PHYSICAL REVIEW D 68, 025016 共2003兲

Effective field theories on noncommutative space-time Xavier Calmet* California Institute of Technology, Pasadena, California 91125, USA

Michael Wohlgenannt† Ludwig-Maximilians-Universita¨t, Theresienstrasse 37, D-80333 Munich, Germany 共Received 9 May 2003; published 21 July 2003兲 We consider Yang-Mills theories formulated on a noncommutative space-time described by a space-time dependent antisymmetric field ␪ ␮ ␯ (x). Using Seiberg-Witten map techniques, we derive the leading order operators for the effective field theories that take into account the effects of such a background field. These effective theories are valid for a weakly noncommutative space-time. It is remarkable to note that already simple models for ␪ ␮ ␯ (x) can help to loosen the bounds on space-time noncommutativity coming from low energy physics. Noncommutative geometry formulated in our framework is a potential candidate for new physics beyond the standard model. DOI: 10.1103/PhysRevD.68.025016

PACS number共s兲: 11.10.Nx, 11.15.⫺q, 12.60.⫺i

I. INTRODUCTION

In recent years, considerable progress towards a consistent formulation of field theories on noncommutative spacetime has been made. The idea that space-time coordinates might not commute at very short distances is nevertheless not new and can be traced back to Heisenberg 关1兴, Pauli 关2兴, and Snyder 关3兴. A nice historical introduction to noncommutative coordinates is given in 关4兴. At that time the main motivation was the hope that the introduction of a new fundamental length scale could help to get rid of the divergencies in quantum field theory. A more modern motivation to study a space-time that satisfies the noncommutative relation ˆ␮

ˆ␯

ˆ␮ˆ␯

ˆ␯ˆ␮

关 x ,x 兴 ⬅x x ⫺x x ⫽i ␪

␮␯

,

␮␯

␪ 苸C

共2兲

which is the analogue to the famous Heisenberg uncertainty relations for momentum and space coordinates. Note that ␪ ␮ ␯ is a dimensional full quantity, dim( ␪ ␮ ␯ )⫽mass⫺2 . If this mass scale is large enough, ␪ ␮ ␯ can be used as an expansion parameter like ប in quantum mechanics. We adopt the usual convention: a variable or function with a hat is a noncommutative one. It should be noted that relations of the type 共1兲 also appear quite naturally in string theory models 关5兴 or in models for quantum gravity 关6兴. It should also be clear that the canonical case 共1兲 is not the most generic case and that other structures can be considered, see, e.g., 关7兴 for a review. In order to consider field theories on a noncommutative space-time, we need to define the concept of noncommutative functions and fields. Noncommutative functions and fields are defined as elements of the noncommutative algebra

共 f 쐓g 兲共 x 兲 ⫽exp

冊

冏

i ␮␯ ⳵ ⳵ ␪ f 共 x 兲g共 y 兲 2 ⳵x␮ ⳵y␯

y→x

共4兲

Intuitively, the star product can be seen as an expansion of the product in terms of the noncommutative parameter ␪ . The star product has the following property:

冕

d 4 x 共 f 쐓g 兲共 x 兲 ⫽ ⫽

冕 冕

d 4 x 共 g쐓 f 兲共 x 兲 d 4x f 共 x 兲g共 x 兲,

共5兲

as can be proven using partial integrations. This property is usually called the trace property. Here f (x) and g(x) are ordinary functions on R4 . Two different approaches to noncommutative field theories can be found in the literature. The first one is a nonperturbative approach 共see, e.g., 关9兴 for a review兲, fields are considered to be Lie algebra valued, and it turns out that only U(N) structure groups are conceivable because the commutator 1 ˆ 쐓ˆ a b ˆ 쐓⌳ ˆ 关⌳ , ⬘ 兴 ⫽ 兵 ⌳a 共 x 兲 , ⌳⬘ b 共 x 兲 其 关 T ,T 兴 2 1 ˆ ˆ ⬘ 共 x 兲兴 兵 T a ,T b 其 ⫹ 关⌳ 共 x 兲쐓⌳ 2 a , b

Email address: [email protected]

0556-2821/2003/68共2兲/025016共11兲/$20.00

冉

i ⫽ f •g⫹ ␪ ␮ ␯ ⳵ ␮ g• ⳵ ␯ f ⫹O 共 ␪ 2 兲 . 2

*Email address: [email protected] †

共3兲

ˆ is the algebra where R are the relations defined in Eq. 共1兲. A of formal power series in the coordinates subject to the relations 共1兲. We also need to introduce the concept of a star product. The Moyal-Weyl star product 쐓 关8兴 of two functions f (x) and g(x) with f (x),g(x)苸R4 , is defined by a formal power series expansion:

共1兲

is that it implies an uncertainty relation for space-time coordinates, 1 ⌬x ␮ ⌬x ␯ ⭓ 兩 ␪ ␮ ␯ 兩 , 2

C xˆ i •••xˆ n 典典 ˆ ⫽ 具具 A , R

68 025016-1

共6兲

©2003 The American Physical Society

PHYSICAL REVIEW D 68, 025016 共2003兲

X. CALMET AND M. WOHLGENANNT

of two Lie algebra valued noncommutative gauge parameters ˆ ⬘ ⫽⌳ ⬘ (x)T a only closes in the Lie algeˆ ⫽⌳ (x)T a and ⌳ ⌳ a

a

bra if the gauge group under consideration is U(N) and if the gauge transformations are in the fundamental representation of this group. But, this approach cannot be used to describe particle physics since we know that SU(N) groups are required to describe the weak and strong interactions. Or at least there is no obvious way known to date to derive the standard model as a low energy effective action coming from a U(N) group. Furthermore it turns out that even in the U共1兲 case, charges are quantized 关10,11兴 and it is thus impossible to describe quarks. The other approach has been developed by Wess and his collaborators 关12–15兴 共see also 关16,17兴兲. The goal of this approach is to consider field theories on noncommutative spaces as effective theories. The main difference from the more conventional approach is to consider fields and gauge transformations which are not Lie algebra valued but which are in the enveloping algebra, ˆ ⫽⌳ 0 共 x 兲 T a ⫹⌳ 1 共 x 兲 :T a T b :⫹⌳ 2 共 x 兲 :T a T b T c :⫹•••, ⌳ a ab abc 共7兲 where : : denotes some appropriate ordering of the Lie algebra generators. One can choose, for example, a symmetrically ordered basis of the enveloping algebra; one then has :T a ªT a and :T a T b ⫽ 21 兵 T a ,T b 其 and so on. The mapping between the noncommutative field theory and the effective field theory on a usual commutative space-time is derived by requiring that the theory be invariant under both noncommutative gauge transformations and under the usual 共classical兲 commutative gauge transformations. These requirements lead to differential equations whose solutions correspond to the Seiberg-Witten map 关18兴 that appeared originally in the context of string theory. It should be noted that the expansion which is performed in that approach is in a sense trivial since it corresponds to a variable change. But, it is well suited for a phenomenological approach since it generates in a constructive way the leading order operators that describe the noncommutative nature of space-time. It also makes clear that, contrary to what one might expect 关19,20兴, the coupling constants are not deformed, but the currents themselves are deformed. We want to emphasize that the two approaches are fundamentally different and lead to fundamentally different physical predictions. In the approach where the fields are taken to be Lie algebra valued, the Feynman rule for the photonelectron-positron interaction is given by ig ␥ ␮ exp共 ip 1 ␣ ␪ ␣␤ p 2 ␤ 兲 ,

