																				δϕ(p_2)
    																				δϕ(p_3)
    																				δϕ(p_4)
    																|_ϕ=0⇒(2π)^4 δ^4(p_1 + p_2 + p_3 + p_4)_conservation of energy-momentum×[-i λ].

												In a typical diagram all (2π)^4 δ(...) factors (but one[we factor it out in the definition of M_αβ, see (<ref>).]) reflecting the energy-momentum conservation at each vertex, are removed by the momentum integration originating from propagators, (<ref>). Due to this, we also omit these factors (see, Table <ref> for examples). 
												
												Given Feynman rules, we can draw all possible diagrams that contribute to a process and evaluate the amplitude. We do not provide the precise prescription here (see textbooks <cit.>  for details)  but just mention the fact that one should keep in mind various symmetry factors and relative  signs that can appear in real calculations.  
											
In order to get probabilities, we have to square matrix elements, 
    |M|^2 = M M^†⇒-0.45
        < g r a p h i c s >

Sometimes we do not care about polarization states of initial or final particles, so we have to sum over final polarization and  average over initial ones. 
That is where spin-sum formulas,
    ∑_s u_s(p_1) u̅_s(p_1) = p̂_1 + m,     ∑_s 
    				       v_s(p_2) v̅_s(p_2) = p̂_2 - m

		 become handy 
			
    M M^†→∑_s,r (u̅_s A v_r) (v̅_r A^†u_s)  =
    							Tr[ (p̂_1 + m) A (p̂_2 - m) A^†].

		As a consequence, one can utilize the well-known machinery for gamma-matrix traces to evaluate probabilities  in an efficient way.


			Let us continue by mentioning that only in tree graphs, such as
												

												
																-0.5 
    < g r a p h i c s >
 
    ⇒     
																				(2π)^4 δ^4
																				(∑_i=1^3 p_i-∑_i=4^6 p_i)
																				[-i λ]^2 
																				  i /q^2 - m^2 + iϵ,



				all the integrations (due to propagators) are “killed” by vertex δ-functions. 
				However, nothing forbids us from forming loops. In this case, we have integrals over unconstrained momenta, in the ϕ^4-theory
			
    -0.5
        < g r a p h i c s >
     :     
    							I_2(k) ≡∫d^4 q/[q^2 + i ϵ] [(k-q)^2 + iϵ]∼∫^∞ |q|^3 d|q|/|q|^4∼ln∞,

			which can lead to divergent (meaningless?) results. This is again a manifestation of UV divergences due to large momenta (“small distances”).  
			
			A natural question arises: Do we have to abandon QFT? Since we still use it, there are reasons not to do this. Indeed, we actually do not know physics up to infinitely small scales and our extrapolation can not be adequate in this case. 
			To make sense of the integrals, we can regularize them, introduce a “cut-off” |q|<Λ,
				
    I_2^Λ(k)  = iπ^2 [ 
    												lnΛ^2/k^2  + 1
    								] + 𝒪(k^2/Λ^2)
    								= i π^2 [ 
    												lnΛ^2/μ^2
    												- lnk^2/μ^2   + 1
    								] + 𝒪(k^2/Λ^2)

				or use another convenient possibility — dimensional regularization, when  d=4 space-time is formally continued to d=4-2ε dimensions:
				
    I_2^4-2ε(k)  = μ^2ε∫ d^4-2ε q/q^2 (k-q)^2	
    								= i π^2 ( 1/ε - lnk^2/μ^2   + 2 )
    				+ 𝒪(ε).

			 
				Both the regularized integrals are now convergent[We do not discuss the issue of possible IR divergences here.] and share the same logarithmic dependence on external momentum k. 
				One can also notice a (one-to-one) correspondence between a logarithmically divergent contribution logΛ^2/μ^2 in (<ref>) and the pole term 1/ε in (<ref>). However, the constant terms are different. How do we make sense of this ambiguity?

				The crucial observation here is that the divergent pieces, which blow up when we try to remove the regulators (Λ→∞ or ε→ 0), are local, depend polynomially on external kinematical parameters. 
				This fact allows us to cancel them by the so-called counterterm (CT) vertices. The latter can be interpreted as new terms in _I. Moreover, in a renormalizable QFT model additional (divergent) contributions have the same form as the initial Lagrangian  and thus  can be “absorbed'	into redefinition of fields and parameters. 

				One can revert the reasoning and assume that the initial Lagrangian is written in terms of the so-called bare (unobservable) quantities. 				The predictions of the model are finite since the explicit dependence of Feynman integrals on the cut-off Λ (or ε) is actually compensated by the implicit dependence of bare fields and parameters. 
				In some sense these quantities represent our ignorance of dynamics at tiny scales. Physical fields and parameters are always “dressed” by clouds of virtual particles.  

				It is obvious that working with bare quantities is not very convenient. One usually makes the dependence on
				Λ (or ε) explicit by introduction of divergent Z-factors
				for bare fields (ϕ_B),  masses  (m_B^2), and couplings (λ_B), 
    _full    = 1/2 (∂ϕ_B)^2 - m_B^2/2ϕ_B^2  + λ_B ϕ_B^4/4! = Z_2/2 (∂ϕ)^2 - Z_m m^2/2 Z_2 ϕ^2 + Z_λλ/4! (Z_2 ϕ^2)^2
        = (∂ϕ)^2/2  - m^2 ϕ^2/2 + λϕ^4/4! 	
    				+ 
    				(Z_2-1)/2 (∂ϕ)^2 - (Z_m Z_2 - 1) m^2/2ϕ^2 + (Z_4 Z_2^2 - 1) λϕ^4/4!_counterterms.

Here ϕ, m and λ denote renormalized (finite) quantities.
				Since we can always subtract something finite from infinity, there is a certain freedom[Different constant terms in (<ref>) and (<ref>) are one manifestation of this fact.] in this procedure.
				So we have to impose additional conditions on Zs, define a renormalization scheme.
				For example, in the minimal (MS) schemes  
				we subtract only the divergent terms, only poles in ε, 
				
				while in the so-called momentum-subtraction (MOM) schemes we require amplitudes (more generally Green functions) 
				to have a certain value at some fixed kinematics.



				As an illustration, let us consider a scattering amplitude 2→2 in the ϕ^4 model calculated in perturbation theory: 

    -0.46
        < g r a p h i c s >
        = 
    				-0.46
        < g r a p h i c s >
    
    				+ 
    				-0.46
        < g r a p h i c s >
    
    				+ permutations
    				+ more loops
        = λ_B(Λ)  
    				- λ_B(Λ)^2/2 (16 π^2)( lnΛ^2/μ^2  - lnk^2/μ^2 + …) 
    				+ …
        =
    				[ λ(μ) + 3/2λ^2(μ)/16 π^2lnΛ^2/μ^2]
    				- λ(μ)^2/2 (16 π^2)( lnΛ^2/μ^2  - lnk^2/μ^2 + …) 
    				+ …
        = 
    				λ(μ)   
    				+ λ(μ)^2/2 (16 π^2)( lnk^2/μ^2 + …) 
    				+ ….

In (<ref>) the tree-level and one-loop diagrams contributing to the matrix element are shown. The corresponding expression in terms of the bare coupling λ_B(Λ) that implicitly depends on the regularization parameter Λ is given in (<ref>). We introduce a renormalized[We use minimal subtractions here and the factor of three comes from the fact that all three one-loop graphs (s, t and u) give rise to the same divergent term.] coupling λ(μ) in (<ref>) to make the dependence explicit: 

    λ_B(Λ) = λ(μ) Z_λ = λ(μ) (1 + 3/2λ(μ)/16π^2lnΛ^2/μ^2 + ...).

