Adler-Bardeen Theorem in the presence of external fields

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Universit`

a degli studi di Pisa

Dipartimento di Fisica

Corso di laurea magistrale in Fisica

Tesi di Laurea Magistrale

Adler-Bardeen Theorem in the

presence of external fields

Relatore:

Prof. Damiano Anselmi

Candidato:

Ivan Garozzo

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Introduction

The quantum field theory is one of the milestones in the modern theoretical physics. From the very beginning, the perturbative framework has revealed to be a prolific approach to compute the physical quantities. This approach rarely allows us to obtain exact quantities to all orders in perturbation theory, with one remarkable exception: certain anomalies. This is due to a powerful theorem known as the Adler-Bardeen theorem. In its original formulation [4, 5] it states that the Adler-Bell-Jackiw axial anomaly is one-loop exact. The theory for which this theorem has been originally proved is the quantum electrodynamics. The proof is based on a diagrammatic analysis that allowed Adler and Bardeen to provide a subtraction scheme where the one-loop exactness of the axial anomaly is a manifest property.

In recent years it has been proved a second statement concerning the Adler-Bardeen theorem [6] in which it is proved that there exists a subtraction scheme where, if gauge anomalies vanish at one-loop, they vanish to all orders. This formulation is crucial since it allows us to prove that the Standard Model is free from gauge anomalies to all orders by just computing them at one loop and showing that they vanish at that order. Nevertheless, if the Adler-Bardeen theorem did not hold, nothing would ensure that the cancellation of the one-loop anomalies implies the anomaly cancellation to all orders.

The aim of this work is to upgrade this second statement of the Adler-Bardeen theorem to the presence of some external, background, fields. The presence of external fields enlarge the structure of the potential anomalies of the theory. In addition to the gauge anomalies, we now may find other two types of anomalies: the ones that are linear in the quantum fields and quadratic in the background fields, and the ones being linear in the background fields and quadratic in the quantum ones. The former represents the potential one-loop exact anomalies to all orders. If we want to avoid violations of the theorem, we have to include the cancellation of the anomalies that are quadratic in the quantum fields and linear in the external ones. In some sense the proof of the Adler-Bardeen theorem with external fields can be regarded as a proof of the ’t Hooft anomaly matching conditions,

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which state that the anomaly coefficient is the same both in the UV and IR regimes of a given theory.

The work is organized as follows:

• In Chapter 2 we introduce anomalies in quantum field theory. After a discussion on the dimensional regularization for chiral theories, we compute the axial anomaly through the celebrated triangle diagram. Then we present the Wess-Zumino consis-tency conditions that constraint the form of the anomaly functional. These condi-tions are employed, using the BRST formalism, to determine the complete anomaly polinomial in D = 4, recovering the famous Bardeen formula for the anomaly in a theory with a non-Abelian symmetry. Moreover, the ’t Hooft anomaly matching conditions are discussed following the original ’t Hooft argument. The chapter con-tains the relation between the axial anomaly and the decay π0 _→ _{2γ. Finally, we}

discuss the cancellation conditions in the Standard Model;

• in Chapter 3 we present the Batalin-Vilkovisky formalism, that will be the language adopted to perform our analysis. For our purposes, this formalism will play a fundamental role since it allows us to keep track of the symmetry properties during the renormalization process of a given theory. After the basic tools have been introduced, namely the master equation, the canonical transformations and the master identity, we calculate the coefficient of the axial anomaly in QED in this formalism. Then, we discuss the properties of chiral Yang-Mills theory, which is fundamental for the proof of the Adler-Bardeen theorem for general chiral gauge theories;

• Chapter 4 contains our original results. First we give a brief review of the original proof provided by Adler and Bardeen. Then, we focus on the discussion of the exter-nal fields, explaining what kind of anomalies may be one-loop exact and motivating why there is the need to enlarge the set of cancellation conditions. Before we enter the technical proof of the theorem we present a simple model to which the theorem applies, namely chiral QED with gauged axial symmetry, which is instructive to un-derstand how the external fields are introduced and to see in a concrete example the role played by the extended set of cancellation conditions. The central part of this chapter is devoted to the proof of the theorem, making use of the renormalization properties of the given theory. In this sense it is crucial to find a suitable subtrac-tion scheme where the gauge anomalies cancellasubtrac-tion and the one-loop exactness of anomalies are manifest to all orders. To this aim, we introduce the DHD

regular-ization, that combines the dimensional regularization, whose cut-off is ε = 4 − D,

where D is the continued complex dimension of spacetime, and the higher-derivative technique, which introduces an energy scale Λ. The scheme in which the theorem is manifest is obtained by letting first ε → 0 and then Λ → ∞. However the physical

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quantities does not depend on the order we take these limits. A change in their order is equivalent to a change in the subtraction scheme, which may require the subtraction of some ad-hoc finite local counterterms from two-loop onwards in order to show the vanishing of gauge anomalies of the one- loop exactness of the ones related to the external fields, spoiling the manifest property of the theorem. In the end, we discuss three models where the theorem provides nontrivial exact quantities. Such examples may inspire interesting extensions of the Standard Model.

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Chapter 2

Anomalies in quantum field theory

The first part of this thesis is devoted to review the concept of anomalies in quantum field theory. In particular we will show the computation of the celebrated

Adler-Bell-Jackiw anomaly, first using a diagrammatic approach and then using the BRST formalism

developed to quantize gauge theories.

Anomalies reflect the break of a symmetry of the classical theory due to the process of quantization. Hence, a way to see the effects of anomalies is to study all features encoding symmetries, global or gauge ones, of a given classical theory. The crucial point is the Noether theorem, encoding the strict relation between symmetries and conservation laws. From such point of view, the anomaly is seen as a breaking term of the conservation law, at the quantum level, obeyed by a conserved current in the classical theory. When a global current is afflicted by an anomaly it is not a problem for the theory, actually sometimes this mechanism is phenomenologically important, being the π0 _{→ γγ} _decay

the most prominent example, as we will discuss later.

We have a completely different story if an anomaly arises for a gauge current, in fact in this case the theory is inconsistent. It is well known that gauge theories are characterized by a redundancy of their degrees of freedom, and the gauge symmetry is required to decouple longitudinal, unphysical degrees of freedom, from the transversal, physical ones. An anomaly in a gauge symmetry is indeed a sign of the inconsistency of the theory.

2.1 Dimensional regularization for chiral theories

The dimensional regularization requires a D−continued extended action. For chiral theo-ries a continued action can be obtained provided a consistent definition for the γ5 matrix

is introduced. The great virtue of the dimensional regularization is that it preserves gauge invariance. Unfortunately it does not admit a naïve definition of γ5. Recall that in four

dimension γ5 satisfies

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Moreover, γ5 matrix is involved in the following identity

Tr[γµγνγργσγ5] = 4εµνρσ. (2.2)

It is not possible to maintain both (2.1) and (2.2) to give rise a consistent definition for

γ5 in a D-dimensional spacetime. A new definition of γ5 matrix in the context of the

dimensional regularization has to be inquired. The key point is that we have to break the spacetime manifold RD _{into R}4

× R−ε_{, where R}4 _{is the usual four dimensional spacetime}

and R−ε _{is the evanescent sector of the spacetime manifold. In most expressions we lose}

the full SO(D−1, 1) invariance. At the same time we require SO(3, 1)×SO(−ε) invariance is not broken. This statement is tantamount to observe that gauge invariance is needed only in D = 4 and it can be lost away from this spacetime dimension. The breaking of spacetime manifold reflects into the split of spacetime indices µ, ν, . . . into barred indices ¯µ, ¯ν, . . . , taking the usual four dimensional values 0, 1, 2, 3, and hatted indices ˆµ, ˆν, . . . that are defined in the evanescent sector of spacetime. Each tensor is now split into its four dimensional and evanescent part, for instance xµ_{is split into x}µ¯ _{and x}µˆ_{, which we can}

equivalently write as ¯xµ _{and ˆx}µ_{. The metric tensor η}

µν, with the choice of mostly minus

signature, is split into ηµ¯¯ν = diag(+1, −1, −1, −1) and ηµˆˆν = −δˆµˆν, and this last expression

has to be used to contract evanescent components. In this construction we assume that the D-dimensional gamma matrices still satisfy the Dirac algebra {γµ, γν}= 2δµν.

