# Renormalization on noncommutative torus

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## Abstract

We study a self-interacting scalar \(\varphi ^4\) theory on the *d*-dimensional noncommutative torus. We determine, for the particular cases \(d=2\) and \(d=4\), the counterterms required by one-loop renormalization. We discuss higher loops in two dimensions and two-loop contributions to the self-energy in four dimensions. Our analysis points toward the absence of any problems related to the ultraviolet/infrared mixing and thus to renormalizability of the theory. However, we find another potentially troubling phenomenon which is a wild behavior of the two-point amplitude as a function of the noncommutativity matrix \(\theta \).

### Keywords

Heat Kernel Star Product External Momentum Diophantine Condition Noncommutativity Parameter## 1 Introduction

One of the motivations for considering quantum field theories on noncommutative spaces was the hope that they may be ultraviolet (UV) finite. It was shown, however, that UV divergences persist on the noncommutative (NC) Moyal plane [1, 2]. Moreover, though certain Feynman diagrams are less UV divergent than in the commutative case, they develop singularities at some special, typically zero, value of the external momenta. When such diagrams appear as subgraphs of higher-order diagrams, the latter diagrams become divergent in a nonrenormalizable manner. This phenomenon [3, 4, 5], called the UV/IR mixing [4], is the main obstacle to renormalization of NC field theories.

It was believed for some time that the UV/IR mixing appears exclusively in Euclidean signature spaces. However, it was demonstrated [6] that similar problems exist in Minkowski spacetime as well.

Various methods were proposed to deal with this problem. Of course, the supersymmetry helps to achieve renormalizability of noncommutative theories [7, 8]. Grosse and Wulkenhaar [9, 10] motivated by the Langmann–Szabo duality [11] proposed to add to the action an oscillator term which breaks translation invariance but ensures renormalizability. Modifications of the momentum dependence of the kinetic term were considered in [12]. Taking the noncommutativity parameter nilpotent [13] also improves renormalization. It has been shown [14] that spontaneous symmetry breaking softens the UV/IR mixing. A fairly recent review is Ref. [15].

In this work, we take a different path. We consider a noncommutative \(\varphi ^4\) theory on a torus. Sensitivity of UV divergences in NC theories to the presence of compact dimension (and even eventual disappearance of such divergences) has been stressed already in [2]; see also [16]. Note, however, that, due to a different implementation of noncommutativity, the existence of a compact dimension in the two-dimensional case considered in [2] guarantees the finiteness of tadpole contributions. This is not the case in the model considered in the present article, where quantum corrections are UV-divergent and must be properly renormalized.

One may get an idea on the structure of counterterms, singularities of the propagators etc. by looking at the heat kernel expansion (see, e.g. [17]). Roughly speaking, the relevant operators^{1} on noncommutative spaces are generalized Laplacians that contain gauge fields and potentials (as usual Laplacians), but these gauge fields and potentials act by left or right Moyal multiplications on the fluctuations \(\delta \varphi \). If the generalized Laplacian contains only left or only right Moyal multiplications, the structure of corresponding heat kernel coefficients is very simple on both NC torus [18] and NC plane [19]—they look almost as their commutative counterparts with star products instead of usual products. For interesting theories, however, the relevant operators contain *both* left and right multiplications. For such operators on the NC plane the structure of heat kernel coefficients is very complicated [20, 21] thus reflecting the presence of the UV/IR mixing. The situation changes drastically on NC torus [22]. If the noncommutativity parameter satisfies the so-called Diophantine condition or is rational, the heat kernel coefficients (and thus the one-loop counterterms) assume a very simple form if written in terms of a suitably defined trace operation on the algebra of smooth functions on the torus. We shall use this observation to formulate our proposal for a (presumably) renormalizable \(\varphi ^4\) theory on NC torus.

Let us stress that the notion of locality does not make much sense in noncommutative theories since the star product itself is nonlocal. Instead of local polynomial actions one has to use traces of polynomials constructed from the fields and their derivatives. There are more different traces on \(\mathbb {T}^d_\theta \) than on \(\mathbb {R}_\theta ^d\). This observation will be crucial for our construction of admissible counterterms.

Here we like to mention several papers that considered quantum field theories on NC torus. In Ref. [7] it was demonstrated that supersymmetric Yang–Mills theory on \(\mathbb {T}^3_\theta \) with rational \(\theta \) is one-loop renormalizable. Pure Yang–Mills theories were considered in [23] at one loop. Some arguments regarding the higher-order behavior were also presented. Relations between NC theories on \(\mathbb {T}^d_\theta \) with rational \(\theta \) and matrix models were studied in [24, 25].

