# Fractional Derivative Regularization in QFT

Authors: Vasily E. Tarasov
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We propose in this paper a 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 noninteger orders is suggested. The regularized loop integrals depend on parameter that is the orderα>0 of the fractional derivative. The regularization procedure is demonstrated for scalar massless fields in φ4-theory on n-dimensional pseudo-Euclidean space-time.


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## Body

## 1. 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 nontrivial limit of theory with an 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, and 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 integrodifferential equations of noninteger orders is a powerful tool to describe the dynamics of systems and processes with power-law nonlocality, long-range memory, and/or fractal properties.Recently, the spatial fractional-order derivatives have been actively used in the space-fractional quantum mechanics [6, 7], the quantum field theory and gravity for fractional space-time [8], and the fractional quantum field theory at positive temperature [9, 10].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 [11]. Then, noninteger powers of d’Alembertian are considered in different works (for example, see Section 28 in [4] and [12–14]). 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 [15–19], 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 [15–18].There are different definitions of fractional derivatives such as Riemann-Liouville, Caputo, Grünwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov, and 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 noninteger orders [20].It should be noted that the use of the fractional derivative of noninteger order is actually equivalent to using an infinite number of derivatives of all integer orders. For example, the Riemann-Liouville derivativeDRLa+α can be represented in the form of the infinite series(1)DRLa+αfx=∑k=0∞Γα+1Γk+1Γα-k+1x-ak-αΓk-α+1dkfxdxkfor 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 noninteger ordersα close to integer values. The use of these fractional derivatives of noninteger 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 spaceRn

### 2.1. Fractional Integration in the Riesz Form

The Riesz fractional integral of orderα for Rn is defined [11] (see also Section 25 of [4]) by equation(2)IRαfP=1Hnα∫RnfQrPQα-ndQ, where P and Q are points of the space Rn, and(3)Hnα=πn/22αΓα/2Γn-α/2.Let us give some properties of the Riesz fractional integral that are proved in [11].(1) The semigroup property of the Riesz fractional integration is(4)IRα1IRα2fP=IRα1+α2fP,whereα1>0,  α2>0,  α1+α2<n.(2) The action of the Laplace operator on the Riesz fractional integral is(5)ΔIRα+2fP=-IRαfP, and we also have(6)ΔkIRα+2kfP=-1kIRαfP,k∈N.(3) The Riesz fractional integration of exponential function does not change this function:(7)IRαeixj=eixj,IRαei∑j=1najxj=∑j=1najxj-α/2ei∑j=1najxj.There is the following important property:(8)IRαfP=-1mIRα+2mΔmfP.This property allows us to use an analytic continuation of IRαfP for negative values of α>-2m, where m∈R (see [11]). In this case, the semigroup property (4) can be used forα1>-2m,  α2>-2m,  α1+α2>-2m, and m∈R. Analogously, we can consider property (6) in the form(9)ΔkIRαfP=-1kIRα-2kfP,k∈N.


### 2.2. Fractional Laplacian in the Riesz Form

For the first time, fractional Laplace operators have been suggested by Riesz in [11] (see also Section 25 of [4]). The fractional Laplacian -Δα/2 in the Riesz form for n-dimensional Euclidean spaceRncan be considered [4] as an inverse Fourier’s integral transform F-1 of kα by(10)-Δα/2φx=F-1kαFφk,where α>0 and x∈Rn.Forα>0, the fractional Laplacian of the Riesz form can be defined [4] as the hypersingular integral(11)-Δα/2φx=1dnm,α∫Rn1zα+nΔzmφzd4z,where m>α and Δzmφz is a finite difference of order m of a field φx with a vector step z∈Rn centered at the point x∈Rn:(12)Δzmφz=∑j=0m-1jm!j!m-j!φx-jz. The constantdnm,αis defined by(13)dnm,α=π1+n/2Amα2αΓ1+α/2Γn+α/2sin⁡πα/2, where(14)Amα=∑j=0m-1j-1m!j!m-j!jα.Note that the hypersingular integral (11) does not depend on the choice of m>α. The Fourier transform F of the fractional Laplacian is given by F{-Δα/2φ}k=kαFφk. This equation is valid for the Lizorkin space [4] and the space C∞Rn of infinitely differentiable functions on Rn with compact support.