共8兲

where p 1 ␮ is the four-momentum of the incoming fermion and p 2 ␯ is the four-momentum of the outgoing fermion. One could hope to recover the Feynman rule obtained in the case where the fields are taken to be in the enveloping algebra, i i ␮␯ ␪ 关 p ␯ 共 k” ⫺m 兲 ⫺k ␯ 共 p” ⫺m 兲兴 ⫺ k ␣ ␪ ␣␤ p ␤ ␥ ␮ , 2 2

共9兲

if an expansion of Eq. 共8兲 in ␪ is performed. However, this is not the case, because some new terms appear in the approach

proposed in 关12–17兴 due to the expansion of the fields in the noncommutative parameter via the Seiberg-Witten map. It is thus clear that the observables calculated with these Feynman rules would be different from those obtained in 关21兴. Note that the two different approaches nevertheless yield the same observables if the diagrams involved only have onshell particles. Unfortunately it turns out that both approaches lead at the one loop level to operators that violate Lorentz invariance. Although it is not clear how to renormalize these models, these bounds might be the sign that noncommutative field theories are in conflict with experiments. If these calculations are taken seriously, one finds the bound ⌳ 2 ␪ ⬍10⫺29 关22兴 共see also 关23兴兲, where ⌳ is the Pauli-Villars cutoff and ␪ is the typical inverse squared scale for the matrix elements of the matrix ␪ ␮ ␯ . In view of this potentially serious problem, it is desirable to formulate noncommutative theories that can avoid the bounds coming from low energy physics. It should nevertheless be noted that the operators discussed in 关22兴, of the type m ␺ ¯␺ ␴ ␮ ␯ ␺␪ ␮ ␯ , are not generated by the theories developed in 关12–17兴 at tree level. On the other hand, the operators generated by the Seiberg-Witten expansion are compatible with the classical gauge invariance and with the noncommutative gauge invariance. It remains to be proven that the operators discussed in 关22兴 are compatible with the noncommutative gauge invariance. If this is not the case, as long as there are no anomalies in the theory, these operators cannot be physical and must be renormalized. It has been shown that in the approach proposed in 关12–17兴, anomalies might be under control 关24兴. There are, nevertheless, bounds ” ␺ which definiin the literature on the operators ␪ ␮ ␯ ¯␺ F ␮ ␯ D tively appear at tree level. One finds the constraint ⌳ NC ⬎10 TeV for the scale where noncommutative physics become relevant 关25兴. This constraint comes again from experiments which are searching for Lorentz violating effects. It is interesting to note that Snyder’s main point in his seminal paper 关3兴 was that noncommuting coordinates can be compatible with Lorentz invariance. But, despite some interesting proposals 关26 –28兴, it is still not clear how to construct a Lorentz invariant gauge theory on a noncommutative space-time. It is not a surprise that theories formulated on a constant background field that select special directions in space-time are severely constrained by experiments since those are basically either type theories. We will formulate an effective field theory for a field theory on a noncommutative space-time which is parametrized by an arbitrary space-time dependent ␪ (x) parameter. But, we will restrict ourselves to the leading order in the expansion in ␪ (x). In this case it is rather simple to use the results obtain in 关12–17兴 to generate the leading order operators. We want to emphasize that it is not obvious how to generalize our results to produce the operators appearing at higher order in the expansion in ␪ . One has to define a new star product which resembles that obtained by Kontsevich in the case of a general Poisson structure on Rn 关29–31兴. We will then study different models for ␪ (x), which allow us to relax the bounds coming from low energy physics experi-

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EFFECTIVE FIELD THEORIES ON NONCOMMUTATIVE . . .

ments. The aim of this work is not to give a mathematically rigorous treatment of the problem. We will only derive the first order operators that take into account the effects of a space-time which is modified by a space-time dependent ␪ (x) parameter. II. A SPACE-TIME DEPENDENT ␪

The aim of this section is to derive an effective Lagrangian for a noncommutative field theory defined on a spacetime satisfying the following noncommutative relation: 关 xˆ ␮ ,xˆ ␯ 兴 ⬅i ␪ˆ ␮ ␯ 共 xˆ 兲 ,

W 共 f 쐓 x g 兲 ⬅W 共 f 兲 •W 共 g 兲 ⫽ ˆf •gˆ .

共10兲

where ␪ˆ (xˆ ) is a space-time dependent bivector field which depends on the noncommutative coordinates. We first need to define the star product 쐓 x . It should be noted that the 쐓 x product is different from the canonical Weyl-Moyal product because ␪ˆ (xˆ ) is coordinate-dependent. ˆ defined as Let us consider the noncommutative algebra A ˆ⫽ A

where Rx are the relations 共10兲, and the usual commutative algebra A⫽C具具 x 1 , . . . ,x 4 典典 . We assume that ␪ˆ ␮ ␯ (xˆ ) is such ˆ possesses the Poincare´-Birkhoff-Witt that the algebra A ˆ be an isomorphism of vector spaces property. Let W:A→A ˆ . The Poincare´-Birkhoffdefined by the choice of a basis in A Witt property insures that the isomorphism maps the algebra of noncommutative functions on the entire algebra of commutative functions. The 쐓 x product extends this map to an algebra isomorphism. The 쐓 x product is defined by

C具具 xˆ 1 , . . . ,xˆ 4 典典 , Rx

共11兲

共12兲

ˆ and exWe first choose a symmetrically ordered basis in A press functions of commutative variables as power series in the coordinates x ␮ , f 共 x 兲⫽

兺i ␣ i •••i 共 x 1 兲 i ••• 共 x 4 兲 i . 1

1

4

4

共13兲

By definition, the isomorphism W identifies commutative monomials with symmetrically ordered polynomials in noncommutative coordinates,

共14兲

ˆ, W:A→A x ␮ 哫xˆ ␮ , x ␮ x ␯ 哫:xˆ ␮ xˆ ␯ :⬅

xˆ ␮ xˆ ␯ ⫹xˆ ␯ xˆ ␮ , 2!

x ␮ x ␯ x ␴ 哫:xˆ ␮ xˆ ␯ xˆ ␴ :⬅

xˆ ␮ xˆ ␯ xˆ ␴ ⫹xˆ ␯ xˆ ␮ xˆ ␴ ⫹xˆ ␮ xˆ ␴ xˆ ␯ ⫹xˆ ␯ xˆ ␴ xˆ ␮ ⫹xˆ ␴ xˆ ␮ xˆ ␯ ⫹xˆ ␴ xˆ ␯ xˆ ␮ 3!

⯗ .

␪ ␳␴ ⳵ ␴ ␪ ␮ ␯ ⫹ ␪ ␮ ␴ ⳵ ␴ ␪ ␯␳ ⫹ ␪ ␯␴ ⳵ ␴ ␪ ␳ ␮ ⫽0.

