Two-Point Functions on Deformed Spacetime

We present a review of the one-loop photon (Π) and neutrino (Σ) two-point functions in a covariant and deformed U(1) gauge-theory on the 4-dimensional noncommutative spaces, determined by a constant antisymmetric tensor θμν, and by a parameter-space (κf,κg), respectively. For the general fermion-pho...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2014
Hauptverfasser:	Trampetić, J., You, J.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2014
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spelling	Trampetić, J. You, J. 2019-02-10T18:13:01Z 2019-02-10T18:13:01Z 2014 Two-Point Functions on Deformed Spacetime / J. Trampetić, J. You // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 75 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R60; 81T10; 81T15 DOI:10.3842/SIGMA.2014.054 https://nasplib.isofts.kiev.ua/handle/123456789/146685 We present a review of the one-loop photon (Π) and neutrino (Σ) two-point functions in a covariant and deformed U(1) gauge-theory on the 4-dimensional noncommutative spaces, determined by a constant antisymmetric tensor θμν, and by a parameter-space (κf,κg), respectively. For the general fermion-photon Sf(κf) and photon self-interaction Sg(κg) the closed form results reveal two-point functions with all kind of pathological terms: the UV divergence, the quadratic UV/IR mixing terms as well as a logarithmic IR divergent term of the type ln(μ²(θp)²). In addition, the photon-loop produces new tensor structures satisfying transversality condition by themselves. We show that the photon two-point function in the 4-dimensional Euclidean spacetime can be reduced to two finite terms by imposing a specific full rank of θμν and setting deformation parameters (κf,κg)=(0,3). In this case the neutrino two-point function vanishes. Thus for a specific point (0,3) in the parameter-space (κf,κg), a covariant θ-exact approach is able to produce a divergence-free result for the one-loop quantum corrections, having also both well-defined commutative limit and point-like limit of an extended object. This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html. J.T. would like to acknowledge J. Erdmenger and W. Hollik for discussions, and Max-PlanckInstitute for Physics, Munich, for hospitality. A great deal of computation was done by using Mathematica 8.0⁴ and tensor algebra package xAct⁵ . en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Two-Point Functions on Deformed Spacetime Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Two-Point Functions on Deformed Spacetime
spellingShingle	Two-Point Functions on Deformed Spacetime Trampetić, J. You, J.
title_short	Two-Point Functions on Deformed Spacetime
title_full	Two-Point Functions on Deformed Spacetime
title_fullStr	Two-Point Functions on Deformed Spacetime
title_full_unstemmed	Two-Point Functions on Deformed Spacetime
title_sort	two-point functions on deformed spacetime
author	Trampetić, J. You, J.
author_facet	Trampetić, J. You, J.
publishDate	2014
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We present a review of the one-loop photon (Π) and neutrino (Σ) two-point functions in a covariant and deformed U(1) gauge-theory on the 4-dimensional noncommutative spaces, determined by a constant antisymmetric tensor θμν, and by a parameter-space (κf,κg), respectively. For the general fermion-photon Sf(κf) and photon self-interaction Sg(κg) the closed form results reveal two-point functions with all kind of pathological terms: the UV divergence, the quadratic UV/IR mixing terms as well as a logarithmic IR divergent term of the type ln(μ²(θp)²). In addition, the photon-loop produces new tensor structures satisfying transversality condition by themselves. We show that the photon two-point function in the 4-dimensional Euclidean spacetime can be reduced to two finite terms by imposing a specific full rank of θμν and setting deformation parameters (κf,κg)=(0,3). In this case the neutrino two-point function vanishes. Thus for a specific point (0,3) in the parameter-space (κf,κg), a covariant θ-exact approach is able to produce a divergence-free result for the one-loop quantum corrections, having also both well-defined commutative limit and point-like limit of an extended object.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146685
citation_txt	Two-Point Functions on Deformed Spacetime / J. Trampetić, J. You // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 75 назв. — англ.
work_keys_str_mv	AT trampeticj twopointfunctionsondeformedspacetime AT youj twopointfunctionsondeformedspacetime
first_indexed	2025-11-24T11:38:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 054, 20 pages Two-Point Functions on Deformed Spacetime? Josip TRAMPETIĆ †‡ and Jiangyang YOU † † Rudjer Bošković Institute, P.O. Box 180, HR-10002 Zagreb, Croatia E-mail: josipt@rex.irb.hr, youjiangyang@gmail.com ‡ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany Received February 24, 2014, in final form May 19, 2014; Published online May 29, 2014 http://dx.doi.org/10.3842/SIGMA.2014.054 Abstract. We present a review of the one-loop photon (Π) and neutrino (Σ) two-point functions in a covariant and deformed U(1) gauge-theory on the 4-dimensional noncommu- tative spaces, determined by a constant antisymmetric tensor θµν , and by a parameter-space (κf , κg), respectively. For the general fermion-photon Sf (κf ) and photon self-interaction Sg(κg) the closed form results reveal two-point functions with all kind of pathological terms: the UV divergence, the quadratic UV/IR mixing terms as well as a logarithmic IR divergent term of the type ln(µ2(θp)2). In addition, the photon-loop produces new tensor struc- tures satisfying transversality condition by themselves. We show that the photon two-point function in the 4-dimensional Euclidean spacetime can be reduced to two finite terms by imposing a specific full rank of θµν and setting deformation parameters (κf , κg) = (0, 3). In this case the neutrino two-point function vanishes. Thus for a specific point (0, 3) in the parameter-space (κf , κg), a covariant θ-exact approach is able to produce a divergence- free result for the one-loop quantum corrections, having also both well-defined commutative limit and point-like limit of an extended object. Key words: non-commutative geometry; photon and neutrino physics; non-perturbative effects 2010 Mathematics Subject Classification: 81R60; 81T10; 81T15 1 Introduction Prior to the late 1990s, the possibility of experimentally testing the nature of quantum gravi- ty was not seriously contemplated because of the immensity of the Planck scale (E = 1.2 × 1019 GeV). Now this view has been significantly alternated: A possibility to have a string scale significantly below the Planck scale in a braneworld scenario [8] become the core of current experimental protocols searching for quantum-gravity phenomena (notably production of black holes) at the Large Hadron Collider at CERN. Almost simultaneously, another possibility arose, where it was pointed out