Self-Quartic Interaction for a Scalar Field in an Extended DFR

Home , Quartic interaction

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I. INTRODUCTION

The understanding of Physics of the early Universe is one the great puzzles of the current science. The idea of combining the fundamental physical constants like ~,c and G to obtain a length scale was given by M. Planck in 1900

~ G −33 ℓP = 3 1.6 10 m , (1.1) r c ∼ ×

where ℓP is the Planck’s length. This relation shows that there is a mixing of quantum phenomena (~), and gravitational world (G) at this length scale. Indeed, it is believed that effects of a quantum gravity theory must emerge next to the Planck’s scale. The introduction of a length scale in a physics theory was only forty seven years after Planck’s idea by constructing a noncommutative (NC) spacetime. The motivation to introduce such scale was the need to tame the ultraviolet divergences in quantum field theory (QFT). The first published work concerning a NC concept of space-time was carried out in 1947 by Snyder in his seminal paper [1]. The central NC idea is that the space-time coordinates xµ (µ = 0, 1, 2, 3) are promoted to operators in order to satisfy the basic commutation relation

[ˆxµ , xˆν ] = i ℓθµν , (1.2)

where θµν is an antisymmetric constant matrix, and ℓ is a length scale. The alternative would be to construct a discrete space-time with a NC algebra. Consequently, the coordinates operators are quantum observable that satisfy the uncertainty relation

∆ˆxµ∆ˆxν ℓθµν , (1.3) ≃ it leads to the interpretation that noncommutativity (NCY) of spacetime must emerge in a fundamental length scale ℓ, i.e. the Planck’s scale, for example. However, Yang [2], a little time later, demonstrated that Snyder’s hopes in cutting off the infinities in QFT were not obtained by NCY. This fact doomed Snyder’s NC theory to years of ostracism. After the important result that the algebra obtained with a string theory embedded in a magnetic background is NC, a new perspective concerning NCY was rekindle [3]. Nowadays, the NC quantum field theory (NCQFT) is one of the most investigated subjects about the description of underlying physics at a fundamental length scale of quantum gravity [4]. The most popular NCY formalism consider θµν as a constant matrix but different from (1.2), this formalism is called canonical NCY and is given by

[xµ, xν]= i θµν . (1.4)

Although it maintains the translational invariance, the Lorentz symmetry is not preserved [5]. For example, in the case of the hydrogen atom, it breaks the rotational symmetry of the model, which removes the degeneracy of the energy levels [39]. To heal this disease a recent approach was introduced by Doplicher, Fredenhagen and Roberts (DFR) [6]. It considers θµν as an ordinary coordinate of the system in which the Lorentz symmetry is preserved. Recently, it has emerged the idea [7] of constructing an extension of this so- called DFR spacetime introducing the conjugate canonical momenta associated with θµν Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 3

[8] (for a review the reader can see [9]). This extended NC spacetime has ten dimensions: four relative to Minkowski spacetime and six relative to θ-space. This new framework is characterized by a field theory constructed in a space-time with extra-dimensions (4 + 6), and which does not need necessarily the presence of a length scale ℓ localized into the six dimensions of the θ-space, where, from (1.4) we can see that θµν now has dimension of length-square. Besides the Lorentz invariance was recovered, and obviously we hope that causality aspects in QFT in this (x + θ) space-time must be preserved too [10]. By following this concept, the algebraic structure of this DFR-extended (which will be called DFR∗ from now on) NC phase-space is enlarged by introducing the momenta operators associated to the coordinates xµ and to the new coordinates θµν . We have the usual µ canonical momenta pµ associated to coordinates x , and for simplicity, we call kµν the anti- symmetrical canonical momentum associated to the new coordinate θµν . These objects are promoted to operators in an also extended Hilbert space [8, 16–18]. All the corresponding operators belong to the same algebra and have the sameH hierarchical level. Recently it was demonstrated [19] that the canonical momentum kµν is in fact connected to the Lorentz invariance of the system. Hence, it is possible to construct a filed theory with this extended phase-space [20]. The addition of the canonical momenta kµν is important if we are interested in the investigation of the propagation of fields along the θ-space. Our objective here is to disclose and analyze new physical aspects that can emerge from the propagation of fields through the θ-direction. In this paper we obtain the one loop corrections to the scalar propagator NC φ4⋆ defined in the DFR∗ space. We propose a NC action with a self quartic interaction φ4⋆, and the Feynman rules, necessary to perturbation theory are obtained. Another target here is to study the primitive divergences of this model and to calculate the one loop correction to the scalar propagator for the cases in which kµν = 0, which is a NC toy model, as we will see, and kµν = 0. The existence of the NC length scale can bring something new to the mass correction6 in the propagator due to the perturbation theory. We will also investigate the influence of a extended Fourier space (pµ,kµν) in the primitive divergence of the propagator. The paper is organized as: the next section is dedicated (for self-containment of this work) to a review of the basics of NC quantum field theory DFR∗ framework, namely, the DFR∗ algebra. In section 3 we write the action of scalar field with φ4⋆ interaction, and consequently, the Feynman rules in the Fourier space. Section 4 is dedicated to the primitive divergences of model. We use the Feynman rules to obtain the expressions of the self-energy at the one loop approximation. We divide it in two subsections, where in the first one we have the case of the self-energy with kµν = 0 (toy model). The second subsection describes the case when kµν = 0, in which the self-energy is calculated to investigate the consequence of this momentum6 in the ultraviolet divergences. To finish, we discussed the results obtained and we depict the final remarks and conclusions.