A function f is thus mapped to ˆf 共 xˆ 兲 ⫽W 关 f 共 x 兲兴 ⫽

兺i ␣ i •••i : 共 xˆ 1 兲 i ••• 共 xˆ 4 兲 i :, 1

1

4

4

共15兲

where the coefficients ␣ I have been defined in Eq. 共13兲. Using the isomorphism W, we can also map ␪ˆ ␮ ␯ (xˆ ), which appears in Eq. 共10兲, to commutative functions ␪ ␮ ␯ (x). We have

␪ˆ 共 xˆ 兲 ⫽

兺k

␤ k 1 •••k 4 : 共 xˆ 1 兲 k 1 ••• 共 xˆ 4 兲 k 4 :

The quantization of a general Poisson structure ␣ has been solved by Kontsevich 关29兴. Kontsevich has shown that it is necessary for ␪ (x) to fulfill the Jacobi identity in order to have an associative star product. To first order, the 쐓 K product is given by the Poisson structure itself. The Kontsevich 쐓 K product is given by the formula i f 쐓 K g⫽ f •g⫹ ␣ i j ⳵ i f • ⳵ j g⫹O 共 ␣ 2 兲 . 2

共16兲

and therefore

␪ 共 x 兲 ⫽W ⫺1 关 ␪ˆ 共 xˆ 兲兴 ⫽ 兺 ␤ k 1 •••k 4 共 x 1 兲 k 1 ••• 共 x 4 兲 k 4 . 共17兲 k

We want to assume that ␪ (x) defines a Poisson structure, i.e., satisfies the Jacobi identity

共18兲

共19兲

A more detailed description can be found in 关29兴 and explicit calculations of higher orders of the 쐓 K product can be found in 关32,33兴. Up to first order, the Kontsevich 쐓 product can be motivated by the Weyl-Moyal product, which is of the same form 共see the Appendix兲. The difference arises in higher order terms where the x dependence of ␪ is crucial. Derivatives will not only act on the functions f and g but also on ␪ (x).

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We are interested in the 쐓 x product to first order and in a ˆ 共14兲. As in Eq. 共19兲, the symmetrically ordered basis of A first order 쐓 x product is determined by ␪ ␮ ␯ (x), which corresponds to a symmetrically ordered basis, cf. Eq. 共12兲, i f 쐓 x g 共 x 兲 ⫽ f •g 共 x 兲 ⫹ ␪ ␮ ␯ 共 x 兲 ⳵ ␮ f • ⳵ ␯ g⫹O 共 ␪ 2 兲 . 2

共20兲

The ordinary integral equipped with this new star product does not satisfy the trace property, since this identity is derived using partial integration, unless ⳵ ␮ ␪ ␮ ␯ ⫽0. We need to introduce a weight function w(x) to make sure that the trace operator defined as Tr ˆf ⫽

冕

d 4 xw 共 x 兲 ˆf 共 x 兲

The field strength is constructed using the gauge potential F ␮ ␯ (x)⫽ ⳵ ␮ A ␯ ⫺ ⳵ ␯ A ␮ ⫹g 关 A ␮ ,A ␯ 兴 and the covariant derivative is given by D ␮ ⫽ ⳵ ␮ ⫺igA ␮ . These are the well-known results already obtained by Yang-Mills a long time ago 关34兴. This classical gauge invariance is imposed on the effective theory, which we will derive. B. Noncommutative gauge transformations

This effective theory should also be invariant under noncommutative transformations defined by ˆ ⫽i⌳ ˆ 共 x 兲쐓 ⌿ ˆ ␦ˆ ⌳ˆ ⌿ x 共 x 兲.

共22兲

Functions carrying a hat have to be expanded via a SeibergWitten map. We now consider the commutator of two nonˆ (x) and ⌺ ˆ (x), commutative gauge transformations ⌳

Tr ˆf gˆ ⫽Tr gˆ ˆf .

ˆ 共 x 兲 쐓 ⌺ˆ 共 x 兲 ⫺⌺ˆ 共 x 兲 쐓 ⌳ ˆ ˆ ⫽关⌳ x x 共 x 兲兴 쐓 x ⌿ 共 x 兲 ˆ 共 x 兲쐓x ⌺ ˆ 共 x 兲兴 쐓 ⌿ ˆ ⫽关⌳ x 共 x 兲. ,

d 4 xw 共 x 兲关 f 共 x 兲 쐓 x g 共 x 兲兴 ⫽ ⫽

冕 冕

d 4 xw 共 x 兲关 g 共 x 兲 쐓 x f 共 x 兲兴 d 4 xw 共 x 兲 f 共 x 兲 g 共 x 兲 .

共23兲

This relation implies ⫺w 共 x 兲 ⳵ i ␪ i j 共 x 兲 ⫽ ⳵ i w 共 x 兲 ␪ i j 共 x 兲 ,

共24兲

which is a partial differential equation for w(x) that can be solved once ␪ i j (x) has been specified. Furthermore, we assume that it is positive and falls to zero quickly enough when ␪ ␮ ␯ (x) is large, so that all integrals are well defined. In the sequel we shall derive the consistency condition for a field theory on a space-time with the structure 共10兲. We shall follow the construction proposed in 关12–14兴 step by step. A. Classical gauge transformations

We consider Yang-Mills gauge theories with the Lie algec a bra 关 T a ,T b 兴 ⫽i f ab c T , where the T are the generators of the gauge group. A field transforms as

␦ ␣ ␺ ⫽i ␣ 共 x 兲 ␺ 共 x 兲 with ␣ 共 x 兲 ⫽ ␣ a 共 x 兲 T a ,

The Lie algebra valued gauge potential transforms as

共26兲

共29兲

In order to fulfill the relation 共29兲, the gauge transformations and thus the fields cannot be Lie algebra valued but must be enveloping algebra valued 关see Eq. 共7兲兴. This is the main achievement of Wess’ approach 关13兴. This is also what allows us to solve the charge quantization problem 关15兴. Since we restrict ourselves to the leading order expansion in ␪ (x), we can restrict ourselves to gauge transformations ˆ ⌳ 关 A 兴 whose x dependence is only coming from the ␣ (x)

␮

gauge potential A ␮ and from the x dependence of the classical gauge transformation ␣ (x), ˆ 共 x 兲. ␦ˆ ⌳ˆ ␺ˆ ⫽i⌳ˆ 关 A ␮ 兴 쐓 x ⌿

共30兲

Subtleties might appear at higher orders in ␪ (x). We assume that ␪ (x) is invariant under a gauge transformation. The operator xˆ is invariant under a gauge transformation. One can as usual introduce covariant coordinates Xˆ ␮ ⫽xˆ ␮ ⫹Aˆ ␮ . The noncommutative field strength can be defined as Fˆ ␮ ␯ ⫽ 关 Xˆ ␮ ,Xˆ ␯ 兴 ⫺ ␪ˆ ␮ ␯ (Xˆ ). These results are very similar to those obtained for the Poisson structure in 关31兴.

共25兲

under a classical gauge transformation. We can consider the commutator of two successive gauge transformations, 共 ␦ ␣ ␦ ␤ ⫺ ␦ ␤ ␦ ␣ 兲 ␺ 共 x 兲 ⫽i ␣ a 共 x 兲 ␤ b 共 x 兲 f ab c ␺共 x 兲.

共28兲

ˆ 共x兲 共 ␦ˆ ⌳ˆ ␦ˆ ⌺ˆ ⫺ ␦ˆ ⌺ˆ ␦ˆ ⌳ˆ 兲 ⌿

We shall not try to construct the function w(x), but assume that it exists and has the following property:

冕

共27兲

共21兲

has the following properties: Tr ˆf ˆf † ⭓0,

␦ ␣ A ␮ 共 x 兲 ⫽ ⳵ ␮ ␣ 共 x 兲 ⫹i 关 ␣ 共 x 兲 ,A ␮ 兴 .

