Noncommutative field theory from angular twist

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Oct 12, 2018 - 1Faculty of Physics, University of Belgrade, Beograd 11000, Serbia. 2Dipartimento di ...... integration path over the energy p0. After that we.
PHYSICAL REVIEW D 98, 085011 (2018)

Noncommutative field theory from angular twist Marija Dimitrijević Ćirić,1,* Nikola Konjik,1,† Maxim A. Kurkov,2,3,‡ Fedele Lizzi,2,3,4,§ and Patrizia Vitale2,3,∥ 1

2

Faculty of Physics, University of Belgrade, Beograd 11000, Serbia Dipartimento di Fisica “Ettore Pancini,” Universit`a di Napoli Federico II, Napoli 80126, Italy 3 INFN, Sezione di Napoli 80126, Italy 4 Departament de Física Qu`antica i Astrofísica and Institut de Cíencies del Cosmos (ICCUB), Universitat de Barcelona, Barcelona 08028, Spain (Received 2 July 2018; published 12 October 2018)

We consider a noncommutative field theory in three space-time dimensions, with space-time star commutators reproducing a solvable Lie algebra. The ⋆-product can be derived from a twist operator and it is shown to be invariant under twisted Poincar´e transformations. In momentum space the noncommutativity manifests itself as a noncommutative ⋆-deformed sum for the momenta, which allows for an equivalent definition of the ⋆-product in terms of twisted convolution of plane waves. As an application, we analyze the λϕ4 field theory at one loop and discuss its UV/IR behavior. We also analyze the kinematics of particle decay for two different situations: the first one corresponds to a splitting of space-time where only space is deformed, whereas the second one entails a nontrivial ⋆-multiplication for the time variable, while one of the three spatial coordinates stays commutative. DOI: 10.1103/PhysRevD.98.085011

I. INTRODUCTION The possibility that space-time is described by a noncommutative geometry is a fascinating idea that has deep physical motivations going back to Bronstein [1] (for a more recent review see [2]). One way to study noncommutative spaces is to study field theories where the product is deformed into a ⋆-product so that the fields do not commute among themselves. The most studied field theory is the one described by the Grönewold-Moyal product [3,4], which adapts to space-time the usual commutation rules of standard quantum mechanics [5], ½xμ ; xν  ¼ iθμν ;

ð1:1Þ

with θμν being a constant with the dimensions of length square. Field theories on these spaces (for a review see [6]) have a peculiar behavior: some ultraviolet divergences are converted to infrared ones, a phenomenon known as UV/IR mixing [7]. This is not only a characteristic of the Grönewold-Moyal product, but in general of all *

[email protected] [email protected] [email protected] § [email protected][email protected] † ‡

½x3 ⋆; x1  ¼ −iθx2 ;

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

2470-0010=2018=98(8)=085011(14)

translationally invariant products [8,9]. These theories are not Poincar´e invariant, but they can be invariant under a twisted symmetry. It was shown in [10,11] that indeed, field theories based on the Grönewold-Moyal ⋆-product are invariant under the twisted θ-Poncar´e symmetry. Another example of a noncommutative (NC) space-time invariant under a quantum symmetry is the κ-Minkowski space-time. It is an example of Lie algebra noncommutativity and it was first introduced [12,13] in the early 90’s. Because of its symmetry properties, i.e., invariance under the κ-Poincar´e Hopf algebra, κ-Minkowski represents an interesting playground for constructing physical models and investigating their behavior under the NC deformation. Indeed, there is a whole family of Lie-algebra based star-products, introduced in [14] (also see [15] for a different approach), among these the one giving rise to the noncommutative space R3λ [16], which has been widely studied [17] in relation with quantum mechanics [18], quantum gravity [19], and quantum field theory [20–25]. In this paper we discuss a scalar field theory for another Lie-algebra kind of noncommutative space which can be described by a Drinfel’d twist [26]. It is the space characterized by the following commutation relations [27,28]:

½x3 ⋆; x2  ¼ iθx1 ;

ð1:2Þ

all other commutators being 0, and with θ being a constant with the dimension of length. The underlying Lie algebra

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Published by the American Physical Society

MARIJA DIMITRIJEVIĆ ĆIRIĆ et al.

PHYS. REV. D 98, 085011 (2018)

can be recognized to be the Euclidean algebra eð2Þ, for which a star-product realization was already found in [14] by means of a Jordan-Schwinger map. One interesting property of the realization considered here relies on the fact that it is possible to exhibit a Drinfel’d twist for it. The main motivation for this work is that, although the commutation relations violate Poincar´e symmetry, the space is the homogenous space of a twisted Poincar´e Hopf algebra. To this, let us define the following Drinfel’d twist:  iθ F ðx; yÞ ¼ exp − ð∂ y3 ðx2 ∂ x1 − x1 ∂ x2 Þ 2  − ∂ x3 ðy2 ∂ y1 − y1 ∂ y2 ÞÞ   iθ ¼ exp ð∂ 3 ∂ − ∂ x3 ∂ φy Þ ; ð1:3Þ 2 y φx where for the last expression we have used cylindrical coordinates x1 ¼ ρx cos φx , x2 ¼ ρx sin φx , and an analogous expression for y. Since the vector fields ∂ φ and ∂ 3 commute, the twist is an admissible one, because it satisfies the cocycle condition, a sufficient requirement for the associativity of the ⋆ product defined by ðf ⋆ gÞðxÞ ¼ F −1 ðy; zÞfðyÞgðzÞjx¼y¼z ¼ fg −

