# Generalized heat kernel coefficients

###### Abstract

Following Osipov and Hiller, a generalized heat kernel expansion is considered for the effective action of bosonic operators. In this generalization, the standard heat kernel expansion, which counts inverse powers of a c-number mass parameter, is extended by allowing the mass to be a matrix in flavor space. We show that the generalized heat kernel coefficients can be related to the standard ones in a simple way. This holds with or without trace and integration over spacetime, to all orders and for general flavor spaces. Gauge invariance is manifest.

PACS numbers: 12.39.Fe 11.30.Rd

Keywords: heat kernel expansion, effective action, renormalization, chiral lagrangians

Let the bosonic operator be

(1) |

where the gauge field and the scalar field are matrices in some flavor space. This kind of operators appear frequently in the computation of the effective action of fermions (see e.g. [1]). In particular the normal parity component of such an effective action is directly related to the determinant of the operator ( being the Dirac operator). A standard technical device to compute the effective action is to split the scalar field into two contributions

(2) |

where is a constant c-number (squared) mass parameter. This allows to carry out an expansion in inverse powers of with coefficients which are homogeneous polynomials constructed with the quantities and . These coefficients are ordered by its scale dimension and thus they are identical to those of the standard heat kernel expansion [2, 3, 4]. These coefficients are very well-known and we refer to [1, 5] for details.

In the context of effective theories of quarks aiming at modeling QCD at low energy [6], Osipov and Hiller [7, 8] consider instead a more general separation of the scalar field

(3) |

where is still a constant (i.e. -independent) but not necessarily a c-number. In general is a matrix in flavor space. In a typical application for flavors, being the constituent quark mass of the -th flavor, and accounts for the deviations of the scalar field from . Obviously in the particular case of degenerated masses the previous case (a c-number) is recovered.

One can try to carry out an expansion in inverse powers of (i.e. a large mass expansion) in this more general setting. A straight approach is to organize the expansion in such a way that each term is again a homogeneous polynomial in and . Technically this can be done by introducing a bookkeeping parameter

(4) |

and then proceed to expand the effective action in powers of . This simple minded approach, however, meets the problem that gauge invariance is not preserved separately by each term of the expansion [7, 8]. (Of course gauge invariance holds for the full effective action functional.) The problem is that we want to regard , as well as its associated effective action, as a functionals where the external fields and are the true variables and and the operator play the role of fixed parameters, i.e.

(5) |

From this point of view, under a gauge transformation ( being a matrix in flavor space)

(6) |

The quantities and are gauge invariant by definition, and transform homogeneously under a similarity transformation and the external fields and transform inhomogeneously:

(7) | |||||

It is clear now that, because does not affect in , the expansion in breaks gauge invariance, i.e. in general will not coincide with at . They do coincide when is a c-number and in this case the standard heat kernel expansion is recovered.

As an aside, we note that manifest gauge invariance is automatic order by order in the context of a strict derivative expansion of the effective action functional (e.g. [9]), that is considering instead . The strict derivative expansion can be viewed as a resummation of the heat kernel expansion to all orders in . Recently explicit closed formulas have been obtained in such an expansion for both the normal and the abnormal parity components of the effective action of fermions coupled to vector, axial, scalar and pseudo-scalar external fields and for an arbitrary flavor group [10]. In the normal parity case the formulas hold for arbitrary space-time dimension through fourth order in the covariant derivatives. In the abnormal parity case the leading order is computed in two and four dimensions.

The problem of obtaining a manifestly gauge invariant inverse mass expansion for matricial has been solved by Osipov and Hiller in [7, 8] and explicit results are presented there for lowest orders in the case of two and three flavors without gauge fields. Presently we reformulate their approach in a way that makes it simple to treat the case of arbitrary flavor group and the introduction of gauge fields. Finally we find a simple relation between the generalized heat kernel coefficients and the standard ones so that no new calculation of these coefficients from scratch is required.