C. Consistency condition and Seiberg-Witten map

As done in 关12–14兴, we impose that our fields transform under the classical gauge transformations according to Eq. 共25兲 and under noncommutative gauge transformation according to Eq. 共28兲. We require that the noncommutative, ˆ and ⌺ˆ fulfill enveloping algebra valued gauge parameters ⌳ the following relation:

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쐓x ˆ ˆ 共 x 兲 ⫽ 共 i ␦ˆ ˆ ⌺ˆ 关 A 兴 ⫺i ␦ˆ ˆ ⌳ ˆ ˆ ˆ 共 ␦ˆ ⌳ˆ ␦ˆ ⌺ˆ ⫺ ␦ˆ ⌺ˆ ␦ˆ ⌳ˆ 兲 쐓 x ⌿ ⌳ ␮ ⌺ 关 A ␮ 兴 ⫹†⌳ 关 A ␮ 兴 , ⌺关 A ␮ 兴 ‡ 兲 쐓 x ⌿ 共 x 兲

ˆ ˆ ⌳⫻⌺ d关A 兴쐓 ⌿ ⬅⌼ ␮ x 共 x 兲,

which defines the noncommutative gauge transformation parameters ⌳ and ⌺. The Seiberg-Witten maps 关18兴 have the remarkable property that ordinary gauge transformations ␦A ␮ ⫽ ⳵ ␮ ⌳ ⫹i 关 ⌳,A ␮ 兴 and ␦⌿⫽i⌳•⌿ induce noncommutative gauge ˆ with gauge parameter ⌳ ˆ as transformations of the fields Aˆ ,⌿ given above,

␦Aˆ ␮ ⫽ ␦ˆ Aˆ ␮ ,

ˆ ⫽ ␦ˆ ⌿ ˆ. ␦⌿

共32兲

␦␭ ␺ 共 x 兲 ⫽i␭ 共 x 兲 ␺ 共 x 兲

共33兲

ˆ ⫽ ␴ 共 x 兲 T a ⫹⌺ 1 :T a T b :⫹O 共 ␪ 2 兲 , ⌺ a ab

␦␭ ␺ 1 关 A ␮ 兴 ⫽i␭ ␺ 1 关 A ␮ 兴 ⫹i⌳ ␭1 ␺ 1 关 A ␮ 兴 1 ⫺ ␪ ␮␯共 x 兲 ⳵ ␮␭ ⳵ ␯␺ 2

1 1 ␺ 1 关 A ␮ 兴 ⫽⫺ ␪ ␮ ␯ 共 x 兲 A ␮ ⳵ ␯ ␺ ⫹i ␪ ␮ ␯ 共 x 兲 A ␮ A ␯ ␺ . 共40兲 2 4 This solution is identical to the one in the case of constant ␪ . The following relation is also useful to build actions: ¯␺ 1 关 A ␮ 兴 ⫽ 共 ␺ 1 关 A ␮ 兴 兲 † ␥ 0

with the understanding that ␭, ␴ , and ␷ are independent of ␪ (x), and ⌳ 1 , ⌺ 1 , and ⌼ 1 are proportional to ␪ (x). Again we restrict ourselves to the leading order terms in ␪ (x). One finds 共34兲

in the zeroth order in ␪ (x) and i ␦␭ ⌺ 1 ⫺i ␦␴ ⌳ 1 ⫹i ␪ ␮ ␯ 共 x 兲 兵 ⳵ ␮ ␭, ⳵ ␯ ␴ 其 ⫹ 关 ␭,⌺ 1 兴 ⫺ 关 ␴ ,⌳ 1 兴 ⬅⌼ 1

1 1 ⫽⫺ ␪ ␮ ␯ 共 x 兲 ⳵ ␯ ¯␺ A ␮ ⫹i ␪ ␮ ␯ 共 x 兲 ¯␺ A ␮ A ␯ . 共41兲 2 4 We shall now consider the gauge potential. It turns out that things are much more complicated in that case than they are when ␪ is constant. We need to introduce the concept of covariant coordinates, as has been done in 关12兴. The noncommutative coordinates xˆ i are invariant under a gauge transformation,

␦ˆ xˆ i ⫽0.

共35兲

共36兲

␦ˆ 共 xˆ i ⌿ˆ 兲 ⫽ixˆ i ⌳ˆ 共 xˆ 兲 ⌿ˆ ⫽i⌳ˆ 共 xˆ 兲 xˆ i ⌿ˆ .

共43兲

To solve this problem, one introduces covariant coordinates Xˆ i 关12兴 such that

1 ⌺ 1 ⫽ ␪ ␮ ␯ 共 x 兲 兵 ⳵ ␮ ␴ ,A ␯ 其 , 4

ˆ ␦ˆ 共 Xˆ i ⌿ˆ 兲 ⫽i⌳ˆ 共 xˆ 兲 Xˆ i ⌿

1 ⌼ 1 ⫽ ␪ ␮ ␯ 共 x 兲 兵 ⳵ ␮ 共 ⫺i 关 ␭, ␴ 兴 兲 ,A ␯ 其 4 solve Eq. 共35兲. This is the usual Seiberg-Witten map in the leading order in ␪ (x). ˆ are also elements of the enveloping The matter fields ⌿ Lie algebra ˆ 关 A 兴 ⫽ ␺ ⫹ ␺ 1 关 A 兴 ⫹O 共 ␪ 2 兲 , ⌿ ␮ ␮

共42兲

ˆ is in general not covariant under a This implies that xˆ i ⌿ gauge transformation,

in the leading order. The Ansa¨tze 1 ⌳ 1 ⫽ ␪ ␮ ␯ 共 x 兲 兵 ⳵ ␮ ␭,A ␯ 其 , 4

共39兲

in the leading order in ␪ (x). The solution is

ˆ ⌳⫻⌺ d ⫽ ␷ a T a ⫹⌼ 1 :T a T b :⫹O 共 ␪ 2 兲 ⌼ ab

关 ␭, ␴ 兴 ⫽i ␷

共38兲

in the zeroth order in ␪ (x), and

ˆ, ⌺ ˆ , and ⌼ ˆ ⌳⫻⌺ d are elements of The gauge parameters ⌳ the enveloping Lie algebra, ˆ ⫽␭ 共 x 兲 T a ⫹⌳ 1 :T a T b :⫹O 共 ␪ 2 兲 , ⌳ a ab

共31兲

共37兲

共44兲

ˆ (xˆ ),Xˆ i 兴 . The Ansatz Xˆ i ⫽xˆ i ⫹Bˆ i (xˆ ) solves the with ␦ˆ Xˆ i ⫽i 关 ⌳ problem if Bˆ i (xˆ ) transforms as

␦ˆ Bˆ i 共 xˆ 兲 ⫽i 关 ⌳ˆ 共 xˆ 兲 ,Bˆ i 共 xˆ 兲兴 ⫺i 关 xˆ i ,⌳ˆ 共 xˆ 兲兴

共45兲

under a gauge transformation. In our case Bˆ i (xˆ ) is not the gauge potential. We need to recall two relations,

where ␺ is independent of ␪ (x) and ␺ 1 is proportional to ␪ (x). Equation 共30兲 becomes 关12–14兴 025016-5

쐓 关 ˆf , x gˆ 兴 ⫽i ␪ i j 共 x 兲 ⳵ i f ⳵ j g⫹O 共 ␪ 3 兲 ,

共46兲

PHYSICAL REVIEW D 68, 025016 共2003兲

X. CALMET AND M. WOHLGENANNT

ˆ 쐓 x ␪ˆ ⫺1 (Xˆ ) 兴 , i.e., ␪ˆ (Xˆ ) is a covariant function of Xˆ . The ⫽i 关 ⌳ ␮i , object under consideration transforms according to

쐓

2 ˆ 兴 ⫽i ␪ i j 共 x 兲 ⳵ ⌳ ˆ 关 x i ,x⌳ j ⫹O 共 ␪ 兲 .

