Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions.The solution cosmological constant problem

We introduce Hausdorff-Colombeau measure in respect with negative fractal dimensions. Axiomatic quantum field theory in spacetime with negative fractal dimensions is proposed.Spacetime is modelled as a multifractal subset of $R^{4}$ with positive and negative fractal dimensions.The cosmological constant problem arises because the magnitude of vacuum energy density predicted by quantum field theory is about 120 orders of magnitude larger than the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff $E$ and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions gives high-energy cutoff on natural way. In order to obtain disered physical result we apply the canonical Pauli-Villars regularization up to $E$. It means that there exist the ghost-driven acceleration of the univers hidden in cosmological constant.

renormalization of QED 3 . One of the greatest challenges in modern physics is to reconcile general relativity and elementary particles physics into a unified theory. Perhaps the most dramatic clash between the two theories lies in the cosmological constant problem [1][2][3][4][5][6] and in the problem of the Dark (i.e., non-luminous and non-absorbing) Matter nature is, arguably, the most widely discussed topic in contemporary particle physics.Naive predictions of vacuum energy from canonical quantum field theory predict a magnitude so high that the expansion of the Universe should have accelerated so quickly that no any structure could have formed. The predicted rate of acceleration resulting from vacuum energy is famously 120 orders of magnitude larger than what is observed. In order to avoid these difficultnes mentioned above we assume that:(i) physics of elementary particles essentially is separated into low/high energy ones, (ii) the standard notion of smooth spacetime is assumed to be altered at a high energy cutoff scale and a new treatment based on QFT in a fractal spacetime with negative dimension is used above that scale . In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions [15] gives high-energy cutoff on natural way.No one knows what dark energy is, but we need it to explain the discovered accelerated expansion of the Universe. The most elegant and natural solution is to identify dark energy with the energy of the quantum vacuum predicted by Quantum Field Theory, but the trouble is that QFT predicts the energy density of the vacuum to be orders of magnitude larger than the observed dark energy density: Recall that it was stressed by Zeldovich [1] that quantum field theory generically demands that cosmological constant or, let us repeat, what is the same, vacuum energy is non-vanishing.Summing the zero-point energies of all normal modes of some quantum field of mass m up to a wave number cut-off /c 2 m, QFT yields [1], [

Sources of Vacuum Energy
It is not excluded experimentally that the number of fermionic and bosonic species in Nature are the same. Moreover it is practically a necessity, because otherwise vacuum energy density would be infinite. Still the masses of bosons and corresponding fermions are different and, with arbitrary relations between their masses, only the leading term, which diverges as the fourth power of the integration limit, would be canceled out. However in some supersymmetric theories with spontaneous symmetry breaking there may be specific relations between masses of different fields which ensure the compensation not only of the leading term but also quadratically and logarithmically divergent terms. This looks as a very strong argument in favor of such models. However the finite terms are not compensated. Moreover in global supersymmetric theories finite contributions into ρvac must be nonzero and by the order of magnitude they are equal to where m susy is the scale of supersymmetry breaking. It is known from experiment that m susy 100 GeV. Correspondingly vac susy 10 8 GeV, i.e. 55 orders of magnitude larger than the permitted upper bound. In more advanced supersymmetric theories which include gravity (the so called supergravity or local supersymmetry) the condition of non-vanishing vacuum energy in the broken symmetry phase is not obligatory. However, if one does not take a special care, the value of vacuum energy in unbroken supergravity models is typically about m Pl 4 10 76 GeV. One can choose in principle the parameters in such a way that this contribution into ρvac is compensated down to zero with the accuracy 10 123 but this demands a fantastic fine-tuning. One more source of vacuum energy is the energy of the scalar (Higgs) field in the theories with spontaneous symmetry breaking.

1.3.New Model of "Nullification" of Vacuum Energy
Several possible approaches to the problem of vacuum energy have been discussed in the contemporary literature, for the review see ref. [5]. They can be roughly devided into four different groups: (1) Modification of gravity on large scales.
(4) Adjustment mechanism. (5) Hidden nonstandard matter sector and corresponding symmetry leading to ρ vac 0. 1.A modification of gravity at large scales should be done in such a way that the general covariance, which ensures vanishing of the graviton mass, is preserved, energy momentum tensor is covariantly conserved, and simultaneously the vacuum part of this tensor, which is proportional to g μ , does not gravitate. This is definitely not an easy thing to do. Possibly due to this reasons there is no satisfactory model of this kind at the present time.
2.Anthropic principle states that the conditions in the universe must be suitable for life, otherwise there would be no observer that could put a question why the universe is such and not another. With cosmological constant which is as large as predicted by natural estimates in quantum theory, life of our type is definitely impossible. Still this point of view does not look very appealing. The situation is similar to the one that existed in the Friedmann cosmology before inflationary resolution of the fundamental cosmological problems has been proposed. There is one more difficulty in the implimenttion of the anthropic principle. Even if we assume that it is effective, there are no visible building blocks to achieve the necessary compensation of vacuum energy. One can say of course that this compensation is not achieved by a physical field but just by a subtraction constant or in other words by a choice of the position of zero on the energy axis. In other words it is assumed that there is some energy coming from nowhere, which exactly cancels out all the contributions of different physical fields. Though formally this is not excluded, it definitely does not look beautiful. 3.Probably the most appealing would be a model based on a symmetry principle which forbids a nonzero vacuum energy. Such a symmetry should connect known fields with new unknown ones. Some of those fields should be very light to achieve the cancellation on the scale 10 3 eV. Neither such fields are observed, nor such a symmetry is known.
4. An adjustment mechanism seems the most promising one at the present time. The idea is similar to the mechanism of solving the problem of natural CPconservation in quantum chromodynamics by the axion field. The axion potential automatically acquires a minimum at the value of the field amplitude that cancels out the CP-odd contribution from the so called theta-term, θGG. Similar mechanism can hopefully kill vacuum energy. Let us assume that there is a very light or massless field coupled to gravity in such a way that it is unstable in De Sitter background and develops the condensate whose energy-momentum tensor is equal by magnitude and opposite by sign to the original vacuum energy-momentum tensor. Though it looks rather promising, it is very difficult, if possible at all, to construct a realistic model based on this idea. 5.Hidden nonstandard matter sector and corresponding symmetry leading to ρ vac 0.
The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark. Of those, roughly 3.6% are identified: cold gas and dust, neutrinos, and black holes. About 23% is dark matter, and the overwhelming majority is some type of gravitationally self-repulsive dark energy.
There is no candidate in the standard model of particle physics.In what way does dark matter extend the standard model? Remark 1.3.1.In order to explain physical nature of dark matter sector we assume that main part of dark matter,i.e., 23% 4. 6% 18% (see Fig.2.3.3) formed by supermassive ghost particles vith masess such that mc 2 . Remark 1.3.2.In order to obtain QFT description of the dark component of matter in natural way we expand now the standard model of particle physics on a sector of ghost particles. QFT in a ghost sector developed in Sect.3.1-3.4 and Sect.4.1-4.8.

