Perturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles
PPerturbation Theory ofNon-Perturbative Yang-Mills Theory
A massive expansion from first principles
Giorgio Comitini
A thesis presented for the Master’s Degree in Physics
Dipartimento di Fisica e Astronomia “E. Majorana”Università degli Studi di CataniaItalyOctober 22, 2019 a r X i v : . [ h e p - t h ] O c t auli asked, "What is the mass of this field B?" I said we did not know.Then I resumed my presentation but soon Pauli asked the same question again.I said something to the effect that it was a very complicated problem, we had workedon it and had come to no definite conclusions. I still remember his repartee:"That is not sufficient excuse". C.N. Yang, Princeton (1954) reface
The main objective of this thesis is to present a new analytical framework for low-energyQCD that goes under the name of massive perturbative expansion. The massive perturba-tive expansion is motivated by the phenomenon of dynamical mass generation, by whichthe gluons acquire a mass of the order of the QCD scale Λ QCD in the limit of vanishingmomentum. It is a simple extension of ordinary perturbation theory that consists in ashift of the expansion point of the Yang-Mills perturbative series with the aim of treatingthe transverse gluons as massive already at tree-level, while leaving the total action of thetheory unchanged. The new framework will be formulated in the context of pure Yang-Mills theory, where the lattice data is readily available for comparison and the perturbativeresults have been perfected by enforcing the gauge invariance of the analytical structureof the gluon propagator.This thesis is organized as follows. In the Introduction we review the definition and maincomputational approaches to QCD and pure Yang-Mills theory, namely, ordinary pertur-bation theory and the discretization on the lattice. In Chapter 1 we address the issue ofdynamical mass generation from a variational perspective by employing a tool known asthe Gaussian Effective Potential (GEP). Through a GEP analysis of Yang-Mills theorywe will show that the massless perturbative vacuum of the gluons is unstable towardsa massive vacuum, implying that a non-standard perturbative expansion that treats thegluons as massive already at tree-level could be more suitable for making calculations inYang-Mills theory and QCD than ordinary, massless perturbation theory. In Chapter 2we set up the massive perturbative framework and use it to compute the gluon and ghostdressed propagators in an arbitrary covariant gauge to one loop. The propagators will beshown to be in excellent agreement with the lattice data in the Landau gauge, despitebeing explicitly dependent on a spurious free parameter which needs to be fixed in order topreserve the predictive power of the method. In Chapter 3 we fix the value of the spuriousparameter by enforcing the gauge invariance of the analytic structure of the gluon propa-gator, as required by the Nielsen identities. The optimization procedure presented in thischapter will leave us with gauge-dependent propagators which are in good agreement withthe available lattice data both in the Landau gauge and outside of the Landau gauge.The contents of Chapter 1-3 are original and were presented for the first time in [71–75,84].I ontents
Introduction 1
Quantum Chromodynamics and pure Yang-Mills theory . . . . . . . . . . . . . . 1Standard perturbation theory for Yang-Mills theory: the non-perturbative natureof the IR regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Yang-Mills theory on the lattice: dynamical mass generation . . . . . . . . . . . . 12 λφ theory as a toy model for mass generation . . . . . . . . . . . . . . . . . . . 191.1.1 Optimized perturbation theory and perturbative ground states . . . . 191.1.2 The perturbative vacuum of a quantum field theory and the GaussianEffective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.1.3 The Gaussian Effective Potential of λφ theory: a toy model for massgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2 The Gaussian Effective Potential of pure Yang-Mills theory . . . . . . . . . . 321.2.1 Mass parameters and the definition of the GEP of Yang-Mills theory . 321.2.2 Computation of the GEP . . . . . . . . . . . . . . . . . . . . . . . . . 341.2.3 Renormalization and gauge invariance . . . . . . . . . . . . . . . . . . 371.2.4 The ghost mass parameter and the variational status of the GEP . . . 381.2.5 The perturbative vacuum of Yang-Mills theory: dynamical mass gen-eration in the gluon sector . . . . . . . . . . . . . . . . . . . . . . . . 41 iorgio Comitini F L . . . . . . . . . . . . . . . . . . . . . . . . 833.2.2 The functions F ( ξ ) , m ( ξ ) and θ ( ξ ) . . . . . . . . . . . . . . . . . . . 903.2.3 Gauge-optimized parameters for the massive expansion and the gauge-independent gluon poles . . . . . . . . . . . . . . . . . . . . . . . . . 943.3 Gauge-optimized one-loop gluon and ghost propagators in an arbitrary co-variant gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.3.1 Gluon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.3.2 Ghost propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Conclusions 105Appendix 107
A. Residues of the gluon propagator: the functions F (cid:48) ( s ) and F (cid:48) ξ ( s ) . . . . . . . . 107B. Gauge-optimized functions F ( ξ ) and m ( ξ ) for the gluon propagator: numer-ical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Bibliography 113 IV ntroduction Quantum Chromodynamics and pure Yang-Mills theory
Quantum Chromodynamics (QCD) is the quantum theory of the strong interactions be-tween the elementary constituents of the hadrons, the quarks and the gluons. It wasformulated in the early 1970s by H. Fritzsch, M. Gell-Mann and H. Leutwyler [1–3] as anextension of the gauge theory of C.N. Yang and R. Mills [4] to the SU(3) color group withthe goal of explaining why the gluons could not be observed as free particles.The concept of the hadrons being composite particles dates back to as early as the 1950s,when the discovery of an ever-increasing number of particles subject to the nuclear inter-actions called for the need of an organizing principle to classify the observed spectrum ofmesons and baryons. The first such principle – termed the Eightfold Way – was put forthby Gell-Mann [5] and independently by Y. Ne’eman [6] in 1961, and was later developedby Gell-Mann himself [7] and G. Zweig [8, 9] into what will come to be known as the quarkmodel (1964). The quark model postulated that all the known hadrons could be consideredas being made up of three kinds of spin- / particles – the u quark, the d quark and the s quark – bound by a yet unidentified interaction of nuclear type. The mesons would bebound states of a quark and an antiquark pair, whereas the baryons would be bound statesof a triplet of quarks or antiquarks. The quark model succeeded in explaining the patternof the hadron masses by organizing the mesons and baryons into multiplets of the flavorSU(3) group. However, it was soon realized that the existence of baryons such as the ∆ ++ or the Ω − – which in the quark model would be made up respectively of three u quarks andthree s quarks in the same quantum state – would violate the Pauli exclusion principle.This issue prompted O.W. Greenberg [10] and M.Y. Han and Y. Nambu [11] to postulatethe existence of a new quantum number for the quarks, termed color charge. Each of thequarks would come in three varieties, known as colors; the mesons would be made up ofquarks of opposite color, whereas the baryons would be made up of quarks of three differ-ent colors. In both cases, the quarks would be no longer in the same state – so that thePauli principle would not be violated – and the resulting hadron would be color-neutral.The strong interactions were postulated to be symmetrical with respect to the continuoustransformation of one color into the other, a feature that would formally imply that thelaws of physics be globally invariant under the action of a SU(3) color group.In the early stages of its formulation, the quark picture was though to be more of a math-ematical device for organizing the spectrum of the observed hadrons, rather than a trulyphysical model for the internal structure of the mesons and baryons. Indeed, the existenceof the quarks was challenged by the fact that such elementary components had never beenobserved as free particles. In 1969 R.P. Feynman had argued [12] that the experimentaldata on hadron collisions was consistent with the picture of the hadrons being made up ofmore elementary point-like components, which he named partons, initially refraining fromidentifying them with Gell-Mann’s quarks. However, the crucial breakthrough came only1 iorgio Comitini later on in the same year, when experiments on the deep-inelastic scattering of electronsfrom protons performed at the Stanford Linear Accelerator Center (SLAC) [13,14] revealedthat the electron differential cross section exhibited a scaling behavior which had alreadybeen studied by J.D. Bjorken [15]. Bjorken himself, together with E.A. Paschos [16], em-ployed Feynman’s parton picture to show that the electrons’ behavior could be explainedby assuming that during each inelastic collision the electron interacted electromagneticallywith just one of the partons contained in the proton.As the evidence for the compositeness of the hadrons accumulated, it remained to be ex-plained why the quarks had never been observed individually. To this end, it was postulatedthat the particles subject to the strong interactions – be they elementary or composite –could only exist as free particles in color-neutral states. The quarks, being colored, wouldbe among the particles that could not be observed if not in combination with one another,forming hadrons. This feature of the strong interactions came to be known as confinement.The precise mechanism by which the interactions between the quarks resulted in their con-finement was (and is still to date) largely unknown.As early as the mid 1960s it had been suggested that, in analogy with Quantum Elec-trodynamics, the interactions between the quarks could be mediated by the exchange ofvector bosons, named gluons. In order to explain why such bosons were not observed inthe experiments, in 1973 Fritzsch, Gell-Mann and Leutwyler [3] proposed that, just like thequarks, the gluons too might carry a color charge, so that they would only be observablein combination with other colored particles. The gluons themselves would be responsiblefor the exchange of the quarks’ color charge, implying that from a mathematical point ofview they would form an octet transforming under the adjoint representation of the SU(3)color group. The concepts of color as the charge associated to the strong interactions andof gluons as a color octet had already been suggested by Han and Nambu in their articleof 1965 [11]; the merit of Fritzsch, Gell-Mann and Leutwyler was in managing to formulatethese ideas in terms of a gauge theory of Yang-Mills type, thus giving birth to the theoryof strong interactions that will later be known as QCD. In what follows we will give a briefdescription of the mathematical formalism and fundamental features of both Yang-Millstheory and QCD.Pure Yang-Mills theory [4] is a quantum field theory of interacting vector bosons sub-ject to a local SU(N) invariance. Its fundamental degrees of freedom are expressed interms of an N A -tuple of vector fields A aµ ( x ) ( a = 1 , . . . , N A ), where N A = N − is thedimension of the Lie group SU(N). At the classical level, it is defined by the Lagrangian L YM = − F aµν F a µν Here F aµν is the field-strength (or curvature) tensor associated to the fields A aµ , F aµν = ∂ µ A aν − ∂ ν A aµ + g f abc A bµ A cν where g is a coupling constant and the structure constants f abc are defined by the commu-tation relations of the su (N) algebra – i.e. the Lie algebra associated to SU(N) – [ T a , T b ] = i f cab T c ( f cab = − f cba ) The generators of su (N) are usually chosen in such a way as to satisfy the trace relation2 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Tr { T a T b } = 12 δ ab It can then be shown that the structure constants satisfy the antisymmetry relations f abc = − f acb = f cab f abc f abd = N δ cd ( f abc = f cab ) By expanding the field-strength tensor in the Yang-Mills Lagrangian, one finds that L YM = −
14 ( ∂ µ A aν − ∂ ν A aµ )( ∂ µ A a ν − ∂ ν A a µ ) − g f abc ∂ µ A aν A b µ A c ν − g f abc f ade A bµ A cν A d µ A e ν The first term in the above equation can be easily recognized as a generalization of theMaxwell Lagrangian to our N A -tuple of vector fields: in the limit of vanishing coupling,Yang-Mills theory describes a set of N A massless vector bosons. If g (cid:54) = 0 the second andthird term cause the bosons to interact: at variance with the photons of electrodynam-ics, the degrees of freedom of Yang-Mills theory do interact with each other. This crucialfeature is ultimately responsible for the richness of both pure Yang-Mills theory and itsextension to the quarks, Quantum Chromodynamics.The Lagrangian of Yang-Mills theory is invariant with respect to the following local SU(N)transformation, parametrized by arbitrary functions χ a ( x ) : A µ ( x ) → (cid:102) A µ ( x ) = U ( x ) (cid:18) A µ ( x ) + ig ∂ µ (cid:19) U † ( x ) U ( x ) = exp (cid:16) iχ a ( x ) T a (cid:17) Here A µ is defined as A µ = A aµ T a and U † is the hermitian conjugate of U . The invarianceof L YM can be easily seen to follow from the corresponding transformation law for thefield-strength tensor, F µν ( x ) → (cid:102) F µ ( x ) = U ( x ) F µν ( x ) U † ( x ) where F µν = F aµν T a . Since F = F aµν F a µν = 2 Tr { F µν F µν } , the invariance of F is asimple consequence of the cyclic property of the trace. Being invariant under an infiniteset of local transformations, Yang-Mills theory is a gauge theory. The implications of thisare twofold. First of all, if A aµ solves the equations of motion derived from the Yang-MillsLagrangian, namely, ∂ µ F aµν + g f abc A b µ F cµν = 0 then, for any choice of the parameter functions χ a , its transform under SU(N) (cid:102) A aµ also does.Since the transformation acts locally rather than globally, the pointwise values of the vectorfields cannot have a genuine physical meaning: some of the local degrees of freedom of thetheory are redundant. When passing to the quantum theory, this redundancy will causeproblems with the definition of the quantum partition function, which will have to be dealtwith by employing the so called Faddeev-Popov quantization procedure. Second of all, ifwe insist that gauge invariance be preserved at the level of the classical action, then Yang-Mills theory cannot be generalized to account for a classical mass for the bosons. Such amass would be incorporated in the theory by adding to its Lagrangian a term of the form3 iorgio Comitini L m = 12 m A aµ A a µ where m is the bosons’ mass. This term, however, is not invariant under local SU(N)transformations: it can be shown that under a gauge transformation δ (cid:18)(cid:90) d d x L m (cid:19) = 2 ig m (cid:90) d d x Tr { U A µ ∂ µ U † } (cid:54) = 0 Therefore, gauge invariance constrains the vector bosons of Yang-Mills theory to be mass-less at the classical level .The quantum dynamics of Yang-Mills theory is defined by the partition function Z [ J ] Z [ J ] = (cid:90) D A exp (cid:26) i (cid:90) d d x L YM + J µa A aµ (cid:27) where J µa is an external source for the gauge field A aµ . From a mathematical point of view,this functional integral is ill-defined: in integrating over all the possible configurations ofthe fields A aµ , we are not taking into account that different configurations may actuallybe equivalent modulo gauge transformations; since the gauge group of L YM is infinite-dimensional , an infinite number of such equivalent configurations exists, resulting in theintegrand being constant over an infinite volume of the configuration space. Therefore, inthis form, Z [ J ] is singular for every J . In order to solve this issue, one can adopt a quanti-zation procedure due to L.D. Faddeev and V. Popov [17]. In the Faddeev-Popov approach,the redundant gauge degrees of freedom of the theory are integrated over in such a way asto insulate the singularity of the partition function into an infinite multiplicative constant C . Since the quantum behavior of the theory is dictated by the derivatives of the logarithmof Z with respect to the source J , such a constant plays no role in the definition of thetheory and can thus be ignored. However, as a result of the integration, the integrand ofthe partition function gets modified as follows: Z [ J ] = C (cid:90) D A det (cid:0) − ∂ µ D µ (cid:1) exp (cid:26) i (cid:90) d d x L YM − ξ ( ∂ µ A aµ ) + J µa A aµ (cid:27) Here ξ is a gauge parameter that can take on any value from zero to infinity and det( − ∂ µ D µ ) – known as a Faddeev-Popov determinant – depends on the vector fields A aµ through thecovariant derivative D µ , which acts on the fields in the adjoint representation of SU(N) as D µ f a = ∂ µ f a + g f abc A bµ f c The Faddeev-Popov determinant is usually expressed in terms of an integral over the con-figurations of a pair of anticommuting fields – known as ghost fields – with values in theadjoint representation, det( − ∂ µ D µ ) = (cid:90) D c D c exp (cid:26) − i (cid:90) d d x c a ∂ µ D µ c a (cid:27) = (cid:90) D c D c exp (cid:26) i (cid:90) d d x ∂ µ c a D µ c a (cid:27) Whether this is still true at the quantum level will be discussed later on in this Introduction. Although SU(N) is finite dimensional as a Lie group, its local action on L YM is parametrized byfunctions χ a ( x ) , rather than by constant parameters. Therefore, the symmetry group of Yang-Mills theoryis actually infinite-dimensional. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles The ghost fields ( c a , c a ) are a mathematical tool for keeping under control the gauge re-dundancy built into Yang-Mills theory, and should not be interpreted as being associatedto any physical particle. As a matter of fact, they do not even obey the spin-statistictheorem, in that they are scalar (i.e. spin-0) fields with fermionic statistics.The Faddeev-Popov quantization procedure leaves us with an effective Lagrangian L forYang-Mills theory, in terms of which its partition function can be expressed as Z [ J, j, j ] = (cid:90) D A D c D c exp (cid:26) i (cid:90) d d x L + J µa A aµ + j a c a + c a j a (cid:27) Explicitly, L = L YM + L g.f. + L ghost = − F aµν F a µν − ξ ∂ µ A aµ ∂ ν A aν + ∂ µ c a D µ c a Observe that this Lagrangian contains a term that causes the ghosts to interact with thevector bosons: the ghost Lagrangian can be expanded as L ghost = ∂ µ c a D µ c a = ∂ µ c a ∂ µ c a + g f abc ∂ µ c a c c A bµ where the first term is just the Klein-Gordon Lagrangian for a massless N A -tuple of com-plex scalar bosons, whereas the second one is an interaction term for the ghosts. This wasto be expected from the fact that the vector bosons interact with one another: if the ghostsare to cancel the unphysical content of the theory due to the gauge redundancy, then theyshould be coupled with the vector bosons in order to counterbalance the effects of theirmutual interaction .Moreover, it should be noticed that, as a result of the Faddeev-Popov procedure, L is notgauge-invariant anymore: neither the ghost Lagrangian L ghost nor the gauge-fixing term L g.f. = − ( ∂ µ A aµ ) / ξ are invariant under SU(N) local transformations. If on the one handthis is a necessary condition in order for the quantum partition function to be well-defined,on the other hand it was shown by I.V. Tyutin [18] and C. Becchi, A. Rouet and R.Stora [19] that gauge invariance is not completely lost at the level of the Faddeev-Popovaction: it has only been replaced by a kind of global supersymmetry called BRST symme-try. In order to see this, re-write the Faddeev-Popov action as L = − F aµν F a µν + ξ B a B a + B a ∂ µ A aµ + ∂ µ c a D µ c a where B a is an auxiliary field – known as the Nakanishi-Lautrup field [20, 21] – whoseequations of motion are B a = − ξ ∂ µ A aµ Since on shell ξ B a B a + B a ∂ µ A aµ = − ξ ∂ µ A aµ ∂ ν A aν Here we have suppressed the uninfluential constant C and added sources j a and j a for the ghosts. For instance, the ghosts of Quantum Electrodynamics do not interact with the photons, since thephotons themselves are not subject to mutual interactions. This is ultimately the reason why the ghostsare not an essential part of QED. iorgio Comitini our two expressions for L indeed coincide. However the new Lagrangian is invariant underthe following global BRST transformation, parametrized by an anticommuting number (cid:15) : δA aµ = (cid:15) D µ c a δc a = − g (cid:15) f abc c b c c δc a = (cid:15) B a δB a = 0 Observe that, as far as the vector fields are concerned, this is just the infinitesimal versionof a local SU(N) transformation parametrized by the ghost fields themselves ( χ a = (cid:15) c a ).BRST symmetry has been a fundamental tool for proving many crucial features of Yang-Mills theory. Among them, we cite the derivation of the non-abelian analogue of the Wardidentities – i.e. the Slavnov-Taylor identites [22, 23] – and the proof of the perturbativerenormalizability of the theory by G. t’ Hooft [24].For N = 3 (so that N A = 8 ), Yang-Mills theory describes the dynamics of an octet ofvector fields that can be readily identified with the gluons of Quantum Chromodynamics.QCD, however, also comprises the quarks. Let us see how the full theory of QuantumChromodynamics is defined.Quantum Chromodynamics [3] is a gauge theory of Dirac fields in the fundamental repre-sentation of SU(3) – the quark fields – minimally coupled to an octet of Yang-Mills vectorfields – the gluon fields. Its Lagrangian L QCD is defined as L QCD = L YM + L quark = − F aµν F a µν + iψγ µ D µ ψ − m ψψ In the above equation, m is the quark mass and D µ is the covariant derivative acting onthe fundamental representation, D µ = ∂ µ − ig A aµ T a where the T a ’s are the generators of su (3) . By expanding the covariant derivative in L QCD ,one finds that L quark = iψγ µ ∂ µ ψ − m ψψ + g ψγ µ T a ψ A aµ The first two terms are just the Dirac Lagrangian for the quark fields. The third termis an interaction between the quarks and the gluon octet, generated by an octet of colorcurrents j µa defined as j µa = ψγ µ T a ψ The classical equations of motion of QCD can be readily derived from L QCD and read ∂ µ F aµν + g f abc A b µ F cµν = − g j aν ( iγ µ D µ − m ) ψ = 0 For simplicity we will be considering only one flavor of quark. Actual QCD contains six flavors ofquark ( u, d, s, c, b, t ) with non-diagonal mass couplings given by the CKM matrix [25, 26]. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles The QCD Lagrangian is invariant with respect to the following local SU(3) transforma-tions, parametrized by arbitrary functions χ a ( x ) : ψ ( x ) → (cid:101) ψ ( x ) = U ( x ) ψ ( x ) A µ ( x ) → (cid:102) A µ ( x ) = U ( x ) (cid:18) A µ ( x ) + ig ∂ µ (cid:19) U † ( x ) where as usual U ( x ) = exp (cid:0) iχ a ( x ) T a (cid:1) . It follows that Quantum Chromodynamics is agauge theory with gauge group SU(3).At the quantum level, QCD is defined by the partition function Z [ J A , J ψ , J ψ ] = (cid:90) D A D ψ D ψ exp (cid:26) i (cid:90) d d x L QCD + J µA a A aµ + J ψ ψ + ψJ ψ (cid:27) Since the boson sector of QCD is that of pure Yang-Mills theory (modulo the interactionwith the quarks), everything we previously said for Yang-Mills theory still applies to QCD.In particular, due to the gauge redundancy, the above partition function is ill-defined.By applying the Faddeev-Popov procedure to Z , we obtain the following gauge-dependentpartition function for QCD: Z [ { J } ] = (cid:90) D A D ψ D ψ D c D c exp (cid:26) i (cid:90) d d x L + J µA a A aµ + J ψ ψ + ψJ ψ + J c c + c J c (cid:27) where L = L YM + L g.f. + L quark + L ghost == − F aµν F a µν − ξ ∂ µ A aµ ∂ ν A aν + iψγ µ D µ ψ − m ψψ + ∂ µ c a D µ c a When re-written in terms of the Nakanishi-Lautrup field B a , this Lagrangian can be shownto be invariant with respect to the global BRST transformation δA aµ = (cid:15) D µ c a δψ = ig (cid:15) c a T a ψδc a = − g (cid:15) f abc c b c c δc a = (cid:15) B a δB a = 0 parametrized by an anticommuting number (cid:15) . By exploiting the BRST symmetry of L one is able to derive the appropriate Slavnov-Taylor identities for QCD and prove itsperturbative renormalizability.Although the primary concern of Quantum Chromodynamics is explaining the dynamicsand interactions between the quarks, pure Yang-Mills theory is still able to account formany of the essential features of the strong interactions by attributing them to the behaviorof the gluons alone. For this reason, over the last fifty years Yang-Mills theory has beena very active field of research. In what follows we will discuss three different approachesto Yang-Mills theory: standard perturbation theory, lattice gauge theory and massiveperturbation theory. 7 iorgio Comitini Standard perturbation theory for Yang-Mills theory: the non-perturbative nature of the IR regime
Since its inception in the 1950s, the primary tool for making calculations in Yang-Millstheory has been perturbation theory. In the (modern) standard perturbative approach,the Faddeev-Popov gauge-fixed action S = (cid:82) d d x L is split into two terms, S = S + S int where S is defined as the zero-coupling limit of S , S = lim g → S and S int = S − S . In terms of S and S int , the Faddeev-Popov partition function can beexpressed as Z = (cid:90) D A D c D c e i S + i S int Z is then expanded in powers of the interaction action S int to yield Z = (cid:90) D A D c D c e i S + ∞ (cid:88) n =0 i n n ! S n int = + ∞ (cid:88) n =0 i n n ! (cid:90) D A D c D c e i S S n int If an average operation (cid:104)·(cid:105) is defined with respect to the zero-order action S as (cid:104)O(cid:105) = (cid:82) D A D c D c e i S O (cid:82) D A D c D c e i S where O is an arbitrary functional of the fields A , c and c , then the partition function canbe further re-written as Z = (cid:18)(cid:90) D A D c D c e i S (cid:19) (cid:32) + ∞ (cid:88) n =0 i n n ! (cid:104)S n int (cid:105) (cid:33) The averages (cid:104)S n int (cid:105) are usually computed in terms of Feynman diagrams, amongst whichthe connected diagrams play a fundamental role: it can be shown that (cid:32) + ∞ (cid:88) n =0 i n n ! (cid:104)S n int (cid:105) (cid:33) = exp (cid:32) + ∞ (cid:88) n =1 i n n ! (cid:104)S n int (cid:105) , conn (cid:33) where (cid:104)S n int (cid:105) , conn is the restriction of (cid:104)S n int (cid:105) to its connected diagrams. In terms of theconnected diagrams, the logarithm of the partition function reads ln Z = ln Z + + ∞ (cid:88) n =1 i n n ! (cid:104)S n int (cid:105) , conn where Z = lim g → Z , yielding an expansion of ln Z in powers of the interaction action,i.e. – since S int is proportional to the coupling constant – in powers of g . For simplicity we set the gluon and ghost sources to zero. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles The zero-order action S describes the dynamics of an N A -tuple of non-interacting mass-less vector bosons, together with two N A -tuples of non-interacting, anticommuting scalarbosons with the wrong statistics. In momentum space, it reads S = (cid:90) d d p (2 π ) d (cid:26) − A aµ δ ab p (cid:20) t µν ( p ) + 1 ξ (cid:96) µν ( p ) (cid:21) A bν + c a δ ab p c b (cid:27) where t µν ( p ) and (cid:96) µν ( p ) are transverse and longitudinal projection tensors, t µν ( p ) = η µν − p µ p ν p (cid:96) µν ( p ) = p µ p ν p (0.1)The bare propagators D ab µν ( p ) and G ab ( p ) associated respectively to the vector and scalarbosons are defined by S = i (cid:90) d d p (2 π ) d (cid:26) A aµ D µν ab ( p ) − A bν + c a G ab ( p ) − c b (cid:27) Therefore D ab µν ( p ) = δ ab (cid:26) − i t µν ( p ) p + i(cid:15) + ξ − i (cid:96) µν ( p ) p + i(cid:15) (cid:27) G ab ( p ) = δ ab ip + i(cid:15) where the + i(cid:15) term ( (cid:15) > ) is introduced in the denominators in order to select the correctintegration contours for the loop integrals of the Feynman diagrams.The interaction action S int , on the other hand, can be expanded to yield S int = (cid:90) d d x (cid:26) − g f abc ∂ µ A aν A b µ A c µ − g f abc f ade A bµ A cν A d µ A e ν + g f abc ∂ µ c a A bµ c c (cid:27) Each of the three terms in S int gives rise to an interaction vertex involving the gluons andghosts. The first one corresponds to a 3-gluon vertex, the second one corresponds to a4-gluon interaction vertex, and the third one corresponds to a ghost-ghost-gluon vertex.The power series that defines ln Z perturbatively is plagued with infinities that arise fromthe divergent loop integrals in the diagrammatic expansion. In the context of Yang-Millstheory and QCD, these divergences are usually cured by a combination of dimensionalregularization [27] and renormalization group (RG) methods [28–30]. In order to absorbthe infinities into finite, renormalized parameters, one is forced to define a scale-dependentrunning coupling constant g ( µ ) whose behavior is determined by the equation µ dgdµ ( µ ) = β ( g ( µ )) The function β ( g ) – called the beta function – can be computed perturbatively to anydesired loop order. To one-loop order in standard perturbation theory it reads [31] β ( g ) = − β g π β = 113 N The same result holds in full QCD with β = 11 − n f / , n f being the number of quark flavors. iorgio Comitini By defining the strong interaction analogue α s of the electromagnetic fine structure con-stant α as α s = g π the one-loop solution to the equation of the running coupling constant can then be put inthe form α s ( µ ) = α s ( µ )1 + β α s ( µ )4 π ln( µ /µ ) where α s ( µ ) is the value of the running coupling α s at some fixed renormalization scale µ . α s ( µ ) is to replace the ordinary coupling constant in RG-improved expressions thatdescribe processes occurring at energy scales of order µ .An alternative expression for α s ( µ ) is obtained by defining an energy scale Λ YM as Λ YM = µ e − π/β α s ( µ ) so that α s ( µ ) = 2 πβ ln( µ/ Λ YM ) This expression is interesting in two respects. First of all, observe that – since ln( µ/ Λ YM ) → + ∞ as µ → ∞ – in the high energy limit the running coupling constant goes to zero .This result, known as asymptotic freedom, was discovered in 1973 by D.J. Gross andF. Wilczek [32] and by H.D. Politzer [33] and is able to explain why, for instance, indeep-inelastic scattering experiments at high momentum transfer the quarks and gluonscontained in the hadrons can be approximately treated as free particles.Second of all – since ln( µ/ Λ YM ) → as µ → Λ YM – at µ = Λ YM the running coupling be-comes infinite . In the literature, the scale at which a coupling constant diverges is knownas a Landau pole [35–37]. Because of the Landau pole, at energy scales µ > Λ YM of thesame order of Λ YM the coupling constant becomes so large that the ordinary perturbativeapproach loses its validity . At scales µ < Λ YM the coupling α s ( µ ) becomes negative –i.e. g ( µ ) becomes imaginary – and ordinary perturbation theory is manifestly ill-defined.In full QCD the Landau pole Λ QCD – also known as the QCD scale – is located at around - MeV. Since this is a quite small scale compared to the energies involved in mod-ern particle physics experiments , the ordinary approach to perturbative QCD has provedsuccessful in explaining much of the experimental data gathered in the last fifty years at thehigh-energy colliders. This success was crucial for establishing that Quantum Chromody-namics is indeed the correct theory of the strong interactions. Critical nuclear phenomenasuch as the binding of the quarks inside the hadrons, or the onset of residual nuclear forcesbetween the nucleons, however, occur at energies that are comparable to the QCD scale.With respect to the description of these phenomena, ordinary perturbation theory is ut-terly ineffective. In order to be able to make predictions about the low-energy behavior of The same behavior is shown by the coupling constant of full QCD – unless the number of quark flavorsis greater than 16. It can be shown that this behavior is not modified by higher order corrections to the beta function –see for instance ref. [34], where results for β ( g ) are reported to order g . It should be noted that since in the perturbative approach the beta function is itself computed pertur-batively, the Landau poles of Yang-Mills theory and QCD may well be artifacts of ordinary perturbationtheory. What the existence of a Landau pole actually tells us is that ordinary perturbation theory becomesinconsistent at energy scales of order Λ YM . As long as they involve processes in which the momentum transfer is not too low. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles the quarks and the gluons, one has to resort to non-perturbative computational methodssuch as lattice gauge theory (to be reviewed in the next section) or the numerical resolutionof an infinite set of Schwinger-Dyson equations (SDEs) [38, 39]. Albeit successful in theirown respect, these methods have the shortcoming of being non-analytical, thus providingnumerical results with no control over the intermediate steps of the calculations.To conclude this brief review of ordinary perturbation theory, we wish to address thetopic that will be the main subject of this thesis, namely, the issue of the mass of thegluons. As we saw in the previous section, by forbidding the inclusion of a mass term forthe gluon fields in the Yang-Mills Lagrangian, gauge invariance constrains the gluons to bemassless at the classical level. However, at the quantum level, the interactions could stillbe responsible for the generation of a dynamical gluon mass [41]. This mass would man-ifest itself in the finiteness of the transverse component of the dressed gluon propagatorevaluated at zero momentum.In ordinary perturbation theory, the dressed gluon propagator (cid:101) D ab µν can be expressed as [31] (cid:101) D ab µν ( p ) = δ ab (cid:26) − i t µν ( p ) p − Π( p ) + ξ − i (cid:96) µν ( p ) p (cid:27) where Π( p ) is the one-particle-irreducible gluon polarization. In the limit of vanishingmomentum, its transverse component (cid:101) D ,T reads (cid:101) D ,T (0) = − i − Π(0) If Π(0) = 0 , as the momentum goes to zero the transverse dressed propagator grows toinfinity. This behavior is typical of massless propagators and is displayed by the bare gluonpropagator D ab µν itself. On the other hand, if Π(0) (cid:54) = 0 , the transverse dressed propagatorremains finite at zero momentum. This is the limiting behavior that characterizes themassive propagators, as exemplified by the ordinary free propagator of a massive particle, ∆( p ) = ip − M Therefore whether a mass is generated or not for the gluons depends on the zero momentumlimit of the gluon polarization.Now, it can be shown that to any finite loop order the gluon polarization of ordinaryperturbation theory vanishes at zero momentum. In the context of pure Yang-Mills theoryor full QCD with massless quarks this is clearly the case, since ordinary perturbationtheory has no intrinsic mass scales and by dimensional analysis Π( p ) must be proportionalto p . In full QCD with massive quarks more elaborate arguments based on gaugeinvariance (or rather BRST invariance) and the structure of the quark-gluon interactionare required to prove this claim [40]. In any case, ordinary perturbation theory is unable todescribe the phenomenon of mass generation: the gluon is constrained to remain masslessto any finite perturbative order. Of course, it could be argued that since mass generationis a low energy phenomenon, no conclusive evidence for its occurrence (or lack thereof) canbe gathered through ordinary perturbation theory. In the next section we will see what anon-perturbative approach like lattice gauge theory has to say with respect to this issue. In good regularization schemes the renormalization scale is contained in logarithmic corrections tothe propagator that cannot modify this behavior. iorgio Comitini Yang-Mills theory on the lattice: dynamical mass generation
Lattice gauge theory [42, 43] is a non-perturbative numerical approach to quantum fieldtheories with a gauge group based on the discretization of spacetime on a finite lattice. Inwhat follows we will briefly review the definition of Yang-Mills theory on the lattice anddiscuss some crucial results which have recently been obtained by the lattice calculations.As a preliminary step in the definition of the lattice approach, we recall that a (finitefour-dimensional cubic) lattice is a set of points of the form a ( n , n , n , n ) ∈ R , where a is the lattice spacing and the n µ ’s are integers that span from zero to a finite number L . As we will see, the dynamical variables of lattice Yang-Mills theory are defined on thelinks that connect the neighboring sites of the lattice.In order to formulate the lattice-equivalent of Yang-Mills theory, one starts by rewritingthe quantum partition function Z of the theory in terms of fields which are defined in Eu-clidean space rather than in Minkowski space. The Euclidean partition function is obtainedfrom Z by replacing everywhere the real time variable t by an imaginary time variable τ defined as τ = it . Since the A at ’s – i.e. the time-components of the Yang-Mills fields –are defined with respect to the real time t , in the Euclidean formulation the latter need tobe replaced by analogous components in imaginary time; this is achieved by substituting A at → iA aτ in the Yang-Mills action. The derivatives with respect to real time too need tobe exchanged with derivatives with respect to imaginary time: in the action we will haveto replace ∂ t → i∂ τ . These redefinitions leave us with a partition function that can be putin the form Z = (cid:90) D A e −S E where S E is a Euclidean action defined as S E = (cid:90) dτ d x F aµν F a µν (cid:12)(cid:12)(cid:12) ∂ t → i∂ τ ,A at → iA aτ The imaginary units are easily seen to drop out from the above equation if we replace theMinkowski metric by the Euclidean metric. The Euclidean Lagrangian L E then reads L E = 14 F aEµν F aµνE = 14 δ µσ δ νλ F aEµν F aEσλ where F E is the field-strength tensor associated to the Euclidean vector fields ( A aτ , A ai ) .From now on we will drop the subscripts E and imply that the Yang-Mills fields, field-strength tensor, metric and action are all defined in Euclidean four-dimensional space.The second step for formulating lattice Yang-Mills theory is to find dynamical variablesthat are appropriate to the discrete structure of the lattice. The hint as to how to dothis comes from the geometrical structure of the gauge fields themselves. Observe that,since the Yang-Mills fields are actually covector fields (i.e. 1-forms) with values in su (N),they can be meaningfully integrated along curves in spacetime to yield elements of the Liealgebra of SU(N). If γ : [0 , → R is such a curve, then we can define (cid:90) γ A = (cid:90) ds dγ µ ds ( s ) A aµ ( γ ( s )) T a By exponentiating this Lie algebra element in a path-ordered fashion, we obtain an element12 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles U γ ( A ) of the group SU(N) that is functionally dependent on A , namely U γ ( A ) = P exp (cid:26) ig (cid:90) γ A (cid:27) Therefore any gauge field A establishes a correspondence between curves in spacetime andgroup elements of SU(N) by associating a U γ ( A ) to each curve γ . In particular, the gaugefield associates SU(N) group elements to each of the links (cid:96) that connect the neighboringlattice points in our discretized spacetime; we shall denote these group elements by U (cid:96) ( A ) .Any arbitrary (cid:96) has the form x + as e µ for s ∈ [0 , , where x is the initial point of thelink and e µ – the direction of the link – can be one of e = (1 , , , , e = (0 , , , , e = (0 , , , or e = (0 , , , . It follows from our general expression for U γ ( A ) that U (cid:96) ( A ) = + iga A µ ( x ) + O ( a ) where A µ = A aµ T a . In particular, in the limit of vanishing lattice spacing, A µ ( x ) = lim a → U (cid:96) − iga where (cid:96) is the link in the direction e µ originating from x . The above expression teaches ushow to recover the gauge field starting from arbitrary group elements defined on the lattice.With such a procedure in our hands, we can seek for a formulation of lattice Yang-Millstheory that has the U (cid:96) ’s, rather than the gauge field, as its dynamical variables. In orderto do so, we take one step back and study the group elements associated by the gaugefields to the closed curves.If γ is a loop – i.e. continuous closed curve – then U γ ( A ) is called the holonomy of γ withrespect to the gauge field A . The holonomies can be related to the field-strength tensor asfollows. Suppose that – as shown in Fig.1 – γ is a loop composed by four rectilinear curvesthat join in succession the points x , x + (cid:15) ξ , x + (cid:15) ξ + (cid:15) ξ , x + (cid:15) ξ and x , where (cid:15) is asmall positive number. Then, by expanding U γ ( A ) in powers of (cid:15) , one finds that [42] U γ ( A ) = + ig(cid:15) F µν ( x ) ξ µ ξ ν − g (cid:15) F µν ( x ) F στ ( x ) ξ µ ξ ν ξ σ ξ τ + O ( (cid:15) ) Figure 1: Integration path for recovering the field-strength tensor from the holonomies.13 iorgio Comitini where F µν = F aµν T a is the field-strength tensor associated to A µ . In particular, if we define U µν ( A ) = U γ ( A ) with ξ = e µ and ξ = e ν unitary vectors in the directions µ and ν , fromthe above expression we obtain U µν ( A ) = + ig(cid:15) F µν ( x ) − g (cid:15) F µν ( x ) F µν ( x )+ O ( (cid:15) ) (no sum over repeated indices)Therefore the holonomy U µν ( A ) contains both the component F µν of the field-strengthtensor and its square ( F µν ) .Now, suppose that (cid:15) = a , the lattice spacing. Then in the limit of vanishing lattice spacingthe components of F can be extracted from the holonomy as F µν ( x ) = lim a → U µν − iga Moreover, recalling that – since T a ∈ su (N) – Tr { T a } = 0 and Tr { T a T b } = δ ab / , by takingthe trace of U µν ( A ) we find thatTr { U µν ( A ) } = N − g a F aµν ( x ) F aµν ( x ) + O ( a ) (no sum over µ, ν indices)This expression brings us to the final step of the definition of the lattice approach, namely,the choice of a discrete action for the group elements U (cid:96) . By summing up both sides ofthe equation with respect to all the possible directions µ and ν , subject to the constraintthat µ < ν in order to avoid the double count of the holonomies, we find that F aµν F a µν = 2 Ng a (cid:88) µ<ν (cid:18) − N Tr { U µν ( A ) } (cid:19) + O ( a ) where this time the indices on the left-hand side are summed over. This result suggeststhe following definition for the lattice action: S W = 2 Ng (cid:88) x, µ<ν (cid:18) − N Tr { U µν } (cid:19) S W is known as the Wilson action [44]; in deriving it from the holonomies we have provedthat it reduces to the Yang-Mills action in the limit of vanishing lattice spacing (and infinitesize of the lattice). In terms of the Wilson action, the lattice partition function Z lat reads Z lat = (cid:90) (cid:89) (cid:96) dU (cid:96) e −S W where the holonomies U µν and group elements U (cid:96) are related by U µν = U (cid:96) U (cid:96) U (cid:96) U (cid:96) , with (cid:96) , (cid:96) , (cid:96) , and (cid:96) the links that make up the loop on which the holonomy is defined. Ob-serve that since the number of links in a finite lattice is itself finite, Z lat is an integral overthe configurations of a finite number of degrees of freedom, and is thus mathematicallywell-defined.For N = 3 , upon introducing appropriate quark variables at the sites of the lattice, theconstruction given above defines the non-perturbative discrete approximation to QuantumChromodynamics known as lattice QCD. Lattice QCD has proven successful in describing14 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles both qualitatively and quantitatively many of the low-energy features of the interactionsbetween the quarks and the gluons. Among the notable results of the lattice approach wecite the derivation of the masses and quantum numbers of the hadrons from the dynamicsof their elementary constituents [45], the description of confinement in terms of gluonicflux tubes [46] and the prediction of the crossover temperature between the confined phaseand deconfined phase of quark-gluon matter [47].Of particular relevance for the purposes of this thesis is the take of the lattice on the issueof mass generation. As we saw in the last section, the generation of a dynamical mass forthe gluons manifests itself in the finiteness of the transverse dressed gluon propagator inthe limit of zero momentum. Through the lattice approach one is able to compute thegluon propagator non-perturbatively, albeit as a function of the Euclidean momentum p E rather than of the Minkowski momentum p . Nonetheless, since p E = ( − ip ) + | p | = − p ,where p is the time-component of the Minkowski momentum, p = ( p , p ) , we have thatthe limits p → and p E → of the Minkowski and Euclidean propagators actually coin-cide. Therefore the question of whether or not the gluons acquire a dynamical mass can beanswered as well by investigating the low momentum behavior of the Euclidean propagatorcomputed on the lattice. In what follows we report the results of ref. [61] (Duarte et al.)for the transverse component of the dressed gluon propagator computed on the lattice inthe Landau gauge in the framework of pure Yang-Mills SU(3) theory.The lattice data of ref. [61] for the gluon propagator is shown as a function of the Eu-clidean momentum in Fig.2. As we can see, as the Euclidean momentum goes to zerothe gluon propagator first changes concavity and then saturates to a finite value of order (300 MeV ) − . This result is of crucial importance, in that it proves that the gluons indeedacquire a dynamical mass in the infrared. This possibility was not unforeseen, as it hadbeen anticipated already in the 1980s by SDE analyses of the Green functions of QCD [41];nonetheless, the lattice calculations were the first approach to give reliable evidence of theoccurrence of mass generation in QCD by making its low-energy regime accessible to thecomputations. D ~ T ( ξ = ) ( p E ) ( G e V - ) p E (GeV) Figure 2: Transverse component of the Euclidean dressed gluon propagator computed onthe lattice in the Landau gauge ( ξ = 0 ). Lattice data points from ref. [61]. For the problem of gauge fixing in lattice gauge theories see for instance [48]. iorgio Comitini To end this introduction we report the lattice data of ref. [61] for the dressed ghost prop-agator (cid:101) G ( p E ) in the Landau gauge. Rather than the propagator itself, in Fig.3 we showthe data points for the ghost dressing function p E (cid:101) G ( p E ) . As we can see, in the limit p E → the dressing function approaches a finite value, implying that the ghost propaga-tor becomes infinite at zero momentum. As discussed in the previous section, this behavioris typical of massless propagators. Therefore the lattice results inform us that – at variancewith the gluons – the ghosts remain massless in the infrared as well as in the ultravioletregime. p E G ~ ( ξ = ) ( p E ) p E (GeV) Figure 3: Lattice ghost dressing function in the Landau gauge ( ξ = 0 ). Data from ref. [61]. For the definition of the ghosts in the lattice approach see for instance [49]. Dynamical mass generation: theperturbative vacuum of Yang-Millstheory from a variational perspective
In recent years, lattice calculations [53–61] have shown that the gluon develops an infrareddynamical mass that prevents its propagator from diverging in the limit of vanishing mo-mentum. In ordinary Yang-Mills and QCD perturbation theory, the phenomenon of dy-namical mass generation cannot be described at any finite perturbative order. The reasonfor this is two-fold. On the one hand, the energy scale for the scaleless pure Yang-Millstheory (or even full QCD with chiral quarks) is set by the spontaneous breaking of scaleinvariance. In perturbation theory the breaking of scale invariance manifests itself withthe appearance of corrections to the Green functions which depend logarithmically on therenormalization scale used to define the theory [31]; in the absence of other energy scales,these terms are not able to generate a dynamical mass for the gluon at any finite order, sothat pure Yang-Mills theory and chiral QCD in the ordinary perturbative approach remainmassless even after the breaking has occurred. On the other hand, even for full QCD withnon-chiral quarks (whose energy scale is set by the interplay between the renormalizationscale and the masses of the quarks), in the absence of explicit gauge-symmetry breakingterms in the original Lagrangian, the Slavnov-Taylor identities [22, 23] constrain the effec-tive action to remain BRST-invariant at any finite order. This implies that the gluon massdoes not get renormalized by the interactions, hence, again, that the gluon cannot acquirea mass at any finite perturbative order [51].The inability to describe the phenomenon of mass generation is a limitation of ordinaryperturbation theory, rather than of Yang-Mills theory itself. As a matter of fact, if weassume that the discretization on the lattice does not fundamentally spoil the symmetriesof the theory, the lattice results lead to the conclusion that either BRST invariance isspontaneously broken at low energies – so that the gluon can freely acquire a dynamicalmass – or that BRST invariance protects the gluon mass from radiative corrections onlyperturbatively – so that non-perturbative approaches (or non-ordinary perturbative ex-pansions) could still be able to describe the phenomenon of dynamical mass generation inYang-Mills theory and QCD.Since dynamical mass generation is an intrinsically low-energy phenomenon, we shouldexpect it to leave traces on the vacuum structure of the theory. As a matter of fact, ithas been proposed that the mechanism for gluon mass generation relies on the presenceof non-perturbative condensates that populate the rich vacuum of Yang-Mills theory andQCD [41]. Since our objective is to develop a non-standard perturbative expansion for17 iorgio Comitini
Yang-Mills theory, the question we wish to ask in this chapter is the following: from aperturbative perspective, which free-particle vacuum state best approximates the true vac-uum of the theory?In order to answer this question, we pursue a simple variational approach that goes underthe name of Gaussian Effective Potential (GEP) [62–70]. The GEP is, roughly speaking,the energy density of a Gaussian state computed to first order in the interactions. Sincethe vacuum states of free field theories are Gaussian states [50], one can easily interpretthese as the free-particle, unperturbed states starting from which one sets up perturbationtheory. Since the width of the Gaussian depends on the mass of the particle, the GEPitself is a function of mass. For an ordinary bosonic field theory, the Jensen-Feynmaninequality [52] states that the GEP is bounded from below by the exact vacuum energy ofthe theory. Therefore, by minimizing the GEP with respect to the mass of the particle,one obtains the best perturbative approximation to the true vacuum of the theory.As we will see in what follows, it turns out that the GEP of Yang-Mills theory is minimizedby a non-zero value of the mass of the gluon. This fact 1. may be interpreted as evidence ofmass generation in Yang-Mills theory, 2. implies that the best perturbative approximationto the vacuum of Yang-Mills theory is attained by massive – rather than massless – gluons,foreshadowing the fact that a non-ordinary perturbation theory which treats the gluons asmassive already at tree-level could lead to predictions which are in better agreement withthe exact, non-perturbative, results.This chapter is organized as follows. In Sec.1.1 we will introduce the concept of optimizedperturbation theory, define the Gaussian Effective Potential for a general quantum fieldtheory and perform the GEP analysis of λφ theory as a toy model for the phenomenon ofmass generation. In Sec.1.2 we will apply the GEP approach to pure Yang-Mills theory andshow that its perturbative vacuum is indeed massive, rather than massless, at far as thetransverse gluons are concerned. In doing so, we will have to deal with subtleties arisingdue to the anticommuting nature of the ghost fields [69]; as we will see, these subtletiescan be explicitly addressed and a variational statement can still be made regarding thelower bound set on the GEP by the exact vacuum energy of the theory.The results of this chapter were presented for the first time in ref. [71] and publishedin ref. [72]. 18 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles λφ theory as a toy model for mass generation Consider an harmonic oscillator of frequency ω perturbed by a quartic potential, H = p ω x λx (1.1)with λ a small parameter. In ordinary perturbation theory one splits H as H ( ω ) + V ,where H ( ω ) = p ω x (1.2)is the Hamiltonian of the unperturbed oscillator of frequency ω . The ground state | ω (cid:105) of H ( ω ) has wavefunction ψ ω ( x ) = (cid:18) ω π (cid:19) / exp (cid:18) − ω x (cid:19) (1.3)and unperturbed energy E (0) g = ω / . To first order in ordinary perturbation theory, theenergy E (1) g ( ω ) of the ground state of H is given by E (1) g ( ω ) = ω (cid:104) ω | V | ω (cid:105) = ω λ ω (1.4)In a non-ordinary formulation of perturbation theory, we may split H as H = H ( ω ) + V (cid:48) ,where V (cid:48) = ( ω − ω ) x λx (1.5)and H ( ω ) is the Hamiltonian of an unperturbed harmonic oscillator of frequency ω . Theground state | ω (cid:105) of H ( ω ) has wavefunction ψ ω ( x ) = (cid:18) ωπ (cid:19) / exp (cid:18) − ωx (cid:19) (1.6)and unperturbed energy ω/ . If we treat V (cid:48) as a perturbation to H ( ω ) , then the energy E (1) g ( ω ) of the ground state of H to first order in perturbation theory is given by E (1) g ( ω ) = ω (cid:104) ω | V (cid:48) | ω (cid:105) = ω ω ω + 3 λ ω (1.7)Notice that since E (1) g ( ω ) = (cid:104) ω | H | ω (cid:105) , the latter is precisely the energy obtained by ap-plying the variational method to the test function ψ ω . Therefore we know that the exactground state energy of the perturbed oscillator is less than or equal to E (1) g ( ω ) , and we canminimize E (1) g ( ω ) with respect to ω to obtain the best estimate of the ground state energy: ∂E (1) g ∂ω ( ω g ) = 14 − ω ω g − λ ω g ⇐⇒ ω g − ω ω g − λ = 0 (1.8)19 iorgio Comitini where we have defined ω g as the frequency that minimizes E (1) g ( ω ) . We see that ω g = ω if and only if λ = 0 , i.e. if the harmonic oscillator is unperturbed. For λ (cid:54) = 0 , we have ω g > ω , so that the best approximation to the ground state energy is given by a Gaussianwith a variance smaller than that of the unperturbed oscillator. This was to be expected,since at high x ’s the quartic potential increases more rapidly than the harmonic potential,thus producing eigenstates which are bound around x = 0 more tightly than those of theunperturbed oscillator. Moreover, as long as λ is sufficiently small, ω g = ω + 3 λω + O ( λ ) (1.9)and ω g is approximately equal to ω . Therefore we can still regard V (cid:48) as a small perturba-tion to H ( ω g ) and our non-ordinary formulation of perturbation theory is still valid with H ( ω g ) as the zero-order Hamiltonian. Since | ω g (cid:105) is closer than | ω (cid:105) to the true groundstate of the perturbed oscillator, we expect the ground state average (cid:104)O(cid:105) g of an arbitraryoperator O to be better approximated in perturbation theory if we compute it by expand-ing perturbatively around the eigenstates of H ( ω g ) , rather than around those of H ( ω ) .For this reason, we call | ω g (cid:105) the perturbative ground state of the theory.This method for computing quantities in quantum mechanics is known as optimized per-turbation theory [62], and can be readily generalized to any quantum system. In a generalsetting, let H be the exact Hamiltonian of some quantum system. H can be arbitrarilysplit as H + H int , where H is an Hamiltonian of which we know the exact eigenstatesand eigenenergies and H int = H − H . If we choose an H such that its eigenstates wellapproximate those of H , then the perturbative series having H as the zero-order Hamilto-nian will lead to more accurate predictions than those obtained by using a different H . Ifthe optimal H is chosen through a variational ansatz, then the perturbative series arisingfrom the split H = H + H int is said to be optimized . The ground state of the optimized H is again called a perturbative ground state or – if the quantum theory is a field theory– a perturbative vacuum . In quantum field theory, ordinary perturbation theory follows from choosing as the zero-order Hamiltonian H the free field Hamiltonian obtained in the limit of vanishing (renor-malized) couplings. For instance, in λφ theory, which has Hamiltonian H = (cid:90) d D x (cid:26) π + 12 |∇ φ | + m φ + λ φ (cid:27) + H c.t. (1.10)with m the pole mass of the scalar propagator and D the number of spatial dimensions,the ordinary choice for H is H = (cid:90) d D x (cid:26) π + 12 |∇ φ | + m φ (cid:27) (1.11) Here H c.t. contains all the relevant renormalization counterterms and vanishes at any perturbativeorder as λ – the renormalized coupling – goes to zero. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles The vacuum states of free field Hamiltonians are Gaussian functionals of the field configu-rations [50]. For fixed spin, the only free parameter of these functionals is the mass of theparticle. For instance, the vacuum wavefunctional of a free real scalar field of mass m in D spatial dimensions is given by [50] ψ [Φ] = N exp (cid:18) − (cid:90) d D x d D y Φ( x ) E m ( x − y ) Φ( y ) (cid:19) (1.12)where N is a normalization constant and E m ( x − y ) = (cid:90) d D k (2 π ) D e i k · ( x − y ) (cid:112) m + | k | (1.13)is the Fourier transform of the energy of the scalar particle. Starting from H and the cor-responding vacuum wavefunctional, we can compute all the relevant quantities in ordinaryperturbation theory by treating H int = H − H as a perturbation.As long as we limit ourselves to free field Hamiltonians and Gaussian wavefunctionals,since, as we said, in this case the only free parameter is the mass of the functional, the ob-vious field-theoretic generalization of ordinary perturbation theory is obtained by choosingas the zero-order Hamiltonian H a free field Hamiltonian with a mass different from thatcontained in the (renormalized) Lagrangian. Then one can optimize the value of the massby requiring it to minimize the vacuum energy of the theory to first order in perturbationtheory – a procedure which is equivalent to applying the variational method to the groundstate of H –, thus obtaining an optimized perturbation theory with a Gaussian perturba-tive vacuum. With some abuse of language, the energy density of the vacuum state of aquantum field theory, computed to first order in its interactions by using as the zero-ordervacuum wavefunctional a Gaussian with free parameters the masses of the particles, iscalled the Gaussian Effective Potential (GEP) [62–70].The GEP is arguably the simplest tool for determining the perturbative ground state ofa field theory. In principle, it may have as additional free parameters the vacuum expec-tation values of the fields; for example, for a scalar particle one may take as the vacuumwavefunctional for computing the GEP ψ [Φ] = N exp (cid:18) − (cid:90) d D x d D y (cid:0) Φ( x ) − (cid:104) Φ (cid:105) (cid:1) E µ ( x − y ) (cid:0) Φ( y ) − (cid:104) Φ (cid:105) (cid:1)(cid:19) (1.14)with an arbitrary mass µ and vacuum expectation value (cid:104) Φ (cid:105) , and compute its GEP. How-ever, since we are only interested in theories whose fields have vanishing vacuum expecta-tion values , in what follows we will limit ourselves to GEP’s whose only free parametersare the masses of the particles.Of particular interest is the case in which the bare masses in the original Hamiltonianare zero. Then the interactions may or may not generate a dynamical mass for the excita-tions of the fields; likewise, the perturbative vacuum of the theory may or may not be theGaussian vacuum of a massive particle. By applying the GEP approach to such theories,one is able to address the issue of mass generation both from a perturbative and from anon-perturbative perspective. If the GEP is found to be minimized by a non-zero value of Since the gluon field is a vector field, a non-zero vacuum expectation value for A aµ would lead to thespontaneous breaking of Lorentz symmetry, which we assume not to occur in any sensible relativistic fieldtheory. iorgio Comitini the mass parameter, implying [63, 64] that the massless vacuum of the theory is unstabletowards a massive vacuum, then one 1. has strong indications of the occurrence of the phe-nomenon of mass generation (non-perturbative aspect of the GEP analysis) and 2. has aneven stronger indication that, since the massless perturbative vacuum is farther away fromthe true vacuum than the massive one, a non-ordinary perturbation theory which treatsthe excitations of the fields as massive already at tree-level may lead to more accuratepredictions than those obtained by ordinary (massless) perturbation theory (perturbativeaspect of the GEP analysis).We now proceed to give a formal definition of the GEP in the Lagrangian framework.The vacuum energy density E of a quantum field theory defined by the action S [ F ] , de-scribing a set of fields which we collectively denote by F , is given by e − i EV d = (cid:90) D F e i S [ F ] (1.15)where V d is the d -dimensional volume of spacetime. If S is polynomial in the fields and itsderivatives, we know how to compute E perturbatively. We set S = S + S int , where S isan action term quadratic in the fields, and expand (cid:90) D F e i S [ F ] = (cid:90) D F e i S [ F ] + ∞ (cid:88) n =0 i n S n int [ F ] n ! = (cid:18)(cid:90) D F e i S [ F ] (cid:19) (cid:32) + ∞ (cid:88) n =0 i n (cid:104)S n int (cid:105) n ! (cid:33) (1.16)so that − i EV d = ln (cid:90) D F e i S [ F ] + ln (cid:32) + ∞ (cid:88) n =1 i n (cid:104)S n int (cid:105) n ! (cid:33) (1.17)In both (1.16) and (1.17), the quantum average (cid:104) · (cid:105) is defined with respect to the zero-order action S . Since exp( i S ) is Gaussian in the field configurations, in order to computethe averages (cid:104)S n int (cid:105) one only needs to evaluate polynomial functional integrals with Gaus-sian kernels; this is usually done by making use of appropriate Feynman rules.In ordinary perturbation theory, one chooses as the zero-order S the free action associatedto the set of fields F , obtained, for instance, by taking the limit of vanishing renormal-ized couplings of the full action S . In a more general setting, we may still define S tobe the free action associated to the fields F , but with arbitrary – rather than on-shell– particle masses, which we collectively denote by m . With this choice, both S and S int = S − S are functions of the mass parameters m . Going back to eq. (1.17) andexpanding ln(1 + x ) = x + O ( x ) , we find − i EV d = ln (cid:90) D F e i S ( m ) + i (cid:104)S int ( m ) (cid:105) + O ( (cid:10) S int (cid:11) ) (1.18)The quantity V G ( m ) , defined by − iV G ( m ) V d = ln (cid:90) D F e i S ( m ) + i (cid:104)S int ( m ) (cid:105) (1.19)is called the Gaussian Effective Potential (GEP). It is the vacuum energy density of thefield theory, computed to first order in its interactions as a function of the tree-level mass22 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles parameters m . Since the GEP is obtained by expanding the vacuum energy density to firstorder in S int rather than to first order in the coupling, V G is an essentially non-perturbativeobject. Since the GEP assumes the zero order action to be Gaussian in the fields, the GEPanalysis addresses the issues of stability and mass generation from a perturbative perspec-tive. If the fields F are c -fields (i.e. if they are not Grassmann-valued), the Jensen-Feynmaninequality [52] can be exploited to show that the exact vacuum energy of the system E setsan upper bound for the GEP V G ( m ) evaluated at any value of m : V G ( m ) ≥ E ∀ m (1.20)This implies that V G ( m ) computed at its minimum is the variational estimate of the vac-uum energy density of the theory. By minimizing the GEP with respect to the massparameters, one obtains the best Gaussian (i.e. free particle-) approximation to the vac-uum of the system, that is, the perturbative vacuum of the theory. Once the perturbativevacuum is known, one can compute the quantities of interest in optimized perturbationtheory by formulating the perturbative series so that S ( m ) – where m is the value thatrealizes the minimum of the GEP – is the zero-order action of the expansion. λφ theory: a toy model formass generation Before moving on to Yang-Mills theory, in order to get acquainted with the formalism, thebasic features of the Gaussian Effective Potential and their connection to the issue of massgeneration, let us define and compute the GEP of λφ theory. The action of λφ theory isgiven by S = (cid:90) d d x (cid:40) ∂ µ φ ∂ µ φ − m p φ − λ φ + L c.t (cid:41) (1.21)where m p is the pole mass of the scalar particle and L c.t. contains the appropriate renor-malization counterterms. The S and S int of ordinary perturbation theory are taken to be S = (cid:90) d d x (cid:40) ∂ µ φ ∂ µ φ − m p φ (cid:41) S int = S − S = (cid:90) d d x (cid:26) − λ φ + L c.t (cid:27) (1.22)In order to define the GEP, we must allow for arbitrary tree-level masses. Hence we chooseas S S ( m ) = (cid:90) d d x (cid:26) ∂ µ φ ∂ µ φ − m φ (cid:27) (1.23)where m is a mass parameter; it follows that S int ( m ) = S − S ( m ) = (cid:90) d d x (cid:40) − m p − m φ − λ φ + L c.t (cid:41) (1.24)From now on, we will work with bare – rather than with renormalized – masses and cou-pling constants: in order to define the renormalized mass and coupling, we are required to23 iorgio Comitini choose a renormalization scheme from the very start; we decide not to do so and ratherto renormalize the GEP a posteriori, according to what divergences may arise from itscomputation. In terms of the bare mass m B and bare coupling λ B , the GEP is given by V G ( m ) = i V d ln (cid:90) D φ e i S ( m ) + 1 V d (cid:90) d d x (cid:26) m B − m (cid:10) φ (cid:11) + λ B (cid:10) φ (cid:11) (cid:27) (1.25) To each term in V G we associate a Feynman diagram. The first, logarithmic term ineq. (1.25) is usually represented as a closed loop with no vertices (first diagram in Fig.4);the quadratic term, being proportional to the spacetime integral of the propagator, isrepresented as a closed loop with a two-point vertex (proportional to m B − m , second di-agram in Fig.4); the quartic term can be interpreted as the integral of the tadpole diagram(Fig.5), and as such it is represented by a double loop with a four-point vertex (the usualfour-point coupling vertex, proportional to λ B , third diagram in Fig.4). These diagramsmay be computed by using appropriate Feynman rules. For better clarity, however, let usdo the computation explicitly in coordinate space. We have (cid:10) φ ( x ) (cid:11) = lim y → x (cid:104) T { φ ( x ) φ ( y ) }(cid:105) = lim y → x D mF ( x − y ) = (cid:90) d d k (2 π ) d ik − m + i(cid:15) = J m (1.26) (cid:10) φ ( x ) (cid:11) = lim y ,y ,y → x (cid:104) T { φ ( x ) φ ( y ) φ ( y ) φ ( y ) }(cid:105) = (1.27) = lim { y i }→ x {D mF ( x − y ) D mF ( y − y ) + D mF ( x − y ) D mF ( y − y ) + D mF ( x − y ) D mF ( y − y ) } == 3[ D mF (0)] = 3 J m where D mF ( x ) is the Feynman propagator (in coordinate space) of the free scalar field ofmass m , D mF ( x ) = (cid:90) d d k (2 π ) d e − ik · x ik − m + i(cid:15) (1.28)Figure 4: Diagrams which contribute to the GEP of λφ theory. From left to right: thelogarithmic contribution, the quadratic contribution, the quartic contribution.24 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Figure 5: The tadpole diagram.and J m is the Euclidean integral defined by J m = (cid:90) d d k E (2 π ) d k E + m (1.29)As for the first term in (1.25), an explicit computation of the Gaussian functional integralleads to ln (cid:90) D F e i S ( m ) = − i V d (cid:90) d d k E (2 π ) d ln( k E + m ) = − i V d K m (1.30)where we have defined the Euclidean integral K m as K m = 12 (cid:90) d d k E (2 π ) d ln( k E + m ) (1.31)By summing up the three contributions with the appropriate coefficients, we find that theGEP of λφ theory is given by V G ( m ) = K m −
12 ( m − m B ) J m + λ B J m (1.32) As it stands, the expression (1.32) for V G ( m ) is ill-defined: both K m and J m are di-vergent integrals which need to be regularized. Let us suppose for the moment that thishas been done. Then, by taking the derivative of V G ( m ) with respect to m , we obtainthe stationarity condition for the GEP: V G ( m ) is extremized by the values m such that ∂V G ∂m ( m ) = − ∂J m ∂m (cid:12)(cid:12)(cid:12)(cid:12) m (cid:26) m − m B − λ B J m (cid:27) = 0 (1.33)where we have used the formal identity ∂K m ∂m = 12 J m (1.34)25 iorgio Comitini Since formally the derivative of J m with respect to m is negative definite, ∂J m ∂m = − (cid:90) d d k E (2 π ) d k E + m ) < (1.35)the derivative of V G is positive for m > m B + λ B J m / and negative for m < m B + λ B J m / :if it exists, the value m = m defined by m = m B + λ B J m (1.36)is a minimum for the GEP. Eq. (1.36) is not new at all: provided that m B (cid:54) = 0 , when thedressed scalar propagator of λφ theory is computed to one loop in ordinary perturbationtheory, one finds that the relation between the bare mass m B and the pole mass m p of thescalar particle is [31] m B = m p − λ J m p (1.37)Therefore, the GEP approach predicts that the vacuum energy density of λφ theory isminimized precisely by the pole mass of the scalar particle computed to one loop order: m = m p .On the other hand, consider what happens in the case of a vanishing bare mass. For m B = 0 , eq. (1.36) reads m = λ B J m (1.38)Eq. (1.38) is known as the gap equation of the GEP. Assuming that it admits a non-zerosolution, by fixing the value of the mass parameter that minimizes V G , the gap equationpredicts that the perturbative vacuum of λφ theory is massive, even if the theory by itselfwas massless. This is at variance with ordinary perturbation theory, which in turn pre-dicts that the propagator of massless λφ theory remains massless even after the quantumcorrections are included .In conclusion, not only through the GEP one is able to derive the perturbative one-looprelation between the bare mass and the pole mass of the massive theory, but the approachalso sheds light on the non-perturbative issue of mass generation in the massless theory.Since we are only interested in the latter case, from now on we will set m B = 0 and studythe behavior of the GEP of λφ theory at vanishing bare mass. Let us now turn to the issue of renormalization in d = 4 . As we will see, perhaps counter-intuitively, different renormalization procedures lead to different conclusions with respectto the issue of mass generation.To begin with, suppose that massless λφ theory is defined with an intrinsic sharp cutoff Λ , so that all the integrals in Euclidean momentum space are convergent and J m and Recall that in λφ theory λ = λ B to one loop order. This prediction is actually renormalization-scheme-dependent. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles ∂J m /∂m are respectively positive and negative definite. An explicit computation showsthat K m = 164 π (cid:26) Λ ln (cid:18) m Λ (cid:19) − Λ m − m ln (cid:18) Λ m + 1 (cid:19)(cid:27) J m = 116 π (cid:26) Λ − m ln (cid:18) Λ m + 1 (cid:19)(cid:27) (1.39) ∂J m ∂m = 116 π (cid:26) − ln (cid:18) Λ m + 1 (cid:19)(cid:27) where we have not yet taken the limit Λ → ∞ in order for the GEP to be defined for all m ’s. According to our calculations, the gap equation reads m = λ B π (cid:26) Λ − m ln (cid:18) Λ m + 1 (cid:19)(cid:27) (1.40)For arbitrarily large λ B ’s, the solution to this equation may be of order Λ or greater (Fig.6).Since Λ is a cutoff, if m is to have any physical meaning at all it must be much smallerthan Λ . This is verified if and only if λ B is sufficiently small, in which case the solution tothe gap equation can be approximated as m ≈ λ B Λ π (1.41) m / Λ λ B Figure 6: Numerical solution to the gap equation of the GEP in the sharp cutoff renor-malization scheme. We have added a term proportional to Λ ln Λ to K m in order to adimensionalize the argument ofthe first logarithm. Such a modification does not spoil our computation, since it amounts to adding anarbitrary, m -independent, constant to the vacuum energy density of the system. iorgio Comitini Therefore, if we regularize the GEP through a cutoff, we find that 1. the solution to thegap equation is physically acceptable only if the bare coupling is sufficiently small, 2. ifthis is the case, then the optimal mass scale is roughly proportional (albeit through a smallproportionality constant) to the cutoff, i.e. m is proportional to the quadratic divergenceof the tadpole diagram.This last feature, in particular, is due to the fact that in λφ theory – at variance withgauge theories – no special symmetry protects the mass of the scalar particle from receivinglarge quantum corrections from the quadratic divergences. If we are to interpret λφ theoryas a toy model for mass generation in Yang-Mills theory, the above solution cannot thenbe deemed satisfactory: one the one hand, it is well known that the quadratic divergencesspoil the renormalizability of gauge theories by contributing with non-renormalizable termsto the masses of the gauge bosons, so that we should prevent them from appearing in ourrenormalized expressions; on the other hand, it is not even clear whether a mass generatedthrough a quadratic divergence can be interpreted as a truly dynamically generated mass.For future reference, we report the leading behavior of the expressions in eq. (1.39) in thelimit m (cid:28) Λ : K m = 164 π (cid:18) − Λ + m m + m ln m Λ (cid:19) J m = 116 π (cid:18) Λ + m ln m Λ (cid:19) (1.42) ∂J m ∂m = 116 π (cid:18) ln m Λ + 1 (cid:19) The considerations of the last paragraph lead us to turn to other renormalization schemesfor the GEP of λφ theory as a model of mass generation. With an eye to Yang-Millstheory, we examine a renormalization scheme known to prevent the gauge bosons fromacquiring a mass due to the quadratic divergences, i.e. dimensional regularization (hence-forth referred to also as dimreg ). Setting (cid:15) = 4 − d , in dimreg we find that J m = − m π (cid:18) (cid:15) − ln m µ + 1 (cid:19) ∂J m ∂m = − π (cid:18) (cid:15) − ln m µ (cid:19) (1.43)where µ = 4 πµ e − γ E is the rationalized mass scale that results from defining the theoryin d (cid:54) = 4 . As for K m , since this integral does not converge even in d = 1 , it is not clearat all what its dimensionally regularized expression should be. However, if we assumeeq. (1.34) to hold also in dimreg, then – modulo an irrelevant m -independent constant –we are naturally lead to define K m as K m = 12 (cid:90) m dm J m = − m π (cid:18) (cid:15) − ln m µ + 12 (cid:19) (1.44)If we now introduce an (cid:15) -dependent mass scale Λ (cid:15) (not to be confused with the cutoff ofthe previous renormalization scheme), defined so that (cid:15) − ln Λ (cid:15) µ + 1 = 0 (Λ (cid:15) = µ e /(cid:15) +1 / ) (1.45) Again, K m is defined modulo an m -independent additive constant with the dimensions of Λ . erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles then we can re-express our three divergent integrals in the form K m = m π (cid:18) ln m Λ (cid:15) − (cid:19) J m = m π ln m Λ (cid:15) ∂J m ∂m = 116 π (cid:18) ln m Λ (cid:15) + 1 (cid:19) (1.46)Observe how radically different these results are from those given by eq. (1.42). Firstof all, the quadratic divergence of J m has disappeared. This is a well known feature ofdimensional regularization, and ultimately the main reason why dimreg is adopted forregularizing the gauge theories. Second of all, while the Λ of eq. (1.42) is a cutoff – hencea very large mass scale –, the Λ (cid:15) of eq. (1.46), defined by eq. (1.45), is either a very largescale for (cid:15) > (i.e. d < ), or a very small scale for (cid:15) < (i.e. d > ). Correspondingly,we have ln m Λ (cid:15) (cid:40) < d < > d > (1.47)in the regions of the GEP in which the mass parameter m has a physical meaning. Itfollows that in dimreg J m and ∂J m /∂m are not always respectively positive and negativedefinite, as implied by the formal definitions (1.29) and (1.35). For this reason, we findourselves in the following interesting situation.Case 1: d < If d < , then J m < , at variance with the formal definition given in(1.29). In particular, the gap equation does not admit non-zero solutions (provided that λ B > , as it should be). On the other hand, since ∂J m /∂m is not a priori negative, thefull equation ∂V G /∂m = 0 admits the solution ∂J m ∂m = 0 = ⇒ m = Λ (cid:15) / √ e (1.48)This m does not depend on the coupling, is of order Λ (cid:15) and is actually a maximum for theGEP. Therefore we must conclude that for d < the GEP does not admit non-zero minima.Case 2: d > If d > , then J m > and the gap equation admits the non-zero solu-tion m = Λ (cid:15) e /α B α B = λ B π (1.49)Since again ∂J m /∂m is not a priori negative, the GEP has a stationary point due to thevanishing of ∂J m /∂m ; for the same reason, we must check explicitly whether the m givenabove is a minimum or a maximum. In order to do so, we replace the mass scale Λ (cid:15) in theGEP with the inverse solution Λ (cid:15) = m e − /α B and express V G ( m ) as a function of m and m . An explicit computation shows that V G ( m ) = m π (cid:18) α B ln m m + 2 ln m m − (cid:19) (1.50)A plot of a normalized version of V G versus m/m is shown in Fig.7 for different values of α B . The third extremum (due to ∂J m /∂m = 0 , the other two being m = 0 and m = m )29 iorgio Comitini is given by m = m e − /α B − / < m . Here the GEP has value V G ( m e − /α B − ) = m e − /α B − π (1 + α B ) > (1.51)On the other hand, for m = m and m = 0 the GEP has values V G (0) = 0 V G ( m ) = − m π < (1.52)Therefore we conclude that for d > the value m = m e − /α B − / is an absolute maxi-mum, the values m = 0 and m = m are relative minima and in particular m = m is anabsolute minimum.The GEP approach clearly predicts the existence of a non-zero minimum for the vacuumenergy density of massless λφ theory in d = 4 + | (cid:15) | . By renormalization, this feature isinherited by the d → + theory: the perturbative vacuum of massless λφ theory, definedin dimreg by letting d → + , is indeed massive, and the scale of the theory is set preciselyby the finite value of m . Of course, since the original classical theory in d = 4 was in-variant under scale transformations, the mass scale of the model comes from the quantummechanical breaking of scale invariance and the actual value of m cannot be predictedfrom first principles: m must be determined a posteriori as a free parameter of the theory.Finally, observe that the value of the GEP at its minimum – i.e. at the only point at whichit has a physical meaning – does not depend on the bare coupling α B and is completelydetermined by the value of m . Therefore, by fixing m , we obtain a fully renormalizedvalue for the first order vacuum energy density of the theory, independent of the regulatorsand bare parameters, as it should be. -1-0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 π V G / m m/m λ B = 0.1 λ B = 0.5 λ B = 1.0 λ B = 1.5 Figure 7: A normalized version of the dimensionally regularized ( d > ) GEP as a functionof m/m for different values of the bare coupling λ B . The relative minimum at m = 0 ,relative maximum at m = m e − /λ B − / and absolute minimum at m = m are clearlyvisible. The value at the minimum does not depend on the coupling.30 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles In the previous paragraphs we have discovered that, depending on the renormalizationscheme used to define the model, the GEP analysis of massless λφ theory leads to verydifferent conclusions with respect to the issue of mass generation.In the presence of a sharp cutoff, the quadratic divergence of the tadpole diagram is foundto generate a non-zero mass for the scalar particle. This is not satisfactory in two respects.First of all, it is not clear whether such a mass should be interpreted as a genuinely dy-namical mass. Second of all, the quadratic divergence is known to break gauge invariance,so that if we are to regard massless λφ theory as a toy model for mass generation inYang-Mills theory, then we must discard results which depend on such a divergence.In dimensional regularization, on the other hand, it seems that taking the limit d → from above or from below leads to two very different theories: whereas mass generation ispredicted to occur in the d → + regularized theory, the same is not true of the d → − regularized one. This state of affairs is known in the literature and was first pointed outby Stevenson in [67], where he showed that mass generation in the dimensionally regu-larized theory with d < occurs if and only if the Z symmetry φ → − φ of the originalLagrangian is spontaneously broken, causing the scalar field φ to acquire a non-vanishingvacuum expectation value (cid:104) φ (cid:105) . Such a breaking is indeed predicted by the GEP equationsthemselves, provided that (cid:104) φ (cid:105) is treated as a free parameter, rather than set to zero fromthe beginning as we did in our analysis. By minimizing the GEP with respect to (cid:104) φ (cid:105) aswell as m , one is able to prove that the massless, symmetric vacuum which we found inour analysis is unstable towards a massive, non-symmetric vacuum. Since we assume theexpectation value of a gauge field to vanish in the vacuum, we must discard this solutiontoo, and conclude that the dimensionally regularized theory in d < is not a suitable modelfor mass generation in Yang-Mills theory. Therefore, of the three proposed regularizationschemes, only dimreg in d > leads to a viable model for our analysis.One may ask how is it that different renormalization schemes lead to different results.With respect to this issue, we take the view that the choice of a renormalization scheme ispart of the definition of the theory, rather than a formal procedure adopted to regularizethe divergences in order to incorporate them into renormalized parameters. As a matter offact, it is common knowledge that – especially when gauge symmetries are involved – notall renormalization schemes are equivalent from a physical point of view, or even from thepoint of view of mathematical consistency. Dimensional regularization has the great ad-vantage of eliminating the symmetry-breaking quadratic divergence of the tadpole diagramfrom the very beginning, thus leading to a perturbative series which can be renormalizedwhile preserving the symmetries of the theory. In the process of doing so, it modifies someof the formal aspects of the expansion. In the GEP approach this is exemplified by thefact that the divergent integrals J m and ∂J m /∂m are not positive and negative definite asthey should formally be, leading to physical consequences which, as we saw, include massgeneration. 31 iorgio Comitini In this section the machinery developed in Sec.1.1.2-3 will be applied to the GEP analysisof Yang-Mills theory. We will start by defining and computing the GEP (Sec.1.2.1-2) andthen we will address the issue of renormalization and gauge invariance (Sec.1.2.3). InSec.1.2.4 the variational status of the Yang-Mills GEP will be investigated in connectionto the anticommuting nature of the ghost fields. In Sec.1.2.5 we will show that the purelygluonic contribution to the GEP is formally identical to the GEP of λφ theory, so thatthe analysis of Sec.1.1.3 can be carried over verbatim to Yang-Mills theory. The masslessperturbative vacuum of the transverse gluons employed in ordinary perturbation theoryis found to be unstable towards a massive vacuum, motivating the massive perturbativeexpansion of Chapter 2. In a general covariant gauge, the Faddeev-Popov gauge fixed action of pure Yang-Millstheory is given by S = (cid:90) d d x (cid:26) − ∂ µ A aν ( ∂ µ A a ν − ∂ ν A a µ ) − ξ ∂ µ A aµ ∂ ν A aν + ∂ µ c a ∂ µ c a + (1.53) − g f abc ∂ µ A aν A b µ A c ν − g f abc f ade A bµ A cν A d µ A e ν − g f abc ∂ µ c a c b A cµ + L c.t. (cid:27) where L c.t. contains the appropriate renormalization counterterms. In ordinary perturba-tion theory, one chooses as the zero-order action S S = (cid:90) d d x (cid:26) − ∂ µ A aν ( ∂ µ A a ν − ∂ ν A a µ ) − ξ ∂ µ A aµ ∂ ν A aν + ∂ µ c a ∂ µ c a (cid:27) = (1.54) = (cid:90) d d p (2 π ) d (cid:26) − A aµ δ ab p (cid:20) t µν ( p ) + 1 ξ (cid:96) µν ( p ) (cid:21) A bν + c a δ ab p c b (cid:27) where t µν ( p ) and (cid:96) µν ( p ) are the transverse and longitudinal projection tensors, t µν ( p ) = η µν − p µ p ν p (cid:96) µν ( p ) = p µ p ν p (1.55)The corresponding gluon and ghost bare propagators D ab µν and G ab are readily determinedto be D ab µν ( p ) = δ ab (cid:26) − i t µν ( p ) p + i(cid:15) + ξ − i (cid:96) µν ( p ) p + i(cid:15) (cid:27) G ab ( p ) = δ ab ip + i(cid:15) (1.56) D ab µν and G ab are massless free particle propagators.In order to define the GEP of Yang-Mills theory, we must add to eq. (1.54) appropriatemass terms for the gluon and ghost fields. Since the gluon propagator has a transverse anda longitudinal component, there is no reason to define a unique mass parameter for thetransverse and longitudinal gluons. Indeed, in momentum space, the most general actionterm for the masses of the gluons and ghosts has the form ∆ S = (cid:90) d d p (2 π ) d (cid:26) A aµ δ ab (cid:20) m t µν ( p ) + 1 ξ m L (cid:96) µν ( p ) (cid:21) A bν − c a δ ab M c b (cid:27) (1.57)32 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles where M is the mass parameter for the ghosts, whereas m and m L are the mass parametersfor the transverse and longitudinal gluons respectively. In principle, we may compute theGEP by using as the zero-order action the sum S + ∆ S , with S given by eq. (1.54). How-ever, non-perturbatively, we know that due to gauge invariance the longitudinal part of thegluon propagator does not get corrected by the interactions [31, 40], so that in particularthe longitudinal gluons cannot develop a mass. By setting m L = 0 from the very start,we obtain the exact, non-perturbative result for the longitudinal gluons. Therefore we willlimit ourselves to study the GEP as a function of the transverse gluon and ghost mass atzero longitudinal gluon mass, and define as the zero-order action S for the computationof the GEP the quantity S = i (cid:90) d d p (2 π ) d (cid:26) A aµ D µνab ( p ) − A bν + c a G ab ( p ) − c b (cid:27) (1.58)where D abµν ( p ) − = i δ ab (cid:26) ( p − m + i(cid:15) ) t µν ( p ) + 1 ξ ( p + i(cid:15) ) (cid:96) µν ( p ) (cid:27) (1.59) G ab ( p ) − = − i δ ab ( p − M + i(cid:15) ) D abµν and G ab being the modified, massive gluon and ghost bare propagators D abµν ( p ) = δ ab (cid:26) − i t µν ( p ) p − m + i(cid:15) + ξ − i (cid:96) µν ( p ) p + i(cid:15) (cid:27) (1.60) G ab ( p ) = δ ab ip − M + i(cid:15) Accordingly, the interaction action S int = S − S reads S int = − (cid:90) d d p (2 π ) d (cid:26) δ ab m t µν ( p ) A aµ ( p ) A bν ( − p ) − δ ab M c a ( p ) c b ( p ) (cid:27) + (1.61) − (cid:90) d d x (cid:26) g B f abc ∂ µ A aν A b µ A c ν + g B f abc f ade A bµ A cν A d µ A e ν + g B f abc ∂ µ c a c b A cµ (cid:27) where the first line comes from the additional mass terms in S and we have expressed thesecond line in function of the bare strong coupling constant g B , rather than its renormal-ized value g , just as we did in Sec.1.1.3 for λφ theory.Since both S and S int depend on m and M , the GEP of Yang-Mills theory is a functionof two mass parameters. Its defining expression is obtained by specializing eq. (1.19) toour choice of S and S int and reads V G ( m, M ) = i V d ln (cid:90) D A D c D c e i S ( m,M ) − V d (cid:104)S int ( m, M ) (cid:105) (1.62)33 iorgio Comitini Let us move on to the explicit computation of V G . Since the vacuum expectation value ofan odd number of field operators with respect to the action of a free theory is zero, theaverage of S int in eq. (1.62) reduces to (cid:104)S int (cid:105) = − (cid:90) d d p (2 π ) d (cid:26) δ ab m t µν ( p ) (cid:68) A aµ ( p ) A bν ( − p ) (cid:69) − δ ab M (cid:68) c a ( p ) c b ( p ) (cid:69) (cid:27) + (1.63) − (cid:90) d d x (cid:26) g B f abc f ade (cid:68) A bµ A cν A d µ A e ν (cid:69) (cid:27) As for the zero-order term of eq. (1.62), we observe that the functional integral of e i S canbe factorized into the product of two integrals, (cid:90) D A D c D c e i S ( m,M ) = (cid:18) (cid:90) D A e i S ( A )0 ( m ) (cid:19)(cid:18) (cid:90) D c D c e i S ( c )0 ( M ) (cid:19) (1.64)where S ( A )0 and S ( c )0 are the A -dependent and c -dependent contributions to S . Therefore,a preliminary expression for the GEP of Yang-Mills theory is given by V G = i V d ln (cid:90) D A e i S ( A )0 ( m ) + i V d ln (cid:90) D c D c e i S ( c )0 ( M ) + (1.65) + 1 V g (cid:90) d d p (2 π ) d (cid:26) δ ab m t µν ( p ) (cid:68) A aµ ( p ) A bν ( − p ) (cid:69) + δ ab M (cid:68) c a ( p ) c b ( p ) (cid:69) (cid:27) + 1 V d g B f abc f ade (cid:90) d d x (cid:68) A bµ ( x ) A cν ( x ) A d µ ( x ) A e ν ( x ) (cid:69) where inside the ghost quadratic average we have exchanged the order of the Grassmannfields. Again, to each of the terms in the equation we may associate a diagram. To thelogarithmic terms we associate closed loops with no vertices (first and second diagram inFig.8), with a wiggly line for the gluon and a dotted line for the ghosts; to the quadraticterms we associate closed loops with a single two-point vertex, proportional to the respec-tive mass parameters squared (third and fourth diagram in Fig.8); to the quartic termwe associate a double loop with a four-point vertex, proportional to g B (last diagram inFig.8). We will now proceed to evaluate these diagrams.Figure 8: Diagrams which contribute to the GEP of Yang-Mills theory.34 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Let us start from the logarithmic contributions to the GEP. In what follows we denoteby det the functional determinant and by
