Non-anomalous `Ward' identities to supplement large-<i>N</i>multi-matrix loop equations for correlations

Akant, Levent; Krishnaswami, Govind S.

doi:10.1088/1126-6708/2007/02/073

Cited by 3 publications

(6 citation statements)

References 32 publications

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“…This is natural due to the emergence of scaling symmetry. ( 16) Some linear combinations of correlations in large-N multi-matrix models vanish because of the presence of hidden non-anomalous symmetries [33]. (17) m Higgs in a SUSY standard model [10] can be naturally small.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

Possible large-Nfixed points and naturalness forO(N) scalar fields

Krishnaswami

2009

J. Phys. A: Math. Theor.

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We try to use scale-invariance and the large-N limit to find a non-trivial 4d O(N ) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions by requiring the effective action for space-time dependent background fields to be finite and scale-invariant when regulators are removed. We find a line of non-trivial UV fixed-points in the large-N limit, parameterized by a dimensionless coupling. They reduce to classical λφ 4 theory whenh → 0. Forh = 0, neither action nor measure is scale-invariant, but the effective action is. Scale invariance makes it natural to set a mass deformation to zero. The model has phases where O(N ) invariance is unbroken or spontaneously broken. Masses of the lightest excitations above the unbroken vacuum are found. We derive a non-linear equation for oscillations about the broken vacuum. The interaction potential is shown to have a locality property at large-N . In 3d, our construction reduces to the line of large-N fixed-points in |φ| 6 theory.We investigate the naturalness concept of 't Hooft [1] applied to 4-dimensional O(N ) scalar fields. By this dogma, if there is a scalar particle very light compared to the microscopic scale at which the model is superseded, it must be for a good reason e.g., a symmetry. We observe that a non-trivial fixed-point in scalar field theory would be enough to make small masses natural. For, setting masses to zero would restore symmetry under rescaling. We try to realize this scenario by developing an idea of Rajeev [2] to find a fixed-point in the large-N limit. Background and motivationsMany discussions of 4d QFT begin with massive λφ 4 theory, but it most likely does not have a non-trivial continuum limit [3,4,5,6]. So we wish to know if there is a non-trivial 4d scalar field theory. Another motivation concerns the UV and naturalness problems in the scalar sector of the standard model of particle physics. The importance of QFTs with controlled UV behavior is well-known: Yang-Mills theories, with a gaussian UV fixed-point provide our best models for strong and weak interactions. In equilibrium statistical mechanics of magnets, the gaussian fixed-point (GFP) controls high energy behavior while the lower energy dynamics is governed by a crossover to the non-trivial Wilson-Fisher fixed-point (WFP) [7]. However, the situation in 4d massive λφ 4 theory, the simplest (but unconfirmed) model for W ± ,Z masses, is less satisfactory. λφ 4 is based on the gaussian IR fixed-point, but doesn't flow to any fixed-point in the UV. Perturbatively, interactions become strong (Landau pole) at energies of O(m exp [16π 2 /3λ]), where m, λ are the parameters of the model in the IR. This is in contrast with asymptotically free theories or theories based on an interacting UV fixed-point which might (in principle) be valid up to a higher energy. Numerical [3,4] and analytic [6,5] calculations suggest that without a UV cutoff, λφ 4 theory is 'trivial' 1 . Unfortunately, the non-trivial WFP in φ 4 theory in 4-ǫ dime...

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Section: Discussion and Open Problemsmentioning

confidence: 99%

Possible large-Nfixed points and naturalness forO(N) scalar fields

Krishnaswami

2009

J. Phys. A: Math. Theor.

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“…So we can think of these as linear equations constraining the higher rank χ K in terms of the lower rank ones which appear quadratically as sources on the right. This structure is reminiscent of the matrix model fSDE: [9,18]. Naively, we expect a large space of solutions to these constraints, since there seem to be a lot more degrees of freedom in the χ I than there are shuffle relations.…”

Section: Matrix Model Analogue Of the Group Of Loopsmentioning

confidence: 97%

“…are expressed in terms of left annihilation operators D j which satisfy D j ξ i 1 ···in = δ i 1 j ξ i 2 ···in or equivalently, [D j G] I = G jI . For more details on the fSDE, we refer to [8,9,18]. In this paper we do not have anything to say about the convergence of matrix integrals.…”

Section: Introductionmentioning

confidence: 99%

“…By a Λ-matrix model [17,8,9,18] we mean a statistical system whose variables are a collection of random hermitian N × N matrices A i , 1 ≤ i ≤ Λ with partition function Z = dAe −N tr S(A) . The integration is over all independent matrix elements.…”

Section: Introductionmentioning

confidence: 99%

“…For more details on the fSDE, we refer to [8,9,18]. In this paper we do not have anything to say about the convergence of matrix integrals.…”

Section: Introductionmentioning

confidence: 99%

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Schwinger–Dyson operators as invariant vector fields on a matrix model analog of the group of loops

Krishnaswami

2008

Journal of Mathematical Physics

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For a class of large-N multi-matrix models, we identify a group G that plays the same role as the group of loops on space-time does for Yang-Mills theory. G is the spectrum of a commutative shuffle-deconcatenation Hopf algebra that we associate to correlations. G is the exponential of the free Lie algebra. The generating series of correlations is a function on G and satisfies quadratic equations in convolution. These factorized Schwinger-Dyson or loop equations involve a collection of Schwinger-Dyson operators, which are shown to be right-invariant vector fields on G, one for each linearly independent primitive of the Hopf algebra. A large class of formal matrix models satisfying these properties are identified, including as special cases, the zero momentum limits of the Gaussian, Chern-Simons and Yang-Mills field theories. Moreover, the Schwinger-Dyson operators of the continuum Yang-Mills action are shown to be right-invariant derivations of the shuffle-deconcatenation Hopf algebra generated by sources labeled by position and polarization.

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Schwinger–Dyson operator of Yang–Mills matrix models with ghosts and derivations of the graded shuffle algebra

Krishnaswami¹

2008

J. Phys. A: Math. Theor.

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We consider large-N multi-matrix models whose action closely mimics that of Yang-Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger-Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G(ξ), are quadratic equations S i G = Gξ i G in concatenation of correlations. The Schwinger-Dyson operator S i is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to concatenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost correlations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger-Dyson operator as a whole is also a derivation of the graded shuffle product. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme.

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Non-anomalous `Ward' identities to supplement large-Nmulti-matrix loop equations for correlations

Cited by 3 publications

References 32 publications

Possible large-Nfixed points and naturalness forO(N) scalar fields

Possible large-Nfixed points and naturalness forO(N) scalar fields

Schwinger–Dyson operators as invariant vector fields on a matrix model analog of the group of loops

Schwinger–Dyson operator of Yang–Mills matrix models with ghosts and derivations of the graded shuffle algebra

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