II. THE DFR∗ NC SPACETIME

A. The algebra

In this section, we will review the main steps published in [8, 16–18]. Namely, we will revisit the basics of the quantum field theory defined in the DFR∗ space. The space-time coordinates xµ = (t, x) do not commute itself satisfying the commutation relation (1.2). Self-quartic interaction for a scalar field in an extended DFR.... 4

The parameters θµν are promoted to coordinates of this spacetime, which has D = 10 and it has six independents spatial coordinates associated to θµν . Consequently, the parameters θµν are promoted to operators θˆµν in the commutation relation. Let us begin with the standard DFR algebra [6] involving only the position operators

[ˆxµ, xˆν ]= iθˆµν , xˆµ, θˆνα = 0 and θˆµν , θˆαβ =0 . (2.1) h i h i notice that we cannot say that θµν are any kind of position operator because, although it is a coordinate in the DFR∗ space, it does not mean that it provide any kind of localization, like xµ do in standard commutative spacetime. It can be shown that this well known standard DFR space is in fact incomplete. As a matter of fact, in [19], one of us showed µν that the existence of the canonical momentum kµν is intrinsically connected with θ and, consequently, with the Lorentz invariance. That is the reason we have called a DFR∗ system (that will be analyzed here) where the momentum associated to θµν is zero, a toy model. In µν other words, if θ are coordinates, there exists kµν. µν Hence, the canonical conjugate momenta operator kˆµν associated with the operator θˆ must satisfy the commutation relation

ˆµν ˆ µν θ , kρσ = iδ ρσ , (2.2) h i where δµν = δµ δν δµ δν . In order to obtain consistency we can write that [8] ρσ ρ σ − σ ρ [ˆxµ, pˆν]= iηµν , [ˆpµ, pˆν]=0 , θˆµν , pˆρ =0 , h i i pˆµ, kˆνρ =0 , [ˆxµ, kˆ ]= δ µσpˆ , (2.3) νρ −2 νρ σ h i and this completes the DFR∗ algebra. It is possible to verify that the whole set of commutation relations listed above is indeed consistent with all possible Jacobi identities and the CCR algebras [10]. The θµν coordinates are constrained by the quantum conditions 1 θ θµν = 0 and ⋆θ θµν = ℓ 4 , (2.4) µν 4 µν P ρσ where ⋆θµν = εµνρσθ and λP is the Planck length. Here we have adopted that (~ = c = ℓ = 1) and consequently the coordinates θµν have dimension of length to the squared. The uncertainty principle (1.2) is altered by

∆ˆxµ∆ˆxν θˆµν , (2.5) ≃h i in which the expected value of the operator θˆ is related to the particles fluctuation position. The last commutation relation of (2.3) suggests that the shifted coordinate operator [39–44] i Xˆ µ =x ˆµ + θˆµν pˆ , (2.6) 2 ν ˆ commutes with kµν. The relation (2.6) is also known as Bopp shift in the literature. The commutation relation (2.3) also commutes with θˆµν and Xˆ µ, and satisfies a non trivial commutation relation withp ˆµ dependent objects, which could be derived from [Xˆ µ, pˆν]= iηµν , [Xˆ µ, Xˆ ν] = 0 (2.7) Self-quartic interaction for a scalar field in an extended DFR.... 5

µ µ and one can note that the propertyp ˆµXˆ =p ˆµxˆ can be easily verified. Hence, we see from these both equations that the shifted coordinated operator (2.6) allows us to recover the commutativity property. The shifted coordinate operator Xˆ µ plays a fundamental role in NC quantum mechanics defined in the (x + θ)-space, since it is possible to form a basis with its eigenvalues. This possibility is forbidden for the usual coordinate operatorx ˆµ since its components satisfy nontrivial commutation relations among themselves (2.1). So, differently fromx ˆµ, we can say that Xˆ µ forms a basis in Hilbert space. The generator of Lorentz group is

Mˆ = Xˆ pˆ Xˆ pˆ + θˆ kˆρ θˆ kˆρ , (2.8) µν µ ν − ν µ νρ µ − µρ ν

and from (2.3) we can write the generators for translations asp ˆµ i∂µ . With these ingredients it is easy to construct the commutation relations → −

[ˆpµ, pˆν]=0 , Mˆ , pˆ = i η pˆ η pˆ , µν ρ µρ ν − µν ρ h i Mˆ , Mˆ = i η Mˆ η Mˆ η Mˆ + η Mˆ , (2.9) µν ρσ µσ ρν − νσ ρµ − µρ σν νρ σµ h i it closes the appropriated algebra, and we can say thatp ˆµ and Mˆ µν are the generators of the DFR∗ algebra. Analyzing the Lorentz symmetry in NCQM following the lines above, we can introduce an appropriate theory, for instance, given by a scalar action. It is well known, however, that elementary particles are classified according to the eigenvalues of the Casimir operators of the inhomogeneous Lorentz group. Hence, let us extend this approach to the Poincarégroup . Considering the operators presented here, we can in principle consider that P 1 1 Gˆ = ω Mˆ µν aµpˆ + b kˆµν , (2.10) 2 µν − µ 2 µν is the generator of some group ′, which has the Poincarégroup as a subgroup. By defining the dynamical transformationP of an arbitrary operator Aˆ in in such a way that δAˆ = i [A,ˆ Gˆ] we arrive at the set of transformations, H

µ µ ν µ δxˆ = ω νxˆ + a ν δpˆµ = ωµ pˆν ˆµν µ ˆρν ν ˆµρ µν δθ = ω ρθ + ω ρθ + b ˆ ρˆ ρˆ δkµν = ωµ kρν + ων kµρ δMˆ µν = ωµ Mˆ ρν + ων Mˆ µρ + aµpˆν aνpˆµ 1 ρ 1 ρ 1 − ˆ µν µ ˆ ρν ν ˆ µρ µρˆ ν νρˆµ δM2 = ω ρM2 + ω ρM2 + b kρ + b k ρ 1 δxˆµ = ωµ xˆν + aµ + bµν pˆ . (2.11) ν 2 ν One can observe that there is an unexpected term in the last equation of (2.11). This is a consequence of the coordinate operator in (2.6), which is a nonlinear combination of operators that act on diﬀerent Hilbert spaces. Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 6

The action of ′ over the Hilbert space operators is in some sense equal to the action of the Poincar´egroup withP an additional translation operation on the (θˆµν ) sector. Its generators, all of them, close in an algebra under commutation. Hence, ′ is a well deﬁned group of transformations. As a matter of fact, the commutation of twoP transformations closes in the algebra [δ2, δ1]ˆy = δ3 yˆ , (2.12) where y represents any one of the operators appearing in (2.11). The parameters composition rule is given by ωµ = ωµ ωα ωµ ωα 3 ν 1 α 2 ν − 2 α 1 ν aµ = ωµ aν ωµ aν 3 1 ν 2 − 2 ν 1 bµν = ωµ bρν ωµ bρν ων bρµ + ων bρµ . (2.13) 3 1 ρ 2 − 2 ρ 1 − 1 ρ 2 2 ρ 1