T r the functional trace. Moreover, we definetransverse and longitudinal bare gluon propagators D T and D L such that D abµν ( p ) = δ ab [ D T ( p ) t µν ( p ) + ξ D L ( p ) (cid:96) µν ( p )] (1.66)We have ln (cid:90) D A e i S ( A )0 = ln det [( D abµν ) / ] = 12 ln det ( D abµν ) = (1.67) = 12 ln (cid:104) det ( δ ab D T t µν ) det ( δ ab ξ D L (cid:96) µν ) (cid:105) == 12 ln det ( δ ab D T t µν ) + 12 ln det ( δ ab ξ D L (cid:96) µν ) == 12 T r ln( δ ab D T t µν ) + 12 T r ln( δ ab D L (cid:96) µν ) + 12 T r ln ξ == − iN A ( d − V d K m − iN A V d K + 12 T r ln ξ where we recall that K m = 12 (cid:90) d d k E (2 π ) d ln( k E + m ) (1.68)and N A = N − , N being the number of colors ( N = 3 for pure gauge QCD). Thelast term in eq. (1.67) is canceled [72] by the logarithm of the ( ξ -dependent) constant C factored out from the partition function by the Faddeev-Popov gauge fixing procedure –see the Introduction. By keeping only the relevant terms and multiplying eq. (1.67) by i/ V d we obtain the zero-order gluon contribution V (0 ,A ) G to the GEP, V (0 ,A ) G = N A ( d − K m + N A K (1.69)As for the ghost loop, we simply have ln (cid:90) D c D c e i S ( c )0 = ln det [( G ab ) − ] = − T r ln( G ab ) = 2 iN A V d K M (1.70)so that the zero-order ghost contribution V (0 ,c ) G to the GEP is given by V (0 ,c ) G = − N A K M (1.71) The quadratic averages are readily evaluated: (cid:68) A aµ ( p ) A bν ( − p ) (cid:69) = V d D abµν ( p ) (cid:68) c a ( p ) c b ( p ) (cid:69) = V d G ab ( p ) (1.72)35 iorgio Comitini If follows that the gluon loop with the two-point vertex contributes to the GEP with aterm V (2 ,A ) G given by V (2 ,A ) G = 12 m (cid:90) d d p (2 π ) d δ ab t µν ( p ) D abµν ( p ) = − N A ( d − m J m (1.73)whereas the contribution V (2 ,c ) G due to the ghost loop with the two-point vertex is V (2 ,c ) G = M (cid:90) d d p (2 π ) d δ ab G ab ( p ) = N A M J M (1.74)Recall that J m = (cid:90) d d k E (2 π ) d k E + m (1.75) As for the quartic average, we have (cid:68) A bµ ( x ) A cν ( x ) A d µ ( x ) A e ν ( x ) (cid:69) = lim { y i }→ x (cid:110) D bcµν ( x − y ) D de µν ( y − y )+ (1.76) + D bd µµ ( x − y ) D ce νν ( y − y )++ D beµν ( x − y ) D cd µν ( y − y ) (cid:111) where D abµν ( x ) is the massive gluon propagator in coordinate space, D abµν ( x ) = (cid:90) d d k (2 π ) d e − ik · x D abµν ( k ) (1.77)When evaluated at x = 0 , the latter can be expressed as lim x → D abµν ( x ) = δ ab (cid:90) d d k (2 π ) d (cid:26) − i t µν ( k ) k − m + i(cid:15) + ξ − i (cid:96) µν ( k ) k + i(cid:15) (cid:27) = (1.78) = δ ab η µν (cid:90) d d k (2 π ) d (cid:26) (cid:18) − d (cid:19) − ik − m + i(cid:15) + ξd − ik + i(cid:15) (cid:27) == − δ ab η µν (cid:18) d − d J m + ξd J (cid:19) Therefore (cid:68) A bµ ( x ) A cν ( x ) A d µ ( x ) A e ν ( x ) (cid:69) = d ( δ bc δ de + δ bd δ ce d + δ be δ cd ) (cid:18) d − d J m + ξd J (cid:19) (1.79)and the gluon double loop contributes to the GEP with a term V (4 ,A ) G given by36 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles V (4 ,A ) G = g B f abc f ade d ( δ bc δ de + δ bd δ ce d + δ be δ cd ) (cid:18) d − d J m + ξd J (cid:19) = (1.80) = N g B N A d ( d − (cid:18) d − d J m + ξd J (cid:19) where we have used f abb = 0 , f abc f abd = N δ cd .By adding up the five contributions V (0 ,A ) G , V (0 ,c ) G , V (2 ,A ) G , V (2 ,c ) G and V (4 ,A ) G , we find ourfinal expression for the GEP of Yang-Mills theory in an arbitrary covariant gauge andrenormalization scheme: V G ( m, M ) = N A ( d − K m + N A K − N A K M − N A ( d − m J m + N A M J M ++ N g B N A d ( d − (cid:18) d − d J m + ξd J (cid:19) (1.81)Observe that V G ( m, M ) is a sum of two terms, the first one depending only on the gluonmass parameter squared m and the second one depending only on the ghost mass param-eter squared M , V G ( m , M ) = V ( A ) G ( m ) + V ( c ) G ( M ) (1.82)where V ( A ) G ( m ) N A = ( d − K m + K − ( d − m J m + N g B d − d (cid:18) J m + ξd − J (cid:19) (1.83) V ( c ) G ( M )2 N A = − (cid:18) K M − M J M (cid:19) (1.84)Therefore the stationary points of the GEP can be determined by separately extremizing V ( A ) G with respect to m and V ( c ) G with respect to M . In order to find the stationary points of the GEP, we must first of all regularize theintegrals K and J by choosing a suitable renormalization scheme. In Sec.1.1.3 we havediscussed the renormalization of the GEP of massless λφ theory. There we saw that incutoff regularization the squared mass generated for the scalar particle is proportional tothe quadratic divergence of the tadpole diagram; we then discarded the scheme with thejustification that quadratic divergences are known to spoil the gauge invariance of gaugetheories. Let us see how this comes about in the GEP analysis.Our computation lead us to the expression (1.81) for the GEP of Yang-Mills theory. Thegauge dependence of the GEP comes entirely from the product ξJ in the last term ofeq. (1.83). In cutoff regularization, 37 iorgio Comitini ξJ = ξ π (cid:90) Λ0 dk E k E = ξ Λ π (cid:54) = 0 (1.85)Therefore in the presence of a sharp cutoff the GEP is explicitly gauge dependent, and thegauge dependence is caused precisely by the quadratic divergence of the tadpole diagram.Of course, a gauge-dependent GEP will have gauge-dependent minima which cannot bephysically meaningful. We must then conclude that the sharp cutoff is not suitable forregularizing the GEP of Yang-Mills theory.On the other hand, consider what happens in dimensional regularization. In dimreg, J m is given by eq. (1.42), J m = − m π (cid:18) (cid:15) − ln m µ + 1 (cid:19) (1.86)By taking the limit m → , we find that J = 0 . It follows that in dimensional regulariza-tion the GEP, as well as its extrema, are gauge-independent.In light of what we just saw, we take the following standpoint on the renormalization of theGEP of Yang-Mills theory (cf. Sec.1.1.3.4). We interpret dimensional regularization as therenormalization scheme which, by removing the quadratic divergence from the equations,preserves the gauge invariance of the theory and allows for the definition of a physicallymeaningful GEP. It is our scheme of choice for the regularization of the GEP of Yang-Millstheory, and the only one that we will consider in what follows. Since in dimreg K m ∝ m – eq. (1.46) –, modulo an inessential m -independent constant, K as well as J vanishesand we are left with the following regularized expressions for the gluonic and ghost contri-butions to the GEP: V ( A ) G ( m ) N A ( d −
1) = K m − m J m + N g B d − d J m (1.87) V ( c ) G ( M )2 N A = − (cid:18) K M − M J M (cid:19) (1.88)In the above equations, K and J are given by the dimensionally regularized expressions ineq. (1.46), where the mass scale Λ (cid:15) is defined by eq. (1.45). As we saw, the GEP of Yang-Mills theory depends on the ghost mass M through theghost contribution V ( c ) G ( M ) , eq. (1.88). Observe that V ( c ) G is independent of the strongcoupling constant g B : as a matter of fact, V ( c ) G is the same GEP that would be obtainedfrom the free ghost Lagrangian alone. This feature of the Yang-Mills GEP challenges theapplicability of the GEP analysis to the ghost sector, leading us to question whether itmakes any sense at all to define a mass parameter for the ghost in the first place. Afterall – as we showed in the Introduction – through the non-perturbative lattice calculationsthe dressed ghost propagator was found to be massless in the infrared as well as in theultraviolet regime. Since however – to our knowledge – the masslessness of the ghosts atlow energies has not yet been proven analytically from first principles, it is still of someimportance to take up this issue explicitly and ask whether it is possible to show through aGEP analysis that the ghosts do not acquire a dynamical mass in the infrared. The answer38 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles to this question turns out to be in the positive. In order to reach this conclusion, we startby studying the stationary points of the ghost GEP.By taking the derivative of V ( c ) G with respect to M we find that ∂V ( c ) G ∂M = N A M ∂J M ∂M (1.89)so that the stationary ghost masses are given by M = 0 and – cf. eq. (1.46) – M = Λ (cid:15) / √ e .The second derivative of the ghost GEP, on the other hand, reads ∂ V ( c ) G ∂ ( M ) = N A (cid:18) ∂J M ∂M + M ∂ J M ∂ ( M ) (cid:19) = N A π (cid:18) ln M Λ (cid:15) + 2 (cid:19) (1.90)Since the second derivative is negative at M = 0 and positive at M = Λ (cid:15) / √ e , M = 0 isa maximum, whereas M = Λ (cid:15) / √ e is a minimum. At first sight this result may seem toimply that the ghosts actually acquire a mass; M = Λ (cid:15) / √ e , however, does not dependon the coupling and as such cannot be interpreted as being due to the interactions. Themaximum M = 0 , on the other hand, would be a far more acceptable result: not only itsbeing zero would explain why the ghost GEP is independent of the coupling, but M = 0 is also known to be the correct, non-perturbative result for the ghosts. So how is it thatfor the ghost sector it looks like maximizing the GEP, rather than minimizing it, is thecorrect procedure to follow in order to obtain physically meaningful results?The answer to this question lies in the non-commuting nature of the ghost fields, as relatedto the variational status of the GEP in Yang-Mills theory. As discussed in Sec.1.1.2, theGEP approach is motivated by the Jensen-Feynman inequality, which shows that for a bosonic field theory the Gaussian Effective Potential is bounded from below by the exactvacuum energy density of the system. However, once a gauge has been fixed through theFaddeev-Popov procedure, Yang-Mills theory becomes a theory of bosonic gluons on theone side and fermionic ghosts – yet with a bosonic kinetic action – on the other side. Forthis reason, one must carefully check whether in the presence of such fields the GEP is stilla consistent variational approximation to the exact vacuum energy density of the system,i.e. whether the latter still constitutes a lower bound for the GEP. We will now go throughthe computations needed to address this issue. A more detailed derivation may be foundin [72], where the variational statement is analyzed and extended to finite temperatures.In its Euclidean formulation, the exact vacuum energy density of Yang-Mills theory E is given by the equation e −V d E = (cid:90) D A D c D c e −S (1.91)where now V d and S are the Euclidean volume and action of the system. The presenceof the anticommuting ghost fields c and c spoils the direct applicability of the Jensen in-equality, which states that for any positive (bosonic) probability measure with associatedaverage (cid:104)·(cid:105) and any function f on the measure space (cid:10) e f (cid:11) ≥ e (cid:104) f (cid:105) [52]. One way to avoidthis obstacle is to go one step backward in the derivation of the Faddeev-Popov Yang-Millsaction and integrate out the ghost fields, so that the functional averages are taken onlywith respect to the gluon configurations. This integration leads to e −V d E = (cid:90) D A e −S (cid:48) det( M F P ) (1.92)39 iorgio Comitini where S (cid:48) is the pure gluonic (albeit gauge-fixed) action, obtained for instance from S bysetting to zero the ghost fields, and det( M F P ( A )) = det( − ∂ · D [ A ]) – D [ A ] being thegauge covariant derivative associated to the gluon field – is the Faddeev-Popov determi-nant. Rewriting the determinant as det( M F P ) = e ln det( M FP ) and splitting the gluonicaction S (cid:48) as S (cid:48) + S (cid:48) int , where S (cid:48) is the massive gluonic free action, we obtain E = V (cid:48) − V d ln (cid:68) e −S (cid:48) int +ln det( M FP ) (cid:69) (cid:48) (1.93)where V (cid:48) = − V d ln (cid:90) D A e −S (cid:48) (1.94)is the gluonic kinetic vacuum energy density – first two terms in eq. (1.87) – and (cid:104)·(cid:105) (cid:48) is theaverage with respect to the (bosonic) Euclidean action S (cid:48) . To the second term in eq. (1.92)we can now apply the Jensen inequality in the form ln (cid:10) e f (cid:11) ≥ (cid:104) f (cid:105) , to find that E ≤ V (cid:48) + 1 V d (cid:10) S (cid:48) int (cid:11) (cid:48) − V d (cid:104) ln det( M F P ) (cid:105) (cid:48) (1.95)Since the second term in this equation contains the purely gluonic interactions to firstorder, the first and second term sum to the gluonic contribution to the GEP, V ( A ) G . Onthe other hand, the third term contains the ghost kinetic vacuum energy density and theghost loops to arbitrarily high order. Therefore, if we define an energy term δ E as δ E = − V d (cid:104) ln det( M F P ) (cid:105) (cid:48) − V ( c ) G (1.96)where V ( c ) G is the ghost contribution to the GEP, we find that E ≤ V ( A ) G + V ( c ) G + δ E (1.97)or V G = V ( A ) G + V ( c ) G ≥ E − δ E (1.98) δ E can actually be shown to be non-negative: rewrite the average in δ E as (cid:104) ln det( M F P ) (cid:105) (cid:48) = Tr (cid:104) ln( M F P ) (cid:105) (cid:48) (1.99)and apply the Jensen inequality in the form ln (cid:104) f (cid:105) ≥ (cid:104) ln f (cid:105) to obtain − V d (cid:104) ln det( M F P ) (cid:105) (cid:48) = − V d Tr (cid:104) ln( M F P ) (cid:105) (cid:48) ≥ − V d Tr ln (cid:104)M F P (cid:105) (cid:48) (1.100)Since M F P is the sum of minus the d’Alembert operator and of an operator which is linearin the gluon field, the latter averages to zero and − V d (cid:104) ln det( M F P ) (cid:105) (cid:48) ≥ − V d Tr ln (cid:104)M F P (cid:105) (cid:48) = − V d Tr ln( − ∂ ) = V ( c ) G (cid:12)(cid:12) M =0 (1.101)Hence, since V ( c ) G attains its maximum at M = 0 , δ E ≥ V ( c ) G (cid:12)(cid:12) M =0 − V ( c ) G ≥ (1.102)40 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Eq. (1.98) with δ E ≥ tells us that the GEP of Yang-Mills theory is not actually boundfrom below by the exact vacuum energy density E , but rather that the lower bound is setby an even lower energy density, E − δ E . Therefore, by minimizing “too much” the GEP,we may drift away from the exact vacuum energy density E and approach E − δ E , whichis the farther away from E the larger is δ E . However, observe that by minimizing δ E thispotential error can be kept under control: if δ E is very small, then E − δ E is not far awayfrom E and by minimizing the GEP we still approach the exact vacuum energy density.Since as we saw δ E ≥ V ( c ) G (cid:12)(cid:12) M =0 − V ( c ) G , in order for δ E to be the smallest the ghost GEP V ( c ) G must approach its maximal value V ( c ) G (cid:12)(cid:12) M =0 . Therefore, the Yang-Mills GEP can stillbe used as a variational tool provided that its ghost contribution is maximized rather thanminimized. In the next section we will give further evidence that the Yang-Mills GEP isa good variational estimate of the exact vacuum energy density of Yang-Mills theory bydiscussing some results which have been obtained in [71, 72] at non-zero temperatures.The modified Jensen-Feynman inequality given by eq. (1.98) leads us to take M = 0 –i.e. the maximum of V ( c ) G – as the result of the GEP analysis of the ghost sector. M = 0 is precisely the result found on the lattice. Since modulo an M -independent constant V ( c ) G (cid:12)(cid:12) M =0 = 0 , in the presence of massless ghosts the GEP of Yang-Mills theory is givenentirely by its gluonic contribution: V G ( m ) = V ( A ) G ( m ) (1.103) Having set the longitudinal gluon mass m L = 0 from the beginning and having found thatby choosing a ghost mass M = 0 the GEP is more closely bound to the exact vacuum en-ergy density, we are left with the following expression for the Gaussian Effective Potentialof Yang-Mills theory: V G ( m ) = N A ( d − (cid:20) K m − m J m + N g B d − d J m (cid:21) (1.104)Following our discussions in Sec.1.1.3.3-4 and Sec.1.2.3, here the divergent integrals K m and J m must be understood as having been regularized in dimreg with d > , so that theirexplicit expressions are given by eq. (1.46).We still need to understand how to treat the d ’s that appear explicitly in eq. (1.104). Thesecome from the number of polarization degrees of freedom of the gluons in d (cid:54) = 4 . Now, ifwe set d = 4 − (cid:15) in eq. (1.104), the O ( (cid:15) − ) and O ( (cid:15) − ) terms in K m , J m and J m , whenmultiplied to the O ( (cid:15) ) and O ( (cid:15) ) terms in ( d − and ( d − /d , would give rise to finiteterms that would spoil the structure of the gap equation. To be specific, the third termof the GEP could not be expressed as a coefficient times the square of the second termdivided by m – as formally implied by eq. (1.104) –, leading to a first derivative of theGEP which is not proportional to ∂J m /∂m times the gap equation. However, observethat we are still free to define dimensional regularization in such a way that the explicit d ’s in the GEP are to be straightforwardly set to four. With this prescription, if we define41 iorgio Comitini a bare coupling constant α B as α B = 9 N g B π = 9 N α sB π (1.105)then we can put the GEP in the final form V G ( m ) = 3 N A (cid:18) K m − m J m + 2 π α B J m (cid:19) (1.106)Apart from a multiplicative factor of N A , eq. (1.106) is formally identical to the GEP ofmassless λφ theory, given by eq. (1.32). In eq. (1.105) we have defined α B precisely so thatthe equations of Sec.1.1.3 can be carried over verbatim to Yang-Mills theory. Thereforewe do not need to explicitly repeat the GEP analysis, and we can limit ourselves to justrestating the results. From eq. (1.106), the following gap equation of Yang-Mills theorycan be derived: m = 8 π α B J m (1.107)In d → + – just as in massless λφ theory –, the latter has a non-zero solution at m = Λ (cid:15) e /α B . In terms of m rather than of the unrenormalized mass scale Λ (cid:15) used toregularize the divergent integrals, the GEP reads V G ( m ) = m π (cid:18) α B ln m m + 2 ln m m − (cid:19) (1.108)As shown in Fig.7, V G ( m ) has a relative minimum at m = 0 , as well as a relative maximumat m = m e − /α B − < m ; since V G | m =0 = 0 while V G | m = m < – cf. eq. (1.52) –, m is the absolute minimum of the GEP. As already discussed in the context of massless λφ theory, since Yang-Mills theory is scale-free at the classical level, the actual value of m cannot be predicted from first principles and must be fixed by experiment. The GEPanalysis only informs us that m is different from zero. Finally, the value of the GEP atits absolute minimum, V G ( m ) = − m π (1.109)does not depend on the bare coupling α B , but only on the position m of the minimum.Since V G ( m ) is the GEP approximation to the vacuum energy density of the system –hence the only value of the GEP with a physical meaning –, from a physical point of viewthe bare coupling disappears, as it should, from the equations, having been absorbed intothe definition of the renormalized mass scale m . On the other hand, the values of theGEP at m (cid:54) = m are not physical, hence their dependence on α B is uninfluential for thephysics of the system.At this point of the analysis, one may wonder whether the non-zero mass derived throughthe GEP should be interpreted as a physically meaningful parameter, rather than an ar-tifact of the variational approach. One way to address this issue is to ask whether thezero-temperature minimum found by the GEP is stable against thermal excitations, i.e. ifit remains non-zero at small but finite temperatures, and whether its evolution with thetemperature leads to physically meaningful predictions.42 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles -0.04-0.035-0.03-0.025-0.02-0.015-0.01-0.005 0 0.005 0 0.2 0.4 0.6 0.8 1 1.2 T = . m T = . m T = . m T = . m T = . m F G / m m / m Figure 9: A normalized version of the finite-temperature Gaussian Effective Potential F G as a function of m/m for different values of the temperature T . The bare coupling constantis kept fixed at the value α B = 0 . .This aspect of the variational approach was explored in ref. [71,72]. At finite temperatures,rather than by the exact vacuum energy density E of the system, the GEP is bound frombelow by the exact thermodynamical free energy density F . The finite-temperature GEPis usually denoted by F G and is a function of both the mass parameter m and the tem-perature T of the system. By minimizing F G ( m , T ) with respect to m at fixed T , oneobtains the optimal mass parameter at temperature T ; as T varies, these parameters definean optimal mass function m ( T ) . Since F G | T =0 = V G , the zero-temperature minimum of F G is precisely the m that we found in this chapter. m can then be used to set thescale of both the temperature T and the finite-temperature minima m ( T ) , so that once m has been fixed no other external parameters are needed to determine m ( T ) , which is thusis an actual prediction from first principles of the variational approach. Moreover, sincethe GEP itself is a variational approximation to the free energy density of the system, byevaluating F G ( m , T ) at its minimum m ( T ) one obtains a variational estimate of the exactfree energy density F ( T ) . From this estimate the entropy and specific heat of the systemcan then be computed as functions of the temperature, allowing for a semi-quantitativecharacterization of the thermodynamics of the gluonic system.In Fig.9 the finite-temperature GEP F G is shown as a function of the mass parameterfor different values of the temperature . As we can see, as the temperature grows fromzero to finite values, the absolute minimum of the GEP remains quite close to m until acritical temperature T c ≈ . m is reached. At T = T c the relative minimum which wasonce at m = 0 becomes dominant, and the optimal mass parameter drops discontinuously At variance with the zero-temperature V G , at its minimum F G still slightly depends on the barecoupling α B . The curve in Fig.9 assumes an optimal value of α B = 0 . , as discussed in [72]. iorgio Comitini to lower values. This discontinuity in the optimal mass causes the computed entropy ofthe system to be discontinuous as well. Therefore, when extended to finite temperatures,the GEP analysis gives clear evidence of the occurrence of a first order phase transitionin the Yang-Mills system. Such a first order transition is indeed known to occur: it is thedeconfinement phase transition of gluonic matter, connecting a low temperature phase inwhich the gluons cannot exist as free particles (confined gluons) to a high temperaturephase in which the gluons are free to propagate (deconfined gluons).The GEP analysis of the thermal behavior ofYang-Mills theory [71, 72] manages to re-produce in a semi-quantitative fashion results which had already been provided by latticecomputations [76–78]. These results rest on the validity of the GEP approach at zero tem-perature – especially in reference to its variational status, discussed in Sec.1.2.4 – and onthe physical significance of the non-zero vacuum minimum m . They lead us to interpret m as a physically meaningful – albeit a priori unfixed – energy scale of Yang-Mills theoryand to take seriously the conclusions of the GEP analysis already at zero temperature.The existence of a non-zero absolute minimum for the vacuum GEP implies that themassless gluon perturbative vacuum is unstable towards a massive vacuum. As discussedat length in Sec.1.1, this fact may be interpreted as evidence for the phenomenon of dy-namical mass generation in the gluon sector.Nonetheless, it must be kept in mind that, being of variational nature, the evidence gath-ered through the GEP approach cannot be understood as a rigorous proof that in thevacuum the gluons acquire a mass through the strong interactions. Furthermore, the GEPapproach is not able to explain the actual mechanism by which the dynamical mass is gen-erated. The only precise statement that we can make based on the GEP analysis is thatthe massless Gaussian (i.e. free-particle) state which is adopted as the zero-order vacuumstate of ordinary perturbation theory is farther away from the true vacuum of the theorythan a massive Gaussian state.This statement leads us to believe that a perturbative formulation of Yang-Mills theorywhich treats the transverse gluons as massive already at tree-level may provide resultswhich are closer to their exact counterpart than those obtained by ordinary perturbationtheory. Since m L = 0 from first principles and M = 0 by maximization of the ghostGEP, in such a formulation the ghosts and longitudinal gluons will need to be treated asmassless.The massive formulation of Yang-Mills perturbation theory will be presented in Chap-ter 2. There we will see that the massive expansion indeed reproduces features of the exactdressed gluon propagator – including the generation of a truly dynamical gluon mass –which cannot be described at any finite order by ordinary, massless perturbation theory,providing fully quantitative predictions in astonishing agreement with the lattice results.This fact reinforces the idea that massive perturbation theory actually may be the way togo when exploring the low energy dynamics of the strong interactions.44 The massive perturbative expansionof Yang-Mills theory in an arbitrarycovariant gauge
In Chapter 1 by a GEP analysis of Yang-Mills theory we were able to reach the conclu-sion that the massless perturbative vacuum of the transverse gluons is unstable towards amassive vacuum. We interpreted this fact as evidence for dynamical mass generation, andargued that because of it a non-ordinary perturbation theory which treats the transversegluons as massive already at tree-level may lead to a perturbative series which more ac-curately captures the low energy dynamics of Yang-Mills theory. It is now time to leavethe simplified realm of the Gaussian Effective Potential and confront ourselves with therichness of the full theory from the perspective of the massive perturbative expansion.The massive perturbative expansion is formally defined by the same procedure used toderive the GEP. One starts from the massless kinetic Lagrangian obtained by sending tozero the strong coupling g in the full Faddeev-Popov gauge-fixed action and adds to itthe relevant mass terms for the gluon and ghost fields. Since the ghosts and longitudinalgluons – either by first principles or variationally – are found to remain massless, from thestandpoint of optimized perturbation theory a single mass term needs to be added to theordinary kinetic Lagrangian of Yang-Mills theory, namely, the mass term for the transversegluons. In order for the total action to remain unchanged, the very same term must then besubtracted from the interaction Lagrangian. As a result of this shift, the Feynman rules ofthe massive perturbative expansion are different from that of ordinary perturbation theoryin two respects. First of all, the additional mass term in the kinetic Lagrangian causes thetransverse component of the bare gluon propagator to be massive, rather than massless.Second of all, the subtraction of the mass term from the interaction Lagrangian gives riseto a new two-gluon vertex which is not present in the ordinary, massless expansion. Theshift introduces a spurious mass parameter m which cannot be fixed by first principles,since the original theory was scale-free at the classical level.Equipped with the Feynman rules of the massive expansion, one can go on and computethe quantities of physical interest by using the usual machinery of perturbation theory. Inthis chapter we will be mainly concerned with the computation of the gluon and ghost two-point functions in momentum space, i.e. of the dressed gluon and ghost propagators, in anarbitrary covariant gauge. As we will see, the massive expansion is able to incorporate thephenomenon of dynamical mass generation for the transverse gluons in a non-trivial way,overcoming the limitations of ordinary perturbation theory. By contrast, in agreementwith the lattice data, the ghost propagator will be shown to remain massless.45 iorgio Comitini Massive perturbation theory provides explicit analytical expressions for the propagatorswhich can then be continued to the whole complex plane and its Euclidean subdomain,where the lattice calculations are defined. The infrared results of the massive expansioncan thus be tested against the predictions of the lattice in a quantitative as well as a quali-tative fashion. In this chapter we will compare our expressions with the lattice data in theLandau gauge, where the data is the most reliable. Since the shift introduces a new free pa-rameter – namely the mass parameter – in the equations, at this stage the comparison hasthe status of a fit, rather than of an actual comparison from first principles . As we willsee, the Landau-gauge Euclidean ghost and gluon propagators turn out to be in astonish-ing agreement with the lattice data, reinforcing the idea that most of the non-perturbativecontent of Yang-Mills theory may actually be incorporated into the gluons’ transverse mass.This chapter is organized as follows. In Sec.2.1 we will define the massive perturbativeexpansion and outline its general features both from a diagrammatic and from an analyticpoint of view. In Sec.2.2 we will define and compute the one-loop ghost propagator inan arbitrary covariant gauge, investigate its asymptotic limits and compare its low energybehavior with the lattice data in the Landau gauge. In Sec.2.3 we will do the same for thegluon propagator, with particular emphasis on the issue of mass generation. Due to theircomplexity, the calculations which lead to the final expression for the gluon propagator aredescribed in much less detail than those presented for the ghost propagator. A thoroughderivation of the one-loop gluon polarization may be found in the Appendix of ref. [74].The results of this chapter were presented and published for the first time in ref. [73–75]. This limitation will be overcome in the next chapter, where the alleged loss of predictivity will beinvestigated in connection to the gauge invariance of the massive expansion. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles In this section we will proceed with the definition of the massive expansion and the deriva-tion of its Feynman rules. We start again from the Faddeev-Popov gauge-fixed action ofYang-Mills theory in an arbitrary covariant gauge, S = (cid:90) d d x (cid:26) − ∂ µ A aν ( ∂ µ A a ν − ∂ ν A a µ ) − ξ ∂ µ A aµ ∂ ν A aν + ∂ µ c a ∂ µ c a + (2.1) − g f abc ∂ µ A aν A b µ A c ν − g f abc f ade A bµ A cν A d µ A e ν − g f abc ∂ µ c a c b A cµ + L c.t. (cid:27) were L c.t. contains the renormalization counterterms and ξ ≥ is the gauge parameter.In order to define a perturbation theory, we must fix a zero-order (kinetic) action S suchthat in momentum space the gluon and ghost bare propagators D abµν and G ab are given by S = i (cid:90) d d p (2 π ) d (cid:26) A aµ D µνab ( p ) − A bν + c a G ab ( p ) − c b (cid:27) (2.2)In the massive perturbative expansion, the ghosts and longitudinal gluons are treated asmassless at tree-level, whereas the transverse gluons are given a mass m ; their bare prop-agators are therefore defined as D abµν ( p ) = δ ab (cid:26) − i t µν ( p ) p − m + i(cid:15) + ξ − i (cid:96) µν ( p ) p + i(cid:15) (cid:27) G ab ( p ) = δ ab ip + i(cid:15) (2.3)and have inverses D µνab ( p ) − = i δ ab (cid:26) ( p − m ) t µν ( p ) + 1 ξ p (cid:96) µν ( p ) (cid:27) G ab ( p ) − = − i δ ab p (2.4)where t µν and (cid:96) µν are transverse and longitudinal projectors, t µν ( p ) = η µν − p µ p ν p (cid:96) µν ( p ) = p µ p ν p (2.5)Plugging eq. (2.4) into eq. (2.2), we obtain the following zero order action for the massiveexpansion: S = (cid:90) d d p (2 π ) d (cid:26) − A aµ δ ab (cid:20) ( p − m ) t µν ( p ) + 1 ξ p (cid:96) µν ( p ) (cid:21) A bν + c a δ ab p c b (cid:27) (2.6)The latter is identical to the kinetic action of ordinary perturbation theory – first line ofeq. (2.1) in momentum space – apart from a single term δ S yielding the transverse gluonmass, δ S = (cid:90) d d p (2 π ) d A aµ m δ ab t µν ( p ) A bν (2.7) δ S needs to be included in the interaction action S int in order for the total action S to47 iorgio Comitini remain unchanged, S = S + S int . Setting S int = S − S , we obtain S int = (cid:90) d d p (2 π ) d (cid:26) − A aµ m δ ab t µν ( p ) A bν (cid:27) + (cid:90) d d x (cid:26) − g f abc ∂ µ A aν A b µ A c ν + (2.8) − g f abc f ade A bµ A cν A d µ A e ν − g f abc ∂ µ c a c b A cµ + L c.t. (cid:27) In eq. (2.8), the coordinate space integral contains the ordinary 3-gluon, 4-gluon andghost-gluon interactions and the ordinary renormalization counterterms; all of them areproportional either to the strong coupling constant g or its square g . The momentumspace integral, on the other hand, contains a new 2-gluon interaction, proportional to thegluon mass parameter squared m . This interaction is due to the shift induced by the term δ S and is not present in ordinary perturbation theory.Each of the interaction terms in S int yields a vertex for the diagrammatic computationof the quantities of physical interest in perturbation theory. The 3-gluon, 4-gluon andghost-gluon vertices are left unchanged by the massive shift; their expressions are thoseof ordinary perturbation theory, Figg.10-12. The 2-gluon vertex can be read out directlyfrom − iδ S ; it is given by Fig.13. Since it was obtained by the same addition/subtractionmechanism which usually defines the renormalization counterterms, we shall refer to it asthe mass counterterm and denote it with a cross. = g f abc [ η µν ( k − p ) ρ + η νρ ( p − q ) µ + η ρµ ( q − k ) ν ] b, νa, µ c, ρkp q Figure 10: 3-gluon vertex a, µc, ρ b, νd, σ = − ig [ f abe f cde ( η µρ η νσ − η µσ η νρ )++ f ace f bde ( η µν η ρσ − η µσ η νρ )++ f ade f bce ( η µν η ρσ − η µρ η νσ ) ] Figure 11: 4-gluon vertex48 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles = − g f abc p µ ca b, µp Figure 12: Ghost-gluon vertexFigure 13: Mass countertermFrom the kinetic action S , through eqq. (2.2) and (2.3), we read out the gluon and ghostbare propagators, to be associated to the internal lines of the Feynman diagrams. Theseare given in Fig.14-15. Figure 14: Bare gluon propagator a b = δ ab ip + i(cid:15) Figure 15: Bare ghost propagatorThe Feynman rules of Figg.10-15 fully define the massive perturbative expansion. Sincethe shift induced by δ S does not modify the total Faddeev-Popov action S – hence thephysical content of Yang-Mills theory –, the massive expansion is non-perturbatively equiv-alent to ordinary perturbation theory. However, due to the massiveness of its bare gluonpropagator and the presence of the new mass counterterm, the truncation of the massiveperturbative series to any finite order yields different results than those of standard, mass-less perturbation theory. 