Equation 共45兲 then becomes 쐓 ␦ˆ Bˆ i 共 x 兲 ⫽ ␪ i j 共 x 兲 ⳵ j ⌳ˆ 共 x 兲 ⫹i 关 ⌳ˆ 共 x 兲 x Bˆ i 共 x 兲兴 . ,

共47兲

Following 关12兴, we expand Bˆ i as follows: Bˆ ⫽ ␪ 共 x 兲 B j ⫹B ⫹O 共 ␪ 兲 . i

ij

1i

3

ˆ 兲… ␦ˆ „⫺i ␪ˆ ␮⫺1i 共 Xˆ 兲 쐓 x 共 Xˆ i 쐓 x ⌿

共48兲

We obtain the following consistency relation for Bˆ i : 1 ␦ ␭ B 1i ⫽ ␪ i j 共 x 兲 ⳵ j ⌳ 1 ⫺ ␪ kl 共 x 兲 兵 ⳵ k ␭ ⳵ l 关 ␪ i j 共 x 兲 B j 兴 2 ⫺ ⳵ k 关 ␪ i j 共 x 兲 B j 兴 ⳵ l ␭ 其 ⫹i 关 ␭,B 1i 兴 ⫹i 关 ⌳ 1 , ␪ i j 共 x 兲 B j 兴 . 共49兲

ˆ ˆ 쐓 ␪ˆ ⫺1 共 Xˆ 兲 쐓 Xˆ i 쐓 ⌿ ⫽⫺i⌳ x ␮i x x . ˆ ␮, We can thus define a covariant derivative D ˆ ⫽⫺i ␪ˆ ⫺1 共 Xˆ 兲 쐓 Xˆ i 쐓 ⌿ ˆ ˆ ␮쐓 x⌿ D x x , ␮i

1 B ⫽⫺ ␪ kl 共 x 兲 兵 B k , ⳵ l 关 ␪ i j 共 x 兲 B j 兴 ⫹ ␪ i j 共 x 兲 F Bl j 其 , 4

共52兲

which transforms covariantly. There is one new subtlety appearing in our case. Note that ⫺1 ˆ ␪ ␮ i (X ) depends on the covariant coordinate Xˆ ␮ . We need to expand ␪ ␮⫺1i (Xˆ ) in ␪ . This is done again via a Seiberg-Witten map. The transformation property of ␪ˆ ␮⫺1␯ implies 쐓

␦␪ˆ ␮⫺1␯ 共 Xˆ 兲 ⫽i 关 ⌳ˆ , x ␪ˆ ␮⫺1␯ 共 Xˆ 兲兴

These equations are fulfilled by the Ansa¨tze 1i

共51兲

⫽⫺ ␪ kl 共 x 兲 ⳵ k ␣ ⳵ l 关 ␪ ␮0 ␯ 共 x 兲兴 ⫺1

共50兲

⫹i†␭, 关 ␪ ␮1 ␯ 共 xˆ 兲兴 ⫺1 ‡⫹•••,

共53兲

where we have used the expansion ␪ˆ ⫺1 (Xˆ )⫽ 关 ␪ 0 (xˆ ) 兴 ⫺1 ⫹ 关 ␪ 1 (xˆ ) 兴 ⫺1 ⫹O( ␪ 2 ) for ␪ˆ ⫺1 (Xˆ ). One finds

1 ⌳ 1 ⫽ ␪ lm 共 x 兲 兵 ⳵ l ␭,B m 其 4 where F Bi j ⫽ ⳵ i B j ⫺ ⳵ j B i ⫺i 关 B i ,B j 兴 . The Jacobi identity 共18兲 is required to show that these Ansa¨tze work. The problem is to find the relation to the Yang-Mills gauge potential A ␮ . If ␪ is constant, the relation is trivial: Bˆ i ⫽ ␪ i ␮ Aˆ ␮ . Our goal is to find a relation between Aˆ ␮ , deˆ ␮ ⫽ ⳵ ␮ ⫺iAˆ ␮ , and Bˆ i such that the covariant defined as D ˆ ␮ transforms covariantly under a gauge transforrivative D mation. ˆ again. It transforms Let us consider the product Xˆ i 쐓 x ⌿ covariantly according to Eq. 共44兲. Let us now consider ˆ ), with ␦␪ˆ ␮⫺1i (Xˆ ) the object ⫺i ␦ˆ ␮⫺1i (Xˆ )쐓 x (Xˆ i 쐓 x ⌿

␦共 ␪ ␮0 ␯ 兲 ⫺1 ⫽0,

共54兲

␦共 ␪ ␮1 ␯ 兲 ⫺1 ⫽⫺ ␪ kl ⳵ k ␭ ⳵ l 关 ␪ ␮0 ␯ 共 xˆ 兲兴 ⫺1 ⫹i†␭, 关 ␪ ␮1 ␯ 共 xˆ 兲兴 ⫺1 ‡. This system is solved by 共 ␪ ␮0 ␯ 兲 ⫺1 ⫽ ␪ ␮⫺1␯ 共 x 兲 ,

共55兲

共 ␪ ␮1 ␯ 兲 ⫺1 ⫽ ␪ i j 共 x 兲 A j ⳵ i ␪ ␮⫺1␯ 共 x 兲 .

Note that this expansion coincides with a Taylor expansion for ( ␪ˆ ␮⫺1␯ )(Xˆ ). The Yang-Mills gauge potential is then given by

ˆ ⫽ ␦ˆ ⫺1 共 Xˆ 兲 쐓 Bˆ i 共 x 兲 쐓 ⌿ ˆ Aˆ ␮ 共 x 兲 쐓 x ⌿ x x ␮i 1 ⫺1 ˆ i ˆ ⫹i 1 ␪ ␣␤ 共 x 兲 ⳵ ␪ ⫺1 共 x 兲 ⳵ 关 Bˆ i 共 x 兲 쐓 ⌿ ˆ ˆ. B 共 x 兲⌿ ⫽ ␪ ␮⫺1i 共 x 兲 Bˆ i 共 x 兲 쐓 x ⌿ ␣ ␮i ␤ x 兴 ⫹共 ␪ ␮i 兲 2

共56兲

One finds ˆ ⫽B 쐓 ⌿ ˆ A ␮쐓 x⌿ ␮ x

共57兲

1 ␣␤ 1 ⫺1 ␯ ⫺1 i ␣ ˆ ⫽ ␪ ⫺1 共 x 兲 B 1i 쐓 ⌿ ˆ ˆ ˆ ␪ 共 x 兲A ␣쐓 x⌿ A ␮1 쐓 x ⌿ x ⫹i ␪ 共 x 兲 ⳵ ␣ ␪ ␮ ␯ 共 x 兲 ⳵ ␤ 关 B 共 x 兲 ⌿ 兴 ⫹ 关 ␪ ␮ i 共 x 兲兴 ␮i 2

共58兲

1 ˆ ⫺ 1 ␪ kl 共 x 兲 兵 A , ⳵ A ⫹F 其 ⌿ ˆ ⫽⫺ ␪ ␮⫺1i 共 x 兲 ␪ kl 共 x 兲 ⳵ l ␪ i j 共 x 兲 兵 A k ,A j 其 ⌿ k l ␮ l␮ 4 4 1 ˆ 兴 ⫹ ␪ kl 共 x 兲 A ⳵ ␪ ⫺1 共 x 兲 ␪ ␯ ␣ 共 x 兲 A ⌿ ˆ ⫹i ␪ ␣␤ 共 x 兲 ⳵ ␣ ␪ ␮⫺1␯ 共 x 兲 ⳵ ␤ 关 ␪ ␯␳ 共 x 兲 A ␳ ⌿ l k ␮␯ ␣ . 2 025016-6

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EFFECTIVE FIELD THEORIES ON NONCOMMUTATIVE . . .