B. DFR∗ quantum mechanics and ﬁeld theory

To sum up, the framework showed above demonstrated that in NCQM, the physical coordinates do not commute and the respective eigenvectors cannot be used to form a basis in = 1 2 [16]. This can be accomplished using the Bopp shift defined in (2.6) with (2.7)H asH consequence.⊕ H So, we can introduce a coordinate basis X′, θ′ = X′ θ′ and p′,k′ = p′ k′ , in such a way that | i | i⊗| i | i | i⊗| i Xˆ µ X′, θ′ = X′µ X′, θ′ and θˆµν X′, θ′ = θ′µν X′, θ′ , (2.14) | i | i | i | i and pˆ p′,k′ = p′ p′,k′ , kˆ p′,k′ = k′ p′,k′ . (2.15) µ| i µ| i µν | i µν| i The wave function φ(X′, θ′)= X′, θ′ φ represents the physical state φ in the coordinate basis defined above. This wave functionh | i satisfies some wave equation| thati can be derived from an action, through a variational principle, as usual. In [16], the author constructed directly an ordinary relativistic free quantum theory. It was assumed that the physical states are annihilated by the mass-shell condition pˆ pˆµ m2 φ =0 , (2.16) µ − | i µ demonstrated through the Casimir operator C1 =p ˆµpˆ (for more algebraic details see [16]). It is easy to see that in the coordinate representation, this originates the NC Klein-Gordon equation. Condition (2.16) selects the physical states that must be invariant under gauge transformations. To treat the NC case, let us assume that the second mass-shell condition

kˆ kˆµν ∆2 φ =0 , (2.17) µν − | i and must be imposed on the physical states, where ∆ is some constant with dimension M 4, which sign and value can be deﬁned if k is spacelike, timelike or null. Analogously the µν Casimir invariant is C2 = kˆµν kˆ , demonstrated the validity of (2.16) (see [16] for details). Both equations (2.16) and (2.17) permit us to construct a general expression for the plane wave solution such as [16] d4p d6k i φ(x′, θ′) := X′, θ′ φ = φ(p,kµν) exp ip x′µ + k θ′µν , (2.18) h | i (2π)4 (2πλ−2)6 µ 2 µν Z e Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 7

where p2 m2 = 0 and k2 ∆2 = 0, and we have used that p X = p x. The length λ−2 is introduced− conveniently in− the k-integration to maintain the ﬁeld· with· dimension of length inverse. Consequently, the k-integration stays dimensionless. In coordinate representation, the operators (ˆp, kˆ) are written in terms of the derivatives ∂ pˆ i∂ and kˆ i , (2.19) µ → − µ µν → − ∂θµν and consequently, both (2.16) and (2.17) are combined into a single equation to give the Klein-Gordon equation in the DFR space for the scalar ﬁeld φ

2 2 ✷ + λ ✷θ + m φ(x, θ)=0 , (2.20)

✷ 1 µν ∂ µν where we have deﬁned θ = 2 ∂ ∂µν and ∂µν = ∂θµν , with η = diag(1, 1, 1, 1). Substituting the wave plane solution (2.18), we obtain the mass invariant − − −

λ2 p2 + k kµν = m2 , (2.21) 2 µν where λ is a parameter with dimension of length defined before, as the Planck length. We µν define the components of the k-momentum k =( k, k) and kµν =(k, k), to obtain the DFR∗ dispersion relation − − e e ω(p, k, k)= p2 + λ2 k2 + k2 + m2 , (2.22) r e e in which ki is the dual vector of the components kij, that is, kij = ǫijkkk (i, j, k = 1, 2, 3, ). It is easy to see that, promoting the limit λ 0 in Eq. (2.22) causes the recovering of → commutativitye [10]. e To propose the action of a scalar field we need to define the Weyl representation for DFR operators. It is given by the mapping

4 6 d p d k · i ·ˆ ˆ (f)(ˆx, θˆ)= f(p,kµν) eip xˆ+ 2 k θ , (2.23) W (2π)4 (2πλ−2)6 Z e ∗ in which (ˆx, θˆ) are the position operators satisfying the DFR algebra, pµ and kµν are the conjugated momentum of the coordinates xµ and θµν , respectively. The Weyl symbol provides a map from the operator algebra to the algebra of functions equipped with a star- product, via the Weyl-Moyal correspondence

fˆ(ˆx, θˆ)g ˆ(ˆx, θˆ) f(x, θ) ⋆g(x, θ) , (2.24) ↔ and the star-product turns out to be the same as in the usual NC case

i µν ′ ′ f(x, θ) ⋆g(x, θ)= e 2 θ ∂µ∂ν f(x, θ)g(x , θ) , (2.25) x′=x

for any functions f and g. The Weyl operator (2.23) has the following traces properties

Tr ˆ (f) = d4xd6θ W (θ) f(x, θ) , (2.26) W h i Z Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 8

and for a product of n functions (f1, ..., fn)

Tr ˆ (f )... ˆ (f ) = d4xd6θ W (θ) f (x, θ) ⋆ ... ⋆ f (x, θ) . (2.27) W 1 W n 1 n h i Z The function W is a Lorentz invariant integration-θ measure. This weight function is introduced in the context of NC ﬁeld theory to control divergences of the integration in the θ-space [7, 12, 14]. It will permit us to work with series expansions in θ, i.e., with truncated power series expansion of functions of θ. For any large θµν it falls to zero quickly so that all integrals are well deﬁned, in that it is assumed the normalization condition

1 = d6θ W (θ)=1 . (2.28) h i Z The function W should be an even function of θ, that is, W ( θ)= W (θ), and consequently it implies that −

θµν = d6θ W (θ) θµν =0 . (2.29) h i Z The non trivial integrals are expressed in terms of the invariant

θ2n = d6θ W (θ)(θ θµν )n , with n Z , (2.30) h i µν ∈ + Z in which the normalization condition corresponding to the case n = 0. For n = 1, we have that θ2 d6θ W (θ) θµν θρλ = h i1[µν,ρλ] , (2.31) 6 Z where 1[µν,ρλ] := (gµρgνλ gµλgνρ)/2 is the identity antisymmetric on index (µν) and (ρλ). For n = 2, we can write that− θ4 d6θ W (θ) θµν θρλθαβθγσ = h i 1[µν,ρλ]1[αβ,γσ] + 1[µν,αβ]1[ρλ,γσ] + 1[µν,γσ]1[ρλ,αβ] . (2.32) 48 Z Using the previous properties an important integration is

