Global symmetry breaking and topological defects

Inthisarticle, we present the fundamentals of Goldstone’s theorem and the breaking of global symmetries. As well as one of the important aspects to take into account in the process of spontaneous symmetry breaking are the possible topological defects generated by the breaking. For this reason, we will expose the concept of topological defect and the calculation of the relic density of the axion, considered as a candidate to dark matter


Introduction
The dynamics of a system [14] involving a quantum eld, Φ (x), is determined in terms of the action dened as where L is the Lagrangian density and Γ a region of spacetime.Thus, any transformation that satises the condition S → S ′ = S, The Hilbert space is a complete, separable, innitedimensional unit vector space.It is also a space constituted by square-integrable (wave) functions ψ (q), that is, whose integral over its domain of denition D, D ψ * (q) ψ (q) dq ≡ ∥ψ (q)∥ 2 , exists.The vector space thus constituted is usually denoted as L 2 (D) and the above integral gives the square of the modulus of ψ (q).
Quantum elds have bundles of energy and momentum, that is, particles.The state of a free particle with quadrimomentum p, mass m, satisfying p 2 = m 2 > 0, spin s, projection σ into the z axis with respect to a reference frame at rest, is created by applying the operator a † σ (p) to the empty state σ, of the system |0⟩ |q, τ ⟩ = a † σ (q) |0⟩ , where p 0 is the relativistic energy of the particle p 0 = + |p| 2 + m 2 .Thus, the states of a system of particles are built from the vacuum state, creating a Hilbert space, called a Fock space 1 .The creation operator is used to compute the inner product ⟨q, τ |p, σ⟩ = ⟨0| aτ (q) a † σ (p) |0⟩ , where aτ (q) is the annihilation operator that satises aτ (q) |0⟩ = 0. Commutation relations are generated by involving the creation and annihilation operators.In the case of a real scalar eld (s = 0), the commutation relations are dened as [a (q) , a (p)] = 0, a † (q) , a † (p) = 0 a (q) , a † (p) = 2p 0 δ (3) (q − p) .(5) Then the inner product has the form ⟨q |p⟩ = 2p 0 δ (3) (q − p) .The representation of each element g that belongs to the group of internal symmetries G is a linear operator U .If the internal symmetry depends on the parameter θ, then the global and local transformations are given by respectively.

Lagrangian Formalism
The variation of the action before the innitesimal transformations where the variation of the hypervolume dierential is given by and the variation of the Lagrangian density with δ (∂µΦ) ̸ = ∂µ (δΦ) given (10).
The total variation of the eld is composed of where the rst term represents the direct variation of the eld δ0Φ = Φ ′ (x) − Φ (x) and the second term arises as a consequence of the variation in x ρ .The variation of the derivative of the eld is given by Under the coordinate transformation, the right-hand side of this equation can be written as where ∂ ′ µ x ν is an element of the Jacobian matrix of the transformation x ′µ −→ x µ , which is given by Thus, we get the variation Therefore, the variation of the action is Innitesimal transformations represent symmetries, if and only if δS = 0.
Thus, the Noether current [5] is dened as and satises the continuity equation Integrating this equation in a nite volume separate the temporal from the spatial derivatives and by virtue of Einstein's notation [16] V as well as by applying Gauss's theorem [17], we nd The volume integral is dened as the charge If there is no input or output ow at the boundary (isolated system), then we get that is, charge is conserved.
Every eld behaves like a scalar eld (δΦ = 0) in space-time translations, therefore, the current density is from which the energy-momentum tensor is dened Using the Minkowsky metric tensor η νρ to raise the subscript ν, T µν = η νρ T µ ρ and integrating the continuity equation in a spatial volume V , we nd d dt Consequently, the associated conserved charge is the quadromoment, which is the generator of innitesimal space-time translations, 3 Global Symmetries Let the quantum eld Φ k (x) be members of a multiplet, the transformation before the element g (θ1, θ2, ..., θN ) from the symmetry group SU (n) results where (U) kl is an element of the matrix that represents the transformation.Since the nite transformation can always be constructed by the composition of innitesimal transformations, the variation under the global innitesimal transformation becomes with θ a as innitesimals and a = 1, 2, 3, ..., n 2 − 1.Thus, we arrive at the currents of Noether we associate the charges It is enough to show that the n 2 − 1 charges QV,a are precisely the generators of the symmetries.For this, a symmetry operation represents a change of base.The components of the eld are transformed by Therefore, for an innitesimal transformation we obtain the expression that indicates how the components are transformed in the group For the proof, we rst dene the conjugate moment and then the commutator is evaluated in equal times, and by virtue of the properties of the Lie brackets Consequently, the commutation relations of the elds Φ k and Π k are needed to proceed.In bosonic elds, these are dened as The commutator results [QV,a, Φm (y)] = − l (Ta) ml Φ l (y) .(40) Finally, the commutator is evaluated by x 0 =y 0 , rearranging terms, on the right side of the equation we have x 0 =y 0 .
After applying the properties of the Lie brackets, we have x 0 =y 0 (Ta) kl Φ l (x) .
By virtue of equations (39) one nds Furthermore, the properties of the Dirac delta function [18] will be used to obtain With the help of the denition of the Lie commutator or in its equivalent form Finally, we have With equations ( 36), ( 40) and ( 41) we conclude that the charges are the generators of the symmetry.If H is the Hamiltonian of the theory, the Heisenberg equa- which expresses the invariance of the theory under transformations of the group G .

