# One loop radiative corrections to the translation-invariant noncommutative Yukawa Theory

###### Abstract

We elaborate in this paper a translation-invariant model for fermions in 4-dimensional noncommutative Euclidean space. The construction is done on the basis of the renormalizable noncommutative translation-invariant theory introduced by R. Gurau et al. We combine our model with the scalar model, in order to study the noncommutative pseudo-scalar Yukawa theory. After we derive the Feynman rules of the theory, we perform an explicit calculation of the quantum corrections at one loop level to the propagators and vertices.

## 1 Introduction

In the last two decades, a lot of work has been devoted to the study of noncommutative quantum field theories. The main idea behind these theories is that at the Planck scale the space-time is no longer commutative, this fact makes the noncommutative geometry an essential ingredient when probing spacetime structure at very small distances doglas , zsabo1 . The original motivation for investigating such theories was the hope of solving the problem of infinities of quantum field theory and the possible formulation of consistent quantum gravity dopl1 , dopl2 . Despite the collective efforts deployed by physicists none of these goals is yet reached.

In fact, instead of the elimination of ultraviolet infinities, the use of noncommutative geometry in quantum field theories gives rise to a new set of problems and makes the short distance behavior of those theories more ambiguous filk -ramsd . The conventional theories became non renormalizable due to the infamous ultraviolet/infrared mixing. Many attempts were made to overcome this UV/IR mixing, but in general, the problem persists.

However, there are few models in which renormalizability was restored. It was achieved by adding a suitable term to the initial action of the theory. The procedure was first used by Grosse and Wulkenhaar gross1 -gross4 to solve the UV/IR mixing of the noncommutative Euclidean theory. In their model they added a harmonic oscillator term which depends explicitly on the Moyal space coordinates , where . It turns out that the model is covariant under Langmann-Szabo duality zsabo2 but also breaks the translation invariance of the action.

Another approach, using the same method, was proposed by R. Gurau et al. rivasseau2008 . This model preserves the translation invariance of the noncommutative theory. The term added to the action is in fact a non local counter-term of the form , which is written in momentum space as . This model is known as the translation-invariant -model. The UV/IR mixing problem was solved by the elimination of the quadratic IR divergence of non-planar diagrams. Both of these scalar models were constructed on the Moyal space and were proven to be renormalizable to all orders in perturbation theory.

The noncommutative fermion theory was also formulated in the case of the Gross-Neveu model fabien . Following the same procedure in Grosse and Wulkenhaar model, the term added to the action is . This model was proven to be renormalizable to all orders in perturbation theory, but unlike the noncommutative models, it still presents a UV/IR mixing even after renormalization. In fact, the Gross-Neveu model is renormalizable even without adding an extra-term.

Motivated by the renormalizable noncommutative translation-invariant -model, and since it has not been extended to fermions, we propose to construct its fermionic version. It is well known in ordinary quantum field theory that the scalar propagator is perceived as the square of the Dirac propagator, indeed we have

(1) |

this means also that the scalar propagator appears naturally in the expression of the Dirac propagator

(2) |

where is the Dirac propagator and the scalar propagator, here expressed in their Euclidean forms.

It seems reasonable to impose the condition (1) in the noncommutative case if we want to have a consistent theory that involves both scalar and fermion fields. Thus, our starting point is the construction of a model in which the modified scalar and fermion propagators are correlated in the same way as in the ordinary quantum field theory. The extra-term in the fermionic action is chosen accordingly.

The consistency of our model relies on the fulfillment of the condition (1), but this does not guarantee its renormalizability. This is why we apply it, in addition to the scalar model, to study the noncommutative pseudo-scalar Yukawa theory. We recall that the Yukawa interaction between a pseudo-scalar field and a Dirac field is represented in the Euclidean space by the action

(3) |

this interaction is used in the standard model to describe the coupling of Higgs particle with fermions. The calculation of the quantum corrections at one loop level enables us to test the consistency of the whole model and its renormalizability. Further, it reveals more about the behavior of these modified models and allows us to improve them if necessary.

We note here, that the method used in the renormalizable models gave an alternative approach to construct noncommutative field theories free of UV/IR mixing. So, it was natural to extend these models to noncommutative gauge field theory, hoping to have the same success. But unfortunately this method failed to solve UV/IR mixing problem, although several promising approaches were made tanasa -gross gt . Currently, there is no explicit procedure to deal with this problem.