2 i µν − hθ i µν d6θ W (θ) e 2 kµν θ = e 48 kµν k , (2.33) Z that is easily demonstrated integrating the exponential series. By the deﬁnition of the Moyal product (2.25) it is trivial to obtain the property

d4xd6θ W (θ) f(x, θ) ⋆g(x, θ)= d4xd6θ W (θ) f(x, θ) g(x, θ) . (2.34) Z Z The physical interpretation of average of the components of θµν, i.e. θ2 , is the deﬁnition of the NC energy scale [12] h i

12 1/4 1 Λ = = , (2.35) NC θ2 λ h i Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 9

in which λ is the fundamental length scale that emerging in the KLEIN-GORDON equation (2.20), and in the dispersion relation (2.22). This approach has the advantage that it is not need to specify the form of the function W , at least for lowest-order processes. The study of Lorentz-invariant noncommutative QED, as Bhabha scattering, dilepton and diphoton production to LEP data led the authors of [13, 14] to the bound

ΛNC > 160 GeV 95% C.L. . (2.36) After the discussion of the W -function we postulate the completeness relations

d4x′d6θ′ W (θ′) X′, θ′ X′, θ′ = 1 , (2.37) | ih | Z and d4p′ d6k′ p′,k′ p′,k′ = 1 . (2.38) (2π)4 (2π)6 | ih | Z Using the previous completeness relations and the integral (2.33), we can obtain

X′, θ′ X′′, θ′′ = δ(4)(x′ x′′) W −1(θ′)δ(6)(θ′ θ′′) , (2.39) h | i − − and

4 2 − λ − ′ p,k p′,k′ = (2π)4δ(4)(p p′) e 4 (kµν kµν ) , (2.40) h | i − where we have used θ2 = 12λ4 from (2.35), and the matrix elements h i X′, θ′ pˆ X′′, θ′′ = i ∂′ δ(4)(x′ x′′)W −1(θ′)δ(6)(θ′ θ′′) , (2.41) h | µ| i − µ − − and ∂ X′, θ′ kˆ X′′, θ′′ = δ(4)(x′ x′′)( i ) W −1(θ′)δ(6)(θ′ θ′′) , (2.42) h | µν| i − − ∂θ′µν − that confirms the differential representation of (2.19). The result (2.40) reveals that the µν canonical momentum kµν associated to θ is not conserved due to introduction of the W -function. Mathematically speaking, in (2.40) we can see that at first sight there is a momentum conservation problem at the vertice. It will be clarified in the future.

III. THE ACTION OF φ4⋆ MODEL AND FEYNMAN RULES

The DFR∗ action of a scalar field with quartic self-interaction φ4⋆, defined at the NC spacetime is given by 1 λ4 1 g S(φ)= d4xd6θ W (θ) ∂ φ ⋆ ∂µφ + ∂ φ ⋆ ∂µν φ m2φ⋆φ φ⋆φ⋆φ⋆φ , 2 µ 4 µν − 2 − 4! Z (3.1) where g is a constant coupling, and using the identity (2.34), this action is reduced to 1 λ4 g S(φ)= d4x (∂ φ)2 + (∂ φ)2 m2φ2 d4xd6θ W (θ)(φ⋆φ)2 . (3.2) 2 µ 2 µν − − 4! Z Z Self-quartic interaction for a scalar field in an extended DFR.... 10

Using the Fourier transform, the free action in momentum space gives us the Feynman propagator [11]

4 2 λ ′ − 4 (kµν +kµν ) ′ 4 4 ′ i e ∆F (p; kµν,k )=(2π) δ (p + p) 4 . (3.3) µν p2 + λ k′ kµν m2 + iε 2 µν − The inﬂuence of the non-commutativity via Moyal’s product is in the interaction term φ4⋆ of (3.2). To obtain the convenient Feynman rule for the vertex, we write the interaction term of (3.2) in the Fourier space

g 4 d4p d6k S (φ)= i i φ(p ,k )(2π)4δ(4) (p + p + p + p ) int −4! (2π)4 (2π)6 i iµν 1 2 3 4 × i=1 Z Y e e i µν d6θ W (θ) F (p ,p ,p ,p ) e 2 (k1µν +k2µν +k3µν +k4µν )θ , (3.4) × 1 2 3 4 Z where φ is Fourier transform of φ and F is a function totally symmetric that changes the momenta (p1,p2,p3,p4) as e 1 p p p p p p p p F (p ,p ,p ,p )= cos 1 ∧ 2 cos 3 ∧ 4 + cos 1 ∧ 3 cos 2 ∧ 4 1 2 3 4 3 2 2 2 2 h p p p p + cos 1 ∧ 4 cos 2 ∧ 3 . (3.5) 2 2 i µν The symbol means the product pi pj = θ pµipνj, if i = j (i, j =1, 2, 3, 4), and pi pj = 0, if i = j. The∧ θ-integral of (3.4)∧ can be calculated with6 help of (2.33) and after∧ some manipulations we obtain

g 4 d4p d6k S (φ)= i i φ(p ,k )(2π)4δ(4) (p + p + p + p ) int −6 4! (2π)4 (2π)6 i iµν 1 2 3 4 × × Z i=1 4 4 Y 4 − eλ µν − λ − e− 2 λ e 4 ksµν ks e 16 (p1µp2ν p1ν p2µ+p3µp4ν p3ν p4µ) cosh kµν (p p + p p ) × 2 s 1µ 2ν 3µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp2ν p1ν p2µ p3µp4ν +p3ν p4µ) cosh kµν (p p p p ) 2 s 1µ 2ν − 3µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp3ν p1ν p3µ+p2µp4ν p2ν p4µ) cosh kµν (p p + p p ) 2 s 1µ 3ν 2µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp3ν p1ν p3µ p2µp4ν +p2ν p4µ) cosh kµν (p p p p ) 2 s 1µ 3ν − 2µ 4ν 4 4 λ 2 − (p1 p4 −p1 p4 +p2 p3 −p2 p3 ) λ µν +e 16 µ ν ν µ µ ν ν µ cosh k (p p + p p ) 2 s 1µ 4ν 2µ 3ν 4 4 − λ − − 2 λ +e 16 (p1µp4ν p1ν p4µ p2µp3ν +p2ν p3µ) cosh kµν (p p p p ) . (3.6) 2 s 1µ 4ν − 2µ 3ν Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 11