The final result (<ref>) is finite (when Λ→∞) and can be confronted with experiment. It seems to depend on an auxiliary scale μ, which inevitably appears in any renormalization scheme. 
The crucial point here is that observables (if all orders of PT are taken into account) actually do not depend on μ. 
Changing μ corresponds to a certain reshuffling of the PT series: some terms from loop corrections are absorbed into the rescaled (running) couplings. 
This allows one to improve  the “convergence”[Actually, the PT series are asymptotic (divergent) and we speak about the behavior of a limited number of first terms here.] of  the series.  


The scale-dependence of the running couplings is governed by renormalization-group equations (RGE). In the considered case we have

    λ(μ_0) →λ(μ),   d/d lnμλ = β_λ(λ),   β_λ = 3/2λ^2 /16π^2+ ...

	The beta-function β_λ can be calculated order-by-order in PT. However,
	the (initial) value λ(μ_0) needed to solve (<ref>) is not predicted and has to be extracted from experiment. 



	It is worth pointing out here that two different numerical values of the renormalized self-coupling, λ_1 and λ_2, 	do not necessarily correspond to different Physics.
	Indeed, if they are fitted from measurements at different scales, μ_0 and μ,
	and are related by means of RGE, they represent the same Physics 
	(see Fig. <ref>).	
	A prominent example is the running of the strong coupling in Quantum chromodynamics (QCD) described by (see <cit.>) 

    β_α_s = -α^2_s/4π(11 - 2/3 n_f) +
    								... + 𝒪(α^7_s),    n_f - number of flavours.

In Fig. <ref> one can see a remarkable consistency between different measurements of α_s(μ) and 
the scale dependence predicted by perturbative QCD.






 §.§ Renormalizable or non-renormalizable?


Let us stress again that the model is called renormalizable if all the divergences that appear in loop integrals
can be canceled by local counterterms due to renormalization of bare parameters and couplings from _full. 
But what happens if there is a divergent amplitude but the structure of the required subtraction does not have  a counter-part in our initial Lagrangian, we do not have a coupling to absorb the infinity? Obviously, we can modify _full and add the required term (and the coupling). 

An example of such a situation can be found in the model with a scalar ϕ (Higgs) coupled to a fermion ψ (top quark) via the Yukawa interaction  characterized by the coupling y

    _I ∋δ_Y = -y ·ψ̅ψϕ.

Let us assume for the moment that we set the self-coupling to zero λ=0 and want to calculate the Higgs-scattering amplitude due to top quarks (see, Fig. <ref>). We immediately realize that the contribution is divergent and without δ_4 = - λϕ^4/4! we are not able to cancel it. Due to this, we are forced to consider the ϕ^4 term in a consistent theory. 



Since we modified _full, we have to re-calculate all the amplitudes. In principle, new terms in _I will generate new diagrams, which can require new interactions to be added to _I.  Will this process terminate? 
In the case of renormalizable models the answer is positive.  We just need to make sure that _I include all possible terms with dimensionless couplings[Remember the T-shirt Lagrangian?], or, equivalently, local dimension-4 operators built from quantum fields and their derivatives.

On the contrary, if one has to add more and more terms to _I, this is a signal of a  non-renormalizable model. 
It looks like that we have to abandon such models since we need to measure an infinite number of couplings to predict something in this situation! However, it should be stressed that non-renormalizable models, contrary to renormalizable ones, involve couplings G_i with negative mass dimension [G_i]<0! Due to this, not all of them are important at low energies, such as

    G_i E^-[G_i]≪ 1.

This explains the success of the Fermi model involving the dimension-6 four-fermion operator  

    - _I =G Ψ̅_p γ_ρΨ_n ·Ψ̅_e γ_ρΨ_ν +

in the description of the β-decay n → p + e^-  + ν̅_e. 
Since the model turns  out to be a harbinger of the modern electroweak theory, let us consider it in more detail and discuss its features, which eventually lead to the construction of the SM.  

In 1957 R. Marshak and G.Sudarshan, R. Feynman and M. Gell-Mann modified the original Fermi theory of beta-decay to incorporate 100 % violation of Parity discovered by C.S. Wu in 1956:

    - _Fermi  = G_F/2 √(2) (J^+_μ J^-_μ + ).

Here the current  

    J^-_ρ  = (V-A)_ρ^nucleons 
    + Ψ_e γ_ρ(1-γ_5)Ψ_ν_e
    + Ψ_μγ_ρ(1-γ_5) Ψ_ν_μ
    							+ ...

is the difference between Vector (V) and Axial (A) parts.  This kind of current-current interactions 
		 can describe not only the proton beta-decay  but also the muon decay μ→ e ν_μν̅_e or the process of ν_e e - scattering.
		 Since the Fermi constant G_F≃ 10^-5 GeV^-1, from simple dimensional grounds we have

    σ(ν_e e →ν_e e) ∝ G_F^2 s,      s=(p_e + p_ν)^2.

With such a dependence on energy we eventually violate unitarity. This is another manifestation of the fact that non-renormalizable interactions are not self-consistent. 

However, a modern view on the Fermi model treats it as an effective field theory <cit.> with 
	certain limits of applicability. It perfectly describes low-energy experiments and one can fit the value of G_F very precisely (see <cit.>). 

The magnitude of G_F tells us something about a more fundamental theory (the SM in our case): around G_F^-1/2∼ 10^2-10^3 GeV there should be some “New Physics” (NP) to cure the above-mentioned shortcomings. Indeed, by analogy with (renormalizable) QED we can introduce mediators of the weak interactions – electrically charged vector fields W_μ^± (see, Fig. <ref>): 

    _Fermi  = - G_F/2 √(2) (J^+_μ J^-_μ + ) 
    				→_I  = g/2 √(2) (W^+_μJ^-_μ + )

with a dimensionless coupling g. Since we know that weak interactions are short-range, the W-bosons should be massive. Given _I  we can calculate the tree-level scattering amplitude due to the exchange of W^± between two fermionic currents:

    T = i(2π)^4  g^2/8 J^+_α[g_αβ - p_αp_β/ M_W^2 /p^2 - M_W^2] J^-_β.

In the limit |p|≪ M_W, (<ref>) reproduces the prediction of the effective theory (Fermi model) if we identify (“match”) 

    (effective theory)  G_F/√(2) = g^2/8 M_W^2  (more fundamental theory).

However, one can see that in the UV region (|p|≫ M_W) the amplitude (<ref>) still has bad behavior, leading to all the above-mentioned problems. To deal with the issue, we utilize gauge symmetry, which will be discussed in the next section.
	


§ GAUGE SYMMETRIES



				We are seeking for a model of weak interactions that has good UV-properties.
				Let us revise how the gauge principle is implemented in QED. First of all, 
				consider  

    _0 = ψ̅( i ∂̂- m ) ψ

			and make the global U(1)-symmetry of _0

    ψ→ψ' = e^i e ωψ

local, ω→ω(x). In this case, the Lagrangian ceases to be invariant[Note that one can use this fact to get an expression for the Noether current J_μ.]:  

    δ_0 = ∂_μω· J_μ,      J_μ = - e ψ̅γ_μψ,

To compensate this term, we add the interaction of the current J_μ with the photon field A_μ:

    _0 → = _0 + A_μ J_μ = ψ̅[i (∂̂+ i e Â) - m] ψ,      A_μ→ A'_μ = A_μ-∂_μω.

The photon A_μ is an example of gauge field. To get the full QED Lagrangian, we should also add a kinetic term for the photon: 

    _QED    = ψ̅( i D̂ - m ) ψ - 1/4 F_μν^2 
    			
    
    								D_μ    = ∂_μ + i e A_μ,     
    				F_μν  = ∂_μ A_ν - ∂_ν A_μ.

