Fractional Derivative Regularization in QFT

In this paper, we propose new regularization, where integer-order differential operators are replaced by fractional-order operators. Regularization for quantum field theories based on application of the Riesz fractional derivatives of non-integer orders is suggested. The regularized loop integrals depend on parameter that is the order alpha>0 of the fractional derivative. The regularization procedure is demonstrated for scalar massless fields in phi^4-theory on n-dimensional pseudo-Euclidean space-time.


Introduction
A characteristic feature of quantum field theories, which are used in high-energy physics, is ultraviolet divergence [1,2]. These divergences arise in momentum space from modes of very high moment, i.e. the structure of the field theories at very short distances. In the narrow class of quantum theories, which are called "renormalizable", the divergences can be removed by a singular redefinition of the parameters of the theory. This process is called the renormalization [3], and it defines a quantum field theory as a non-trivial limit of theory with a ultraviolet cut-off.
The renormalization requires the regularization of the loop integrals in momentum space. These regularized integrals depend on parameters such as the momentum cut-off, the Pauli-Villars masses, the dimensional regularization parameter, which are used in the corresponding regularization procedure. This regularized integration is ultraviolet finite.
We suggest a regularization procedure based on fractional-order derivatives of the Riesz type. In momentum space these derivatives are represented by power functions of momentum.
Fractional calculus and fractional differential equations [4,5] have a wide application in mechanics and physics. The theory of integro-differential equations of non-integer orders is a powerful tool to describe the dynamics of systems and processes with power-law non-locality, long-range memory and/or fractal properties.
Recently the spatial fractional-order derivatives have been actively used in the space-fractional quantum mechanics [7,8], the quantum field theory and gravity for fractional space-time [9], the fractional quantum field theory at positive temperature [10,11].
Fractional calculus allows us to take into account fractional power-law nonlocality of continuously distributed systems and classical fields. Using the fractional calculus, we can consider space-time fractional differential equations in the quantum field theory. For the first time, the fractional-order Laplace and d'Alembert operators have been suggested by Riesz in [12]. Then non-integer powers of d'Alembertian are considered in different works (for example, see Section 28 in [4] and [13,14,15]). The fractional Laplace and d'Alembert operators of the Riesz type are the basis for the fractional field theory in multidimensional spaces.
As it was shown in [16,17,18,19,20], the continuum equations with fractional derivatives of the Riesz type can be directly connected to lattice models with long-range properties. A connection between the dynamics of lattice system with long-range properties and the fractional continuum equations are proved by using the transform operation [16,17,18,19].
There are different definitions of fractional derivatives such as Riemann-Liouville, Caputo, Grünwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov, Riesz [4,5]. Unfortunately all these fractional derivatives have a lot of unusual properties. For example, the well-known Leibniz rule does not hold for differentiation of non-integer orders [26].
It should be noted that the use of the fractional derivative of non-integer order is actually equivalent to using an infinite number of derivatives of all integer orders. For example, the Riemann-Liouville derivative RL D α a+ can be represented in the form of the infinite series for analytic (expandable in a power series on the interval) functions on (a, b), (see Lemma 15.3 of [4]). Therefore, an application of fractional derivatives means that we take into account a contribution of derivatives of all integer orders with the power-law weight.
In the suggested FD-regularization we use fractional derivatives of non-integer orders α close to integer values. The use of these fractional derivatives of non-integer orders α close to integer values in the regularization procedure means that we consider a weak nonlocality, i.e. a slight deviation from the locality.
2 Fractional Laplacian for Euclidean space R n

Fractional integration in the Riesz form
The Riesz fractional integral of order α for R n is defined [12] (see also Section 25 of [4]) by equation where P and Q are points of the space R n , and Let us give some properties of the Riesz fractional integral that are proved in [12]: 1. The semi-group property of the Riesz fractional integration where α 1 > 0, α 2 > 0, α 1 + α 2 < n.
2. The action of the Laplace operator on the Riesz fractional integral and we also have 3. The Riesz fractional integration of exponential function does not change this function There is the following important property This property allows us to use an analytic continuation of R I α f (P ) for negative values of α > −2m, where m ∈ R (see [12]). In this case, the semigroup property (4) can be used for α 1 > −2m, α 2 > −2m, α 1 + α 2 > −2m, and m ∈ R. Analogously, we can consider the property (6) in the form