that distant astrophysical objects with rapid time variations could provide the most sensitive opportunities to probe very high energy scales, i.e., almost the near- Planck scale physics [7]. Another route to search for quantum-gravity effects involves a spontaneous breaking of Lorentz symmetry in string theory, when a tensor field acquires a vacuum expectation value (vev). Unlike the case of scalars, these tensor vevs do carry spacetime indices, causing the interaction repre- sented by the Standard Model (SM) fields coupled to these vevs to depend on the direction or velocity of the said fields. Stated differently, these background vevs bring about the breakdown of Lorentz symmetry. This entails a distinctive fact of most of Lorentz violating (LV) theories on the existence of preferred reference frames, where the equations of motion take on the simplest ?This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html mailto:josipt@rex.irb.hr mailto:youjiangyang@gmail.com http://dx.doi.org/10.3842/SIGMA.2014.054 http://www.emis.de/journals/SIGMA/space-time.html 2 J. Trampetić and J. You form. In contrast to the notion of the motionless aether from the end of the 19th century, we have a rather unique example of such a frame in modern cosmology today: the frame in which the Cosmic Microwave Background Radiation (CMBR) looks isotropic. From the determination of the detailed spectrum of the CMBR dipole (generally interpreted as a Doppler shift due to the Earth’s motion), our velocity with respect to that frame, of order of 10−3 c, can be inferred. An eligible way to infer the preferred reference frame predicted by generic quantum gravity frameworks, is to study dispersion relations for propagating particles. Instead of propagating (in a vacuum) with the speed of light, in Lorentz violating theories one expects an energy-dependent velocity v(E) for massless particles. This is a consequence of the loss of Lorentz covariance in the dispersion relations for propagating particles, with the implication that a specific form v(E) can be at best valid only in one specific reference frame. Thus, a preferred reference frame in which the equation of motions possess the simplest form is singled out. This opens up a unique possibility to study constraints on violations of Lorentz invariance. The modification of the photon velocity of the form v(E) would induce time lag for photons of different energies, which could be subsequently detected if such particles can propagate at cosmological distances. Such an alternation of the photon velocities has already been obtained in Loop quantum gravity (being another popular approach to quantum gravity) [63, 64] as well as in heuristic models of space- time foam inspired by string theory [7]. One of the most striking observation regarding spontaneous Lorentz breaking via tensor vevs in the string theory framework is that it can be formulated as deformed field theories. Specifically, a low-energy limit is identified where the entire boson-string dynamics in a Neveu- Schwartz condensate is described by a minimally coupled supersymmetric gauge theory on noncommutative (NC) space [67] such that the mathematical framework of noncommutative geometry/field theory [25, 26, 49, 70] does apply. In such a scenario, noncommutative Dirac– Born–Infeld (DBI) action is realized as a special limit of open strings in a background Bµν field, in which closed string (i.e. gravitational) modes are decoupled, leaving only open string interactions. Since in string theory Bµν field is a rather mild background, the antisymmetric tensor θµν governing spacetime noncommutative deformations is not specified, and therefore the scale of noncommutativity could, in principle, lie anywhere between the weak and the Planck scale [15, 26, 70, 71]. It is thus of crucial importance to set a bound on this scale from experi- ments [71]. It is important to stress the invariance of the theory under coordinate changes, i.e., the invariance under an observer transformation (where the coordinates of the observer are boosted or rotated). This transformation is not related to the concept of Lorentz violation since in this transformation the properties of the background fields transform to a new set of coordinates as well. On the other hand, an invariance under active or particle transformation, where both fields and states are being transformed, is now broken by the background fields themselves, leading to the concept of Lorentz violation. In a simple model of the noncommutative spacetime we consider coordinates xµ as the Her- mitian operators x̂µ [45], [x̂µ, x̂ν ] = iθµν , \|θµν \| ∼ Λ−2 NC, (1.1) where θµν is a constant real antisymmetric matrix of dimension length2, and ΛNC being the scale of noncomutativity. It is straightforward to formulate field theories on such noncommutative spaces as a deformation of the ordinary field theories [15, 26, 70, 71]. The noncommutative deformation is implemented by replacing the usual pointwise product of a pair of fields φ(x) and ψ(x) by the star(?)-product in any action: φ(x)ψ(x) −→ (φ ? ψ)(x) = φ(x)ψ(x) +O(θ, ∂φ, ∂ψ). Two-Point Functions on Deformed Spacetime 3 The specific Moyal–Weyl ?-product is relevant for the case of a constant antisymmetric non- commutative deformation tensor θµν and is defined as follows: (φ ? ψ)(x) = e i 2 θµν∂ηµ∂ ξ νφ(x+ η)ψ(y + ξ) ∣∣ η,ξ→0 ≡ φ(x)e i 2 ←− ∂µθµν −→ ∂νψ(x). (1.2) The ?-product has also an alternative integral formulation, making its non-local character more transparent. The coordinate-operator commutation relation (1.1) is then realized by the star(?)- commutator of the usual coordinates [x̂µ, x̂ν ] = [xµ ?, xν ] = iθµν , implying the following spacetime uncertainty relations ∆xµ∆xν ≥ 1 2 \|θµν \|. The above procedure introduces in general the field operators ordering ambiguities and breaks ordinary gauge invariance. Since commutative local gauge transformations for the D-brane effective action do not com- mute with ?-products, it is important to note that the introduction of ?-products induces field operator ordering ambiguities and also breaks ordinary gauge invariance in the naive sense. However both the commutative gauge symmetry and the deformed noncommutative gauge sym- metry describe the same physical system, therefore they are expected to be equivalent. This disagreement is remedied by a set of nonlocal and highly nonlinear parameter redefinitions called Seiberg–Witten (SW) map [67]. This map promotes not only the noncommutative fields and composite operators of the commutative fields, but also the noncommutative gauge transfor- mations as the composite operators of the commutative gauge fields and gauge transforma- tions. Through this procedure, deformed gauge field theories can be defined for arbitrary gauge groups/representations. Consequently, building semi-realistic deformed particle physics models are made much easier. It is reasonable to expect that the new underlying mathematical structures in the NC gauge field theories (NCGFT) could lead to profound observable consequences for the low energy physics. This is realized by the perturbative loop computation first proposed by Filk [28]. There are also famous examples of running of the coupling constant in the U(1) NCGFT in the ?