Here we introduce a covariant derivative D_μ and a field-strength tensor F_μν. One can check that (<ref>) is invariant  under

    ψ   →ψ'  = e^i e ω(x)ψ
    
    				A_μ   → A_μ'  = A_μ - ∂_μω
    
    				D_μψ   → D'_μψ'  = e^i e ω(x) D_μψ.


The second Noether theorem <cit.> states that theories possessing gauge symmetries are redundant, some degrees of freedom are not physical. This makes quantization non-trivial. To deal with this problem in QED, one adds a gauge-fixing term to the free vector-field Lagrangian:

    _0(A)  = -1/4 F_μν^2 - 1/2ξ(∂_μ A_μ)^2≡ - 1/2 A_μ K_μν A_ν.


This term allows one to obtain the photon propagator by inverting[If we omit the gauge-fixing term, we will not be able to invert the quadratic form.]
				K_μν:

    ⟨0| T A_μ (x) A_ν(y) |0⟩  = 
    				∫d^4 p/(2π)^4-i [ g_μν - (1-ξ) p_μ p_ν/p^2]/p^2 + i ϵ
    				e^-i p(x-y)

The propagator now involves an auxiliary parameter ξ.  It controls the propagation of unphysical longitudinal polarization ϵ^L_μ∝ p_μ. The polarization turns out to be  harmless in QED since the corresponding terms drop out 
			of physical quantities, due to current conservation

    e_μ^L J_μ∝ p_μ J_μ =  0    [we have no source for unphysical γ].

			One can see that the propagator has good UV behaviour and falls down as 1/p^2 for large p. 
			The gauge symmetry of QED is U(1). It is Abelian  since the order of two transformations is irrelevant (see Fig. <ref>). 
			However, if we want to apply the gauge principle to the case of EW interactions, we have to generalize U(1) to the  Non-Abelian case. Let us consider 
the SU(n) group, i.e., unitary n × n matrices U_ij depending on n^2-1 parameters ω^a and having U=1:

    ψ_i →ψ'_i = U_ij(ω) ψ_j,    U(ω) = e^i gt^aω^a.

	In general, different transformations do not commute in the non-Abelian case. This fact is reflected in commutation relations for the group generators t^a, which obey the su(n)-algebra: 
	
    [t^a, t^b] = i f^abc t^c,      f^abc -  structure constants .

												For constant ω^a the transformation (<ref>) is a symmetry of  the Lagrangian

    _0 = ψ̅_i ( i∂̂- m ) ψ_i,      i=1,...,n

												describing n free fermions in the fundamental representation of SU(n). 

								In order to make the symmetry local, we introduce a 
												(matrix) covariant derivative depending on n^2-1 gauge fields W^a_μ: 

    (D_μ)_ij = ∂_μδ_ij - i gt_ij^a W^a_μ.

The transformation properties of W^a_μ should guarantee that for space-time dependent ω^a(x) the covariant derivative of ψ transforms in the same way as the field itself:

    D'_μψ' =   U (ω) (D_μψ),    U(ω) = e^i g t^a ω^a.

		One can find that 

    W^a_μ→ W^'a_μ    =  W^a_μ + ∂_μω^a + g f^abc  W^b_μω^c 
    						
        = W^a_μ + (D_μ)^abω_b,      (D_μ)^ab≡∂_μδ^ab - i g( - i f^abc )W_μ^c,

where we introduce the covariant derivative (<ref>) D_μ^ab  with generators (t^c)^ab = -i f^cab in the adjoint representation. The field-strength tensor for each component of W^a_μ is given by the commutator 

    [ D_μ, D_ν]  = - i g t^a ℱ_μν^a,   ℱ_μν^a  = ∂_μ W_ν^a - ∂_μ W_ν^a + g f^abc W_μ^a W_ν^b.

Contrary to the U(1) case, ℱ_μν^a contains an additional term  quadratic in W^a_μ.  Due to this,  the gauge symmetry predicts not only interactions between fermions ψ (or fields in the fundamental representation of the gauge group) and W^a_μ  but also self-interactions of the latter (the gauge fields are “charged” under the group).
	
Combining all the ingredients, we can write down the following Lagrangian for an SU(n) gauge (Yang-Mills) theory : 

    = ψ̅( i D̂ - m ) ψ - 1/4ℱ^a_μνℱ^a_μν 
    				= _0 + _I, 
    _0     =  ψ̅( i ∂̂- m ) - 1/4 F^a_μν F^a_μν,    F_μν^a = ∂_μ W^a_ν - ∂_ν W^a_μ, 
    _I     = g ψ̅_α^iγ^μ_αβ t_ij^aψ^j_β W^a_μ
    				
    				- g/2 f^abc W^b_μ W^c_ν F^a_μν 
    				 - g^2/4 f^abc f^ade W^a_μ W^b_ν W^d_μ W^e_ν.

For illustration purposes we explicitly specify all the indices in the first term of interaction Lagrangian _I:  the Greek ones correspond to Dirac (α, β) and Lorentz (μ) indices, while the Latin ones belong to different representations of SU(n): i,j – fundamental, a – adjoint.  One can also see that the strength of all interactions in _I is governed by the single dimensionless coupling g. 



	To quantize a Yang-Mills theory, we generalize the QED gauge-fixing term and write, 
    _gf = -1/2ξ(F^a)^2,      F^a = ∂_μ W^a_μ
	
		with F^a being a gauge-fixing function. This again introduces unphysical states in 
		the W^a_μ propagator. However,  contrary to the case of QED, the fermionic current J_μ^a = g ψ̅t^a γ_μψ is not conserved and can produce longitudinal W^a_μ. Nevertheless, the structure of vector-boson self-interactions guarantees that at tree level amplitudes involving unphysical polarizations for external W^a_μ vanish (see, Fig. <ref>).




Unfortunately, this is not sufficient to get rid of unphysical states in loops. To deal with the problem in a covariant way, one introduces the so-called Fadeev-Popov ghosts c̅_a and c_a. They are anticommuting “scalars” and precisely cancel the annoying contribution.  The Lagrangian for the fictitious particles is related to the gauge-fixing function F_a(W_μ) = ∂_μ W_μ^a via

    _ghosts    = -c̅^a ∂F_a(W^ω)/∂ω_b c^b
    			= 	
    			-c̅^a ∂_μ D^ab_μ c^b  
        = 
    			-c̅^a ∂^2 c^a - g f^abc (∂_μc̅^a) c^b A_μ^b.

The ghosts are charged under SU(n)  and interact with gauge fields  in the same way as the unphysical modes. However, there is an additional minus sign for the loops involving anticommuting ghosts (see, Fig. <ref>) that leads to the above-mentioned cancellations.  
 




§ GAUGE THEORY OF ELECTROWEAK INTERACTIONS





 §.§ Fermion couplings to gauge bosons


In the SM we use the gauge principle to introduce EW interactions. Indeed, we utilize  

    SU(2)_L⊗ U(1)_Y

gauge group that has four generators or, equivalently, four gauge bosons. Three of them, W_μ,  belong to weak-isospin SU(2)_L, while the photon-like B_μ mediates weak-hypercharge U(1)_Y interactions. The SM fermions are charged under the group (<ref>). To account for the (V-A) pattern only left fermions interact with W_μ and form SU(2)_L doublets: 

    L = [ ν_l; l^- ]_L, 
    								Q = [ q_u; q_d ]_L,      q_u=u,c,t; q_d=d,s,b; l=e,μ,τ .


Since the generators of SU(2) are just the Pauli matrices, we immediately write the following expression for the corresponding covariant derivative

    D^L_μ
    								   = 
    								[ ∂_μ - i/2(g W^3_μ + g' Y^f_L B_μ)                    -i g/√(2)W_μ^+;                    -i g/√(2)W_μ^- ∂_μ + i/2(g W^3_μ - g' Y^f_L B_μ) ].

