Some Aspects of Supersymmetric Field Theories with Minimal Length and Maximal Momentum

We consider a real scalar field and a Majorana fermion field to construct a supersymmetric quantum theory of free fermion fields based on the deformed Heisenberg algebra $[x,p]=i\hbar\big(1-\beta p+2\beta^{2}p^{2}\big)$, where $\beta $ is a deformation parameter. We present a deformed supersymmetric algebra in the presence of minimal length and maximal momentum.


Introduction
Physics in extremely high energy regions is particularly of interest to particle physicists. One of the promising idea in the realm of high energy physics is the idea of supersymmetry (SUSY) (see for instance [1] and references therein). This is a symmetry that relates or mixes (unites) fermions and bosons. Supersymmetry has the potential to solve some outstanding mysteries in particle physics. One important problem that stands out is called the hierarchy problem. It is believed that the mass of the Higgs boson, m h , is much smaller than the fundamental Planck mass, m p = hc G ≈ 10 19 GeV/c 2 . On the other hand, when we discuss gravity in the context of the quantum field theory, it is expected that there is a minimal measurable length that restricts resolution of adjacent spacetime points up to the Planck length. This is actually a common address of all quantum gravity candidates (see [2] and references therein). Incorporation of this idea in quantum field theory provides a theory that is naturally regularized in the ultraviolate regime [3,4,5]. Recently in the context of the Doubly Special Relativity it has been revealed that a test particle's momentum cannot be arbitrarily imprecise leading nontrivially to the result that there is a maximal momentum for test particles [6]. This idea can be explained also through modified dispersion relations as a phenomenological outcome of foam-like structure of spacetime at the Planck scale [7,8]. String theory which has a characteristic scale √ α, is one of the most successful theoretical framework which overcomes the difficulty of ultra-violet divergence in quantum theory of gravity. Therefore, if we construct a field theory which captures some stringy nature and/or includes stringy corrections, it would play a pivotal role in investigating physics in high energy regions even near the Planck scale. One way to discuss these corrections is deforming the standard Heisenberg uncertainty principle to a generalized uncertainty principle (GUP). A GUP that predicts a minimal observable length can be written as follows This GUP leads to the following algebraic structure where β = β 0 /(M pl c) 2 = β 0 l 2 pl /h 2 . If GUP is realized in a certain string theory context, β takes a value of the order of the string scale, β ∼ α. In the presence of both minimal length and maximal momentum, the GUP can be given as follows In this framework the following algebraic structure can be deduced [9] [x, p] = ih(1 − βp + 2β 2 p 2 ) .
In this context, we can define the generalized momentum operator as With these preliminaries, in this paper we construct a quantum theory of free fermion fields based on the deformed Heisenberg algebra (4). We respect the supersymmetry in this context as a guiding principle. This is because string theory contains this symmetry and it provides a useful tool to understand physics in ultra-violet regime.
Because of existence of natural cutoffs as minimal length and maximal momentum, the supersymmetry is deformed in this context. From the fermionic part of the system, we propose an action of fermionic field based on the GUP in a spacetime with arbitrary number of dimensions. We construct the deformed supersymmetric algebra in this context. Finally, we compute some scattering amplitudes such as the pair annihilation amplitude to see how these important quantities can be calculated in the presence of natural cutoffs. We note that experimental limits and the general phenomenology perspective on this issue are not discussed here. One can see Ref. [10] for this purpose. We note also that while the study of SUSY in relation to GUP is a relatively young subject, a bit more has been done for other quantum-spacetime intuitions. For instance, in Ref. [11] the authors have studied the basic twist quantization of osp(1|2) and kappadeformation of D = 1 superconformal mechanics. The organization of the paper is as follows: in Section 2 we construct a scalar field theory based on the GUP (4). Section 3 is devoted to the issue of supersymmetry in the presence of natural cutoffs as minimal length and maximal momentum. In Section 4 we compute scattering amplitude of the pair annihilation in the presence of natural cutoffs. The paper follows by a summary and conclusion in section 5.