### 2.3. Fractional Laplacian in the Riesz-Trujillo Form

Using property (8), we can define the fractional Laplace operator by the equation(15)ΔRTα/2fP=-1mIR2m-αΔmfP, where m>α/2 and m∈R. This form of definition of the fractional Laplacian has been suggested by Trujillo in 2012 (see also [21]).It is important that the semigroup property holds for these operators(16)ΔRTα1/2ΔRTα2/2fP=ΔRTα1+α2/2fP,where0<α1<2m,  0<α2<2m, and  α1+α2<2m.We can see that(17)ΔRTkfP=-1kΔkfP, where k∈N. In Riesz’s notation [11], this equation means that(18)IR-2kfP=-1kΔkfP. We also have a generalization in the form(19)IRα-2kfP=-1kΔkIRαfP.Note that the value ofm can be chosen sufficiently large to fulfill all the conditions on the fractional order α in property (16).


## 2.1. Fractional Integration in the Riesz Form

The Riesz fractional integral of orderα for Rn is defined [11] (see also Section 25 of [4]) by equation(2)IRαfP=1Hnα∫RnfQrPQα-ndQ, where P and Q are points of the space Rn, and(3)Hnα=πn/22αΓα/2Γn-α/2.Let us give some properties of the Riesz fractional integral that are proved in [11].(1) The semigroup property of the Riesz fractional integration is(4)IRα1IRα2fP=IRα1+α2fP,whereα1>0,  α2>0,  α1+α2<n.(2) The action of the Laplace operator on the Riesz fractional integral is(5)ΔIRα+2fP=-IRαfP, and we also have(6)ΔkIRα+2kfP=-1kIRαfP,k∈N.(3) The Riesz fractional integration of exponential function does not change this function:(7)IRαeixj=eixj,IRαei∑j=1najxj=∑j=1najxj-α/2ei∑j=1najxj.There is the following important property:(8)IRαfP=-1mIRα+2mΔmfP.This property allows us to use an analytic continuation of IRαfP for negative values of α>-2m, where m∈R (see [11]). In this case, the semigroup property (4) can be used forα1>-2m,  α2>-2m,  α1+α2>-2m, and m∈R. Analogously, we can consider property (6) in the form(9)ΔkIRαfP=-1kIRα-2kfP,k∈N.


## 2.2. Fractional Laplacian in the Riesz Form

For the first time, fractional Laplace operators have been suggested by Riesz in [11] (see also Section 25 of [4]). The fractional Laplacian -Δα/2 in the Riesz form for n-dimensional Euclidean spaceRncan be considered [4] as an inverse Fourier’s integral transform F-1 of kα by(10)-Δα/2φx=F-1kαFφk,where α>0 and x∈Rn.Forα>0, the fractional Laplacian of the Riesz form can be defined [4] as the hypersingular integral(11)-Δα/2φx=1dnm,α∫Rn1zα+nΔzmφzd4z,where m>α and Δzmφz is a finite difference of order m of a field φx with a vector step z∈Rn centered at the point x∈Rn:(12)Δzmφz=∑j=0m-1jm!j!m-j!φx-jz. The constantdnm,αis defined by(13)dnm,α=π1+n/2Amα2αΓ1+α/2Γn+α/2sin⁡πα/2, where(14)Amα=∑j=0m-1j-1m!j!m-j!jα.Note that the hypersingular integral (11) does not depend on the choice of m>α. The Fourier transform F of the fractional Laplacian is given by F{-Δα/2φ}k=kαFφk. This equation is valid for the Lizorkin space [4] and the space C∞Rn of infinitely differentiable functions on Rn with compact support.