				The right fermions[In what follows we do not consider right-handed neutrino and refer again to <cit.>.] are SU(2)_L singlets and do not couple to W_μ: 
				
    D^R_μ  = ∂_μ - i g' Y^f_R/2B_μ.

				The covariant derivatives involve two gauge couplings g, g' corresponding to SU(2)_L and U(1)_Y, respectively. Different Y^f_L/R denote weak hypercharges of the fermions and up to now the values are not fixed. Let us put some constraints on Y^f_L/R. The first restriction comes from the SU(2)_L symmetry, Y^u_L = Y^d_L≡ Y^Q_L, and Y^ν_L = Y^e_L ≡ Y^L_l.

				One can see that the EW interaction Lagrangian
				
    _W = _NC + _CC,

				in addition to the charged-current interactions of the form  

    ^l_CC  = g/√(2)ν̅^e_Lγ_μ W^+_μ e_L +  = g/2 √(2)ν̅_e 
    				γ_μ W_μ^+ (1 - γ_5) e +

also involves 
neutral-current interactions 

    ^l_NC    = 
    				ν̅_L^e γ_μ( 
    				1/2 g W^3_μ + Y^l_L/2 g' B_μ) ν^e_L 
    				
    				+
    				e̅_L γ_μ( 
    				-1/2 g W^3_μ + Y^l_L/2 g' B_μ) e_L 
    				
    				+ g' e̅_R γ_μY^e_R/2 B_μ e_R.

It is obvious that we have to account for QED in the SM and should predict a photon field that couples to fermions with the correct values of the electric charges. Since both W^3_μ and B_μ are electrically neutral, they can mix

    W_μ^3     = -Z_μcosθ_W  + A_μsinθ_W
    
    				B_μ    = -Z_μsinθ_W  + A_μcosθ_W.

Here we introduce the Weinberg angle θ_W. One can try to fix sinθ_W and 
various Y^f_L/R from the requirement that, A_μ has the same interactions as the photon in QED. Indeed, given fermion electric charges Q_f (see Fig. <ref>) in the units of the elementary charge , one can derive the following relations: 

    g sinθ_W    =  (Q_ν - Q_e) =  (Q_u - Q_d),  
    
    				g' Y^l_L cosθ_W    =  (Q_ν + Q_e)  = - , 
    
    				g' Y^Q_L cosθ_W    = (Q_u + Q_d)  = 1/3, 
     
    				g' Y^f_R cosθ_W    = 2  Q_f,      f=e, u, d.

		As a consequence, = g sinθ_W and, e = 3 g' Y_L^Q cosθ_W, so that 	

    Y_L^l  = - 3 Y_L^Q,    
    				Y_R^e  = - 6 Y_L^Q,   
    				Y_R^u  =   4 Y_L^Q,   
    				Y_R^d  = - 2  Y_L^Q

are fixed in terms of one (arbitrary chosen) Y_L^Q. It is convenient to normalize the U(1)_Y coupling g' so that e=g' cosθ_W, so Y_L^Q = 1/3. 
As a consequence, the photon field couples to the electric charge Q_f of a fermion f. The latter is related to the weak hypercharge 
and the third component of weak isospin T^f_3 via the Gell-Mann–Nishijima formula:

    _NC   ∋f̅[ ( g T^f_3 sinθ_W + g' Y^L_f/2cosθ_W )  P_L +
    							 ( g' Y^R_f/2cosθ_W )  P_R
    			 ]
    								γ_μ f
    				A_μ
        = e f̅( T_3 + Y/2) γ_μ f A_μ  = e Q_f f̅γ_μ f A_μ,

where in (<ref>) we assume that T_3 and Y are operators, which give T_3^f and Y_L^f, when 
acting on left components, and T_3 = 0 and  Y_R^f=2 Q_f for right fermions.

The relations (<ref>) allow one to rewrite the neutral-current Lagrangian as 

    _NC  =  J^A_μ A^μ   + g/cosθ_W J_μ^Z Z_μ,

where the photon A_μ and a new Z-boson couple to the currents of the form

    J^A_μ    =  ∑_f Q_f f̅γ_μ f,      
    								J^Z_μ  = 1/4∑_f f̅γ_μ( v_f - a_fγ_5) f, 
    v_f     = 2 T_3^f - 4 Q_f sin^2θ_W,     a_f = 2 T_3^f,

where T_3^f = ±1/2 for left up-type/down-type fermions. For example, in the case 
of u-quarks, Q_u = 2/3, T_3^u = 1/2, so

    v_u = 1 - 8/3sin^2 θ_W,      a_u = 1.

For completeness, let us give the expression for  the charged-current interactions in the EW model

    _CC  =  g/√(2)(J_μ^+ W^+μ + J^-_μ W^-μ),   
    				J^+_μ  = 1/2∑_f  f̅_u γ_μ(1 - γ_5) f_d,

where f_u(f_d) is the up-type (down-type) component of an SU(2)_L doublet f. 
The corresponding interaction vertices are given in Fig. <ref>. It is worth emphasizing that in the SM the couplings between fermions and gauge bosons exhibit Universality.


It turns out that it was a prediction of the electroweak SM that there should be an additional neutral gauge boson Z_μ.  Contrary to the photon, the Z-boson also interacts with neutrinos.  This crucial property was used in the experiment called Gargamelle at CERN, where in 1973 the discovery was presented (Fig. <ref>). 
About ten years later both W and Z were directly produced at Super Proton Synchrotron (SPS) at CERN. Finally, in the early 90s a comprehensive analysis 
of the e^+ e^- → f f̅ process, which  was  carried out at the Large Electron Proton (LEP) Collier (CERN) and at the Standford Linear Collider (SLAC) confirmed the SM predictions for the Z couplings to fermions (<ref>).



It is also worth mentioning the fact that the (hyper)-charge assignment (<ref>)
satisfies very non-trivial constraints related to cancellation of gauge anomalies.  Anomalies correspond to situations when a symmetry of the classical Lagrangian is violated at the quantum level. 
A well-known example is Axial or Chiral or Adler–Bell–Jackiw(ABJ) anomaly when the classical conservation 
law for the axial current J_μ^A is modified due to quantum effects:

    J_μ^A     = Ψ̅γ_μγ_5 Ψ,
    				    ∂_μ J^A_μ  = 2 i m Ψγ_5 Ψ  + α/2π F_μνF̃_μν_anomaly ,     F̃_μν = 1/2 ϵ_μνρσ F_ρσ.

			The FF̃-term appears due to loop diagrams
			presented in Fig. <ref>. 
			
			
	

There is nothing wrong when the anomalous current J_μ^A corresponds 
to a global symmetry and does not enter into .  
It just implies that 
a classically forbidden processes may actually occur in the quantum theory.  For example, it is the anomaly in the global axial flavour symmetry that is responsible for the decay π→γγ. 
On the contrary, if an axial current couples to a gauge field, anomalies break gauge invariance, 
thus  rendering the corresponding QFT inconsistent. 
In the SM left and right fermions (eigenvectors of γ_5) have different SU(2)_L × U(1)_Y quantum numbers, leaving space for potential anomalies.  
However, since we have to take into account all fermions which  couple to a gauge field, there is a possibility that contributions from different species cancel each other due to a special assignment of fermion charges.  Indeed, in the case of chiral[that distinguish left and right fermions] theories,  anomalies are proportional to (γ_5 = P_R - P_L) 
	
    Anom∝Tr [t^a,{t^b,t^c}]_L - Tr[t^a,{t^b,t^c}]_R,

	where t^a are generators of the considered symmetries and the traces are over left (L) or right (R) fields. In the SM the requirement that all anomalies should be zero imposes the following
	conditions on fermion hypercharges:
	
	
	
    0      =    2 Y^Q_L  - Y_R^u - Y_R^d , 
    										     U(1)_Y - SU(3)_c - SU(3)_c,  
    