At this point we define γ5 as the product of the gamma matrices defined in the physical

sector of spacetime

γ5 = γ1γ2γ3γ4 = 1

4!εµνρσγµγνγργσ ≡ 1

4!εµ¯¯ν ¯ρ¯σγµ¯γ¯νγρ¯γ¯σ. (2.3)

Once we have introduced γ5 in the same fashion as in four dimensions, it is quite obvious

that the usual anticommutation rule {γ5, γµ}= 0 gives rise to the anticommutator for γµ¯,

while it reduces to a commutator in the case of γµˆ

{γ5, γµ¯}= 0, [γ5, γµˆ] = 0. (2.4)

In the usual fashion we define left and right handed projectors

PR =

1 + γ5

2 , PL=

1 − γ5

2 . (2.5)

The definition of the continued γ5 matrix makes it evident that the dimensional

regu-larization has some troubles when it is applied to chiral theories. In particular, when a theory has a symmetry involving γ5, since dimensional regularization does not

mani-festly preserve this symmetry, we may potentially have an anomaly. In the context of the dimensional regularization a potential anomaly may come from an evanescent operator

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2.2. TRIANGLE DIAGRAM AND ANOMALY

which, simplifying a pole 1/ε, gives rise to a finite term surviving the physical limit.

2.2 Triangle Diagram and Anomaly

We start by describing how an anomaly appears in the quantization of a theory in the simplest case, provided by the axial symmetry in QED.

The QED action in euclidian space is

S = Z 1 4Fµν2 + Z ¯ ψ/∂ψ+ Z ıe ¯ψ /Aψ, (2.6)

where ¯ψ, ψ are fermionic fields, Aµ is the electro-magnetic potential, and Fµν is the field

strength Fµν = ∂µAν − ∂νAµ. This theory is invariant under a local U(1) transformation

¯

ψ0 = e−ıΛψ,¯ ψ0 = eıΛψ, A0_µ= Aµ− ∂µΛ, (2.7)

being Λ ≡ Λ(x) a spacetime dependent parameter. Furthermore, S is invariant under a global symmetry involving γ5, the so called axial symmetry

ψ0 = eıαγ5_ψ, _ψ¯0 = ¯_ψeıαγ5_. (2.8)

In D = 4 the axial transformation is a symmetry of QED because of the γ5

anticommu-tation relation (2.1). The Noether current associated to the axial symmetry is

J5µ = ¯ψγ5γµψ, (2.9)

being a conserved current in D = 4 thanks to (2.1). The axial symmetry would be automatically broken by the presence of a fermion mass term ¯ψψ. However, our aim is not

to take into account explicit symmetry breaking since it appears also at the classical level, rather we focus on a quantum breaking of the symmetry. At this point of the discussion we need to introduce how the dimensional regularization works when the theory contains chiral fermions. In particular, we may wonder if the conservation rule ∂µJ5µ = 0 is true

even for the continued D-dimensional action. To this end, let us compute the divergence of the current having in mind the definition of γ5 in dimensional regularization

∂µJ5µ = ∂µ( ¯ψγ5γµψ) = Dµ( ¯ψγ5γµψ) = ¯ψγ5Dψ/ + (Dµψ¯)γ5γµψ

= ¯ψγ5Dψ −/ (Dµψ¯)γµγ5ψ+ (Dµψ¯){γ5, γµ}ψ, (2.10)

splitting γµ in the anticommutator into barred and hatted indices we get ∂µJ5µ = ¯ψγ5Dψ −/ (Dµψ¯)γµγ5ψ+ (Dµˆψ¯){γ5, γµˆ}ψ+ (Dµ¯ψ¯){γ5, γµ¯}ψ,

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using the definition of γ5 we have ∂µJ5µ = 2(Dµˆψ¯)γ5γµˆψ+ ¯ψγ5E_ψ¯+ Eψγ5ψ, (2.11) where E_ψ¯ = δlS δ ¯ψ = /Dψ, Eψ = δrS δψ = −Dµψγ¯ µ, (2.12)

are the equation of motions for the fermions. Hence we found that the axial current is on shell conserved apart for the evanescent operator (Dµˆψ¯)γ5γµψˆ , which may lead to

an anomaly. The next step is to analyze the effect of this evanescent operator from a diagrammatic point of view, which amounts to compute some possibile loop corrections to the conservation law ∂µJ5µ = 0. Recall that the axial current, involving a product

of two fields in the same spacetime point is a composite operator, and it is the same for its divergence. In particular we need to study insertions of ∂µJ5µ inside correlation

functions. This can be achieved introducing in the lagrangian a new vertex given exactly by the divergence of the axial current

− L5∂µJ5µ, (2.13)

where L5 is an external source from which we get insertions of ∂µJ5µ by taking functional

derivatives with respect to it. Because of the presence of this extra vertex we have a new corresponding Feynman rule

p q

L5

= iγ5(/p − /q)

. (2.14)

We want to study the correlation function

h∂ · J5(x)Aµ(y)Aν(z)i, (2.15)

and in particular we focus on the one-loop contributions, given by the diagrams

∂ · J5 Aµ(k1) Aν(k2) ∂ · J5 Aµ(k1) Aν(k2)

The two diagrams differ only for the exchange µ ↔ ν and k1 ↔ k2, so we can

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computation is the dimensional one. The first diagram reads: Iµν(−k1− k2, k1, k2) = (ıe)2 Z dDp (2π)D Tr[γ5(/k1+ /k2)(/p + /k2)γν/pγµ(/p − /k1)] p2(p − k 1)2(p + k2)2 , (2.16)

External momenta do not need to be taken in the full continued spacetime manifold since they are not integrated, so we can consider them purely four dimensional. Using the following identity

γ5( /¯k1+ /¯k1) = γ5(/p + /¯k2) + (/p − /¯k1)γ5−2γ5/ˆp, (2.17)

inside Iµν we obtain three terms

Iµν(−k1− k2, k1, k2) = Iµν(1)(−k1− k2, k1, k2) + Iµν(2)(−k1− k2, k1, k2) + Iµν(3)(k1, k2), (2.18) where I(1) µν(−k1− k2, k1, k2) = −e2 Z dDp (2π)D Tr[γ5γν/pγµ(/p − /¯k1)] p2(p − k 1)2 = e2_f_(D)ε ¯ µ¯ν ¯ρ¯σ Z dDp (2π)D pρ¯pσ¯− pρ¯¯k2¯σ p2(p − k 1)2 = 0, (2.19)

because εµ¯¯ν ¯ρ¯σpρ¯pσ¯ = 0, and the second integral vanishes since it is proportional to pρ¯.

