Supersymmetric nonperturbative formulation of the WZ model in lower dimensions

A nonperturbative formulation of the Wess-Zumino (WZ) model in two and three dimensions is proposed on the basis of momentum-modes truncation. The formulation manifestly preserves full supersymmetry as well as the translational invariance and all global symmetries, while it is shown to be consistent with the expected locality to all orders of perturbation theory. For the two-dimensional WZ model, a well-defined Nicolai map in the formulation provides an interesting algorithm for Monte Carlo simulations.


Introduction
In this Letter, we propose a nonperturbative Euclidean formulation of a dimensional reduction of the Wess-Zumino (WZ) model [1] to three and two dimensions. 1 On a nonperturbative formulation of the WZ model, there exist many preceding studies, mostly based on spacetime or spatial lattices . 2 (For exact renormalization group approaches to the WZ model, see Refs. [48,49].) The desired features of our present proposal are: (I) full supersymmetry (SUSY) as well as the translational invariance and all global symmetries are manifestly preserved, (II) it is amenable to nonperturbative studies by, for example, Monte Carlo simulations, (III) the formulation of the two-dimensional (2D) N = (2, 2) WZ model possesses a well-defined Nicolai map [50,51] (see also Refs. [52,53]). On the other hand, the locality and the reflection positivity are not manifest in our formulation and we will show that there is actually no problem concerning these in lower-dimensional models, at least to all orders of perturbation theory. Therefore, we propose our formulation as a nonperturbative definition of the WZ model in lower dimensions, although its nonperturbative validity still remains to be examined by using, for example, numerical simulations.
Our idea is very simple. The off-shell super multiplets in the WZ model (the chiral and anti-chiral multiplets) are expressed by the chiral and anti-chiral superfields, Φ and Φ † . 3 In the momentum space, Φ(p, θ,θ) = e −θσµθpµ Ã (p) + √ 2θψ(p) + θθF (p) , Φ † (p, θ,θ) = e θσµθpµ Ã * (p) + √ 2θψ(p) +θθF * (p) , (1.1) as they satisfy the chiral constraints,DαΦ(p) = 0 and D αΦ † (p) = 0, where covariant spinor derivatives are given by D α = ∂/∂θ α − σ µααθα p µ andDα = −∂/∂θα + θ α σ µαα p µ . On such off-shell multiplets in the momentum space, SUSY is linearly realized and super transformations generated by do not mix momentum modes with different momenta. This fact suggests that one can regularize the functional integral of the model by restricting possible momenta of off-shell super multiplets. Any restriction on momenta does not break SUSY. 4 From a perspective of the Euclidean rotational symmetry in the infinite volume, it would be preferable to take a rotational invariant restriction 3 We follow the notational conventions of Ref. [54], except that we consider Euclidean field theory in terms of momentum modes. (The Fourier transformation is over repeated indices is always meant and the Greek index µ runs from 0 to d − 1, where d ≤ 4 is the spacetime dimension. Although we consider the system with a single chiral multiplet for notational simplicity, the generalization to cases with multi chiral multiplets is straightforward. 4 It is crucial that we restrict the momentum of off-shell super multiplets on which SUSY is linearly realized. If the super transformations are non-linear in field variables, such as in supersymmetric gauge theories in the Wess-Zumino gauge, super transformations mix modes with different momenta and restriction on possible momenta breaks SUSY. In this aspect, our formulation differs from a formulation of such as p 2 ≡ p µ p µ ≤ Λ 2 , where Λ is an UV cutoff. Thus, one may define a regularized partition function of the WZ model by To make this expression fully well-defined, we may assume that the system is put in a Euclidean box of size L and the momentum p is discrete, p µ = (2π/L)n µ , where n µ ∈ Z. The Euclidean action S in the momentum space reads where the symbol * denotes the convolution supersymmetric gauge theories in Ref. [55], in which SUSY is expected to be exact only in the limit that the momentum cutoff is removed. Note that the present model has no gauge symmetry; it is clear that any restriction on possible momenta breaks local gauge invariance.