    									0     =  N_c Y^Q_L + Y^l_L,  
    						 		        U(1)_Y - SU(2)_L - SU(2)_L,  
    
    									0     =  N_c [2 (Y^Q_L)^3   - (Y_R^u)^3 - (Y_R^d)^3 ]
    					+  [ 2 (Y^l_L)^3 - (Y_R^e)^3], 
    										    U(1)_Y - U(1)_Y - U(1)_Y, 
    
    									0     =  N_c [2 Y^Q_L   - Y_R^u - Y_R^d ]
    					      +  [ 2 Y^l_L - Y_R^e], 
    										    U(1)_Y - grav. - grav. ,

	
	
	where, in addition to the EW gauge group, we also consider strong interactions of quarks 
	that have N_c=3 colours[In the SM coloured quarks belong to the fundamental representation of the corresponding gauge group SU(3)_c.]. 
	While the first three conditions come
	from the SM interactions, the last one (<ref>) is due to the coupling to gravity. Other anomalies are trivially zero. One can see that the hypercharges introduced in (<ref>) do satisfy the equations. It is interesting to note that 
	contributions due to colour quarks miraculously cancel those of leptons and the cancellation works within a single generation. This put a rather strong restriction on possible new fermions that can couple to the SM gauge bosons: new particles should appear in a complete generation (quarks + leptons) in order not to spoil anomaly cancellation within the SM. Moreover, the anomaly cancellation condition can select viable models that go beyond the SM (BSM). 



 §.§ Properties of the EW gauge bosons


Due to the non-Abelian nature of the SU(2)_L group, the gauge fields W_i
have  triple and quartic self-interactions (see (<ref>)). Since W_3 
is a linear combination of the Z-boson and photon, the same is true for Z and γ.
In Fig. <ref>, self-interaction vertices for the EW gauge bosons are depicted.


The triple vertices WWγ and WWZ predicted by the SM were tested at LEP2 in the e^+ e^- → W^+ W^- process (Fig. <ref>) and agreement with the SM predictions was found. Subsequent studies at hadron colliders  (Tevatron and LHC) aimed at both quartic and triple gauge couplings (QGC and TGC, respectively) also show consistency with the SM and put limits on possible deviations (so-called anomalous TGC and QGC). 
	


Since we do not observe Z-bosons flying around like photons, Z_μ should have a non-zero mass M_Z and similar to W^± give rise to Fermi-like interactions between neutral currents J^μ_Z at low energies. 
The relative strength of the charged and neutral current-current interactions 
(J^Z_μ J^μ_Z)/(J^+μ J^+_μ) can be measured by the parameter ρ:

    ρ≡M_W^2/M_Z^2 cos^2θ_W.

Up to now, we do not specify any relations between M_Z and M_W. Due to this, the value of ρ can, in principle, be arbitrary. 
However, it is a prediction of the full SM that ρ≃ 1 (see below). 



The fact that both W and Z should be massive poses 
a serious problem for theoretical description of the EW interactions. 
First of all, the naive introduction of the corresponding mass terms 
breaks the gauge symmetry (<ref>). 
For example, m_W^2 W^+_μ W^-_μ is forbidden due to W_μ→ W_μ 
+ ∂_μω + .... 
One can also mention an issue with unitarity, which arises in the scattering of longitudinal EW bosons due to gauge self-interactions in Fig. <ref>. 

In addition, the symmetry also forbids explicit mass terms for fermions, since 
m_μ (μ̅_L μ_R + ), which accounts for muon mass, 
mixes left and right fields that transform differently under the electroweak group (<ref>). 
In the next section, we discuss how these problems can be solved 
by coupling the SM fermions and gauge bosons to the scalar (Higgs) sector (see also <cit.>). 



 §.§ Spontaneous symmetry breaking and gauge-boson masses

We need to  generate masses for W_μ^± and Z_μ (but not for A_μ) without explicit breaking of the gauge symmetry. Let us consider for simplicity scalar 
electrodynamics:

    = ∂_μϕ^†∂_μϕ 
    				- V(ϕ^†ϕ) 
    				- 1/4 F_μν^2 
    				+ i (ϕ^†∂_μϕ - ϕ∂_μϕ^†) A_μ + ^2 A_μ A_μϕ^†ϕ≡_1,

which is invariant under U(1)

    ϕ→ e^i ω(x)ϕ,    A_μ→ A_μ + ∂_μω.

In (<ref>) a complex scalar ϕ interacts with the photon A_μ. 
We can use polar coordinates to rewrite the Lagrangian in terms of new variables

    = 1/2 (∂_μρ)^2 + ^2 ρ^2/2( A_μ - 1/∂_μθ)
    				( A_μ - 1/∂_μθ) - V(ρ^2/2) - 1/4 F_μν^2, 
        = 1/2 (∂_μρ)^2 + e^2 ρ^2/2 B_μ B_μ 				- V(ρ^2/2) - 1/4 F_μν^2(B),

where ρ is gauge invariant, while the U(1) transformation (<ref>) gives rise to  a shift in θ:
		
    ϕ = 1/√(2)ρ(x) e^i θ(x),
    						  ρ→ρ ,   θ→θ + e ω.

One can also notice that B_μ≡ A_μ - 1/∂_μθ is 
also invariant! Moreover, since F_μν(A) = F_μν(B), we can completely get rid of θ. As a consequence, the gauge symmetry becomes “hidden” when the system is described by the variables B_μ(x) and ρ(x).  

If in (<ref>) we replace our dynamical field ρ(x) by a constant ρ→ v=const, we get the mass term for B_μ. This can be achieved by considering 
the potential V(ϕ) of the form (written in terms of initial variables)

    V  = μ^2ϕ^†ϕ + λ (ϕ^†ϕ)^2.

	One can distinguish two different situations (see Fig. <ref>):
	
					
  * μ^2>0 — a single minimum with ϕ=0; 
					
  * μ^2<0 — a valley of degenerate minima with ϕ≠ 0.
	

In both cases we solve EOM for the homogeneous (in space and time) field. When μ^2>0 the 
solution is unique and symmetric, it does not transform under U(1). In the second case, in which we are interested here, the potential has non-trivial minima 

    .∂ V/∂ϕ^†|_ϕ=ϕ_0 = 0 
    								⇒ϕ_0^†ϕ_0 = -μ^2/2 λ  = v^2/2 > 0
    				⇒ϕ_0 = v/√(2) e^i β,

which are related by global U(1) transformations (<ref>) that change β→β + e ω. 
So, in spite of the fact that we do not break the symmetry explicitly,
it is spontaneously broken (SSB) due to a particular choice of our solution (β). 

In QFT we interpret ϕ_0 as a characteristic of our vacuum state, as a vacuum expectation value (vev) or condensate of the quantum field
:

    ϕ_0 = ⟨0|ϕ(x) |0⟩β=0=v/√(2).

				Since we want to introduce particles as excitations above the vacuum, we have to shift the field:

    ϕ(x) = v + h(x)/√(2) e^i ζ(x)/v,     ⟨0| h(x) |0⟩ = 0,   ⟨0|ζ(x) |0⟩ = 0.

As a consequence, (<ref>) can be rewritten as

    = 1/2 (∂_μ h)^2 + ^2 v^2/2( 1 + h/v)^2 B_μ B_μ
    								- V(h) - 1/4 F_μν^2(B) ≡_2, 
    
    								V(h)     = -|μ|^2/2( v + h )^2  + λ/4 (v + h)^4 = 
    				2 λ v^2/2 h^2 + λ v h^3 + λ/4 h^4 - λ/4 v^4.