The paper is organized as follows:
In Sec.2, classical Zel'dovich approuch [1] to cosmological constant problem revisited. In Sec.2.1, we summarize the aspects of the cosmological constant problem that are relevant to this work. In Sec.2.2, we summarize the model of cosmological dynamics in the presence of a vacuum energy that was introduced in [1][2][3][4], and how it attempts to resolve the problem.
In Sec.2.3, dark matter nature is considered. We argue that dark matter sector essentially formed only by super massive ghost particles.The Standard Model of fundamental interactions is extendent on a ghost sector. In Sec.3, Pauli-Villars ghosts as physical dark matter is considered. In Sec.3.1,Pauli-Villars renormalization of the 4 4 field theory by using Pauli-Villars host fields is considered.New physical interpretation of the scalar Pauli-Villars host fields is given. In Sec.3.2, Pauli-Villars renormalization of the the QED 3 by using Pauli-Villars ghost fields is considered. New physical interpretation is given. In Sec. 3 In Sec.4.6, the general structure of the -operation in a ghost fields sector via Colombeau generalized functions is given. In Sec.4.7, the renormalization Group in a ghost sector is considered. In Sec.4.8, dimensional regularization and the MS scheme in a ghost sector is given. In Sec.5, the higher-derivative-quantum-gravity is considered as physical quantum-gravity theory below high energy cutoff The renormalizable models of quantum-gravity which we have considered in this section, many years mistakenly regarded only as constructs for a study of the ultraviolet problem of quantum gravity. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Minkowski space-time. In canonical case they do have only some promise as phenomenological models.However, for their unphysical behavior may be restricted to arbitrarily large energy scales mentioned above by an appropriate limitation on the renormalized masses m 2 and m 0 .Actually, it is only the massive spin-two excitations of the field which give the trouble with unitarity and thus require a very large mass. The limit on the mass m 0 is determined only by the observational constraints on the static field. In Sec.6, Hausdorff-Colombeau measure and associated negative Hausdorff-Colombeau dimensions is considered successfully. In Sec.6.1, we provide fractional integration in negative dimensions on natural way via Colombeau generalized functions. In Sec.6.2, Using Hausdorff measure with associated positive Hausdorff dimension the rigorous definition of the Colombeau-Feynman path integral in D 4 from dimensional regularization is given. In Sec.6.3, we provide Hausdorff-Colombeau measure and associated negative Hausdorff-Colombeau dimensions. In Sec.6.4, we provide the main properties of the Hausdorff-Colombeau metric measures with associated negative Hausdorff-Colombeau dimensions. In Sec.7, we provide scalar quantum field theory in spacetime with Hausdorff-Colombeau negative dimensions. In Sec.7.1,the equation of motion and Hamiltonian in spacetime with Hausdorff-Colombeau negative dimensions is considerd. In Sec.7.2,propagator of a free scalar quantum field in configuration space with Hausdorff-Colombeau negative dimensions is considerd. In Sec.7.3,Green's functions corresponding to a self-interecting scalar quantum field in spacetime with Hausdorff-Colombeau negative dimensions is considerd. In Sec.7.4,saddle-point evaluation of the Colombeau-Feynman path integral corresponding to a self-interecting scalar quantum field in negative dimensions is considerd successfully. In Sec.7.5,an criteria of the power-counting renormalizability of P D _ scalar quantum field theory in negative dimensions D 0 is considerd successfully. In Sec.7.6,we have proved power-counting renormalizability of Einstein gravity in negative dimensions. In Sec.7.7,an criteria of thepower-counting renormalizability of Ho rava gravity in negative dimensions. In Sec.8,the solution cosmological constant problem is considerd successfully. In Sec.8.1,Zeropoint energy density corresponding to Einstein-Gliner-Zel'dovich vacuum with tiny Lorentz invariance violation is considerd. In Sec.8.2,Zeropoint energy density corresponding to a non-singular Gliner cosmology is considerd. In Sec.8.3, Zeropoint energy density in models with supermassive physical ghost fields is considerd. In Sec.9, we compare the classical and non classical assumptions that are made in the different formulation of the cosmological constant problem.
In Sec.9.1,we briefly review the canonical assumptions that are made in the usual formulation of the cosmological constant problem. In Sec.9.2,we list the modified assumptions that are made in this paper. In Sec.9.3, In Sec.9.4, In Sec.9.5, In Sec.9.6,semiclassical Möller-Rosenfeld gravity via aprouch proposed in this paper is considerd. We conclude that Moller-Rosenfeld equation holds again in a good approximation. In Sec.9.7,we briefly discussed higher-derivative quantum gravity at energy scale and corresponding controlable tiny violetion of the unitarity condition. We conclude with the physical significance of the new results in Sec.9-10.
2. Zel'dovich approach to cosmological constant problem by using Pauli-Villars regularization revisited.Ghost particles as physical dark matter.
2.1.The formulation of the cosmological constant problem.
The cosmological constant problem arises at the intersection between general relativity and quantum field theory, and is regarded as a fundamental unsolved problem in modern physics. Remind that a peculiar and truly quantum mechanical feature of the quantum fields is that they exhibit zero-point fluctuations everywhere in space, even in regions which are otherwise 'empty' (i.e.devoid of matter and radiation).This vacuum energy density is believed to act as a contribution to the cosmological constant appearing in Einstein's field equations from 1917, where R and R refer to the curvature of space-time, g is the metric, T the energymomentum tensor, where T is the energy-momentum tensor of matter. In modern cosmology it is assumed that the observable universe was initially vacuumlike, i.e., the cosmological medium was non-singular and Lorentz invariant. In the earlier, non-singular Friedmann cosmology the Friedmann universe comes into being during the phase transition of an initial vacuumlike state to the state of 'ordinary' matter [2], [3].
The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the cosmological principle; empirically, this is justified on scales larger than~100 Mpc. The cosmological principle implies that the metric of the universe must be of the form Robertson-Walker metric [2]. Robertson-Walker metric reads ds 2 dt 2 a 2 t dr 2 1 kr 2 r 2 d 2 sin 2 d 2 .
2. 1. 5 For such a metric, the Ricci curvature scalar is R 6k and it is said that space has the curvature k.The scaling factor a t rescales this curvature for a given time t, producing a curvature k t k/a t . The scaling factor a t is given by two independent Friedmann equations for modeling a homogeneous, isotropic universe reads where p is pressure and is a density of the cosmological medium. For the case of the vacuumlike cosmological medium equation of state reads [1], [2], [3], [4]: By virtue of Friedman's equations (2.1.6) in the Universe filled with a vacuum-like medium, the density of the medium is preserved, i.e.
const, but the scale factor a t grows exponentially. By virtue of continuity, it can be assumed that the admixture of a substance does not change the nature of the growth of the latter, and the density of the medium hardly changes. This growth, interpreted by analogy with the Friedmann models as an expansion of the universe, but almost without changing the density of the medium! -was named inflation. The idea of inflation is the basis of inflation scenarios [2].
Non-singular cosmology [2], [3], [4] suggests that the initial state of the observable universe was vacuum-like, but unstable with respect to the phase transition to the ordinary non-Lorentz-invariant medium. This, for example, takes place if, by virtue of the equations of state of the medium, a fluctuation decrease in its density d violates the condition of vacuum-like degeneration, p or, which is the same, 3p 2 0, replacing it with According to Friedman's equations, it corresponds to an accelerated expansion of the cosmological medium, accompanied by a drop in its density, which makes the process irreversible [2]. The impulse for expansion in this scenario, the vacuum-like environment, is not reported to itself (bloating), but to the emerging Friedmann environment.
In review [5], Weinberg indicates that the first published discussion of the contribution of quantum fluctuations to the cosmological constant was a 1967 paper by Zel'dovich [6].In his article [1] Zel'dovich emphasizes that zeropoint energies of particle physics theories cannot be ignored when gravitation is taken into account, and since he explicitly discusses the discrepancy between estimates of vacuum energy and observations, he is clearly pointing to a cosmological constant problem. As well known zeropoint energy density of scalar quantum field,etc.is divergent vac m 2 c 2 3 0 p 2 m 2 c 2 p 2 dp .