The purpose of this paper is to set up the stage for renormalization on NC torus and to discuss basic features of this process. First, we write down the model and introduce new counterterms for Diophantine and rational \(\theta \). We analyze in detail two- and four-point functions at one loop. In \(d=2\), the only superficially divergent diagrams are the one-loop two-point functions. We demonstrate that the insertion of these diagrams (together with counterterms) into internal lines of other diagrams does not lead to any divergences, so that there is no UV/IR mixing (at least in its classical formulation [4]) on \(\mathbb {T}^2_\theta \). In \(d=4\), we analyze the two-loop self-energy diagrams. All our findings, though do not contain a complete proof, strongly suggest that the introduction of new counterterms does make the \(\varphi ^4\) theory on NC torus in \(d=2\) and \(d=4\) renormalizable.

The counterterms depend in a very essential way on the number theory nature of \(\theta \). But not only this, we show that also renormalized two-point functions (too) strongly depend on \(\theta \). More precisely, we compare the two-point functions in \(d=2\) for two close values \(\theta _1\) and \(\theta _2\) of the noncommutativity matrix, one being rational, and the other irrational (Diophantine). We find that the typical variation of the two-point function is \(\sim \) \( \ln ||\theta _1 -\theta _2||\). However, this does not necessarily mean that the theory has no prediction power. We discuss the implications and possible ways out in the Conclusions of the article.

The paper is organized as follows. The next section contains the definitions that will be used throughout the text. In Sect. 3 we consider the two-point functions at one-loop order and analyze their renormalization and variation with \(\theta \). Section 4 is dedicated to four-point functions at one loop. Higher loops in \(d=2\) are considered in Sect. 5 and two-loop two-point functions in \(d=4\) in Sect. 6. Our results are discussed in Sect. 7. The behavior of some double sums is analyzed in Appendices A and B.

## 2 The model

*d*-dimensional noncommutative (NC) torus \(\mathbb {T}^d_\theta \) with unit radii; see [26]. The algebra \(\mathcal {A}_\theta \) of smooth functions on \(\mathbb {T}^d_\theta \) is formed by the Fourier-type series

*p*. The unitary \(U_p\) satisfy

*formal*expansion in the noncommutativity parameter, i.e., it is not convergent.

^{2}

*C*and \(\beta \), such that

^{3}

## 3 One-loop renormalization of self-energy diagrams

*n*-point function; the NC torus is not an exception to this fact.

*p*—determined by the numerical character of \(\theta \)—so we need an appropriate definition of this series that provides a regularization of its divergences.

*K*represents the modified Bessel function. The sum \(S_1(p)\), originally defined in (16), is then given—for \(n=1\) and any \(p\notin \mathcal {Z}_\theta \)—by the convergent series in the r.h.s. of (21).

*p*belongs to the set \(\mathcal {Z}_\theta \), then the term in the series (19) with \(k=-\theta p\) does not present the exponential decrease for small

*t*so the integration must be performed for \(\mathrm{Re}(\epsilon )>-1+d/2n\). If we separate this term we get, after integration in

*t*,

*p*belongs to \(\mathcal {Z}_\theta \) so we need to introduce new mass terms in the action

*p*. Note that, upon quantum corrections, the mass of the field takes a different value for field components with momentum in \(\mathcal {Z}_\theta \). In particular, for irrational \(\theta \) the new term (26) in the action can be written as

Although the one-loop correction to the self-energy for any value of the external momentum is rendered finite by the mass renormalizations, the sum of all these contributions—implicit in the effective action—can be seen to be convergent, for irrational \(\theta \), only under the Diophantine condition [22].

*p*-independent value. Therefore, the renormalized two-point function reads

One may find some similarities between this situation and the one in the matrix model approach to noncommutivity, where the effective action behaves quite irregularly for some relations between parameters of the theory [28].