In order to present the formalism let and denote two matrices (or operators) in some space , such that the combination transforms by a similarity transformation, represents the term which is defined to be invariant under gauge transformations and transforms inhomogeneously (thus, , and generalize , and respectively). Consider now the gauge covariant quantity where is some arbitrary function such as e.g. the logarithm. (Note that itself is a c-number although its argument, and thus its value, can be a matrix.) Loosely speaking what we are seeking is to obtain an expansion for small (or large ) that generalizes the usual Taylor expansion valid for c-number and (or more generally, valid when and commute), but in such a way that each term of the expansion is separately gauge covariant. This can be achieved as follows. Let the coefficients be defined by the set of relations

(8) |

In this formula the notation represents an average of a matrix in , namely

(9) |

where denotes the trace operation in . The coefficients are matrices and are recursively defined by the formula. The depend both on and in general. Of course when is a c-number is simply .

Since is a basis of functions we can take linear combinations in the previous formula and write more generally

(10) |

where is an arbitrary function and is its -th
derivative.^{1}^{1}1As usual in quantum field theory, we will be
happy if the formulas hold in the sense of asymptotic series. They are
not required nor expected to be convergent. Note that the
coefficients do not depend on the function .

Three crucial properties of these coefficients can be established without explicit computation:

(i) The depend on and but this dependence is such that remains unchanged if is replaced by where is any c-number:

(11) |

This is because the shift introduced by can be absorbed by a redefinition of .

(ii) The coefficients are gauge covariant since is covariant and is invariant for all :

(12) |

(iii) For the traced quantity

(13) |

Then, the vanish for vanishing (except which equals ):

(14) |

This latter property distinguishes this expansion from other possible expansions which also enjoy the properties (i) and (ii), for instance

(15) |

The correct choice of among other possible choices is a merit of [7, 8].

At lowest orders

(16) | |||||

where we have introduced the following notation

(17) |

For the traced coefficients

(18) | |||||

The property (i) noted above is manifest since depends on only (and is independent of ). Gauge covariance is also obvious since comes as a combination of powers of , and this matrix transforms covariantly.

For subsequent application in the effective action problem some properties of the will be needed. First note that by taking a first order variation with respect to , either in (8) or (10), and using the identity

(19) |

it follows that

(20) |

which is well-known in the context of the heat kernel expansion [1]. Next note that from their definition (8) and the property (11) above, which allows to use instead of , one has to all orders

(22) | |||||

for some c-number coefficients which do not depend on . The allowable dependence of these coefficients on and can be delimited by using (20) which immediately implies that

(23) |

Therefore (defining ) eq. (22) can be given the following sharper form:

(24) |

The recurrence (22) then takes the form

(25) |

and for lowest orders yields

(26) | |||||

Note that this differs from the usual cumulant expansion beyond third order.

A further identity will be needed to relate the standard and generalized heat kernel coefficients. Let

(27) |

Here refers to the dependence on of written as , and so . Then the following identity holds

(28) |

This is easily proved as follows

(29) | |||||

Let us now turn to the application of the previous results to compute
the generalized heat kernel expansion. As is well-known the effective
action of the complex bosonic field with Klein-Gordon operator
is (where refers to functional
trace). This and related functionals can be obtained once the “current”
is known.^{2}^{2}2It is understood
that the current is known for the whole family of operators
for any complex , then

(30) |

where

(31) |

( being the spacetime dimension) and are the (diagonal) heat kernel coefficients. They are polynomials of dimension constructed with , and their covariant derivatives and they do not explicitly depend on . At lowest orders

(32) | |||||

(Our convention is that of [11, 5] which differs from that of [1] by a factor ) The lowest order integrals are ultraviolet divergent and so some renormalization is understood. Because the corresponding heat kernel coefficients are polynomials, this renormalization translates into the standard polynomial ambiguity in the effective action (and current etc) in its ultraviolet divergent contributions.

Quite naturally, in the general case of arbitrary the generalized heat kernel coefficients are defined as [7, 8]

(33) |

where now

(34) |

and the average refers to flavor space.