Similar reasons hold for I(2)

µν(k1, k2). Hence we are left only with

Iµν(−k1− k2, k1, k2) = 2e2 Z dDp (2π)D Tr[γ5/ˆp(/p + /¯k2)γν¯/pγµ¯(/p − /¯k1)] p2(p − k 1)2(p + k2)2 . (2.20)

Let us observe the presence of a /ˆp in the numerator. Since an invariant tensor with only an hatted index does not exist, there has to be another /p projected onto the evanescent sector of spacetime. Then, we get

Iµν(−k1− k2, k1, k2) = −2e2f(D)εµ¯¯ν ¯ρ¯σδα ˆˆβ Z dDp (2π)D pαpβ(pρ¯pσ¯− pρ¯k1¯σ+ pρ¯k2¯σ + k1 ¯ρk2¯σ) p2(p − k 1)2(p + k2)2 , (2.21) where we have used ˆp2 _{= δ}

ˆ

α ˆβpαpβ. The only non vanishing contribution comes from the

last term Iµν(−k1− k2, k1, k2) = −2e2f(D)εµ¯¯ν ¯ρ¯σδ_{α ˆ}_ˆ_βk1 ¯ρk2¯σ Z dDp (2π)D pαpβ p2(p − k 1)2(p + k2)2 . (2.22)

Now we concentrate on the integral. Let us note that we are dealing with an integral which has a logarithmic divergence, whose divergent part does not depend either on k1

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or k2, as we now show. The derivative of the integral with respect to k1 ∂ ∂k1ρ Z dDp (2π)D pαpβ p2(p − k 1)2(p + k2)2 = −2Z _(2π)dDp_D pαpβ(p + k2)ρ p2(p − k 1)2(p + k2)4 , (2.23)

is convergent because of a straightforward application of power counting. This result ensures that the divergent part of the integral does not depend on the external momenta, not even on some mass parameters. To this extent we can set k1 ¯ρ = k2¯σ = 0 to compute

the divergent part of Iµν, inserting a mass parameter m, needed to regularize the integral

in the infrared limit. The result for the divergent part of the integral is the following

Z dDp (2π)D pαpβ (p2+ m2)6 = δαβ 32π2_ε + O(1). (2.24)

Substituting this expression into (2.22) we obtain Iµν(−k1− k2, k1, k2) div = − e2 16π2_εf(D)εµ¯¯ν ¯ρ¯σδα ˆˆβδαβk1 ¯ρk2¯σ. (2.25)

At this point we note that something anomalous happens. The previous expression con-tains the contraction δα ˆˆβδαβ = −ε, which simplifies the divergence giving a finite result.

The divergent part of the correlation function (2.15) is obtained inserting a factor 2 in the previous expression in order to count the second diagram displayed in (2.2). Taking the physical limit ε → 0 we have

Iµν(−k1− k2, k1, k2) _div = e2 2π2εµνρσk1ρk2σ. (2.26)

The corresponding term into the the Γ functional is

∆Γ = −_4πe2₂ Z dk1dk2εµνρσk1ρk2σL5(−k1− k2)Aµ(k1)Aν(k2). (2.27)

Turning back to the coordinate space we get ∆Γ = − e2

16π2 Z

d4xL5(x)εµνρσFµν(x)Fρσ(x). (2.28)

Finally, the one loop contribution to h∂ · J5i is obtained taking a functional derivative

of ∆Γ with respect to L5, with an extra sign due to the fact that we have included the

vertex with the axial current as −L5∂ · J5

h∂ · J5i _one-loop = e2 16π2εµνρσFµν(x)Fρσ(x). (2.29)

This result shows that the divergence of the axial current is no longer conserved at the quantum level. One may wonder if the contribution which spoils the conservation of the

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axial current can be cancelled by introducing a local counterterm into the action. We could define a new axial current

J_5µ0 = J5µ−

e2

8π2εµνρσAνFρσ, (2.30)

such that now we have

h∂ · J₅0i _one-loop= 0. (2.31) Indeed, J0

5µ is a conserved current, although it is not gauge invariant. Hence, to recover

the axial symmetry we have to lose gauge invariance. However, the axial symmetry is only a global symmetry of the theory and can be lost without any consequence, instead gauge invariance is required for the consistency of the theory, ensuring that only physical degrees of freedom propagate. The conclusion is that the axial symmetry in QED is anomalous, and the anomaly manifests itself spoiling the conservation of the axial current.

Before the end this section we comment on the anomaly coefficient. In fact, as it will be discussed in the cancellation of gauge anomalies in the Standard Model, the vanishing of anomalies depends on the matter content of the theory, reflected in the anomaly coefficient. Consider again (2.2) and let us imagine we are in the presence of non-Abelian gauge fields, where color indices have to be taken into account. Denoting the correlation function giving rise to the anomaly by

h∂ · Ja 5(x)A a µ(y)A c ν(z)i, (2.32)

the trace over the color indices gives

Tr[TaTbTc+ TaTcTb] = Tr[{Ta, Tb}Tc]. (2.33)

and we call this expression the symmetric trace, denoted by Dabc. Usually a theory may

show an anomaly if it is chiral, which means that left and right handed fermions do not combine to give a Dirac fermion. It will be clear from the explicit computation of the anomaly coefficient for the QED only with a left-handed fermion performed in section (3.6), that left and right handed particles contribute to the complete anomaly coefficient with opposite signs. Thus, (2.33) gives

TrL[{Ta, Tb}Tc] − TrR[{Ta, Tb}Tc], (2.34)

and this is the expression that allows us to compute the anomaly coefficient only given the gauge group and the matter content of a theory.

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2.3 Wess-Zumino Consistency Conditions

The anomaly functional satisfies some integrability conditions, known as the Wess-Zumino

consistency conditions, which were investigated for the first time in [1]. To derive these

conditions we first show that the anomaly functional can be expressed as the covariant derivative of the variation with respect of gauge fields of the effective action Γ, constraining its form. It is worth deriving first the integrability conditions in their original form. Then, in the next section, we are going to explain the importance of the Wess-Zumino conditions in order to determine the expression of the complete anomaly functional, a task better achieved employing the BRST formalism.

Consider the effective action once the integration over fermionic fields has been per-formed, which depends only on the gauge fields Aa

µ, and compute its gauge variation δΓ(A) = Z d4_xδΓ(A) δAa µ δAa_µ= Z d4_xδΓ(A) δAa µ DµΛa= Z d4_x_Λa _−D µ δΓ(A) δAa µ ! , (2.35)

where in the last step the covariant derivative has been integrated by parts. Let us observe that the anomaly functional corresponds to the gauge variation of the effective action once one has integrated over fermionic fields, hence it is possible to recognize in the previous expression

Aa_{(x) = G}a_(x)Γ(A), _(2.36)

in which we have defined the operator Ga_(x): Ga(x) = −Dµ

δ δAa

µ(x)

. (2.37)

After a little bit of algebra it is possible to show that Ga_{(x) satisfies the same Lie algebra}

as the theory does, i.e.

[Ga_{(x), G}d_{(y)] = gf}adc_Gc_{(x)δ(x − y).} _(2.38)

Applying this result to the effective action Γ(A) we finally get the consistency conditions

Ga(x)Ab(y) − Gb(y)Aa(x) = fabcδ(x − y)Ac(x). (2.39)

2.4 BRST formalism

We now introduce the BRST formalism for the quantization of gauge theories. After the basic tools for the quantization have been developed, we are going to discuss how anomalies appear in this formalism. The framework developed in this section is closely related to the one used for proof of the Adler and Bardeen theorem in the presence of

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2.4. BRST FORMALISM

external fields, namely the Batalin-Vilkovisky formalism.