iθ ð∂ f∂ g − ∂ 3 f∂ φ gÞ þ Oðθ2 Þ: ð1:4Þ 2 φ 3

Let us recall that the existence of a twist greatly simplifies the construction of a differential calculus and the definition of noncommutative gauge and field theories [29]. This space is a variation of the previously mentioned κ-Minkowski space where the commutation relations are similarly of Lie-algebra type: ½x0 ; xi  ¼ i 1κ xi , all other commutators vanishing. Field theory on the κ-Minkowski space-time is very much studied in the literature. In particular, scalar field theory and UV/IR mixing were discussed in [30–33], while NC gauge theory and coupling with fermions were introduced in [34,35]. Besides the already quoted [14], where all three-dimensional Lie algebras are analyzed, and related star products are introduced, Lie-algebra based NC spaces and the corresponding field theories were studied also in [36] who considered nilpotent algebras (our case is solvable but not nilpotent) and [37], which did not consider our noncommutative case. A common trait with the latter paper is the intertwining of UV/IR mixing with the violation of translational invariance. Field theories based on the suð2Þ case as an instance of a simple algebra, also in connection with the UV/IR mixing, have been studied in [20,23]. Relations (1.2) can be substituted by an equivalent set with x3 substituted by the time coordinate x0 . This was the choice in the original paper [28]. From an algebraic point of view there is no difference. Conceptually the two choices

are clearly different: the former corresponds to a splitting of space-time where time remains commutative, while the latter singles out time as a noncommuting operator. The computational difference appears as soon as one considers loop diagrams (see the remark in Sec. III). The main result of the paper is the persistence of the UV/IR mixing at one loop for an interacting scalar field theory, λϕ⋆4 , which, although not retaining Poincar´e symmetry of its commutative analogue, is however invariant under twisted Poincar´e transformations. The paper is organized as follows. In Sec. II we introduce the noncommutative algebra under consideration and compute the ⋆-convolution of plane waves in terms of ⋆-sums of momenta. We thus review the twisted Poincar´e algebra introduced in [28] and relate our ⋆-sum to the deformed coproduct there defined. In Sec. III we introduce the λϕ⋆4 model and carefully study it in momentum space, at one loop. We find a deformed conservation of momenta that entails a deformation of the nonplanar correction to the propagator, with respect to the commutative case. We conclude that the phenomenon of UV/IR mixing, which was first discovered for Moyal noncommutativity, persists in the case under consideration. In order to better understand the phenomenon, in Sec. IV we pass to the three-dimensional case and repeat the analysis. We find that again, the planar diagram does not change with respect to the commutative case, while the nonplanar contribution exhibits mixing. As another application of our findings with respect to the deformed sum of momenta, we study in Sec. V the kinematics of particle decays at the tree level. We conclude with a short summary and an appendix where the main calculations are performed. II. ANGULAR NONCOMMUTATIVITY The Abelian twist (1.3) is a special example of a more general twist introduced in [38–40]. The NC differential geometry induced by (1.3) was constructed in [28], where also the NC field theory of a scalar field in the ReissnerNordström background was investigated. We recall that the twist (1.3) is Abelian, being based on two commuting vector fields, ∂ x3 and ∂ φ . Its form is reminiscent of the Moyal twist for the Moyal algebra ½x1 ⋆; x2  ¼ iθ, where the two vector fields are ∂ x1 and ∂ x2 , but one should refrain from introducing the star commutator ½x3 ⋆; φ ¼ iθ since φ is not a well-defined continuous function. The following relations hold: ½x3 ⋆; ρ ¼ 0; ½x3 ⋆; eiφ  ¼ −θeiφ ; ½x3 ⋆; fðx0 ; x3 ; ρ; φÞ ¼ iθ∂ φ f:

ð2:1Þ

For field theory it is useful to calculate the ⋆-product of two plane waves. This enables the construction of the

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NONCOMMUTATIVE FIELD THEORY FROM ANGULAR TWIST product in Fourier transform as some kind of twisted convolution. We recall that for the Gronewöld-Moyal product (1.1) the analogous formula is eip·x  eiq·x ¼ μν eiðpþqÞ·xþpμ θ qν . While the explicit calculations are performed in the appendix, we reproduce here the relevant results. We have e−ip·x ⋆ e−iq·x ¼ e−iðpþ⋆ qÞ·x ;

ð2:2Þ

where the ⋆-sum of the 4-momenta is defined as follows: p þ⋆ q ¼ Rðq3 Þp þ Rð−p3 Þq;

1

0 0     B θt θt B 0 cos sin B 2 2 B     RðtÞ ≡ B B θt θt B 0 − sin cos @ 2 2 0 0 0

0

1

C 0C C C C: C 0C A

ð2:4Þ

1

The matrix R corresponds to a rotation matrix in the ðp1 p2 Þ plane. The angle of rotation is proportional to the noncommutativity parameter, and to the momenta involved. It reduces to the identity in the commutative limit θ → 0 as well as in the low momentum limit. It is remarkable that, even though after the first two steps, (A10) and (A11), one obtains expressions that are singular at θq2 3 ¼ π2 þ πk, k ∈ Z, the final expression (2.3) contains no singularities. It is easy to see that the ⋆-sum is noncommutative, but associative1 and satisfies p þ⋆ ð−pÞ ¼ 0

e

⋆e

−iq·x

−ir·x

⋆e

¼e

−iðpþ⋆ qþ⋆ rÞ·x

;

ð2:6Þ

with2 p þ⋆ q þ⋆ r ¼ Rðr3 þ q3 Þp þ Rð−p3 þ r3 Þq þ Rð−p3 − q3 Þr: ð2:7Þ By induction it is easily shown that 1

þ⋆ … þ⋆ p

ðNÞ

 X N X X ðkÞ  ðkÞ ¼ R − p3 þ p3 pðjÞ : j¼1

1≤k