Scalar Field Theory
Generalization of the Heisenberg algebra to d-dimensions, where rotational symmetry is preserved and there are both a minimal length and maximal momentum as natural cutoffs, is as follows Here i and j run from 1 to d, where in three dimensions p = p x i + p y j + p z k that i, j and k are unit vectors of cartesian coordinates and Hereafter, we use indices i and j for spatial coordinates and a and b for all spacetime coordinates. Now the Jacobi identity determines the full algebra of the theory which is given as follows (see the work by K. Nozari and A. Etemadi in Ref. [6])

Some Aspects of Supersymmetric Field Theories with Minimal Length and Maximal Momentum4
The presence of the 1 p is a trace of the existence of the maximal momentum in this setup. Note also that Eq. (9) reflects the noncommutative nature of the spacetime manifold in Planck scale. Here L ij are angular momentum like operators defined as Since operators p i 's commute with each other and we have not assumed the existence of a minimal uncertainty in momentum (see Ref. [12] for possible extension in this regard), we can construct our theory in momentum space representation. In momentum space representation, momentum operators are diagonalized simultaneously and we do not distinguish eigenvalues of momentum p i from operator p i . In which follows, we seth = 1 for simplicity. The Lagrangian in d + 1 dimensional spacetime and in the presence of a minimal observable length and a maximal momentum can be written as follows Note that the difference from ordinary quantum field theory is the extra prefactor (1 − βp + 2β 2 p 2 ) −1 in the Lagrangian. Using the Bjorken-Johnson-Low prescription, from behavior of T * -product through the following relations where In this relation, E(p) ≡ Z(p) + m 2 where Z(p) is an arbitrary even function whose explicit form has no influence on the arguments. One can see from equation (13) that a deformation prefactor, (1 − βp + 2β 2 p 2 ), of Heisenberg algebra in the first quantization (6) also appears in canonical commutation relation of the second quantized field theory.
In the fermion field case, we encounter a difficulty in the construction of the second quantized Hilbert space which dose not appear in a scalar field system. In which follows we use the idea of supersymmetry to construct a quantized field theory of fermions. In fact, we construct a quantum Field Theory of fermions, which is consistent with the above scalar field theory, and by using supersymmetry prescription with a minimal length and maximal momentum scale.

Supersymmetry and GUP
Supersymmetric quantum field theory with minimal length has been studied by Shibusa [13]. In which follows, we generalize the pioneer work of Shibusa to the more general case that there are both minimal length and maximal momentum as natural cutoffs.
As we have mentioned in the introduction, supersymmetry proposes that to each fermion there exists a boson and vice versa. Thus in two and three-dimensional spacetime, a system with a real scalar and a Majorana fermion has a special symmetry between a boson and a fermion, namely SUSY. Following Ref. [13], our notation for two and three-dimensional spacetime is as follows: in those dimensional spacetime (with signature (−, +) or (−, +, +)) the Lorantz group has a real (Majorana) two-component spinor representation ψ α . For instance, in three-dimensional spacetime, we define a representation of Gamma matrices by Pauli matrices as follows [13]: We note that the spinor indices in this case are lowered or raised by charge conjugation matrix C αβ ≡ Γ 0 and its inverse matrix C −1 . The generalized supersymmetric algebra in the presence of a minimal length and a maximal momentum and its action on a scalar field φ, a Majorana fermion ψ and an auxiliary field F with parameter ǫ α is as follows where Υ and λ i are functions of GUP deformation parameter β and momentum. These factors reduce to unity in the limit of β → 0 and will be determined later by consistency condition. The closeness of algebra requires Now the supersymmetric Lagrangian can be written as a sum of separate Lagrangians, that is,