The vertex V (4) of the φ4⋆ interaction in the momentum space is given by g V (4)(p , ,p ; k , ,k )= (2π)4δ(4) (p + p + p + p ) 1 ··· 4 1µν ··· 4µν −6 1 2 3 4 × 4 4 4 − λ µν − λ − − 2 λ e 4 ksµν ks e 16 (p1µp2ν p1ν p2µ+p3µp4ν p3ν p4µ) cosh kµν (p p + p p ) × 2 s 1µ 2ν 3µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp2ν p1ν p2µ p3µp4ν +p3ν p4µ) cosh kµν (p p p p ) 2 s 1µ 2ν − 3µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp3ν p1ν p3µ+p2µp4ν p2ν p4µ) cosh kµν (p p + p p ) 2 s 1µ 3ν 2µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp3ν p1ν p3µ p2µp4ν +p2ν p4µ) cosh kµν (p p p p ) 2 s 1µ 3ν − 2µ 4ν 4 4 − λ − − 2 λ +e 16 (p1µp4ν p1ν p4µ+p2µp3ν p2ν p3µ) cosh kµν (p p + p p ) 2 s 1µ 4ν 2µ 3ν 4 4 − λ − − 2 λ +e 16 (p1µp4ν p1ν p4µ p2µp3ν +p2ν p3µ) cosh kµν (p p p p ) , (3.7) 2 s 1µ 4ν − 2µ 3ν where we have deﬁned the sum of the k-momenta as

µν µν µν µν µν ks = k1 + k2 + k3 + k4 . (3.8) The expressions (3.3) and (4.2) are the Feynman rules of the φ4⋆ DFRA model. For radiative corrections of the perturbative series, lines and vertex are represented in the momentum space by those expressions. Clearly, the vertex expression shows that the external total momentum associated to extra dimension θ is not conserved, that is, ksµν = 0, while the total usual momentum pµ is conserved. But notice that we are stating that the6 physical momentum is not conserved in θ-space. We are talking about the vertex expression for the momentum.

IV. RADIATIVE CORRECTIONS

In this section we will analyze carefully the radiative corrections that are aﬀected by the NCY property of θ-space. However, we have to remember that the kµν = 0 case concerns the ∗ DFR spacetime. The objective here is to compare with DFR phase-space where kµν = 0. This will be carried out in the next section. 6

A. The self-energy propagator when kµν = 0

Now we use the Feynman rules in momentum space (3.3) and (3.7) to obtain the radiative corrections to the propagator and vertex of the φ4⋆ model. For simplify we begin analyzing the divergences at the one loop approximation when the θ conjugate momentum kµν = 0 in the Feynman rules of the previous section. The propagator is analogous to the usual NC commutative case i ∆ (p; k =0)= , (4.1) F µν p2 m2 + iε − Self-quartic interaction for a scalar field in an extended DFR.... 12 and the vertex expression is reduced to g V (4)(p , ,p ; k =0)= (2π)4δ(4) (p + p + p + p ) 1 ··· 4 µν −6 1 2 3 4 × 4 4 − λ − − 2 − λ − − 2 e 16 (p1µp2ν p1ν p2µ+p3µp4ν p3ν p4µ) + e 16 (p1µp2ν p1ν p2µ p3µp4ν +p3ν p4µ) × 4 4 h − λ − − 2 − λ − − 2 +e 16 (p1µp3ν p1ν p3µ+p2µp4ν p2ν p4µ) + e 16 (p1µp3ν p1ν p3µ p2µp4ν +p2ν p4µ) 4 4 − λ − − 2 − λ − − 2 +e 16 (p1µp4ν p1ν p4µ+p2µp3ν p2ν p3µ) + e 16 (p1µp4ν p1ν p4µ p2µp3ν +p2ν p3µ) . (4.2) i We just restrict the calculation to the one loop approximation. The first non-trivial correction to the scalar field propagator comes from the diagram

q q q q 2 Σ1(m ,p) = + p p p p − − where 1 d4p d4p d4p d4p i Σ (m2,p)= 1 2 3 4 1 2 (2π)4 (2π)4 (2π)4 (2π)4 p2 m2 + iε × Z 2 − (2π)4δ(4)(p + p )(2π)4δ(4)(p p)V (4)(p ,p ,p ,p ) , (4.3) × 2 3 1 − 1 2 3 4 and after some calculation we have that

4 4 g d q i − λ − 2 Σ (m2,p)= 2+ e 4 (qµpν pµqν ) . (4.4) 1 −6 (2π)4 q2 m2 + iε Z − h i where we can see the NC factor λ and the contribution of NCY. The previous expression suggests us to write it as

2 2 2 Σ1(m ,p) = Σ1(pn)(m ) + Σ1(np)(m ,p) , (4.5) where Σ1(pn) is the self-energy of the planar (pn) diagram

g d4q i Σ (m2)= , (4.6) 1(pn) −6 (2π)4 q2 m2 + iε Z −

while Σ1(np) is the non-planar (np) diagram

4 4 g d q i − λ 2 2− · 2 Σ (m2,p)= e 2 (q p (q p) ) . (4.7) 1(np) −3 (2π)4 q2 m2 + iε Z − where from Eqs. (4.23) and (4.7) we can see that the non-planar (np) diagram is the one aﬀected by NCY, i.e., we have a non-planar diagram in DFR∗ phase-space. The expression planar is very similar to the self-energy of the φ4 commutative at the one loop approximation. It has an integral that diverges in the ultraviolet limit, that is, when Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 13

k . For future convenience, we use the method of Schwinger’s parametrization to |evaluate|→∞ the integrals (4.23) and (4.7), in which it is introduced the integral

∞ i 2 2 = ds eis (q −m +iε) , (4.8) q2 m2 + iε − Z0 where s is the proper-time parameter that has length to the squared dimension. The self- energy planar term is easily calculated to give the s-integration