Gauge symmetries
If the parameters of the global transformation are considered to be space-time dependent, we are dealing with a local transformation In a global transformation we have If the theory is claimed to be invariant under a local transformation, the covariant derivative is introduced where the n 2 − 1 quantum elds W a µ are called gauge elds and g is the coupling constant.We demanding then, we get the innitesimal transformation of the elds This transformation is known as the gauge transformation of W a µ , which only depends on the group G and not on how the elds Φ k , called material elds, are transformed.Considering an innitesinal global transformation, the gauge elds are transformed by and since the structure constants generate the adjoint representation, we have By generalizing the electromagnetic tensor, the Yang-Mills eld [21] is dened as is directly transformed into the adjoint representation.The conclusion is that for a Lagrangian density where the last contribution contains the kinetic energy terms and the self-interactions of the non-abelian gauge elds W a µ .
In general, the Lagrangian density ) and the interactions between the particles Lint (Φ k (x) , ∂µΦ k (x)).Therefore, the material elds become the fundamental representation and the norm elds the adjoining representation.Also, the symmetries determine the interactions between the elds by means of the gauge principle [22].By promoting a global symmetry to a local one, the introduced gauge elds induce the terms of the interactions, therefore, these terms are the mediators of the interactions.

Spontaneous breaking of symmetry
Since for a global symmetry the generators satisfy these do not necessarily leave the vacuum state invariant.
The Wigner-Weyl symmetry leaves the vacuum state invariant and, consequently, the mass spectrum of the particles in the theory is degenerate.The Nambu-Goldstone symmetry does not leave the vacuum state invariant, and therefore the particle spectrum must contain a particle with zero mass, known as the Goldstone boson.The Nambu-Goldstone type symmetry is a spontaneously broken symmetry.
When considering the expectation value of the vacuum state to (36), we obtain In a Wigner-Weyl symmetry we have that the generators consequently In a spontaneously broken symmetry Goldstone bosons are a consequence of Goldstone's theorem [23], which is based on the conservation of currents ∂µJ µ a (x) = 0.
(57) Expression ( 53) is rewritten in terms of the current Since the current can be written in terms of the translation operators of the Poincare group thus, these operators leave invariant the state of the vacuum, then If we insert By dening the coecient and since with ∆ 3 V − → P as an approximation of the Dirac delta, the expression (61) becomes Since the left-hand side is independent of y 0 , it follows that Thus, the massless states n G correspond to the Goldstone bosons.The number of Goldstone bosons is equal to the number of generators that do not annihilate the vacuum state.
To satisfy the norm principle, the global transformation must be promoted to a local one.In the Nambu-Goldstone case, the Goldstone bosons associated with the spontaneous symmetry-breaking generators disappear via a local gauge transformation and the corresponding gauge elds gain mass.6 Goldstone's theorem If we consider the global invariant Lagrangian density U (1) composed of a complex scalar eld Φ (x) and its complex conjugate Φ * (x) then (67) The described system of the Lagrangian density (67) transforms with the Lagrangian density composed of the two real elds φ1 and φ2, which are related to Φ and Φ * through from Equation ( 68) has the symmetry O (2), that is, the Lagrangian density (68) is invariant when considering the following transformation so that the symmetry U (1) is equivalent to the symmetry O (2).
In quantum eld theory, excitations of particles in a eld are dened as quantum uctuations of the eld around the lowest energy state, i. e., the state related to vacuum.The constant value of the eld corresponding to the lower energy state is called the vacuum expectation value (VEV), i. e. ⟨0| Φ |0⟩ ≡ Φ0.To nd the spectrum of the particle, the potential is expanded around the minimum corresponding to the lowest energy state and requires the following condition for the vacuum Thus, through equations ( 72) and ( 73), it is obtained to the single vacuum with φ01 = φ02 = 0.The mass matrix becomes diagonal in this case which means that φ1 and φ2 have equal masses m as can be seen from equation (71).
