Euclidean lattice simulation for dynamical supersymmetry breaking

Abstract

The global supersymmetry is spontaneously broken if and only if the ground-state energy is strictly positive. We propose to use this fact to observe the spontaneous supersymmetry breaking in euclidean lattice simulations. For lattice formulations that possess a manifest fermionic symmetry, there exists a natural choice of a hamiltonian operator that is consistent with a topological property of the Witten index. We confirm validity of our idea in models of the supersymmetric quantum mechanics. We then examine a possibility of a dynamical supersymmetry breaking in the twodimensional N = (2, 2) super Yang-Mills theory with the gauge group SU(2), for which the Witten index is unknown. Differently from a recent conjectural claim, our numerical result tempts us to conclude that supersymmetry is not spontaneously broken in this system. PACS numbers: 11.15.Ha, 11.30.Pb, 11.30.Qc It is widely believed that supersymmetry is relevant in particle physics beyond the standard model and it is spontaneously broken by some mechanism. If supersymmetry is not spontaneously broken at the tree level of the loop expansion, it remains so to all orders of the loop expansion. There still exists, however, a possibility that supersymmetry is spontaneously broken nonperturbatively. Precise study of such dynamical supersymmetry breaking remains elusive because we have no universal framework that defines supersymmetric (especially gauge) theories at a non-perturbative level.Generally, the Witten index Tr(−1) F [1], where F is the fermion number operator, provides an important clue. One can infer that the dynamical supersymmetry breaking does not occur in a wide class of supersymmetric models for which the Witten index can be computed to be non-zero. However, the Witten index is not a panacea. There exist physically interesting models for which it is very difficult to determine the Witten index and, in some cases, the index itself might be ill-defined due to a gapless continuous spectrum.In this letter, we consider a possibility to observe the dynamical supersymmetry breaking in euclidean lattice simulations, in the light of recent developments on lattice formulation of supersymmetric theories [2,3,4,5]. The conceptually clearest way to observe the spontaneous supersymmetry breaking would be to examine the degeneracy of bosons' and fermions' mass spectra in two-point correlation functions. Here, we propose an alternative method that is based on the following fact. The global supersymmetry is spontaneously broken if and only if the ground-state energy is strictly positive [6]. In princi- * Electronic address: kanamori-i@riken.jp † Electronic address: hsuzuki@riken.jp ‡ Electronic address: fumihiko˙sugino@pref.okayama.lg.jp ple, therefore, one can judge whether the supersymmetry breaking takes place or not if the ground-state energy can be computed. Let us first recall that the thermal average of the hamiltonian H with the inverse temperature β is expressed by the euclidean functional integral as [22] Tr He −βH Tr e −βH = aPBC dµ He −S aPBC ...

A method for measuring the Witten index using lattice simulation

“…The fact that σ[D] 0,P has a close relation to the partition function and thus the Witten index has been pointed out in [19]. See also [42] for a related numerical result.…”

“…We use the Hybrid Monte Carlo algorithm. See [19] for the implementation for this system 10 . We fix the physical volume L phys = 1.…”

“…To study the breaking of supersymmetry using lattice simulation, a Hamiltonian approach was used in [18]. The present author with H. Suzuki and F. Sugino proposed a method for detecting the spontaneous supersymmetry breaking using lattice simulation in [19], where they measured the vacuum energy. They also pointed out a possible relation between spontaneous supersymmetry breaking and the existence of a sign problem in the simulation, which comes from the phase of the Pfaffian of the fermion bilinear operator.…”

Ginsparg-Wilson Formulation of 2D N = (2,2) SQCD with Exact Lattice Supersymmetry

“…Actually, numerical investigation of the two-dimensional lattice SYM models has been developed in [48,49,50,25]. In particular, ref.…”

Supersymmetry on the lattice

“…33 The one and two dimensional Wess-Zumino models also serve as useful arenas for investigating spontaneous supersymmetry breaking. 16,17,34 In addition to serving as toy models for understanding the general problem of lattice supersymmetry low dimensional models can also have interesting applications in gauge-gravity duality -finite temperature super Yang-Mills models in one, two and three dimensions and in the planar limit are conjectured to be dual to certain supergravity theories containing black holes. Moving away from the planar limit allows the Yang-Mills simulations to tell us something about quantum and string corrections in these theories.…”