For definiteness we develop the BRST method in the case of Yang-Mills theory, al-though the considerations are valid for any theory that possesses a local symmetry. It is well known that a perturbative approach to a gauge theory makes sense iff one introduces a gauge fixing term in the action. In fact to get the propagator of gauge fields one needs to invert the operator defining the quadratic action. Nonetheless, in the case of a gauge theory this operator has zero modes, corresponding to the gauge transformations. Hence, the propagator of gauge fields is well defined only after the action has been gauge fixed. Obviously, the introduction of a gauge fixing breaks, by definition, the gauge symmetry. It was noted by Becchi, Rouet and Stora in [2] that the original action, supplemented by a ghost term, possesses a global invariance under a symmetry whose parameter is of fermionic statistic.

Start with the Yang-Mills action

Sc= −

1 4

Z

d4xF_µνa Faµν, (2.40)

and denote the gauge fixing function as Ga_(A

µ) = 0, for instance, in the case of the Lorentz

gauge this takes the form Ga_(A

µ) = ∂µAaµ = 0. The gauge fixing term in the action is

introduced with the aim of a Lagrange multiplier field, Ba_{, also called Nakanishi-Lautrup}

field

Sgf = −

Z

d4xBaGa(Aµ). (2.41)

As discussed previously, the gauge fixed action Sc+ Sgf is clearly not gauge invariant.

Nonetheless the gauge fixed action is invariant under the BRST symmetry, with a constant Grassman odd parameter. The BRST variation of the gauge fields is the same as the usual gauge transformation except for the replacement

Λa_{(x) → θC}a_(x), _(2.42)

where Ca_{(x) are the ghost fields, i.e. bosons with a fermionic statistic. The gauge field}

transformation is now

δAa_µ = DµΛa → θDµCa. (2.43)

Ghosts are introduced with an additional quantum number, the ghost number, such that gh(C) = 1, physical fields having zero ghost charge. We introduce the BRST operator s such that

δΛφ= δθCφ= θ(sφ), (2.44)

for instance, in the case of the transformation of Aa

µ we get sA a

µ = DµCa. Observe that

because of the definition of s, it raises the ghost number of a given field of one unit, hence it has gh(s) = 1. The BRST transformations are taken to be nilpotent, in other words

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applying twice the transformation on any field we get zero

s2 = 0. (2.45)

In order to achieve the nilpotency, the transformation of the ghosts is taken to be

sCa= −1

2fabcCbCc. (2.46)

Finally, we introduce the antighosts ¯Ca such that their BRST transformation yield the Ba field, while Ba has a trivial transformation

s ¯Ca= Ba, sBa = 0, (2.47)

where gh( ¯C) = −1. Then, the construction of the complete BRST invariant action

proceeds straightforwardly

S = Sc+ Sgf + Sgh, (2.48)

Sgh being the action for ghost fields. It is worth noting that Sc is itself BRST invariant

because of the original gauge invariance. In order to ensure the invariance of the complete action we use the nilpotency of the BRST operator. In fact Sgf+ Sgh arises as follows

Sgf+ Sgh= Z d4_{x s}¯ CaGa(Aµ) , (2.49) implying that s(Sgf+ Sgh) = 0.

Let us introduce some terminology. We define a BRST exact object Φ if it has the form Φ = sφ, while it is said to be BRST closed if sΦ = 0. Note that the closed objects belong to the kernel of the BRST operator Ker(s) = {Φ : sΦ = 0}, instead the exact ones belong to the imagine Im(s) = {Φ : Φ = sφ}. Obviously, we have Im(s) ⊂ Ker(s) because of the nilpotency property, in fact given an exact term it is automatically closed

Φ = sφ → sΦ = s2_φ_{= 0.} _(2.50)

Following these definitions, Sgf+ Sgh is BRST exact. For definiteness, in the case of the

Lorentz gauge, the variation s¯

CaGa(Aµ)

leads to the ghost action

Sgh= −

Z

d4x ¯Ca∂µDµCa. (2.51)

In the introduction of this section we motivated the need for the gauge fixing as a funda-mental tool in order to make perturbation theory meaningful. Moreover, there is another fundamental reason to perform the gauge fixing. In the path integral formulation of a field theory we need to integrate over all possible field configurations, however the

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re-2.4. BRST FORMALISM

dundancy given by the gauge transformations implies to the integration over equivalent configurations gives infinity. To avoid such an issue we need to define carefully physical states, choosing only a configuration among the ones differing by a gauge transformation. This goal can be achieved by introducing the so called BRST cohomolgy, which allows us to define the quotient space

H(s) = Ker(s)

Im(s) , (2.52)

H(s) is the cohomology space and contains the physical states of the theory. In other

words, we define the physical states Ψ as those that satisfy

sΨ = 0 with the identification Ψ1 ∼Ψ2 = Ψ1+ sΩ. (2.53)

The states satisfying these requirements are exactly the ones contained in H(s). It is in this framework that anomalies are better described. In particular, we are going to display an equivalent version of the Wess-Zumino consistency conditions in term of the BRST cohomology.

Consider first the BRST variation of a functional depending only on the gauge fields, for instance the effective action Γ(A)

sΓ(A) = Z d4x(sAa_µ) δ δAa µ Γ(A) =Z d4x(DµCa) δ δAa µ Γ(A) =Z d4xCa −Dµ δ δAa µ !

Γ(A) =Z d4x CaAa_{(x) ≡ A(C, A),} _(2.54)

where we emphasized that the anomaly depends on the ghosts (linearly) and on the gauge fields. We now state that the Wess-Zumino conditions in terms of the BRST formulation are given by

sA(C, A) = 0. (2.55)

The equivalence of this statement with (2.39) is proved by noting that

sA(C, A) = s

Z

d4x Ca(x)Aa(x, A) =

Z

d4x[(sCa(x))Aa(x, A) − Ca(x)(sAa(x, A))]

=Z d4x −1 2fabcCb(x)Cc(x)Aa(x, A) − Cb(x) Z d4 y Cc(y)Gc(y)Ab(x, A) =Z d4x d4y −1 2Cb(x)Cc(y) _h

fabcδ(x − y)Aa(x, A)+ (2.56)

+Gb_{(y)A(x, A)}c_{− G}c_{(x)A(y, A)}bi

, (2.57)

where in the last step we have used the anticommutation of Cb_(x)Cc_{(y) to antisymmetrize}

the term Gc_(y)Ab_{(x, A). Thus, using (2.39) it follows that sA(C, A) = 0. Finally, we}

are able to connect anomalies and BRST cohomology classes. From the Wess-Zumino consistency conditions it follows that A(C, A) is a BRST exact functional at ghost number

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one. Recall that an anomaly cannot be removed by adding some local counterterms to the action. As we discussed in the construction of the BRST invariant Yang-Mills action, each term in the action has to be BRST exact. This translates into the condition that an anomaly must be identified up to local exact terms

A(C, A) ∼ A(C, A) + sF (A), (2.58)

where F (A) is some local functional of the gauge fields. These considerations lead to the definition of the anomaly functional as the non-trivial cohomology classes at ghost number one in the space of local functionals.

The characterization that has been given of the Wess-Zumino consistency conditions is fundamental in the determination of higher order terms in the anomaly functional. In fact, in section (2.2), we determined only the quadratic term in the gauge fields of the anomaly functional; however no one ensures that this is the complete anomaly. In four dimensions, diagrams with more than three photons are convergent, therefore they do not contribute to the anomaly, but we still have the one with three photons and one gauge current.