(2) On the other hand, in which the vacuum state is non-unique is called Nambu-Goldstone phase.
In this case, for example, in continuously or innitely degenerate vacuum states with φ01 ̸ = 0 and/or φ02 ̸ = 0, for the potential with m 2 = −µ 2 µ 2 > 0 and λ > 0 The minimum of V is in the following partial derivatives leading to the condition In other words, all points on a circle with radius v = µ 2 /λ in the plane (φ1, φ2) correspond to the minimum of V, that is, the vacuum state is already nonunique but has O (2) symmetry.From expression (75) we obtain Therefore, if we choose a point (φ01 = v, φ02 = 0) as a physical vacuum, the mass matrix can be expressed as Therefore, φ ′ 1 = φ1 − v corresponds to a particle with mass m 2 = 2λv 2 , while φ ′ 2 = φ2 is massless.φ ′ 2 is called the "Goldstone boson".If these new elds are used, the Lagrangian density (68) is expressed The Lagrangian density no longer has O (2) symmetry, although the Lagrangian density (68) does.That is, the symmetry of the Lagrangian density has been broken by the symmetry breaking of the vacuum state.This symmetry is called a hidden symmetry or spontaneous symmetry breaking (SSB).
In short, starting from the Lagrangian density with a global symmetry, negative parameter m 2 = −µ 2 and breaking the symmetry of the vacuum states by choosing a particular point among the states of symmetrically degenerate vacuum, it was found that a massless particle (Goldstone boson) appeared.This mechanism is called Goldstone's theorem [23] [24] [25].In general, massless Goldstone bosons [26] appear depending on symmetry properties when a global symmetry is spontaneously broken.
7 Finite temperature eects The usual methods used in quantum eld theories are adequate to describe situations in a vacuum, such as occur in accelerators.However, in the primitive Universe the conditions are quite dierent, being characterized by a high-temperature plasma, with an energy density that cannot be neglected.In these conditions, it is necessary to nd other methods, halfway between thermodynamics and quantum eld theories, that allow realistic calculations to be made under these conditions where the environment is characterized by a thermal bath.These methods are developed by nite temperature eld theory [27].

Eective potential
The function that contains all the nite temperature effects is the eective potential, V ef f β .This is composed of the classical potential in vacuum, V0, plus the term that describes the quantum and temperature eects V β where ϕc ≡ ϕ (x) is the constant value in space that the eld takes, for a translational invariant theory.
There are two formalisms to calculate the second term of equation ( 84), that is, the potential V β , both giving the same results, at least to rst order.One formalism is the imaginary time method to describe equilibrium situations, and the other is the real time method, with which certain non-equilibrium systems can be investigated.Taking into account the contributions of the diagrams to 1 loop, it follows that the eective potential where β ≡ 1/T and ω is where m 2 (ϕc) is the curvature of the potential, also called displaced mass The rst part of the integral (85) accounts for the quantum corrections in vacuum, giving the eective potential at zero temperature, V1| T =0 .The part of (85) that depends on temperature can be written as where the bosonic thermal function, JB, has been dened as In this way, the eective potential at 1 loop consists of the following parts The function JB [27] admits an expansion for high temperatures JB m 2 /T 2 = − where ζ is the Riemann zeta function and a b = 16π 2 exp 3 2 − 2γE , with γE ≃ 0.577 the Euler-Mascheroni constant.
With these approximations we have that, at high temperatures, the part of the eective potential to a loop that depends on temperature approaches where the terms that are constant with the eld have been neglected.