∞ ig ds 2 Σ (m2)= e−is (m −iε) . (4.9) 1(pn) 96π2 s2 Z0 This s-integral evidently diverges when s = 0. Then we introduce a cutoﬀ function regulator 2 given by e−1/i4sΛ , where Λ is a momenta cutoﬀ parameter

∞ ig ds 2 2 −1 Σ (m2, Λ) = e−is (m −iε)−(i4sΛ ) , (4.10) 1(pn) 96π2 s2 Z0 in which the divergence is naturally recovered in the limit Λ , and the function regulator is one. Using the Handbook integral [46], we obtain the regularized→∞ function

gm2 Λ m Σ (m2, Λ/m)= K , (4.11) 1(pn) −24π2 m 1 Λ where K1 is the Bessel function of the second kind. When Λ/m 1, we expand the Bessel function to obtain the result ≫ gm2 Λ2 gm2 m Σ (m2, Λ/m) ln , (4.12) 1(pn) ≈ −24π2 m2 − 48π2 Λ that has a quadratic divergence summed to the logarithmic one in Λ . In the non-planar expression (4.7), we have an exponential function→∞ in the numerator, that suggest one to introduce the Schwinger’s parametrization (4.8) in (4.7), so we obtain

∞ 4 4 g − 2− d q 2− λ 2 2− · 2 Σ (m2,p)= ds e is (m iε) eisq 2 (q p (q p) ) . (4.13) 1(np) −3 (2π)4 Z0 Z

These quadri-integral can be written explicitly in terms of the integrals in components (q0, q)

∞ 3 4 4 g 2 d q λ µ q2 λ q p 2 2 −is (m −iε) − is− 2 pµp + 2 ( · ) Σ1(np)(m ,p)= ds e e 3 −3 0 (2π) × Z Z ∞ 4 dq λ p2 2− 4 q·p 0 e is+ 2 q0 λ p0( )q0 , (4.14) × 2π Z−∞

where the q0-integral is a Gaussian one that can be computed to give us

1/2 ∞ 2 µ ig π −is (m2−iε) Σ1(np)(m ,p )= ds 4 e −6π is + λ p2 × Z0 2 ! 4 4 4 λ µ λ µ 2 λ 2 is− 2 pµp 3 − is− pµp q + (q·p) q 2 2 4 2 d is+ λ p ! e 2 . (4.15) × (2π)3 Z Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 14

Using the spherical coordinates, we calculate the angular integrals to obtain the radial integral

2 g i ∞ e−is(m −iε) Σ (m2,pµ)= ds 1(np) 2 4 1/2 −6 λ p 0 is λ p pµ × | | Z − 2 µ 4 1/2 ∞ 2 4 2 λ µ λ µ 2 is p p dq − is− 2 pµp q λ 2 µ e erfi q p − 4 , (4.16) × (2π)2  2 | | is + λ p2  Z0 2 !   where erfi is the imaginary error function [46]. It is not difficult to see that this integral µ µ does diverges in the k-integration (θ-space) when pµp is null or time-like, that is, pµp 0. µ µ ≥ In contrast, it is well defined when pµp is space-like, that is, pµp < 0. For this condition, we obtain

∞ 4 −3/2 ig ds 2 λ Σ (m2,p pµ)= e−is(m −iε) 1+ i p pµ , (4.17) 1(np) µ 48π2 s2 2s µ Z0 µ in which pµp < 0. Making the s-integral, we obtain the result

gm2 √π 1 λ4 Σ (m2,p pµ)= U , 0; m2p pµ , (4.18) 1(np) µ 24π2 λ4m2p pµ 2 − 2 µ µ where U is a conﬂuent hypergeometric function (Kummer’s function). When we expand the µ − previous result around the pµp 0 , it emerges a ultraviolet divergence summed up to infrared divergence term in pµpµ =0→

2 4 2 2 µ g 1 gm λ m µ lim Σ1(np)(m ,pµp )= 1+ γ + ln pµp . (4.19) p→0− 12π2 λ4p pµ − 48π2 − 8 µ This result reveals itself as being a mix of ultraviolet/infrared (UV/IR) divergences in DFR∗ NC models. It is immediate to compare this conclusion with the usual NC case when θµν is just a real parameter

2 2 2 µ g 1 gm m lim Σ1(np)(m ,pµp )= + ln , (4.20) p→0 96π2 p pµ 96π2 4p pµ µ µ where p := θ pν. This non-planar expression also exhibits a mixing (UV/IR) when p pµ = µ µν e e e e µ 0. The two NC formalisms exhibit an analogous (UV/IR) mixing. e Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 15

B. The self-energy propagator when k = 0 µν 6 The case of kµν = 0 is more complicated since we must use the Feynman rules in the momentum space (3.3)6 and (3.7). The correction to one loop in the propagator is illustrated by the diagram

q,k2 q,k3 q,k2 q,k3 2 µ Σ1(m ,p ; k1µν,k4µν)= + p,k µν p,k µν p,k µν p,k µν 1 − 4 1 − 4 where the self-energy is given by 4 λ 2 4 4 4 4 6 6 − 4 (k2µν +k3µν ) 2 µ 1 d p1 d p2 d p3 d p4 d k2 d k3 i e Σ1(m ,p ; k1 + k4)= 2 2 (2π)4 (2π)4 (2π)4 (2π)4 (2π)6 (2π)6 p2 + λ k kµν m2 + iε × Z 2 2 2µν 3 − (2π)4δ(4)(p + p )(2π)4δ(4)(p p)V (4)(p , ,p ; k , ,k ) . (4.21) × 2 3 1 − 1 ··· 4 1µν ··· 4µν By integrating the delta functions, the previous expression is reduced to

6 6 4 g d k2 d k3 − λ 2 Σ (m2,pµ; k + k )= e 4 (k2µν +k3µν ) 1 1 4 −6 (2π)6 (2π)6 × Z 4 4 λ 2 d q i − 4 (k1µν +k2µν +k3µν +k4µν ) e 2 × (2π)4 q2 + λ k kµν m2 + iε × Z 2 2µν 3 4 − − λ − 2 2+ e 4 (qµpν pµqν ) cosh λ4(k + k + k + k )qµpν , (4.22) × 1µν 2µν 3µν 4µν where we rewriteh it as a sum of a planar and non-planar parts, as we did before,i so