-product formalism [53], and the exhibition of fascinating dynamics due to the cele- brated ultraviolet/infrared (UV/IR) phenomenon, without [56, 60], and with the Seiberg–Witten map [38, 40, 41, 57, 66] included. Precisely, in [15, 60, 56] it was shown for the first time how UV short distance effects, considered to be irrelevant, could alter the IR dynamics, thus becoming known as the UV/IR mixing. Some significant progress on UV/IR mixing and related issues has been achieved [16, 17, 32, 50, 58] while a proper understanding of loop corrections is still sought for. More serious efforts on formulating NCQFT models with potential phenomenological influen- ce have started for about a decade ago. Strong boost came from the Seiberg–Witten map [67] based enveloping algebra approach, which enables a direct deformation of comprehensive phe- nomenological models like the standard model or GUTs [9, 12, 24]. It appeared then relevant to study ordinary gauge theories with the additional couplings inspired by the SW map/defor- mation included [9, 12, 24]. To include a reasonably relevant part of all SW map inspired couplings, one usually calls for an expansion and cut-off procedure, that is, an expansion of the action in powers of θµν [9, 12, 24, 36, 52, 74]. Next follows theoretical studies of one loop quantum properties [10, 13, 14, 19, 21, 22, 23, 30, 48, 51, 54, 55], as well as studies of some new physical phenomena, like breaking of Landau–Yang theorem, [3, 4, 5, 20, 59, 61, 65], etc. It was also observed that 4 J. Trampetić and J. You allowing a deformation-freedom via varying the ratio between individual gauge invariant terms could improve the renormalizability at one loop level [23, 48]. The studies on phenomenology (possible experimental signal/bounds on noncommutative background) started parallel to the pure theoretical developments of NCQFTs. The majority of the accelerator processes had been surveyed up to the second power in θµν [3, 4, 5, 61]. The processes involving photons in noncommutative U(1) gauge theory now involve corrections to the known processes, since the new couplings, of which the most distinctive being the various photon self couplings, now emerge in the noncommutative background even at tree level. Such couplings might give rise to novel processes (normally forbidden in the standard theory) or to provide new channels in the already known processes. The formulation of the SW-mapped actions has recently been made exact with respect to the noncommutative parameter θµν [57, 66, 75], offering thus an opportunity to compute various processes across full energy scale [42, 43]. Accordingly, one should no longer rely on the expansion in powers in θ, which could be especially beneficial in case when the quantum gravity scale is not so tantalizing close to the Planck scale. Thus, in this and several prior work(s) we formulate the θ-exact model action employing formal powers of fields [11, 31, 37, 42, 43, 47, 57, 62, 75], aiming at the same time to keep the nonlocal nature of the modified theory. Introduction of a nonstandard momentum dependent quantity of the type sin2(pθk/2)/(pθk/2)2 into the loop integrals makes these theories drastically different from their θ-expanded cousins, being thus interesting for pure field theoretical reasons. The deformation-freedom parameters (ratios of weight-parameters of each gauge invariant terms in the actions) are found to be compatible with the θ-exact action therefore included to study their possible effects on divergence cancelation(s). In this review we present closed forms for fermion-loop and photon-loop corrections to the photon and the neutrino two-point functions using dimensional regularization technique and we combine parameterizations of Schwinger, Feynman, and modified heavy quark effective theo- ry parameterization (HQET) [33]. Both two-point functions were obtained as a function of unspecified number of the integration dimensions D. Next we specify gauge field theory dimen- sion 4, and discuss the limits D → 4. The review is structured as follows: In the following section we describe generalized defor- mation freedom induced actions, and we give the relevant Feynman rules. Sections 3 and 4 are devoted to the computation/presentation/discussions of photon and neutrino two-point func- tions containing the fermion and the photon loop. Section 5 is devoted to discussions and conclusions. 2 The model construction The main principle that we are implementing in the construction of our θ-exact noncommutative model is that electrically neutral matter fields will be promoted via hybrid SW map deforma- tions [39] to the neutral noncommutative fields that couple to photons and transform in the adjoint representation of U?(1). We consider a U(1) gauge theory with a neutral fermion which decouples from the gauge boson in the commutative limit. We specify the action and deforma- tion as a minimal θ-exact completion of the prior first order in θ models [12, 23, 24, 59, 65], i.e. the new (inter-)action has the prior tri-particle vertices as the leading order. In the tree-level neutrino-photon coupling processes only vertices of the form ψ̄aψ contribute, therefore an expansion to lowest nontrivial order in aµ (but all orders in θ) is enough. There are at least three known methods for θ-exact computations: The closed formula derived using deformation quantization based on Kontsevich formality maps [47], the relationship between open Wilson lines in the commutative and noncommutative picture [57, 62], and direct recursive computations using consistency conditions. For the lowest nontrivial order a direct deduction Two-Point Functions on Deformed Spacetime 5 from the recursion and consistency relations δΛAµ = ∂µΛ− i[Aµ ?, Λ] ≡ Aµ[aµ + δλaµ]−Aµ[aµ], δΛΨ = i[Λ ?, Ψ] ≡ Ψ[aµ + δλaµ, ψ + δλψ]−Ψ[aµ, ψ], Λ[[λ1, λ2], aµ] = [Λ[λ1, aµ] ?, Λ[λ2, aµ]] + iδλ1Λ[λ2, aµ]− iδλ2Λ[λ1, aµ], with the ansatz Λ = Λ̂[aµ]λ = ( 1 + Λ̂1[aµ] + Λ̂2[aµ] +O ( a3 )) λ, Ψ = Ψ̂[aµ]ψ = ( 1 + Ψ̂1[aµ] + Ψ̂2[aµ] +O ( a3 )) ψ, is already sufficient. Capital letters denote noncommutative objects, small letters denote com- mutative objects, hatted capital letters denote differential operator maps from the latter to the former. In particular, here Ψ̂[aµ] and Λ̂[aµ] are gauge-field dependent differential operators that we shall now determine: Starting with the fermion field Ψ, at lowest order we have i[λ ?, ψ] = Ψ̂[∂λ]ψ. Writing the ?-commutator explicitly as [φ ?, ψ] = φ(x) ( ei ∂xθ∂y 2 − e−i ∂xθ∂y 2 ) ψ(y) ∣∣∣∣ x=y = 2iφ(x) sin ( ∂xθ∂y 2 ) ψ(y) ∣∣∣∣ x=y = iθij ( ∂φ(x) ∂xi ) sin (∂xθ∂y 2 ) ∂xθ∂y 2 ( ∂ψ(y) ∂yj )∣∣∣∣ x=y , we observe that Ψ̂[aµ] = −θijai ?2 ∂j , will fulfill the consistency relation. The generalized star-product ?2, appearing in the above, is defined, respectively, as [38, 40, 57, 66]: φ(x) ?2 ψ(x) = sin ∂1θ∂2 2 ∂1θ∂2 2 φ(x1)ψ(x2) ∣∣∣∣ x1=x2=x . Here ?