1. 10
In order avoid difficultnes mentioned above, in article [1] Zel'dovich has applied canonical Pauli-Villars regularization [7], [8] and formally has obtained an finite result (his formulas [1], Eqs. (VIII.12)-(VIII.13) p.228) where m (the ultra-violet cut-of ) is taken equal to the proton mass. Zel'dovich notes that since this estimate exceeds observational bounds by 46 orders of magnitude it is clear that "...such an estimate has nothing in common with reality". In his paper [1], Zel'dovich wroted:" Recently A.D. Sakharov proposed a theory of gravitation, or, more precisely, a justification GR equations based on consideration of vacuum fluctuations.In this theory, the essential assumption is that there is some elementary length L or the corresponding limiting momentum p 0 /L. Shorter lengths or for large impulses theory is not applicable. Sakharov gets the expression of gravitational constant G through L or p 0 (his formula [1],Eq.(IX.6)) This expression has been known since the days of Planck, but it was read "from right to left": gravity determines the length L and the momentum p 0 . According to Sakharov, L and p 0 are primary. Substitute Eq.(IX. 6) in the expression Eq.(IX.4) (see [1]), we get That is expressions that the first members (in the formulas [1],Eqs.(VIII.10)-(VIII. 11)) which are vanishes (with p 0 ).Thus, we can suggest the following interpretation of the cosmological constant: there is a theory of elementary particles, which would give (according to the mechanism that has not been revealed at the present time) identically zero vacuum energy, if this theory were applicable infinitely, up to arbitrarily large momentum; there is a momentum p 0 , beyond which the theory is nont aplicable; along with other implications, modifying the theory gives different from zero vacuum energy; general considerations make it likely that the effect is portional p 0 2 .Clarification of the question of the existence and magnitude of the cosmological constant will also be of fundamental importance for the theory of elementary particles". In contrast with Zel'dovich paper [1] we assume that Poincaré group is deformed at some fundamental high-energy cutoff [9], [10], [11] in accordance on the basis of the following deformed Poisson brackets where μ, , 0, 1, 2, 3, 1, 1, 1, 1 and is a parameter identified as the ratio between the high-energy cutoff and the light speed. The corresponding to (2.1.16) momentum transformation reads [11] and coordinate transformation reads [11] t t ux/c 2 It is easy to check that the energy E c , identified as the high-energy cutoff , is an invariant as it is also the case for the fundamental length l c/E / . Remark 2.1.2. Note that the transformation (2.1.17) defined in p-space and the transformation (2.1.18) defined in x-space becomes Lorentz for small energies and momenta and defines a large invariant energy l 1 . The high-energy cutoff is preserved by the modified action of the Lorentz group [9], [10].
This meant that the canonical concept of metric as quadratic invariant collapses at high energies, being replaced by the non-quadratic invariant [9]: or by the non-quadratic invariant where l 1 , a, b 0, 1, 2, 3. x , x i n 0 N n с n c n S x x , PV,n 2 .

43
Assume now that   18.In order to avoid these difficultnes mentioned above we assume that (i) physics of elementary particles is separated into low/high energy ones, (ii) the standard notion of smooth spacetime is assumed to be altered at a high energy cutoff scale and a new treatment based on QFT in a fractal spacetime with negative dimension is used above that scale (iii) instead Eqs.(2.2.54) we assume now that where is a high-energy cutoff [5]. In order to avoid these difficulties we apply instead Einstein-Hilbert action (2.2.55) the gravitational action which include terms quadratic in the curvature tensor  [13]. The requirement that the graviton propagator behave like p 4 for large momenta makes it necessary to choose the indefinite-metric vector space over the negative-energy states.These negative-norm states cannot be excluded from the physical sector of the vector space without destroying the unitarity of the S matrix, however, for their unphysical behavior may be restricted to arbitrarily large energy scales by an appropriate limitation on the renormalized masses m 2 and m 0 . Remark 2.2.21.We assum that m 0 c eff , m 2 c eff . Remark 2.2.22.The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff, see for example [14]. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions [15] gives high-energy cutoff on natural way.
2.3.Dark matter nature. A common origin of the dark energy and dark matter phenomena.
Dark matter is a hypothetical form of matter that is thought to account for approximately 85% of the matter in the universe, and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet undiscovered subatomic particles.Its presence is implied in a variety of astrophysical observations, including gravitational effects that cannot be explained unless more matter is present than can be seen. For this reason, most experts think dark matter to be ubiquitous in the universe and to have had a strong influence on its structure and evolution. Dark matter is called dark because it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible to the entire electromagnetic spectrum, making it extremely difficult to detect using usual astronomical equipment  Telescope in Chile, suggests that dark matter may be less dense and more smoothly distributed throughout space than previously thought. An international team used data from the Kilo Degree Survey (KiDS) to study how the light from about 15 million distant galaxies was affected by the gravitational influence of matter on the largest scales in the Universe. The results appear to be in disagreement with earlier results from the Planck satellite.
This map of dark matter in the Universe was obtained from data from the KiDS survey, using the VLT Survey Telescope at ESO's Paranal Observatory in Chile. It reveals an expansive web of dense (light) and empty (dark) regions. This image is one out of five patches of the sky observed by KiDS. Here the invisible dark matter is seen rendered in pink, covering an area of sky around 420 times the size of the full moon. This image reconstruction was made by analysing the light collected from over three million distant galaxies more than 6 billion light-years away. The observed galaxy images were warped by the gravitational pull of dark matter as the light travelled through the Universe. Some small dark regions, with sharp boundaries, appear in this image. They are the locations of bright stars and other nearby objects that get in the way of the observations of more distant galaxies and are hence masked out in these maps as no weak-lensing signal can be measured in these areas [16][17].  The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark. Of those, roughly 3.6% are identified: cold gas and dust, neutrinos, and black holes. About 23% is dark matter, and the overwhelming majority is some type of gravitationally self-repulsive dark energy.  Let's remind that in the Standard Model (SM) of fundamental interactions besides the gauge interactions and the quartic interaction of the Higgs fields there are also Yukawa type interactions of the fermion fields with the Higgs field. These interactions are also renormalizable and is characterized by the Yukawa coupling constants, one for each fermion field. The peculiarity of the SM is that the masses of the fields appear as a result of spontaneous symmetry breaking when the Higgs field develops a vacuum expectation value. As a result the masses are not independent but are expressed via the coupling constant multiplied by the vacuum expectation value. Another property of the Standard Model is that it has the gauge group SU c 3 SU L 2 U Y 1 which is spontaneously broken to SU c 3 U EM 1 . In the theories with spontaneously broken symmetry, according to the Goldstone theorem there are massless particles, the goldstone bosons.
where the following notation for the covariant derivatives is used

