Yang-Mills theories and confinement vs. Gravity and geometry

Transcription

1 Yang-Mills theories and confinement vs. Gravity and geometry Antônio D. Pereira, 1, Rodrigo F. Sobreiro, 1, and Anderson A. Tomaz 1, 1 UFF Universidade Federal Fluminense, Instituto de Física, Campus da Praia Vermelha, Avenida General Milton Tavares de Souza s/n, , Niterói, RJ, Brasil. (Dated: July 2015) These lectures contains a dense material about gauge theories and the first order formalism of gravity. A scenario where quantum gravity is realized as a Yang-Mills theory is discussed with some detail. In this scenario, gravity as a geometrodynamical effect is an emergent phenomenon. Lectures presented at the X Escola do Centro Brasileiro de Pesquisas Físicas July Electronic address: aduarte@if.uff.br Electronic address: sobreiro@if.uff.br Electronic address: tomaz@if.uff.br

2 2 Contents INTRODUCTION 4 PART I: GAUGE THEORIES AND THE GRIBOV PROBLEM 5 I. Gauge theories and the gauge principle 5 A. Electromagnetism 5 B. Covariant formulation and U(1) symmetry 7 1. Definitions 7 2. Maxwell equations and gauge fixing 8 C. Non-Abelian gauge theories 9 1. Lie groups basics 9 2. Gauge field, field strength and Yang-Mills action 11 II. Quantization of gauge theories 12 A. Faddeev-Popov quantization 13 B. BRST quantization 15 C. Renormalization, physical states and unitarity 16 D. Asymptotic freedom and confinement 17 III. Gribov ambiguities 18 A. Statement of the problem 18 B. Gribov s solution 19 C. The Gribov-Zwanziger action and BRST soft breaking 22 D. The Refined Gribov-Zwanziger action 23 PART II: GRAVITY AS A GAUGE THEORY 25 IV. Equivalence principle and geometrodynamics 25 A. General relativity in the metric formalism 25 B. First order formalism of gravity Vierbein Spin-connection Curvature, torsion and hierarchy identities Einstein-Hilbert action 30 C. Gravity as a gauge theory 31 V. Generalized theories 31 A. Lovelock theory 31 B. Mardones-Zanelli theory 32 VI. Quantization attempts 33 PART III: EFFECTIVE GRAVITY FROM YANG-MILLS THEORIES 34 VII. General ideas 34 VIII. SO(5) Yang-Mills theory 36 A. Projection and rescaling 36 B. Contraction and symmetry breaking 37 IX. Induced gravity 38 X. Quantum aspects 38 A. Weinberg-Witten theorems, emergent gravity and spin-1 gravitons 38 B. Cosmological constant problem 39 C. Quantum predictions Running parameters Numerical estimates 41

3 XI. Classical aspects 42 A. Classical field equations 42 B. Spherical symmetry static solutions 43 C. Cosmology The Λ-CDM model High curvature regime 45 D. Dark stuff 48 3 A. Differential forms Exterior product and p-forms Exterior derivative The Hodge dual operator Integration of differential forms 50 B. Geometrical aspects of gauge theories Principal bundles and connections Universal bundles Gravity geometrical structure 53 Acknowledgments 54 References 54

4 4 INTRODUCTION Gauge theories[1 6] have been extremely successful in describing at least three of the four known fundamental interactions in Nature, at classical and quantum level. Except for gravity, the Standard Model is the state of the art of quantum field theory and particle physics where Yang-Mills theories [7, 8] is the main framework. Gravity, on the other hand, is not described by interacting fields but by a geometrodynamical theory for the spacetime itself [9 13]. In despite of the success of both type theories in describing Nature, some open issues remain to be understood. In particular, from the theoretical point of view, confinement and quantum gravity are, perhaps, the most important open problems of these theories. Confinement refers to a specific behavior of the strong sector of the Standard model, known as Quantum Chromodynamics (QCD) which describes, essentially, the dynamics of quarks and gluons. The SU(3) Yang Mills (YM) gauge theory perfectly describes this dynamics at high energies where quarks and gluons are almost free. It displays the so called asymptotic freedom characterizing the increasing of the coupling parameter as the energy decreases. Thus, as lower the energy is, as higher the force between quarks and gluons is. The low energy sector is highly non-perturbative and very difficult to be handled. Nevertheless, V. N. Gribov has found in in the late seventies [14, 15] that, at this region, the theory lacks of a consistent quantization at the non-perturbative level. This is a technical problem, known as Gribov ambiguities, which must be considered for quantum consistency. Remarkably, the improvement of the quantization leads naturally to confinement evidences in a highly non-trivial way. On the other hand, gravity, as a geometrodynamical phenomenon, lacks for a quantum version (even perturbatively). In many aspects, the principles of general relativity seem to be incompatible to the principles of quantum field theory, dooming their union to failure. Many alternative approaches to describe gravity at quantum levels have been proposed for almost one century. Widely known examples are string theory [16 19], loop quantum gravity [20, 21], asymptotic safety [22, 23], Hořava-Lifshitz gravity [24 26], the vast scenarios of emergent gravities [27 40] and so on. In particular, emergent gravities possibly solve the problem by stating that gravity as a geometrodynamical phenomenon can not be quantized. Gravity should be an emergent feature of a more fundamental theory that have nothing to do with the spacetime dynamics at quantum level. In these lectures we discuss an interesting analogy between non-perturbative QCD and gravity [37 42]. In this proposal, gravity emerges from a Yang-Mills theory for the gauge symmetry group SO(5) in four dimensions. Thus, we will discuss the details of Yang-Mills theories and the Gribov problem as well as the first order formalism of gravity in order to reach the goal of understanding how gravity can emerge from a quantum gauge theory. Specifically, quantum gravity will be described by a Yang-Mills action in flat space while classical gravity will be obtained from it as a geometrodynamical theory for the spacetime and both sectors should not intersect. Before that, a dense discussion about Yang-Mills theories and the first order theory of gravity are discussed. The idea of these lectures is to provide a complete view of gauge theories and gravity in a conceptual way. Due to the vast amount of results known in these theories, many details are omitted in favor of conceptual discussions. A lot of technical passages are left as exercises of all kinds of difficulty levels. These exercise are recommended for students interested in explore the field. Previous notions in quantum field theory (QFT), differential geometry, group theory and general relativity is advisable, but not mandatory. Along the text many references are suggested and two appendices (about differential forms and fibre bundles) is included for self-consistency. At the end of the course, the student should acquire a lot of conceptual information about current research on gauge theories, confinement, gravity and quantum gravity.