Fractional Laplacian in the Riesz form
For the first time fractional Laplace operators have been suggested by Riesz in [12] (see also Section 25 of [4]). The fractional Laplacian (−∆) α/2 in the Riesz form for n-dimensional Euclidean space R n can be considered [4] as an inverse Fourier's integral transform where α > 0 and x ∈ R n . For α > 0, the fractional Laplacian of the Riesz form can be defined [4] as the hypersingular integral where m > α, and (∆ m z ϕ)(z) is a finite difference of order m of a field ϕ(x) with a vector step z ∈ R n and centered at the point x ∈ R n : Note that the hypersingular integral (12) does not depend on the choice of m > α. The Fourier transform F of the fractional Laplacian is given by This equation is valid for the Lizorkin space [4] and the space C ∞ (R n ) of infinitely differentiable functions on R n with compact support.

Fractional Laplacian in the Riesz-Trujillo form
Using the property (9), we can define the fractional Laplace operator by the equation where m > α/2 and m ∈ R. This form of definition of the fractional Laplacian has been suggested by J.J. Trujillo in 2012 (see also [25]). It is important that the semi-group property holds for these operators where 0 < α 1 < 2m, 0 < α 2 < 2m, α 1 + α 2 < 2m.
We can see that where k ∈ N. In the Riesz's notation [12], this equation means that We also have a generalization in the form Note that the value of m can be chosen sufficiently large to fulfill all the conditions on the fractional order α in the property (14).

Fractional d'Alembertian for pseudo-Euclidean space
For the first time the fractional d'Alembertian has been suggested by Riesz in [12] (see also [25]). Note that in the Riesz paper [12], the d'Alembertian is denoted by ∆ and it is called for the Laplace operator for R n 1,n−1 . We will use the generally accepted notation. For the pseudo-Euclidean space-time R n 1,n−1 , we use and the operator The Riesz fractional integal is defined as where H n (α) is defined by the equations (see Eq. 20 and 88 of [12]) This expression of H n (α) does not coincide with the Euclidean case (3). To ensure the convergence of the integral, we should assume that the parameter α satisfies the condition α−n > −2, i.e. α > n − 2.
There is the following important property (see Eq. 67 of Chapter 3 in [12]) that allows us to use an analytic continuation of R I α f (P ) for negative values of α if Using the property (22), we can define the fractional d'Alembertian for n-dimensional pseudo-Euclidean space-time R n 1,n−1 by the equation where m > (α + n − 2)/2 and m ∈ R.

Scalar field in pseudo-Euclidean space-time
Let us consider a scalar field ϕ(x) in the n-dimensional pseudo-Euclidean space-time R n 1,n−1 that is described by the field equation where is the d'Alembert operator, ϕ(x) is a real field, and x ∈ R n 1,n−1 is the space-time vector with components x µ , where µ = 0, 1, 2, ..., n. Suppose that the scalar field ϕ(x) has a source J(x), and we have put = 1. Field equation (26) with the Lagrangian The solution of (28) can be represented in the form where Σ F (x − y) is the so-called Feynman propagator, obeying Here δ(x) is the Dirac delta-function. It is easy to see that Σ F (x − y) has the Fourier representation We can consider ϕ 4 theory with an interaction constant g. It is known that Σ F (0) is a divergent quantity, which modifies the free particle propagator and contributes to the selfenergy. In momentum space it corresponds to the loop integral There are nth powers of q in the numerator and two in the denominator, so the integral diverges quadratically at large q, i.e. it is ultra-violet divergent. Expression (32) corresponds to a oneloop diagram Feynman that has the order g. Another divergent diagram is the O(g 2 ) graph. The corresponding expression of the Feynman integral is Here there are nth powers of q in numerator and the four powers in denominator. For n = 4, we have four powers of q in both numerator and denominator, so we get a logarithmic divergence.