Some Aspects of Supersymmetric Field Theories with Minimal Length and Maximal Momentum6
where L B , L F and L aux are Bosonic, Fermionic and auxiliary fields Lagrangian respectively. Following [13], we generalize the Lagrangian by introduction of factors ζ i , which are functions of deforming parameter β and momentum. These functions can be determined as follows. Starting with Lagrangian where d is the number of the spatial coordinates. By integrating out the field F , we obtain the Lagrangian for the scalar and Majorana fields as follows On the other hand, the invariance of the Lagrangian (21) under supersymmetry variation (17)-(19) in the presence of the minimal length and maximal momentum leads to Using the conditions (20) and (23), only ζ 1 and ζ 2 need to be determined. Note also that factor λ 1 can be absorbed into normalization of F . In which follows we set λ 1 = 1 for F to be an auxiliary field. Noether's current for supersymmetric Lagrangian (22) can be calculated and the supersymmetric charge is obtained to be Now the Hamiltonian of this system can be written as follows Using the Bjorken-Johnson-Low prescription, from behaviors of T * -product between fields, as has been explained in the previous section, we obtain canonical relations as follows Therefore, the Hamiltonian (25) now can be written as follows where π(p, t) ≡ ∂ t φ(−p, t) and i = 1, 2, ..., d. There is another condition that can be used to determine ζ 1 and ζ 2 and comes from the free energy of the supersymmetric vacuum. From algebra (16), the supersymmetric state has zero energy where T r B and T r F represent trace in bosonic and fermionic Hilbert space respectively. This relation leads to the following condition Finally we can write ζ 1 = 1 − βp + 2β 2 p 2 −1 as one can read from (11). Thus we have Therefore, we have constructed the quantized field of fermions as follows in consequence, the supersymmetric algebra in the presence of both minimal length and maximal momentum is deformed from the usual one as follows These results can be generalized to higher dimensions easily. Finally we note that the Lagrangian now takes the following form We see that in the presence of quantum gravity effects (as minimal length and maximal momentum) there is an extra, universal prefactor 1−βp+2β 2 p 2 − 1 2 in comparison with the usual fermion action, regardless of existence or absence of supersymmetry. We note that the major difference between our framework and the formalism presented in Ref. [13] is the difference in the mentioned prefactor. Our prefactor contains both minimal length and maximal momentum simultaneously. Otherwise, the algebraic structure of the two supersymmetric algebras are the same.

Some Scattering Amplitudes in the Presence of Minimal Length and Maximal Momentum
In a perturbative expansion one usually refines the calculations using corrections that are becoming smaller and smaller. The reduction in importance is quantified by the power of the perturbation parameter, which is usually the coupling strength. At some point we are adding refinements that are too small to be measured and we know we can stop adding refinements. In a perturbation expansion, the amplitude M for a given process can be computed using an expansion of the type [14] where g kn is the coupling constant. In which follows we are going to see the effect of minimal length and maximal momentum as natural cutoffs on the scattering amplitude of the annihilation-creation process with spin-0 boson. In the presence of just a minimal measurable length as ultra-violet cutoff, momentum generalizes to p → p(1 + βp 2 ). So we have We have discard the remaining delta function which is that p i are defined in the geometry of the process as shown in figure 1. Simplifying relation (38), we find By solving the integral, we obtain The probability for the process to occur is given by |M| 2 . Now we calculate the scattering amplitude in the presence of both minimal length and maximal momentum. In this case, p → p(1 − βp + 2β 2 p 2 ) and we have which can be simplified to find By solving this integral, we find the following result for scattering amplitude in the presence of both minimal length and maximal momentum where by definition R is the root of 2z 2 + 7z + 7 − 32β 32 m b = 0. Now the probability for the process to occur in the presence of these natural cutoffs is given by |M| 2 .

Summary
In this paper we have constructed a quantum theory of free fermion fields based on the deformed Heisenberg algebra that contains both a minimal measurable length and a maximal momentum for test particles. Our strategy was to respect the supersymmetry in this context as a guiding principle. This is because string theory contains this symmetry and it provides a useful tool to understand physics in ultra-violet regime. Due to existence of natural cutoffs as minimal length and maximal momentum, the supersymmetry is deformed in this context. From the fermionic part of the system, we proposed an action of fermionic field based on the GUP in a spacetime with arbitrary number of dimensions. We have constructed the deformed supersymmetric algebra in this context. Finally, we have computed the scattering amplitude for pair annihilation to see how these important quantities can be calculated in the presence of natural cutoffs as minimal length and maximal momentum.