The paper is organized as follows: in the next Section we define our model and derive its Feynman rules. In Section 3 we perform an explicit Feynman graph calculations at one loop level in order to evaluate the radiative contributions to the scalar and the fermion propagators and Yukawa and vertices. The Section 4 is devoted to remarks and conclusions.

## 2 The Model

The realization of noncommutative modified models cited above was achieved by the substitution of the ordinary product between fields by the Weyl-Moyal star product akfor

(4) |

This approach is considered to be the simplest way to construct a noncommutative field theory, the coordinates fulfill, the commutation relation

(5) |

where is the deformation matrix, it is assumed to have a simple block-diagonal form

(6) |

here is the deformation parameter, it is taken to be real and gives the measure of the strength of noncommutativity. Throughout this paper we use the Euclidean metric, the Feynman convention and the notation .

The free scalar action of the translation-invariant -model is rivasseau2008

(7) |

in the Euclidean space. In the expression (7), the parameter is a real dimensionless constant. The modified scalar propagator in momentum space is then

(8) |

In order to recover the modified scalar propagator from the square of the fermion propagator, as in the commutative theory (1), we propose to modify the free fermion action in the following way

(9) |

where is a real dimensionless constant. We have added an extra-term to the original fermion action which reads in momentum space as .

The Yukawa theory in four dimensions Euclidean space includes the self interaction in order to be renormalized, the noncommutative interaction action is thus

(10) |

with the use of the trace property of the star product micu&jab2001

(11) |

the pseudo scalar Yukawa action reduces to

(12) |

where and

The total action of our model reads

(13) |

where and are the dressed fields and and are the bare fields, we used as usual the substitution

(14) |

The counter-terms action is then

(15) |

where the renormalization factors are

(16) |

The different actions written above are used next to derive the Feynman rules for the propagators and vertices.

### 2.1 Propagators

The noncommutative free theory is the same as the commutative one micu&jab2001 , the action remains unchanged, and this is due to the relation

(17) |

Even when the actions are modified by adding some extra-terms the propagators are calculated using the same techniques as the ordinary quantum field theory. The modified scalar propagator in momentum space (8) is written as

(18) |

where . It is possible to rewrite this propagator, under a more suitable form schweda1 , in order to evaluate the Feynman integrals by the use of the usual mathematical techniques

(19) |

where , and if we use Schwinger’s exponential parametrization, with and , the propagator is then

(20) |

The modified fermion propagator is calculated from the action (9), we obtain

(21) |

where . This propagator fulfills the condition (1), or in this case

(22) |

thereafter, the modified scalar propagator is naturally recovered in the expression of

(23) |

as a consequence, the fermion propagator reproduces the same ”damping” behavior for vanishing momentum as the modified scalar propagator schweda1

(24) |

### 2.2 Vertices

The Feynman rule in momentum space for the self interaction vertex is given by micu&jab2001

(25) |

where

The Feynman rule for the Yukawa interaction vertex in momentum space is calculated from the Yukawa action (12) using the Fourier transformation of the fields

(26) |

thus

(27) |

where is a phase factor

(28) |

in the last expression we used the notation. The Yukawa interaction in our model is represented by two coupling constants and , the commutative coupling constant is recovered when : where .

We note, finally, that the modification of the ordinary commutative vertices is a natural consequence of the introduction of Moyal star product, unlike the propagators which are modified artificially by adding the extra-terms to the actions.

### 2.3 Counter-terms

The renormalized Feynman rules can be deduced easily from the counter-terms action (15), they are written in momentum space as

(29) |

We note from the counter-terms that the constants and could receive corrections in order to eliminate IR divergences of the form and , respectively. The Feynman rules for propagators and vertices are now established, they are used in the next section to evaluate the one loop quantum corrections. The Feynman rules for counter-terms were also given, they will be used in the renormalization process which will be discussed in a forthcoming paper.

## 3 One loop corrections

We are going to determine, in this section the relevant corrections for the 1PI two-point functions, for the scalar and fermion field, and the three and four-point functions at one loop level using dimensional regularization method. We use the results of the multiscale analysis rivasseau2008 to eliminate the subleading logarithmic singularities of non-planar graphs for vanishing momentum, because they represent a mild divergence schweda1 . Therefore, we keep in our results only the UV divergences of the planar integrals and the leading quadratic IR divergences of the non-planar integrals.

