A Simple Method for Correlation Functions via the Effective Actions in the Renormalization Group Framework

Bałaban, Tadeusz; O’Carroll, Michael

doi:10.1006/aphy.1997.5690

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…Indeed, in the context of the RG, considering models which are small perturbations of the free field, the generating function of field correlations and the correlations can be obtained through a formula which involves the effective actions generated applying the RG transformations to the partition function (see e.g. [13]). However, unfortunately, in the case of gauge fields, this question, as well as the incorporation of fermion fields and the verification of the Osterwalder-Schrader-Seiler axioms [4], have never been completely analyzed up to now, for d = 4.…”

Section: Introductionmentioning

confidence: 99%

On Yang-Mills Stability Bounds and Plaquette Field Generating Function

Veiga¹,

O’Carroll²

2022

Preprint

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We consider the Yang-Mills (YM) QFT with group U (N ). We take a finite lattice regularization Λ ⊂ aZ d , d = 2, 3, 4, with a ∈ (0, 1] and L (even) sites on a side. Each bond has a gauge variable U ∈ U (N ). The Wilson partition function is used and the action is a sum of gauge-invariant plaquette (minimal square) actions times a d−4 /g 2 , g 2 ∈ (0, g 2 0 ], 0 < g 2 0 < ∞. A plaquette action has the product of its four variables and the partition function is the integral of the Boltzmann factor with a product of U (N ) Haar measures. Formally, when a ց 0 our action gives the usual YM continuum action. For free and periodic b.c., we show thermodynamic and stability bounds for a normalized partition function of any YM model defined as before, with bound constants independent of L, a, g. The subsequential thermodynamic and ultraviolet limit of the free energy exist. To get our bounds, the Weyl integration formula is used and, to obtain the lower bound, a new quadratic global upper bound on the action is derived. We define gauge-invariant physical and scaled plaquette fields. Using periodic b.c. and the multi-reflection method, we bound the generating function of r−scaled plaquette correlations. A normalized generating function for the correlations of r scaled fields is absolutely bounded, for any L, a, g, and location of the external fields. From the joint analyticity on the field sources, correlations are bounded. The bounds are new and we get a −d for the physical two-plaquette correlation at coincident points. Comparing with the a ց 0 singularity of the physical derivative massless scalar free field two-point correlation, this is a measure of ultraviolet asymptotic freedom in the context of a lattice QFT. Our methods are an alternative and complete the more traditional ones.

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Section: Introductionmentioning

confidence: 99%

On Yang-Mills Stability Bounds and Plaquette Field Generating Function

Veiga¹,

O’Carroll²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed, in the context of the renormalization group, considering models which are small perturbations of the free field, the generating functional of field correlations and the correlations can be obtained through a formula which involves the effective actions generated applying the renormalization group transformations to the partition function (see e.g. [12]). However, in the case of gauge fields, this question, as well as the incorporation of fermion fields, have never been analyzed up to now.…”

Section: Introductionmentioning

confidence: 99%

On Yang-Mills Stability and Plaquette Field Generating Functional

O'Carroll,

da Veiga

2020

Preprint

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We consider the pure Yang-Mills relativistic quantum field theory in an imaginary time functional integral formulation. The gauge group is taken to be G = U(N ). We use a lattice ultraviolet regularization, starting with the model defined on a finite hypercubic lattice Λ ⊂ aZ d , d = 2, 3, 4, with lattice spacing a ∈ (0, 1] and L ∈ N sites on a side. The Wilson partition function is used where the action is a sum over four lattice bond variables of gauge-invariant plaquette (lattice minimal squares) actions with a prefactor a d−4 /g 2 , where we take the gauge coupling g ∈ (0, g 2 0 ], 0 < g0 < ∞. In a recent paper, for free boundary conditions, we proved that a normalized model partition function satisfies thermodynamic and ultraviolet stable stability bounds. Here, we extend the stability bounds to the Yang-Mills model with periodic boundary conditions, with constants which are also independent of L, a, g. Furthermore, we also consider a normalized generating functional for the correlations of r ∈ N gauge-invariant plaquette fields. Using periodic boundary conditions and the multireflection method, we then prove that this generating functional is bounded, with a bound that is independent of L, a, g and the location and orientation of the r plaquette fields. The bounds factorize and each factor is a single-bond variable, single-plaquette partition function. The number of factors is, up to boundary corrections, the number of non-temporal lattice bonds, such as (d − 1)L d . A new global quadratic upper bound in the gluon fields is proved for the Wilson plaquette action.

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A Simple Method for Correlation Functions via the Effective Actions in the Renormalization Group Framework

Cited by 2 publications

References 14 publications

On Yang-Mills Stability Bounds and Plaquette Field Generating Function

On Yang-Mills Stability Bounds and Plaquette Field Generating Function

On Yang-Mills Stability and Plaquette Field Generating Functional

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