Figure 2.1: Square diagram contributing to the O(A3_{) of the anomaly functional.}

The goal of the next section is to introduce first the basic tools to manage differential forms and then to apply the consistency conditions determining the complete anomaly polynomial, that will contain a quadratic and cubic term, since higher order terms would correspond to convergent diagrams.

2.4.1 Anomaly polynomial in D

= 4

Before we start the derivation of the complete anomaly polynomial, leading us to the Bardeen formula, a few rules to deal with gauge theories in terms of differential forms are needed. We first define the gauge field one form A

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2.4. BRST FORMALISM

and the corresponding field strength

F = 1

2Fµνdxµdxν, (2.60)

where for simplicity we omit the wedge symbol, since no confusion may arise taking into account the odd nature of the coordinate differentials. For instance, let us compute

A2 = A ∧ A = AµAνdxµdxν =

1

2[Aµ, Aν]dxµdxν, (2.61)

that allows us to write the field strength as

F = 1

2Fµνdxµdxν =

1

2(∂µAν − ∂νAµ+ g[Aµ, Aν]) = dA + gA2, (2.62)

Consider, for instance, the chiral anomaly in the case of chiral QED with a left handed fermion, which reads

AL = −g3 12π2 Z d4_{x ε}µνρσ_Tr[∂ µC(Aν∂ρAσ)]. (2.63)

We want to write this expression in terms of differential forms. Integrating by parts and taking into account that the integration measure can be written as εµνρσ_d4_x ₌

dxµ_dxν_dxρ_dxσ _{one gets} AL= g3 3π2 Z Tr[CdAdA], (2.64)

where the numerical factor has changed since

∂µAνdxµdxν = 2dA. (2.65)

In order to impose the Wess-Zumino consistency conditions in the next section we need a few more rules to manage non-Abelian gauge theories in the language of differential forms. First, we choose the ghost fields to anticommute with the differentials

Cdxµ= −dxµC. (2.66)

It follows that the BRST operator anticommutes with the derivative

sd = −ds. (2.67)

It is easy to see that the gauge transformations for the gauge fields and the ghosts are given by

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We are finally ready to employ the Wess-Zumino consistency conditions in order to derive the complete anomaly polinomial in D = 4. The first step is to consider (2.64) and compute its gauge variation

sA= g 3 3π2 Z sTr[CdAdA] = g 3 3π2 Z

Tr[(sC)dAdA + Cd(sA)dA + CdA(sA)]. (2.69) After some algebraic rearrangements we get

sA= g

3

3π2 Z

Tr[dCdCA2_]. _(2.70)

It is worth stressing that we have neglected all the exact terms that have been appeared along the way. Now, in order to get the complete anomaly polinomial we add all the possibile terms up to the quartic order in A with a generic coefficient

A= g

3

3π2 Z

Tr[C(dAdA + αA2_{dA + βAdAA + γdAA}2_{+ δA}4_)] _(2.71)

and impose sA = 0, in such a way that the variation of the terms we have added cancels the variation of the Abelian term. Let us first analyze the variation of the last term

sTr[CA4] = Tr[(sC)A4− C(sA4_{)] = Tr[C}2_A4_{] + O(C, dC),} _(2.72)

where for simplicity we put all the terms containing C and dC in O(C, dC) since they do not play a role in the following argument. Note that no other terms, except Tr[A4_{], can}

generate Tr[C2_A4_{] in the variation. Hence we conclude that δ = 0. The variation of the}

remaining terms gives

αs[Tr(CA2dA)] = αTr[CdCAdA − CAdCdA] + αTr[C2A2dA + dCCA3+ dCACA2], βs[Tr(CAdAA)] = βTr[CdCdAA − dCCAdA] + βTr[C2AdAA + CAdCA2+ dCACA2], γs[Tr(CdAA2)] = γTr[−dCCdAA + CdAdCA] + γTr[C2dAA2− CdCA3_{+ CAdCA}2_].

(2.73) Let us observe that we have separated the terms with two derivatives from the ones with only one derivative. Recall that (2.70) contains only one term with two derivatives, then all the terms with one derivative need to cancel among themselves. The only way this can happen is that for a suitable choice of the coefficients α, β, γ some of them sum up to an exact form. Indeed, such a choice exists, and it is given by

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2.5. ’T HOOFT ANOMALY MATCHING CONDITIONS

for which the terms with one derivative sum up to Tr[d(C2

A3)]. Rearranging the remaining

terms, the consistency conditions yield to

Tr[dCdCA2_{] − 2αTr[dCdCA}2_{] = 0,} _(2.75)

from which it follows that

a= 1

2. (2.76)

Having obtained this coefficient and turning back to (2.71) we have A= g 3 3π2 Z TrCd AdA +1 2A3 , (2.77)

that is the famous expression for the chiral anomaly in the case of a non-Abelian theory.

2.5 ’t Hooft anomaly matching conditions

In the early 1980’s ’t Hooft [8] was looking for some generalized version of QCD in which the chiral symmetry is only partially broken, leaving room for few chiral bound states. Unfortunately, he did not find any model reproducing the quark-lepton observed spectrum. However, ’t Hooft discovered that anomalies could provide, in a sense we will explain in a moment, a bridge between the high and low energy formulation of a given theory, i.e. encoding some non perturbative property.

Suppose we have a Gc gauge theory described by a lagrangian L invariant under a

global group Gf, which we may call flavour group, in analogy with the SU(Nf) group

in QCD. Denote with AU V the value of the triangle diagram with three currents Jµa,

associated to the global symmetry group. Let us stress that the non vanishing value of this diagram actually does not represent an issue for the consistency of the theory since it contributes to the three point function of global currents and it does not involve any

internal field. Among the possible anomalies we have not mentioned the ones associated

to Gc× Gf since in its original arguments ’t Hooft takes the only part of the gauge group

that does not contain these anomalies.

To study a generalized QCD ’t Hooft studies the dynamics of a new theory in which the flavour group is now gauged, so the gauge group is enhanced to Gc× Gf. By

construc-tion this theory is not consistent because of the presence of AU V, which now becomes a

true gauge anomaly. Hence, to ensure the consistency of the theory we need to cancel this anomaly. This goal can be achieved by introducing some fermionic spectator fields cou-pling only to the gauge fields associated to Gf such that they exactly cancel the anomaly

As+AU V = 0. Now, let us look at the low energy regime of this theory, where we only see

the Gf gauge fields coupled to the massless bound states that appear because of the

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at energies below the mass scale at which the color binding occurs we cannot see the original fermionic spectrum of the theory, i.e. the quarks, while we only see the hadronic spectrum. Now, the crucial observation is that the fermionic spectrum must arrange in such a way the anomaly cancels, since unitarity has to be obeyed also in the low-energy regime of this theory. Thus we have AIR+ As = 0. Putting this condition together with

the anomaly cancellation in the high-energy regime of the theory we get the relation

AIR = AU V, (2.78)

stating that the coefficients of the anomaly in the UV and IR limit have to match. This relation is known as the ’t Hooft matching anomaly condition. The original theory can be recovered by taking the limit gf → 0, in which the spectator fields completely decouple,

and the relation (2.78) is still valid. In the end, the ’t Hooft matching anomaly conditions state that the coefficients of the anomalies related to global currents are scale invariant quantities, so their values match in the UV and IR regimes.

