On general ultraviolet properties of a class of confining propagators

The European Physical Journal C, Mar 2016

We study the ultraviolet properties of theories whose fundamental fields display a confining, Gribov-type, propagator. These are propagators that exhibit complex poles and violate positivity, thus precluding a physical propagating particle interpretation. We show that the properties of this type of confining propagators do not change the ultraviolet behavior of the theory, in the sense that no new ultraviolet primitive divergences are generated, thus securing the renormalizability of these confining theories. We illustrate these properties by studying a variety of models, including bosonic and fermionic confined degrees of freedom. The more intricate case of super-Yang–Mills with \(\mathcal{N} =1\) supersymmetries in the Wess–Zumino gauge is taken as example in order to prove these statements to all orders by means of the algebraic renormalization set up.

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On general ultraviolet properties of a class of confining propagators

Eur. Phys. J. C On general ultraviolet properties of a class of confining propagators M. A. L. Capri 0 M. S. Guimaraes 0 I. F. Justo 0 L. F. Palhares 0 S. P. Sorella 0 0 Departamento de Física Teórica, Instituto de Física , UERJ, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro 20550-013 , Brazil We study the ultraviolet properties of theories whose fundamental fields display a confining, Gribov-type, propagator. These are propagators that exhibit complex poles and violate positivity, thus precluding a physical propagating particle interpretation. We show that the properties of this type of confining propagators do not change the ultraviolet behavior of the theory, in the sense that no new ultraviolet primitive divergences are generated, thus securing the renormalizability of these confining theories. We illustrate these properties by studying a variety of models, including bosonic and fermionic confined degrees of freedom. The more intricate case of super-Yang-Mills with N = 1 supersymmetries in the Wess-Zumino gauge is taken as example in order to prove these statements to all orders by means of the algebraic renormalization set up. 1 Introduction The quantization of non-abelian gauge theories is still an open and rich subject for quantum field theorists. A general framework which takes into account the non-perturbative phenomena of gluon and quark confinement and of chiral symmetry breaking is still lacking. Needless to say, these issues represent a major challenge for our current understanding of non-abelian gauge theories in the non-perturbative infrared regime. A successful approach to investigate these topics within the context of the Euclidean quantum field theory is provided by the Gribov framework1. In his seminal work [ 4 ], Gribov pointed out that the Faddeev–Popov gauge-fixing procedure is plagued by the existence of Gribov copies. In a path-integral formulation this manifests in the fact that the Faddeev–Popov operator develops zero modes. For example, in the Landau gauge, ∂μ Aaμ = 0, the Faddeev–Popov operator is given by Mab = −(∂2δab − g f abc Acμ∂μ). For sufficiently strong coupling constant, zero modes of this operator start to appear, rendering ill-defined the Faddeev-Propov procedure. To deal with this issue, Gribov proposed to restrict the domain of integration in the path integral to a region in field space where the eigenvalues of the Faddeev–Popov operator are strictly positive. This region is known as the Gribov region , being defined as It is useful to observe here that the inverse of the Faddeev– Popov operator, (M−1)ab, yields precisely the propagator G(k, A) of the Faddeev–Popov ghosts in the presence of an external gauge field Aaμ [ 1–4 ], namely G(k, A) = (N 2 − 1) 1 δab k|(M−1)ab|k . This property can be employed to implement the restriction to the Gribov region by requiring the absence of poles in the ghost propagator for any non-zero value of the ghost external momentum k. This requirement is known as the Gribov nopole condition. More precisely, following [ 4 ], one can always represent the exact ghost propagator in the presence of an external gauge field as 1 For reviews on the Gribov issues, see [ 1–3 ] and references therein. (1) (2) 1 G(k, A) = k2 (1 + σ (k, A)), where σ (k, A) is the ghost form factor. Using the general properties of the diagrammatic expansion of quantum field theory, we can write 1 G(k) = G(k, A) conn = k2 (1 + σ (k, A) conn) 1 1 = k2 (1 − σ (k, A) 1P I ) where “conn” stands for the connected set of diagrams and 1 P I denotes the 1-particle irreducible ones. It can be shown that σ (k, A) 1P I is a decreasing function of k [ 1–4 ]. Therefore, the condition that the ghost propagator has no poles for any non-zero value of the ghost external momentum can be expressed as a condition for the maximum value of the ghost form factor, i.e. σ (0, A) 1P I = 1. Equation (5) expresses the no-pole condition. In [ 5 ], it has been shown that an exact closed expression for σ (0, A) can be obtained, being proportional to the horizon function H ( A) of Zwanziger’s formalism [ 1 ]: . The no-pole condition (5) is thus equivalent to H ( A) 1P I = V 4(N 2 − 1), which is called the horizon condition. The relevance of the horizon function H ( A) relies on the fact that the restriction of the domain of integration in the functional integral to the Gribov region can be effectively implemented by adding to the Yang–Mills action the quantity H ( A). More precisely, it turns out that, in the thermodynamic limit [ 1,6–8 ], the partition function of the theory with the cut-off at the Gribov region is given by Z = = D Aδ(∂ A) det(Mab)e−SY M D Aδ(∂ A) det(Mab)e−(SY M +γ 4 H(A)−γ 4V 4(N 2−1)), where the massive parameter γ is known as the Gribov parameter [ 1–4 ]. It is not a free parameter, being determined in a self-consistent way by the horizon condition (7), which can be rewritten as a stati (...truncated)


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M. A. L. Capri, M. S. Guimaraes, I. F. Justo, L. F. Palhares, S. P. Sorella. On general ultraviolet properties of a class of confining propagators, The European Physical Journal C, 2016, pp. 141, Volume 76, Issue 3, DOI: 10.1140/epjc/s10052-016-3974-3