Locality and finiteness
The above description sounds too good to be true. Actually, it is not clear whether definition (1.3) (in the limit that the UV and IR cutoffs are removed, Λ → ∞ and L → ∞) is consistent with the expected locality in the target theory. The reflection positivity is a related issue. Now, if the model is massive, W (Φ) = (1/2)mΦ 2 + · · · , the free super propagators are given by where the momentum contained in D α andDα is p. In prescription (1.3), the functional integral is defined in terms of momentum modes. One may then define the field variable in the real space by . Then free super propagators in the real space are proportional to (2.5) 5 In this subsection and in Appendix, where the issue of locality is addressed, we set L → ∞ because the notion of locality becomes transparent only in this limit.
where J ν (z) (K ν (z)) denotes the (modified) Bessel function. In the right-hand side, the first term is the standard massive propagator which dumps exponentially ∼ e −|m||x−y| . The second term is the cutoff effect and its amplitude dumps only in the inverse powers of |x − y|. Therefore, when Λ is kept fixed, the second term dominates the first for |x − y| large. Nevertheless, since the second term is integrable at p → ∞, the second term with |x − y| kept fixed vanishes as Λ → ∞. Free propagators in the real space thus restore the expected locality for Λ → ∞.
Next, we consider the effect of interaction. In prescription (1.3), only momentum modes with p 2 ≤ Λ 2 appear and, in perturbation theory, this restriction can be taken into account by substituting all factors 1/(p 2 +|m| 2 ) in Eqs.
, where Θ(x) denotes the step function. The locality is not obvious in general with this prescription as illustrated in Appendix. Nevertheless, if a convergence property of a Feynman integral is good enough, the value of the Feynman integral must be independent of the regularization as Λ → ∞; then the issue of locality and reflection positivity should not matter. Since our formulation preserves manifest SUSY, we expect a better convergence property of Feynman integrals compared with formulations which do not have manifest SUSY. 6 Since SUSY is manifest with prescription (1.3), one can determine the superficial degrees of divergence on the basis of the super Feynman rule (see Ref. [54]). In the WZ model in d dimensions, the superficial degrees of divergence ω(Γ ) of a super diagram Γ is given by (cf. Sec. 6.6 of Ref. [61]) For the four-dimensional (4D) system, d = 4, the perturbative renormalizability requires the superpotential is cubic For the three-dimensional (3D) system, d = 3, if the superpotential is cubic and, since again the tadpoles identically vanish owing to SUSY, all Feynman diagrams have strictly negative superficial degrees of divergence.
For the two-dimensional (2D) system, d = 2, for any (polynomial) superpotential, we have ω(Γ ) = − i V i − C and again all Feynman diagrams have strictly negative superficial degrees of divergence.
The above counting shows that, in 3D N = 2 WZ model with the cubic superpotential and in 2D N = (2, 2) WZ model with arbitrary superpotential, all 1PI diagrams have strictly negative superficial degrees of divergence. Combined this with the power-counting theorem [62,63], we see that all Feynman integrals in these lower-dimensional models are absolutely convergent. We then intuitively expect that, owing to this good convergence property, the correct (finite) value of Feynman diagrams is reproduced with prescription (1.3) in the Λ → ∞ limit. Then there will be no need to worry about the locality and the reflection positivity for Λ → ∞.
This natural expectation is rigorously confirmed by the following

Lemma 1 For any Feynman integral
where k i are loop momenta and p collectively denotes external momenta, that is absolutely convergent for any fixed p, where I ′ F (k, p; Λ) is a modified integrand that is defined by substituting all propagators 1/(ℓ 2 i + |m| 2 ) in the original integrand PROOF. From the definition of I ′ F (k, p; Λ), 9) where N denotes the number of propagators and propagators' momenta ℓ i are linear combinations of k and p. For fixed p, for sufficiently large Λ, there exists a region containing the origin of R Ld of size Λ ′ , We then take the Λ → ∞ limit on the both sides of Eq. (2.10). The most left-hand side of (2.9) becomes |I F (p) − lim Λ→∞ d d k 1 · · · d d k L I ′ F (k, p; Λ)|. In the right-hand side of Eq. (2.10), we may then take the Λ ′ → ∞ limit and this leads to lim is absolutely convergent. This shows Eq. (2.8).
To summarize, to all orders of perturbation theory, definition (1.3) provides a valid formulation of 3D N = 2 WZ model with the cubic superpotential and of 2D N = (2, 2) WZ model with arbitrary superpotential. 7 We thus propose to use Eq. (1.3) as a nonperturbative definition of these models. As already noted, all symmetries in the target theory are manifest and, moreover, the formulation is amenable to nonperturbative Monte Carlo simulations.