3. 3
The Yukawa part of the Lagrangian which is needed for the generation of the quark and lepton masses is also chosen in the gauge invariant form and contains arbitrary Yukawa couplings (we ignore the neutrino masses, for simplicity) Here there are two arbitrary parameters: m 2 and . The ghost fields and the gauge fixing terms are omitted.The Lagrangian of the SM contains the following set of free parameters: (1) 3 gauge couplings g s , g, g , (2) 3 Yukawa matrices y L , y D , y U , The mass matrices have to be diagonalized to get the quark and lepton masses. The explicit mass terms in the Lagrangian are forbidden because they are not SU left 2 symmetric. They would destroy the gauge invariance and, hence, the renormalizability of the Standard Model. To preserve the gauge invariance we use the mechanism of spontaneous symmetry breaking which, as was explained above, allows one to get the renormalizable theory with massive fields. The Feynman rules in the SM include the ones for QED and QCD with additional new vertices corresponding to the SU 2 group and the Yukawa interaction, as well as the vertices with goldstone particles if one works in the renormalizable gauge. We will not write them down due to their complexity, though the general form is obvious.
Let us consider now the one-loop divergent diagrams in the SM. Besides the familiar diagrams in QED and QCD discussed below in section IV.5 one has the diagrams presented in Fig.2  where, for simplicity, we ignored the mixing between the generations and assumed the Since the masses of all the particles are equal to the product of the gauge or Yukawa couplings and the vacuum expectation value of the Higgs field, in the minimal subtraction scheme the mass ratios are renormalized the same way as the ratio of couplings. To find the renormalization of the mass itself, one should know how the v.e.v. is renormalized or find explicitly the mass counter-term from Feynman diagrams. In this case, one has also to take into account the tad-pole diagrams shown in Fig.2

3. 13
The result for the t-quark can be obtained by replacing b by t.  In the extendended Standard Model of fundamental interactions besides the gauge interactions and the quartic interaction of the Higgs ghost fields there are also Yukawa type interactions of the fermion ghost fields with the Higgs ghost field. These interactions are also renormalizable and is characterized by the Yukawa coupling constants, one for each fermion field. Another property of the Standard Model is that it has the gauge group SU c 3 SU L 2 U Y 1 which is spontaneously broken to SU c 3 U EM 1 . In the theories with spontaneously broken symmetry, according to the Goldstone theorem there are massless particles, the goldstone ghost bosons.
The Lagrangian of the extendended Standard Model in a ghost sector consists of the following three parts: The gauge part is totally fixed by the requirement of the gauge invariance leaving only the values of the couplings as free parameters . . , H Bare and where the following notations for the covariant derivatives are used

3. 19
The Yukawa part of the Lagrangian in a ghost sector, which is needed for the generation of the ghost quark and ghost lepton masses is also chosen in the gauge invariant form and contains arbitrary Yukawa couplings (we ignore the neutrino masses, for simplicity) where H i 2 H .At last the ghost Higgs part of the Lagrangian contains the Higgs potential which is chosen in such a way that the ghost Higgs field acquires the vacuum expectation value and the potential itself is stable Here there are two bare parameters m 2 , .
The Lagrangian of the SM in a ghost sector contains the following set of the bare parameters: As a result, the gauge group is spontaneously broken down to The etc.,

3. 29
for some 0, 1 . Remind that vacuum energy density for free scalar quantum field with a wrong statistic is:

3. 41
Definition 2.3.2.We will call the function f as a full continuous Pauli-Villars masses distribution. Definition 2.3.3.We define now: and we will call it as discrete Pauli-Villars masses distribution of the bosonic ghost matter and and we will call it as discrete Pauli-Villars masses distribution of the fermionic ghost matter. Remark 2.3.9.We rewrite now the Eqs.(2.3.36)-(2.3.37) in the following equivalent form where j i i M, i 1 1, 2, . . . , M. Remark 2.3.10.We assume now that: where obviously 0.

1. 31
obviously breaks down (see Remark 2.2.8) and therefore the Eq.(3.1.32) is not holds

1. 32
But therefore the Eq.(3.1.29) also is not holds and Pauli-Villars procedure completely breaks down.  3 . What is wrong with Pauli-Villars renormalization of QED 3 . New physical interpretation Pauli-Villars ghost fields. 3 . What is wrong with Pauli-Villars renormalization of QED 3 .

3.2.1.Pauli-Villars renormalization of QED
Let us consider now the conditions that must be required on the masses and coupling constants of the regulator fields such that a regularized closed fermion loop in 2 1 dimensions is rendered finite in the calculations.Thus corresponding Feynman integral is proportional to so,for large |p|,its integrand behaves like |p| n , whereas for n 4 the integral (3.2.1) diverges as 0 p 2 dp p n~ 0 dp p n _ 2 . We apply now the momentum -cutoff regularization |p| and have to replaced ill defined formal expression (3.2.1) by the well defined Integral

2. 2
Remark 3.2.1.Note that -cutoff regularization meant lattice QED 3 on a lattice at length scales a 1 . However quantity behaves as p n 2 dp and therefore gives unphysical result which strictly depends on parametr .

Remark 3.2.2.
In order obtain physical result we apply now Pauli-Villars renormalization.
The integrand p, m in (3.2.2) can be written as p, m k m k a n k p n k .

2. 3
Therefore, in making the canonical substitution p, m i 0 and where n s is the number of auxiliary spinor fields, we must impose in the vacuum polarization case (n 2) the following conditions in order to eliminate the linear and logarithmic divergences, respectively. Let us calculate of the vacuum polarization tensor in spinor QED 3 . In the standard notation, the regularized expression for the vacuum polarization tensor reads We choose now both the electron mass and that of the auxiliary field M 1 to be positive. Using now the canonical Feynman parametrization M i

2. 9
If we carry out the momentum integrations in Eqs.(3.2.9), it is straightforward to arrive at 3 M k 2 0, as expected by gauge invariance.Let us take now c 1 1, c 2 , c j 0 , j 2, where the parameter can assume any real value except zero and unity, so that condition given by Eq.

2. 11
Note that for M 1 , M 2 m from Eq.

2. 21
so that the integrand p, k 1 , . . . , k n 1 , m i, behaves like p, k 1 , . . . , k n 1 , m i, P n p P 2n p m i, P n 1 p P 2n p m i, 2 P n p P 2n p P n 2 p P n p P 2n 2 p P 2n p m i, 3 P n 1 p P 2n p P n 3 p P n 1 p P 2n 2 p P 2n p k m i, k p n k .

2. 22
Therefore, in making the substitution in Colombeau integral (3.2.18) where m 0, m for any 0, 1 and where n s is the number of auxiliary spinor fields, we obtain

2. 24
where in order to eliminate the infinite large Colombeau generalized quantities ln 1 , 1 , . . . , q , respectively from Colombeau integral (3.2.25),i.e., make it finite in canonical sense. Using now the canonical Feynman parametrization in the following Colombeau form

2. 27
and performing the momentum shift p

2. 29
Carry out the momentum integrations in Eqs. (3.2.29), it is straightforward to arrive at 3 M k 2 , 0, as expected by gauge invariance.Let us take now c 1 1, c 2 , c j 0 , j 2, where the parameter can assume any real value except zero and unity, so that condition given by Eq.
The standard approach to regularization of non-Abelian gauge theories is dimensional regularization but this of course is inherently perturbative [27]. However, the ordinary PV renormalization of non-Abelian theories fails. Gauge invariance is violated, is blocking any hope of BRST invariance, which confounds proofs of renormalizability. Remark 3.3.1.Note that the existence of interesting non-perturbative phenomena in gauge theories requires the introduction of a non-perturbative regularization. Discretization of space-time leads in a natural way to lattice regularizations which preserve gauge invariance and have a non-perturbative meaning. The construction of a non-perturbative gauge invariant regularization of gauge theories in a continuum space-time has been a challenging problem in gauge theories. A natural candidate has always been a gauge invariant generalization of Pauli-Villars methods involving high derivatives in the action.
Recall that the euclidean action of Yang-Mills theory is given by Recall that the high covariant derivatives method proposed in papers [28]- [30] proceeds by two steps. The Yang-Mills action is replaced by its regularized version f bc a A c and λ is an arbitrary real constant.Then the partition function for the regularized action in α 0 -gauge reads In this way, provided n 2, all 1PI diagrams with more than one loop acquire a negative degree of divergence by power counting.However, the degree of divergence of one-loop 1PI diagrams is unchanged by the addition of covariant derivatives.Therefore the theory is not completely regularized by the simple fact of adding higher covariant derivatives to the action as for the case of scalar field theories,however, that due to the regular behaviour of the gluonic propagator the contributions in the effective action to the ghost two point function and gluon-ghost vertex are finite at one loop. This implies that one loop divergences exclusively arise in diagrams with only external gluon lines, and are given by the following product of determinants Note that the that all the determinants in (3.3.6) are explicitly gauge invariant. This fact can be understood as a consequence of the absence of divergent radiative corrections to the interaction of Faddeev-Popov ghost fields, which also implies that the BRST symmetry is only renormalized by finite radiative corrections. Remark 3.3.3.Note that gauge invariance is not lost when we add mass terms in (3.3.6).
Then, it seems natural to use these determinants as the Pauli-Villars counterterms that subtract divergences at one loop in a gauge invariant way. This is the Slavnov approach introduced in Ref.