## 2.3. Fractional Laplacian in the Riesz-Trujillo Form

Using property (8), we can define the fractional Laplace operator by the equation(15)ΔRTα/2fP=-1mIR2m-αΔmfP, where m>α/2 and m∈R. This form of definition of the fractional Laplacian has been suggested by Trujillo in 2012 (see also [21]).It is important that the semigroup property holds for these operators(16)ΔRTα1/2ΔRTα2/2fP=ΔRTα1+α2/2fP,where0<α1<2m,  0<α2<2m, and  α1+α2<2m.We can see that(17)ΔRTkfP=-1kΔkfP, where k∈N. In Riesz’s notation [11], this equation means that(18)IR-2kfP=-1kΔkfP. We also have a generalization in the form(19)IRα-2kfP=-1kΔkIRαfP.Note that the value ofm can be chosen sufficiently large to fulfill all the conditions on the fractional order α in property (16).


## 3. Fractional d’Alembertian for Pseudo-Euclidean Space-TimeR1,n-1n

For the first time, the fractional d’Alembertian has been suggested by Riesz in [11] (see also [21]). Note that, in Riesz’s paper [11], the d’Alembertian □ is denoted by Δ and it is called for the Laplace operator for R1,n-1n. We will use the generally accepted notation.For the pseudo-Euclidean space-timeR1,n-1n, we use(20)rQP2=rPQ2=x1-y12-∑j=2nxj-yj2, and the operator(21)□=∂2∂x12-∑j=2n∂2∂xj2.The Riesz fractional integral is defined as(22)IRαfP=1Hnα∫R1,n-1nfQrPQα-ndQ, where Hnα is defined by the equations (see Eqs. 20 and 88 of [11])(23)Hnα=πn-2/22α-1Γα2Γα+2-n2=πn-1/2Γα+2-n/2ΓαΓα+1/2. This expression of Hnα 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 [11]):(24)IRαfP=IRα+2m□mfP,which allows us to use an analytic continuation of IRαfP for negative values of α if(25)α>n-2-2m.Using property (24), we can define the fractional d’Alembertian forn-dimensional pseudo-Euclidean space-time R1,n-1n by the equation(26)□α/2fP=IR2m-α□mfP,where m>α+n-2/2 and m∈R.It is important that the semigroup property,(27)□α1/2□α2/2fP=□α1+α2/2fP,holds for operators (26) if 0<α1<2m+2-n,  0<α2<2m+2-n, and α1+α2<2m+2-n.The fractional d’Alembertian□α/2 for n-dimensional pseudo-Euclidean space-time R1,n-1n can be considered as an inverse Fourier’s integral transform F-1 of kα by(28)□α/2φx=F-1kαFφk,whereα>0andx∈R1,n-1n.


## 4. Scalar Field in Pseudo-Euclidean Space-Time

Let us consider a scalar fieldφx in the n-dimensional pseudo-Euclidean space-time R1,n-1n that is described by the field equation(29)□+m2φx=Jx,where □ is the d’Alembert operator, φxis a real field, and x∈R1,n-1n is the space-time vector with components xμ, where μ=0,1,2,…,n. Suppose that the scalar field φx has a source Jx, and we have put ħ=1. Field equation (29) follows from stationary action principle, δSφ=0, where the action Sφ has the form(30)Sφ=∫R1,n-1ndnxLφxwith the Lagrangian(31)Lφx=-12φx□+m2φx.The solution of (31) can be represented in the form(32)φx=-∫ΣFx-yJydny, where ΣFx-y is the so-called Feynman propagator, obeying(33)□+m2-iεΣFx-yx=-δx. Here, δx is the Dirac delta function. It is easy to see that ΣFx-y has the Fourier representation(34)ΣFx-yx=∫dnk2πne-ikxk2-m2+iε.We can considerφ4 theory with an interaction constant g. It is known that ΣF0 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(35)gΣF0=g∫dnq2πn1q2-m2.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 ultraviolet divergent. Expression (35) corresponds to a one-loop Feynman diagram that has the order g. Another divergent diagram is the Og2 graph. The corresponding expression of the Feynman integral is(36)g2∫dnq12πndnq22πnδq1+q2-p1-p2q12-m2q22-m2=g2∫dnq2πn1q2-m2p1+p2-q2-m2.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.