6 6 4 g d k2 d k3 − λ 2 Σ (m2; k + k )= e 4 (k2µν +k3µν ) 1(pn) 1 4 −3 (2π)6 (2π)6 × Z 4 4 λ 2 d q i − 4 (k1µν +k2µν +k3µν +k4µν ) e 2 , (4.23) × (2π)4 q2 + λ k kµν m2 + iε Z 2 2µν 3 − and 6 6 4 g d k2 d k3 − λ 2 Σ (m2,pµ; k + k )= e 4 (k2µν +k3µν ) 1(np) 1 4 −6 (2π)6 (2π)6 × Z 4 4 λ 2 d q i − 4 (k1µν +k2µν +k3µν +k4µν ) e 2 × (2π)4 q2 + λ k kµν m2 + iε × Z 2 2µν 3 4 − − λ 2 2− · 2 e 2 (q p (p q) ) cosh λ4(k + k + k + k )qµpν , (4.24) × 1µν 2µν 3µν 4µν respectively. Notice that in this case, both planar and non-planar parts are aﬀected by NCY. The planar self-energy has a ultraviolet divergence in the q-integration, and consequently, we use the same technique of the previous section to calculate it, so we obtain ∞ ig ds 2 Σ (m2; k + k )= e−is(m −iε) 1(pn) 1 4 48π2 s2 × Z0 6 6 4 4 2 d k2 d k3 − λ 2− λ 2 λ µν e 4 (k2µν +k3µν ) 4 (k1µν +k2µν +k3µν +k4µν ) +is 2 k2µν k3 . (4.25) × (2πλ−2)6 (2πλ−2)6 Z Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 16

The Gaussian integrals can be calculated to construct the planar sector as

2 4 2 ∞ 2 6 −is(m −iε) λ 2 1+i2λ /s − (k1µν +k4µν ) 2 ig ds λ e 4 1+i4λ2/s Σ1(pn)(m ; k1 +k4)= e ,. (4.26) 48π8 s2 s (1 + i4λ2/s)3 Z0 and we can see that the previous result has mass dimension to the squared, which agrees with the dimension of the self energy. This s-integral has a strong ultraviolet divergence when s = 0, due to the term s−8. Hence, it needs a cutoﬀ regulator ultraviolet to turn it well deﬁned. Concerning the non-planar expression (4.24), we can use the Schwinger’s parametrization to write this integral in terms of exponentials

∞ 6 6 2 µ g −is(m2−iε) d k2 d k3 Σ1(np)(m ,p ; k1 + k4)= ds e −2 6 −2 6 −6 0 (2πλ ) (2πλ ) × 4 Z 4 Z 2 − λ 2− λ 2 λ µν e 4 (k2µν +k3µν ) 4 (k1µν +k2µν +k3µν +k4µν ) +is 2 k2µν k3 4 × 4 × d q 2− λ 2 2− · 2 eisq 2 (q p (p q) ) cosh λ4 (k + k + k + k ) qµpν . (4.27) × (2π)4 1µν 2µν 3µν 4µν Z where we have written the hyperbolic cosine in this expression in terms of exponentials to simplify this integral as the sum

1 1 4 µ ν cosh λ4 (k + k + k + k ) qµpν = eσ λ (k1µν +k2µν +k3µν +k4µν )q p , (4.28) 1µν 2µν 3µν 4µν 2 σ=−1 X with σ = 0. So, we have that 6 1 ∞ g 2 Σ (m2,pµ; k + k )= ds e−is(m −iε) 1(np) 1 4 −12 × σ=−1 0 X Z 4 4 6 6 4 d q isq2− λ q2p2−(p·q)2 d k2 d k3 − λ 2 e 2 ( ) e 4 (k2µν +k3µν ) × (2π)4 (2πλ−2)6 (2πλ−2)6 × Z 4 Z 2 − λ 2 λ µν 4 µ ν e 4 (k1µν +k2µν +k3µν +k4µν ) +is 2 k2µν k3 +σλ (k1µν +k2µν +k3µν +k4µν )q p . (4.29) ×

After the calculation of the Gaussian integrals in (k2,k3), we obtain

1 ∞ 2 6 −is(m2−iε) 2 µ g λ e Σ1(np)(m ,p ; k1 + k4)= ds −12 πs 2 3 × σ=−1 0 (1 + i4λ /s) X Z 4 4 4 η λ 2 2 λ 2 2 2 4 µ ν − (k1 +k4 ) d q isq − ξ q p −(p·q) +λ η (k1 +k4 )q p e σ 4 µν µν e 2 ( ) µν µν , (4.30) × (2π)4 Z where we have deﬁned the complex constants

i2λ2/s 1+ i2λ2/s ξ(σ) := 1 σ2 and η(σ) := σ , (4.31) − 1+ i4λ2/s 1+ i4λ2/s Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 17 in which ξ(σ = 1) = (1+ i2λ2/s)(1+ i4λ2/s)−1, and η(σ = 1) = ξ that agrees with ± ± ± the sum deﬁned in (4.28). By writing it in terms of integrals (q0, q) we have that

1 ∞ 2 6 −is(m2−iε) 2 µ g λ e Σ1(np)(m ,p ; k1 + k4)= ds −12 πs 2 3 × σ=−1 0 (1 + i4λ /s) X Z 4 3 4 4 η λ 2 λ µ 2 λ 2 4 µj j − (k1 +k4 ) d q − is− ξpµp q + ξ(q·p) −λ ηpµ(k1+k4) q e σ 4 µν µν e 2 2 × (2π)3 × Z ∞ 4 dq λ p2 2− 4 q·p − µ0 0 e is+ 2 ξ q0 λ [ξ( )p0 ηpµ(k1+k4) ]q0 , (4.32) × 2π Z−∞ where the q0-integral is a Gaussian one that after computation we have that 1/2 1 ∞ 2 6 2 µ ig λ π Σ1(np)(m ,p ; k1 + k4)= ds 4 −24π πs is + λ ξp2 × σ=−1 Z0 2 ! 2 X −is m −iε 4 8 4 −1 ( ) η λ 2 λ 2 µ0 2 λ 2 e − 4 (k1µν +k4µν ) − 4 η [pµ(k1+k4) ] is+ 2 ξp e σ × (1 + i4λ2/s)3 × 3 4 4 4 4 −1 d q − − λ µ q2 λ q·p 2 − λ µ λ p2 e is 2 ξpµp + 2 ξ( ) is 2 ξpµp is+ 2 ξ × (2π)3 × Z 4 4 −1 4 λ µ0 λ 2 j µ j j λ η 2 ξpµ(k1+k4) is+ 2 ξp p0p −pµ(k1+k4) q e . (4.33) × We have used the spherical coordinates to reduce the previous integrals to the radial integral