### 3.1 Two-point function

#### 3.1.1 Scalar propagator

The diagrammatic expansion of represents the quantum corrections for the scalar field propagator, at one loop level we have the tadpole and the fermion loop graphs to evaluate, the first one is represented by the integral

(30) |

where the integration is in a -dimension Euclidian space. We evaluate the divergent part of the planar and non-planar integral using the dimensional regularization method, the result is

(31) |

where is the modified Bessel function. Thereafter we put , where , this reveals the UV divergence of the planar part

(32) |

the non-planar integral depends on external momentum and it is finite for , however it reveals a leading quadratic IR for

(33) |

The total divergence of the tadpole integral is then

(34) |

We note that the UV divergence is different by a numeric factor from the commutative case. This difference is due to the scalar vertex which adds a factor (see reference micu&jab2001 ) and the extra-term in the propagator numerator which adds a factor .

The second contribution to the scalar two-point function comes from the fermionic loop. After performing a trace over the fermion loop, the integral representing the second graph reads

(35) |

this can be divided, as usual, into planar and non-planar integrals, following the same terms order in the last expression, we have

(36) |

and

(37) |

where each integral and is evaluated separately. For the planar integrals, we have the following results:

-the integrals , and presents an UV divergences, their divergent parts give the contribution

- the integrals and are finite, but after integration the products and are proportional to and then vanish.

- the integrals and are finite for .

The non-planar integrals are finite for , but they could reveal an IR divergence when , we have the following results:

- the integrals , and presents an IR divergences, their divergent parts give the contribution

- the integrals , and are finite, but after integration the products and are proportional to and then vanish.

- the integrals , and are finite for .

The fermion contribution to the scalar two-point function reads

(38) |

where denote functions that result from the finite integrals, they are analytic for , this notation is used thereafter. We note here that the UV divergence in (38) is the same as the commutative case where .

Thus, the total one loop contribution to the scalar field propagator is

(39) |

and are given in the expressions (34) and (38), respectively. There is, as expected, a leading quadratic IR divergence resulting from the non-planar integrals beside the ordinary UV divergence. The additional term is finite for , it is a result of the fermionic extra-term, thus it vanishes for .

#### 3.1.2 Fermion propagator

The quantum corrections for the fermion field propagator at one loop level are given by the integral

(40) |

this can be divided into planar and non-planar integrals, following the same terms order in the last expression, we have

(41) |

and

(42) |

We calculate each integral apart and using the same procedure as in the tadpole integral, we obtain

(43) |

we note that the UV divergence, in the last relation, is the same as the commutative theory with . We have also here an additional term which is finite for .

### 3.2 Three-point function

The one loop quantum corrections to Yukawa vertex are given by only one graph, namely

(44) |

where represents the product of the phase factors of the Yukawa vertices

(45) |

the first term is independent of , therefore it appears as a factor of the planar integrals while the other terms enter in the non-planar integrals. The function represents the product of the fermions propagators numerators with the matrix

(46) |

the use of dimensional power counting, reveals that all the terms of this function contribute in a convergent integrals except the one with , the resulting divergence, from the planar and non-planar integrals, is then logarithmic. Thus, the evaluation of the divergent parts of the integral (44) gives

(47) |

where are analytic functions for resulting from the finite integrals. In order to recover results of the UV divergence of the commutative case we have to make the substitution

### 3.3 Four-point function

In order to evaluate the four-point function at one loop level, we have to include all the contributions that give and corrections to the vertex.

#### 3.3.1 with coupling

The scalar one loop contributions to the vertex comes from the following graphs

(48) |

which are evaluated from these integrals

(49) |

where is the product of the two vertices.

These integrals were evaluated in schweda1 by introducing a cut-off, we find equivalent results using the dimensional regularization method. These diagrams present a logarithmic UV divergence

(50) |

which is different by a numeric factor from the commutative case where .

#### 3.3.2 with Yukawa coupling

The contributions to the vertex come in this case from the Yukawa interaction, it represent the fermion corrections to the scalar coupling constant. In order to have an effective contribution to the vertex, from the fermion loop, we need to recover in our final result the extra-factor of from the product of the Yukawa phase factors . If we consider only the permutations of external momenta, as in the commutative theory, then we will have only six diagrams to evaluate, namely

(51) |

However, in the noncommutative case, there are also the phase factors coming from the Yukawa vertices which depend explicitly on internal momenta. This means that when we expand the four-point function at one loop level we will have more diagrams to evaluate. In fact there only two different permutations of internal momentum for each one of the last six diagrams, it can be represented as follow

(52) |

As a result, the diagrammatic expansion of the four-point function is represented by twelve diagrams, the integral corresponding to each diagram have the generic form