				One can see that the Lagrangian (<ref>) describes a massive vector field B_μ with m_B^2 = e^2 v^2 and a massive scalar h with m_h^2 = 2 λ v^2. We do not break the symmetry explicitly. It is again hidden in the relations between couplings and  masses. This is the essence of the Brout-Englert-Higgs-Hagen-Guralnik-Kibble mechanism <cit.>.
				
				The Lagrangians _1 (<ref>) and _2 (<ref>) describe the same Physics  but written in terms of different quantities (variables). Expression (<ref>) involves 
				a complex scalar ϕ with 2 (real) degrees of freedom (DOFs)
				and a massless gauge field (A_μ) also having 2 DOFs.  It is manifestly gauge invariant  but not suitable 
				for perturbative expansion (ϕ has imaginary mass).	
				
				On the contrary, in _2 the gauge symmetry is hidden[One can also say that _2 corresponds to the unitary gauge, no unphysical “states” in the particle spectrum.] and it is written in terms of physical DOFs, a real scalar h (1 DOF) and a massive vector B_μ (3 DOFs).  In a sense, one scalar DOF (ζ) is “eaten” by the  gauge field to become massive. It is important to note that the postulated gauge symmetry allows us to avoid the consequences of the Goldstone theorem, which states that if the vacuum breaks a global continuous symmetry there is a massless boson (Nambu-Goldstone) in the spectrum[any non-derivative interactions violate the shift symmetry ζ→ζ + e v ω for ω=const]. This boson is associated with `oscillations” along the valley, in the broken direction (see Fig. <ref>).  However, due to the local character of symmetry, χ is not physical anymore, its disappearance (or appearance, see below) reflects the redundancy, which was mentioned above. 
						
				In  Sec. <ref>, we demonstrated that the massive-vector propagator has rather bad UV behavior and is not very convenient for doing calculations in PT.  It looks like we gain nothing  from the gauge principle. But it is not true. 
				We can write the model Lagrangian in the Cartesian coordinates
		ϕ = 1/√(2)( v + η + i χ):
			
    _3
         =
            -1/4F_μνF_μν 
    				+e^2v^2/2 A_μ A_μ 
    				+1/2∂_μχ ∂ _μχ 
    				- ev A_μ∂_μχ 
                +1/2∂_μη ∂_μη
                -2v^2λ/2η^2 
          +   v^4λ/4
        
           +  e A_μχ∂_μη 
           - e A_μη∂_μχ
           - vλη (η^2 +χ^2)
           - λ/4(η^2 + χ^2)^2
          + e^2/2 A_μ A_μ (2 v η +η^2 + χ^2).

		The “free” part (<ref>) of _3  seems to describe
		5 real DOFs: a massive scalar η, a massless (would-be Nambu-Goldstone) boson χ and a massive A_μ. However, 
		there is a mixing between the longitudinal component of A_μ and χ that spoils this naive counting (unphysical χ is “partially eaten” by A_μ) .

		In spite of this subtlety, _3 is more convenient for calculations in PT. To quantize the model, one can utilize the gauge-fixing freedom and add the following expression to _3 
		
    δ_g.f. = - 1/2 ξ( ∂_μ A_μ + e v ξχ)^2 = - 1/2 ξ( ∂_μ A_μ)^2  
    - e v χ∂_μ A_μ - e^2 v^2 ξ/2χ^2.

It removes 
the mixing from (<ref>) and introduces a mass for χ, m_χ^2 = (e^2 v^2) ξ. In addition, the vector-boson propagator in this case looks like

    ⟨0| T A_μ (x) A_ν(y) |0⟩  = 
    				∫d^4 p/(2π)^4-i [ g_μν - (1-ξ) 
    				p_μ p_ν/p^2- ξ m_A^2]/p^2 - m_A^2 + i ϵ
    				e^-i p(x-y),    m_A = e v.

One can see that for ξ→∞ we reproduce (<ref>), while for finite ξ the propagator behaves like 1/p^2 as  p→∞, thus  making it convenient for PT calculations. 

It should be mentioned that contrary to _2  the full Lagrangian corresponding to _3 involves also unphysical ghosts, which do not decouple in the considered case. Nevertheless, it is a relatively small price to pay for the ability to perform high-order calculations required to obtain high-precision predictions.

Let us switch back to the SM. We have three gauge bosons that should become massive. As a consequence, three symmetries should be broken by the SM vacuum 
to feed hungry W^±_μ and Z_μ with (would-be) Goldstone bosons 

    SU(2)_L × U(1)_Y → U(1)_em.

		The photon should remain massless and correspond to the unbroken electromagnetic U(1)_em.
 This can be achieved by considering an SU(2)_L doublet of scalar fields:
				
    Φ =1/√(2)exp(i ζ_j(x)σ^j/2v)
    																				[        0; v + h(x) ]
    																,      Φ_0 ≡⟨0|Φ|0⟩ = 1/√(2)[ 0; v ],

												where we decompose Φ(x) in terms of three (would-be) Goldstone bosons ζ_j  and a Higgs h. 												 The Pauli matrices σ_j represent broken generators of SU(2)_L.
												Let Φ also be charged under U(1)_Y:
												
    Φ→exp(i g σ^i/2ω_a + i g' Y_H/2ω' ) Φ.

								We do not want to break U(1)_em spontaneously so the vacuum characterized by the vev Φ_0 should be invariant under U(1)_em, has no electric charge Q
								
    e^i e Q θΦ_0  = Φ_0 →  Q Φ_0 = 0.

								The operator Q is a linear combination of diagonal generators of SU(2)_L × U(1)_Y, T_3= σ_3/2 and Y/2:

    Q Φ_0 = (T_3 + Y/2) Φ_0 =  1/2[  1 + Y_H        0;        0 -1 + Y_H ][      0; v/√(2) ]?= 0.

As a consequence, to keep U(1)_em unbroken, we should set Y_H = 1.	Since Φ transforms under the EW group, we have to
				introduce gauge interactions for the Higgs doublet to make sure that the scalar sector respects the corresponding local symmetry: 

    _Φ = (D_μΦ)^† (D_μΦ)  - V(Φ), 
      with   
    V(Φ) = m_Φ^2 
    				Φ^†Φ + λ(Φ^†Φ)^2.

For m_Φ^2<0 the symmetry is spontaneously broken. In the unitary gauge (Goldstone bosons are gauged away: in (<ref>) we put ζ_j=0)  the first term in (<ref>) can be cast into 

    |D_μΦ|^2 
       =
      1/2(∂_μ h)^2 
    	+ g^2/8(v+h)^2 |W_μ^1+iW_μ^2|^2	+ 1/8(v+h)^2 (g W_μ^3 - g' Y_H B_μ)^2
       =
      1/2(∂_μ h)^2 
    	+ g^2/4(v+h)^2 W^+ W^-	  [ √(2) W^± = W^1_μ∓ i W^2_μ]
        + 1/8(v+h)^2 [Z_μ (g cosθ_W + g' sinθ_W) 
    + A_μ ( g sinθ_W  - g' cosθ_W)]^2
        = 1/2(∂_μ h)^2 + M_W^2 (1 + h/v)^2 W^+ W^- 
    	+ M_Z^2/2(1 + h/v)^2 Z_μ Z_μ,

where we require the photon to be massless after SSB, 
    g sinθ_W  - g' cosθ_W  = 0 
    				  ⇒  sinθ_W = g'/√(g^2 + g'^2),
    				cosθ_W = g/√(g^2 + g'^2)

			and, consequently,
			
    g cosθ_W  + g' sinθ_W  = √(g^2 + g'^2),      = g sinθ_W = g' cosθ_W =  g g'/√(g^2 + g'^2).

The masses of the Z and W-bosons are proportional to the EW gauge couplings 

    M^2_W = g^2 v^2/4,   M_Z^2 = (g^2 + g'^2) v^2/4.

One can see that the Higgs-gauge boson vertices (Fig. <ref>) are related to the masses M_W and M_Z.