Phase transitions
One of the most relevant consequences of nite temperature eects is the inuence they exert on phase transitions.The key point is that at nite temperature the fundamental state of the system does not correspond to the minimum of V0 (ϕ), but is the value that minimizes the potential V β ef f (ϕ), given in equation (84).Thus, the state that denes the vacuum is a function of temperature, ⟨ϕ (T )⟩.
In the context of cosmology, the Kirzhnits eect [28] is very important.In the Big Bang theory, initially the Universe is at very high temperatures, so it is to be expected that, due to temperature eects, the symmetries are not broken.In this way, for a discrete symmetry, the minimum of the corresponding potential will be found at ⟨ϕ (T )⟩ = 0.For a certain critical temperature value, Tc, the minimum at ϕ = 0 will become metastable and a phase transition will occur.
There are two types of phase transitions: rst order and second order.In the case of rst order transitions, for T ≫ Tc the potential only has a minimum at ϕ = 0.
As the temperature decreases, but still with T > Tc, a local minimum develops at ϕ ̸ = 0.When the temperature reaches its critical value, the minima are degenerate and for T < Tc the minimum at ϕ ̸ = 0 becomes the absolute minimum.In this case, between the local minimum at ϕ = 0 and the global minimum at ϕ ̸ = 0 there is a potential barrier, so a classical transition cannot occur between the two minima.This has to be developed by quantum tunneling, which implies the formation of real vacuum bubbles that rapidly expand up to speeds close to light.On the other hand, in the second order transitions, said potential barrier does not appear between the minima and the transition takes place gradually.Therefore, the origin ϕ = 0 goes from being a global minimum for T > Tc to a local minimum for T < Tc.
An important point, especially at the calculation level, in the study of phase transitions produced by temperature eects is that the rst order (a loop) of the usual perturbation theory, in powers of coupling constants, loses its validity for temperatures above the critical temperature [29].Then it is necessary to replace this theory of perturbations by an improved expansion, where diagrams of more than one loop are taken into account.
In this improved theory an innite number of diagrams are summarized for the order of the expansion, which is known as Daisy summarization [27].The eect of this summarization is to introduce a shift in the mass that depends on the temperature where Π ∝ T 2 is the self-energy corresponding to the propagator when they have taken into account all the diagrams with loops added to the propagator to the rst order in temperature.Obviously, this Π function depends on the potential being considered.
Since we believe that the Universe evolved through such a symmetry breaking sequence.There are many questions we can ask ourselves about the cosmological implications of the regime.