10
is, then, free of divergences at one loop provided the s j parameters are chosen so that Remark 2.2.4.Note that Pauli-Villars conditions do not involve the masses as it is usually the case. This is due to gauge invariance and the high derivative terms in the action that make finite the terms depending on m.
The problem is that Pauli-Villars determinants det 1/2 m do not converge formally to a constant, as they should, when the cutoff is removed. In fact, we have that [31] lim det 1/2 that depends on A through the delta functional D q x . These difficulties mentioned above has been resolved in paper [31].
The regularization method of Pauli-Villars (PV) subtraction is of long standing in quantum field theory.In the more common dimensional regularization the properties of for instance the Dirac algebra are dimension dependent (and in particular the treatment of γ 5 is not unambiguous), and hence problems may arise in the study of chiral phenomena.
Recall that Pauli-Villars regularization requires that for each particle of mass m a new ghost particle of mass M PV be added with either the wrong statistics or the wrong-sign kinetic term. These new particles are designed to cancel exactly loop amplitudes with physical particles at asymptotically large loop momentum. For example, one can write down a Pauli-Villars Lagrangian for QED 4 , which works at the 1-loop level, as ren PV with A gh the ghost photon and gh the ghost electron and F gh A gh A gh . We assume that both the ghost photon and ghost electron have bosonic statistics; the ghost photon has a wrong-sign kinetic term.For example, ren PV leads to a Feynman-gauge ghost-photon propagator of the form

Remark 3.4.1.(i)
Since this has the opposite sign from the photon propagator, it will cancel the photon's contribution, for example, to the electron self-energy loop for loop momenta k M. The ghost electron propagator is the same as the regular electron propagator; however, ghost electron loops do not get a factor of 1 (since they are bosonic) and therefore cancel regular electron loops when k M PV . (ii) At the end of the calculation the limit M PV is implied. For example from Eq. The sign of the residue of the propagator is normally dictated by unitaritya particle whose propagator has the sign in Eq.(2.3.2) has negative norm, and would generate probabilities greater than 1. So, A gh cannot create or destroy physical on-shell particles. Thus, fields such as A gh are said to be associated with Pauli-Villars ghosts. Remark 3.4.3.Indeed, the introduction of Pauli-Villars ghosts is much more clearly a deformation in the UV, relevant at energy scales of order the Pauli-Villars mass M PV or larger, than analytically continuing to 4 dimensions. Remark 3.4.4.In order to avoid difficulties with unitarity mentioned above, we assume that:(i) physics of elementary particles is separated into low/high energy ones, (ii) the standard notion of smooth spacetime is assumed to be altered at a high energy cutoff scale and a new treatment based on QFT in a fractal spacetime with negative dimension is used above that energy scale M PV m (iii) at the end of the calculation the limit M PV is not implied.For example instead Eq.
So that the integrand in (3.4.7) behaves like |k| 3 the integral (3.4.7) diverges. Remark 3.4.5.In fact, the causal Green function (3.4.8) of the QED 4 is the classical Schwartz distribution which is defined on a test smooth functions. It has the δ-function like singularities and needs an additional definition for the product of several such functions at a single point.The discussed above diagram (see Fig.3.4.1) is precisely this product. Remark 3.4.6.In his handbook [8] N. N. Bogoliubov argue that a problem of the ultraviolet divergences arises exactly from the Schwartz Impossibility Theorem [32].In fact N. N.Bogoliubov argue that these problems has only purely mathematical nature.However this Bogoliubov statement completely wrong but holds from Bogoliubov time until nowodays. Remark 3.4.7.In particular N. N. Bogoliubov wroted [8]: "We thus see that the purely formal rules for dealing with products of causal functions, which we adopted earlier, lead to a meaningless result in this case.This is essentially a manifestation of the fact that we did not define the product of singular functions as an integrable singular function. In order to solve the problem of determining the coefficients of the chronological product T x 1 x 2 as integrable improper functions, we use the method of transition to the limit similar to that used in §18 (see [8], §18). In order to do this, we first consider an auxiliary fictitious case in which the field operator functions satisfy commutation relations in which the causal c -functions are replaced by reg c ." Remark 3.4.8.Recall that classical Schwartz distribution is defined as linear functionals on a test smooth functions [32]. Schwartz distributions may be multiplied by real numbers and added together, so they form a real vector space. Schwartz distributions may also be multiplied by infinitely differentiable functions, but it is not possible to define a product of general distributions that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by Schwartz (1954), and is usually referred to as the Schwartz Impossibility Theorem [32]. Remark 3.4.9.Note that:(i) by using linear homomorphism (a) D c S c x 2 i adapted to the Lorentz invariance of the Schwartz distributions D c 0 x 2 and S c x 2 we can embed these distributions into Colombeau algebra G x 4 :

4. 11
Therefore term of the scattering matrix corresponding to the diagram of Fig.3

4. 15
where M PV is arbitraly large but finite Pauli-Villars mass. Remark 3.4.11.Note that in contrast with canonical formal approuch [8], [33][34][35][36]  The most popular in gauge theories is the so-called dimensional regularization. In this case, one modifies the integration measure: d 4 q 2 d 4 2 q where μ is a parameter of dimensional regularization with dimension of a mass.In this case, all the ultraviolet and infrared singularities manifest themselves as pole terms in .Consider the earlier discussed example see Fig.3.1.3 and using the Euclidean representation rewrite it formally in D-dimensional space where we assume that the dimension D is such that the integral exists. In this case this is 2 and 3. The main formula

1. 5
We see that the classical ultraviolet divergence now takes the rigorous mathematical form of the infinite large Colombeau generalized number 1 , see [23].
We present below the main Colombeau integrals needed for the one-loop calculations. They can be obtained via the analytical continuation from the integer values of D. We will write them down directly in the pseudo-Euclidean space. First note that

2. 2
The fact that the Colombeau integral (4.2.2) well defined but contains infinite Colombeau generalized number 1 .

The vertex:
Here one also has only one diagram but the external momenta can be adjusted in several ways (see Fig.4.2.2). As a result the total contribution to the vertex function consists of three parts I s, t, u I 1 s I 1 t I 1 u , where we introduced the commonly accepted notation for the Mandelstam variables (we assume here that the momenta p 1 and p 2 are incoming and the momenta p 3 and p 4 are outgoing) s p 1 p 2 2 p 3 p 4 2 , t p 1 p 3 2 p 2 p 4 2 , u p 1 p 4 2 p 2 p 3 2 , and the integral equals which holds in arbitrary noncritical dimension D and any power of the propagator as follows [32]: In the case of the integral (4.