The derivative term is more complex than it is usually, 1 ⫺1 ik ˆ ⫽⳵ ⌿ ˆ ˆ ⫹ i ␪ ␣␤ 共 x 兲 ⳵ ␪ ⫺1 共 x 兲 ⳵ 关 ␪ ␯␳ 共 x 兲 ⳵ ⌿ ˆ ␪ 共 x 兲⳵ k⌿ ⫺i ␪ˆ ␮⫺1i 共 Xˆ 兲 쐓 x x i 쐓 x ⌿ ␮ ⫹ 关 ␪ ␮ i 共 x 兲兴 ␣ ␮␯ ␤ ␳ 兴 ⫹•••. 2

共59兲

ˆ and the modified derivative are not Hermitian. We will have to take this into account when we build the Note that A ␮1 쐓 x ⌿ actions in the next section. III. ACTIONS

In this section, we shall concentrate on the actions of quantum electrodynamics and of the standard model on a background described by a ␪ which is space-time–dependent. The main result is that the leading order operators are the same as in the constant ␪ case, if one substitutes ␪ by ␪ (x). New operators with a derivative acting on ␪ (x) also appear. A. QED on an x-dependent space-time

An invariant action for the gauge potential is 1 S g ⫽⫺ Tr 4

冕

w 共 x 兲 Fˆ ␮ ␯ 쐓 x Fˆ ␮ ␯ d 4 x,

共60兲

where Fˆ ␮ ␯ is defined as ˆ ␮쐓xD ˆ ␯ 兴 ⫽i 关 ⫺i ␪ˆ ␮⫺1i 共 Xˆ 兲 쐓 x Xˆ i 쐓 x ⫺i ␪ˆ ␯⫺1 ˆ ˆi Fˆ ␮ ␯ ⫽i 关 D i 共 X 兲쐓 xX 兴 . , ,

共61兲

For the matter fields, we find S m⫽

冕

¯ˆ ˆ d 4 x, ˆ ␮ ⫺m 兲 ⌿ w共 x 兲⌿ 쐓 x共 i ␥ ␮D

共62兲

ˆ ⫽( ⳵ ⫺iAˆ )쐓 ⌿ ˆ ˆ ␮⌿ where D ␮ ␮ x . We can now expand the noncommutative fields in ␪ (x) and insert the definition for the 쐓 x product. The Lagrangian for a Dirac field that is charged under a SU(N) or U(N) gauge group is given by ¯ˆ ˆ ⫽m ¯␺ ␺ ⫹ i m ␪ ␮ ␯ 共 x 兲 D ¯␺ D ␺ , 쐓 x⌿ m⌿ ␮ ␯ 2

共63兲

¯ˆ ˆ ⫽ ¯␺ i ␥ ␮ D ␺ ⫺ 1 ␪ ␮ ␯ 共 x 兲 D ¯␺ ␥ ␳ D D ␺ ⫺ i ␪ ␮ ␯ 共 x 兲 ¯␺ ␥ ␳ F D ␺ ⫹terms with derivatives acting on ␪ ˆ ␮⌿ ⌿ 쐓 xi ␥ ␮D ␮ ␮ ␯ ␳ ␳␮ ␯ 2 2

共64兲

and the gauge part is given by i 1 1 Fˆ ␮ ␯ 쐓 x Fˆ ␮ ␯ ⫽F ␮ ␯ F ␮ ␯ ⫹ ␪ ␮ ␯ 共 x 兲 D ␮ F ␳␴ D ␯ F ␳␴ ⫹ ␪ ␮ ␯ 共 x 兲 兵兵 F ␳ ␮ ,F ␴␯ 其 ,F ␳␴ 其 ⫺ ␪ ␮ ␯ 共 x 兲 兵 F ␮ ␯ ,F ␳␴ F ␳␴ 其 2 2 4 i ⫺ ␪ ␮ ␯ 共 x 兲关 A ␮ , 兵 A ␯ ,F ␳␴ F ␳␴ 其 兴 ⫹terms with derivatives acting on ␪ . 4

共65兲

The terms involving a derivative acting on ␪ will be written explicitly in the action. They can be cast in a very compact way after partial integration and some algebraic manipulations. The following two relations can be useful in these algebraic manipulations: ␣␳ ⳵ ␮ w 共 x 兲 ⫽ ␪ ␳⫺1 ␮ 共 x 兲⳵ ␣␪ 共 x 兲w共 x 兲, ⫺1 ⳵ ␣ ␪ ␮⫺1␯ 共 x 兲 ⫽⫺ ␪ ␮⫺1␳ 共 x 兲关 ⳵ ␣ ␪ ␳␴ 共 x 兲兴 ␪ ␴␯ 共 x 兲.

One notices that some of the terms with a derivative acting on ␪ are total derivatives, 025016-7

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PHYSICAL REVIEW D 68, 025016 共2003兲

X. CALMET AND M. WOHLGENANNT

冕

w 共 x 兲 ⳵ ␮ 关 ␪ ␮ ␯ 共 x 兲 ⌫ ␯ 兴 d 4 x⫽⫺ ⫽

冕

冕⳵

␮ 关 w 共 x 兲兴 ␪

␮␯

共 x 兲⌫ ␯d 4x

w 共 x 兲 ⳵ ␮ 关 ␪ ␮ ␯ 共 x 兲兴 ⌫ ␯ d 4 x

共68兲

using partial integration and where the last step follows from the property 共24兲. These terms, therefore, do not contribute to the action. For the action we use partial integration, the cyclicity of the trace, and the property 共68兲 and obtain to first order in ␪ (x)

冕

冕

w 共 x 兲 ¯␺ 共 i ␥ ␮ D ␮ ⫺m 兲 ␺ d 4 x⫺

⫺

1 2

⫹

1 4

冕 冕

w 共 x 兲 Fˆ ␮ ␯ 쐓 x Fˆ ␮ ␯ d 4 x⫽⫺

1 4

冕

¯ˆ ˆ d 4 x⫽ ˆ ␮ ⫺m 兲 ⌿ w共 x 兲⌿ 쐓 x共 i ␥ ␮D

1 1 ⫺ Tr 2 4 G

冕

1 4

冕

w 共 x 兲 ␪ ␮ ␯ 共 x 兲 ¯␺ F ␮ ␯ 共 i ␥ ␮ D ␮ ⫺m 兲 ␺ d 4 x

w 共 x 兲 ␪ ␮ ␯ 共 x 兲 ¯␺ ␥ ␳ F ␳ ␮ iD ␯ ␺ d 4 x ⫺1 w 共 x 兲 ␪ ␮␣ ␺ ␥ ␮ D ␴ ␺ d 4 x⫹H.c., 共 x 兲 ␪ ␳␤共 x 兲 ⳵ ␤␪ ␣␴共 x 兲 D ␳¯