Observe that the t’ Hooft argument does not represent a proof of the fact that the anomalies are the same in the UV and IR regimes of a given theory. Moreover, the anomalies that are mixed in the Gcand Gf gauge fields are not analyzed, while they will

play a crucial role in the Adler-Bardeen theorem with external fields. Actually, one has to prove that the anomaly is a one-loop exact quantity, from which it follows that it is also a scale invariant quantity. This feature is exactly the aim of the Adler-Bardeen theorem.

2.6 Anomalies and experiments

Up to now we have discussed the meaning of anomalies in quantum field theory and we displayed in detail two different ways in which the axial anomaly can be computed. However, one may wonder if anomalies are just a mathematical issue concerned with the quantization of a given classical theory, or if they actually represent something real in our world.

Historically, anomalies were discovered trying to solve a well-known puzzle in elemen-tary particle physics: the decay of the neutral pion π0 _{into two photons γ. We now want}

to give a detailed report on the strict relation between the aforementioned decay and the so-called Adler-Bell-Jackiw anomaly, which is nothing but the axial anomaly.

π

0

(q)

γ(ε

1

, k

1

)

γ(ε

2

, k

2

)

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2.6. ANOMALIES AND EXPERIMENTS

The transition matrix element for the decay is hγ(ε1, k1), γ(ε2, k2)|π0(q)i = (2π4)δ4(q − k1− k2)ε

µ

1(k1)ε

µ

2(k2)Γµν(k1, k2, q), (2.79)

where the Lorentz structure of Γµν(k1, k2, q) has to reflect the negative parity of the neutral

pion

Γµν(k1, k2, q) = Γ(q2)εµνρσk1αk

β

2. (2.80)

The application of the LSZ reduction formula to the previous amplitude gives Γµν(k1, k2, q) = e2(−q2+ m2π)

Z

d4y d4z eik2y−iqz_hϕ

π(z)Jν(y)Jµ(0)i. (2.81)

Defining the three point function Tµνρ= hJ5ρ(z)Jµ(y)Jν(x)i, the previous amplitude may

be related to

qρTµνρ =

Z

d4y d4zeiky−iqzh∂(z)ρJ5ρ(z)Jµ(y)Jν(x)i, (2.82)

by the PCAC hypothesis, namely the partially conserved axial current, according to which the pion field couples to the axial-vector current

∂µJ₅aµ = fπm2πϕ a

π, (2.83)

where fπ = 93 MeV and a = 1, 2, 3 is an index running in the fundamental representation

of SU(2) (for a = 3 we get the neutral pion). Observe that in the case of massless pions we get the classical axial-vector current conservation, which is equivalent to say that the pions are the Goldstone bosons of the broken SU(3)A symmetry of the QCD lagrangian.

From the PCAC hypothesis it follows that the relation between qρ_T

µνρ and Γµν is qρTµνρ = fπm2π e2(m2 π− q2) Γµν. (2.84)

Actually, this relation leads to a paradox recognized by Sutherland and Veltman. In fact, since there are no states between the pion and the vacuum, Tµνρ does not contain any

pole as q → 0. It follows that qρ_T

µνρ → 0 as q → 0. The immediate consequence of

this observation is that the amplitude Γµν(k1, k2, q → 0) = 0, meaning that the neutral

pion cannot decade into two photons, although the decay rate had been measured and it results non vanishing. The solution was found by inserting the axial anomaly in the conservation law of the axial-vector current

∂µJ 3µ 5 = fπm2πϕaπ + e2 16π2ε µνρσ_F µνFρσ, (2.85) where J₅3µ= ¯ψγµγ5 σ3 2 ψ. (2.86)

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Now, the pion amplitude in the soft limit is completely determined by the anomaly term Γµν(k1, k2, q →0) = e2_c 2π2_f π εµνρσk1ρk2σ, (2.87)

where c is a coefficient to be determined in the quark model. Let us note that the decay rate involves the computation of the triangle diagram at which vertices we have two electromagnetic currents

Jµ= ¯ψγµQψ, (2.88)

and one axial-vector current

J_5µ(3) = ¯ψγµγ5

λ3

2 ψ, (2.89)

being Q the charge matrix of (u, d, s) quarks and λ3_/_{2 the third Gell-Mann matrix}

Q= 1 3      2 0 0 0 −1 0 0 0 −1      , λ3 =      1 0 0 0 −1 0 0 0 0      . (2.90)

Because of the currents in the triangle diagram it involves as coefficient the trace over the generators Tr " QQλ 3 2 # = 1₆. (2.91)

Furthermore, there is a factor of Nc, namely the number of colors in the quark model,

taking account that there is a triangle diagram for each color. The computation of the decay rate proceeds straightforward using the standard QFT techniques and it gives

Γ(π0 _{→ γγ}_{) =} Nc 3 2 α2m2 π0 64π3_f2 π =Nc 3 2 ×7.63eV. (2.92)

The experimental rate is

Γ(π0 _{→ γγ}₎

exp = 7.37 ± 1.5eV. (2.93)

Comparing the experimental value of the decay rate with the one computed one may infer that they are in agreement only taking Nc= 3, proving evidence that QCD is based on a SU(Nc= 3) gauge group. From this discussion it becomes clear the relevance of anomalies

in quantum field theory: not only they allow to solve the puzzling decay π0 _{→ γγ}_{, but}

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2.7. ANOMALY CANCELLATION IN THE STANDARD MODEL

2.7 Anomaly cancellation in the Standard Model

The Standard Model is the quantum field theory which describes the electroweak and strong interactions. It is based on the gauge group

SU(3)c× SU(2)L× U(1)Y, (2.94)

being L the lepton number and Y the hypercharge. The fermions of the first generation are listed in the following table

SU(3) SU(2) U(1) QL= uL dL ! 3 2 1/3 uR ¯3 1 4/3 dR ¯3 1 −2/3 L= νL eL ! 1 2 −1 eR 1 1 −2

The quarks, represented by u and d, transform in the fundamental representation of

SU(3)c, while leptons ν and e belong to the fundamental representation of SU(2)L.

From the above discussion we have seen that the anomaly cancellation depends upon the symmetric trace Dabc. Thus, to check that the Standard Model is anomaly free we

need to check if this is true for all possible combinations of generators Ta, Tb and Tc over

all the generators of the Standard Model group. Recall that the generators of SU(3)c and SU(2)Lare traceless, so taking all the combinations we can neglect terms with one SU(3)c

or SU(2)Lcurrent since, the result identically vanishes. The only non trivial combinations

we need to check are the following • SU(3)3

c: The fermionic fields that have non-trivial interactions under SU(3) are a

doublet transforming in the representation 3 and two particles transforming in the

¯

3, so they provide the representation

3 ⊕ 3 ⊕ ¯3 ⊕ ¯3, (2.95)

which is real, so Dabc vanishes. Clearly, there is no need to take into account the SU(3) singlets because the generators for the trivial representation vanish. There

is another way to understand in a physical fashion why the anomaly is zero for three SU(3)c currents, which uses that left and right handed fermions have the

same strong interactions, so the sum of the triangle diagrams with left and right contributions exactly cancel out.