An important consequence of the SM gauge symmetry and the existence of the Higgs boson is the
 unitarization of massive vector-boson scattering. 
By means of simple power counting, one can easily convince oneself that 
the amplitude for (longitudinal) W-boson scattering originating from the quartic vertex in Fig. <ref> scales with energy as E^4/M_W^4. This kind of dependence will eventually violate unitarity for E≫ M_W. 
However, in the SM, thanks to gauge symmetry, QGC and TGC are related. This results in  
E^2/M_W^2 behavior when  Z/γ exchange is taken into account.  
Moreover, since the gauge bosons couple also to Higgs, we need to include the corresponding contribution to the total amplitude. It turns out that it is this contribution that cancels the E^2 terms and 
saves unitarity in the WW-scattering. 
Obviously, this pattern is a consequence of the EW symmetry breaking in the SM and can be modified by  the presence of New Physics. 
Due to this, experimental studies of vector boson scattering (VBS) play a role in proving overall consistency of the SM. 
	

Having in mind (<ref>), one can derive the relation

    G_F     = 1/√(2) v^2⇒ v ≃  246 GeV,

		which gives a numerical estimate of v.  
One can also see that due to (<ref>) we have (at the tree level)

    ρ =M_W^2/M_Z^2 cos^2θ_W = 1.

Let us emphasize that it is a consequence of the fact that the SM Higgs is a weak doublet with unit hypercharge. 
Due to this, ρ≃ 1 imposes important constraints on possible extensions of the SM Higgs sector. 
For example, we can generalize expression (<ref>) to account for n scalar (2 I_i + 1)-plets (i=1,...,n) that transform under SU(2)_L and have hypercharges Y_i. 
In case they acquire vevs v_i, which break the EW group, we have 

    ρ = ∑_i (I_i (I_i +1) - Y_i^2) v_i^2/∑_i 2 Y_i^2 v_i^2.

Consequently, any non-doublet (with total weak isospin I_i≠ 1/2) vev leads to a deviation from ρ=1.




 §.§ Fermion-higgs interactions and masses of quarks and leptons
 
Since we fixed all the gauge quantum numbers of the SM fields, 
it is possible to construct the following gauge-invariant Lagrangian:   

    _Y = - y_e  ( +1  L̅  +1  Φ ) 
    													  -2  e_R  
    									- y_d ( -1/3  Q̅ +1  Φ ) 
    													  -2/3  d_R  
    													 - y_u ( -1/3  Q̅ -1  Φ^c  ) 
    													 4/3  u_R  
    													 + ,

which involves dimensionless Yukawa couplings y_f. It describes interactions between the Higgs field Φ, left fermion 
doublets (<ref>) and right 
singlets. In (<ref>) we also indicate weak hypercharges of the corresponding fields. One can see that combinations of two doublets,  (Q̅Φ) , are invariant under SU_L(2) but have a non-zero charge under U(1)_Y. The latter is compensated by hypercharges of right fermions.
In addition, U(1)_Y symmetry forces us to use a charge-conjugated Higgs doublet Φ^c = iσ_2 Φ^* with Y=-1 to 
account for Yukawa interactions involving u_R.

In the spontaneously broken phase with non-zero Higgs vev, the Lagrangian _Y can be written in the following simple form:

    - _Y =  ∑_f y_f v /√(2)(1 + h/v) f̅ f = 
    				∑_f m_f (1 + h/v) f̅ f	
    				,    f = u, d, e,

		where unitary gauge is utilized. One can see that SSB generates fermion masses m_f 
		and, similarly to (<ref>), relates them to the corresponding couplings of the Higgs boson h (see Fig.<ref>a).
	
		
		It is worth noting that (<ref>) is not the most general renormalizable Lagrangian involving the SM scalars and fermions. One can introduce flavour indices and non-diagonal complex Yukawa matrices y^ij_f to account for a possible mixing between the SM fermions, 
    _Yukawa = - y^ij_l  (L̅_i Φ) l_jR - y^ij_d (Q̅_i Φ) d_jR - y^ij_u (Q̅_i Φ^c) u_jR + .
	
Substituting Φ→Φ_0 we derive the expression for fermion mass matrices m_f^ij = y_f^ijv/√(2), which can be diagonalized by suitable unitary rotations of left and right fields.  In the SM the Yukawa matrices (<ref>) are also diagonalized by the same transformations. This leads again (in the unitary gauge) to (<ref>)  but with the fields corresponding to the mass eigenstates. 
The latter do not coincide with weak states, which enter into _W (<ref>).
However, one can rewrite _W in terms of mass eigenstates. Due to large flavour symmetry of weak interactions[In the SM the symmetry is U(3)^5 and corresponds 
				to flavour rotations of left doublets, Q and L, and right singlets, u_R, d_R and l_R. Neutrinos are assumed to be massless.] , 
this introduces a single mixing matrix (the Cabibbo–Kobayashi–Maskawa matrix, or CKM), which manifests itself in the charged-current interactions _CC. 
A remarkable fact is that three generations are required to have 𝒞𝒫 violation in the quark sector. Moreover, a single CKM with only four physical parameters 
(angles and one phase) proves to be very successful in accounting for plethora of phenomena involving transitions between different flavours. 
We will not discuss further details but refer to the dedicated lectures on Flavor Physics <cit.>. 




§ THE SM: THEORY VS. EXPERIMENT




Let us summarize and 
write down the full SM Lagrangian as 

    _SM    = 
    	_Gauge(g_s, g, g')  
    	
    						
    	+ _Yukawa(y_u, y_d, y_l) 
    						
    	+ _Higgs(λ, m_Φ^2) 
    	+ _Gauge-fixing 
      + _Ghosts.

The Yukawa part _Yukawa is given in (<ref>), while _Higgs = - V(Φ) is the Higgs potential from (<ref>).  After SSB the corresponding terms give rise to the Higgs couplings to the SM fermions (Fig.<ref>a) and Higgs self-interactions (Fig.<ref>b). The former are diagonal in the mass basis.  The kinetic term for the Higgs field is included in 

    _Gauge    = -1/4G_μν^a G_μν^a_SU(3)_c
    				-1/4W_μν^i W_μν^i_SU(2)_L
    				-1/4B_μνB_μν_U(1)_Y 
    	+ (D_μΦ)^† (D_μΦ) 
    	
        + L̅_̅i̅  i D̂  L_i 
    		 +  Q̅_̅i̅  i D̂  Q_i_SU(2)_L doublets 
    		 + l̅_Ri  i D̂  l_R_i 
    	+ u̅_Ri  i D̂ u_R_i 
    	+ d̅_Ri  i D̂  d_R_i_SU(2)_L singlets ,

where for completeness we also add the colour  group SU(3)_c responsible for the strong force.
	The first three terms in (<ref>) introduce gauge bosons for the SM gauge groups and in the non-Abelian case account for self-interactions of the latter (Fig. <ref>).
	The fourth term in (<ref>) written in the form (<ref>)
	accounts for gauge interactions of the Higgs field (Fig. <ref>).
	Finally, (<ref>) gives rise to interactions between gauge bosons and the SM fermions (see, Fig. <ref>). 
	
	
	




The SM Lagrangian (<ref>) depends on 18 physical[We do not count unphysical gauge-fixing parameters entering into _Gauge-fixing and _Ghosts.] parameters — 17 dimensionless couplings (gauge, Yukawa, and scalar self-interactions) and only 1 mass parameter m_Φ^2 (see Table. <ref>). 
It is worth emphasizing here that there is certain freedom in the definition of input parameters.
In principle, one can write down the SM predictions for a set of 18 observables (physical particle masses or cross-sections at fixed kinematics) that can be measured in experiments.  With the account of loop corrections the predictions become non-trivial functions of all the Lagrangian parameters. By means of PT it is possible to invert these relations  and express these primary parameters in terms of the chosen measured quantities. This allows us to predict other observables in terms of a finite set of measured observables[One can even avoid the introduction of renormalizable parameters and use bare quantities at the intermediate step.]. 