49 iorgio Comitini As we saw in the last section, the shift induced by the gluon mass term δ S produces a new2-gluon vertex, the mass counterterm, which is not part of the Feynman rules of ordinaryperturbation theory. Due to the new vertex, the loop expansions of massive perturbationtheory contain additional Feynman diagrams, called crossed diagrams . These are diagramswhich differ from those of ordinary perturbation theory by the presence of one or moremass counterterms in the internal gluon lines of the loops. As an example, consider thegluon tadpole diagram, Fig.16. In ordinary perturbation theory there is a single tadpole:the first diagram on the left in the figure (no mass counterterms in the internal gluon line).In massive perturbation theory, on the other hand, there are an infinite number of tadpolediagrams, each with an increasing number of mass counterterms.Figure 16: Tadpole diagrams of the massive perturbative expansion. The internal gluonline of the tadpole may contain an arbitrarily high number of mass counterterms.Since δ S – unlike the gluon-gluon and ghost-gluon interactions – is not proportional to thestrong coupling constant, diagrams which only differ by the number of mass countertermsin any of their internal lines are of the same order in g . The non-perturbative nature ofthe massive expansion emerges precisely in this feature of the perturbative series: as far asthe mass counterterms are concerned, in massive perturbation theory the correspondencebetween the number of vertices and the perturbative order in the coupling g is lost.One of the tasks of the crossed diagrams is to counterbalance the effects of the tree-levelgluon mass m . At the level of the full action S , this is very clear: by definition, the masscounterterm in the interaction Lagrangian exactly cancels out the corresponding term inthe kinetic Lagrangian. At the level of the perturbative series, the same cancellation wouldoccur by the following mechanism. Consider some internal gluon line with a number n ≥ of mass counterterms in an arbitrary Feynman diagram. To such a line we would associatethe expression D ( p ) · (cid:2) − im t ( p ) · D ( p ) (cid:3) n (2.9)where p is the momentum of the line and we have suppressed the tensor indices of boththe gluon propagator D ( p ) and the mass counterterm − im t ( p ) . Now, since t ( p ) · D ( p ) = t ( p ) · (cid:18) − ip − m t ( p ) + − iξp (cid:96) ( p ) (cid:19) = − ip − m t ( p ) (2.10)as long as n is non-zero, the n -th power of the quantity in parentheses in eq. (2.9) reads50 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles [ − im t ( p ) · D ( p )] n = (cid:18) − m p − m t ( p ) (cid:19) n = (cid:18) − m p − m (cid:19) n t ( p ) (2.11)Therefore, for n ≥ , D ( p ) · [ − im t ( p ) · D ( p )] n = − ip − m (cid:18) − m p − m (cid:19) n t ( p ) (2.12)In the full perturbative series, there will be diagrams that are identical to that which weare considering, except for the number of mass counterterms in the very same internal line;this number goes from zero to infinity. By summing up these diagrams, we obtain a singleresummed diagram whose internal gluon line is given by ∞ (cid:88) n =0 D ( p ) · [ − im t ( p ) · D ( p )] n = − ip − m ∞ (cid:88) n =0 (cid:18) − m p − m (cid:19) n t ( p ) + ξ − ip (cid:96) ( p ) (2.13)where the longitudinal term comes from the order zero summand. Since ∞ (cid:88) n =0 (cid:18) − m p − m (cid:19) n = 11 + m p − m = p − m p (2.14)the sum reduces to ∞ (cid:88) n =0 D ( p ) · [ − im t ( p ) · D ( p )] n = − ip t ( p ) + ξ − ip (cid:96) ( p ) (2.15)which is precisely the bare propagator of a massless gluon. Therefore, by summing up allthe crossed diagrams of the perturbative series, we obtain a new set of resummed diagramswith no mass counterterms in which the massive gluon internal lines have been replacedby massless lines. These are precisely the diagrams of ordinary perturbation theory. Ofcourse, since this procedure restores the ordinary perturbative series, in massive perturba-tion theory we are not interested in the resummation of the whole set of crossed diagrams.Actually, this simple calculation teaches us an important lesson: if on the one hand, in thespirit of perturbation theory, a sufficient number of diagrams must be summed in orderto approach the exact result, on the other hand the resummation of too many crosseddiagrams brings us closer to massless perturbation theory and must thus be avoided.Clearly in the massive expansion some general prescription is needed in order to fix thenumber of crossed diagrams that are to be included in the perturbative series at any fixedloop order. Such a prescription, unfortunately, has not yet been established in full gener-ality. Nonetheless, it turns out that the principle of renormalizability, together with theprinciple of minimality, are still sufficient to fix the number of mass counterterms to beincluded in expansions carried out to one loop. In order to motivate this statement, weneed to take a preliminary step and describe a useful method for the computation of dia-grams with an arbitrary number of mass counterterms. Then we will discuss the effect ofthe mass counterterm on the divergences of the massive perturbative expansion and drawour conclusions.Suppose that we want to sum some set of Feynman diagrams in which the number of51 iorgio Comitini mass counterterms goes from zero to a fixed integer N . Let us focus on a single diagramwith n ≥ mass counterterms in one of its internal gluon lines. As we saw, to this line wemay associate the expression D ( p ) · [ − im t ( p ) · D ( p )] n = − ip − m (cid:18) − m p − m (cid:19) n t ( p ) (2.16)Since p − m ) n +1 = 1 n ! ∂ n ∂ ( m ) n p − m (2.17)the right hand side of eq. (2.16) may be re-expressed as − ip − m (cid:18) − m p − m (cid:19) n t ( p ) = ( − m ) n n ! ∂ n ∂ ( m ) n (cid:18) − ip − m t ( p ) (cid:19) = (2.18) = ( − m ) n n ! ∂ n ∂ ( m ) n D ( p ) Therefore D ( p ) · [ − im t ( p ) · D ( p )] n = ( − m ) n n ! ∂ n ∂ ( m ) n D ( p ) (2.19)It follows that to any internal gluon line with n ≥ mass counterterms we can associatethe n -th derivative of the massive propagator with respect to the mass parameter squared,multiplied by ( − m ) n /n ! . Going back to our set of Feynman diagrams, let R be the sumof the subset of diagrams which have no mass counterterms in their internal gluon lines .From eq. (2.19) it is easy to see that the quantity R n = ( − m ) n n ! ∂ n R ∂ ( m ) n (2.20)is precisely the sum of the subset of diagrams with n mass counterterms (not necessarilyon the same gluon lines). Hence the sum R ( N ) of the full set of diagrams, with zero to N mass counterterms on any of their internal lines, may be expressed as R ( N ) = N (cid:88) n =0 R n = N (cid:88) n =0 ( − m ) n n ! ∂ n R ∂ ( m ) n = R − m ∂ R ∂m + m ∂ R ∂ ( m ) + · · · (2.21)Therefore, in order to compute R ( N ) , we only need to know R and its derivatives withrespect to m up to the order N . This is the procedure which we will follow in Sec.2.2-3to obtain the ghost and gluon dressed propagators.Of particular interest to us are the first two terms of the sum in eq. (2.21). Suppose that R contains a term which is linear in m , R = m L + Q , where L is an m -independentcoefficient and Q is a remainder. Then R = − m ∂ R ∂m = − m L − m ∂ Q ∂m (2.22) It is easy to see that the above equation holds also for n = 0 . These are precisely the diagrams of ordinary perturbation theory, albeit computed with massive ratherthan massless bare gluon propagators. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles so that R + R = R − m ∂ R ∂m = Q − m ∂ Q ∂m (2.23)Therefore, even if R and R may separately contain linear terms in m , their sum does not:the linear term in R is precisely the same as that in R but with an opposite sign, leadingto their reciprocal cancellation in the sum. This is especially important in two respects.First of all, as we will see in Sec.2.3, it turns out that because of this mechanism the masssquared m in the denominator of the bare transverse gluon propagator gets cancelled bythe quantum correction due to the simplest of the crossed diagrams, the single countertermwith no loops. It follows that the mass of the gluon, if any, must come from the loops of theperturbative series, instead of being a trivial consequence of the shift of the Fadeev-Popovaction. For instance, the massive perturbative approach would not predict the generationof a mass for the photons of Quantum Electrodynamics. Second of all, the existence of themass scale m in the massive expansion may invalidate the perturbative series by givingrise to the so-called mass divergences , i.e. divergences proportional to m , which cannotbe removed from the series by making use of the renormalization counterterms of masslessperturbation theory. However, thanks to the cancellation of the linear terms in m , it turnsout that – at least to one loop –, once the mass counterterms are taken into account, suchdivergences disappear from the series and the renormalizability of the theory is not spoiled.The potential appearance of mass divergences provides a criterion for fixing the minimumnumber of crossed diagrams that must be included in the perturbative series at a given looporder: such a number must be high enough for the cancellation of the mass divergences tooccur. For the one-loop ghost and gluon dressed propagators, Sec.2.2-3, this criterion fixesthe minimum number of crossed diagrams to be equal to one. The maximum number ofcrossed diagrams, on the other hand, is not constrained by the principle of renormalizabil-ity alone: since each mass counterterm multiplies the internal line on which it is attachedby a factor of ( p − m ) − , the loop integrals become less and less divergent as the numberof mass counterterms in their internal lines is increased, and eventually become finite.In order to fix the maximum number of crossed diagrams at a given loop order, one of thesimplest criteria is given by the principle of minimality. Suppose that we want to computesome quantity in massive perturbation theory to a given order. Due to the mass scale m ,the uncrossed loops in the expansion will give rise to non-renormalizable mass divergencesand a minimum number of crossed diagrams will have to be included in order to preservethe renormalizability of the series. One or more of these diagrams will have a maximaltotal number of vertices, be they mass counterterms, gluon-gluon vertices or ghost-gluonvertices. Such diagrams arise from the average (cid:10) S N V int (cid:11) in the perturbative series , where N V is the maximal total number of vertices in the diagram. Therefore, in order to beconsistent with the perturbative order in the powers of the interaction action, we mayinclude all the diagrams with a number of mass counterterms such that the total numberof vertices is less than or equal to N V .These criteria will be adopted for the computation of the one-loop ghost and gluon prop-agators in Sec.2.2-3. It turns out that the uncrossed diagrams of the ghost propagatordo not contain mass divergences, whereas those of the gluon propagator do. In order toremove the mass divergences of the uncrossed gluon diagrams, we will need to includecrossed diagrams with a total number of vertices up to three. Then, to be consistent withthe order in the number of vertices, we will do the same for the ghost propagator. Here (cid:104)·(cid:105) , as in Chapter 1, denotes the average with respect to the zero-order massive action. iorgio Comitini As a final remark, we wish to illustrate the general mechanism by which ordinary pertur-bation theory and the standard perturbative results are recovered in the high energy limitof the massive expansion. From eq. (2.13) we see that when we resum a finite number ofmass counterterms in an internal gluon line, we obtain an effective line that propagatesthe gluons according to the expression N (cid:88) n =0 D ( p ) · [ − im t ( p ) · D ( p )] n = − ip − m N (cid:88) n =0 (cid:18) − m p − m (cid:19) n t ( p ) + ξ − ip (cid:96) ( p ) (2.24)where N is the number of resummed mass counterterms. At high momenta ( | p | (cid:29) m )we have that p − m → p and for n ≥ the n -th power of the expression in parenthesesbecomes negligible with respect to the zero-order summand. Accordingly, lim | p |→∞ N (cid:88) n =0 D ( p ) · [ − im t ( p ) · D ( p )] n = − ip t ( p ) + ξ − ip (cid:96) ( p ) = D ( p ) (2.25)which implies that at high momenta the gluons effectively propagate through the ordinary,massless propagator.As a result of this, the shift of the kinetic action does not modify the high energy behaviorof the theory and in the UV, as we will explicitly see in Sec.2.2-3, the expressions derived inmassive perturbation theory match the standard perturbative results. The mass parameter m sets the scale at which the predictions of the massive expansion diverge from those ofordinary perturbation theory. 54 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles The dressed ghost propagator (cid:101) G ab ( p ) is defined as (cid:101) G ab ( p ) = (cid:90) d d x e ip · x (cid:104) Ω | T (cid:8) c a ( x ) c b (0) (cid:9) | Ω (cid:105) (2.26)where T {·} is the time ordering meta-operator and | Ω (cid:105) is the exact vacuum state of thetheory. (cid:101) G ab ( p ) is related to the one-particle-irreducible ghost self-energy Σ ab ( p ) = δ ab Σ( p ) through the equation (cid:101) G ab ( p ) = δ ab ip − Σ( p ) (2.27)To one loop, a single irreducible uncrossed diagram contributes to the ghost self-energy:the first diagram in Fig.17. This diagram will turn out not to contain mass divergences, sothat from the point of view of renormalizability no crossed diagrams need to be includedin the ghost self-energy. However, as we will see, the uncrossed diagrams of the one-loop gluon propagator do contain divergences proportional to m . To one-loop, a single masscounterterm is enough to remove such divergences, and by the principle of minimality wewill need to include in the expansion crossed diagrams up to an order of three in the totalnumber of vertices. Therefore, in order to be consistent with the maximum total numberof vertices for both the gluon and the ghost propagator, the one-loop ghost self-energy willneed to contain any crossed diagram with a total number of vertices up to three. The onlysingle-loop diagram with this property is the second diagram in Fig.17. Thus the one-loopghost self-energy, as computed in the massive expansion to order three in the interaction,is the sum of the two diagrams in figure.Figure 17: Diagrams which contribute to the one-loop ghost self-energy. Let us denote by Σ ab ( p ) the uncrossed loop in Fig.17. Using the Feynman rules of massiveperturbation theory, we find that Σ ab ( p ) = ig f dca f bcd (cid:90) d d q (2 π ) d ( p − q ) µ p ν ( p − q ) + i(cid:15) (cid:26) t µν ( q ) q − m + i(cid:15) + ξ (cid:96) µν ( q ) q + i(cid:15) (cid:27) = (2.28) = − iN g δ ab (cid:90) d d q (2 π ) d p − q ) (cid:26) q p − ( p · q ) q ( q − m ) + ξ p · q ( p · q − q ) q (cid:27) == δ ab (cid:0) Σ A ( p ) + Σ B ( p ) (cid:1) iorgio Comitini where N is the number of colors ( N = 3 for pure gauge QCD) and Σ A ( p ) = − iN g (cid:90) d d q (2 π ) d q p − ( p · q ) q ( p − q ) ( q − m ) (2.29) Σ B ( p ) = − iN g ξ (cid:90) d d q (2 π ) d p · q ( p · q − q ) q ( p − q ) (2.30)The triple denominator in the integral Σ A ( p ) may be transformed into a double denomi-nator by using the identity q ( q − m ) = 1 m (cid:18) q − m − q (cid:19) (2.31)which allows us to put Σ A ( p ) in the form Σ A ( p ) = − iN g m (cid:90) d d q (2 π ) d (cid:26) q p − ( p · q ) ( p − q ) ( q − m ) − q p − ( p · q ) ( p − q ) q (cid:27) (2.32)By introducing a Feynman parameter x , changing variables of integration and discardingany term which is zero by Lorentz symmetry, we obtain Σ A ( p ) = − iN g m (cid:90) dx (cid:90) d d q (2 π ) d (cid:26) q p − ( p · q ) [ q − xm + x (1 − x ) p ] − [ m → (cid:27) == − i N ( d − g d p m (cid:90) dx (cid:90) d d q (2 π ) d (cid:26) q [ q − xm + x (1 − x ) p ] − [ m → (cid:27) (2.33)where [ m → denotes the limit m → of the first term in parentheses. In dimensionalregularization, (cid:90) d d q (2 π ) d q ( q − ∆) = − i ∆(4 π ) d − d Γ (cid:18) − d (cid:19) (cid:18) ∆4 πµ (cid:19) d/ − → (2.34) → i ∆(4 π ) (cid:20) (cid:15) − ln ∆ µ + 12 (cid:21) where µ and µ = 4 πµ e − γ E are the MS and MS mass scales resulting from the definitionof the theory in d (cid:54) = 4 and we have set d = 4 − (cid:15) . Therefore, with ∆ = xm − x (1 − x ) p (2.35)the dimensionally regularized expression for Σ A ( p ) reads Σ A ( p ) = 3 N g π ) p m (cid:90) dx (cid:26) ∆ (cid:20) (cid:15) − ln ∆ µ + 13 (cid:21) − [ m → (cid:27) = (2.36) = 3 N g π ) p (cid:26) (cid:15) − ln − p µ + 13 − m (cid:90) dx (cid:18) ∆ ln ∆ − p − [ m → (cid:19)(cid:27) erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Upon explicitly evaluating the integral in parentheses, we find that Σ A ( p ) = α p (cid:18) (cid:15) − ln m µ − g ( s ) + 53 (cid:19) (2.37)where s = − p /m is an adimensionalized momentum variable, g ( s ) is a function given by g ( s ) = (1 + s ) s ln(1 + s ) − s ln s − s (2.38)and we have defined an effective coupling constant α by α = 3 N α s π α s = g π (2.39)As for the integral Σ B ( p ) , by introducing a Feynman parameter x , changing variables ofintegration and discarding any term which is zero by Lorentz symmetry, we obtain Σ B ( p ) = − iN g ξ p (cid:90) dx (1 − x ) (cid:90) d d q (2 π ) d (cid:0) − (2 + d ) x (cid:1) q /d + x (1 − x ) p ( q + x (1 − x ) p ) (2.40)In dimensional regularization, (cid:90) d d q (2 π ) d q ( q − ∆) = i (4 π ) d (cid:18) − d (cid:19) (cid:18) ∆4 πµ (cid:19) d/ − → (2.41) → i (4 π ) d (cid:18) (cid:15) − ln ∆ µ (cid:19) The integrand term proportional to p , on the other hand, yields a finite integral, since (cid:90) d d q (2 π ) d q − ∆) = − i (4 π ) (2.42)Therefore, the dimensionally regularized expression of Σ B ( p ) reads Σ B ( p ) = N g π ) ξ p (cid:90) dx (1 − x ) (cid:26)(cid:0) − (6 − (cid:15) ) x (cid:1) (cid:18) (cid:15) − ln ∆ µ (cid:19) + 2 x (cid:27) = (2.43) = − N g π ) ξ p (cid:26) (cid:15) − ln m µ −
43 + 2 (cid:90) dx (1 − x )(1 − x ) ln ∆ m (cid:27) where ∆ = − x (1 − x ) p . Upon explicitly computing the integral in the above equation,we find that Σ B ( p ) = − α ξ p (cid:18) (cid:15) − ln m µ − ln s (cid:19) (2.44)Finally, by summing up Σ A and Σ B we obtain the following expression for the uncrossedghost self-energy: 57 iorgio Comitini Σ ( p ) = Σ ,d ( p ) + Σ ,f ( p ) (2.45)where Σ ,d ( p ) = α p (cid:18) − ξ (cid:19) (cid:18) (cid:15) − ln m µ (cid:19) (2.46)is a divergent contribution, while Σ ,f ( p ) = − α p ( g ( s ) − − ξ ln s ) (2.47)is a finite contribution. Let Σ ab ( p ) denote the crossed diagram in Fig.17. Following our discussion in Sec.2.1.2, Σ ab ( p ) may be computed by taking a derivative of Σ ab ( p ) with respect to the mass pa-rameter squared m according to the law Σ ab ( p ) = − m ∂∂m Σ ab ( p ) = − δ ab m ∂∂m Σ ( p ) (2.48)The derivative of the divergent part of Σ ( p ) yields a finite term, ∂∂m Σ ,d ( p ) = − α p m (cid:18) − ξ (cid:19) (2.49)In order to compute the derivative of the finite part of Σ ( p ) , we may use chain differen-tiation to convert the derivative with respect to m into a derivative with respect to theadimensional momentum variable s , ∂∂m = ∂ ( − p /m ) ∂m ∂∂ ( − p /m ) = − sm ∂∂s (2.50)We then find that ∂∂m Σ ,f ( p ) = α p m (cid:0) sg (cid:48) ( s ) − ξ (cid:1) (2.51)Adding up the contributions due to the divergent and finite parts of Σ ( p ) , we obtain Σ ( p ) = − α p (cid:0) sg (cid:48) ( s ) − (cid:1) (2.52)where the derivative of g ( s ) reads g (cid:48) ( s ) = (1 + s ) ( s − s ln(1 + s ) − ln s + 2 s + 2 s (2.53)58 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles By adding up the crossed and uncrossed diagrams, we obtain the total ghost self-energy Σ ab ( p ) = δ ab (cid:16) Σ d ( p ) + Σ f ( p ) (cid:17) (2.54)where Σ d ( p ) = α p (cid:18) − ξ (cid:19) (cid:18) (cid:15) − ln m µ (cid:19) (2.55)is a divergent contribution and Σ f ( p ) = − α p (cid:0) g ( s ) + sg (cid:48) ( s ) − − ξ ln s (cid:1) (2.56)is a finite contribution. By explicitly evaluating the sum g ( s ) + sg (cid:48) ( s ) , Σ f ( p ) can be putin the form Σ f ( p ) = − α p (cid:18) G ( s ) − − ξ
12 ln s (cid:19) (2.57)where G ( s ) = 112 [ g ( s ) + sg (cid:48) ( s )] = 112 (cid:20) (1 + s ) (2 s − s ln(1 + s ) − s ln s + 1 s + 2 (cid:21) (2.58) As anticipated earlier in this section, neither of the two diagrams of the one-loop ghostself-energy contains mass divergences. The divergent part of the self-energy, given byeq. (2.55), is identical to its analogue in ordinary perturbation theory [40], and may beabsorbed in the standard ghost field-strength renormalization counterterm, Σ ct ( p ) = − δ c p (2.59)where δ c = Z c − . If we define the renormalization counterterm so that δ c = α (cid:26)(cid:18) − ξ (cid:19) (cid:18) (cid:15) − ln m µ (cid:19) + 83 + 4 g (cid:27) (2.60)where g is an arbitrary renormalization constant, then the sum Σ d ( p ) + Σ ct ( p ) adds up to Σ d ( p ) + Σ ct ( p ) = − α p (cid:18)
23 + g (cid:19) (2.61)and we are left with a finite renormalized self-energy given by Σ R ( p ) = − α p (cid:18) G ( s ) + g − ξ
12 ln s (cid:19) (2.62)The constant g depends on the renormalization conditions for the ghost propagator.59 iorgio Comitini Plugging Σ( p ) = Σ R ( p ) into eq. (2.26), we obtain the following renormalized expressionfor the one-loop ghost propagator in an arbitrary covariant gauge: (cid:101) G ab ( p ) = δ ab (cid:101) G ( p ) (2.63)where (cid:101) G ( p ) = ip [1 + α ( G ( s ) + g − ξ ln s/ (2.64)A further simplification is achieved by absorbing the one in parentheses into a new arbitraryconstant G , defined as G = g + 1 α (2.65)in terms of which the ghost propagator reads (cid:101) G ( p ) = iZ G p ( G ( s ) + G − ξ ln s/ Z G = 1 α (2.66)Since in general the dressed propagators are defined modulo a multiplicative normalizationfactor, Z G may actually be given an arbitrary value.Through eq. (2.66) we managed to express the dressed ghost propagator in terms of twoadimensional renormalization constants: a multiplicative constant – Z G – and an additiveconstant – G . The coupling constant g – having been reabsorbed into the definition of Z G and G – has completely disappeared from our equations and we are left with a propagatorwhich depends only on the renormalization conditions and on the energy scale set by themass parameter m . Having computed the ghost propagator, we are now in a position to test our results againstthe predictions of both ordinary perturbation theory and the lattice calculations.Let us start from the high energy behavior of the propagator. In the UV, ordinary per-turbation theory is effective in describing the dynamics of Yang-Mills theory and – asdiscussed in Sec.2.1.2 – the expressions derived in the massive expansion should match thestandard results. By taking the high momentum limit ( | s | (cid:29) ) of the function G ( s ) , wefind that lim | s |→∞ G ( s ) = 14 ln s + 13 (2.67)Hence in the UV the renormalized ghost self-energy – eq. (2.62) – reads Σ R ( p ) = − α p (cid:20)(cid:18) − ξ (cid:19) ln (cid:18) − p m (cid:19) + 43 + g (cid:21) ( | p | (cid:29) m ) (2.68)By inspection of eq. (2.55), we see that the logarithm in the above expression has the samecoefficient as the logarithm in the divergent part Σ d of the unrenormalized self-energy. Thiswas to be expected since, if in the high energy limit the massive expansion is to reduce60 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles to ordinary perturbation theory, then for sufficiently large momenta the mass parameter m has to disappear from the equations. As a matter of fact, if we add back the divergentpart of the self-energy to eq. (2.68), we find that lim | p |→∞ Σ( p ) = 3 N α s π p (cid:18) − ξ (cid:19) (cid:18) (cid:15) − ln − p µ (cid:19) + α s p C (2.69)where C is an additive renormalization constant. This is precisely the result of ordinaryperturbation theory, as reported for instance in [40]. Eq. (2.69) establishes that the ghostpropagator, as computed in the massive expansion, has the correct high-energy behavior.In the infrared regime, most of our knowledge on the dynamics of Yang-Mills theory comesfrom the lattice calculations. These inform us that in the limit of vanishing momentumthe exact ghost propagator grows to infinity, implying that the ghosts – at variance withthe gluons – do not acquire a mass through the interactions. In order to investigate the p → limit of the ghost propagator, we need to send s → in the function G ( s ) , lim s → G ( s ) = 524 (2.70)and plug the result back into eq. (2.66). By doing so, we find that the ghost propagatorhas the following zero-momentum limit: (cid:101) G ( p ) = iZ G p [5 /
24 + G − ξ ln( − p /m ) /
12] ( p → (2.71)Since p ln( − p ) → as p → , in the limit of zero-momentum the denominator vanishesand the propagator grows to infinity. Therefore – in accordance with the lattice results– the massive expansion predicts that the ghosts remain massless even after the radiativecorrections are included. In passing, notice that while in the Landau gauge ( ξ = 0 ) p (cid:101) G ( p ) is finite and non-zero as p → , in other gauges p (cid:101) G ( p ) → due to the logarithmic term.We can now go one step further and test the predictions of the massive expansion againstthe results of the lattice calculations. As discussed in the Introduction, the lattice datais defined in Euclidean space, obtained from Minkowski space by complexifying the timecomponents of the Minkowski vectors and restricting them to imaginary values. In mo-mentum space, starting from the Minkowski momentum p = ( p , p ) , we assume that p can take on any value in the complex plane and then ask that p be imaginary, so that p = ip for some real p . Then to the complexified Minkowski momentum p = ( ip , p ) we associate an ordinary Euclidean momentum p E = ( p , p ) which has Euclidean norm p E = ( p ) + | p | = − p , the latter being the Minkowski norm of p .By complexifying the time component of the momentum variable of a Minkowski propa-gator, restricting to imaginary values and multiplying the propagator by i , we obtain theEuclidean version of the propagator, which can be computed non-perturbatively on thelattice. For the ghosts, this amounts to defining (cid:101) G ( p E ) = i (cid:101) G ( p ( p E )) (2.72) (cid:101) G ( p E ) being the Euclidean ghost propagator. Going back to eq. (2.66), by replacing p with − p E and multiplying by i , we obtain the one-loop approximation to (cid:101) G ( p E ) in theform 61 iorgio Comitini (cid:101) G ( p E ) = Z G p E ( G ( s ) + G − ξ ln s/ (2.73)where now s = p E /m . The Euclidean ghost propagator is expressed in terms of the samerenormalization constants Z G and G that define the propagator in Minkowski space.Since – especially for the ghosts – the lattice calculations are the most reliable in the Lan-dau gauge ( ξ = 0 ), in what follows we will limit ourself to this gauge and compare ourresults with the lattice data of ref. [61] (Duarte et al.), which were already presented inthe Introduction. In the Landau gauge, (cid:101) G ( ξ =0) ( p E ) = Z G p E ( G ( s ) + G ) (2.74)In the above expression, there are three parameters to be fixed, namely Z G , G and themass parameter m . The propagators of ordinary perturbation theory, by contrast, onlydepend on two parameters: the coupling constant and a multiplicative renormalizationconstant (together with some renormalization conditions). In Sec.2.2.1 we have absorbedthe coupling constant into the renormalization constants, so that the number of adimen-sional free parameters of the ghost propagator is still equal to two; however, due to theshift of the kinetic action, we are still left with the spurious mass parameter, which cannotbe fixed by first principles since classically Yang-Mills theory does not possess any massscales. The presence of an extra free parameter in the equations, at face value, may seem tocause the massive expansion to lose its predictive power. Nonetheless, in the next chapterwe will see that in order for gauge invariance to be preserved in the massive expansion,one of the three free parameters of the dressed gluon propagator needs to be fixed to aspecific value, resulting in a reduction in the number of adimensional free parameters whichactually compensates for the arbitrariness of m .Here and in the following section we do not concern ourselves with the issue of the predic-tivity of the massive expansion. Our only interest is determining whether our results agreewith the lattice data for some choice of the values of the free parameters, a task which canbe easily carried out by fitting the data to our functions and checking that the resultingfit well reproduces the data. Since the value of the mass parameter m – as we will see inthe next section – has a strong influence on the infrared behavior of the gluon propagator,it is best to fit it on the basis of the lattice data for the gluons, rather than on that of theghosts. For the fit of the ghost propagator, we fix m = 0 . GeV, which is the value thatwe will find for the Landau gauge gluon propagator in Sec.2.3.2.In Fig.18 we show the lattice data of ref. [61] for the Landau gauge Euclidean ghost dressingfunction p E (cid:101) G ( ξ =0) ( p E ) together with the one-loop results of massive perturbation theory,given by eq. (2.74) mutiplied by p E . At fixed m = 0 . GeV, the values of Z G and G which were found to best fit the data are reported in Tab.1. G Z G m (GeV) . . . Table 1: Parameters found from the fit of the lattice data of ref. [61] for the Landau gaugeghost propagator in the range − GeV at fixed m = 0 . GeV.62 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles p E G ~ ( ξ = ) ( p E ) p E (GeV)Lattice: Duarte et al.Massive expansion: Fit Figure 18: Ghost dressing function in the Landau gauge ( ξ = 0 ) as a function of theEuclidean momentum. Solid line: one-loop approximation in massive perturbation theorywith the parameters of Tab.1. Squares: lattice data from ref. [61].Setting m = 0 . GeV and G = 0 . in eq. (2.74) we obtain a dressing function p E (cid:101) G ( ξ =0) ( p E ) – hence a propagator (cid:101) G ( ξ =0) ( p E ) – which is in excellent agreement with thelattice data for the ghosts in the infrared regime, showing that the massive perturbativeexpansion indeed is capable of accurately reproducing the low energy dynamics of theghosts. 