# One-loop counterterms for the dimensional regularization of arbitrary Lagrangians.

###### Abstract

We present master formulas for the divergent part of the one-loop effective action for an arbitrary (both minimal and nonminimal) operators of any order in the 4-dimensional curved space. They can be considered as computer algorithms, because the one-loop calculations are then reduced to the simplest algebraic operations. Some test applications are considered by REDUCE analitical calculation system.

pacs nombers 11.10.Gh, 04.62.+v

## 1 Introduction.

Progress in the quantum field theory and quantum gravity in particular depends much on the development of methods for the calculation of the effective action. For a lot of problems the analyses can be confined to the one-loop approximation. In this case

(1) |

where latin letters denote the whole set of indexes.

Finding its divergent part is sometimes a rather complicated technical problem, especially in the curved space. Unfortunately, the usual diagram technique is not manifestly covariant. A very good tool to make calculations in the curved space-time is the Schwinger-DeWitt proper time method [1, 2]. It allows to make manifestly covariant calculations of Feynman graphs if the propagator depends on the background metric [3]. This approach was successfully applied to obtain one-loop counterterms in theories with the simplest second and forth order operators . We should also mention other covariant methods, that in principle allow to get the divergent part of effective action [4, 5, 6].

A different approach was proposed by t’Hooft and Veltman [7]. Instead of calculating Feynman graphs for each new theory, they made it only once for a rather general case. Nevertheless, if in (1) is not a minimal second order oparetor, their results can not be used directly.

In this paper we extend t’Hooft and Veltman approach to the most general case. We construct the explicit expression for the divergent part of the one-loop effective action without any restrictions to the form and the order of the operator in the 4-dimentional space-time. Then the divergent part of the one-loop effective action can be found only by making the simplest algebraic operations, for example, by computers.

Our paper is organized as follows.

In Sec. 2 we briefly remind t’Hooft-Veltman diagram technique and introduce some notations and definitions.

In Sec. 3 we consider a theory with an arbitrary minimal operator in the curved space-time. The main result here is an explicit expression for the one-loop contribution to the divergent part of the effective action.

In Sec. 4 we describe a method for the derivation of a master formula for an arbitrary nonminimal operator on the curved background and present the result.

Sec. 5 is devoted to the consideration of some particular cases. We show the agreement of our results with the earlier known ones. Here we proove the correctness of the method and that is why we do not consider here new applications.

In Sec. 6 we give a summary of our results and discuss prospects of using computers for the automatization of calculations.

In Appendix A we describe in details the derivation of the one-loop counterterms for an arbitrary minimal operator.

In Appendix B we illustrate the general method presented in Sec. 4 by calculating the simplest Feynman graphs for an arbitrary nonminimal operator.

Appendix C is devoted to the derivation of some useful identities.

## 2 Diagramic approach in the background field method

We will calculate the divergent part of the one-loop effective action for a general theory by the diagram tecknique. First we note, that

(2) |

is a differential operator depending on the background field . Its most general form is

(3) |

where is a covariant derivative:

(4) |

Here is the Cristoffel symbol

(5) |

and is a connection on the principle bundle.

Commuting covariant derivatives we can always make symmetric in the greek indexes. This condition is very convenient for the calculations, so we will assume it to be satisfied.

The operator is called minimal if and is a totally symmetric tensor, built by : where

(6) |

If an operator can not be reduced to this form, we will call it nonminimal one.

Commuting covariant derivatives we can rewrite a minimal operator in the form:

(7) |

where .

The one-loop effective action (1) can be presented as a sum of one-loop diagrams, say, for a minimal operator, as follows:

(8) |

where

(9) |

and we omit an infinite numerical constant.

Terms can be found by a series expansion of the operator in powers of and .

In the momentum space the propagator is . The form of the vertexes is rather evident, for example, the vertex with the external line of type can be written as

(10) |

Similar notations we will use for other expressions, for example,

(11) |

Numerical factors for the Feynman graphs can be easily found by (2).

The number of diagrams in (2) is infinite, but most of them are convergent. Really, it is easy to see that the degree of divergence of a one-loop graph with legs of type, legs of type, legs of type, legs of type and so on () in the flat space (, ) is

(12) |

Therefore, there are only a finite number of the divergent diagrams. They are presented at the Fig. One-loop counterterms for the dimensional regularization of arbitrary Lagrangians.. (We excluded divergent graphs, that give zero contribution to the effective action, for example, some tadpole ones.)