5 5 PART I: GAUGE THEORIES AND THE GRIBOV PROBLEM In this part we focus on the construction of gauge theories. Starting with Maxwell equations as an Abelian gauge theory, we define gauge field, gauge transformations and the gauge principle. A few concepts in Lie groups and Lie algebras are introduced in order to discuss non-abelian gauge theories, which are the main theme of this part. Quantization is discussed with some detail within Faddeev-Popov s method. Then, BRST symmetry and its consequences, including BRST quantization, are carried out. The concepts of physical states, renormalizability, unitarity, asymptotic freedom and confinement are established. All these aspects are mainly found in quantum field theory text books, see for instance [1 4, 6, 43]. The issue of Gribov ambiguities is then addressed: a review since Gribov s seminal work back in 1978 is compiled with the most important results and concepts. The main references about the Gribov problem are [14, 15, 44 50]. I. GAUGE THEORIES AND THE GAUGE PRINCIPLE A. Electromagnetism The most known example of a gauge theory is, perhaps, electromagnetism [1, 51, 52]. In its non-covariant form, electromagnetism is described by Maxwell equations in vacuum, E = ρ ɛ o, B = 0, E + B t = 0, E B ɛ o µ o t = µ o j, (1) where E and B are the electric and magnetic fields, respectively; ɛ o is the electric permittivity of the vacuum and µ o is the magnetic permeability of the vacuum; the charge density is ρ while j = ρv is the current density with velocity v. Maxwell equations can be simplified by the introduction of potential fields. In fact, the second equation (magnetic Gauss law - first homogeneous equation) states that a solution for B is B = A, (2) where A is the vector potential field. Using this solution at Faraday law s (second homogeneous equation) it is possible to show that the solution for the electric field is 1 E = φ A t, (3) where φ is the electric scalar potential. Remarkably, the vector and scalar potentials can always be redefined in such a way that the electric and magnetic fields are left invariant (and so are the Maxwell equations, obviously). These redefinitions are called gauge transformations: φ φ ξ t, A A + ξ, (4) with ξ = ξ(r, t) being the gauge transformation parameter. As opposed to coordinate transformations (for example, Lorentz transformations), gauge transformations are transformations that leave spacetime coordinates intact, it is a type of transformation that affects only fields. The fields φ and A are recognized as the gauge fields. The electric and 1 The fact that the homogeneous equations are used to define the potentials are not accidental. These equations have a deep geometrical meaning, which we will briefly discuss in this section and along this part.

6 magnetic fields are said to be gauge invariants. In terms of the gauge fields, the Maxwell equations (1) read (Exercise 1.1) 6 2 A + ɛ o µ o 2 A t 2 2 φ t A = ρ, ɛ ( ) o + φ A + ɛ o µ o = µ o j. (5) t We notice that the homogeneous equations are automatically satisfied through (2) and (3). Thus, (5) are obtained from the substitution of these solutions into the inhomogeneous Maxwell equations. Although equations (5) appear complicated with respect to the original equations (1), it is important to stress out that E and B carry six components of freedom while φ and A carry only four. The fact that the gauge fields can be redefined up to a scalar function establishes that they are not observables because there is an infinite number of equivalent fields leading to the same pair of electric and magnetic fields, recognized as the actual physical fields. Thus, no matter how one chooses ξ to solve a problem, the corresponding electric and magnetic fields will always be the same. Moreover, to solve a specific problem, it is possible to choose a unique class of gauge fields in order to simplify the equations. This procedure is called gauge fixing and is performed by the introduction of a constraint for the gauge potentials. For instance, the Coulomb gauge is characterized by transverse vector potentials, A = 0, leading to 2 2 A A + ɛ o µ o t 2 2 φ = ρ ɛ o, = µ o j ɛ o µ o φ, (6) t Then, the scalar potential is the solution of the Poisson equation and is an instantaneous field, a non-physical feature, while the vector potential is the solution of a inhomogeneous wave equation. Another example (Exercise 1.2) is the Landau gauge 2 φ, A + ɛ o µ o t = 0, which leads to inhomogeneous wave equations for both, φ and A. The difference between physical and non-physical quantities can be summarized by the so called gauge principle Gauge Principle: Physical observables are described by gauge invariant quantities. Since E and B are gauge invariants, anything constructed with them are also gauge invariant, for instance, the electromagnetic energy density ɛ o E 2 /2 + B 2 /2µ o. In this way, although physical, neither E or B are fundamental. The fundamental fields are φ and A, in the sense that they are indivisible. Hence, a gauge theory is constructed from the gauge fields (as fundamental objects) of the theory. However, due to the gauge symmetry, they can not be observables. Instead, any observable must be constructed from gauge fields and has to be gauge invariant. A very beautiful example (Exercise 1.3) of the gauge principle is the Aharonov-Bohm effect [53, 54]. Let us consider a long thin solenoid with constant magnetic field given by B = B oˆk inside the solenoid and 0 otherwise. A vector potential whose curl produces such magnetic field is A = Φ/(2πr) ˆϕ where ˆϕ is the polar angle in cylindrical coordinates and Φ = B ds. (7) S is the magnetic flux through a surface whose boundary is a curve around the solenoid. Then, an electron beam is split in two, with each sub-beam passing on different sides of the solenoid (This can be done by the the double-slit apparatus). It can be shown that the interference pattern between both sub-beams on a screen will be shifted by a factor ( ) e δ = exp A dl. (8) C with respect to the case with no solenoid. In (8), e is the electron charge and is the Planck constant. Since this is an observable phenomenon, it should be consistent with the gauge principle. Indeed, (8) is gauge invariant: e A dl e A dl + e ξ dl. (9) C C C 2 Also known as Lorentz gauge.