-product (1.2) is associative but noncommutative, while ?2 is commutative but nonasso- ciative. The ?-commutator can then be rewritten as the following ?2-products [φ ?, ψ] = iθij∂iφ ?2 ∂jψ. (2.1) The gauge transformation Λ can be worked out similarly, namely 0 = [λ1 ?, λ2] + iΛ̂[∂λ1]λ2 − iΛ̂[∂λ2]λ1 = 1 2 ( [λ1 ?, λ2]− [λ2 ?, λ1] ) + iΛ̂[∂λ1]λ2 − iΛ̂[∂λ2]λ1, and hence Λ̂1 = −1 2 θijai ?2 ∂j . The gauge field aµ requires slightly more work. The lowest order terms in its consistency relation are −∂µ ( 1 2 θijai ?2 ∂jλ ) − i[λ ?, aµ] = A2 µ[aµ + ∂µλ]−A2 µ[aµ], 6 J. Trampetić and J. You where A2 is the a2 order term in the expansion of A as power series of a. Using the relation (2.1), the left hand side can be rewritten as −1 2θ ij∂µai ?2 ∂jλ− 1 2θ ijai ?2 ∂µ∂jλ− θij∂iλ ?2 ∂jaµ, where the first term comes from −1 2θ ij∂µai ?2 aj , while the third one comes from −θijai ?2 ∂jaµ. After a gauge transformation, the sum of the first and third terms equals the second term. Ultimately, we obtain SW map solutions up to the O ( a2 ) order: Aµ = aµ − 1 2 θνρaν ?2 (∂ρaµ + fρµ) +O ( a3 ) , Ψ = ψ − θµνaµ ?2 ∂νψ +O ( a2 ) ψ, Λ = λ− 1 2 θµνaµ ?2 ∂νλ+O ( a2 ) λ, (2.2) with fµν being the commutative Abelian field strength fµν = ∂µaν − ∂νaµ. The resulting expansion defines in the next section the one-photon-two-fermion and the three-photon vertices, θ-exactly. 2.1 Actions We start with the minimal NC model of a SW type U?(1) gauge theory on Euclidean space- time. Here in the starting action, the minimal refers on the number of fields/gauge field strengths/covariant derivatives: two gauge fields, two gauge field strengths, one covariant deriva- tive and three fields for gauge-fermion interactions. Thus we have Smin = ∫ −1 2 Fµν ? Fµν + iΨ̄ ? /DΨ, (2.3) with the coupling constant to be set as e = 1, and with the following definitions of the non- Abelian NC covariant derivative and the field strength, respectively: DµΨ = ∂µΨ− i[Aµ ?, Ψ] and Fµν = ∂µAν − ∂νAµ − i[Aµ ?, Aν ]. All the fields in this action are images under (hybrid) Seiberg–Witten maps of the corresponding commutative fields aµ and ψ. In the original work of Seiberg and Witten and in virtually all subsequent applications, these maps are understood as (formal) series in powers of the noncom- mutativity parameter θµν . Physically, this corresponds to an expansion in momenta and is valid only for low energy phenomena. Here we shall not subscribe to this point of view and instead interpret the noncommutative fields as valued in the enveloping algebra of the underlying gauge group. This naturally corresponds to an expansion in powers of the gauge field aµ and hence in powers of the coupling constant e. At each order in aµ we shall determine θ-exact expressions. In the following we discuss the model construction for the photon and the massless fermion case. Since we have set e = 1, to restore the coupling constant one simply substitutes aµ by eaµ and then divides the gauge-field term in the Lagrangian by e2. Coupling constant e, carries (mass) dimension (4− d)/2 in the d-dimensional field theory. The expansion in powers of the commutative gauge field content is motivated by the obvious fact that in perturbative quantum field theory one can sort the interaction vertices by the number of external legs and this is equivalent to the number of field operators in the corresponding interacting terms. For any specific process and loop order there exists an upper limit on the number of external legs. So if one expands the noncommutative fields with respect to the formal power of the commutative fields which are the primary fields in the theory up to an appropriate order, the relevant vertices in a specific diagram will automatically be exact to all orders of θ. The minimal gauge invariant nonlocal interaction (2.3) includes the gauge boson self-coupling as well as the fermion-gauge boson coupling, denoted here as Sg and Sf , respectively: Smin = SU(1) + Sg + Sf . Two-Point Functions on Deformed Spacetime 7 In the next step we expand the action (2.3) in terms of the commutative gauge parameter λ and fields aµ and ψ using the U(1) SW map solutions (2.2). This way, the photon self-interaction up to the lowest nontrivial order is obtained [38, 41]: Sg = ∫ ifµν ? [aµ ?, aν ] + ∂µ ( θρσaρ ?2 (∂σaν + fσν) ) ? fµν +O ( a4 ) = ∫ θρτfµν ( 1 4 fρτ ?2 fµν − fµρ ?2 fντ ) +O ( a4 ) . (2.4) The the lowest order photon-fermion interaction (first three terms of equation (2.7) from [38]) reads as follows Sf = ∫ ψ̄γµ[aµ ?, ψ] + i(θij∂iψ̄ ?2 aj)/∂ψ − iψ̄ ? /∂(θijai ?2 ∂jψ) + ψ̄O ( a2 ) ψ = − ∫ iθρτ ψ̄γµ ( 1 2 fρτ ?2 ∂µψ − fµρ ?2 ∂τψ ) + ψ̄O ( a2 ) ψ. (2.5) Note that actions for the gauge and the matter fields obtained above, (2.4) and (2.5) respectively, are nonlocal objects due to the presence of the (generalized) star products. 2.2 General deformed actions: Sf and Sg It is easy to see that each of the interactions (2.4) and (2.5) contains two U(1) gauge invariant terms, therefore one could vary the ratio between them without spoiling the gauge invariance. Prior studies have also indicated that varying these ratios can improve the one-loop behav- ior of the model [23, 38, 41, 42, 48]. For this propose we introduce further two-dimensional deformation-parameter-space (κf , κg). The deformation parameter κf in the photon-gauge boson interaction can be so chosen that it realizes the linear superposition of two possible nontrivial noncommutative deformations of a free neutral fermion action proposed in [38, 41, 42]. Its existence was already hinted in the θ-expanded expressions in [65] but not fully exploited in the corresponding loop computation before. The pure gauge action Sg deformation κg was first presented in the non-Abelian gauge sector action of the NCSM and NC SU(N) at first order in θ, Sθg [23, 48]. This could be realized by generalizing the standard SW map expression for linear in θ gauge field strength into the following form [73]: F θµν(κg) = fµν + θρτ ( κ−1 g fµρfντ − aρ∂τfµν ) +O ( θ2 ) . The gauge transformation for the noncommutative field strength δλFµν = i[Λ ?, Fµν ] will still be satisfied at its leading order in θ. We have observed in prior studies [37, 41, 73] that the above deformation can be made θ-exact and adopted it here Fµν(κg) = fµν + θρτ ( κ−1 g fµρ ?2 fντ − aρ ?2 ∂τfµν ) +O ( a3 ) . (2.6) Using the relation (2.1) we can see that δλFµν(κg) = i[λ ?, fµν ]+O ( a2 ) λ, which represents again the desired field strength consistency at the corresponding order. Thus, starting with (2.3), (2.6) and (2.5), followed by an appropriate field strength redefinition fµν → κgfµν , and finally after an overall rescaling κ−2 g , we can write the generalized manifestly gauge invariant actions with minimal number of fields1: SU(1) = ∫ −1 2 fµνf µν + iψ̄/∂ψ, (2.7) 1It could be simpler if we have associated κg with fµρ in (2.8) as the κf in (2.9), we choose the other way around to unify our result with the prior works [23, 37, 41, 48, 73]. 