Topological defects
Another important aspect to take into account in the process of spontaneous symmetry breaking are the possible topological defects generated by the breaking.Topological defects [30] [31] [32] are concentrations of energy that are generated after the spontaneous breaking of a symmetry, in case the resulting vacuum has a non-trivial topology.To understand non-trivial topology, let's look at a couple of examples: domain walls [33] [34] and cosmic strings [35] [36].The existence and stability of these objects will be dictated by topological considerations, with numerical simulations being the only treatment that allows studying their evolution.
As a rst example, we consider the breaking of symmetry in a potential with a symmetry under reection Z2, that is, an invariance before the change ϕ −→ −ϕ.
Given the breaking of this symmetry, the expected value of the eld in a vacuum can take the value ⟨ϕ⟩ = +v, or the value ⟨ϕ⟩ = −v.Considering that the theory was translational invariant, it has been assumed that all space is in the same ground state, an assumption that is not true.There will be regions where ⟨ϕ⟩ = +v and others where ⟨ϕ⟩ = −v, there being no reason for the eld to take one or the other value.Then, since between the regions where ⟨ϕ⟩ is dierent, the eld has to pass from ϕ = +v to ϕ = −v continuously, there must be regions in space in which ϕ = 0, that is, zones of false vacuum.
These zones are two-dimensional regions, with a certain thickness, that separate the dierent boundaries where the expected value of the eld has dierent values.These zones are called domain walls and they appear whenever a discrete symmetry is broken.In this case, the vacuum is said to have a non-trivial topology because it consists of two disconnected states.If we look at the shape of the potential after the break, we can see that in areas where ϕ = 0, the energy is greater than in the rest of the space.
As a conclusion, these domain walls are two-dimensional regions with a high concentration of energy.
Another example of a topological defect is that of cosmic strings [35] [36].In this case, the symmetry to consider would be, for example, the N = 2 case of the O (N ) symmetry, respected by the potential (103) (see appendix A).A particular case would be a complex eld, ϕ = |ϕ| exp (iθ).In that case, once the symmetry is broken, the expectation value of the eld in vacuum can be found at any point on a circle of radius v.That is, the expected value of the module of the eld is xed at ⟨|ϕ|⟩ = v, but the value of its phase can be any.Thus, there is a phase that depends on the position, θ ( − → r ), which can be dierent at each point in space.However, the ϕ eld can only take on a single value at each point in space, so the change in phase, ∆θ, around any closed path must be a multiple of 2π.We take, for example, a path in which ∆θ = 2π.As we compress the path into a point, Theta cannot continuously vary from ∆θ = 2π to ∆θ = 0.For this reason, there will be a point in said path where the time phase is undened, the only possibility of this undened being that ⟨|ϕ|⟩ = 0.This false vacuum point on the considered path is part of a onedimensional false vacuum tube.These tubes are called cosmic strings and would be the one-dimensional analogue to domain walls.In this case, the topology of the vacuum is non-trivial because there is no unique way to map the manifold that denes it to a circle.As in the case of domain walls, in the region where ⟨|ϕ|⟩ = v, the energy is greater than the rest of space, so these strings are one-dimensional concentrations of energy.
Other examples of topological defects are magnetic monopoles [37] [38] and textures [39].The former are zero-dimensional analogues to strings and domain walls and appear in breaking spherical symmetries.Textures are delocalized objects of higher dimensions, which appear when breaking the most complicated symmetry groups.In addition, there can also be topological defects that are a combination of those described, such as [40] domain walls linked by cosmic strings, strings that end in monopoles, etc.
The specic eects vary depending on the topological defect considered.For example, domain walls and monopoles are cosmologically catastrophic.Any model in which there is a production of these types of defects has to face the problem that defects evolve in such a way that they contradict dierent irrefutable observational facts of the Universe, such as producing too large anisotropies in the background of radiation or predicting densities of Ω ∼ 10 11 ρc.Therefore, the models must be discarded.
However, cosmic strings and possibly textures are much more benign.Among other characteristics, they can play the role of seeds that give rise to the formation of the large-scale structure that is currently observed, as well as the anisotropies of the radiation background.
Another important contribution would be to the dark matter of the Universe.
The spontaneous breakage of many theories predicts the existence of one or more types of topological defects.
These objects are inherently nonperturbative and probably cannot be produced in the high-energy collisions that take place in terrestrial accelerators.Instead, they can be produced in phase transitions that occur in the early Universe.For this reason, these defects are considered to be traces of the rst moments of the Uni- to the CP problem of strong interactions, so if the axion nally does not exist, an alternative solution to this problem will have to be found [42].Axion physics has very precise properties, all of which depend on a single free parameter, fa, the symmetry breaking scale of PQ.
The importance of the axion is mainly due to the possibility that it is the main component of the Dark Matter of the Universe [43] [44] [45] [46].