## 5. Regularization

Regularization is a method of isolating the divergences. There are several techniques of regularization. The most intuitive one is to introduce a lattice regularization [22].(1) The Pauli-Villars regularization modifies the quadratic terms by subtraction of the same quadratic terms with a much larger mass:(37)12∂μφ∂μφ-12m2φ2-12∂μϕ∂μϕ-12M2ϕ2. The Pauli-Villars regularization is a contribution of another field (the Pauli-Villars fieldϕ) with the same quantum numbers as the original field, but having the opposite statistics. For example, in the φ4-theory, the Pauli-Villars field ϕ is used, and the Pauli-Villars regularization changes the propagator as(38)1p2+m2⟶1p2+m2-1p2+m2e-p2/Λ2p2+M2.(2) The Gaussian cut-off regularization is implemented by the modified kinetic term in the form(39)12∂μφe□/Λ2∂μφ, which changes the propagator(40)1p2+m2⟶e-p2/Λ2p2+m2.(3) The higher derivative regularization also modifies the quadratic terms such as(41)12∂μφ1-□Λ2∂μφ+12m2φ1-□Λ2φ. Then, the propagator is modified:(42)1p2+m2⟶1+p2Λ21p2+m2.


## 6. 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:(43)□⟶□α/2. Here, we use the dimensionless coordinates.The fractional derivative regularization modifies the quadratic terms such as(44)-12φx□+m2φx⟶-12φx□α/2+m2φx.Using the fractional d’Alembertian for pseudo-Euclidean space-timeR1,n-1n, we can replace field equation (29) by the equation(45)□α/2+m2φx=Jx,where □α is the fractional d’Alembert operator, φx is a real field, and x∈R1,n-1n is the space-time vector with components xμ, where μ=0,1,2,…,n.The solution of (45) is(46)φx=-∫ΣFx-yJydny, where ΣFx-y is the Feynman propagator, obeying(47)□α/2+m2-iεΣFx-yx=-δx. Using (28), it is easy to see that ΣFx-yx has the Fourier representation(48)ΣFx=∫dnk2πne-ikxk2α/2-m2+iε.In this case,ΣF0 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(49)gΣF0=g∫dnq2πn1q2α/2-m2.There are nth powers of q in the numerator and α in the denominator. Another divergent diagram is the Og2 graph. The corresponding expression of the Feynman integral is(50)g2∫dnq12πndnq22πnδq1+q2-p1-p2q12α/2-m2q22α/2-m2=g2∫dnq2πn1q2α/2-m2·1p1+p2-q2α/2-m2.Here, there are nth powers of q in numerator and the 2α powers in denominator.