	
However, it is not always practical to strictly follow this procedure. For example, 
due to confinement we are not able to directly probe the strong coupling g_s and usually treat it as a scale-dependent parameter (4π) α_s = g_s^2 defined in the modified minimal-subtraction (MS) scheme. 
It is customary to use the value of α^(5)_s(M_Z)=0.1181±0.011 at the Z-mass scale as an input for theoretical predictions. 
A convenient choice of other input parameters is presented in Table.<ref>. 

It is mostly dictated by the fact that the parameters from the “practical” set are measured with better precision than the others. 

At the tree level one can write 

    α_s  = g_s^2/4 π,     (4 π) α = g^2 g'^2/(g^2 + g'^2),      M^2_Z = (g^2 + g'^2) v^2/4,  
    G_F  = 1/√(2) v^2,     M^2_h = 2 λ v^2 = 2 |m_Φ|^2,      m_f = y_f v/√(2).

The relations  are modified at higher orders in PT and perturbative corrections turn out to be mandatory if one wants to confront theory predictions <cit.> with high-precision experiments. 
A simple example to demonstrate this fact comes from the tree-level “prediction” for the W-mass M_W. 
From (<ref>) and (<ref>) we can derive 

    G_F/√(2) = πα/2 M_W^2 (1 - M_W^2/M_Z^2).

Plugging recent PDG <cit.> values

    α^-1  = 137.035 999 139(31),   M_Z = 91.1876(21) GeV,   G_F = 1.166 378 7(6) × 10^-5 GeV^-2,

in (<ref>), one can predict 

    M_W^tree  = 80.9387(25) GeV,

where only uncertainties due to the input parameters (<ref>) are taken into account. 

			Comparing M_W^tree with the measured value M_W^exp = 80.379(12) GeV, one can see that
			our naive prediction is off by about 47σ! Of course, this is not the reason to abandon the SM.
			We just need to take radiative corrections into account (see, Fig.<ref>). Among other things the latter allows one to connect phenomena at 
			different scales in the context of a single model. 

			A modern way to obtain the values of the SM parameters is to perform a global fit 
			to confront state-of-the-art SM predictions with high-precision experimental data. Due to quantum effects, we can even probe New Physics that can contribute to the SM processes at low energies via virtual states. 
			
Indeed, LEP precision measurements interpreted in the context of the SM were used in a multidimensional parameter fits to predict the mass of the top quark M_t (”New Physics”), prior to its discovery at the Tevatron. After M_t was measured it was included in the fit as an additional constraint, and the same approach led to the prediction of a light Higgs boson. In Fig.<ref>, the famous blue-band plot by the LEP Electroweak Working Group (LEPEWWG <cit.>) is presented. It was prepared a couple of months before the official announcement of the Higgs-boson discovery. One can see that the best-fit value 
			corresponding to Δχ_min^2=0 lies just about 1σ below the region not excluded by LEP and LHC.


			Obviously, at the moment the global EW fit is overconstrained and can be used to test overall consistency of the SM. In Fig. <ref> we present the comparison between measurements of different (pseudo)observables O^meas and the SM predictions O^fit corresponding to the best-fit values of fitted parameters. 
			Although there are several quantities where pulls, deviations between the theory and experiment, reach more than two standard deviations, the average situation should be considered as extremely good. 
			A similar conclusion can be drawn from the recent Figs. <ref> and <ref>, in which experimental results for various cross-sections measured by ATLAS and CMS are compared with the SM predictions.   


			In case one is interested in the behavior of the SM at ultra-high energies, 
			it is more convenient to get back to the primary parameters and use the renormalization group to estimate how they change with scale. In Fig. <ref>, the scale dependence of the SM parameters  is presented.  

One can see that the gauge couplings tend to converge to a single value at about 10^13-15 GeV, thus  providing a hint for Grand Unification. Another important consequence of this kind of studies is related to the EW vacuum (meta)stability (see, <cit.>). In Fig. <ref>, it manifests itself at the scale μ≃ 10^10 GeV, at which the self-coupling λ becomes negative, making the tree-level potential unbounded from below.    











§ CONCLUSIONS



Let us summarize and discuss briefly the pros and cons of the SM. 
The Standard Model  has many nice features: 
				
								
  * it is based on Symmetry principles: Lorentz + SU(3)_C×SU(2)_L×U(1)_Y gauge symmetry;
								
  * it is renormalizable and unitary;  
								
  * the structure of all interactions is fixed (but not all couplings are tested experimentally);
								
  * it is an anomaly-free theory;
								
  * it can account for rich Flavour Physics (see <cit.>);
								
  * three generations allow 𝒞𝒫-violation (see <cit.>); 
								
  * it can be extended to incorporate neutrino masses and mixing (see <cit.>); 
								
  * it allows making systematic predictions for a wide range of phenomena at different scales; 
								
  * all predicted particles have been discovered experimentally;
								
  * it survives stringent experimental tests.
				
			Due to this, the SM is enormously successful (Absolutely Amazing Theory of Almost Everything).
			Since it works so well, any New Physics should reproduce it in the low-energy limit. Unfortunately, contrary to the Fermi-like non-renormalizable theories, 
			the values of the SM parameters do not give us obvious hints for a New Physics scale.  But why do we need New Physics if the model is so perfect?  
			It turns out that we do not understand, why the SM works so well.  
			For example, one needs to clarify the following:


				
  * What explains the pattern behind Flavour Physics (hierarchy in masses and mixing, 3 generations)?
				
  * Is there a symmetry behind the SM (electric) charge assignment?  
				
  * What is the origin of the Higgs potential?
				
  * What is the origin of accidental Baryon and Lepton number symmetries?
				
  * Why is there no CP-violation in the strong interactions (strong CP problem)[The 
								SM Gauge group allows such a term in the SM Lagrangian, 
								∋θ_CP1/16π^2 F^a_μνF̃^a_μν. But it turns out that θ_CP = 0.]?
				
  * Why is the Higgs-boson mass so low? (Hierarchy/Naturalness  problem, see <cit.>)  
				
  * Is it possible to unify all the interactions, including gravity?

			In addition, there are phenomenological problems that are waiting for solutions and probably require introduction of some New Physics:  
			
							
  * Origin of neutrino masses (see <cit.>);
							
  * Baryon asymmetry (see <cit.>);	
							
  * Dark matter, Dark energy, Inflation (see <cit.>);
							
  * Tension in (g-2)_μ, b→ sμμ, b→ c lν;
							
  * Possible problems with Lepton Universality of EW interactions (see <cit.>).
			

			In view of the above-mentioned issues we believe that the SM is not an ultimate theory 
			(see <cit.>) and enormous work is ongoing to prove the existence of some New Physics. 
			In the absence of a direct signal a key role is played by precision measurements,
			which can reveal tiny, yet significant,  deviations from the SM predictions.
			The latter should be accurate enough (see, <cit.>) to compete with modern and future experimental precision <cit.>. 

			To conclude, one of the most important tasks in modern high-energy physics is to find the scale  at which the SM breaks down. There is a big chance that some new physical phenomena will eventually manifest themselves in the ongoing or future experiments, thus  allowing us to single out viable model(s) in the enormous pool of existing NP scenarios.  



§ ACKNOWLEDGEMENTS
    
I would like to thank the organizers for the invitation to participate in the School as a lecturer. I am also grateful to the School participants for interesting and illuminating discussions. The support from the Grant of the Russian Federation Government, Agreement No. 14.W03.31.0026 from 15.02.2018 is kindly acknowledged.

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