Cosmological production
There are dierent mechanisms that cause the production of axions, which we will group into thermal and nonthermal mechanisms, the latter being the ones that predominate.Among the non-thermal mechanisms we can basically nd two classes, those originated in the oscillations of the axion eld and those that are produced from the decay of the topological defects generated in the spontaneous breaking of the symmetry PQ [47] [48].
The cosmological history [15] [49] of the axion begins at temperatures T ∼ fa, when the U (1) P Q symmetry breaks spontaneously.In principle, all possible expected values for the axion eld, ⟨a⟩, are equally likely, but naturally ⟨a⟩ is expected to be of the order of the scale of PQ.
The vacuum expectation value (VEV) for the axion eld is subjected to the evolution equation where H is the Hubble parameter and ma (t) is the mass of the axion, which evolves over time (as a function of temperature and temperature as a function of time).In the interval fa > T ≫ ΛQCD, the Hubble parameter is much larger than ma and ⟨a⟩ remains constant, since H acts as a friction.The mass of the axion is suppressed at high energies and it is on the QCD scale that it begins to be relevant, quickly becoming ma ≃ H. From this moment on, the friction induced by H is inecient and the eld of the axion begins to evolve.Thus, for a temperature T ∼ ΛQCD we have moment from which the axion eld begins to oscillate around ⟨a⟩ = 0.
The mass of the axion increases adiabatically as T decreases, so the oscillation is approximately sinusoidal with an amplitude that decreases with time [50] [51] ⟨a⟩ ≃ A (t) cos (ma (t) t) , to which corresponds a density of axions and an energy density ρa = mana. (98) From these reasonings, we can estimate the current energy density of axions produced from these oscillations ρ0 ∼ m 2 a A 2 0 ∼ ma0 maA 2 i T0 ΛQCD 3 .
(99) ma0 it must be interpreted as the mass of the axion at zero temperature, given by ma = 0.6eV 10 7 GeV /fa .Approximately it can be accepted that ma0 ≃ Λ 2 QCD /fa.Taking mai from formula (99) and considering that Ai ∼ ⟨a⟩ i ∼ fa, we nd so we see that the relic axion density increases with fa.
A detailed calculation of these oscillations [52] yields an axion density where Ca is a constant between 0.5 and 10.
Another source of non-thermal axion production is the decay of topological defects.As seen above, the spontaneous breaking of a global symmetry gives rise to a series of topological defects.
The last production mechanism to be discussed is thermal processes [53].Taking into account the interactions of the axions with quarks and photons, in [54] it is found that the axions were in thermal contact with the photon bath if fa < 10 Finally, there are dierent mechanisms that cause the production of axions, which we will group into thermal and non-thermal mechanisms, the latter being the ones that predominate.Among the non-thermal mechanisms we can basically nd two classes, those originated in the oscillations of the axion eld and those that are produced from the decay of the topological defects generated in the spontaneous breaking of the symmetry PQ.
Appendix A: Sigma model The Lagrangian density of the linear sigma model contains a set of N real scalar elds where there is a sum at i in each factor of ϕ i 2 . The Lagrangian density (102) is invariant under symmetry ϕ i → R ij N ϕ j , for any matrix RN , orthogonal and N × N .The group of transformations ϕ i → R ij N ϕ j , is the group of rotations in N dimensions, also called O (N ), Ndimensional orthogonal group.
The classical conguration of the eld that represents the ground state, ϕ i 0 , is the one that minimizes the potential for any value ϕ i 0 that satises ϕ i This condition only xes the length of the vector ϕ i 0 , its direction being arbitrary.Without losing generality, we can choose that the vector ϕ i 0 points in the N -th direction ϕ i 0 = (0, 0, ..., 0, v) = 0, 0, ..., 0, µ √ λ . (105) For N = 2, the potential V ϕ i has the typical shape of the Mexican hat, where the minimum can be found at any point on a circle of radius v.It is convenient to redene the eld ϕ i (x), introducing the elds π i (x) and η (x) where i = 1, 2, 3, ..., N − 1.
Introducing the change (106) in (102), the Lagrangian density takes after spontaneous symmetry breaking Analyzing the new Lagrangian density (107), we observe that the spectrum consists of a massive η eld and a set of N − 1 massless elds π i .Symmetry O (N ) original is hidden, leaving the symmetry subgroup explicit O (N − 1), that transforms the elds π i , that is, π i → R ij N π j .The massive eld η describes oscillations of the eld ϕi in the radial direction, where the potential has a non-zero second derivative.The massless elds π i describe oscillations of ϕ i in the tangential directions, along the valley of the potential.This valley is a (N − 1)-dimensional surface, where all N − 1 directions have no slope and are equivalent, which shows the symmetry O (N − 1).
In the example used, initially we had N (N − 1) /2 continuous symmetries, corresponding to the dierent orthogonal axes about which a rotation can be performed O (N ), in N dimensions.After the spontaneous break, the subgroup remained O (N − 1), containing (N − 1) (N − 2) /2 continuous symmetries.Thus, the dierence N − 1 will be the number of symmetries that have been broken.This is precisely the number of massless particles that have appeared in theory, as predicted by Goldstone's theorem.
In nature, one can nd many examples of spontaneously broken global symmetries that give rise to Goldstone bosons.One of them is family symmetry.This symmetry related to the number and properties of the families of the standard model.