7 Since the last term vanishes identically, the shift (8) is gauge invariant. Moreover, due to Stokes theorem ln δ = A ds = B ds = Φ, (10) S where S is any surface with boundary C. Thus, the shift of the interference pattern, although given in terms of A, is a gauge invariant quantity. This illustrates the fact that, even though A is not physical, it can generate observable phenomena through gauge invariant objects constructed from it and starting from postulates which are based on gauge concepts. S 7 B. Covariant formulation and U(1) symmetry We will now be a bit more formal in constructing the classical electromagnetism as a gauge theory by writing it down in a manifestly covariant form, since it is a natural relativistic theory. 1. Definitions We define Greek indexes as collective indexes for space and time, α, β,... {0, 1, 2, 3}, where the 0th coordinate stands for time. The derivative operator will be written in a simplified form as µ = / x µ. Moreover, from now on, we adopt natural units 3 and the 4-dimensional Euclidean 4 metric will be assumed, i.e., δ µν diag(1, 1, 1, 1). We consider a four dimensional gauge field A µ and its gauge transformation A µ A µ + 1 e µξ. (11) This definition coincides with the gauge transformations (4) for A 0 = φ (Exercise 1.4) and the factor 1/e is introduced for convenience. The gauge transformation (11) is a gauge transformation of a gauge connection under the action of an element of the U(1) group, i.e., the group of complex numbers with unit modulus. In fact, (11) can be rewritten as A µ A µ + 1 e e ξ µ e ξ. (12) where 5 e ξ U(1). Formally, the gauge field is a connection over an abstract topological space 6 which we will call gauge space. Thus, a covariant derivative can be defined as D µ = µ iea µ. (13) where e is the coupling parameter which is typically associated with the electron charge in QED. In general, covariant derivatives do not commute and define the curvature of the gauge space (commonly known as the field strength) (Exercise 1.5): [D µ, D ν ] = ief µν = ie ( µ A ν ν A µ ). (14) where the derivatives are supposed to act over a generic charged field, for instance, a spinor field describing the electron. It is simple to show that the field strength is gauge invariant (Exercise 1.6). The tensor F µν is obviously anti-symmetric. Thus, it can be described by an anti-symmetric 4 4 matrix whose components coincide with the electric and magnetic fields (Exercise 1.7), 0 E 1 E 2 E 3 E F µν 1 0 B 3 B 2 E 2 B 3 0 B (15) 1 E 3 B 2 B From now on, unless the contrary is said, we will employ natural units c = = 1. 4 Minkowskian metric would be preferable for perturbative treatments. However, we will deal with non-perturbative phenomena whether Wick s rotation is not established. Thus, it is often chosen to start from Euclidean metric, where we can actually perform explicit computations in quantum field theory. 5 Any complex number with unit norm can be written in this form. 6 See Ap. B.

8 The field strength is clearly gauge invariant, a property which makes it an observable of the theory, as it should be due to its gauge invariant components Maxwell equations and gauge fixing In the covariant formulation, Maxwell equations (1) read ν F µν = j µ, ɛ µναβ α F µν = 0, (16) where j µ is the four-current density, j µ (ρ, ρv). The first set of the Maxwell equations above coincide with the inhomogeneous equations of (1) (Exercise 1.8) while the second set coincide with the homogeneous equations of (1) (Exercise 1.9). In fact, analogously to (2) and (3), the solution of the second equation of (16) is (14). Moreover, this is a topological equation because the solution is the definition of the gauge potential, i.e. the definition of the gauge potential automatically satisfies this equation. These equations are recognized as the electromagnetic Bianchi identities. Maxwell equations (16) can be rewritten in terms of the gauge field (Exercise 1.10): the first of (16) leads to µ ν A ν A µ = j µ, (17) while the second is automatically satisfied, as discussed. A useful way to simplify the equations (17) is to fix a gauge, which means that we can impose a constraint over A µ in order to select a class of gauge potentials among all possible gauge potentials that could lead to a consistent field strength solution. Typicality, the linear covariant gauges are imposed: µ A µ = f, where f is a spacetime function. The special case f = 0 is recognized as the Landau gauge while the non-covariant gauge i A i = 0 is the Coulomb gauge. In the case of the Landau gauge, for instance, Maxwell equations (17) simplify to A µ = j µ, (18) It is very useful to define an action for electromagnetism. A typical choice is ( ) 1 ( α ) S M = d 4 x 4 F µνf µν j µ A µ + d 4 x 2 b2 + b µ A µ, (19) where b is the Lautrup-Nakanishi field [43] which is a Lagrange multiplier enforcing the linear covariant gauge fixing. It is a straightforward exercise (Exercise 1.11) to show that the minimization of this action with respect to A µ leads to a gauge fixed Maxwell equations and the minimization with respect to b leads to the gauge fixing condition 7. In summary, electrodynamics, as described by the action (19), can be obtained by postulating: Gauge principle with respect to the U(1) group - This postulate naturally leads to the existence of a gauge connection and a gauge curvature. (Euclidean) Lorentz invariance of the action - This postulate ensures the principle of relativity. Moreover, Lorentz transformations form a group whose Euclidean version is the group of four-dimensional rotations in spacetime, the SO(4) group. Further requirements as locality 8 and absence of higher order derivatives can be employed in a consistent way in order to avoid unwanted terms in (19). Any field coupled to A µ must respect all above postulates. 7 In field theory, the fields equations can be obtained through the generalized Euler-Lagrange equations: ( ) L Φ A L µ ( µφ A = 0. ) where Φ A is a generic field with A being a collective index. 8 Roughly speaking, a local quantity in QFT is an object that only depends on a single spacetime point.