8 J. Trampetić and J. You Sg(κg) = ∫ θρτfµν (κg 4 fρτ ?2 fµν − fµρ ?2 fντ ) , (2.8) Sf(κf ) = − ∫ iθρτ ψ̄γµ ( 1 2 fρτ ?2 ∂µψ − κffµρ ?2 ∂τψ ) . (2.9) Since Sg(κg) and Sf(κf ) are both gauge invariant by themselves, one can incorporate either one or both of them into the full Lagrangian. The above actions were obtained by a θ-exact gauge-invariant truncation of a U?(1) model up to tri-leg vertices. Such an operation is achievable because the U(1) gauge transformation after deformation preserves the number of fields within each term. Motivation to introduce deformation parameters κg and κf was, besides the general gauge invariance of the action, to help eliminating one-loop pathologies due to the UV and/or IR diver- gences in both sectors. The parameter-space (κf , κg) represents a measure of the deformation- freedom in the matter Sf(κf ) and the gauge Sg(κg) sectors, respectively. We should clarify, that we are interested in the general gauge invariant interactions induced by the θµν background instead of the strictly Moyal–Weyl star-product deformation of the commutative gauge theories and its Seiberg–Witten map extension. We relax the constraint that a deformation should be Moyal–Weyl type for the hope that such variation could provide certain additional control on the novel pathologies due to the noncommutativity, which had indeed occurred in the θ-expanded models studied before, and as we will discuss later, in our θ-exact model as well. We still con- strained our model building by requiring that each of the gauge invariant interaction terms arises within a Seiberg–Witten map type deformation, only their linear combination ratios κg and κf are allowed to vary. This is all explained in full details in [38, 41, 42]. Each parameter bears the origin from the corresponding θ-expanded theory [23, 48, 65]. By straightforward reading-out procedure from Sg (2.8) we obtain the following Feynman rule for the triple-photon vertex in momentum space: Γµνρκg (p, k, q) = F (k, q)V µνρ κg (p, k, q), F (k, q) = sin kθq 2 kθq 2 , (2.10) where momenta p, k, q are taken to be incoming satisfying the momentum conservation p+k+q = 0 [41]. The deformation freedom ambiguity κg is included in the vertex function: V µνρ κg (p, k, q) = −(pθk)[(p− k)ρgµν + (k − q)µgνρ + (q − p)νgµρ] − θµν [pρ(kq)− kρ(pq)]− θνρ[kµ(pq)− qµ(pk)]− θρµ[qν(pk)− pν(kq)] + (θp)ν [gµρq2 − qνqρ] + (θp)ρ[gµνk2 − kµkν ] + (θk)µ[gνρq2 − qνqρ] + (θk)ρ[gµνp2 − pµpν ] + (θq)ν [gµρp2 − pµpρ] + (θq)µ[gνρk2 − kνkρ] + (κg − 1) ( (θp)µ[gνρ(kq)− qνkρ] + (θk)ν [gµρ(qp)− qµpρ] + (θq)ρ[gµν(kp)− kµpν ] ) . (2.11) The above vertex function (2.11) is in accord with corresponding Feynman rule for triple neutral gauge-boson coupling in [20]. From Sf (2.9) the fermion-photon vertex reads as follows Γµκf (k, q) = F (k, q)V µ κf (k, q) = F (k, q) [ κf ( /k(θq)µ − γµ(kθq) ) − (θk)µ/q ] , (2.12) where k is the photon incoming momentum, and the fermion momentum q flows through the vertex, as it should [41]. Two-Point Functions on Deformed Spacetime 9 Figure 1. Fermion-loop contribution to the photon two-point function. 3 Photon two-point function 3.1 Computing photon two-point function using dimensional regularization Employing the parametrization given in this section we illustrate the way we have performed the computation of the integrals which differ from regular ones by the existence of a non-quadratic kθp denominators. The key point was to introduce the HQET parametrization [33], represented as follows 1 an1 1 an2 2 = Γ(n1 + n2) Γ(n1)Γ(n2) ∫ ∞ 0 in1yn1−1dy (ia1y + a2)n1+n2 . To perform computations of our integrals, we first use the Feynman parametrization on the quadratic denominators, then the HQET parametrization help us to combine the quadratic and linear denominators. For example 1 k2(p+ k)2 1 kθp = 2i ∫ 1 0 dx ∫ ∞ 0 dy [( k2 + iε ) (1− x) + ( (p+ k)2 + iε ) x+ iy(kθp) ]−3 . After employing the Schwinger parametrization, the phase factors from (2.12) can be absorbed by redefining the y integral. This way we obtain 2− eikθp − e−ikθp k2(p+ k)2(kθp) · {numerator} = 2i ∫ 1 0 dx ∫ 1 λ 0 dy ∫ ∞ 0 dλλ2e−λ ( l2+x(1−x)p2+ y2 4 (θp)2 ) · {y odd terms of the numerator}, with loop-momenta being l = k + xp + i 2y(θp). By this means the y-integral limits take the places of planar/nonplanar parts of the loop integral. For higher negative power(s) of kθp, the parametrization follows the same way except the appearance of the additional y-integrals which lead to finite hypergeometric functions2. Following [39], we are enabled to follow the general procedure of dimensional regularization in computing one-loop two point functions. Thus we start the computations with respect to general integration dimension D, next we set the D → 4 limits and perform the full analysis of the one-loop two point functions behavior. Photon two-point function: fermion-loop. The fermion-loop contribution is read out from Fig. 1. Dimensional regularization than gives photon polarization tensor full expression Πµν κf (p)D = − trµ4−D ∫ dDk (2π)D Γµκf (−p, p+ k) i(/p+ /k) (p+ k)2 Γνκf (p, k) i/k k2 , where the momentum structure and dependence on the parameter κf is encoded. Performing a large amount of computations we have found the following structure: Πµν κf (p)D = 1 (4π)2 [( gµνp2 − pµpν ) F κf 1 (p) + (θp)µ(θp)νF κf 2 (p) ] , (3.1) 2See http://functions.wolfram.com/07.32.06.0031.01. http://functions.wolfram.com/07.32.06.0031.01 10 J. Trampetić and J. You Figure 2. Photon-loop contribution to the photon two-point function. while the full details of the loop-coefficients F κf 1,2(p) computations are given in [41]. It is straight- forward to see that each term in the tensor structure (3.1) does satisfy the Ward identity by itself, therefore pµΠµν κf (p)D = pνΠµν κf (p)D = 0, ∀ (D,κf ). In the limit D → 4− ε, the loop-coefficients can be expressed in the following closed forms: F κf 1 (p) = −κ2 f 8 3 [ 2 ε + lnπeγE + ln ( µ2(θp)2 )] + 4κ2 fp 2(θp)2 ∞∑ k=0 (k + 2)(p2(θp)2)k 4kΓ[2k + 6] × [ (k + 2) ( ln ( p2(θp)2 ) − ψ(2k + 6)− ln 4 ) + 2 ] , (3.2) F κf 2 (p) = κf 8 3 p2 (θp)2 [ κf − 8 ( κf + 2 ) 1 p2(θp)2 ] − 4κ2 fp 4 ∞∑ k=0 (p2(θp)2)k 4kΓ[2k + 6] × [ (k + 1)(k + 2) ( ln ( p2(θp)2 ) − 2ψ(2k + 6)− ln 4 ) + 2k + 3 ] , (3.3) with γE ' 0.577216 being Euler’s constant. The above expressions for F κf 1,2(p) contain both contributions, from the planar as well as from the non-planar graphs. All of the divergences arising from the fermion-loop (Fig. 1) could be removed by the unique choice κf = 0, as in that case the whole general amplitude (3.1) vanishes for any integration dimensions D [41]. Evaluation of the four-dimensional θ-exact fermion-part contribution to the photon polari- zation tensor, i.e. the fermion-loop photon two-point function (3.1) yields two already known tensor structures [18, 34, 35]. The loop-coefficients F κf 1,2(p), on the other hand, exhibit nontrivial κf dependence. Namely, in the limit κf → 0 =⇒ F κf 1 (p) = F κf 2 (p) = 0, thus the photon polarization tensor (3.1) vanishes, while κf = 1 appears to be identical to the non SW-map model. Fermion-loop contains UV and