## 4 One-loop renormalization of four-point functions

*s*-channel (\(p_1,p_2\) entering the same vertex) are given by

*L*(

*p*,

*q*) is defined as the analytic extension to \(\epsilon =0\) of

*u*,

*v*to collect both propagators into a single denominator, we use the Schwinger proper time representation and then the Poisson resummation formula; the result reads

*t*so it can be integrated in some neighborhood of \(\epsilon =0\);

*L*(

*p*,

*q*) is thus finite for \(q\notin \mathcal {Z}_\theta \). On the other hand, for \(q\in \mathcal {Z}_\theta \), integration in

*t*gives

*L*(

*p*,

*q*) that determine the contributions of the diagrams displayed in Figs. 3, 4, 5, 6, 7, and 8 can then be written, for \(q\in \mathcal {Z}_\theta \), as

*p*) for \(d=4\). In higher dimensions the residue depends on

*p*.

## 5 Higher loops at two dimensions

Before analyzing higher order of perturbation series on \(\mathbb {T}^2_\theta \) let us briefly recall the UV/IR mixing problem on noncommutative plane \(\mathbb {R}^d_\theta \). The nonplanar diagrams on \(\mathbb {R}^d_\theta \) behave better in the ultraviolet than their commutative counterparts since \((p\theta )^{-1}\) (with *p* being an external momentum) serves as an effective ultraviolet cutoff. However, the divergences are restored in the commutative limit, \(\theta \rightarrow 0\), implying also a singularity at \(p\rightarrow 0\). According to Ref. [4], these singularities cause troubles with the convergence of loop integrals at zero momenta if nonplanar diagrams are inserted into internal lines of other diagrams. Note that in two dimensions 1PI diagrams are at most logarithmically divergent, so that the IR singularity may also be at most \(\ln |p|\). This singularity is rather mild. Thus one does not expect much troubles with the UV/IR mixing in \(d=2\). For this reason, our consideration of the two-torus will also be rather sketchy.

*p*. For \(p\not \in \mathcal {Z}_\theta \), there may be a growing contribution to \(S_1(p)\), which comes from the momentum \(k_p\) in (22) that minimizes \(|k+\theta p|\). It reads

*p*|. Therefore, the renormalized 2-point function on \(\mathbb {T}^2_\theta \) has a logarithmic singularity, but in contrast to \(\mathbb {R}^2_\theta \) this singularity is UV rather than IR. The UV singularities on the quantum plane, discussed in [29], were found more severe than on the commutative plane. However, the singularities on \(T^2_\theta \) are very mild. Indeed, if one inserts the renormalized diagram of Fig. 2 into an internal line with the momentum

*p*of some other diagram one gets (at large |

*p*|) a multiplier of \(\ln |p|\) from the diagram itself and \((p^2+m^2)^{-1}\) from an extra propagator. The overall contribution behaves as \(\ln |p| \cdot (p^2+m^2)^{-1}\) and does even improve the convergence of larger diagram.

We saw that in the \(\varphi ^4\) theory on \(\mathbb {T}^2_\theta \) (i) all superficially divergent diagrams can be renormalized by the counterterms that we have proposed, and (ii) the insertion of renormalized superficially divergent diagrams does not lead to any problems with convergence. Hence, there is no UV/IR mixing in this model, and it is likely renormalizable.

## 6 Two-loop self-energy at four dimensions

In this last section we describe the diagrams that contribute to \(\Sigma _2(p)\)—the second order correction to the self-energy—in the four-dimensional torus. For corresponding analysis on \(\mathbb {R}^4_\theta \) one may consult Ref. [30]. In this section we restrict ourselves to the case of a pure irrational (Diophantine) noncommutativity parameter. Therefore, \(\mathcal {Z}_\theta =\{ 0\}\). We analyze the divergences of two-loop diagrams and point out the main difficulties one finds in computing the remaining double sums; in the course of this analysis we will see the importance of the Diophantine condition on the matrix \(\theta \). Let us also remark that, since we are interested in the renormalizability of two-point functions, we will neglect divergent contributions which are either independent or quadratic in the external momentum *p* for they can be removed by mass and field renormalizations of order \(O(\lambda ^2)\).

*p*-independent or quadratic in

*p*.

The two-loop diagrams can be built in two different ways: either by inserting the one-loop self-energy \(\Sigma _1(p)\) into the internal propagator of a planar or a nonplanar tadpole (see Figs. 9, 10, respectively), so that both external legs enter the same vertex, or by attaching each external leg to a different vertex so that the two loops share a common internal momentum, as in Figs. 11, 12, 13, and 14.

*T*(

*p*) is defined as the analytic extension to \(\epsilon =0\) of the series

*p*-dependent, is exactly canceled by (50).