9 9 C. Non-Abelian gauge theories From now on, due to the intricacies of non-abelian theories (which will become evident as we proceed), we will neglect the coupling with external sources j µ or other fields. 1. Lie groups basics Our aim in this section is to provide a brief overview of what we need about Lie groups and group theory. It is not our intent to provide detailed discussion on this matter. For detailed texts about Lie Groups, see for instance [55 57] and also [1] for an extended, yet summarized, introduction. To start with, let us enunciate the axioms of a group. Consider a set G of elements u and an operation, then G is said to be a group if its elements obey: Closure: If u 1 and u 2 G then u 3 = u 1 u 2 G. Associativity: If u 1, u 2 and u 3 G, then (u 1 u 2 ) u 3 = u 1 (u 2 u 3 ). Identity: u o = 1 G u 1 = 1 u = u u G. Inverse: u 1 G u 1 u = u u 1 = 1 u G. Moreover, If G is infinite and continuous, the group is said to be a Lie Group. If [u 1, u 2 ] = 0 u 1 and u 2 G, the group is said to be an Abelian group, otherwise it is said to be a non-abelian group. The notion of subgroup will also be important along this lectures: If G is a group and H G is a smaller set than G, then H is a subgroup of G if H is also a group, i.e. if the elements of H obeys the four group postulates. Lie groups are typically, but not exclusively, N N matrix groups (the case N=1 is a group formed by scalars, like U(1)). A useful property of Lie groups is that their elements can always be written as exponentials u = e ξ, where ξ is the gauge parameter. When the gauge parameter is a function of the spacetime coordinates, the group is said to be a local group, otherwise the group is said to be a global group. The algebra of the group is the tangent space of the group at the identity[1]. For any element near the identity, we can write u 1 + ξ. We can expand ξ through a matrix basis T A as ξ = ξ A T A where the capital Latin indexes A, B,..., H {1, 2,..., dim G}. The exponent ξ is said to belong to the algebra of G. It should be clear that if is multiplication, its action with respect to ξ and T A is addition. It is important to have in mind that, in general, the dimension of the group does not coincide with the dimension of its elements, dim G N. The basis elements T A are called generators. The characterization of the algebra is given by the commutation rules of the generators, [T A, T B ] = f ABC T C, (20) where the skew-symmetric object f ABC is composed by the structure constants of the group. If f ABC = 0, then the group is obviously Abelian. Moreover, the structure constants obey the Jacobi relations (which can be derived from (20)) As a supplementary condition, the generators are normalized as f ABC f CDE + f ADC f CEB + f AEC f CBD = 0. (21) Tr(T A T B ) = 1 2 δab. (22) A representation of a group is the realization of the elements u G as a linear operator U(u) acting on a linear space. Thus, given the elements of V, denoted by ψ i, a representation is characterized by ψ i = U ij ψ j. (23)

10 10 It is demanded that U(u 1 u 2 ) = U(u 1 )U(u 2 ), U(u 1 ) = [U(u)] 1, U(1) = 1. (24) Clearly, this is a mapping u U which will be reflected on the generators of G. Hence, (24) implies that where a R. Thus, we also have U(T A + T B ) = U(T A ) + U(T B ), U(aT A ) = au(t A ), U([T A, T B ]) = [U(T A ), U(T A )], (25) U(u 1 + ξ) 1 + U(ξ) = 1 + ξ A U(T A ). (26) In gauge theories, the most relevant representations are the fundamental representation and the adjoint representation. For the fundamental representation, let us consider G as a group of N N matrices. Then, an N-dimensional vector space V with elements ψ I, transform under G as ψ I = U IJ ψ J, where I, J, K,... 1, 2, 3,... N. In this case, the elements U are simply N N matrices. The adjoint representation is a direct realization of the algebra of G. The action of an element u in the adjoint representation on an element of the algebra ξ is given by Ad(u)ξ = u 1 ξu. Then, u 1 ξu is required to belong to the adjoint representation as well. For u 1 + η it is then clear that (Exercise 1.12) Ad(η)ξ = [ξ, η]. Thus, if ξ and η are simply the generators, we have that Ad(T B )T A = f ABC T C. On the other hand, the definition of a representation (23) yields [T A ] BC = f ABC, (27) which means that in the adjoint representation the dimension of the elements coincide with the group dimension. It is a straightforward exercise (Exercise 1.13) to check that both, fundamental and adjoint, representations respect (24) and (25). Let us discuss a few important groups. We already discussed the unitary Abelian group U(1) which is the group of all complex numbers with unitary modulus, i.e. zz = 1. Taking z = e iξ, where ξ R, it is almost trivial to check all four group axioms (Exercise 1.14). The generalization of the unitary group to matrix groups is realized by the group of unitary N N complex matrices, i.e. uu = 1. This group is denoted by U(N). From its definition we have that det u = ±1 and dim U(N) = N 2 1. The restriction for positive determinants excludes discrete transformations such as reflections. This is the so called special unitary group, denoted by SU(N). It is easy to verify that unitary groups respect the group axioms (Exercise 1.15). Geometrically, the elements u SU(N) are rotations of complex vectors. The group of N N orthogonal real matrices is denoted by O(N) and obeys uu T = 1 where u T is the transpose of u. Obviously, det u = ±1. It is clear that u O(N) represent rotations and reflections of real vectors in an N-dimensional Euclidean space and hence dim O(N) = N(N 1)/2. The restriction to pure rotations leads to the special orthogonal groups SO(N). Again, the verification of the group axioms is very easy (Exercise 1.16). The algebra of SO(N) is related to the Euclidean metric. In fact, in the adjoint representation, the generators are matrices anti-symmetric in their indexes, T ab = T ba, where a, b, c,..., h {1, 2, 3,..., N(N 1)/2}, obeying [T ab, T bc ] = f abcdef T ef, f abcdef = 1 2 [( δ ad δ be δ ae δ bd) δ cf + ( δ ac δ be δ ae δ bc) δ df ]. (28) All examples mentioned above assume to be matrix multiplication. The group of translations, commonly denoted by R N, is an example of a group whose operation is addition. Hence, given an N-dimensional vector space V, the elements u R N are also N-dimensional vectors in the fundamental representation. The action of the group elements on the elements of V is given by ψ = ψ + u. In this case, the identity is the null vector ψ i o = 0 and the inverse is u. Closure and associativity can be directly verified (Exercise 1.17). Because translations commute, the algebra of translations is characterized by [P A, P B ] = 0. (29)