ig 1 ∞ λ2 6 1 Σ (m2,pµ; k + k )= ds 1(np) 1 4 4 1/2 −48 0 πs λ µ × σ=−1 Z is 2 ξpµp 2 X − −is m −iε 4 8 4 −1 ( ) η λ 2 λ 2 µ0 2 λ 2 e − 4 (k1µν +k4µν ) − 4 η [pµ(k1+k4) ] is+ 2 ξp e σ × (1 + i4λ2/s)3 × 2 4 2 4 4 −1 4 4 −1 λ η λ µ0 λ 2 j µ j λ 2 λ µ ξpµ(k1+k4) is+ ξp p0p −pµ(k1+k4) is+ ξp is− ξpµp 2ξp2 2 2 2 2 e × 1/2 ∞ 2 4 × 2 dq − − λ µ 2 e is 2 ξpµp q × λ4ξp2 (2π)2 × Z0 4 1/2 ξ is λ ξp pµ 2 p − 2 µ erﬁ qλ λ4 ×   | | 2 is + ξp2 !  2 4 4 µ0 j  µj λ p2 λ ξpµ (k1 + k4) p0p 2pµ (k1 + k4) is + 2 ξ +λ2η − 4 4 4 p ξ is λ ξp pµ is + λ ξp2  | | 2 − 2 µ 2 q 4 1/2 ξ is λ ξp pµ +erﬁ qλ2 p − 2 µ  | | 2 λ4 2 is + 2 ξp !

4 µ0 j  µj λ4 2 λ ξpµ (k1 + k4) p0p 2pµ (k1 + k4) is + 2 ξp λ2η − . (4.34) 4 4 − 4 p ξ is λ ξp pµ is + λ ξp2  | | 2 − 2 µ 2  q   Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 18

µ It is not diﬃcult to see that the previous integral has a good behavior when pµp < 0, while µ it is divergent if pµp > 0. After a tedious calculus, the radial integral can be manipulated in order to help us to use the Handbook’s integral to obtain the result

∞ 2 6 4 −3/2 2 µ ig ds λ λ µ Σ1(np)(m ,p ; k1 + k4)= 2 2 1+ i ξpµp 48π 0 s πs 2s × 2 Z −is(m −iε) 4 8 2 4 −1 e − λ 2− λ 2 µ0 λ p2 e 4 ξ(k1µν +k4µν ) 4 ξ [pµ(k1+k4) ] is+ 2 ξ × (1 + i4λ2/s)3 × 2 4 4 4 −1 4 4 −1 λ ξ λ λ 2 µ0 j µ j λ 2 λ µ ξ is+ ξp p (k1+k4) p0p −p (k1+k4) is+ ξp is− ξp p 2p2 2 2 µ µ 2 2 µ e 2 × 4 4 4 −1 4 2 4 −×1 λ ξ λ λ 2 µ0 j µj λ 2 λ µ ξ is+ ξp pµ(k1+k4) p0p −pµ(k1+k4) is+ ξp is− ξpµp 2p2is 2 2 2 2 e . (4.35) ×

When the external k-momenta are null, that is, k1 = k4 = 0, it is direct to see that the result (4.35) reduces to the expression (4.17), with the factor (1 + i4λ2/s)−3 due to the integrations (k2,k3) in the loop. Hence, we can write the integral in (4.35) as ig ∞ ds λ2 6 Σ (m2,pµ; k + k =0)= 1(np) 1 4 48π2 s2 πs × Z0 4 −3/2 λ −3 2 1+ i ξp pµ 1+ i4λ2/s e−is(m −iε) . (4.36) × 2s µ Notice, that NCY introduces a huge diﬃculty concerning the solution of this integral. We can see that the integral variable s is present in every term of the integral. So, there is mix of high power of s together with square root of s and more than one term with negative high power of s, which means high power of s in the denominator. And the numerator cannot be eliminated by the denominator because s is inside non trivial expressions with the NC parameter. Unfortunately, the solution of this integral requires numerical computation work, which is out of the scope of this paper.

V. CONCLUSIONS AND FINAL REMARKS

The quest to understand the Early Universe physics has many candidates and one of them is NCY which has its measure linked to the Planck scale. This θ-parameter, nowadays, has two features. It can be constant, which causes the Lorentz invariance breaking, or as a µν coordinate of a NC phase-space that is formed by (x, p, θ ,kµν) where kµν is the canonical µν momentum conjugate to θ . It can be demonstrated that kµν is also connected to the Lorentz invariance [19]. Considering θµν as a coordinate we have in the current literature, two formulations that are directly related to each other. The standard DFR considers θµν as a coordinate and the ∗ so-called DFR-extended, i.e., DFR , which recognize the existence of kµν. As we saw in this work, we can construct an interactive QFT in this DFR∗. Having said that, in this work we have analyzed the self-quartic interaction for a scalar ﬁeld through Feynman diagrams formalism and the vertices calculation were accomplished. After obtaining these Feynman rules, we computed the radiative corrections to one loop order for the propagator of the model. The objective here is to analyze the mixing IR/UV divergences. Self-quartic interaction for a scalar ﬁeld in an extended DFR.... 19

Concerning the radiative corrections we have investigated two cases: kµν = 0 and kµν = 0. The ﬁrst one, although it is not an element of the DFR∗ phase-space, as we have explained,6 ∗ it is useful to compare with the kµν = 0 case, namely, kµν = 0 is a toy model in DFR phase- space. Hence, both the self-energy6 of planar and non-planar diagrams presents diﬀerence concerning the NC parameter presence. The non-planar diagram is infected with NCY. The second case, kµν = 0, presents both planar and non-planar scenarios contaminated with NCY. 6

VI. ACKNOWLEDGMENTS

EMCA would like to thank CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico), Brazilian scientific support agency, for partial financial support.

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