*U*,

*V*are defined as the analytic extensions to \(\epsilon =0\) of the series

*U*(

*p*, 0) corresponds to the diagram of the ordinary commutative case, its divergence is a quadratic polynomial in

*p*. According to Appendix A, the sum \(U(p,p,\epsilon )\) behaves as

^{4}in

*p*. Therefore, the nonlocal non-trace-like divergence introduced by

*U*(

*p*,

*p*) completely cancels (51).

*p*plus a term proportional to \(\delta _p\).

The double sums in (60) can all be treated in a unified way. The divergences of these sums in \(\mathbb {Z}^8\) at \(\epsilon =0\) arise from the fact that the denominator increases only with a sixth power at infinity. Nevertheless, the twisting factor \(\mathrm{e}^{2\pi i\,k\theta l}\) contributes to regularize the series. This certainly happens in the continuum case where the corresponding integration in \(\mathbb {R}^8\) is finite. However, in the discrete case there exist four-dimensional subspaces—with null measure in \(\mathbb {R}^8\)—for which the twisting factor vanishes.

*p*, while the first one is proportional to \(S_2(0,\epsilon )\); see (18). We conclude that

*k*,

*l*as long as \(|k-\theta l|\) does not decrease too fast, which is guaranteed by the Diophantine condition on \(\theta \). This implies in particular

Although our analysis is far from being exhaustive, we believe it strongly suggest that the \(\varphi ^4\) theory on \(\mathbb {T}_\theta ^4\) is renormalizable.

## 7 Conclusions

In this paper, we analyzed the renormalization of a scalar field theory on \(\mathbb {T}_\theta ^d\) with a quartic self-interaction after the introduction of a new type of nonlocal (but trace-like) interactions suggested by the previous heat kernel calculations [22]. At one loop our analysis is complete. We also argued that in two dimensions no problems appear at higher orders as well. In four dimensions, we checked the renormalization of self-energy at two-loop order relying on our understanding of the behavior of double sums, which we were able to reconfirm by rigorous methods for all diagrams but one. Our findings strongly suggest that the \(\varphi ^4\) theory on \(\mathbb {T}^2_\theta \) and \(\mathbb {T}^4_\theta \) is renormalizable. The renormalization always strongly depends on the Diophantine character of the noncommutativity matrix \(\theta \). We cannot exclude completely that some more elaborate multiple-trace counterterms will be needed, though their algebraic nature is less clear than that of the ones listed (11). To check this, one has to calculate the two-loop four-point functions.

On the technical side, it is important to develop the theory of regularized multiple sums with twisting factors. To the best of our knowledge, such sums have not been considered in the mathematics literature so far (see, e.g. [31]).

- 1.
Since the plane waves \(U_p\) do not commute even classically, see (2), they probably do not form a good basis. Therefore, the correlation functions of plane waves may be of little physical relevance by themselves. The problem is then to find a physically motivated basis of states that will ensure a kind of “smooth” dependence of the correlation functions on \(\theta \).

- 2.
One can try to achieve a meaningful answer by smearing the correlation functions over a small vicinity of a given \(\theta \). The key issue is to find an appropriate measure.

- 3.
Finally, perhaps one can extend the model to fix \(\theta \) sharply to certain value, e.g. by some topological considerations.

## Footnotes

- 1.
For bosonic theories, these are the operators

*L*appearing in the second variation of classical action, \(S_2=\int (\delta \varphi ) L(\varphi ) (\delta \varphi )\), with \(\delta \varphi \) being a fluctuation and \(\varphi \)—a background field. - 2.
We use \(\delta _p\) to denote 1 if \(p=0\), and 0 otherwise.

- 3.
- 4.
More precisely, “quad. pol.” should have the form \(a+bp^2\), i.e. the coefficient in front of \(p_\mu p_\nu \) has to be proportional to the unit matrix.

- 5.
In other words, we assume that the NC torus has two sets of noncommuting coordinates with the same noncommutativity parameter.

## Notes

### Acknowledgments

DVV was supported in part by FAPESP, Project 2012/00333-7, CNPq, Project 306208/2013-0 and by the Tomsk State University Competitiveness Improvement Program. DD acknowledges support from CONICET and UNLP (Proj. 11/X615), Argentina. PP acknowledges support from CONICET (PIP 1787/681), ANPCyT (PICT-2011-0605) and UNLP (Proj. 11/X615), Argentina.

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