63 iorgio Comitini The dressed gluon propagator (cid:101) D abµν ( p ) is defined as (cid:101) D abµν ( p ) = (cid:90) d d x e ip · x (cid:104) Ω | T (cid:8) A aµ ( x ) A bν (0) (cid:9) | Ω (cid:105) (2.75) (cid:101) D abµν ( p ) is related to the one-particle-irreducible gluon polarization Π abµν ( p ) through theequation (cid:101) D abµν ( p ) = δ ab (cid:26) − ip − m − Π T ( p ) t µν ( p ) + ξ − ip − ξ Π L ( p ) (cid:96) µν ( p ) (cid:27) (2.76)where Π T and Π L are the transverse and longitudinal components of the polarization, Π abµν ( p ) = δ ab { Π T ( p ) t µν ( p ) + Π L ( p ) (cid:96) µν ( p ) } (2.77)Because of gauge invariance, the exact longitudinal component of the gluon polarization Π L is known to vanish [31, 40], so that the exact longitudinal gluon propagator (cid:101) D L reads (cid:101) D L ( p ) = ξ − ip (2.78)If in ordinary perturbation theory this constraint can be shown to be fulfilled to any fixedloop order [40, 51], in the massive expansion the massiveness of the bare gluon propagatorin general causes the approximate longitudinal polarization to be non-zero until the fullset of crossed diagrams is resummed. In the Landau gauge ( ξ = 0 ) this poses no prob-lem, since due to the ξ ’s in eq. (2.76) the dressed gluon propagator of the Landau gaugeremains transverse anyway. In different gauges, however, Π L (cid:54) = 0 may lead to an incorrectapproximation to the exact longitudinal dressed propagator, eq. (2.78). With respect tothis issue, we take the view that, since the exact longitudinal polarization is known tovanish, there is no actual need to compute it perturbatively: we may set it to zero fromthe very start. From a computational point of view, this may formally be achieved byresumming the infinite set of crossed diagrams limited to the longitudinal component ofthe polarization, thus reverting back to ordinary perturbation theory where we know theconstraint Π L = 0 to hold at any fixed loop order. In what follows, we will assume thatsuch a resummation has been performed and that Π L = 0 . We will therefore limit ourselvesto the computation of the transverse component of the gluon polarization Π T , droppingthe subscript T , referring to it simply as the gluon polarization and setting (cid:101) D abµν ( p ) = δ ab (cid:26) − ip − m − Π( p ) t µν ( p ) + ξ − ip (cid:96) µν ( p ) (cid:27) (2.79)To one loop, the irreducible uncrossed diagrams which contribute to the gluon polarizationare the first three in Fig.19. The gluon tadpole ( a ) and the gluon loop ( a ) will be shownto contain mass divergences which need to be eliminated by including in the polarizationanalogous crossed diagrams ( b and b ). Since the crossed gluon loop (diagram b ) is oforder three in the total number of vertices, we will also include in the polarization thedoubly crossed tadpole ( c ), which contains the same number of vertices.64 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Figure 19: Loop diagrams which contribute to the one-loop gluon polarization.Figure 20: Single-counterterm diagram.Up to order three in the number of vertices, there is one last irreducible diagram, theloopless, single-counterterm diagram in Fig.20. From a theoretical point of view, this is amostly important diagram. By multiplying the mass counterterm vertex by − i , we obtainits contribution Π abct µν to the gluon polarization, Π abct µν ( p ) = − δ ab m t µν ( p ) Π ct ( p ) = − m (2.80)If we now split the one-loop gluon polarization Π into the sum Π ct + Π loops , where Π loops contains the contributions due to the six diagrams in Fig.19, since m + Π ct ( p ) = 0 , wefind that (cid:101) D abµν ( p ) = δ ab (cid:26) − ip − Π loops ( p ) t µν ( p ) + ξ − ip (cid:96) µν ( p ) (cid:27) (2.81)Therefore, once the single-counterterm diagram has been included in the polarization, thetree-level gluon mass m produced by the shift of the kinetic action disappears from thepropagator. Thus the mass of the gluon, if any, does not trivially result from the shift, butrather has to come from the loops of the polarization: the mass derived in the frameworkof massive perturbation theory is a truly dynamically generated mass. It is non-zero if inthe limit of vanishing momentum Π loops ( p ) approaches a finite, non-zero value, so that thepropagator remains finite at p = 0 instead of growing to infinity . In order to reach aconclusion with respect to the issue of mass generation in Yang-Mills theory, we need toinvestigate the zero-momentum limit of Π loops . Let us denote with Π ab (1) µν ( p ) the contribution to the gluon polarization due to the ghostloop – diagram in Fig.19. Π ab (1) µν ( p ) may be put in the form Π ab (1) µν ( p ) = iN g δ ab (cid:90) d d q (2 π ) d q µ ( q − p ) ν q ( q − p ) (2.82) It can be shown, for instance, that in the framework of massive QED perturbation theory the photonloop polarization vanishes at zero momentum, so that the photons do not acquire a mass. iorgio Comitini Since this diagram does not contain internal gluon lines nor mass counterterms, the aboveexpression is identical to its analogue in ordinary perturbation theory. An explicit com-putation in dimensional regularization shows that the transverse component of Π ab (1) µν ( p ) , Π (1) ( p ) , is given by Π (1) ( p ) = α p (cid:18) (cid:15) − ln m µ − ln s + 2 (cid:19) (2.83)where µ is the MS mass scale, (cid:15) = 4 − d , s = − p /m is an adimensionalized momentumvariable and α = 3 N α s π α s = g π (2.84)as in Sec.2.2.The gluon tadpole – diagram a in Fig.19 – contributes to the self-energy with themomentum-independent term Π ab (2 a ) µν ( p ) = iN ( d − g d δ ab η µν (cid:90) d d q (2 π ) d q − m (2.85)which in dimensional regularization yields a transverse projection Π (2 a ) ( p ) = − α m (cid:18) (cid:15) − ln m µ + 16 (cid:19) (2.86)As anticipated earlier in this section, the tadpole diagram contains a non-renormalizablemass divergence which will be removed from the polarization by an opposite divergence inthe singly-crossed tadpole – diagram b .Finally, the contribution due to the gluon loop – diagram a in Fig.19 – is given by Π ab (3 a ) µν ( p ) = Π ab (3 a ) µν ( p ) + ξ Π ξ ab (3 a ) µν ( p ) + ξ Π ξξ ab (3 a ) µν ( p ) (2.87)where the transverse components of the the three polarization terms can be put in the form Π a ) ( p ) = − iN g (cid:90) d d q (2 π ) d q ⊥ F ( q, p )( q − m [( q + p ) − m ] F ( q, p )12 = q + p q + p ( q + p ) − p q ⊥ q + p ) q (2.88) q ⊥ = q − ( q · p ) p Π ξ (3 a ) ( p ) = − iN g (cid:90) d d q (2 π ) d (cid:26) F ξ ( q, p )( q − m )( q + p ) + F ξ ( q, p ) q [( q + p ) − m ] (cid:27) F ξ ( q, p ) = (3 q − q ⊥ )( q − p ) q ( q + p ) (2.89) F ξ ( q, p ) = 3 q + 12 p + 12 q · p − q ⊥ (cid:18) p + 2 q · pq + p ( q + p ) q (cid:19) erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Π ξξ (3 a ) ( p ) = − iN g (cid:90) d d q (2 π ) d F ξξ ( q, p ) q ( q + p ) (2.90) F ξξ ( q, p ) = p q ⊥ q ( q + p ) The integral in eq. (2.90) is convergent and turns out to be equal to a finite constant times ξ p . Since these constants can be absorbed in the gluon field-strength renormalizationcounterterm (see ahead), we will not keep track of it. A straightforward albeit tedious cal-culation then leads to the following expression for the transverse component of the gluonloop: Π (3 a ) ( p ) = α (cid:20)(cid:18) ξ (cid:19) m + (cid:18) − ξ (cid:19) p (cid:21) (cid:18) (cid:15) − ln m µ (cid:19) + (2.91) + α p (cid:18) s − s + 2263 + s ln s − l A ( s ) − l B ( s ) + C (cid:19) ++ α ξ p (cid:18) (1 + s )(1 − s ) s ln(1 + s ) + s ln s − s + C ξ (cid:19) where l A ( s ) and l B ( s ) are logarithmic functions given by l A ( s ) = ( s − s + 12) (cid:18) ss (cid:19) / ln (cid:18) √ s − √ s √ s + √ s (cid:19) (2.92) l B ( s ) = 2(1 + s ) s ( s − s + 1) ln(1 + s ) (2.93)and C and C ξ are irrelevant finite constants. Just as the tadpole, the gluon loop contains anon-renormalizable mass divergence which is removed from the polarization by an oppositedivergence in the crossed gluon loop – diagram b . The three crossed diagrams in Fig.19 may be computed by deriving the uncrossed dia-grams a and a with respect to the mass parameter squared m (cfr. Sec.2.1.2). To bespecific, denoting with Π (2 b ) , Π (2 c ) and Π (3 b ) the transverse components of the polarizationterms due to the respective diagrams, we have Π (2 b ) = − m ∂∂m Π (2 a ) Π (2 c ) = m ∂ ∂ ( m ) Π (2 a ) (2.94) Π (3 b ) = − m ∂∂m Π (3 a ) Again, when needed, we may replace the derivative with respect to m by a derivative with67 iorgio Comitini respect to s = − p /m through chain differentiation, ∂∂m = − sm ∂∂s (2.95)An explicit calculation leads to the following expressions for the crossed polarization terms: Π (2 b ) ( p ) = 3 α m (cid:18) (cid:15) − ln m µ − (cid:19) (2.96) Π (2 c ) ( p ) = 3 α m (2.97) Π (3 b ) ( p ) = − α (cid:18) ξ (cid:19) m (cid:18) (cid:15) − ln m µ (cid:19) + (2.98) + α p (cid:18) − s + 63 s + 50 + s + 2 s ln s − s l (cid:48) A ( s ) − s l (cid:48) B ( s ) (cid:19) + − α ξ p (cid:18) (1 − s ) ( s + 2 s + 3) s ln(1 + s ) − s ln s − s − s − s (cid:19) where the derivatives of the logarithmic functions read l (cid:48) A ( s ) = (cid:114) ss (cid:26) s − s + 20 s − s ln (cid:18) √ s − √ s √ s + √ s (cid:19) − s − s + 12 s (cid:27) (2.99) l (cid:48) B ( s ) = 2(1 + s ) s (2 s − s + 20 s −
3) + 2(1 + s ) s ( s − s + 1) (2.100) By adding up the contributions due to the six loops in Fig.19, we find the following ex-pression for the transverse component Π loops of the one-loop gluon polarization: Π loops ( p ) = Π d loops ( p ) + Π f loops ( p ) (2.101)where Π d loops ( p ) = α (cid:18) − ξ (cid:19) p (cid:18) (cid:15) − ln m µ (cid:19) (2.102)is a divergent term and Π f loops ( p ) = − α p ( F ( s ) + ξF ξ ( s ) + C ) (2.103)is a finite term, with C an irrelevant finite constant. In Π f loops the functions F ( s ) and F ξ ( s ) are defined as F ( s ) = 58 s + 172 [ L a ( s ) + L b ( s ) + L c ( s ) + R a ( s ) + R b ( s ) + R c ( s )] (2.104) F ξ ( s ) = 14 s − (cid:20) s ln s − − s )(1 − s ) s ln(1 + s ) + 3 s − s + 2 s (cid:21) (2.105)68 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles where the functions L a ( s ) , L b ( s ) , L c ( s ) , R a ( s ) , R b ( s ) and R c ( s ) are given by L a ( s ) = 3 s − s − s − s (cid:114) ss ln (cid:18) √ s − √ s √ s + √ s (cid:19) L b ( s ) = 2(1 + s ) s (3 s − s + 11 s −
2) ln(1 + s ) (2.106) L c ( s ) = (2 − s ) ln sR a ( s ) = − ss ( s − s + 12) R b ( s ) = 2(1 + s ) s ( s − s + 1) (2.107) R c ( s ) = 2 s + 2 − s Once the crossed diagrams are included in the polarization, the mass divergences aris-ing from the gluon tadpole and the gluon loop cancel and we are left with a renormalizabledivergence – eq. (2.102) – which is identical to its analogue in ordinary perturbation the-ory [31, 40]. As usual, this divergence may be absorbed in the gluon field-strength renor-malization contribution to the polarization, Π abct µν ( p ) = − δ p δ ab t µν ( p ) (2.108)by defining the field-strength renormalization counterterm δ = Z − to be equal to δ = α (cid:18) − ξ (cid:19) p (cid:18) (cid:15) − ln m µ (cid:19) + α ( f − C ) (2.109)where f is an arbitrary renormalization constant. With δ as above, the renormalizedtransverse component of the gluon polarization reads Π R ( p ) = − α p ( F ( s ) + ξF ξ ( s ) + f ) (2.110)Therefore, setting Π loops = Π R in eq.(2.81), we find that the renormalized expression forthe transverse component of the gluon propagator is given by (cid:101) D T ( p ) = − ip [1 + α ( F ( s ) + ξF ξ ( s ) + f )] (2.111)Finally, by defining an arbitrary renormalization constant F as F = f + 1 α (2.112)69 iorgio Comitini we can put the propagator in the form (cid:101) D T ( p ) = − iZ D p ( F ( s ) + ξF ξ ( s ) + F ) Z D = 1 α (2.113)where, since the propagator is defined modulo an arbitrary multiplicative factor, Z D maybe given an arbitrary value.Through eq. (3.12) we have managed to express the gluon propagator in terms of twoadimensional renormalization constants: a multiplicative constant – Z D – and an additiveconstant – F . The coupling constant g has disappeared from the equation and we areleft with a propagator which depends only on the renormalization conditions and on theenergy scale set by the mass parameter m . Having computed the one-loop gluon propagator, we can now proceed to investigate itsUV and IR behavior and compare our results with the predictions of ordinary perturbationtheory and the lattice computations.The UV limit of the gluon propagator is obtained by sending | s | → ∞ in the functions F ( s ) and F ξ ( s ) defined by eqq. (2.104)-(2.107). An explicit calculation shows that lim | s |→∞ F ( s ) = 1718 + 1318 ln s lim | s |→∞ F ξ ( s ) = −
16 ln s − (2.114)Thus, going back to eq. (2.110), we find that in the UV lim | p |→∞ Π R ( p ) = − α p (cid:18) − ξ (cid:19) ln − p m + α s p C (2.115)where C is an irrelevant constant. The coefficient of the logarithmic term in Π R is thesame as that of the divergence in Π d loops – eq. (2.102) –, as needed to eliminate the massparameter m from the expressions in the high energy limit. By adding back the divergenceto the renormalized polarization, we find that Π loops ( p ) = N α s π (cid:18) − ξ (cid:19) p (cid:18) (cid:15) − ln − p µ (cid:19) + α s p C ( | p | (cid:29) m ) (2.116)which is the very same result of ordinary perturbation theory, as reported for instancein ref. [40]. Therefore, as anticipated by our discussion in Sec.2.1.2, the one-loop gluonpropagator of the massive expansion has the correct, ordinary UV behavior.In the IR, as discussed at length in the Introduction and in Chapter 1, the transversegluons are reported by the lattice calculations to acquire a dynamical mass. The gluonmass manifests itself in the finiteness of the gluon propagator at vanishing momentum, atvariance with the typical behavior of massless propagators which grow to infinity as p → .In the limit of vanishing momentum ( s → ), the functions F ( s ) and F ξ ( s ) both tend toinfinity as /s : 70 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles lim s → F ( s ) = 58 s lim s → F ξ ( s ) = 14 s (2.117)Since in the propagator – eq. (3.12) –, the functions F and F ξ are multiplied by a factor of p , the zero-momentum limit of the transverse component of the dressed gluon propagatoris finite and reads (cid:101) D T (0) = − iZ D − M ξ (2.118)where M ξ is a gauge-dependent mass scale defined by M ξ = 5 m (cid:18) ξ (cid:19) (2.119)Therefore, in accordance with the lattice data, the massive perturbative expansion indeedpredicts that the transverse gluon propagator acquires a mass in the infrared regime.As discussed in the introduction to this section, the mass derived in the framework ofthe massive expansion is dynamical in nature: it does not automatically result from themassive shift of the kinetic action, but rather is generated by the loops of the polarization.Let us see how this comes about. By taking the zero-momentum limit of the loop diagramsin Fig.19 we find that Π (1) (0) = 0Π (2 a ) (0) + Π (2 b ) (0) + Π (2 c ) (0) = − α m (2.120) Π (3 a ) (0) + Π (3 b ) (0) = α (cid:18) ξ (cid:19) m where we have summed the diagrams which only differ by the number of mass countertermsin order to get rid of the spurious mass divergences. At zero momentum, the ghost loop –diagram in Fig.19 – vanishes, hence it does not contribute to the gluon mass. The gluontadpoles and the gluon loops – diagrams a - c and a - b in Fig.19 –, on the other hand,do not vanish and sum to α times M ξ . Therefore we conclude that the mass of the gluonsis generated both by the gluon tadpoles and by the gluon loops.The ability to predict a non-zero mass for the gluon propagator is a necessary condition forthe validity of any result on the low energy dynamics of the gluons in Yang-Mills theory.Having shown that in massive perturbation theory the one-loop transverse gluon propa-gator is massive as it should be, we can now proceed to compare our predictions with thelattice data. For the comparison we will again use the data reported in ref. [61] by Duarteet al., already presented in the Introduction. By analytically continuing the Minkowskimomentum p to the Euclidean momentum p E – equivalently, by setting p = − p E – and bymultiplying the Minkowski propagator by − i , we obtain the transverse gluon propagatorin Euclidean space (cid:101) D T ( p E ) : (cid:101) D T ( p E ) = Z D p E ( F ( s ) + ξF ξ ( s ) + F ) (2.121) The minus sign in − i is due to the vector nature of the gluon field: as we go from Minkowski spaceto the Euclidean space we must replace everywhere the Minkowski metric η µν with the Euclidean metric − δ µν , causing a change of sign in the tensor propagator D µν ( p ) . iorgio Comitini where now s = p E /m . The Euclidean propagator is expressed in terms of the same renor-malization constants of the Minkowski propagator, namely Z D and F . Again, since thelattice data is the most reliable in the Landau gauge ( ξ = 0 ), in this section we will limitour comparison to this gauge. Setting ξ = 0 in eq. (2.121) we obtain (cid:101) D ( ξ =0) T ( p E ) = Z D p E ( F ( s ) + F ) (2.122) Z D and F , together with the value of the mass parameter m , are free parameters whichwe need to fit starting from the lattice data. We observe that, like the ghost propagator– see Sec.2.2.2 –, the gluon propagator too depends on a free parameter in excess of thetwo required for the perturbative results to be true predictions from first principles. Whatwe will see in the next chapter is that gauge invariance constrains the renormalizationconstant F to take a specific value, thus reducing the number of free parameters back totwo (the constant Z D and the mass m ) and making our results as predictive as they canbe in the context of perturbation theory. For the moment, we treat F as a free parameterto be fitted to the lattice data, just as we did for the constant G in the ghost propagator.The fitted value of F will be shown to be in perfect agreement with the one dictated bygauge invariance in the next chapter.In Fig.21 we show the lattice data of ref. [61] for the Landau gauge transverse gluonpropagator together with the one-loop results of massive perturbation theory, given byeq. (2.122). The parameters which were found to best fit the data in the momentum range − GeV are reported in Tab.2. D ~ T ( ξ = ) ( p E ) ( G e V - ) p E (GeV)Lattice: Duarte et al.Massive expansion: Fit Figure 21: Transverse gluon propagator in the Landau gauge ( ξ = 0 ) as a function of theEuclidean momentum. Solid line: one-loop approximation in massive perturbation theorywith the parameters of Tab.2. Squares: lattice data from ref. [61].72 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles F m (GeV) Z D − . . . Table 2: Parameters found from the fit of the lattice data of ref. [61] for the Landau gaugeEuclidean transverse gluon propagator in the range − GeV.As we can see, by choosing m = 0 . GeV and F = − . we obtain a propagatorwhich is in astonishing agreement with the lattice data already at one loop, proving thatthe massive expansion is able to describe the low energy dynamics of the gluons not onlyqualitatively through the generation of a dynamical mass for the gluons, but also quanti-tatively, to an unexpected degree of accuracy.In the next chapter we will extend our comparison with the lattice to covariant gaugesup to ξ = 0 . , where some data – albeit less accurate than that of the Landau gauge – isalready available. 73 iorgio Comitini In this chapter we have computed the one-loop ghost and gluon propagators in the frame-work of massive perturbation theory and compared our results with the predictions ofordinary perturbation theory (UV regime) and – limited to the Landau gauge – of thelattice calculations (IR regime).In massive perturbation theory, the one-loop ghost and gluon propagators can be expressedin terms of five free parameters: two multiplicative renormalization constants – Z G and Z D –, two additive renormalization constants – G and F – and a mass parameter – m .The mass parameter is a spurious free parameter, in that it is artificially introduced inthe perturbative series by the shift of the Faddeev-Popov action which defines the massiveexpansion. It cannot be fixed from first principles since Yang-Mills theory is scale-free atthe classical level.The one-loop ghost propagator was found to remain massless in any covariant gauge. In thehigh-energy limit, its expression was shown to match the results of ordinary perturbationtheory. In the low-energy limit, an appropriate choice of the renormalization constantsand of the mass parameter – cf. Tab.1 – leads to a Euclidean propagator which accuratelyreproduces the lattice data of ref. [61] (Duarte et al.). The value of the additive renor-malization constant which was found to best fit the data at fixed m = 0 . GeV is G = 0 . – cf. Tab.1.The transverse component of the gluon propagator was found to non-trivially acquire amass in any covariant gauge due to the non-vanishing zero-momentum limit of the gluontadpoles and of the gluon loops. The high-energy limit of the propagator was shown tomatch the predictions of ordinary perturbation theory. In the low-energy limit, the propa-gator was found to reproduce the lattice data of ref. [61] (Duarte et al.) to an astonishingdegree of accuracy. The parameters which were found to best fit the data – cf. Tab.2 –are F = − . and m = 0 . GeV.The asymptotic analysis of the one-loop ghost and gluon propagators shows that massiveperturbation theory is capable of describing both the UV and IR behavior of Yang-Millstheory by the use of elementary quantum field theoretic methods. In the framework of themassive expansion, the non-perturbative content which was responsible for the failure ofordinary perturbation theory is effectively incorporated in the dynamical mass generatedfor the gluons through the loops of the massive series. Our results suggest that the break-down of the ordinary perturbative methods at low energies may be due to a bad choice ofthe expansion point of the perturbative series, rather than to an intrinsic limitation of themethods themselves. In the next chapter we will perfect the massive perturbative methodand restore the predictivity of the expansion, which was weakened by the introduction ofthe spurious mass parameter m . This is the value of m obtained from the fit of the gluon propagator, see ahead. The analytic structure of the gluonpropagator and the Nielsenidentities: gauge invariance and thepredictivity of the massive expansion
As we saw in Chapter 2, through massive perturbation theory one is able to derive one-loop ghost and gluon propagators which have the correct asymptotic behavior and – by anappropriate choice of free parameters – quantitatively reproduce the lattice data to a highdegree of accuracy. Nonetheless, the theory presented in the previous chapter is incompletein at least two respects. First of all, the shift of the kinetic action that defines the massiveexpansion introduces a spurious mass parameter which cannot be fixed from first principlesand must be provided as an external input. The need for an external input, at face value,results in a loss of the predictive power of the theory, which is not desirable if one wants tointerpret the massive perturbation theory as a true approach form first principles. Secondof all, the fact that the massive expansion treats the gluons as massive at tree-level makes itunclear to which extent gauge invariance – or rather BRST invariance – is preserved at thelevel of its perturbative series. If non-perturbatively, by leaving the Faddeev-Popov action S unchanged, the massive expansion is guaranteed to preserve gauge symmetry, from thepoint of view of perturbation theory the massiveness of the bare gluon propagator and thepresence of the gluon mass counterterm may lead to results in which BRST invariance isnot automatically implemented to any finite perturbative order.The aim of this chapter is to present a common solution to both of these issues. Thekey idea is that gauge invariance imposes strict constraints on the analytic structure ofthe dressed gluon propagator, which in massive perturbation theory are not automaticallysatisfied due to the massiveness of the propagator. By enforcing these constraints in themassive expansion, one is able to fix the value of the renormalization constant F thatdefines the gluon propagator. Once F has been fixed from first principles, we are leftwith a gluon propagator which complies with BRST invariance and that depends on twofree parameters only – a multiplicative renormalization constant Z D and the value of themass parameter m –, two being the correct number of free parameters for a propagatorcomputed from first principles.Gauge invariance constrains the analytical structure of the gauge-dependent gluon prop-agator through the Nielsen identities. In the context of non-abelian gauge theories, thepropagators associated to the gauge bosons – and, more generally, any of the Green func-tions of the theory – depend on the gauge in a non-trivial fashion. The Nielsen identities75 iorgio Comitini are equations that describe how the Green functions change as functions of the gauge pa-rameter ξ . They are a consequence of the BRST invariance of the Faddeev-Popov action,and can be derived by ordinary functional methods. Through the Nielsen identities onecan prove that, even though the gluon propagator depends on the gauge, the position ofits poles in the complex plane is gauge-independent, i.e. it is fixed regardless of the valueof ξ . In ordinary perturbation theory, this requirement is trivially met: the gluon prop-agator possesses a single pole at p = 0 , which is obviously a gauge-independent value.In massive perturbation theory, on the other hand, dynamical mass generation causes theanalytic structure of the propagator to be much richer: as we will see, the gluon propagatorturns out to have two non-zero complex-conjugated poles, whose position depends on thevalue of the renormalization constant F . In order to comply with gauge invariance, F cannot be arbitrary, but must be given the value that fixes the gluon poles in the correctposition. In addition, the Nielsen identities can be used to show that the residues of thegluon propagator at its poles also are gauge-invariant. Ultimately, this condition will allowus to completely determine the gluon propagator as a function of ξ and to prove that therequirement of gauge invariance alone is able to restore the predictivity of the massiveexpansion by fixing the value of the renormalization constant F .This chapter is organized as follows. In Sec.3.1 we discuss the Nielsen identities anddescribe the method that will allow us to fix the value of F by enforcing the gauge invari-ance of the pole structure of the gluon propagator. In Sec.3.2. we investigate the analyticstructure of the gluon propagator in the Landau gauge as F is tuned across the real num-bers, apply the method laid out in the previous section and present our results. The valueof F singled out by our requirements of gauge invariance will be shown to be very closeto that which was obtained in Sec.2.3.2 by fitting the propagator to the lattice data. InSec.3.3 we discuss a renormalization-scheme-dependent method presented in ref. [84] forfixing the value of the ghost additive renormalization constant G in the Landau gauge.The results of this chapter were presented and published for the first time in ref. [74].76 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles In Chapter 2 we saw that the gluon propagator computed in massive perturbation theorydepends on the gauge chosen for the definition of the theory, in that it contains terms whichare proportional to the gauge parameter ξ . This is a general, non-perturbative feature ofthe Green functions of non-abelian gauge theories: although the quantities which can bemeasured in the laboratory are gauge-invariant, the basic building blocks from which theyare made of typically depend on the gauge. In order for the gauge dependence to ultimatelyfactor out from the equations, the Green functions of any gauge theory must satisfy specificconstraints, which are formulated in terms of the so-called Nielsen identities.The Nielsen identities [79, 80] are non-perturbative equations which relate the gauge de-pendence of a given Green function to the content of other Green functions. They arederived by ordinary functional methods starting from the BRST invariance of the Fadeev-Popov action. In this chapter we will not go through their explicit derivation, for which werefer to [81] (Breckenridge et al.). Rather, we will use the identities to prove two resultswhich are of fundamental importance in the analysis of the massive perturbative expan-sion, namely, the gauge invariance of the position and residues of the poles of the gluonpropagator.In a general covariant gauge parametrized by ξ , the gauge dependence of the transversecomponent of the dressed gluon propagator (cid:101) D T ( p , ξ ) is described by the following Nielsenidentity (cf. eq.