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• SU(3)2

cU(1)Y: the only contributions come from QL, uR and dR

3 · 2 · 1₃−3 · 4 3−3 · 2 3 = 0. (2.96) • SU(2)3

L: recall that generators of SU(2) in the fundamental representation are the

Pauli matrices σa/2, which obey the following identity

{σa, σb}= 2δabI, (2.97)

from which it follows

Dabc= Tr[σa{σb, σc}] = 0. (2.98)

• SU(2)2

LU(1)Y: here the contributions are given by QL and LL

3 · 2 · 1₃ + 1 · 2 · (−1) = 0. (2.99) • U(1)3

Y: for three U(1) currents we obtain a vanishing Dabc

3 · 2 ·1 3 3 −3 · 4 3 3 −3 · −2 3 3 + 2 · (−1)3₋_{1 · (−2)}3 _{= 0.} _(2.100)

This analysis exhausts the proof that the gauge symmetries in the Standard Model are non anomalous, at least at one-loop. In our approach, all the hypercharges have been given a

priori. Actually, one may wonder why the hypercharges have to be exactly the values in

the table. It is possible to pursue a more general approach by assigning arbitrary weak charges to the particles. Now the request for a consistent theory is anomaly cancellation. Performing the computations we find that the only physical solution is the one in which the hypercharges are the ones from which we started. This shows that the anomaly cancellation is a strong constraint on a theory. So far we limited our analysis on the Standard Model, leaving the possibility of an interaction with the gravitational field. We can investigate if switching on the gravitational field may have a non-trivial effect on the gauge symmetries. The result is that an anomalous triangle with two energy momentum tensor and a gauge current arises, being proportional to

Tr[T ]εµνρσ_R

µνκλRρσκλ. (2.101)

This represents the so called mixed gravitational anomaly. The presence of this anomaly can spoil the gauge symmetry, so we need that it vanishes. Since for SU(3)c and SU(2)L

the trace of a generator automatically vanishes, the only non-trivial contribution may come from a U(1)Y current. Computing the trace of the U(1)Y generators over the

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2.7. ANOMALY CANCELLATION IN THE STANDARD MODEL fermions we get 3 · 2 · 1₃ −3 · 4 3 −3 · −2 3 −+2 · (−1) − 1 · (−2) = 0 (2.102) showing that the mixed gravitational anomaly vanishes.

In principle, one could construct other triangle diagrams using the energy-momentum tensor:

• T3

µν: this anomaly vanishes since left and right handed fermions gives opposite

contributions;

• TµνJ2: in this situation we have two different cases. If the divergence acts on

the energy-momentum tensor the anomaly represents a violation in the traslational invariance, which cannot be anomalous because the dimensional regularization pre-serve this symmetry of the theory. The second case is related to the action of the divergence on the J current, leading to a vanishing anomaly in the quadratic approximation since the expression

∂ · Ja∼ εµνρσRγδαβF a

λτ, (2.103)

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Chapter 3

The Batalin-Vilkovisky Formalism

3.1 Introduction

In this section we introduce a powerful tool which allows the quantization of gauge the-ories in the functional language: the Batalin-Vilkovisky formalism. With the aim of this formalism we can quantize a gauge theory in a systematic way, studying its

renormaliza-tion, and other properties, to all orders in perturbation theory. A key point for our goal is

that with the Batalin-Vlikovisky formalism it is possible to analyze possible breaking of symmetries of the classical theory due to the process of quantization, i.e. anomalies. In some sense, this formalism is an upgrading of the BRST method to funcional methods. We are going to use this formalism in its lagrangian formulation, even though it was initially derived in the hamiltonian case by Batalin-Fradkin-Vilkovisky. Although this method works for general cases in which the gauge symmetry is reducible and the gauge algebra closes only on shell1_{, for the applications we need we can concentrate only on irreducible}

gauge symmetries, where the gauge algebra closes off shell.

3.2 Batalin-Vilkovisky quantization

The starting point to describe the formalism, which we will also call canonical gauge

formalism, is to consider a classical action Sc(φ), where φ collects all the classical fields φi = (Aa_µ, ¯ψ, ψ, ϕ), (3.1)

and a set of local transformations with a gauge parameter Λ(x)

δΛφi = Ric(φ, Λ), (3.2)

1_{On shell closure means that the algebra of gauge transformations does not close because of the}

presence of terms proportional to the equations of motion, hence the algebra closes on shell. Supergravity is an example of theory with an algebra which closes only on shell.

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under which the action is invariant δΛS = Z δΛφi δSc δφi = Z Ri(φ, Λ)δSc δφi = 0. (3.3)

Provided that the main goal is the quantization of the theory, we want to introduce a fermionic field Ca_{, which are nothing but the Faddeev-Popov ghosts. Using the linearity}

of the gauge transformations on the gauge parameter we can replace Λ(x) = θC(x), where

θ is a constant Grassmann number, writing the transformations as

Ri_{(φ, Λ) = θR}i_{(φ, C).} _(3.4)

We can enlarge the set of fields including also the ghosts just introduced Φα _{= (A}a

µ, C a

, ¯ψ, ψ, ϕ), (3.5)

For each field we introduce a source2 _{with an opposite statistic}

Kα = (Kµa, K a

C, Kψ¯, Kψ, Kϕ). (3.6)

Hence, defining the statistic of a field or a source ε to be zero for bosons or one for fermions we are led to the relation

εΦα = ε_K

α + 1 mod 2. (3.7)

We are now ready to introduce the key tool of the Batalin-Vilkovisky formalism, which is called the antiparentheses. Given two functionals X ≡ X(Φ, K) Y ≡ Y (Φ, K), we define their antiparetheses as (X, Y ) ≡Z dD x ( δrX δΦα(x) δlY δKα(x) − δrX δKα(x) δlY δΦα(x) ) , (3.8)

where left and right derivatives act as dX(Φ, K) = ∂rX dΦ dΦ + ∂rX dK dK = dΦ ∂lX ∂Φ + dK ∂lX ∂K. (3.9)

First, we observe that locality is not spoiled by the antiparentheses, so if X and Y are local, this property is still valid for (X, Y ). Using only these definitions we can easily

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3.2. BATALIN-VILKOVISKY QUANTIZATION

prove that the antiparentheses satisfy the following properties

(Y, X) = −(−1)(εX+1)(εY+1)(X, Y ), (3.10)

(−1)(εX+1)(εY+1)(X, (Y, Z)) + cyclic permutations = 0, (3.11)

ε(X,Y ) = εX + εY + 1, (3.12)

gh(X, Y ) = ghX + ghY + 1. (3.13)

If we apply (3.10) to the antiparentheses of two bosonic (B) or fermionic funcionals (F ) we immediately get (B, B) = 2Z dDx δrB δΦα(x) δlB δKα(x) = −2Z δrB δKα(x) δlB δΦα(x) , (F, F ) = 0. (3.14)

The property (3.11) is the so called graded Jacobi identity, and if we take X = Y = Z we obtain, for every functional X

(X, (X, X)) = 0, (3.15)

this identity is especially useful to analyze anomalies.

In the Batalin-Vilkovisky language we define the extended action S(Φ, K) as the so-lution of the master equation

(S, S) = 0, (3.16)

solved with the boundary conditions

S(Φ, 0) = Sc(φ), − δrS(Φ, K) δKi _K=0 = Ri_{(φ, C).} _(3.17)

The minimal solution of the master equation is

S(Φ, K) = Sc(φ) + SK(Φ, K) = Sc(φ) −

Z

Rα_(Φ)K

α. (3.18)

Since only gauge algebras which closes off shell are considered, it is always possible to find a frame variable where the solution of the master equation is linear in K. In particular we now show that (SK, SK) = 0 in every dimension D. Let us consider:

0 =(S, S) = 2Z Rα_(Φ)δlS δΦα = 2 Z " Ri_{(φ, C)}δlS δφi + R a C(Φ) δlS δCa # =2Z " Ri_{(φ, C)}δlSc δφi + R i_{(φ, C)}δlSK δφi + R a C(Φ) δlSK δCa # , (3.19)

where in the former term we separated S = Sc+ SK, while in the last term we used that

only SK contains ghosts. Notice that the term involving the classical action, independent

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linear term in K gives 0 = 2Z " Ri(φ, C)δlSK δφi + R a C(Φ) δlSK δCa # = 2Z Rα(Φ)δlSK δΦα. (3.20)

We can recognize this expression also calculating the antiparentheses (SK, SK) = 2

Z

Rα(Φ)δlSK

δΦα = 0, (3.21)

because of the previous expression.