1 w 共 x 兲 F ␮ ␯ F ␮ ␯ d 4 x⫹ t 1 8

1 ⫺ t1 2

冕

冕

共69兲

w 共 x 兲 ␪ ␴␳ 共 x 兲 F ␴␳ F ␮ ␯ F ␮ ␯ d 4 x

w 共 x 兲 ␪ ␴␳ 共 x 兲 F ␮ ␴ F ␯␳ F ␮ ␯ d 4 x⫹terms with derivatives acting on ␪ ,

共70兲

where t 1 is a free parameter that depends on the choice of the matrix Y 共see 关15兴兲. We have not calculated explicitly the terms with derivatives acting on ␪ for the gauge part of the action. These terms are model-dependent as they depend on the choice of the matrix Y. These terms will be calculated explicitly in a forthcoming publication. We used the following notations: ˆ (n) ˆ (n) ⬀g ⌿ G⌿ n

1

1 Tr 2 Fˆ ␮ ␯ 쐓 x Fˆ ␮ ␯ ⫽ N G

and

N

e2

兺 n⫽1

g 2n

共 q (n) 兲 2 Fˆ ␮(n)␯ 쐓 x Fˆ (n) ␮ ␯

共71兲

and ˆ (n) ⬅eq (n) Fˆ (n) ⌿ ˆ (n) . Fˆ ␮ ␯ ⌿ ␮␯

共72兲

The usual coupling constant e can be expressed in terms of the g n by Tr

N

1 G

Q ⫽ 2

2

1

1

兺 2 共 q (n) 兲 2 ⫽ 2e 2 . n⫽1 g

共73兲

n

B. The standard model on an x-dependent space-time

The noncommutative standard model can also be written in a very compact way following 关15兴, S NCSM⫽

冕

3

d x w共 x 兲 4

兺 i⫽1

¯ ˆ (i) 쐓 iD ˆ ˆ (i) ⌿ x ” ⌿L ⫹ L

冕

¯ 4 4 ˆ (i) 쐓 iD ˆ ˆ ␮␯ ˆ ˆ (i) ⌿ 兺 R x ” ⌿ R ⫺ 冕 d x w 共 x 兲 Tr 2 F ␮ ␯ 쐓 x F ⫹ 冕 d x w 共 x 兲 i⫽1 G 3

d x w共 x 兲 4

1

2 ˆ ⌽ ˆ † ˆ␮ˆ ˆ † ˆ ˆ † ˆ ˆ † ˆ ⫻ 关 ␳ 0共 D ␮ 兲 쐓 x ␳ 0 共 D ⌽ 兲 ⫺ ␮ ␳ 0 共 ⌽ 兲 쐓 x ␳ 0 共 ⌽ 兲 ⫺␭ ␳ 0 共 ⌽ 兲 쐓 x ␳ 0 共 ⌽ 兲 쐓 x ␳ 0 共 ⌽ 兲 쐓 x ␳ 0 共 ⌽ 兲兴 ⫹

冉

3

⫻ ⫺ 3

⫺

兺

i, j⫽1 3

⫺

兺

兺

i, j⫽1

i, j⫽1

¯ ˆ 兲兴 쐓 eˆ ( j) 其 ⫺ W i j 兵 关 Lˆ L(i) 쐓 x ␳ L 共 ⌽ x R

3

¯ˆ 兲 † 쐓 Q ˆ ( j) 共 G †u 兲 i j 兵¯uˆ R(i) 쐓 x 关 ␳ Q¯ 共 ⌽ x L 兴其⫺

冊

兺

i, j⫽1 3

兺

i, j⫽1

3

ˆ 兲 † 쐓 Lˆ ( j) 兴 其 ⫺ 共 W † 兲 i j 兵¯eˆ R(i) 쐓 x 关 ␳ L 共 ⌽ x L

兺

i, j⫽1

冕

d 4x w共 x 兲

¯ˆ (i) C 兲兴 쐓 uˆ ( j) 其 ¯ 共⌽ G iuj 兵 关 Q x R L 쐓 x␳ Q

¯ˆ (i) ˆ ˆ ( j) G idj 兵 关 Q L 쐓 x ␳ Q 共 ⌽ 兲兴 쐓 x d R 其

¯ ˆ 兲†쐓 Q ˆ ( j) 共 G †d 兲 i j 兵 dˆ R(i) 쐓 x 关 ␳ Q 共 ⌽ x L 兴其 .

共74兲 025016-8

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EFFECTIVE FIELD THEORIES ON NONCOMMUTATIVE . . .

The notations are the same as those introduced in 关15兴. The only difference is the introduction of the weight function w(x). The expansion is performed as described in 关15兴. There are new operators with derivatives acting on ␪ (x), but the terms suppressed by ␪ (x) that do not involve derivatives on ␪ are the same as those found in 关15兴. One basically has to replace ␪ by ␪ (x) in all the results obtained in 关15兴.

simply implies that there is an energy threshold for the effects of space-time noncommutativity. In that case, the vertex studied in Eq. 共75兲 becomes

␦4 共 b ␮ ⫺q ␮ ⫺k ␮ ⫹ p ␮ 兲 ␪ 共 b 0 ⫺⌳ R 兲 ⫻

冉

共76兲

C. Feynman rules

We shall concentrate on the vertex involving two fermions and a gauge boson which is modified by ␪ (x). One finds

冕

d 4 xe [⫺ix

␮ (b ⫺q ⫺k ⫹p )] ␮ ␮ ␮ ␮

冉

冊

i i ␮␯ ␪ 关 p ␯ 共 k” ⫺m 兲 ⫺k ␯ 共 p” ⫺m 兲兴 ⫺ k ␣ ␪ ␣␤ p ␤ ␥ ␮ , 2 2

i ␮␯ ˜␪ 共 b 兲关 p ␯ 共 k” ⫺m 兲 2

冊

i ⫺k ␯ 共 p” ⫺m 兲兴 ⫺ k ␣˜␪ ␣␤ 共 b 兲 p ␤ ␥ ␮ , 2

共75兲

where ˜␪ is the Fourier transform of ␪ (x). This is the lowest order vertex in g and ␪ (x) which is model independent, i.e., independent of t 1 共see Fig. 1兲. It is clear that the dominant signal is a violation of the energy-momentum conservation, as some energy can be absorbed in the background field or released from the background field. Similar effects will occur for the three-gauge-boson interaction and for the twofermion–two-gauge-boson interactions. IV. MODELS FOR ␪ „x…

The function ␪ (x) is basically unknown. It depends on the details of the fundamental theory which is at the origin of the noncommutative nature of space-time. Recently, noncommutative theories with a nonconstant noncommutative parameter have been found in the framework of string theory 关35– 38兴. But, since we do not know what will eventually turn out to be the fundamental theory at the origin of space-time noncommutativity, we can consider different models for ␪ (x). One particularly interesting example for ˜␪ (b) is a Heaviside step function times a constant antisymmetric tensor ˜␪ ␮ ␯ (b) ⫽ ␪ (b 0 ⫺⌳ R ) ␪ ␮ ␯ . The main motivation for such an Ansatz is that mentioned in 关15兴; the noncommutative nature of spacetime sets in only at short distances. A Heaviside function

FIG. 1. Correction to the two-fermion gauge boson vertex.