Relation to a lattice formulation based on the SLAC derivative
As noted in Introduction, the SUSY invariance holds with any restriction on possible momenta of super multiplets. For example, since the rotational symmetry is in any case broken in a finite box, one may adopt a "cubic" restriction −Λ ≤ p µ ≤ Λ for all µ, rather than the "spherical one" p 2 ≤ Λ 2 , and 11) where the action S is again given by Eq. (1.4). This prescription shares desired features with Eq. (1.3), such as SUSY is manifest and the locality is restored in the Λ → ∞ limit for lower-dimensional models (to all orders of perturbation theory; Lemma 1 can appropriately be modified for the cubic momentum restriction above). Prescription (2.11) is, however, nothing but a lattice formulation of the WZ model in Ref. [11] that is based on the SLAC lattice derivative [46,47]. 8 In fact, if one expresses the lattice formulation in Ref. [11] 7 Power-counting shows that if we generalize the Kähler potential to an arbitrary real function Φ † Φ → K(Φ † , Φ), new logarithmic divergences may appear and the present argument does not apply. 8 See also Refs. [15,16] and Refs. [37,41,43] for related formulations.  p, θ,θ), where x denotes the lattice point, one ends up with Eq. (2.11) with the identification Λ ≡ π/a (a is the lattice spacing).
Since two prescriptions (1.3) and (2.11) should be essentially equivalent for Λ → ∞, our proposal (1.3) is basically equivalent to the lattice formulation in Ref. [11] on the basis of the SLAC derivative. 9 The SLAC derivative is not usually adopted in lattice (gauge) theory, because the locality could be violated [64]. See also Refs. [2,65]. This is also the case with our prescription for 4D WZ model; as discussed in Appendix, the consistency of prescription (1.3) with locality is not clear for 4D WZ model. However, as we have discussed so far, the prescription can be consistent with the locality in lower-dimensional models and we can expect the same for prescription (2.11).
Thus, from this perspective, our contribution in the present Letter is merely in that we gave a strong affirmative argument for the applicability of the formulation in Ref. [11] to 3D N = 2 and 2D N = (2, 2) WZ models, dimensional reductions of the original 4D N = 1 WZ model. On the other hand, we have to say that its validity for 4D WZ model itself is still not clear, unfortunately.
Since SUSY is manifest in our formulation, we could repeat the argument in, for example, Sec. 2 of Ref. [71]. For the argument there, important fermionic symmetries are (in the notation of Ref. [54]) These nilpotent symmetries imply that, among correlation functions of scalar fieldsÃ(p), only those of zero momentum modesÃ(0) can be nontrivial. This follows from the fact thatÃ(p) with p = 0 areQ˙1 orQ˙2 exact andÃ(p) are closed underQ˙1 andQ˙2. 10 Moreover, since the anti-holomorphic part of the superpotential isQ˙1 orQ˙2 exact, correlation functions ofÃ(0) depend on parameters in the superpotential only holomorphically (i.e., they depend on g but not on g * ). It is interesting that our prescription provides a solid basis for these arguments which assume a supersymmetric regularization. In the context of the Landau-Ginsburg model for the superconformal field theory, one has to consider the massless (or critical) limit. Since in this limit perturbation theory suffers from severe infrared divergences, it must be important to formulate the system nonperturbatively.
Note that, in functional integral (1.3), the momentum of auxiliary fieldsG(p) andG * (p) is restricted to p 2 ≤ Λ 2 . Therefore, in Eq. The existence of the Nicolai map provides a quite interesting simulation algorithm for the present system. See Ref. [20] for actual implementation of this idea in a discretized real space. One first generates a set of Gaussian random numbers with the unit covariance; this gives a configuration of {Ñ(p),Ñ * (p)}. Then one inverts Nicolai map (3.5) by numerical means. There may exist several inverse images and one must in principle find all of them. This provides configuration(s) of {Ã(p),Ã * (p)}. Repeating these steps, one obtains a statistical ensemble of {Ã(p),Ã * (p)}. Correlation functions of the fermion field can also be obtained from bosonic ones without inversion (by assuming that SUSY is not spontaneously broken). A great advantage of this algorithm is that, compared with conventional methods based on the Markov process, there is (in principle) no autocorrelation between configurations in the obtained ensemble. 12 Another important point to note is that the fermion determinant in the present system is generally complex and thus conventional methods may fail owing to the sign problem, while the algorithm based on the Nicolai map seems to be free from this difficulty. In the near future, we hope to carry out Monte Carlo simulations of 2D N = (2, 2) WZ model on the basis of the Nicolai map in the present momentum-space formulation.

