An alternative lattice field theory formulation inspired by lattice supersymmetry

Abstract

Abstract:We propose an unconventional formulation of lattice field theories which is quite general, although originally motivated by the quest of exact lattice supersymmetry. Two long standing problems have a solution in this context: 1) Each degree of freedom on the lattice corresponds to 2 d degrees of freedom in the continuum, but all these doublers have (in the case of fermions) the same chirality and can be either identified, thus removing the degeneracy, or, in some theories with extended supersymmetry, identified with different members of the same supermultiplet. 2) The derivative operator, defined on the lattice as a suitable periodic function of the lattice momentum, is an addittive and conserved quantity, thus assuring that the Leibniz rule is satisfied. This implies that the product of two fields on the lattice is replaced by a non-local "star product" which is however in general nonassociative. Associativity of the "star product" poses strong restrictions on the form of the lattice derivative operator (which becomes the inverse Gudermannian function of the lattice momentum) and has the consequence that the degrees of freedom of the lattice theory and of the continuum theory are in one-to-one correspondence, so that the two theories are eventually equivalent. We can show that the non-local star product of the fields effectively turns into a local one in the continuum limit. Regularization of the ultraviolet divergences on the lattice is not associated to the lattice spacing, which does not act as a regulator, but may be obtained by a one parameter deformation of the lattice derivative, thus preserving the lattice structure even in the limit of infinite momentum cutoff. However this regularization breaks gauge invariance and a gauge invariant regularization within the lattice formulation is still lacking.

Progress and prospects of lattice supersymmetry

“…Ref. [12] introduces a different non-local 'star product' that preserves the Leibniz rule, with the consequence that the lattice spacing no longer acts as a regulator. Despite the complicated intricacies of these constructions, so far their applicability appears limited to systems without gauge invariance, and only in (0+1) dimensions [10,13] or on infinite lattices [12].…”

“…[12] introduces a different non-local 'star product' that preserves the Leibniz rule, with the consequence that the lattice spacing no longer acts as a regulator. Despite the complicated intricacies of these constructions, so far their applicability appears limited to systems without gauge invariance, and only in (0+1) dimensions [10,13] or on infinite lattices [12].As a consequence of broken supersymmetry, quantum effects in the lattice calculation will generate supersymmetry-violating operators. These include, in particular, relevant operators for which counterterms will have to be fine-tuned in order to recover the supersymmetric QFT of interest in the a → 0 continuum limit that corresponds to removing the UV cutoff a −1 .…”

Direct computational approach to lattice supersymmetric quantum mechanics

“…The partition function Z ′ with the action S ′ is also expressed as the form (3.6) with modified T ′ and S ′ . We have 21) and the one-point local function can be estimated without using an impurity matrix. Also in the case, the cost reduces by diagonalizing S ′ and T ′ .…”

“…There have been many attempts to define SUSY QM and low dimensional Wess-Zumino model on the lattice [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The proposed lattice models have provided not only further understanding of SUSY models but also testing grounds for lattice formulations of higher dimensional SUSY theories [22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”

Lattice study of supersymmetry breaking in N=2 supersymmetric quantum mechanics

“…In order to control the artificial SUSY breaking due to the lattice cut-off, various studies have been carried out so far [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. In N = 2 supersymmetric quantum mechanics (SUSY QM), lattice models with non-local SLAC derivative and improved Wilson term and one-exact supersymmetric action have been studied [14,16,27,28].…”