11 It is possible to define a more general group by considering rotations and translations in a single larger group, the so called Euclidean Poincaré groups ISO(N) = SO(N) R N, where is a semi-direct product 9. The action of the Euclidean Poincaré group on a vector ψ V is ψ = Λψ + u, (30) 11 where ψ is written as a N-dimensional column vector, Λ = Λ ab T ab SO(N) and u = u a P a R N. To show that ISO(N) is a group is also an easy exercise, yet bigger than the previous ones (Exercise 1.18). In the same way, the properties of the Lorentz group SO(1, 3) (which is the group of four-dimensional orthogonal matrices preserving the Minkowskian metric) and the general linear group GL(N, R) (which is the group of all N N real matrices) can be easily checked (Exercise 1.19). In what follows, we will stick mainly to Lie groups which are semi-simple and compact Gauge field, field strength and Yang-Mills action The generalization from QED to non-abelian gauge theories relies on its extension to a non-abelian group, G. Following the Abelian case, we start by defining the gauge field as an algebra valued connection of the group space in the adjoint representation, A µ = A A µ T A. The generators are chosen to be anti-hermitian, (T A ) = T A, so [T A, T B ] = f ABC T C. The respective covariant derivative which acts on objects with adjoint group indexes is D AB µ = δ AB µ κf ABC A C µ, (31) where κ is the coupling parameter (analogous to e in QED). Equation (31), in a component independent notation, can be written as D µ = µ + κ[a µ, ]. (32) It is easy to show that (32) reduces to (31) (Exercise 1.20). The gauge transformations are now given by A µ u (A 1 µ + 1 ) κ µ u, (33) where u G. It is convenient to consider infinitesimal transformations through u 1 + ξ, which gives (Exercise 1.21) A A µ A A µ + 1 κ DAB µ ξ B. (34) From (33) and (34) we can see that gauge transformations are no longer linear as in the Abelian case. This non-linear character is the main reason why non-abelian theories are much more complicated than Abelian ones. Again, by commuting covariant derivatives, one finds the field strength (curvature of the gauge space): where [D µ, D ν ] = κf µν. (35) F µν = µ A ν ν A µ + κ[a µ, A ν ]. (36) Two very important observations are in order now: First, the field strength has a non-linear piece, not present at the Abelian case. This feature means that the gauge field interacts with itself, for instance, gluons interact among each other (a property not observed in photons). Second, the field strength is not a gauge invariant object (Exercise 1.22), F µν u 1 F µν u. (37) 9 The semi-direct product is a direct product for which one sector is a normal subgroup, which is the case of R N. See also the next footnote 10 Roughly speaking, a compact group is a bounded and closed group. Semi-simple group is a group that can be decomposed in simple groups. Simple groups are groups with no normal subgroups, except for the identity and the group itself. Finally, a normal subgroup N of G is characterized by g 1 Ng N. For the actual formal definitions we refer to [55 57]

12 In fact, this is a typical transformation rule of matter fields in the adjoint representation. So, the electromagnetic analogues can not be associated with observables. The field strength can be written in components of the algebra, F µν = F A µνt A, as (Exercise 1.23) F A µν = µ A A ν ν A A µ + κf ABC A B µ A C ν. (38) 12 To construct a consistent action, we evoke: The gauge principle for the group G; Euclidean Lorentz invariance; Locality; Absence of higher order derivatives. It turns out that the simple generalization of (19) is indeed gauge invariant (Exercise 1.24): S Y M = d 4 x 1 4 F µνf A µν A, (39) where we have omitted the gauge fixed part (see next section) and a possible interacting term (as previously announced). This action is known as Yang-Mills action [6, 7]. The non-linear character of the field strength leads to non-linear field equations which means that this kind of theories are physically richer as well as much complicated when compared to Abelian cases. In fact, variation of the Yang-Mills action with respect to the gauge field leads to (Exercise 1.25) On the other hand, because of (38), a topological identity exists, D µ F A µν = 0. (40) ɛ µναβ D ν F A αβ = 0, (41) which is recognized as the Bianchi identity and generalizes the homogenous electromagnetic field equations in (16). II. QUANTIZATION OF GAUGE THEORIES Til now, we have defined only the classical quantities of gauge theories. To study the quantum aspects of Yang- Mills theories we have to promote the fields to quantum operators or, equivalently, write down a consistent functional integral of the exponentiated classical action. We will follow the second option here [6]. If we naively try to consider 11 Z = DAe S Y M, (42) we will face inconsistencies right from the beginning. The main problem relies on the gauge symmetry: The functional integral sums over all possible field configurations in order to provide a probability of an event or an expectation value. However, the gauge symmetry establishes that there is an infinity number of equivalent fields related through gauge transformations. Therefore, each field is actually been infinitely overcounted. Hence, the probabilities are not being conserved. In fact, if one tries to compute the propagators of the theory (take the linearized version eq. (40) and try to find the associated Green function), a divergent Green function is found (Exercise 1.26). Thus, even the free theory is inconsistent (this is also true for QED). The solution is to consider only one configuration among all equivalent fields (those connected through gauge transformations) - for each infinite set of gauge fields related through gauge transformations, we must choose only one representative. This procedure is called gauge fixing and is implemented by imposing a constraint over the gauge field. The gauge fixing for the Abelian case was already discussed here, at classical level - It is easily generalized to the non-abelian case, also at classical level. At quantum level however, it is not a trivial task. The way to do this in the path integral is through the Faddeev-Popov trick [58], which we discuss in the next section. 11 Since we are not going to perform any practical computation at this level, we omit the classical Schwinger sources [6].