where ␪ (b 0 ⫺⌳ R ) is the Heaviside step function. In other words, the energy of the decaying particle has to be above the energy ⌳ R corresponding to the distance R. Note that we now have two scales, namely the noncommutative scale ⌳ NC included in ␪ and the scale corresponding to the distance where the effects of noncommutative physics set in, ⌳ R . A small scale of, e.g., 1 GeV for ⌳ R is sufficient to get rid of all the constraints coming from low energy experiments and in particular from experiment that are searching for violations of Lorentz invariance. This implies that heavy particles are more sensitive to the noncommutative nature of space-time than the light ones. It would be very interesting to search for a violation of energy conservation in the top quark decays since they are the heaviest particles currently accessible. Clearly, there are certainly models that are more appropriate than a Heaviside step function. This issue is related to model building and is beyond the scope of the present paper. Our aim was to give a simple example of the type of model that can help to loosen the experimental constraints. Another interesting possibility is that ␪ ␮ ␯ transforms as a Lorentz tensor: ␪ ␮ ␯ (x ⬘ )⫽⌳ ␳␮ ⌳ ␴␯ ␪ ␳␴ (x), in which case the action we have obtained is Lorentz invariant. It is nevertheless not clear which symmetry acting on ␪ (xˆ ), i.e., at the noncommutative level, could reproduce the usual Lorentz symmetry once the expansion in ␪ is performed. There are nevertheless examples of quantum groups, where a deformed Lorentz invariance can be defined 关39,40兴. Note that if ␪ (x) develops a vacuum expectation value, Lorentz invariance is spontaneously broken. V. CONCLUSIONS

We have proposed a formulation of Yang-Mills field theory on a noncommutative space-time described by a space-time–dependent antisymmetric tensor ␪ (x). Our results are only valid in the leading order of the expansion in ␪ . It is nevertheless not obvious that these results can easily be generalized. The basic assumption is that ␪ (x) satisfies the Jacobi identity. This insures that the star product is associative. We have generalized the method developed by Wess and his collaborators to the case of a nonconstant field ␪ , and we have derived the Seiberg-Witten maps for the gauge transformations, the gauge fields, and the matter fields. The main difficulty is to find the relation between the gauge potential of the covariant coordinates and the Yang-Mills gauge potential. As expected, new operators with derivative acting on ␪ are generated in the leading order of the expansion in ␪ . But,

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most of them drop out of the action because they correspond to total derivatives. The main difference between the constant ␪ case is that the energy momentum at each vertex is not conserved from the particles point of view, i.e., some energy can be absorbed or created by the background field. One can consider different models for the deformation ␪ . It is interesting to note that already a simple model can help to avoid low energy physics constraints. This implies that noncommutative physics becomes relevant again as a candidate for new physics beyond the standard model in the TeV region. ACKNOWLEDGMENTS

One of us 共X.C.兲 would like to thank J. Gomis, M. Graesser, H. Ooguri, and M. B. Wise for enlightening discussions. He would also like to thank P. Schupp for a useful discussion. The authors are very grateful to B. Jurco and J. Wess for interesting discussions. APPENDIX: THE 쐓 x PRODUCT

In this appendix, we shall derive the 쐓 x product using the deformation quantization. We want to emphasize the fact that this approach only works in the leading order in ␪ (x). In that case it is rather straightforward to apply the formalism developed in 关12–14兴 with minor modifications, which we shall describe. We shall follow the usual procedure 共see, e.g., 关7兴兲. Let us ˆ defined as consider the noncommutative algebra A 1 4 ˆ ˆ C具具 x , . . . ,x 典典 /Rx , where Rx is the relation 共10兲 and the usual commutative algebra A⫽C具具 x 1 , . . . ,x 4 典典 . Let W:A ˆ be an isomorphism of vector spaces. The 쐓 x product is →A defined by W 共 f 쐓 x g 兲 ⬅W 共 f 兲 •W 共 g 兲 ⫽ ˆf •gˆ .

共A1兲

We now consider the 쐓 x product of two functions f and g, W共 f 쐓 xg 兲⫽

1

冕

d 4 k exp共 ik j xˆ j 兲˜f 共 k 兲

共A2兲

d 4 k exp共 ⫺ik j x j 兲 f 共 x 兲 .

共A3兲

共 2␲ 兲2

with ˜f 共 k 兲 ⫽

1 共 2␲ 兲2

冕

关1兴 Letter of Heisenberg to Peierls 共1930兲, Wolfgang Pauli, Scientific Correspondence, edited by Karl von Meyenn 共SpringerVerlag, Berlin, 1985兲, Vol. II, p. 15. 关2兴 Letter of Pauli to Oppenheimer 共1946兲, Wolfgang Pauli, Sci-

共 2␲ 兲4

冕

d 4 kd 4 p exp共 ik j xˆ j 兲

⫻exp共 ip j xˆ j 兲˜f 共 k 兲˜g 共 p 兲 .

共A4兲

The coordinates are noncommutating. The Campbell-BakerHausdorff formula e A e B ⫽e A⫹B⫹(1/2)[A,B]⫹(1/12)[[A,B],B]⫺(1/12)[[A,B],A]⫹••• 共A5兲 is thus need to evaluate this expression. This is where a potential problem arises. The commutator of two noncommutative coordinates is, in our case, by assumption not constant and it is not obvious whether the Campbell-BakerHausdorff formula will terminate. But, as already mentioned previously, we are only interested in the leading order noncommutative corrections and we thus neglect the higher order in ␪ terms which will involve derivatives acting on ␪ (x). In the leading order in ␪ we have

冉

冊

i exp共 ik j xˆ j 兲 exp共 ik j xˆ j 兲 ⫽exp i 共 k i ⫹ p i 兲 xˆ i ⫺ ␪ i j 共 x 兲 k i p j ⫹••• 2 共A6兲 and W ⫺1 关 ␪ˆ i j 共 xˆ 兲兴 ⫽ ␪ i j 共 x 兲 ⫹O 共 ␪ 2 兲 . One thus finds f 쐓 xg共 x 兲⫽

冕

共A7兲

冉

i d 4 kd 4 pexp i 共 k i ⫹ p i 兲 xˆ i ⫺ ␪ i j 共 x 兲 k i p j ⫹••• 2

⫻˜f 共 k 兲˜g 共 p 兲 ,

冊

共A8兲

where we define the 쐓 x product in the following way:

冏

⳵ f 共 x 兲 ⳵g共 y 兲 i f 쐓 x g⬅ f •g⫹ ␪ ␮ ␯ 共 x 兲 2 ⳵x␮ ⳵y␯

In general, we do not know how to construct this new star product, but since we are only interested in the leading order operators, all we need is to define the new star product in the leading order and this can be done easily, as described in 关12–14兴, by considering the Weyl deformation quantization procedure 关41兴, ˆf ⫽W 共 f 兲 ⫽

1

⳵ f 共 x 兲 ⳵g共 x 兲 i ⬅ f •g⫹ ␪ ␮ ␯ 共 x 兲 , 2 ⳵x␮ ⳵x␯

y→x

共A9兲

neglecting higher order terms in ␪ that are unknown and taking the limit y→x. It is interesting to note that it corresponds to the leading order of the star product defined for a Poisson structure 关29–31兴. We want to insist on the fact that the results presented in this appendix cannot be generalized to higher order in ␪ . This can be done using Kontsevich’s method, which is unfortunately much more difficult to handle.

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