2. 6
The four-point vertex in the one-loop approximation reads These counter-terms correspond to additional vertices shown in Fig.4 correspondingly.Notice that the obtained expressions have no Colombeau infinities but contain the dependence on the regularization parameter 2 which was absent in the initial theory. The appearance of this dependence on a dimensional parameter is inherent in any regularization and is called the dimensional transmutation,i.e., an appearance of a new scale in a theory. We write the counter-term in the following way Writing the "bare" Lagrangian in the same form as the initial one but in terms of the "bare" fields and couplings  The corresponding Colombeau integral reads One has to transform each of the propagators into coordinate space, multiply them and transform back to momentum space. This reduces to writing down the corresponding transformation factors. Thus where the Euler constant and ln 4 are omitted. The appeared ultraviolet divergence, the pole in , can be removed via the introduction of the (quasi) local Colombeau counter-term where the wave function renormalization constant Z 2, in the MS scheme is obtained by taking the infinite large part of the Colombeau integral with the opposite sign

3. 4
After that the propagator in the massless case reads

3. 5
The vertex: In the given order there are two diagrams (remind that in the massless case the tad-poles equal to zero) shown in Fig.4.3.2.

3. 6
In the same order of 3 one gets additional diagrams presented in Fig.4.3.3.

3. 7
Notice that after the subtractions of subgraphs the Colombeau singular part is local, i.e. in momentum space does not contain ln p 2 . The terms with the single pole 1/ are absent since the diagram can be factorized into two diagrams of the lower order. The contribution of a given diagram to the vertex function equals

3. 9
The second diagram with the crossed terms contains 6 different cases. Consider one of them. Since we are interested here in the singular parts contributing to the renormalization constants, we perform some simplification of the original integral. We use a very important property of the minimal subtraction scheme that the renormalization constants depend only on dimensionless coupling constants and do not depend on the masses and the choice of external momenta. Therefore, we put all the masses equal to zero, and to avoid artificial infrared divergences, we also put equal to zero one of the external momenta. Then the diagram becomes the propagator type one: (1/48 is the combinatorial coefficient). Since putting one of the momenta equal to zero we reduced the diagram to the propagator type, we can again use the advocated method to calculate the massless integral. Therefore As one can see, in this case we again have the second order pole in and, accordingly, the single pole with the logarithm of momentum. The reason of their appearance is the presence of the divergent subgraph. Here we again have to look at the counter-terms of the previous order which eliminate the divergence from the one-loop subgraph. The subtraction of divergent subgraphs (the R-operation without the last subtraction) looks like and hence the radiative corrections finite. In the case of nonzero mass, one should also add the mass counter-term.  [32]. Schwartz distributions may be multiplied by real numbers and added together, so they form a real vector space. Schwartz distributions may also be multiplied by infinitely differentiable functions, but it is not possible to define a product of general distributions that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by Schwartz (1954), and is usually referred to as the Schwartz Impossibility Theorem [32]. Remark 4.3.5.In coordinate space the large values of momenta correspond to the small distances. Hence, the ultraviolet divergences allow for the singularities at small distances.
Indeed, the simplest divergent loop diagram (Fig.3.1.3) in coordinate space is the product of two propagators. Each Euclidean propagator x r D 4 , x 4 , r x is uniquely defined in momentum as well as in coordinate space, but the square of the propagator has already an ill-defined Fourier transform, it is ultraviolet divergent. The reason is that the square of the propagator is singular as r 2 0 and behaves like r 2 1/r 4 .

3. 29
In fact, the causal Green function of the QFT is the classical Schwartz distribution which is defined on a test smooth functions. It has the δ-function like singularities and needs an additional definition for the product of several such functions at a single point.The discussed above diagram (see Fig.3 (see [21][22]) of the square of the propagator well defined and C 2 r 2 ; G p 4 . there is no any problem arises from the Schwartz Impossibility Theorem, (ii) classical ill-defined ultraviolet divergences replased by well defined infinite large Colombeau generalized numbers. Remark 4.3.10. Thus by using Colombeau algebra G x 4 (see [21][22]) of the Colombeau generalized functions instead classical Schwartz distribution and Colombeau generalized numbers (see [21]- [22], [23]- [26]) there is no any problem arises from the Schwartz Impossibility Theorem.

4.4.Quantum electrodynamics in a ghost sector via Colombeau generalized functions.
Let us consider now the calculation of the diagrams in the gauge theories. We start with quantum electrodynamics. The QED 4 Lagrangian has the form where the electromagnetic stress tensor is F A A , and the last term in   where the "-" sign comes from the fermion loop and q q . We set now D 4 2 , 0, 1 . where p 2 ; is given by eq.(4.4.7). Notice that the radiative corrections are always proportional to the transverse tensor P g p p /p 2 . This is a consequence of the gauge invariance and follows from the Ward identities Let us consider now the electron self-energy graph Fig.4.4.2.b). The corresponding formal expression is

14
The integral over k is straightforward and reads

15
As one can see, the first integral is finite Colombeau quantity and the second one is logarithmically divergent. Expanding in series in , 0, 1 we obtain

16
Quantum electrodynamics (4.4.1) is a renormalizable theory; hence, the Colombeau counter-terms repeat the structure of the Lagrangian. They can be written as

17
The term that fixes the gauge is not renormalized. In the leading order of perturbation theory we calculated the corresponding diagrams with the help of dimensional regularization mentioned above. Their infinite large Colombeau parts with the opposite sign give the proper Colombeau renormalization constants. They are, respectively,

25
The gauge invariance connects the vertex Green function and the fermion propagator (the Ward identity), which leads to the identity Z 1, Z 2, .

4.5.Quantum chromodynamics in a ghost sector via Colombeau generalized functions.
Consider now the non-Abelian gauge theories and, in particular, QCD. The Lagrangian of QCD has the form where the stress tensor of the gauge field is now F a A a A a gf abc A b A c and the last terms represent the Faddeev-Popov ghosts. The complications which appear in non-Abelian theories are caused by the presence of many vertices with the same coupling as it follows from the gauge invariance. Hence, they have to renormalize the same way, i.e there appear new identities, called the Slavnov-Taylor identities. The full set of the counter-terms in QCD are

5. 4
This results in the relations between the renormalized and the bare fields and couplings

5. 4
The last line of equalities follows from the requirement of identical renormalization of the coupling in various vertices and represents the Slavnov-Taylor identities for the singular parts.The explicit form of the renormalization constants in the lowest approximation follows from the one-loop diagrams of QCD. Aa usual, one has to take the singular part with the opposite sign. For instance one has in the MS scheme:

5. 9
The relations between the renormalized and the bare fields and couplings reads
The structure of the counter-terms as functions of the field operators depends on the type of a theory. According to the canonical classification [8], [20],the QFT theories are divided into three classes: superrenormalizable (a finite number of divergent diagrams), renormalizable (a finite number of types of divergent diagrams) and non-renormalizable (a infinite number of types of divergent diagrams). Accordingly, in the first case one has a finite number of counter-terms; in the second case, a infinite number of counter-terms but they repeat the structure of the initial Lagrangian, and in the last case, one has an infinite number of structures with an increasing number of the fields and derivatives. Remark 5.6.1.In the case of renormalizable and superrenormalizable theories, since the Colombeau counter-terms repeat the structure of the initial Lagrangian, the result of the introduction of counter-terms can be represented as i.e., L Bare is the same Lagrangian L but with the fields, masses and coupling constants being the "bare" ones related to the renormalized quantities by the multiplicative equalities. . In some cases the renormalization can be nondiagonal and the renormalization constants become matrices.The renormalization constants are not unique and depend on the renormalization scheme. This arbitrariness, however, does not influence the observables expressed through the renormalized quantities. We will come back to this problem later when discussing the group of renormalization. In the gauge theories Z i, may depend on the choice of the gauge though in the minimal subtraction scheme the renormalizations of the masses and the couplings are gauge invariant.In the minimal schemes the renormalization constants do not depend on dimensional parameters like masses and do not depend on the arrangement of external momenta in the diagrams. This property allows one to simplify the calculation of the counter-terms putting the masses and some external momenta to zero, as it was exemplified above by calculation of the two-loop diagrams. In making this trick, however, one has to be careful not to create artificial infrared divergences. Since in dimensional regularization they also have the form of poles in , this may lead to the wrong answers.In renormalizable theory the finite Green function is obtained from the "bare" one, i.e., is calculated from the "bare" Lagrangian by multiplication on the corresponding Colombeau renormalization constant p 2 , 2 , g , Z , 1/ , g , Bare p 2 , 1/ , g Bare , 4. 6. 3 where in the n-th order of perturbation theory the "bare" parameters in the RHS of the Eq.(4.6.3) have to be expressed in terms of the renormalized ones with the help of relations (4.6.2.a) or (4.6.2.b) taken in the (n 1)-th order. The remaining constant Z , creates the counter-term of the n-th order of the form L Z , 1 O , , where the Colombeau generalized operator O , reflects the corresponding Green function. If the Green function is finite by itself (for instance, has many legs), then one has to remove the divergences only in the subgraphs and the corresponding renormalization Colombeau constant Z , 1. Note that since the propagator is inverse to the operator quadratic in fields in the Lagrangian, the renormalization of the propagator is also inverse to the renormalization of the 1-particle irreducible two-point correspondingly. The propagator renormalization constant is also the renormalization constant of the corresponding field, but the fields themselves, contrary to the masses and couplings, do not enter into the expressions for observables.
We would like to stress once more that the R-operation works independently on the fact renormalizable or non-renormalizable the theory is. In local theory the counter-terms are local anyway. But only in renormalizable theory the counter-terms are reduced to the multiplicative renormalization of the finite number of fields and parameters.
One can perform the R-operation for each diagram separately. For this purpose one has first of all to subtract the divergences in the subgraphs and then subtract the divergence in the diagram itself which has to be local. This serves as a good test that the divergences in the subgraphs are subtracted correctly. In this case the R-operation can be symbolically written in a factorized form where G is the initial diagram, M is the -subtraction operator (for instance, subtraction of the -singular part of the regularized diagram) and the product goes over all divergent subgraphs including the diagram itself. By a subgraph we mean here the 1-particle irreducible diagram consisting of the vertices and lines of the diagram which is UV divergent. The 1-particle irreducible is called the diagram which can not be made disconnected by deleting of one line.
We have demonstrated above the application of the R-operation to the two-loop diagrams in a scalar theory. Consider some other examples of diagrams with larger number of loops shown in Fig.4.6.1. They appear in the 4 4 theory in the three-loop approximation. In order to perform the R-operation for these diagrams one first has to find out the divergent subgraphs. They are shown in Fig.4.6.2. Let us use the factorized representation of the R-operation in the form of (4.6.5). For the three chosen diagrams one has, respectively, Here, as before, the graph surrounded with the dashed circle means its singular part and the remaining graph is obtained by shrinking the singular subgraph to a point. Let us demonstrate how the R -operation works for the diagram Fig.4.6.1a). Since the result of the R -operation does not depend on external momenta, we put two momenta on the diagonal to be equal to zero so that the integral takes the propagator form. Then we can use the method based on Fourier-transform, as it was explained above. One has Combining all together we get Note the cancellation of all nonlocal contributions. The singular part after the R -operation is always local.
The realization of the R -operation for each diagram G allows one to find the contribution of a given diagram to the corresponding counter-term and, in the case of a renormalizable theory, to find the renormalization constant equal to where K means the -extraction of the -singular part. Adding the contribution of various diagrams we get the resulting counter-term of a given order and, accordingly, the renormalization constant.

4.7.Renormalization Group in a ghost sector.
The procedure formulated above allows one to eliminate the ultraviolet divergences and get the finite expression for any Green function in any local quantum field theory. In renormalizable theories this procedure is reduced to the multiplicative renormalization of parameters (masses and couplings) and multiplication of the Green function by its own renormalization constant. This is true for any regularization and subtraction scheme. Thus, for example, in the canonical cutoff regularization and dimensional regularization via Colombeau generalized functions the relation between the "bare" and renormalized Green functions for standard matter sector looks like The invariant charge , being RG-invariant, obeys the RG equation without the anomalous dimension and plays an important role in the formulation of the renormalization group together with the effective charge. In some cases, for instance in the MOM subtraction scheme, the effective and invariant charges coincide.
The usefulness of solution (4.7.16) is that it allows one to sum up an infinite series of logs coming from the Feynman diagrams in the infrared (t ) or ultraviolet (t ) regime and improve the usual perturbation theory expansions. This in its turn extends the applicability of perturbation theory and allows one to study the infrared or the ultraviolet asymptotics of the Green functions.
To demonstrate the power of the RG, let us consider the invariant charge in a theory with a single coupling and restrict ourselves to the massless case. Let the perturbative expansion be

7. 22
The function in the one-loop approximation is given by

7. 23
Notice that the coefficient b of the logarithm in Eq.(4.7.22) coincides with that of the function. Alternatively the function can be defined as the derivative of the invariant charge with respect to logarithm of momentum g p 2 d dp 2 p 2 2 , g p 2 2

7. 24
This definition is useful in the MOM scheme where the mass is not considered as a coupling but as a parameter and the renormalization constants depend on it. We will come back to the discussion of this question below when considering different definitions of the mass. According to Eq.(4.7.16) (with vanishing anomalous dimension) the RG-improved expression for the invariant charge corresponding to the perturbative expression (4.7.22) is: where we have put in eq.(4.7.16) p 2 2 and then replaced t by t ln p 2 / 2 . The effective coupling is a solution of the characteristic equation The solution of this equation is g t, g g 1 bt g .

7. 27
Being expanded over t, the geometrical progression (4.7.27) reproduces the expansion (4.7.22); however, it sums the infinite series of terms of the form g n t n . This is called the leading log approximation (LLA) in QFT. To get the correction to the LLA, one has to consider the next term in the expansion of the function. Then one can sum up the next series of terms of the form g n t n 1 which is called the next to leading log approximation (NLLA), etc. This procedure allows one to describe the leading asymptotics of the Green functions for t .Let us consider now the Green function with non-zero anomalous dimension. Let its perturbative expansion be . . .

7. 28
Then in the one-loop approximation the anomalous dimension is g c g .

7. 29
Again the coefficient of the logarithm coincides with that of the anomalous dimension. In analogy with Eq.(4.7.24) the anomalous dimension can be defined as a derivative with respect to the logarithm of momentum g p 2 d dp 2 ln

7. 31
This gives for the Green function the improved expression 1 ct . . .