13 13 A. Faddeev-Popov quantization The question here is: How to introduce a gauge fixing in a consistent way in (42)? The idea is to integrate only over fields obeying a certain constraint. Let us consider the linear covariant gauges µ A µ = f. The trick is to write the unit in a non-trivial way 1 = (A) Duδ(f µ A u µ), (43) where A u µ is the gauge orbit, which is defined through A u µ = u (A 1 µ + 1 ) κ µ u, (44) while A µ is a fixed, yet arbitrary, gauge configuration. It should be clear that the integral in (43) is gauge invariant because of the closure property of G. This means that (A) = (A u ). Therefore, Z = DAe S Y M (A) (A) Duδ(f µ A u µ), (45) From the gauge invariance of S Y M, (A) and DA, we can write Z = DA u e S Y M (A u) (A u ) Duδ(f µ A u µ), (46) Thus, performing the change of variables A u µ u ( A u µ + 1 κ µ) u 1 = A µ, we achieve Z = DAe S Y M (A) (A) Duδ(f µ A µ ). (47) It remains to determine (A). This is easily done by changing the integration variable of (43) from u to f = µ A u µ f: ( ) ( ) δu δu 1 (A) = Duδ(f) = Dfδ(f) det = det, (48) δf δf f=0 which can be evaluated for infinitesimal gauge transformations u 1 + ξ (Exercise 1.27), ( ) δf (A) = det = δξ f=0 1 κ det( µd µ ). (49) Thus which can be normalized to finally obtain Z = DAe S Y M det( µ D µ )δ(f µ A µ ) 1 Du. (50) κ Z = DAe S Y M det( µ D µ )δ(f µ A µ ). (51) This expression is supposedly free of gauge ambiguities. However, the determinant and the functional delta is not in a form which we can recognize a gauge fixing, as in (19). In fact, these terms can be very complicated to handle because they may ruin a simple algorithm to perform explicit perturbative computations such as the Feynman rules. Fortunately, both terms can be easily exponetiated with the help of additional fields. Let us start with the functional delta: Because gauge invariant quantities should not depend on f, we can integrate over f with a Gaussian weight 12, Df exp ( 1 α Tr d 4 xf 2 )δ(f µ A µ ) = exp [ 1 α ] d 4 xtr( µ A µ ) 2, (52) 12 The sign of the Gaussian seems to be wrong. However, (22) still have to be employed, correcting the sign at the final expression.

14 14 where α is a non-negative number 13. It will be useful for the next sections to rewrite (52) as a normalized functional integral of an auxiliary bosonic algebra-valued field b (Exercise 1.28), [ ] 1 exp d 4 xtr( µ A µ ) 2 Db exp [Tr (2b µ A µ + αbb)]. (53) α The field b is recognized as the Lautrup-Nakanishi field [43]. Just like the Abelian case, it plays the role of a Lagrange multiplier enforcing the gauge fixing. The determinant is also simple lo exponentiate, it is a determinant with positive unit power, thus, it can be substituted by a Gaussian functional integral over Grassmann variables [6]. Recalling that µ D µ is an dim G dim G matrix operator, we need a set of dim G 2 independent variables, thus [ ] det( µ D µ ) = DcDc exp c µ D µ c, (54) where c and c are independent Grassmann variables. The local version of (51) is then Z = DADbDcDc e S Y MF P. (55) where the action which must be considered for quantum theory is with A few observations are in order: S Y MF P = S Y M + S gf, (56) [ ( S gf = d 4 x b A µ A A µ + α 2 ba) + c A µ Dµ AB c B]. (57) The classical field equation of b results in the gauge fixing condition for f a = αb a (Exercise 1.29). Therefore, with the help of b a, it is more evident that the gauge fixed action actually enforces the proposed constraint; The cases α = 0 and α = 1 are recognized as the Landau and Feynman gauges, respectively; The algebra-valued fields c = c A T A and c = c A T A are the Faddeev-Popov ghosts and anti-ghost fields. They are fermionic fields with no Lorentz indexes (scalars), i.e., they are spin-0 bosons obeying Fermi statistics (because of the Faddeev-Popov determinant). In principle, they should violate the spin-statistics theorem [59] and, consequently, a causality violation is expected. However, what happens is that these fields are actually non-physical because they do not appear as asymptotic states [6], i.e., they are not observables of the theory. Nevertheless, together with the gauge fixing, they play an important role in canceling non-physical degrees of freedom of the gauge fields, ensuring the unitarity of the theory. In fact, there is a class of gauge fixings where the Faddeev-Popov fields decouple, the so called axial gauge n µ A µ = 0, where n µ is a fixed vector [4]; An important consequence of (56) is that it displays the Faddeev-Popov discrete symmetry, which means that S Y MF P is invariant by c c and c c. This symmetry characterizes the ghost number, see Table I. Since it is a symmetry of the action, all observables should also be invariant under the Faddeev-Popov symmetry, which means that this symmetry prevents ghost states to appear in external legs of any diagram - so, causality is safe indeed; In the case where G is Abelian, such as in QED, the Faddeev-Popov determinant can be eliminated by simple normalization of Z (Exercise 1.30). Thus, QED is naturally ghost-free. The Yang-Mills action (56) has no mass parameters, i.e. the theory is massless and conformal at classical level. It is important at this point to expend a few words about the mass dimensions of the fields in four-dimensional spacetime. Because = 1, the action should be dimensionless in natural units. Thus, since [x] = 1 in mass dimension, [ d 4 x] = 4 and [ ] = 1. As a consequence, [L] = 4. A simple algebraic analysis leads to the second column of Table I (Exercise 1.31); 13 In a way, the arbitrariness of f is replaced by α.