## 7. FD Regularization for Massless Theory

For simplification, we will consider a massless scalar field theory. For this case, expression (36) with m=0 has the form(51)g2∫dnq12πndnq22πnδq1+q2-p1-p2q12α/2q22α/2=g2∫dnq2πn1q2α/2·1p1+p2-q2α/2.The denominators in the integrand of (36) are combined by using Feynman’s parametric integral formula(52)1AaBb=Γa+bΓaΓb∫01dzza-11-zb-1Az+B1-za+b.Changing the variablesq→p and p1+p2→q, integral (51) takes the form(53)Iq=g2∫dnp2πn1p2α/2·1p-q2α/2.Using formula (52), we get(54)1p2α/2·1p-q2α/2=∫01dzzα/2-11-zα/2-1p2-2pq1-z+q21-zα.As a result, we have(55)Iq=g2∫01zα/2-11-zα/2-1dz∫dnp2πn1p2-2pq1-z+q21-zα.Usingp=p0,r and the polar coordinates, we get that momentum integral of (55) is(56)In,α,q=∫dnpp2-2pQ+M2α=2πn-1/2Γn-1/2∫-∞+∞dp0∫0∞rn-2drp02-r2-2pQ+M2α,where(57)Q=q1-z,M2=q21-z.This integral is Lorentz invariant, so we evaluate it in the frameqμ=μ,0. Then, 2pQ=2μ1-zp0. Changing variables to pμ′=pμ-qμ1-z, which implies that(58)p02-2q1-zp0=p02-2μ1-zp0=p0′2-q21-z2, we have(59)In,α,q=2πn-1/2Γn-1/2∫-∞+∞dp0′∫0∞rn-2drp0′2-r2+q2z1-zα,where we use q21-z-q21-z2=q2z1-z.Using the beta function(60)2∫0∞t2x-11+t2x+y=Bx,y=ΓxΓyΓx+y, we get(61)∫0∞sβdss2+M2α=Γ1+β/2Γα-1+β/22M2α-1+β/2Γα,where we can use(62)M2=-p0′2-q2z1-z,β=n-2.Using (61) with (62), (59) gives(63)In,α,q=πn-1/2Γα-n-1/2Γα∫-∞+∞dp0′-p0′2-q2z1-zα-n-1/2=-1-α+n-1/2πn-1/2Γα-n-1/2Γα∫-∞+∞dp0′p0′2+q2z1-zα-n-1/2. Using (61), we get(64)In,α,q=iπn/2Γα-n/2Γα1-q2z1-zα-n/2.As a result, we obtain(65)In,α,q=∫dnpp2-2pq1-z+q21-zα=iπn/2Γα-n/2Γα1-q2z1-zα-n/2.Using (65), expression (55) has the form(66)Iq=ig2πn/2Γα-n/2Γα∫01zα/2-11-zα/2-1dz-q2z1-zα-n/2.Then, we have(67)Iq=ig2πn/2Γα-n/2Γα1-q2α-n/2∫01zn-α/2-11-zn-α/2-1dz.Forn=4 and α=2+ε, we get(68)1-q2α-n/2=-1n/2-α1+εln⁡q2+Oε2. Here, we can use(69)-1n/2-α=-1-ε=eiπε=1+iπε.We can introduce the function(70)An,α=∫01zn-α/2-11-zn-α/2-1dz.For n=4 and α=2+ε, expression (70) takes the form(71)A4,2+ε=∫01z1-z-ε/2dz=1-ε2∫01ln⁡z1-zdz+Oε2.Using(72)∫01ln⁡z1-zdz=-2, (71) is written as(73)A4,2+ε=1+ε+Oε2.For 4-dimensional space-time,(74)Γα-n/2Γα=1α-1α-2.For n=4 and α=2+ε, (74) gives(75)Γα-n/2Γα=11+εε=1ε-1.Then, integral (67) with n=4 and α=2+ε has the form(76)Iq=ig2π21ε-11+ε1+iπε1+εln⁡q2.As a result, we get(77)Iq=ig2π2ε+ig2π2ln⁡q2-π3g2+Oε.Remarks 1.
For the special cases, we have(78)A2,1=π,A4,1=π8,A4,2=1,A3,1=1.
The well-known asymptotic expression for the gamma function is(79)Γε=1ε-γ+12γ2+π26ε+Oε2ε⟶0.
For 2-dimensional space-time,(80)Γα-n/2Γα=1α-1. If α=1+ε, then(81)Γα-n/2Γα=1ε.
For 4-dimensional space-time (n=4),(82)Γα-n/2Γα=1α-1α-2. If α=1+ε, then(83)Γα-n/2Γα=1εε-1=-1ε-1. If α=2+ε, then(84)Γα-n/2Γα=11+εε=1ε-1.Remark 2.
By changing variables top′=p-q1-z, we see that the denominator in the integrand of (54) is the square of p′2-m2+q2z1-z. But dnp′=dnp, so relabeling p′→p, expression (51) becomes(85)Iq=g2∫01zα/2-11-zα/2-1dz∫dnp2πn1p2+q2z1-zα.


## 8. Conclusion

In this paper, we proposed a new regularization of quantum field theories, which is based on the Riesz fractional derivatives of noninteger orderα>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 noninteger order [11, 21] 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 [9, 10, 22–24]. It should be noted that we can use the Riesz fractional derivatives [11, 21], the fractional Laplacian, and d’Alembertian in the lattice quantum field theories [22]. Using the exact discretization of the fractional Laplacian d’Alembertian [19], we can consider exact discrete analogs [25] of the quantum field theories. The fractional derivatives of the Riesz type [11, 21] are directly connected to the long-range properties and nonlocal interactions [17, 18, 25]. 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.



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