The extension of this results to a nonminimal operator will be made below.

## 3 Effective action for the theory with minimal operator.

Now we should calculate the divergent part of diagrams presented at the Fig. One-loop counterterms for the dimensional regularization of arbitrary Lagrangians.. We will do it using dimensional regularization. So, in order to find the divergent part of an integral

(13) |

it is necessary to expand the function into series, retain only logarithmically divergent terms and perform the integration according to the following equations [8]

(14) |

where is a unit vector () and

(15) |

Using this prescription one can easily find the divergent part of the diagrams at the Fig One-loop counterterms for the dimensional regularization of arbitrary Lagrangians.. The calculations are presented in details in the appendix A.

Collecting the results for all graphs, we obtain the divergent part of the one-loop effective action for the minimal operator (2) in the flat space:

(16) |

where

(17) |

In order to extend this result to the curved space-time we first consider a minimal operator in the form (2).

In this case we can not calculate all divergent graphs, because their number is infinite. (The matter is that the degree of divergence does not depend on the number of vertexes and there are infinite number of such vertexes too). Nevertheless if we note that the answer should be invariant under the general coordinate transformations, the result can be found by calculating only a finite number of graphs. Really, we should replace derivatives in (3) by the covariant ones and add expressions, containing curvature tensors and . The most general form of additional terms is

(18) |

where

(19) |

(We take into account that the expression is a total derivative and may be omitted).

Then the coefficients - can be found by calculating the diagrams presented at the Fig. One-loop counterterms for the dimensional regularization of arbitrary Lagrangians.. They conform to the first nontrivial approximation in the counterterm expansion in powers of weak fields and .

In the appendix A we present detailed calculation of the coefficients and . The other ones were found in the same way. After rather cumbersome calculations we obtain the following formula for the divergent part of the one-loop effective action for a minimal operator (2) in the curved space:

(20) |

It is more convenient sometimes to use a different form of the operator:

(21) |

where was defined in (6).

By the help of the same method we found the following answer for the divergent part of the one-loop effective action:

(22) |

## 4 Effective action for theory with nonminimal operator.

Let us consider a theory with an arbitrary nonminimal operator (2) first in the flat space (, ). In this case

(23) |

where is given in (2). There are the following differences from the minimal operator:

1. For the nonminimal operator the propagator in the momentum space is , where

(24) |

(We assume, that exists. Usually it can be made by adding gauge fixing terms to the action).

2. For a nonminimal operator depends on some additional fields besides and . It will be considered in details below.

The divergent graphs are the same as in the case of a minimal operator (see Fig. One-loop counterterms for the dimensional regularization of arbitrary Lagrangians.), but because of the difference of the propagators the calculations are also different. They are considered in the appendix B. The result is

(25) |

where we use the following notations (compare with (3))

(26) |

etc.

The generalization of this result to the curved space-time can also be made in the frames of the weak field approximation. We should substitute derivatives in (4) by the covariant ones and add some terms, containing curvature tensors. The additional terms can be found by calculating Feynman diagrams, that conform to the first terms of their expansion over weak fields. Nevertheless, in this case there are some difficulties, for example, now we do not know and, therefore, how it depends on . Thus, we do not know expressions for vertexes in the weak field limit.

In order to overcome this difficulty we will use the following trick. Suppose, that does not depend on , but depend on some external fields . Also, we impose a condition

(27) |

that is, of cause, satisfied if depends only on . From (4) we conclude that should be considered as a weak field.

Thus, unlike the minimal operator, besides the diagrams with external and lines, presented at the Fig One-loop counterterms for the dimensional regularization of arbitrary Lagrangians. we should consider also graphs with external lines, presented at the Fig One-loop counterterms for the dimensional regularization of arbitrary Lagrangians..

Computing this diagrams we obtain expressions, containing . Substituting it by (4), we found the result depending on and . The calculation of graphs, presented at the Fig One-loop counterterms for the dimensional regularization of arbitrary Lagrangians. gives results, that can not be written as a weak field limit of a covariant expression. Nevertheless, the covariant result should be found by adding a contribution of diagrams with external -lines.

Therefore, for a nonminimal operator we can not calculate terms, containing and separately.

We illustrate the above discussion in the appendix B by calculating the simplest group of diagrams.

Summing up the results for all graphs, we find the divergent part of the one-loop effective action for an arbitrary nonminimal operator

(28) |

where

(29) | |||

(30) | |||

(31) | |||