The minimal solution of the master equation can be extended in order to include the gauge fixing. First, we need to enlarge the set of fields and sources by introducing a contractible pair3 _{made of the antighosts ¯}

Ca and the Lagrange multipliers Ba, whose

transformations are

δ ¯Ca= Ba, δBa = 0. (3.22)

The complete set of field and sources is Φα_{= (A}a µ, C a_{, ¯}_Ca_{, B}a_{, ¯}_{ψ, ψ, ϕ}_{), K} α = (Kµa, K a C, K a ¯ C, K a B, Kψ¯, Kψ, Kϕ). (3.23)

The extended solution, included the gauge fixing term, reads

S(Φ, K) = Sc(φ) + SK(Φ, K) + (SK,Ψ), (3.24)

where Ψ(Φ) is the gauge fermion, a functional with dimension [Ψ] = D − 1 and of ghost number ghΨ = −1, that includes the gauge fixing conditions. It is possibile to choose linear gauge fixing Ga_{(A) conditions, like the Lorenz gauge G}α_{(A) = ∂ · A}a_{, arising from}

the following choice of the gauge fermion

Ψ(Φ) =Z C¯a Ga(A) − ξ 2Ba

!

. (3.25)

Given any functional X, taking its antiparentheses with the action and using Jacobi identity, one finds

(S, (S, X)) = −1

2(X, (S, S)) = 0, (3.26)

because of the master equation. From this observation we see the nilpotency of the map (S, X). In particular, the action of the map (S, X) on field and sources is

(S, Φα_{) = R}α_(Φ), _{(S, K}α_{) =} δrS

δΦα. (3.27)

3_{Two fields x and y are said to form a contractible pair if δx = y, δy = 0, and there is no other field}

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3.3. A SIMPLE EXAMPLE: QED

From the second equation we can observe that the sources are algebraically related to the field equations of motion δrS/δΦα.

3.3 A simple example: QED

To be concrete we want to give a brief example of the Batalin-Vilkovisky formalism applying it to the quantization of QED, being one of the simplest and instructive cases. We start with the classical action

Sc(φ) = 1 4 Z F_µν2 + Z ¯ ψ(/∂ + ıe /A)ψ, (3.28)

which is invariant under a local U(1) symmetry

δAµ= ∂µΛ, δ ¯ψ = ieΛ ¯ψ, δψ = −ieΛψ. (3.29)

Following the prescription of the Batalin-Vilkovisky formalism let us introduce a ghost field C which replaces the gauge parameter Λ(x)

δAµ = ∂µC, δ ¯ψ = −ie ¯ψC, δψ= −ieCψ. (3.30)

We miss the transformation of the ghost, but in the Abelian case we get RC = 0. The

solution of the master equation is

S(Φ, K) = Sc(φ) −

Z

dD_x_(∂

µCKµ− ie ¯ψCK_ψ¯− ieKψCψ+ BK_C¯) (3.31)

To add the gauge fixing term consider the gauge fermion introduced in the (3.92), and compute (SK,Ψ) (SK,Ψ) = Z dD x −λ 2B2+ B∂ · A − ¯C2C ! . (3.32)

Adding this term to S(Φ, K) we obtain the quantum gauge fixed action of QED.

3.4 Canonical transformations

We define a canonical transformation as a field and source transformation Φα0_{(Φ, K),}

Kα0(Φ, K), (3.33)

which preserves the antiparentheses (X0_(Φ0

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The generator of canonical transformations is a fermionic functional F(Φ, K0_{) such that} Φα0 ₌ δF δK0 α , Kα = δF δΦα. (3.35)

In the previous expressions we do not need to specify if the derivative on F is left or right because, being F a fermionic functional, each derivative acts in the same way. The most general form of the generating functional can be written separating the terms that do not depend on sources plus everything else

F(Φ, K0) = Ψ(Φ) +

Z

K_α0Uα(Φ, K0) (3.36)

Plugging this expression into the (3.33) we obtain Φα0 _{= U(Φ, K}0_), Kα = δΨ(Φ) δΦα + Z K_β0 δrU β δΦα . (3.37)

Actually, sources have been introduced in the formalism in order to keep the gauge sym-metry under control, but at the end of the computations we set them to zero. Thus we can consider the expression we have just found for F(Φ, K0_{), Φ}α0 _{and K}

α at K0 = 0

Φα0 _{= U(Φ, 0),} _K α=

δΨ(Φ)

δΦα . (3.38)

Hence we see that the most generic canonical transformation can be obtained from a field redefinition, given by U(Φ, 0), and the most general gauge fixing.

3.5 Master identity

In quantum field theory, given the action of a theory, we introduce the generating

function-als of the correlation functions. As usual, the generating functional of all the correlations

is given by Z(K, J) = Z [dΦ] exp−S(Φ, K) + Z ΦαJα = exp W(J, K), (3.39) where W(J, K) is the generating functional of the connected correlations. Performing a Legendre transform of W(J, K) with respect to the sources J, K being spectators, we can introduce the generating functional for the one particle irreducible (1PI) correlation functions

Γ(Φ, K) = −W(J, K) +Z Φα

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3.5. MASTER IDENTITY

This functional satisfies an important identity in the context of the Batalin-Vilkovisky formalism, the so called master identity

(Γ, Γ) = h(S, S)i. (3.41)

Proof. Recall that in dimensional regularization the functional integration measure is

in-variant under each local field redefinition. Actually, from the diagrammatic point of view, the Jacobian of a pertubative local field redefinition is the identity plus terms which are integrals of the momenta p in dD_p_{, which vanish in dimensional regularization. Starting}

from the functional integral (3.39), let us perform the following field transformation Φα _→_Φα_{+ θ(S, Φ}α_{) = Φ}α_{− θ} δrS

δKα, (3.42)

where θ is a constant Grassmann number. The change in the action is

S(Φ, K) → S Φ − θδrS δΦα, K ! = S(Φ, K) − θZ δrS δKα δlS δΦα,= S(Φ, K) − 1 2θ(S, S) (3.43) where the expansion stops at the first order in θ since, being an anticommuting variable, all its powers vanish. Hence, substituting in (3.39) we get

Z(J, K) = Z [dΦ] exp −S(Φ, K) + Z Φα Jα− θ 2(S, S) − θ Z δ rS δKαJ α ! = Z(J, K) 1 − θ₂h(S, S)i + θ Z *δ rS δKα + Jα ! . (3.44)

Since the value of the functional integral has to be unchanged we obtain h(S, S)i = −2 Z * δ rS δKα + Jα. (3.45)

Writing explicitly the average of δrS/δKα it is easy to see that

* δrS δKα + = −δrW δKα = δrΓ δKα , (3.46)

where (4.161) has been used. Substituting back into the expression for h(S, S)i and using that Jα = −δlΓ/δΦα, we have h(S, S)i = −2 Z _δrΓ δKα δlΓ δΦα = (Γ, Γ). (3.47)