Acknowledgements
We would like to thank Tetsuo Horigane, Hiroki Kawai, Yoshio Kikukawa and Fumihiko Sugino for enlightening and enjoyable discussions. This work was initially motivated by Jun Saito's talk at the YITP workshop, "Development of Quantum Field Theory and String Theory" (YITP-W-09-04). The work of H.S. is supported in part by a Grant-in-Aid for Scientific Research, 18540305.

Note added
After completing this work, we became aware of a paper by Georg Bergner [77] in which a lattice Monte Carlo simulation of the SUSY quantum mechanics (QM) [72] was carried out on the basis of a "full supersymmetric model". This lattice formulation is just the supersymmetric lattice formulation of Ref. [11] applied to SUSY QM. Since SUSY QM is UV finite with a supersymmetric regularization, a variant of Lemma 1 ensures the restoration of locality in this formulation to all orders of perturbation theory.
A Locality in 4D N = 1 WZ model where the one-loop Feynman integral is given by In the second equality, we have re-organized the integrand to address the locality. First, the integral in Eq. (A.3) is absolutely convergent even without the regularization factor Θ(Λ 2 − k 2 )Θ(Λ 2 − (k + p) 2 ) and, according to Lemma 1, we may simply discard the factor Θ(Λ 2 − k 2 )Θ(Λ 2 − (k + p) 2 ) → 1 in the limit Λ → ∞. Therefore, in this limit, Eq. (A.3) becomes nothing but the finite part of the Feynman integral which is given by the BPHZ subtraction scheme applied to the logarithmically divergent integral in the original un-regularized theory. Next, Eq. (A.4) ] is an ultraviolet diverging part that would be obtained in the conventional momentum-cutoff regularization. This is a part subtracted by a local counterterm (the wave function renormalization, in the present case) in the BPHZ renormalization.
We have thus observed that, in the Λ → ∞ limit, Eqs. (A.3) and (A.4) reproduce the correct finite part and a divergent local term corresponding to the BPHZ subtraction scheme. These two terms are thus consistent with the expected locality. On the other hand, if Eq. (A.5) does survive in the Λ → ∞ limit, the expected locality would be violated because Eq. (A.5) could not be a smooth function of the external momentum p; in other words, Eq. (A.5) could not be interpreted by an insertion of local operators.
To estimate Eq. (A.5), we note that the factor Θ( is non-zero only in a region which is sandwiched in between two 3-spheres with the radius Λ, one has its center at k = 0 and another at k = −p. The 4-volume of this region V(p; Λ) is given by On the other hand, we have simple bounds on the factor 1/(k 2 + |m| 2 ) 2 in Eq. (A.5), from the consideration of the integration region, when the external momentum p is kept fixed in the limit, because V(p; Λ) = O(Λ 3 ) in such a limit. Eq. (A.5) therefore vanishes in the Λ → ∞ limit and the expected locality is reproduced. This shows that prescription (1.3) is consistent with the locality at least in the one-loop level.
Nevertheless, it is still not clear whether this leads to a breakdown of locality. If the Λ → ∞ limit of Eq. (A.10) is constant in q, then the contribution could be removed by a local counter term-a finite wave-function renormalization.
On the other hand, if the limit of Eq. (A.10) is a nontrivial function such as ∼ ln(|q|/Λ), the contribution cannot be removed by local counterterms and the expected locality is broken. Unfortunately, we have not been able to find which is really the case.