logarithmic divergence in F κf 1 (p) for κf 6= 0, while the quadratic UV/IR mixing could be removed by setting κf = 0,−2 in F κf 2 (p). Finally, it is important to stress that there is an additional fermion-loop tadpole diagram contribution to the photon 2-point function, arising from 2-photon-2-fermion (ψ̄a2ψ) interaction vertices [38, 42]. However, it was shown in [66], that this tadpole diagram vanishes due to the internal Lorentz structure. Photon two-point function: photon-loop. The photon-loop computation involves a sin- gle photon-loop integral contribution to the photon polarization tensor from Fig. 2. Using dimensional regularization, in [41] we have computed the integral, Πµν κg (p)D = 1 2 µ4−D ∫ dDk (2π)D Γµρσκg (−p;−k, p+ k) −igρρ′ k2 Γνρ ′σ′ κg (p; k,−k − p) −igσσ ′ (p+ k)2 , as a function of deformation freedom κg ambiguity. We obtained the following compact form of the photon-loop contribution to the photon two-point function in D-dimensions, Πµν κg (p)D = 1 (4π)2 {[ gµνp2 − pµpν ] B κg 1 (p) + (θp)µ(θp)νB κg 2 (p) + [ gµν(θp)2 − (θθ)µνp2 + p{µ(θθp)ν} ] B κg 3 (p) Two-Point Functions on Deformed Spacetime 11 + [ (θθ)µν(θp)2 + (θθp)µ(θθp)ν ] B κg 4 (p) + (θp){µ(θθθp)ν}B κg 5 (p) } . (3.4) Each term of the above tensor structure satisfies Ward identities by itself, i.e. pµΠµν κg (p)D = pνΠµν κg (p)D = 0,∀(D,κg). The structure of the photon-loop contribution to the photon polarization tensor contains various previously unknown new momentum structures. It is much reacher with respect to earlier non SW map θ-exact results based on ?-product only [18, 34]. These higher order in θ (θθθθ types) terms suggest certain connection to the open/closed string correspondence [47, 67] (in an inverted way). We consider such connection plausible given the connection between noncommutative field theory and quantum gravity/string theory. All coefficients B κg i (p) can be expressed as sum over integrals over modified Bessel and generalized hypergeometric functions. A complete list of coefficients B κg i (p) including compu- tations, as a functions of dimension D, is given in full details in [41]. Next we concentrate on the divergent parts in the limit D → 4− ε, and in the IR regime only B κg 1 (p) ∼ ( 2 3 (κg − 3)2 + 2 3 (κg + 2)2 p 2(tr θθ) (θp)2 + 4 3 ( κ2 g + 4κg + 1 )p2(θθp)2 (θp)4 ) × [ 2 ε + ln(µ2(θp)2) ] − 16 3 (κg − 1)2 1 (θp)6 ( (tr θθ)(θp)2 + 4(θθp)2 ) , (3.5) B κg 2 (p) ∼ ( 8 3 (κg − 1)2 p 4(θθp)2 (θp)6 + 2 3 ( κ2 g − 2κg − 5 )p4(tr θθ) (θp)4 + 2 3 ( 25κ2 g − 86κg + 73 ) × p2 (θp)2 )[ 2 ε + ln ( µ2(θp)2 )] − 16 3 (κg − 3)(3κg − 1) 1 (θp)4 + 32 3 (κg − 1 )2 1 (θp)8 ( (tr θθ)(θp)2 + 6(θθp)2 ) , (3.6) B κg 3 (p) ∼ −1 3 ( κ2 g − 2κg − 11 ) p2 (θp)2 [ 2 ε + ln ( µ2(θp)2 )] − 8 3(θp)4 ( κ2 g − 10κg + 17 ) , (3.7) B κg 4 (p) ∼ −2(κg + 1)2 p4 (θp)4 [ 2 ε + ln ( µ2(θp)2 )] − 32p2 3(θp)6 ( κ2 g − 6κg + 7 ) , (3.8) B κg 5 (p) ∼ 4 3 ( κ2 g + κg + 4 ) p4 (θp)4 [ 2 ε + ln ( µ2(θp)2 )] + 64p2 3(θp)6 (κg − 1)(κg − 2). (3.9) Note that all B κg i (p) coefficients are computed for abitrary κg and the notation ∼ means that in the above equations we have neglected all finite terms. We observe here the presence of the UV divergences as well as a quadratic UV/IR mixing in all B κg i ’s. Up to the 1/ε terms, the UV divergence is at most logarithmic, i.e. there is a logarithmic ultraviolet/infrared term representing a soft UV/IR mixing. The results (3.5)–(3.9) in four dimensions for arbitrary κg show power type UV/IR mixing, therefore diverge at both the commutative limit (θ → 0) and the size-of- the-object limit (\|θp\| → 0). Inspecting (3.5) to (3.9) together with general structure (3.4) we found decouplings of UV and logarithmic IR divergences from the power UV/IR mixing terms. The latter exists in all B κg i (p)’s. The logarithmic IR divergences from planar and nonplanar sources appear to have identical coefficient and combine into a single lnµ2(θp)2 term. Finally it is important to stress that no single κg value is capable of removing all novel divergences. 3.2 Photon-loop with a special θµν in four dimensions In our prior analysis we have found that in theD → 4−ε limit the general off-shell contribution of photon self-interaction loop to the photon two-point function contains complicated non-vanishing UV and IR divergent terms with existing and new momentum structures, regardless the κg values 12 J. Trampetić and J. You we take. To see whether there exists certain remedy to this situation we explore two conditions which have emerged in the prior studies. First we tested the zero mass-shell condition/limit (p2 → 0) used in θ-expanded models [55]. Inspection of equations (3.2)–(3.3) and (3.5)–(3.9) show some simplification but not the full cancelation of the pathological divergences. Such condition clearly appears to be unsatisfactory. Next we have turned into the other one, namely the special full rank θµν choice θµν ≡ θµνσ2 = 1 Λ2 NC  0 −1 0 0 1 0 0 0 0 0 0 −1 0 0 1 0  = 1 Λ2 NC ( iσ2 0 0 iσ2 ) ≡ 1 Λ2 NC iσ2 ⊗ I2, (3.10) with σ2 being famous Pauli matrix. This constraints was used in the renormalizability studies of 4d NCGFT without SW map [16, 17]. Note also that this θµνσ2 is full rank and thus breaks in general the unitarity if one performs Wick rotation to the Minkowski spacetime [29]. This choice, in 4d Euclidean spacetime, induces a relation (θθ)µν = − 1 Λ4 NC gµν . The tensor structures (3.4), with restored coupling constant e included then reduces into two parts and we obtain the same Lorentz structure as we did from the fermion-loop (3.1): Πµν κg (p)4 ∣∣∣θσ2 = e2 (4π)2 {[ gµνp2 − pµpν ] B κg a (p) + (θp)µ(θp)νB κg b (p) } . Neglecting the IR safe terms we have found that the B κg a (p) and B κg b (p) exhibits divergent structures [41]: B κg a (p) ∼ 4(κg − 3)2 3 ( 2 ε + ln ( µ2(θp)2 )) + 16 3 (κg − 3)(κg + 1) p2(θp)2 , B κg b (p) ∼ 2p2 (κg − 3)(7κg − 9) (θp)2 ( 2 ε + ln ( µ2(θp)2 )) − 16 3 (κg − 3)(7κg − 5) (θp)4 , which can all be eliminated by choosing the deformation freedom point κg = 3. A careful evaluation of the full photon-loop at this point exhibits a simple structures B κg=3 a (p) = 2 [ 56 3 + I ] , B κg=3 b (p) = − p2 (θp)2 9[8− I], where in [41] we have shown that I = 0. Thus, for special choice (3.10) in the D → 4− ε limit, and at κg = 3 point, we have found B κg=3 a (p) = 112 3 , B κg=3 b (p) = −72 p2 (θp)2 . Summing up the contributions to the photon polarization tensor. To simplify the tremendous divergent structures in the loop-coefficients F κf 1,2(p)’s, and B κg 1,...,5(p)’s at D → 4 − ε, we have been forced to probe two additional constraints: One which appears to be ineffective is the zero mass-shell condition/limit p2 → 0, due to the uncertainty on its own validity when quantum corrections present. The other constraint, namely setting θµν to a special full ranked value θµνσ2 (3.10), reduces the number of different momentum structures from five to two. Then all divergences and the IR safe contributions disappear at a deformation parameter-space unique point (κf , κg) = (0, 3) leaving, Πµν (0,3)(p) ∣∣θσ2 ≡ [Πµν κf=0(p) + Πµµ κg=3(p) ]∣∣θσ2 = e2p2 π2 [ 7 3 ( gµν − pµpν p2 ) − 9 2 (θp)µ(θp)ν (θp)2 ] , as the only one-loop-finite contribution/correction to the photon two-point function. Two-Point Functions on Deformed Spacetime 13 Figure 3. Bubble-graph contribution to the neutrino two-point function. 