(3.6d) in ref. [81]): ∂ (cid:101) D − T ∂ξ ( p , ξ ) = H T ( p , ξ ) (cid:101) D − T ( p , ξ ) (3.1)In this equation, H T is the transverse component of a Green function H abµν defined as H abµν ( p, ξ ) = (cid:90) d d x d d y e ip · ( x − y ) (cid:68) T (cid:110) D µ c a ( x ) A bν ( y ) c c (0) B c (0) (cid:111)(cid:69) ξ (3.2)where B a is an auxiliary field – the Nakanishi-Lautrup field – introduced to enforcethe off-shell BRST invariance of the Faddeev-Popov action and (cid:104)·(cid:105) ξ denotes the vacuumexpectation value computed in the gauge ξ . For our purposes, the explicit form of H T is irrelevant. What matters to us is that the position of the poles of H T , in general, isdifferent from that of the gluon propagators’. This is important for the following reason.Suppose that as ξ varies the gluon propagator has a pole at p = p ( ξ ) , so that (cid:101) D − T ( p ( ξ ) , ξ ) = 0 (3.3)for every ξ . Then, by taking the total derivative of the above equation with respect to ξ ,we find that ∂ (cid:101) D − T ∂ξ ( p ( ξ ) , ξ ) + dp dξ ( ξ ) ∂ (cid:101) D − T ∂p ( p ( ξ ) , ξ ) = 0 (3.4) See for instance the Introduction to this thesis. iorgio Comitini However, from eq. (3.1) we know that ∂ (cid:101) D − T ∂ξ ( p ( ξ ) , ξ ) = H T ( p ( ξ ) , ξ ) (cid:101) D − T ( p ( ξ ) , ξ ) = 0 (3.5)where the vanishing of the right-hand side is due to eq. (3.3) together with the fact that H T ( p , ξ ) was assumed not to have poles – hence to be finite – at p ( ξ ) . Therefore thepartial derivative with respect to ξ in eq. (3.4) vanishes and, since the partial derivative of (cid:101) D − T with respect to p in general is different from zero, we are left with dp dξ ( ξ ) = 0 (3.6)Eq. (3.6) informs us that if the gluon propagator possesses a pole, then the position of thispole must be gauge-invariant: because of eq. (3.1) we must have p ( ξ ) = p ( ξ ) for anygauges ξ and ξ .Actually, we can go one step further and show that the residues of the gluon propagator atits poles must also be gauge-invariant. In order to do this, we start by taking the partialderivative of eq. (3.1) with respect to p . By exchanging the order of the derivatives onthe left-hand side, we obtain ∂ (cid:101) D − T ∂ξ∂p ( p , ξ ) = ∂H T ∂p ( p , ξ ) (cid:101) D − T ( p , ξ ) + 2 H T ( p , ξ ) (cid:101) D − T ( p , ξ ) ∂ (cid:101) D − T ∂p ( p , ξ ) (3.7)Since (cid:101) D − T vanishes at the poles and since the other terms on the right-hand side are finite,when computed at p = p ( ξ ) eq. (3.7) simply reads ∂ (cid:101) D − T ∂ξ∂p ( p ( ξ ) , ξ ) = 0 (3.8)Now, recall the definition of the residue of a function f at one of its poles z , R ( f, z ) = lim z → z f ( z )( z − z ) = lim z → z z − z f ( z ) − f ( z ) = (cid:20) ddz f ( z ) (cid:21) − z = z (3.9)By applying this definition to (cid:101) D T , we can express the inverse of the residue of the propa-gator at p ( ξ ) as R − ( (cid:101) D T , p ( ξ )) = ∂ (cid:101) D − T ∂p ( p ( ξ ) , ξ ) (3.10)By taking the total derivative of the above equation with respect to ξ we find that ddξ R − ( (cid:101) D T , p ( ξ )) = ∂ (cid:101) D − T ∂ξ∂p ( p ( ξ ) , ξ ) + dp dξ ( ξ ) ∂ (cid:101) D − T ∂ ( p ) ( p ( ξ ) , ξ ) = 0 (3.11)where the vanishing of the right-hand side is due to eqq. (3.6) and (3.8). Therefore, theresidues of the gluon propagator at its gauge-invariant poles are themselves gauge-invariant.Through the Nielsen identities one is able to show that – despite the gluon propagatorbeing gauge-dependent – the position and the residues of its poles are gauge-invariant. Inthe absence of mass generation, this result – at least as far as the position of the poles78 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles is concerned – is trivial: a massless gluon propagator has a single pole at p = 0 , whichis obviously a gauge-invariant position. If a dynamical mass is generated for the gluons,on the other hand, the poles may be found at any location in the complex plane. Hence,in the presence of mass generation, the Nielsen identities impose strict new constraints onthe analytic structure of the gluon propagator. The gauge invariance of the position and residues of the poles of the gluon propagator isa non-perturbative feature of Yang-Mills theory which is not guaranteed to be retained atany finite order in massive perturbation theory. The aim of this section is to provide amethod for reducing the number of free parameters of the massive expansion by exploitingthis apparent weakness of the massive approach.Let us investigate under which conditions the poles of the gluon propagator derived inChapter 2 have a gauge-invariant position. In Sec.2.3.1 we saw that the one-loop trans-verse gluon propagator of the massive expansion can be expressed as (cid:101) D T ( p ) = − iZ D p ( F ( s ) + ξF ξ ( s ) + F ) (3.12)where Z D and F are arbitrary renormalization constants, s = − p /m and the functions F ( s ) and F ξ ( s ) are defined in eqq. (2.104)-(2.107). In Sec.2.3.2 we showed that the propa-gator is finite at p = 0 . Hence (cid:101) D T , as is obvious in the presence of mass generation, doesnot have a massless pole. Since the value p = 0 is excluded, a necessary and sufficientcondition for (cid:101) D T to have a pole at p = p is that F ( − p /m ) + ξ F ξ ( − p /m ) + F = 0 (3.13)We remark that what we are actually looking for are the poles of the analytic continuation of (cid:101) D T to the whole complex plane, so that p in general is complex and the above equation is acomplex equation. Now, the Nielsen identities inform us that eq. (3.13) should be verifiedfor every ξ , since p is a gauge-independent value. It is easy to see that this conditioncannot be satisfied unless F , and perhaps even the mass parameter m , are functions ofthe gauge. F – being a renormalization constant – can actually be given different valuesin different gauges. This is because, first of all, the renormalization conditions may bechosen to be different in different gauges and, second of all, because the propagator itself isgauge-dependent, so that equal renormalization conditions for different gauges can be onlyimplemented by choosing different F ’s. Therefore, in general, we should regard F as beinga gauge-dependent constant: F = F ( ξ ) . As for the mass parameter, we observe that therole of m in the framework of the massive expansion is to absorb the non-perturbativecontent of the theory by introducing a mass scale in the perturbative series; for a gauge-dependent propagator, this task may require the scale itself to be different in differentgauges. Therefore, in general, m also may also be regarded as being a gauge-dependentparameter: m = m ( ξ ) .With F and m dependent on the gauge, the condition for the gauge invariance of thepole p reads F ( − p /m ( ξ )) + ξ F ξ ( − p /m ( ξ )) + F ( ξ ) = 0 ∀ ξ (3.14)79 iorgio Comitini For arbitrary F ( ξ ) and m ( ξ ) , this condition will not be met. Therefore, in general, theone-loop gluon propagator computed in massive perturbation theory does not comply withthe Nielsen identities. However, observe that if we knew in advance the position of the pole,then the above equation could be solved to find the functions F ( ξ ) and m ( ξ ) which realizethe gauge invariance of its position. Indeed, since eq. (3.14) is a complex equation, thevanishing of its real and imaginary parts at fixed p is in principle sufficient to determinethe two real functions. Knowing F ( ξ ) and m ( ξ ) would then be equivalent to knowingthe gluon propagator in any gauge, with no dependence left on any free parameter . Theproblem is, of course, that we do not actually know p . Therefore what we need to ask is:is it possible to determine the position of the poles starting from the requirement of thegauge invariance alone? The answer turns out to be yes, as long as we are willing to retainthe value of m in some fixed gauge as a free parameter.To prove our claim, we start by observing that, in order to determine the position of thepoles of the propagator, we only need to know the value of F and m in some specificgauge. For instance, we may pick the Landau gauge ( ξ = 0 ) and solve for p the equation F ( − p /m L ) + F L = 0 (3.15)where m L = m (0) and F L = F (0) are the mass parameter and additive renormalizationconstant in the Landau gauge. If F L is known, then p = m L z , where z solves theequation F ( − z ) + F L = 0 (3.16)Of course, since Yang-Mills theory is scale-free at the classical level, the value of the massparameter m L cannot be determined from first principles. However, once m L has beenprovided as an external input, we are left with a solution p that depends exclusively onthe value of the additive renormalization constant F L in the Landau gauge: the solutionto eq. (3.15) takes the form of an F L -dependent pole function p ( F L ) such that F ( − p ( F L ) /m L ) + F L = 0 (3.17)Once some value for F L has been fixed and eq. (3.15) has been solved to find the corre-sponding pole, we can then plug the solution back into eq. (3.14) to determine the functions F ( ξ ) and m ( ξ ) for any value of ξ . This procedure leaves us with gauge-dependent func-tions that depend only on the value of F L which was used in the first place to computethe position of the pole, with m L providing the scale for the mass parameter in an arbi-trary gauge. By construction, the functions F ( ξ ) and m ( ξ ) so obtained realize the gaugeinvariance of the position of the poles of the gluon propagator, which now depends on m L and F L only. Luckily, however, this is not the end of the story. As a matter of fact, aswe saw in the last section, the Nielsen identities constrain not only the positions of thepoles, but also their residues to be gauge-invariant. This constraint can be turned to ouradvantage as follows.Once F ( ξ ) and m ( ξ ) have been obtained by picking some value for F L , we can go onand compute the residue of the gluon propagator at the pole p ( F L ) as a function of thegauge. By virtue of eq. (3.10) and of the vanishing of the inverse propagator at the pole,the residue can be computed as Except for the multiplicative renormalization constant Z D . Not to be confused with the longitudinal mass parameter of Chapter 1. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles R p ( ξ ) = m ( ξ ) p iZ D F (cid:48) ( − p /m ( ξ )) + ξ F (cid:48) ξ ( − p /m ( ξ )) (3.18)where F (cid:48) ( s ) and F (cid:48) ξ ( s ) are the derivatives of F ( s ) and F ξ ( s ) . By the Nielsen identities,we should have R p ( ξ ) = R p ( ξ ) for any two gauges ξ and ξ . However, observe thatthe multiplicative renormalization constant Z D can actually be given arbitrary values inarbitrary gauges: we could set Z D = Z D ( ξ ) for any positive real function Z D ( ξ ) withoutviolating the principles of renormalization. It follows that the residues of the propagatorcan actually be defined only modulo a positive real function, so that the Nielsen identitiescannot ensure that the modulus |R p ( ξ ) | of the residue is the same in different gauges.Nonetheless, since the phase of the residue cannot be modified by a change of Z D , theNielsen identities still apply to the phase of the residue, constraining the latter to begauge-invariant and the residue to be of the form R p ( ξ ) = − i |R p ( ξ ) | e iϕ (3.19)for some real, gauge-independent phase ϕ .In massive perturbation theory, for an arbitrary choice of the free parameters eq. (3.19)is not satisfied. Indeed, if we denote by θ p ( ξ ) the difference between the phases of theresidues at p in the gauge ξ and in the Landau gauge, namely, θ p ( ξ ) = Arg (cid:40) R p ( ξ ) R p ( ξ = 0) (cid:41) = Arg (cid:40) F (cid:48) ( − p /m L ) F (cid:48) ( − p /m ( ξ )) + ξ F (cid:48) ξ ( − p /m ( ξ )) (cid:41) (3.20)where m ( ξ ) is computed by the procedure described on the previous page starting fromthe value of F L that produced p , a numerical evaluation of the phases informs us that θ p ( ξ ) in general is not equal to zero, as it should be as a consequence of the Nielsen iden-tities. Therefore, in the framework of massive perturbation theory, the Nielsen identitiesimpose a very strict condition on the value of F L : F L must be such that, once thequantities p , F ( ξ ) and m ( ξ ) have been derived by the procedure of the previous page,the poles of the gauge-dependent propagator resulting from these quantities have residueswith gauge-invariant phases. Equivalently, F L must realize the vanishing of the function θ p ( ξ ) associated to the poles of the propagator. Observe that a value of F L with such aproperty is not by any means guaranteed to exists. Since we are dealing with an approxi-mation to the full theory, we should not expect θ p ( ξ ) to be exactly zero for any value of F L . Nonetheless, in the next section we will see that there actually are values of F L forwhich | θ p ( ξ ) | is as small as three parts in one thousand. For these values the phases ofthe residues of the gluon propagator are are gauge-invariant for all practical purpose.In summary, the strategy that we will adopt in the next section to obtain the optimalvalue of F L and, together with it, the position p of the pole, the optimal gauge-dependentrenormalization parameter F ( ξ ) and the optimal gauge-dependent mass parameter m ( ξ ) is the following: 1. we will choose some arbitrary value for F L and solve eq. (3.15) for thegauge-invariant position p of the pole, which we will express as an adimensional complex The factor of − i comes from the definition of the gluon propagator in Minkowski space. We remark that in the context of this derivation the Landau gauge does not have any special status:it is just our gauge of choice for the definition of the mass parameter m which needs to be supplied tothe theory as an external input. iorgio Comitini number z times an external mass parameter m L , 2. we will solve eq. (3.14) for differentvalues of ξ in order to obtain the functions F ( ξ ) and m ( ξ ) in an arbitrary gauge, 3. wewill use these functions to compute θ p ( ξ ) . The optimal value of F L will be the one whichresults in a θ p ( ξ ) that is as close as possible to zero, implying that for this value the gaugeinvariance of both the position and the phases of the residues of the poles of the gluonpropagator is achieved. 82 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles F L In order to carry out the programme of Sec.3.1.2, the first thing we need to do is studythe analytic structure of the gluon propagator in the Landau gauge as a function of F L ,the additive renormalization constant at ξ = 0 . What we are interested in is the numberand position of the poles of the propagator as F L is tuned across the real numbers. Re-call that in the Landau gauge the poles of the propagator are found by solving the equation F ( − p /m L ) + F L = 0 (3.21)where m L is the mass parameter at ξ = 0 , to be provided as an external input. Since F ( s ) is a complicated function of s , eq. (3.21) can only be solved numerically. However, in orderto figure out the general structure of its solutions, we can start from a graphical analysisof the zero set of the sum on its left hand side.Eq. (3.21) can be expressed in terms of its real and imaginary parts as R e { F ( − p /m L ) } + F L = 0 I m { F ( − p /m L ) } = 0 (3.22)The imaginary part of the equation does not depend on F L ; therefore, regardless of thevalue of F L , the poles of the propagator are constrained to lie in the one-dimensional sub-set of the complex plane defined by I m { F } = 0 . This subset is shown in Fig.22. -4 -3 -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) Figure 22: Zero set of the imaginary part of F ( − p /m L ) . The positive real axis is anartifact of the graphical algorithm and is not actually part of the zero set.83 iorgio Comitini The zero set of I m { F ( − p /m L ) } consists of two concentric subsets with the topology ofa circle, together with the negative real axis ( F ( s ) is real for positive s ). The positivereal axis is not actually part of the set: since F ( s ) contains a natural logarithm of s , F ( − p /m L ) has a branch cut for positive real p ’s; the small imaginary discontinuity ofthe function across the branch cut causes the graphical algorithm to erroneously includethe positive real axis in the zero set of the function.The zero set of I m { F } is symmetric with respect to the real axis. This is a consequence ofthe identity F ( s ) = F ( s ) – where the bar denotes complex conjugation – which holds for F since the latter is a sum of products of logarithms, square roots and rational functions, allof which have real coefficients. The same identity also tells us that if p solves eq. (3.21),then p does too: the gluon propagator has complex conjugate poles, which must thus bereal or come in conjugate pairs.In order to be poles of the propagator, the zeros of I m { F } must also be zeros of R e { F } + F L .The latter condition depends on F L , whose value therefore influences the position of thepoles. Actually, it turns out that the number of poles also depends on the value of F L .Again, this can be shown through a graphical analysis. -12 -10 -8 -6 -4 -2 0 2Re(p /m )-4-3-2-1 0 1 2 3 4 I m ( p / m ) F = -2.80 -12 -10 -8 -6 -4 -2 0 2Re(p /m )-4-3-2-1 0 1 2 3 4 I m ( p / m ) F = -2.50-12 -10 -8 -6 -4 -2 0 2Re(p /m )-4-3-2-1 0 1 2 3 4 I m ( p / m ) F = -2.20 -12 -10 -8 -6 -4 -2 0 2Re(p /m )-4-3-2-1 0 1 2 3 4 I m ( p / m ) F = -2.06 Figure 23: Case 1 – zero sets of the real and imaginary part of F ( − p /m L ) + F L for F L = − . , − . , − . , − . . The propagator has two negative real poles.84 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Case 1: F L (cid:47) − . .The zero sets of the real and imaginary part of F ( − p /m L ) + F L are shown in Fig.23for different values of F L < − . . The poles of the propagator are found at the intersec-tion of the two sets.For F L (cid:47) − . , the propagator has two poles on the negative real axis. This is an imme-diate consequence of F ( s ) being real and greater than approximately . for positive real s : as F L is tuned below − . , the graph of F ( − p /m L ) + F L (for p ∈ R − ) intersects thehorizontal axis in two points, as shown in Fig.24. Since F ( − p /m L ) tends to + ∞ bothin the p → and in the p → −∞ limit, this behavior is easily seen to be shared bypropagators with arbitrarily negative F L . -1-0.5 0 0.5 1 1.5 2 2.5 3-14 -12 -10 -8 -6 -4 -2 0 F (- p / m ) + F p /m F = -2.80F = -2.50F = -2.20F = -2.06 Figure 24: Case 1 – graph of the function F ( − p /m L ) + F L for negative real p at F L = − . , − . , − . , − . .Case 2: F L (cid:39) − . , F L < .The zero sets of the real and imaginary part of F ( − p /m L ) + F L are shown in Fig.25for different values of F L ∈ ] − . , . The intersections at p = 0 and I m ( p ) = 0 , R e ( p ) > are not actual poles: we know F ( s ) to be infinite at s = 0 , which excludes thesolution p = 0 , and we know F ( s ) to have a branch cut for negative real s , which excludesthe positive real solutions.As F L is tuned above approximately − . , the function F ( − p /m L )+ F L becomes strictlypositive for negative real p , so that the negative real poles of Case 1 disappear. In theirplace, the propagator develops two complex conjugate poles. In this range of F L , thereal part of the conjugate poles can be either negative or positive, implying that the realpart of (cid:112) p can be either less or greater than its imaginary part. The value of F L forwhich the real part of the poles is zero – equivalently, for which R e ( (cid:112) p ) = I m ( (cid:112) p ) –is F L ≈ − . . As F L approaches zero, the zero set of R e { F ( − p /m L ) + F L } starts toshrink. 85 iorgio Comitini -3 -2 -1 0 1 2 3 4 5 6Re(p /m )-6-4-2 0 2 4 6 I m ( p / m ) F = -2.04 -3 -2 -1 0 1 2 3 4 5 6Re(p /m )-6-4-2 0 2 4 6 I m ( p / m ) F = -1.50-3 -2 -1 0 1 2 3 4 5 6Re(p /m )-6-4-2 0 2 4 6 I m ( p / m ) F = -1.20 -3 -2 -1 0 1 2 3 4 5 6Re(p /m )-6-4-2 0 2 4 6 I m ( p / m ) F = -0.90-3 -2 -1 0 1 2 3 4 5 6Re(p /m )-6-4-2 0 2 4 6 I m ( p / m ) F = -0.50 -3 -2 -1 0 1 2 3 4 5 6Re(p /m )-6-4-2 0 2 4 6 I m ( p / m ) F = -0.05 Figure 25: Case 2 – zero sets of the real and imaginary part of F ( − p /m L ) + F L for F L = − . , − . , − . , − . , − . , − . . The propagator has two complex conjugatepoles.Case 3: F L > , F L (cid:47) +0 . .The zero sets of the real and imaginary part of F ( − p /m L ) + F L are shown in Fig.26for different values of F L ∈ ] 0 , .
23 [ . Again, the intersections at p = 0 and I m ( p ) = 0 , R e ( p ) > are not actual poles.The R e { F + F } = 0 set intersects both of the circular subsets of I m { F + F } = 0 .Therefore the propagator has four complex poles, conjugated in pairs.86 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +0.05 -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +0.20 Figure 26: Case 3 – zero sets of the real and imaginary part of F ( − p /m L ) + F L for F L = +0 . , +0 . . The propagator has two pairs of complex conjugate poles. -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +0.26 -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +0.75-2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +2.25 -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +7.25 Figure 27: Case 4 – zero sets of the real and imaginary part of F ( − p /m L ) + F L for F L = +0 . , +0 . , +2 . , +7 . . The propagators has two complex conjugate poles.87 iorgio Comitini Case 4: F L (cid:39) +0 . , F L (cid:47) +9 . .The zero sets of the real and imaginary part of F ( − p /m L ) + F L are shown in Fig.27for different values of F L ∈ ] 0 . , .
20 [ . The intersections at p = 0 and I m ( p ) = 0 , R e ( p ) > are not actual poles.As F L is tuned above approximately 0.23, one of the two connected components of R e { F + F } = 0 ceases to intersect the zero set of I m { F + F } , then shrinks to zero and disap-pears: in this range the propagator has two complex conjugate poles. As F L approachesapproximately . the second connected component of R e { F + F } = 0 also shrinks tozero.Case 5: F L (cid:39) +9 . .The zero sets of the real and imaginary part of F ( − p /m L ) + F L are shown in Fig.28for a single value of F L (cid:39) . . The equation R e { F ( − p /m L ) + F L } = 0 has no solutions.Therefore the propagator has no poles. -2 -1 0 1 2 3 4Re(p /m )-2-1.5-1-0.5 0 0.5 1 1.5 2 I m ( p / m ) F = +10.00 Figure 28: Case 5 – zero sets of the real and imaginary part of F ( − p /m L ) + F L for F L = +10 . . The zero set of R e { F ( − p /m L ) + F L } is empty, therefore the propagatorhas no poles.The graphical analysis of the zero sets of F ( − p /m L ) + F L informs us that the topol-ogy of the poles of the gluon propagator is very much dependent on the value of F L . As F L is tuned from arbitrarily negative values to arbitrarily positive values, the propaga-tor goes from having two real negative poles (Euclidean poles), to having two complexconjugate poles, to having two pairs of complex conjugate poles, to having two complexconjugate poles again, to having no poles at all. In principle, any of these solutions mightyield the true analytic structure of the gluon propagator. However, we have reasons to be-88 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles lieve that only the values of Case 2, namely, F L ∈ ] − . , , are reasonable parametersfor the propagator. The motivations for this are the following.First of all, consider Case 1: F L (cid:47) − . . The poles of Case 1 are real negative be-cause for these values of F L the function F ( s ) + F L evaluated at positive real s goes frombeing positive, to being negative, to being positive again as s is tuned from to + ∞ –cf. Fig.24. Observe that s ∈ R +0 is precisely the domain of definition of the Euclideanpropagator, which therefore, in this range of F L , not only becomes infinite at two finitevalues of the Euclidean momentum, but is also negative in some momentum interval. Thisis not a reasonable behavior for the Euclidean propagator. Therefore we must concludethat ] − ∞ , − .
05 [ is not an acceptable range of values for F L .As for Cases 3 and 4, we observe that the poles resulting from F L > lie in regions of I m { F + F } = 0 which have a fairly bizarre shape and are quite flattened against thebranch cut on the positive real axis. Now, since we are working with an approximationto the exact propagator, we might expect the true zero set of the imaginary part of theinverse propagator to be somewhat shifted from I m { F + F } = 0 . Due to the closenessof the aforementioned regions to the real axis, a small shift originating in higher orderradiative corrections to the propagator may cause the F L > poles to completely disap-pear: because of their position and of the shape of the set to which they belong, such polescannot be guaranteed to be genuine, so much as an artifact of the massive expansion. Inaddition to this, we could also argue that the values F L > fail to reproduce the Euclideanpropagator computed on the lattice: in Sec.2.3.2 we found that the value of F L which bestfits the lattice data in the Landau gauge is F L = − . , which is quite far away from thepositive values of Cases 3 to 5.These considerations lead us to conclude that the most sensible range of values for theadditive renormalization constant F L is that of Case 2, namely, ] − . , . As we willsee in the following section, this range contains values of F L for which the phase difference θ ( ξ ) is smaller than . in absolute value.Having restricted the optimal range of F L to ] − . , , we can now move on to thenumerical solution of eq. (3.15). The position of the poles as a function of F L is reportedin Tab.3 for different values of the renormalization constant in the aforementioned range. F L p /m L − . − . ± . i − . − . ± . i − . − . ± . i − . − . ± . i − .
25 0 . ± . i − .
00 0 . ± . i − .
75 0 . ± . i − .
50 0 . ± . i − .
25 0 . ± . i − .
10 0 . ± . i Table 3: Poles of the gluon propagator as a function of F L ∈ ] − . , .89 iorgio Comitini I m ( p / m ) Re(p /m ) Figure 29: Poles of the gluon propagator as a function of F L ∈ ] − . , . Left to right: F = − . , − . , − . , − . , − . , − . , − . , − . , − . , − . .In Fig.29 we show the location of the poles of Tab.3 in the complex plane . As F L is tuned from − . to − . , the real part of the complex conjugate poles grows from − . to . (in units of m L ); their imaginary part, on the other hand, grows in abso-lute value from . to . at F L = − . and then falls back to . . These poles arenot guaranteed to have residues whose phases – in compliance with the Nielsen identities –are gauge-invariant. In order to be able to compute the phases of the residues as a functionof the gauge parameter ξ , we first need to know the gauge-dependent parameter functions F ( ξ ) and m ( ξ ) . These will be derived in the next section. F ( ξ ) , m ( ξ ) and θ ( ξ ) Having found the location of the poles of the gluon propagator as F L is tuned across theinterval ] − . , , we are now in a position to compute the gauge-dependent parameterfunctions F ( ξ ) and m ( ξ ) associated to such poles. Recall that the latter are defined asthe renormalization and mass parameters which lead to the poles in the gauge ξ being inthe same position of those in the Landau gauge. Equivalently, F ( ξ ) and m ( ξ ) can bedefined as the functions which solve the equation F ( − p /m ( ξ )) + ξ F ξ ( − p /m ( ξ )) + F ( ξ ) = 0 ∀ ξ (3.23)where p has been computed by fixing some value for the Landau gauge renormalizationparameter F L . Observe that since ξ , F ( ξ ) and m ( ξ ) are real, and since F ( s ) = F ( s ) , F ξ ( s ) = F ξ ( s ) , this equation yields the same solutions with both p and p as an input. Only the pole with a positive imaginary part is shown in the figure. erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles Therefore its solutions do not depend on the choice of the pole, as it should be .Eq. (3.23) can be solved by isolating its real and imaginary parts. Since F ( ξ ) is real, theimaginary part of the equation depends on m ( ξ ) only and reads I m { F ( − p /m ( ξ )) + ξ F ξ ( − p /m ( ξ )) } = 0 (3.24)This equation can be solved for m ( ξ ) alone, without having to worry about F ( ξ ) . Once m ( ξ ) is known, one can compute F ( ξ ) as F ( ξ ) = − R e { F ( − p /m ( ξ )) + ξ F ξ ( − p /m ( ξ )) } (3.25)Of course, both m ( ξ ) and F ( ξ ) will depend on the value of F L which has been used tocompute p in the first place.In Fig.30 we show the gauge-dependent mass parameter m ( ξ ) – as obtained by numericallysolving eq. (3.24) – as a function of ξ for different values of F L in the range ] − . , .As F L is increased from − . to − . at constant ξ , the value of m ( ξ ) first increasesand then decreases. For F L between approximately − . and − . , the functions m ( ξ ) computed at different F L ’s are almost indistinguishable from one another. F L = − . (red line in figure) is the value for which the mass parameter is both stationary with re-spect to F L at any fixed ξ and closer to m L . This feature can be shown [74] to be a directconsequence of the phase of the residue being gauge independent (or almost so) for theaforementioned value of F L (see ahead). m ( ξ ) decreases with the gauge for every valueof F L . m ( ξ ) / m ξ F = -1.50F = -1.25F = -1.00F = -0.876F = -0.75F = -0.50F = -0.25 Figure 30: Gauge-dependent mass parameter m ( ξ ) as a function of ξ for different valuesof F L in the range ] − . , . In the presence of two pairs of complex conjugate poles, the solutions of eq. (3.23) depend on thechoice of one of the two pairs, so that both pairs cannot, in general, be simultaneously gauge-invariant.This gives us further confidence that Case 3 in Sec.3.2.1 must be discarded. iorgio Comitini -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 F = -1.50F = -1.25F = -1.00F = -0.876F = -0.75F = -0.50F = -0.25 F ( ξ ) ξ Figure 31: Gauge-dependent renormalization constant F L ( ξ ) as a function of ξ for differentvalues of F L in the range ] − . , .In Fig.31 we show the gauge-dependent renormalization constant F L ( ξ ) – as computedfrom eq. (3.25) – as a function of ξ for different values of F L in the range ] − . , . As ξ increases from zero to higher values, the function F ( ξ ) deviates from its value F L in theLandau gauge in a non-monotonic fashion. For F L in the range [ − . , − .
75 ] , F ( ξ ) isnearly equal to F L up to and beyond the Feynman gauge ( ξ = 1 ).The solutions presented in Figg.30-31 for the functions F ( ξ ) and m ( ξ ) show some formof stationarity for values of F L in the range around − . to about − . , suggesting thatsomething interesting might be happening in this interval. As we will see in a moment,this is indeed the case.Having computed F ( ξ ) and m ( ξ ) for different values of F L , we are now in possessionof a family of propagators which in any gauge – modulo an arbitrary multiplicative gauge-dependent constant Z D ( ξ ) – depend only on the mass parameter and additive renormal-ization constant m L and F L in the Landau gauge: (cid:101) D T ( p , ξ ) = − iZ D ( ξ ) p (cid:0) F ( − p /m ( ξ )) + ξF ξ ( − p /m ( ξ )) + F ( ξ ) (cid:1) (3.26)These propagators comply with the Nielsen identities in that their poles are by construc-tion gauge-invariant. It is now time to see whether for any of the values of F L in the range ] − . , the phases of the residues at their poles also are gauge-invariant. Recall thedefinition of θ p ( ξ ) from Sec.3.1.2: θ p ( ξ ) = Arg (cid:40) F (cid:48) ( − p /m L ) F (cid:48) ( − p /m ( ξ )) + ξ F (cid:48) ξ ( − p /m ( ξ )) (cid:41) (3.27) θ p ( ξ ) is simply the phase difference between the residue at p in the gauge ξ and that inthe Landau gauge. From eq. (3.27) one can easily prove that θ p ( ξ ) = − θ p ( ξ ) .92 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles ϑ ( ξ ) ξ F = -1.50F = -1.25F = -1.00F = -0.876F = -0.75F = -0.50F = -0.25 Figure 32: Phase difference of the residues θ ( ξ ) as a function of ξ for different values of F L in the range ] − . , .Since the two complex conjugate poles have opposite phase differences, in what followswe will drop the label p and denote by θ ( ξ ) the phase difference at the pole with a pos-itive imaginary part. Explicit analytic expressions for the functions F (cid:48) ( s ) and F (cid:48) ξ ( s ) thatappear in the definition of θ ( ξ ) can be found in Appendix A.In Fig.32 we show the phase difference θ ( ξ ) – as computed from eq. (3.27) – as a functionof ξ for different values of F L in the range ] − . , . As we can see, for arbitrary valuesof F L in said range, the phase of the residue in the gauge ξ can be quite different fromthat of the Landau gauge. | θ ( ξ ) | can be as high as ◦ or more unless F L is chosen inthe stationarity interval ] − . , − .