15 15 Fields A b c c κ Dimension Ghost number TABLE I: Quantum numbers of the Yang-Mills fields in four-dimensional spacetime. B. BRST quantization A remarkable feature of the gauge fixed action (56) is that, in despite of the broken gauge symmetry, a new symmetry shows up, the so called BRST symmetry [6, 43, 60, 61], due to C. Becchi, A. Rouet, R. Stora and, independently, I. V. Tyutin. BRST symmetry is characterized by a fermionic operator s, acting on the fields as sa A µ = Dµ AB c B, sc A = κ 2 f ABC c B c C, sc A = b A, sb A = 0. It is a straightforward exercise to show that (Exercise 1.32) (58a) (58b) (58c) (58d) ss Y MGF = 0. (59) From (58a - 58d) it is clear that BRST symmetry is actually a supersymmetry because it transforms bosons in fermions and vice versa. Moreover, the BRST operator is nilpotent, s 2 = 0 (Exercise ), a very welcome feature. From (58a - 58d) and Table I we also have that [s] = 0 and that s has ghost number 1. Hence, when acting over an object, the BRST operator increases its ghost number by 1. It turns out that this very beautiful symmetry plays a fundamental role in non-abelian gauge theories. As briefly discussed in the Appendix B, s has a deep geometrical meaning, it is the exterior derivative in the functional space of all gauge configurations along a gauge orbit. Moreover, the Faddeev-Popov ghost field is a Maurer-Cartan form in this very same space while the gauge field is a connection over a more primitive fiber bundle structure [2, 62, 63], which we have called gauge space. In fact, the Faddeev-Popov determinant corresponds to a Riemannian metric in the gauge functional space. Thus, independently of the Physics (independently of the chosen action), A, c and s exist as fundamental mathematical abstract structures. This fundamental appeal is the origin of the so called BRST quantization method where the BRST symmetry is taken as the fundamental principle to construct the invariant action. The main ingredients are the gauge field A, the ghost field c and the BRST operator s. The anti-ghost c and the Lautrup-Nakanishi b fields are introduced in order to account for the gauge fixing and also to ensure the Faddeev-Popov discrete symmetry. Before we proceed, let us enunciate two important results about the BRST operator: The fact that s is a nilpotent operators allows one to define its cohomology, the BRST cohomology. The BRST cohomology problem is the problem of finding the solution of 15 sθ g n = 0, (60) where Θ g n is typically and integrated object with specific ghost number g and dimension n. The solution of (60) is [43] Θ g n = Υ g n + sθ g 1 n sυg n = 0 and Υ g n s(something). (61) In (61), Υ g n is the non-trivial part of the cohomology and sθ g 1 n is the trivial sector of the cohomology Tip: Use the Jacobi identities (21). 15 This is actually the fundamental cohomology problem for any nilpotent operator. 16 It is commonly used the terminology BRST exact object for trivial objects and BRST closed object for non-trivial objects s(exact) = 0 exact = s(something), s(closed) = 0 closed s(something). (62)

16 BRST doublets consist on pair of fields transforming through sx = Y and sy = 0. Clearly, c and b are BRST doublets. A general result is that BRST doublets belong exclusively to the trivial sector of the cohomology [43]. In practice, any BRST doublet can be introduced because they are harmless to the physical content of the theory. Thus, to find the BRST invariant action it is imposed that S is a local polynomial in the fields and their derivatives (with no higher order derivatives), with dimension bounded by 4, vanishing ghost number, Euclidean spacetime isometry, obeying 16 ss = 0. (63) Condition (63) is actually the BRST equivalent of the gauge principle. Therefore, following (63), the solution is then given by S = S 0 + s 1 ss 0 = 0, S 0 s(something). (64) where 1 is a local integrated polynomial in the fields and their derivatives, with dimension bounded by 4 and ghost number 1. The fact that BRST doublets always belong to the trivial sector combined to the Faddeev-Popov symmetry yield that c, b and c are not allowed at S 0, i.e. S 0 = S 0 (A). Hence, it can be shown that (Exercise ) (see [43]) S 0 = S Y M, ( 1 = d 4 x c A µ A A µ + α 2 ba + βf ABC c B c C). (65) where α and β are dimensionless gauge parameters. For the particular case β = 0, (64) reduces to the Faddeev-Popov result(56) (Exercise 1.35). In fact, (57) can be written as a BRST exact object ( S gf = s d 4 x c A µ A A µ + α 2 ba). (66) which means that the gauge fixing does not interfere with the physical observables. An example of a non-physical quantity is the gauge field propagator (Exercise 1.36), where the transverse and longitudinal projectors read A A µ ( p)a B µ (p) = δ AB 1 p 2 [T µν + αl µν ], (67) T µν L µν = δ µν p µp ν p 2, = p µp ν p 2. (68) The gauge field propagator is then clearly gauge dependent and describes massless non-physical states. C. Renormalization, physical states and unitarity There are three very important consequences of BRST symmetry: renormalizability, definition of physical states and unitarity. Renormalizability states that all divergences can be consistently eliminated by a suitable redefinition of the fields and parameters. This can be performed through the algebraic renormalization techniques [43]. Essentially, BRST symmetry is imposed for the quantum action Γ, defined through e Γ = Z. (69) 17 This is not a trivial exercise. For the non-trivial part, write down a linear combination of all possible integrated polynomial terms using A,, κ, and f ABC. Impose the requirements above listed and then apply s. The result must vanish, a property that will provide specific values for the coefficients of the linear combination. The final result should be the Yang-Mills action. The trivial part is a bit more complicated due to the higher number of fields (which includes A as well).