4 Neutrino two-point function One-loop contributions as a function of κf receive the same spinor structure as in [38]. We now reconfirm that by using the action (2.3) together with the Feynman rule (2.12), out of four diagrams in Fig. 2 of [38], only the non-vanishing bubble-graph is considered in this manuscript. Thus, in the present scenario a contribution from Fig. 3 reads Σκf (p)D = −1 (4π)2 [ γµp µN κf 1 (p) + γµ(θθp)µN κf 2 (p) ] . (4.1) Complete loop-coefficients N κf 1,2(p), as a functions of an arbitrary dimensions D are given in [41]. For arbitrary κf in the limit D → 4− ε we have obtained the following loop-coefficients: N κf 1 (p) = κf {[ 2κf + ( κf − 1 )(2 ε + lnπeγE + ln ( µ2(θp)2 ))] − p2(θp)2 4 ∞∑ k=0 (p2(θp)2)k 4kk(k + 1)(2k + 1)2(2k + 3)Γ[2k + 4] × [ k(k + 1)(2k + 1)(2k + 3) ( κf (2k + 3)− 1 )( ln ( p2(θp)2 ) − 2ψ(2k)− ln 4 ) + 3 + 28k + 46k2 + 20k3 − κf ( 2k + 3 )2( 1 + 8k + 8k2 )] + (tr θθ) { p2 (θp)2 [ 2 ε + 2 + γE + lnπ + ln ( µ2(θp)2 ) + 8 ( κf − 1 ) 3κf (θp)2p2 ] − p4 4 ∞∑ k=0 (p2(θp)2)k 4kk(k + 1)(2k + 1)2(2k + 3) [ k(k + 1)(2k + 1)(2k + 3) × ( ln ( p2(θp)2 ) − 2ψ(2k)− ln 4 ) − 2k ( 14 + k(23 + 10k) ) − 3 ]} + (θθp)2 { 2 p2 (θp)4 [ 2 ε + 1 + γE + lnπ + ln ( µ2(θp)2 ) + 16 ( κf − 1 ) 3κf (θp)2p2 ] + p4 2(θp)2 ∞∑ k=0 k(p2(θp)2)k (k + 1)(2k + 1)2(2k + 3)Γ[2k + 4] [ (k + 1)(2k + 1)(2k + 3) × ( ln ( p2(θp)2 ) − 2ψ(2k)− ln 4 ) + 16k2 − 34k − 17 ]}} , N κf 2 (p) = −κf p2 (θp)2 { 4 + ( κf − 1 )[2 ε + lnπeγE + ln ( µ2(θp)2 )] − 16 ( κf − 1 ) 3(θp)2p2 − p2(θp)2 4 ∞∑ k=0 (p2(θp)2)k 4kk(k + 1)(2k + 1)2(2k + 3)Γ[2k + 4] [ k(1 + k)(1 + 2k)(3 + 2k) × ( 1 + 3kf + 2 ( kf + 1 ) k )( ln ( p2(θp)2 ) − 2ψ(2k)− ln 4 ) − 3− 9κf − 4k ( 7 + 21κf + ( 20 + 43κf + 2k ( 11 + 4k + 4κf (4 + k) )) k )]} . 14 J. Trampetić and J. You In the expressions for N κf 1,2(p) contributions from both the planar as well as the non-planar graphs are present. For any κf 6= 1 our neutrino two point function receive UV, and power as well as logarithmic UV/IR mixing terms. First we analyze the choice κf = 1. For D = 4 − ε in the limit ε → 0, we obtain the final expression as Σκf=1(p) = −e2 (4π)2 γµ [ pµN1(p) + (θθp)µ p2 (θp)2 N2(p) ] , (4.2) with restored coupling constant e included. Here N1,2(p) coefficients are as follows N1(p) = p2 ( tr θθ (θp)2 + 2 (θθp)2 (θp)4 ) A+ [ 1 + p2 ( tr θθ (θp)2 + (θθp)2 (θp)4 )] B, (4.3) A = 2 ε + ln(µ2(θp)2) + ln(πeγE) + ∞∑ k=1 ( p2(θp)2/4 )k Γ(2k + 2) ( ln p2(θp)2 4 + 2ψ0(2k + 2) ) , (4.4) B = −8π2N2(p) = −2 + ∞∑ k=0 ( p2(θp)2/4 )k+1 (2k + 1)(2k + 3)Γ(2k + 2) × ( ln p2(θp)2 4 − 2ψ0(2k + 2)− 8(k + 1) (2k + 1)(2k + 3) ) . (4.5) It is to be noted here that the spinor structure proportional to γµ(θp)µ is missing in the final result. This conforms with the calculation of the neutral fermion self-energy in the θ-expanded SW map approach [27]. The 1/ε UV divergence could in principle be removed by a properly chosen counterterm. However (as already mentioned) due to the specific momentum-dependent coefficient in front of it, a nonlocal form for it is required3. It is important to stress here that amongst other terms contained in both coefficients N1,2(p), there are structures proportional to( p2(θp)2 )n+1( ln ( p2(θp)2 ))m , ∀n and m = 0, 1. The numerical factors in front of the above structures are rapidly-decaying, thus series are always convergent for finite argument, as we demonstrate in [38]. Turning to the UV/IR mixing problem, we recognize a soft UV/IR mixing term represented by a logarithm, Σ UV/IR κf=1 = −e2 (4π)2 /pp2 ( tr θθ (θp)2 + 2 (θθp)2 (θp)4 ) · ln ∣∣µ2(θp)2 ∣∣. Instead of dealing with nonlocal counterterms, we take a different route here to cope with various divergences besetting (4.2). Since θ0i 6= 0 makes a NC theory nonunitary [29], we can, without loss of generality, chose θ to lie in the (1, 2) plane θµνspec = 1 Λ2 NC  0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0 0  , (4.6) yielding tr θθ (θp)2 + 2 (θθp)2 (θp)4 = 0, ∀ p. (4.7) 3Any quadratic counterterm for the neutral fermion would be gauge invariant since the neutral fermion field is invariant under the U(1) gauge transformation. Two-Point Functions on Deformed Spacetime 15 With (4.7), Σspec κf=1, in terms of Euclidean momenta, receives the following form: Σspec κf=1(p) = −e2 (4π)2 γµ [ pµ ( 1 + tr θθ 2 p2 (θp)2 ) − 2(θθp)µ p2 (θp)2 ] N2(p). (4.8) By inspecting (4.5) one can be easily convinced that N2(p) is free from the 1/ε divergence and the UV/IR mixing term, being also well-behaved in the infrared, in the θ → 0 as well as θp→ 0 limit. We see, however, that the two terms in (4.8), one being proportional to pµ and the other proportional to (θθp)µ, are still ill-behaved in the θp→ 0 limit. If, for the choice (4.6), P denotes the momentum in the (1,2) plane, then θp = θP . For instance, a particle moving inside the noncommutative plane with momentum P along the one axis, has a spatial extension of size \|θP \| along the other. For the choice (4.6), θp→ 0 corresponds to a zero momentum projection onto the (1,2) plane. Thus, albeit in our approach the commutative limit (θ → 0) is smooth at the quantum level, the limit when an extended object (arising due to the fuzziness of space) shrinks to zero, is not. We could surely claim that in our approach the UV/IR mixing problem is considerably softened; on the other hand, we have witnessed how the problem strikes back in an unexpected way. This is, at the same time, the first example where this two limits are not degenerate (or two limits do not commute). Next we analyze the choice κf = 0. Using the Feynman rule (2.12) for κf = 0 and for general θ, we find the following closed form contribution to the neutrino two point function (from diagram Σ1 in [38, 40]): Σκf=0(p) = e2 (4π)2 /p [ 8 3 1 (θp)2 ( tr θθ (θp)2 + 4 (θθp)2 (θp)4 )] . (4.9) It is important to stress that we have found that diagram Σ2 = 0 in all κf -cases, while diag- rams Σ3 and Σ4 vanish due to charge conjugation symmetry, see [38, 40]. There is no alternative dispersion relation in degenerate case (4.6), since the factor that multiplies /p in (4.9), does not dependent on the time-like component p0 (energy). Considering neutrino two point function (4.1), our results extends the prior works [38, 40] by completing the behavior for general κf . Here we discuss some novel behaviors associated with general κf . The neutrino two point function does posses power UV/IR mixing phenomenon for arbitrary values of κf , except κf = 1. In the limit κf → 0 all UV, IR divergent terms as well as constant terms in N κf 1,2(p) vanish; what remains are only the power UV/IR mixing terms. The UV divergence can be localized using the special θ value [38, 40] in N κf 1 (p) but not in N κf 2 (p). The UV and the power IR divergence in N κf 2 (p) can be removed by setting κf = 1. Summing up, choice κf = 1 eliminates some of divergences, but not all of them. Imposing the special θµνσ2 reduces the contribution to quadratic UV/IR mixing into a single term from N κf 2 (p), which has two zero points κf = 0, 1. Only κf = 0 can induce full divergence cancelations, by removing the whole Σ(p), i.e. we have Σκf=0(p) ∣∣θσ2 = 0. 