75 [ , where it is less than about ◦ . The stationarityinterval contains a value for which the phase difference is exceedingly small up to andbeyond the Feynman gauge ( ξ = 1 ): for F L = − . (red line in figure) one finds that | θ ( ξ ) | < . · − . For all practical purposes, this difference is so small that we can safelystate that the phases of the residues of the gluon propagator computed for this value of F L are gauge-invariant, in compliance with the Nielsen identities.We conclude that from the perspective of gauge invariance F L = − . is the optimalvalue of the additive renormalization constant in the Landau gauge. Notice that this valueis within about from that which in Chapter 2 was found to best fit the lattice data inthe Euclidean space, namely F = − . . Numerical values for the function θ ( ξ ) computed at F L = − . can be found in Appendix B. iorgio Comitini In the previous section we have established that from the point of view of gauge invariance F = − . is the optimal value of the gluon additive renormalization constant in theLandau gauge: starting from this value and employing the procedure laid out in Sec.3.1.2one obtains gauge-dependent functions F ( ξ ) and m ( ξ ) which yield a gluon propagatorthat complies with the Nielsen identities. This section will be devoted to giving somequantitative details about the functions F ( ξ ) and m ( ξ ) and the position and phases ofthe residues of the poles computed by using the optimal value of F L . In what follows m L is the value of the mass parameter in the Landau gauge, which must be provided as anexternal input.Let us start from the poles of the propagator. In Tab.4 we report the position z = p /m L and the phase ϕ of the residue of the gluon poles computed at F L = − . . As in Sec.3.1.2the phase is defined modulo the − i of the propagator in Minkowski space. F L = − . z = 0 . ± . i z = ± . ± . i ϕ = ± . Table 4: Position and phases of the residues of the gluon poles for F L = − . .The gauge-dependent mass parameter m ( ξ ) computed at F L = − . is shown in Fig.33as a function of ξ . The numerical data – presented in Appendix B – can be polynomiallyinterpolated to yield the following approximate expression for m ( ξ ) : m ( ξ ) ≈ m L (cid:0) − . ξ + 0 . ξ (cid:1) (3.28)This approximation is very precise up to and beyond the Feynman gauge ( ξ = 1 ). ( ξ )/m = 1 - 0.39997 ξ + 0.064141 ξ m ( ξ ) / m ξ Figure 33: Optimal mass parameter m ( ξ ) as a function of ξ computed at F L = − . .Dashed red line: polynomial interpolation given by eq.(3.28).94 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles -0.88-0.878-0.876-0.874-0.872-0.87 0 0.2 0.4 0.6 0.8 1F ( ξ ) = -0.8759 - 0.01260 ξ + 0.009536 ξ + 0.009012 ξ F ( ξ ) ξ Figure 34: Optimal additive renormalization constant F ( ξ ) as a function of ξ computedat F L = − . . Dashed red line: polynomial interpolation given by eq.(3.29).The gauge-dependent renormalization constant F ( ξ ) computed at F L = − . is shownin Fig.34 as a function of ξ . The numerical data – presented in Appendix B – can bepolynomially interpolated to yield the following approximate expression for F ( ξ ) : F ( ξ ) ≈ − . − . ξ + 0 . ξ + 0 . ξ (3.29)Again, this approximation is very precise up to and beyond the Feynman gauge ( ξ = 1 ).In the next section we will use the values and functions presented above to investigatethe behavior of the optimized gluon propagator in different gauges. Our results will beshown to be in good agreement with the available lattice data for the Euclidean gluonpropagator in gauges different from the Landau gauge.95 iorgio Comitini Our analysis of the gauge invariance of the poles of the gluon propagator equipped us withprecise values for the position and residues of the poles and with accurate approximationsfor the optimal gauge-dependent functions F ( ξ ) and m ( ξ ) . It is now time to put to useour results and investigate the behavior of the optimized gluon propagator as a functionof both the momentum and the gauge.Let us start from the principal part of the propagator. Recall that the principal partof an analytic function f is the part of its series expansion that contains its poles. If f haspoles at { z k } k with residues {R k } k , then its principal part f | PP is defined as f ( z ) (cid:12)(cid:12)(cid:12) PP = (cid:88) k R k z − z k (3.30)Due to the Nielsen identities, the principal part of the gluon propagator, namely (cid:101) D T ( p ) (cid:12)(cid:12)(cid:12) PP = − i |R| e iϕ p − p + − i |R| e − iϕ p − p (3.31)is constrained to be gauge-invariant modulo an arbitrary gauge-dependent multiplicativefactor. The invariance of (cid:101) D T | PP suggests that the latter may play an important role in thedefinition of physically meaningful quantities; therefore, it is worth to study its behaviorand contrast it with that of the full propagator. Since the gluon propagator has a branchcut at positive p , in order to investigate the momentum-dependence of its principal partwe switch to Euclidean space. Here (cid:101) D T | PP reads m D ~ T ( p E ) p E /m L Full propagatorPrincipal part
Figure 35: Optimized gluon propagator in the Landau gauge ( F = − . ). Black line:full propagator. Red line: principal part of the propagator, with poles and residues givenby Tab.4. |R| = 0 . . .96 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles (cid:101) D T ( p E ) (cid:12)(cid:12)(cid:12) PP = |R| e iϕ p E + p + |R| e − iϕ p E + p (3.32)Of course, |R| depends on the value of the multiplicative renormalization constant Z D . Anumerical evaluation shows that for Z D = 1 the modulus of the residue is |R| = 0 . .In Fig.35 we show the optimized gluon propagator in the Landau gauge together withits gauge-invariant principal part (cid:101) D T | PP , both computed at Z D = 1 . As we can see, atlow momenta the full propagator is slightly suppressed with respect to its principal part.Nonetheless, (cid:101) D T | PP still makes up for the largest part of the propagator. Actually, asshown in Fig.36, most of the difference between (cid:101) D T | PP and the full propagator can beabsorbed into the normalization of the former, to the extent that in practical applicationsone may as well replace the propagator by a normalized version of its principal part.By adding up the contributions due to the two poles in eq.(3.32), we find that (cid:101) D T | PP canbe put in the Gribov-Zwanziger form [83] (cid:101) D T ( p E ) (cid:12)(cid:12)(cid:12) PP = Z GZ p E + M p E + M p E + M Z GZ = 2 R e {R} (3.33)where, with p = M + iγ t = I m {R} / R e {R} (3.34)the mass scales M , M and M are defined as M = M − γ + 2 M γt M = 2 ( M − γ ) M = ( M + γ ) (3.35)In Tab.5 we report the optimized values of the three scales. Observe that t = tan ϕ , sothat with ϕ = 1 . we have t = 3 . . m D ~ T ( p E ) p E /m L Full propagatorPrincipal part (norm.)
Figure 36: Same plot as Fig.35, with the principal part normalized by a factor of . ..97 iorgio Comitini M /m L = 3 . M /m L = 0 . M /m L = 1 . Table 5: Gribov-Zwanziger parameters for the principal part of the gluon propagator.The next thing we want to do is confront our results for the optimized gluon propaga-tor with the predictions of the lattice. In Chapter 2 we saw that the Euclidean gluonpropagator computed in the massive expansion well reproduces the lattice data in theLandau gauge provided that the parameters given in Tab.2 (Sec.2.3.2) are used for its def-inition. Now, the value of F which was obtained from the fit of the lattice data, namely F = − . , falls within from the optimized value obtained by enforcing the Nielsenidentities, F = − . . Therefore we expect that by fixing F = − . and fitting theother free parameters of the propagator – Z D and m – to the lattice data, we will obtaina curve which is in good agreement with the lattice results. This is indeed the case, aswe show in Fig.37 by displaying the lattice data of ref. [61] together with the results ofmassive perturbation theory both for the fitted value of F and for its gauge-optimizedvalue. The value of m which is found to best fit the data at F = − . is m = 0 . GeV – see Tab.6. This is to be compared with the value obtained by freely fitting F to thelattice data, namely m = 0 . GeV (Tab.2), which is just 0.3% lower. Since the optimizedparameters are extremely close to the fitted parameters, the optimized propagator is nearlyundistinguishable from the fitted propagator. This major result validates the method ofoptimization by gauge invariance and illustrates the extent to which the massive expansionis able to make predictions from first principles about the infrared behavior of Yang-Millstheory. In Tab.7 we report the dimensionful position of the optimized gluon poles obtainedby using m = 0 . GeV as the value of the mass parameter in the Landau gauge. D ~ T ( ξ = ) ( p E ) ( G e V - ) p E (GeV)Lattice: Duarte et al.Mass. exp.: FitMass. exp.: Optimized Figure 37: Gluon propagator in the Landau gauge. Black solid line: massive expansion with F fitted from the lattice data. Green dashed line: massive expansion with F optimizedby gauge invariance ( F = − . ). Data points: lattice propagator from ref. [61]..98 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles F m (GeV) Z D − .
876 0 . . Table 6: Parameters found from the fit of the lattice data of ref. [61] to the gluon propagatorof the massive expansion at fixed F = − . in the range − GeV. F L = − . p = (0 . ± . i ) GeV p = ( ± . ± . i ) GeVTable 7: Position of the gluon poles for F L = − . and m L = 0 . . m L was fixed byfitting the optimized gluon propagator to the lattice data of ref. [61].As for the gauge dependence of the optimized propagator, in order to test our results wewill again resort to a comparison with the lattice data. It must be noted, however, thatthe lattice calculations for Yang-Mills theory are not yet very accurate outside the Landaugauge; therefore any comparison for ξ (cid:54) = 0 must be made with caution. In what follows wewill employ the lattice data of ref. [82] (Bicudo et al.).In Fig.38 we show the lattice data of ref. [82] for the Euclidean gluon propagator in thecovariant gauges ξ = 0 . , . , together with the corresponding propagators computed inthe massive expansion and optimized by gauge invariance. In addition, we also show ouroptimized results for the Feynman gauge ( ξ = 1 ) propagator, for which no lattice data isyet available. The comparison between different gauges is made by renormalizing the valueof the propagator at the scale µ = 4 . GeV. As we can see, our propagators follow thegeneral trend of the lattice. As the gauge is increased at fixed, low Euclidean momenta thepropagator is suppressed. At higher momenta, on the other hand, the propagator becomesless and less dependent from the gauge.In Fig.39 we show the lattice data of ref. [82] for the ratio of the Euclidean gluon propagatorin the gauges ξ = 0 . , . to the propagator in the Landau gauge, together with thepredictions of the optimized massive perturbation theory up to the Feynman gauge ( ξ = 1 ).The large errors to which the lattice calculations are still subject in gauges different fromthe Landau gauge are clearly displayed in the figure. Nonetheless, again, our predictionscan be seen to follow the general trend of the lattice.99 iorgio Comitini D ~ T ( p E ) p E2 (GeV ) ξ = 0.0 ξ = 0.5 ξ = 1.0 ξ = 0.0 ξ = 0.5 Figure 38: Euclidean gluon propagator in different gauges, renormalized at µ = 4 . GeV.Data points from ref. [82] (Bicudo et al.). . D ~ T ( p E ) / D ~ T ξ = ( p E ) p E (GeV) ξ = 0.1 ξ = 0.5 ξ = 1.0 Figure 39: Ratio of the Euclidean gluon propagator in different gauges to the propagatorin the Landau gauge, renormalized at µ = 4 . GeV. Data points from ref. [82] (Bicudoet al.). .100 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles
In this section we will discuss a renormalization-scheme-dependent method for fixing thevalue of the ghost additive renormalization constant G starting from the value of thegluon renormalization constant F . For simplicity, we will limit our discussion to theLandau gauge ( ξ = 0 ); our results can be straightforwardly generalized to an arbitrarycovariant gauge. The contents of this section have been recently presented in ref. [84].In Sec.2.2.1 and 2.3.1 we saw that in the massive expansion the Landau gauge ghost andtransverse gluon dressed propagators can be expressed as – cf. eqq.(2.64) and (2.111) – (cid:101) G ( p ) = ip [1 + α ( G ( s ) + g )] (3.36) (cid:101) D T ( p ) = − ip [1 + α ( F ( s ) + f )] (3.37)where α = 3 N α s / π and g and f are renormalization-scheme-dependent additive renor-malization constants related to the constants G and F by the equations G = 1 α + g F = 1 α + f (3.38)In an arbitrary renormalization scheme, g and f are defined by the values (cid:101) G ( − µ ) and (cid:101) D T ( − µ ) of the propagators at the renormalization scale p = − µ : g ( µ ) = 1 α ( µ ) (cid:104)(cid:0) iµ (cid:101) G ( − µ ) (cid:1) − − (cid:105) − G ( µ /m ) (3.39) f ( µ ) = 1 α ( µ ) (cid:104)(cid:0) − iµ (cid:101) D T ( − µ ) (cid:1) − − (cid:105) − F ( µ /m ) (3.40)where α ( µ ) is the coupling at the scale µ in the chosen renormalization scheme. Fromthe above equations it follows that, as soon as we fix the renormalization conditions forthe propagators – i.e. the values of (cid:101) G ( − µ ) and (cid:101) D T ( − µ ) –, g ( µ ) and f ( µ ) are notindependent from one another, but rather are related to each other by the value of therenormalized coupling. Indeed, eq. (3.40) allows us to express the inverse coupling as α ( µ ) = (cid:104)(cid:0) − iµ (cid:101) D T ( − µ ) (cid:1) − − (cid:105) − (cid:0) F ( µ /m ) + f ( µ ) (cid:1) (3.41)which can be plugged back into eq. (3.39) to yield g ( µ ) = (cid:0) iµ (cid:101) G ( − µ ) (cid:1) − − (cid:0) − iµ (cid:101) D T ( − µ ) (cid:1) − − (cid:0) F ( µ /m ) + f ( µ ) (cid:1) − G ( µ /m ) (3.42)In what follows, in order to fix the value of g starting from the value of f – hence G starting from F , cf. eq. (3.38) –, we will adopt a renormalization scheme termed ScreenedMOMentum subtraction scheme (SMOM) [84]. The SMOM scheme is defined by setting (cid:101) G ( − µ ) = i − µ (cid:101) D T ( − µ ) = − i − µ − m (3.43)101 iorgio Comitini where m is the gluon mass parameter, so that at the scale µ the ghost and gluon propa-gators are given respectively the values of a bare massless propagator and a bare massivepropagator. The mass parameter in (cid:101) D T ( − µ ) prevents the latter from diverging at lowrenormalization scales, hence the name of the scheme. In the SMOM scheme we have [ iµ (cid:101) G ( − µ )] − = 1 [ − iµ (cid:101) D T ( − µ )] − = 1 + m µ (3.44)It follows from eqq. (3.41), (3.42) and (3.44) that the SMOM coupling constant can beexpressed in terms of F ( µ /m ) and f ( µ ) as α SMOM ( µ ) = m µ (cid:0) F ( µ /m ) + f ( µ ) (cid:1) − (3.45)whereas g ( µ ) is related to the value of the gluon function G ( s ) at the renormalizationscale µ by the simple relation g ( µ ) = − G ( µ /m ) (3.46)In terms of the renormalization constants G and F at the scale µ – cf. eq. (3.38) –, theabove equations read α SMOM ( µ ) = (cid:18) m µ (cid:19) [ F ( µ /m ) + F ( µ )] − (3.47) G ( µ ) = (cid:18) m µ (cid:19) − [ F ( µ /m ) + F ( µ )] − G ( µ /m ) (3.48)Eqq. (3.47) and (3.48) completely fix the value of the SMOM coupling α SMOM and ghostrenormalization constant G at the scale µ starting from the knowledge of the ghost renor-malization constant F at the scale µ .In this chapter the value of F in the Landau gauge was optimized by the requirementof the gauge invariance of the position and phases of the residues of the poles of the gluonpropagator. Our derivation did not assume any specific renormalization condition for thegluon propagator and thus provided us with a renormalization-scheme-independent valuefor F , namely, F = − . . Now, from eq. (3.48) it is clear that if F does not dependon the renormalization scale, then the constant G does. Therefore, in the framework ofthe gauge-optimized massive perturbation theory renormalized in the SMOM scheme, theghost constant G is predicted to be dependent on the renormalization scale.In ref. [84] it was shown that a renormalization-scale-dependent G , in general, spoilsthe multiplicative renormalizability of the ghost propagator. This apparent drawback ofthe gauge-optimized SMOM framework was exploited to fix the value of G accordingto Stevenson’s principle of minimal sensitivity [85]. The principle of minimal sensitivitystates that the best approximation to a Green function renormalized in a momentum-subtraction-like scheme in which some parameter is required to be independent from therenormalization scale µ is obtained by choosing the renormalization scale in such a waythat the aforementioned parameter is less sensitive to a variation of µ . In other words, therenormalization scale should be chosen so that the parameter is stationary with respectto µ . In particular, in the gauge-optimized SMOM framework, the principle of minimalsensitivity requires that we renormalize the propagators at the scale µ = µ (cid:63) such that G (cid:48) ( µ (cid:63) ) = 0 , where the prime denotes a derivative with respect to µ .102 erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles = − ← G = 0.14524 G ( t ) t = µ / m Figure 40: SMOM function G ( µ ) as a function of the adimensionalized renormalizationscale squared µ /m for the optimal value F = − . . The minimum G ( µ (cid:63) ) = 0 . is found at the renormalization scale µ (cid:63) = 1 . m .In Fig.40 we display the SMOM function G ( µ ) – as defined by eq. (3.48) – computed atfixed F = − . . G ( µ ) can be seen to have a pronounced minimum at a renormalizationscale µ (cid:63) ≈ m . A numerical evaluation shows that µ (cid:63) = 1 . m and G ( µ (cid:63) ) = 0 . [84].Observe that the optimal renormalization scale µ (cid:63) derived by the principle of minimalsensitivity is remarkably close to the value of the gluon mass parameter. Moreover, theoptimal value G = G ( µ (cid:63) ) is within 0.8% from the value G = 0 . which was foundto best fit the lattice data in Chapter 2 (see Tab.1 in Sec.2.2.2). Since the optimized mass m was also found to be very close to the fitted mass – cf. Sec.3.3.1 –, we expect theoptimized ghost propagator to accurately reproduce the lattice data. This is confirmedby Fig.41, where we plot the Euclidean Landau gauge ghost dressing function p E (cid:101) G ( p E ) obtained by setting G = G ( µ (cid:63) ) = 0 . and m = 0 . GeV – cf. Tab.8 – togetherwith the lattice results of ref. [61] and the fitted dressing function. As we can see, thefitted dressing function and the optimized dressing function are almost indistinguishablefrom one another. Therefore a comparison with the lattice data validates the method ofoptimization by minimal sensitivity. m (GeV) G Z G .
656 0 . . Table 8: Parameter Z G found from the fit of the lattice data of ref. [61] for the Landaugauge ghost propagator in the range − GeV at fixed m = 0 . GeV and G = 0 . .103 iorgio Comitini p E G ~ ( ξ = ) ( p E ) p E (GeV)Lattice: Duarte et al.Mass. exp.: FitMass. exp.: Optimized Figure 41: Ghost dressing function in the Landau gauge. Black solid line: massive expan-sion with G and m fitted from the lattice data. Green dashed line: massive expansionwith G optimized in the SMOM scheme by the principle of minimal sensitivity and m optimized by gauge-invariance. Data points: lattice dressing function from ref. [61].104 onclusions In this thesis we have addressed the issue of dynamical mass generation from differentperspectives and presented a new perturbative framework for making computations inlow-energy Yang-Mills theory, the massive perturbative expansion.In Chapter 1 we performed a GEP analysis of the perturbative vacuum of Yang-Mills the-ory. The Yang-Mills GEP was computed at zero longitudinal gluon mass, in compliancewith the non-perturbative transversality of the gluon one-particle-irreducible polarization.By analyzing the influence of the ghost contribution to the GEP, we have concluded that inorder for the Gaussian Effective Potential at the gluon minimum to be a good variationalestimate of the exact vacuum energy density of Yang-Mills theory the mass of the ghostsmust be set to zero. By solving the gap equation of the GEP we have discovered thatthe massless perturbative vacuum of the transverse gluons is unstable towards a massivevacuum, giving evidence for the occurrence of dynamical mass generation and indicatingthat a non-standard perturbation theory that treats the transverse gluons as massive al-ready at tree-level could be more suitable for making calculations in low-energy Yang-Millstheory. In such a framework, both the ghosts and the longitudinal gluons are to be treatedas massless.In Chapter 2 we formulated the massive perturbative framework and used it to computethe ghost and gluon dressed propagators to one loop. Massive perturbation theory wasdefined by a shift of the expansion point of the Yang-Mills perturbative series, achievedby adding a mass term for the transverse gluons to the kinetic Yang-Mills Lagrangianand subtracting the same term from the interaction Lagrangian. While leaving the totalaction and physical content of the theory unchanged, the shift was shown to modify theFeynman rules of the diagrammatic expansion by replacing the massless transverse baregluon propagator of ordinary perturbation theory with a massive propagator and givingrise to a new two-gluon vertex, proportional to the gluon mass parameter squared m . Theone-loop ghost and gluon propagators computed in the massive framework were shown tobe free of spurious, non-renormalizable mass divergences and to have the correct UV be-havior, matching the results of ordinary perturbation theory in the high-energy limit. Atlow energies, the gluon propagator was shown to develop a dynamical mass by way of anon-trivial mechanism that involves the gluon loops and the gluon tadpoles. The ghostpropagator, on the other hand, was shown to remain massless. In the IR we compared ourresults with the Euclidean lattice data in the Landau gauge and found that the two agreeto a high degree of accuracy. In Chapter 2 we also discussed the issue of the predictivityof the massive framework. We concluded that one of the free parameters of the massiveexpansion must be fixed in order to make its results as predictive as those obtained instandard perturbation theory.In Chapter 3 we advanced a method for optimizing the massive expansion by the require-ment of the gauge-invariance of the position and phases of the residues of the poles of thedressed gluon propagator, in compliance with the Nielsen identities. This requirement was105 iorgio Comitini shown to be sufficient to fix the value of the gluon additive renormalization constant in theLandau gauge to F = − . , thus reducing the number of free parameters of the gluonpropagator computed in the massive expansion back to two and restoring the predictivityof the massive approach. The optimization procedure left us with an expression for thedressed gluon propagator in a general covariant gauge whose principal part – modulo agauge-dependent multiplicative renormalization factor – is gauge-invariant. A comparisonwith the lattice data allowed us to fix the optimal value of the gluon mass parameter to m = 0 . GeV and the position of the complex conjugate poles of the gluon propagatorto p = ( ± . ± . i ) GeV. The optimal value of the gluon renormalization constant F was then used to determine the optimal value of the gluon additive renormalizationconstant G in the SMOM renormalization scheme according to Stevenson’s principle ofminimal sensitivity. The optimal value of G was found to be G = 0 . . The optimizedvalues of the gluon and ghost renormalization constants were shown to agree with thoseobtained from a fit to the lattice data to less than 1%.The massive perturbative expansion is able to provide a clear picture of the infrared be-havior of Yang-Mills theory from first principles and without the need of any externalphenomenological parameter other than the mass scale of the theory. The main achieve-ment of the massive framework is the prediction that at low energies the gluons acquirea dynamically generated transverse mass of the order of the QCD scale. Although thisresult had already been anticipated through SDE methods [41], first proved numericallythrough lattice calculations [53–61] and built into phenomenological massive models ofQCD [86–88], the massive approach to perturbation theory is the first analytical methodto prove it from first principles, employing only elementary quantum-field-theoretic tech-niques.The analytic expressions derived in the massive framework can be continued to the com-plex plane both to study the behavior of the propagators in the Euclidean space and toseek for complex poles which, being gauge invariant, might be directly related to physicalobservables [89]. As a matter of fact, a knowledge of the analytic structure of the gluonpropagator alone is sufficient to gain significant insight on the low-energy dynamics of thegluons. For instance, the existence of two complex conjugate poles for the gluon propaga-tor – which is predicted by the gauge-optimized massive expansion – was shown in [74] toresult in a violation of the positivity of the gluon spectral function, a feature which in theliterature has been linked to the problem of gluon confinement [90].The success of the massive perturbative expansion in accurately reproducing the latticedata in the Euclidean space gives us confidence in the general validity of the method andleads us to present it as a viable framework for doing calculations in low energy Yang-Millstheory. 106 ppendix A. Residues of the gluon propagator: the functions F (cid:48) ( s ) and F (cid:48) ξ ( s ) In the framework of massive perturbation theory in order to compute the residues of thegluon propagator at its poles one needs to know the derivatives of the functions F ( s ) and F ξ ( s ) which we have defined in Sec.2.3.1. These are given by F (cid:48) ( s ) = − s + 172 [ L (cid:48) a ( s ) + L (cid:48) a ( s ) + L (cid:48) a ( s ) + R (cid:48) ( s )] (A.1) F (cid:48) ξ ( s ) = s + 6 s − s ln(1 + s ) −
16 ln s + (1 − s )(1 − s )6 s (1 + s ) + 13 s − s − (A.2)where L (cid:48) a ( s ) = 6 s − s − s + 80 s + 144 s ( s + 4) (cid:114) s + 4 s ln (cid:18) √ s + 4 − √ s √ s + 4 + √ s (cid:19) (A.3) L (cid:48) b ( s ) = 4(1 + s ) s (3 s − s + 10 s − s + 3) ln(1 + s ) (A.4) L (cid:48) c ( s ) = − s ln s (A.5) R (cid:48) ( s ) = 12 s + 106 s − s (A.6)In terms of F (cid:48) ( s ) and F (cid:48) ξ ( s ) , the residue R p ( ξ ) of the gluon propagator at p in the gauge ξ can be expressed as – cf. Sec.3.1.2 – R p ( ξ ) = m ( ξ ) p iZ D F (cid:48) ( − p /m ( ξ )) + ξ F (cid:48) ξ ( − p /m ( ξ )) (A.7)where m ( ξ ) is the gauge-dependent mass parameter in the gauge ξ .107 iorgio Comitini B. Gauge-optimized functions F ( ξ ) and m ( ξ ) for the gluonpropagator: numerical data In this Appendix we report the numerical values of the gauge-dependent functions F ( ξ ) and m ( ξ ) , as obtained by the optimization procedure described in Sec.3.1.2 for the value F L = − . . The third column in the table contains the phase difference θ ( ξ ) betweenthe residue at the pole in the gauge ξ and that in the Landau gauge ( ξ = 0 ). ξ m ( ξ ) /m L F ( ξ ) θ ( ξ )0 .
00 1 . − . . .
01 0 . − . − . .
02 0 . − . − . .
03 0 . − . − . .
04 0 . − . − . .
05 0 . − . − . .
06 0 . − . − . .
07 0 . − . − . .
08 0 . − . − . .
09 0 . − . − . .
10 0 . − . − . .
11 0 . − . − . .
12 0 . − . − . .
13 0 . − . − . .
14 0 . − . − . .
15 0 . − . − . .
16 0 . − . − . .
17 0 . − . − . .
18 0 . − . − . .
19 0 . − . − . .
20 0 . − . − . .
21 0 . − . − . .
22 0 . − . − . .
23 0 . − . − . .
24 0 . − . − . .
25 0 . − . − . .
26 0 . − . − . .
27 0 . − . − . .
28 0 . − . − . .
29 0 . − . − . erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles ξ m ( ξ ) /m L F ( ξ ) θ ( ξ )0 .
30 0 . − . − . .
31 0 . − . − . .
32 0 . − . − . .
33 0 . − . − . .
34 0 . − . − . .
35 0 . − . − . .
36 0 . − . − . .
37 0 . − . − . .
38 0 . − . − . .
39 0 . − . − . .
40 0 . − . − . .
41 0 . − . − . .
42 0 . − . − . .
43 0 . − . − . .
44 0 . − . − . .
45 0 . − . − . .
46 0 . − . − . .
47 0 . − . − . .
48 0 . − . − . .
49 0 . − . − . .
50 0 . − . − . .
51 0 . − . − . .
52 0 . − . − . .
53 0 . − . − . .
54 0 . − . − . .
55 0 . − . − . .
56 0 . − . − . .
57 0 . − . − . .
58 0 . − . . .
59 0 . − . . .
60 0 . − . . .
61 0 . − . . .
62 0 . − . . .
63 0 . − . . .
64 0 . − . . iorgio Comitini ξ m ( ξ ) /m L F ( ξ ) θ ( ξ )0 .
65 0 . − . . .
66 0 . − . . .
67 0 . − . . .
68 0 . − . . .
69 0 . − . . .
70 0 . − . . .
71 0 . − . . .
72 0 . − . . .
73 0 . − . . .
74 0 . − . . .
75 0 . − . . .
76 0 . − . . .
77 0 . − . . .
78 0 . − . . .
79 0 . − . . .
80 0 . − . . .
81 0 . − . . .
82 0 . − . . .
83 0 . − . . .
84 0 . − . . .
85 0 . − . . .
86 0 . − . . .
87 0 . − . . .
88 0 . − . . .
89 0 . − . . .
90 0 . − . . .
91 0 . − . . .
92 0 . − . . .
93 0 . − . . .
94 0 . − . . .
95 0 . − . . .
96 0 . − . . .
97 0 . − . . .
98 0 . − . . .
99 0 . − . . erturbation theory of non-perturbative Yang-Mills theory: a massive expansion from first principles ξ m ( ξ ) /m L F ( ξ ) θ ( ξ )1 .
00 0 . − . . .
01 0 . − . . .
02 0 . − . . .
03 0 . − . . .
04 0 . − . . .
05 0 . − . . .
06 0 . − . . .
07 0 . − . . .
08 0 . − . . .
09 0 . − . . .
10 0 . − . . .
11 0 . − . . .
12 0 . − . . .
13 0 . − . . .
14 0 . − . . .
15 0 . − . − . .
16 0 . − . − . .
17 0 . − . − . .
18 0 . − . − . .
19 0 . − . − . .
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