17 17 The quantum action can be perturbatively expanded around the classical action, Γ = Γ 0 + ɛγ 1 + ɛ 2 Γ (70) where Γ 0 = S Y MF P, ɛ is a small expansion parameter and Γ n, for n > 0, are integrated local polynomials in the fields and their derivatives bounded by dimension four, vanishing ghost number and Euclidean Lorentz invariance. Imposing 18 sγ = 0 and using sγ 0 = 0, it is possible to show that, at first order, ɛγ 1 can be reabsorbed in Γ 0 by means of multiplicative redefinition of the fields and parameters, ensuring that Γ (1) = Γ 0 + ɛγ 1 has the same form as Γ 0. The method is recursive and can be applied to all orders. Thus, Yang-Mills theories are renormalizable to all orders in perturbation theory, at least in a large class of covariant gauges. The gauge principle is used to determine physical observables at the classical level. In non-abelian theories, the BRST symmetry must be taken in favor of gauge symmetry because of the presence of the Faddeev-Popov ghosts. Thus, (56) can also be considered at classical level. Therefore, since the BRST symmetry is postulated to be the fundamental symmetry of Yang-Mills theories (it generalizes the gauge symmetry), it should be used to define the physical states of the quantum theory. For that, it is used the nilpotent BRST charge Q brst which is the generator of BRST symmetry. From (Exercise ), Q brst = d 3 j0 brst, (71) we can define the physical quantum states as BRST invariants, Q brst ψ = 0, (72) where ψ is required to have vanishing ghost number [64, 65] and to respect all symmetries displayed by S Y MF P. However, following the cohomological prescription (60) and (61), ψ will have a non-trivial piece ψ phys and a trivial redundancy ψ nphys, namely Q brst ψ phys = 0 ψ phys Q brst something, Q brst ψ nphys = 0 ψ nphys = Q brst something. (73) It is clear that ψ nphys has a vanishing norm (Exercise 1.38). Thus, the actual physical states are those belonging to the non-trivial BRST cohomology in the space of vanishing ghost number states. Unitarity of the S-matrix ensures the probability conservation in physical processes [6, 64, 65]. BRST symmetry plays a fundamental role in that issue, guaranteeing that states with negative norm do not appear as asymptotic states, i.e. they are not observables. The demonstration relies on the definition of physical states, Ward identities 20 and commutation relations involving Q brst. D. Asymptotic freedom and confinement The renormalization of the coupling parameter κ leads to the concept of asymptotic freedom [6, 66 68], which means that the coupling parameter is higher as lower is the energy scale. In fact, at first order in perturbation theory, it is found that [66, 67] κ 2 = 16π 2 11N 3 ln µ2 Λ 2 qcd, (74) 18 All symmetries must be imposed as well. 19 Noether s theorem [6] establishes that j µ = allfields L ( µφ A ) δφa. 20 Ward-Takahashi identities are functional equations describing the symmetries of an action. At quantum level, their role is to provide recurrence relations between Green funtions.

18 18 where N is the Casimir of the gauge group (This is the dimension of the vector space of the fundamental representation), µ is the energy scale and Λ qcd is the renormalization group cutoff [68]. The renormalization group cutoff is a phenomelogical parameter which is associated with the pion mass. From (74) it is clear that as µ, κ 0. Thus, as higher the energy, more weakly coupled to each other are the gauge fields and perturbation theory is very consistent. At this level, the gauge fields are almost free. On the other hand, as µ Λ qcd, µ. In this case, the gauge fields become strongly coupled and perturbation theory does not hold anymore, i.e. the theory become highly non-perturbative. The point µ = Λ qcd is known as the Landau pole and indicates a phase transition of the theory which is recognized as the hadronization phenomenon [69]. Below µ = Λ qcd the Yang-Mills action must be replaced by a suitable theory where the excitations are hadrons and glueballs 21. The concept of confinement emerges from the behavior (74) together with the global G symmetry 22. The essence of the global symmetry is to forbid any algebra-valued to be a physical state, Q brst ψ A > 0. Only states carrying no group indexes would allowed at the physical sector. This property should hold at all energy scales 23. Even at high energies, where the coupling is very small and the gauge excitations barely interact among each other, is also impossible to observe them due to its algebra-valued character. At this regime, considering also quarks, the theory describes the so-called quark-gluon plasma, a phase in which quarks and gluons are virtually free, even though it is impossible to isolate a single quark or gluon. By decreasing the energy, the coupling increases and the interactions between gluons (and quarks) become stronger and stronger til the point of hadronization at the Landau pole. Then, hadrons and glueballs, which are experimentally observables, are in fact states with no gauge charge. III. GRIBOV AMBIGUITIES In this section, we will introduce the problem pointed out by V. N. Gribov in his seminal paper [14]. The essence of the problem lies on the fact that the Faddeev-Popov method, described in Sect. II, is not sufficient to fix the gauge redundancy completely. It means that, after applying the Faddeev-Popov gauge fixing procedure, a residual symmetry remains: there still are configurations obeying the same gauge fixing and related through gauge transofrmations. Such spourious configurations are called Gribov copies and, although present in a wide class of gauges, their elimination is not fully understood. The first (partial) attempt to deal with such copies was realized by Gribov himself, and it was worked out in more generality by Zwanziger later, [70]. A local and renormalizable action, free from infinitesimal Gribov copies (copies that are related to another configuration through an infinitesimal gauge transformation) was established, the so-called Gribov-Zwanziger action. This action was firstly constructed in the Landau gauge and it has a very interesting feature: the term which is responsible to guarantee the elimination of copies breaks the BRST symmetry (58a-58d) in a soft manner 24. Further non-perturbative effects as the presence of condensates can be taken into account in this scenario giving rise to the Refined Gribov-Zwanziger action, [71], which provides gluon and ghost propagators in very good agreement with lattice results. In this section, we provide an introduction to these developments in a very concise form. For further details, we refer to more complete treatments which are listed along the text. A. Statement of the problem As discussed in Sect. II, the quantization of gauge theories requires the imposition of a condition over the gauge fields. Although the Faddeev-Popov framework (or BRST quantization) provides a very efficient way to fix a gauge in perturbation theory, it fails at the non-perturbative level. The reason is the following: The application of the FP procedure tacitly assumes that the gauge fixing condition has a unique solution for each gauge orbit (44). This assumption does not hold when we enter the non-perturbative sector and spurious configurations are not eliminated from the path integral quantization. As a matter of illustration, we consider pure Yang-Mills action in four-dimensional Euclidean space with SU(N) gauge group 25 and we impose the Landau gauge condition over the gauge fields, which 21 This is actually an open problem in theoretical physics. 22 In QCD this symmetry is the color symmetry associated with the SU(3) global group. In electroweak interactions the SU(2) U(1) local and global symmetries are broken to the U(1) group due to spontaneous symmetry breaking. Hence, confinement is not a feature of electroweak interactions. 23 There are no experimental evidence of confinement violation whatsoever. 24 A breaking is called soft if it is an operator with dimension lower than spacetime dimension. Otherwise, it is a hard breaking. 25 The group SU(N) is chosen to fit with the existing literature. The generalization to a generic semi-simple Lie group G can be straightforwardly done.