5 Discussion and conclusion In this review we present a θ-exact quantum one-loop contributions to the photon (Π) and neu- trino (Σ) two point functions and analyze their properties. In principle the quantum corrections in NCQFTs are extremely profound, revealing a structure of pathological terms far beyond that found in ordinary field theories. For practical purposes, perturbative loop computation was the most intensively studied for the Moyal–Weyl (constant θµν) type deformation [16, 17, 28, 32, 50], 16 J. Trampetić and J. You for its preservation of translation invariance allows a (modified) Feynman diagrammatic calcu- lation. Much of efforts went in taming divergences related to the ultraviolet-infrared connec- tion/mixing, see for example [15, 38, 40, 41, 56, 60]. The UV/IR mixing, built-in as a new principle in all NCQFT models, and closely related to the Black Hole Complementarity and/or Holographic principle [44, 68, 69, 72], does reverse the well-established connection (via the uncer- tainty principle) between energy and size. The existence of such pathological terms in NCQFTs raises a serious concern on the renormalizability/consistency of the theory. Although no sat- isfactory resolution for this issue had been achieved so far, very recently it has been observed that certain control over novel divergences may be obtained by certain gauge invariant variation of the SW mapped action [41]. The anomalous structures in the two point function further suggests possible modifications to obtain trouble-free and meaningful loop-results, necessary for studying the particle propagation [38]. Such effects were largely left untouched in litera- ture so far, mostly due to the prior concern on the consistency/renormalizability from a purely theoretical viewpoint. After having defined and explained the full noncommutative action-model origin of the de- formation parameter-space (κf , κg), we obtained the relevant Feynman rules. Our method, ex- tending the modified Feynman rule procedure [28], yields the one-loop quantum corrections for arbitrary dimensions in closed form, as function of the deformation-freedom parameters κf , κg, as well as momentum pµ and noncommutative parameter θµν . Full parameter-space freedom is kept in our evaluation here. Following the extended dimensional regularization technique we expressed the diagrams as D-dimensional loop-integrals and identify the relevant momentum structures with corresponding loop-coefficients. We have found that total contribution to pho- ton two-point function satisfies the Ward–(Slavnov–Taylor) identity for arbitrary dimensions D and for any point in the (κf , κg) parameter-space: pµΠµν (κf ,κg)(p)D = pµ ( Πµν κf (p)D + Πµν κg (p)D ) = pν ( Πµν κf (p)D + Πµν κg (p)D ) = 0. The one-loop photon polarization tensor in four dimensions contains the UV divergence and UV/IR mixing terms dependent on the freedom parameters κf and κg. The introduction of the freedom parameters univocally has a potential to improve the situation regarding cancellation of divergences, since certain choices for κf and κg could make some of the terms containing singularities to vanish. We observe the following general behavior of one-loop two-point functions in the D → 4− ε limit: The total expressions for both the photon and the neutrino two point functions contain the 1/ε ultraviolet term, the celebrated UV/IR mixing power terms as well as the logarithmic (soft) UV/IR mixing term. The 1/ε divergence is always independent of the noncommutative scale. The logarithmic terms from the ε-expansion and the modified Bessel function integral sum into a common term ln(µ2(θp)2), which is divergent in the infrared \|p\| → 0, in the size- of-the-object \|θp\| → 0 limit, as well as in the vanishing noncommutativity θ → 0 limit. Thus, the existences of UV/IR mixings for both, photons and neutrinos respectively, in 4d spaces deformed by spacetime noncommutativity at low energies, suggests that the relation of quantum corrections to observations [1, 2, 6, 44, 46] is not entirely clear. However, in the context of the UV/IR mixing it is very important to mention a complementary approach [1, 2] where NC gauge theories are realized as effective QFT’s, underlain by some more fundamental theory such as string theory. It was claimed that for a large class of more general QFT’s above the UV cutoff the phenomenological effects of the UV completion can be quite successfully modeled by a threshold value of the UV cutoff. So, in the presence of a finite UV cutoff no one sort of divergence will ever appear since the problematic phase factors effectively transform the highest energy scale (the UV cutoff) into the lowest one (the IR cutoff). What is more, not only the full scope of noncommutativity is experienced only in the range delimited by the two cutoffs, but for the scale of NC high enough, the whole standard model can be placed below the IR cutoff [44]. Two-Point Functions on Deformed Spacetime 17 Thus, a way the UV/IR mixing problem becomes hugely less pressing, making a study of the theory at the quantum level much more reliable. We have demonstrated how quantum effects in the θ-exact Seiberg–Witten map approach to NC gauge field theory reveal a much richer structure for the one-loop quantum correction to the photon and fermion two-point functions (and accordingly for the UV/IR mixing problem) than observed previously in approximate models restricting to low-energy phenomena. Our analysis can be considered trustworthy since we have obtained the final result in an analytic, closed-form manner. We believe that a promising avenue of research would be using the enormous freedom in the Seiberg–Witten map to look for other forms which UV/IR mixing may assume. Two alternative forms have been already found [38]. Finally, our approach to UV/IR mixing should not be confused with the one based on a theory with UV completion (ΛUV <∞), where a theory becomes an effective QFT, and the UV/IR mixing manifests itself via a specific relationships between the UV and the IR cutoffs [1, 2, 6, 44, 46]. In conclusion, our main result in four-dimensional space is that we have all pathological terms under full control after the introduction of the deformation-freedom parameter-space (κf , κg) and a special choice for θµν . Namely, working in the 4d Euclidean space with a special full rank of θµνσ2 and setting the point (κf , κg) = (0, 3), the fermion plus the photon-loop contribution to Πµν (κf ,κg)(p) contain only two finite terms, i.e. all divergent terms are eliminated. 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Two-Point Functions on Deformed Spacetime

Institution

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