Before embarking in the task of computing these generalized coefficients from scratch, it is advisable to rest a moment and consider what result is to be expected. The formulas regarding the expansion of are fairly general (no assumption was made on the vector space ), but of course they are formal due to ultraviolet divergences when applied to the operator . Technically a very definite problem in these calculations is the lack of cyclic property of the trace when the operator (or ) is involved. In order to avoid these complications, let us temporarily neglect the contributions from derivatives. In this case equals and this corresponds to and in the previous formulas. Expanding the current yields and in the standard expansion (30) and and in the generalized case (33). is given in eq. (24) as a definite combination of but with shifted by . In our case corresponds to

(35) |

Therefore, for terms without derivatives, we obtain a simple relation between standard and generalized heat kernel coefficients, which is just a translation of eq. (24), namely

(36) |

where denotes the usual heat kernel coefficient but using everywhere

(37) |

instead of . In addition, the quantities are given by the same formulas (25,26) with .

Because the simple relation (36) is perfectly well-defined and sensible also in presence of covariant derivatives it can be conjectured that it holds in general. In fact this is the case, as will be shown subsequently. This is our main result. To lowest orders

(38) | |||||

The analogous relation holds for the traced and integrated (over ) coefficients needed for the effective action. We have verified that the results in [7] for the traced coefficients in SU(2) are reproduced. We remark that the replacement should be done everywhere in (i.e. in terms with derivatives too). The gauge covariance (in ) is obvious in since is itself covariant. Another remark is that the formula (22) is not sufficient to obtain the result, in fact it does not even guarantee gauge invariance (in ), and the more detailed formula (24) is needed.

Let us now turn to the proof of the relation (36). The main observation is that we do not really need to compute the generalized coefficients but only to relate them to the standard ones. Therefore our strategy will be to start the computation of the coefficients and at some point recognize that the relation (36) will be obtained.

There is an abundant literature on the computation of the heat kernel coefficients in various settings [5]. Here we will use a method convenient for our present purposes. The first step is to use the method of symbols [12] to express the current

(39) |

where is the state with zero momentum , and so . This allows to deal with the ultraviolet divergence but explicit gauge covariance is lost. Explicit gauge invariance is only recovered after integration over .

In order to apply our formulas, we identify

(40) |

regarded as operators in the space of position (spanned by ) and flavor. is a c-number parameter. Because is -independent the averages or are all in flavor space and well-defined. In particular . Further we define

(41) |

A direct application of (10) and (28) then gives

(42) | |||||

(43) |

In this formula the operator acts on the . It is given in (27) and the are constructed with .

A calculation of the coefficients would now proceed from (43) as follows (see e.g. [9]): (i) expanding the binomial, (ii) using angular averaging in momentum space, (iii) using integration by parts in momentum space (this step groups together terms with a same dimension where it counts only the dimension carried by and but not that carried by the momentum), and (iv) bringing the expression to an explicit form where the operators appear in covariant derivatives (i.e. in commutators) only. These manipulations produce terms where all operators are purely multiplicative (all derivative operators are already inside commutators) and so equivalent to ordinary functions of , thus the matrix element simple evaluates that function at . This yields the heat kernel coefficients in this approach. (We have explicitly computed , and using this method to verify that no subtleties arise.) However this is not necessary: it can be observed that when is a c-number vanishes and the formula becomes

(44) |

All the manipulations (i-iv) just described can be carried out here and we know that the final result is just the standard heat kernel expansion quoted in (30). Then when these very manipulations are used in (43) they will produce the same result except that is replaced by (the fact that involves an average over flavor does not make any difference). That is,

(45) |

Finally, using

(46) |

produces

(47) |

and (36) follows.

In summary, developing ideas put forward by Osipov and Hiller, we have
presented a general formalism to treat the problem of expanding
functionals above non c-number operators while preserving full gauge
invariance.^{3}^{3}3A similar construction has been considered in
the context of quantum gravity by Floreanini and Percacci
[13]. We have shown that it is not restricted to
formal applications (finite dimensional spaces) since it holds too in
presence of ultraviolet divergences. This formalism has been applied
to obtain a simple relation (36) between the standard and
the generalized heat kernel coefficients introduced in
[7, 8].

## Acknowledgments

I would like to thank C. García Recio for comments on the manuscript. This work was supported in part by funds provided by the Spanish DGICYT grant no. PB98-1367 and Junta de Andalucía grant no. FQM-225.

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