Regularization
Regularization is a method of isolating the divergences. There are several techniques of regularization. The most intuitive one is to introduce a lattice reguariazation [21]. 1. The Pauli-Villars regularization modifies the quadratic terms by subtraction off the same quadratic terms with a much larger mass The Pauli-Villars regularization is a contribution of another field (the Pauli-Villars field φ) with the same quantum numbers as the original fieeld, but having the opposite statistics. For example, in the ϕ 4 -theory, the Pauli-Villars field φ is used, and the Pauli-Villars regularization changes the propagator as 2. The Gaussian cutoff regularization is implemented by the modified kinetic term in the form which changes the propagator 1 3. The higher derivative regularization also modifies the quadratic terms such as Then the propagator is modified (39)

Regularization by fractional derivatives
We propose a new regularization, where integer-order differential operators are replaced by fractional-order operators. For example, we can replace , which is the d'Alembert operator in the field equations, by the fractional d'Alembert operator α : Here we use the dimensionless coordinates. The fractional derivative regularization modifies the quadratic terms such as Using the fractional d'Alembertian for pseudo-Euclidean space-time R n 1,n−1 , we can replace field equation (26) by the equation where α is the fractional d'Alembert operator, ϕ(x) is a real field, and x ∈ R n 1,n−1 is the space-time vector with components x µ , where µ = 0, 1, 2, ..., n.
The solution of (42) is where Σ F (x − y) is the Feynman propagator, obeying Using (25), it is easy to see that Σ F (x − y)(x) has the Fourier representation In this case, Σ F (0) is a divergent quantity, which modifies the free particle propagator and contributes to the self-energy. In momentum space it corresponds to the loop integral There are nth powers of q in the numerator and α in the denominator. Another divergent diagram is the O(g 2 ) graph. The corresponding expression of the Feynman integral is Here there are nth powers of q in numerator and the 2α powers in denominator.

FD-regularization for massless theory
For simplification, we will consider a massless scalar field theory. For this case, expression (33) with m = 0 has the form The denominators in the integrand of (33) are combined by using the Feynman's parametric integral formula 1 Changing the variables q → p and p 1 + p 2 → q, the integral (48) takes the form Using formula (49), we get As a result, we have Using p = (p 0 , r) and the polar coordinates, we get that momentum integral of (52) is where This integral is Lorentz invariant, so we evaluate it in the frame q µ = (µ, 0). Then 2pQ = 2µ(1 − z)p 0 . Changing variables to p ′ µ = p µ − q µ (1 − z), which implies that we have where we use Using the beta function we get where we can use Using (58) with (59), equation (56) gives As a result, we obtain Using (61), expression (52) has the form Then we have For n = 4 and α = 2 + ε, we get 1 Here we can use We can introduce the function For n = 4 and α = 2 + ε, expression (66) takes the form equation (67) is written as A(4, 2 + ε) = 1 + ε + O(ε 2 ).

Conclusion
In this paper, we proposed a new regularization of quantum field theories, which is based on the Riesz fractional derivatives of non-integer order α > 0. The fractional derivatives of noninteger order are characterized by nonlocality, the proposed FD regularization is actually a regularization by space-time nonlocality.
Note that the Riesz fractional derivatives of non-integer order [12,25] can be used not only for FD regularization. An application of fractional calculus in quantum field theory allows us to take into account space and time nonlocality [28,29,10,11,21]. It should be noted that we can use the Riesz fractional derivatives [12,25], the fractional Laplacian and d'Alembertian in the lattice quantum field theories [21]. Using the exact discretization of the fractional Laplacian d'Alembertian [20], we can consider exact discrete analogs [27] of the quantum field theories. The fractional derivatives of the Riesz type [12,25] are directly connected to the long-range properties and nonlocal interactions [18,19,27]. Because of this, this tool can be used not only for regularization, but also for constructing new quantum theories in high-energy physics and condensed matter physics.