( 2 )
is a symmetry transformation.The relations between symmetries and conservation laws is described by the Noether's theorem.For each symmetry transformation there is a current density, called the Noether current, which is conserved if the symmetry is exact.Each corresponding charge, denoted by the operator Q, , this leads to conservation laws for that charge.In electrodynamics and quantum chromodynamics, local gauge symmetry is responsible for the conservation of color and electric charges.Symmetry transformations are described in terms of groups.To each symmetry corresponds an abstract element of a symmetry group G .This group is represented by a set of operators G.These are like basis change operators on the Hilbert space of quantum elds

( 6 ) 3 p
When considering a system of particles represented by plane waves, that is, free particles, the wave function of each of them has the form exp (±ipx).Thus the real scalar eld for the free particle is written in terms of a 2p 0 exp (−ipx) a (p) + exp (−ipx) a † (p) .

( 8 )
Symmetries can be classied into space-time and internal symmetries.Space-time symmetries transform the coordinates of a point in space-time and into the eld in a particular way, depending on its class (scalar, vector or spin).Internal symmetries only transform the eld.Both classications can be divided into global and local symmetries.Thus, global space-time symmetries are associated with special relativity and local ones with general relativity; the global internal symmetries are associated with the approximate symmetries of avor, isospin, among others, while the local ones with the color norm and electroweak symmetries.
under a global transformation, it can be invariant under a local gauge transformation by introducing the gauge elds W a µ , which enter the covariant derivatives DµΦ k and the Yang-Mills eld Fµν .The Lagrangian density is

2 V ∂φa∂φ b 0 ≡ m 2 ab
where ∆φa = φa − φ0a, ∆φ = φ b − φ 0b and Φ0 = (φ01, φ02) is the VEV of Φ = (φ1, φ2), i. e. φ0a = ⟨0| φ0 |0⟩ (a = 1, 2).Since the potential V has a minimum in Φ = Φ0, the second term of the right side of equation (70) is zero.The ∂ in the third term of the right side of equation (70) is called the mass matrix and is diagonalized to generate the spectrum of the particle mass.Now, two possible cases of V are possible.(1) The rst case corresponds to the single vacuum state and is called Wigner phase.Considering the parameters m 2 and λ for the potential in expression (67) as positive m 2 > 0 and λ > 0, verse.Although they are not minimum energy congurations, monopoles, strings, and domain walls (boundary domain) are topologically stable and, as predicted by the Kibble mechanism[41], their production is inevitable at phase transitions.One of the most important consequences of these objects is that they crucially aect the evolution of the Universe.The concrete eects vary depending on the topological defect considered.For example, domain walls and monopoles are cosmologically catastrophic.Any model in which there is production of these types of defects has to face the problem that the defects evolve in such a way that they contradict dierent irrefutable observational facts of the Universe, such as producing excessively large anisotropies in the Cosmic Microwave Background.10The axionThe axion is one of the best-known pseudo Goldstone bosons, although experiments looking for it have so far failed to detect it.Its motivation is theoretical and arises as a consequence of the spontaneous rupture of Peccei-Quinn (PQ).This symmetry is the most elegant solution 18GeV holds.As a conclusion, it is found that in the early Universe there is thermal production of axions if it is true that fa < 10 12 GeV .
ories predicts the existence of one or more types of topological defects.These objects are inherently nonperturbative and probably cannot be produced in the highenergy collisions that take place in terrestrial accelerators.Instead, they can be produced in phase transitions that occur in the early Universe.For this reason,