7. 32
Thus, one again reproduces the perturbative expansion, but expression (4.7.32) again contains the whole infinite sum of the leading logs. To get the NLLA, one has to take into account the next term in eq.(4.7.29) together with the next term of expansion of the function. All the formulas can be easily generalized to the case of multiple couplings and masses.
The effective coupling in a ghost sector By virtue of the central role played by the effective coupling in RG formulas, consider it in more detail. The behaviour of the effective coupling is determined by the function. Qualitatively, the function can exhibit the behaviour shown in Fig.4.7.1. We restrict ourselves to the region of small couplings.In the first case, the -function is positive. Hence, with increasing momentum the effective coupling unboundedly increases. This situation is typical of most of the models of QFT in standard matter sector in the one-loop approximation when g b g 2 and b 0. The solution of the RG equation for the effective coupling in this case has the form of a geometric progression (4.7.27). In the second case, the -function is negative and, hence, the effective coupling decreases with increasing momentum. This situation appears in the one-loop approximation when b 0, which takes place in the gauge theories. Here we also have a pole but in the infrared region.
In the third case, the -function has zero: at first, it is positive and then is negative. This means that for small initial values the effective coupling increases; and for large ones, decreases. In both the cases, with increasing momentum it tends to the fixed value defined by the zero of the -function. This is the so-called ultraviolet stable fixed point. It appears in some models in higher orders of perturbation theory.

4.8.Dimensional regularization and the MS scheme in a ghost sector
Consider now the calculation of the g function and the anomalous dimensions in some particular models within the dimensional regularization and the minimal subtraction scheme. Note that in transition from dimension 4 to 4 2 the dimension of the coupling is changed and the "bare" coupling acquires the dimension g B, 2 . That is why the relation between the "bare" and renormalized coupling contains the factor 2 g B, 2 Z g g . 4. 8. 1 Hence, even before the renormalization when Z g 1, in order to compensate this factor the dimensionless coupling g should depend on . Differentiating Eq.(4.8.1) with respect to 2 one gets i.e., In the MS scheme the renormalization constants are given by the pole terms in 1/ expansion and so does the bare coupling. They can be written as

8. 6
Equalizing the coefficients of equal powers of , one finds

8. 7
One sees that the coefficients of higher poles c n g , n 2 are completely defined by that of the lowest pole c 1 g and the function. In its turn the -function is also defined by the lowest pole. To see this, consider Eq.(4.7.20). Differentiating it with respect to ln 2 one has g n 1 a n g n g g 1 d dg n 1 a n g n 0.

8. 8
Equalizing the coefficients of equal powers of , one finds g g d dg a 1 g a 1 g , 4. 8. 9 and g d dg a n g a n g g d dg a n 1 g , n 2.

8. 10
Thus, knowing the coefficients of the lower poles one can reproduce all the higher order divergences. This means that they are not independent, all the information about them is connected in the lowest pole. In particular, substituting in (4.8.10) the perturbative expansion given by Eq.(4.8.5) one can solve the recurrent equation and find for the highest pole term a nn g a 11 n g , 4. 8. 11 i.e. in the leading order one has the geometric progression g Bare, g 2 1 1 g a 11 g , 4. 8. 12 which reflects the fact that the effective coupling in the leading log approximation (LLA) is also given by a geometric progression (4.8.12).The pole equations are easily generalized for the multiple couplings case, the higher poles are also expressed through the lower ones though the solutions of the RG equations are more complicated. Consider now some particular models and calculate the corresponding -functions and the anomalous dimensions.
The 4 4 theory

Standard matter sector
We remind that standard matter sector of the 4 theory defined by the inequality

8. 17
One can see from Eqs.(4.8.17) that the first coefficient of the -function is 3/2, i.e., the 4 theory in standard sector belongs to the type of theories shown in Fig.4.7.1a). In the leading log approximation (LLA) one has a Landau pole behaviour. In the two-loop approximation (NLLA) the -function gets a non-trivial zero and the effective coupling possesses an UV fixed point like the one shown in Fig.4.7.1). However, this fixed point is unstable with respect to higher orders and is not reliable. Here we encounter the problem of divergence of perturbation series in quantum field theory, they are the so-called asymptotic series which have a zero radius of convergence.

8. 19
The renormalization constants in the MS scheme up to two loops are given by Eq.

5.1.The Higher-Derivative Theories of Gravitation.Green's functions.
Adding quadratic products of the curvature tensor to the gravitational action leads to field equations in which some terms involve four derivatives. While it is not the purpose of this paper to investigate the novel consequences of these classical field equations, a brief summary of some of the salient features is in order to give a grounding to the following discussion of renormalization.
Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields and of the BRS transformations [13].The generic expression of the action reads where the curvature tensor and the Ricci is defined by R and R R correspondingly, 2 32 G, we used the signature . The convenient definition of the gravitational field variable in terms of the contravariant metric density reads Analysis of the linearized radiation shows that there are eight dynamical degrees of freedom in the field. Two of these excitations correspond to the familiar massless spin-2 graviton. Five more correspond to a massive spin-2 particle with mass m 2 . The eighth corresponds to a massive scalar particle with mass m 0 . Although the linearized field energy of the massless spin-2 and massive scalar excitations is positive definite, the linearized energy of the massive spin-2 excitations is negative definite. This feature is characteristic of higher-derivative models, and poses the major obstacle to their physical interpretation.
In the quantum theory, there is an alternative problem which may be substituted for the negative energy. It is possible to recast the theory so that the massive spin-2 eigenstates of the free-fieid Hamiltonian have positive-definite energy, but also negative norm in the state vector space.
These negative-norm states cannot be excluded from the physical sector of the vector space without destroying the unitarity of the S matrix. The requirement that the graviton propagator behave like p 4 for large momenta makes it necessary to choose the indefinite-metric vector space over the negative-energy states.
The presence of massive quantum states of negative norm which cancel some of the divergences due to the massless states is analogous to the Pauli-Villars regularization of other field theories. For quantum gravity, however, the resulting improvement in the ultraviolet behavior of the theory is sufficient only to make it renormalizable, but not finite.
The gauge choice which we adopt in order to defining the quantum theory is the canonical harmonic gauge: h 0. Corresponding Green's functions are then given by a generating functional

1. 3
Here F F h , F r and the arrow indicates the direction in which the derivative acts. N is an normalization constant. C is the Faddeev-Popov ghost field, and C is the antighost field. Notice that both C and C are anticommuting quantities. D is the operator which generates gauge transformations in h , given an arbitrary spacetime-dependent vector x corresponding to x x and where In the functional integral (5.1.3), we have written the metric for the gravitational field as dh without any local factors of g det g . Such factors do not contribute to the Feynman rules because their effect is to introduce terms proportional to 4 0 d 4 x ln g into the effective action and 4 0 is set equal to zero in dimensional regularization.
In calculating the generating functional (5.1.3.) by using the loop expansion, one may represent the -function which fixes the gauge as the limit of a Gaussian, discarding an infinite normalization constant In this expression, the index has been lowered using the flat-space metric tensor . For the remainder of this paper, we shall adopt the standard approach to the covariant quantization of gravity, in which only Lorentz tensors occur, and all raising and lowering of indices is done with respect to flat space. The graviton propagator may be calculated from I sym 1 Such divergences do cancel when they are connected to tree diagrams whose outermost lines are on the mass shell, as they must if the S matrix is to be made finite without introducing counterterms for them. However, they greatIy complicate the renormalization of Green's functions. We may attempt to extricate ourselves from the situation described in the last paragraph by picking a different weighting functional. Keeping in mind that we want no part of the graviton propagator to fall off slower than p 4 for large momenta, we now choose the weighting functional [12]  where e is any four-vector function.The corresponding gauge-fixing term in the effective action is The graviton propagator resulting from the gauge-fixing term (5.1.7) is derived in [13].
For most values of the parameters and in I sym it satisfies the requirement that all its leading parts fall off like p 4 for large momenta. There are, however, specific choices of these parameters which must be avoided. If 0, the massive spin-2 excitations disappear, and inspection of the graviton propagator shows that some terms then behave like k 2 . Likewise, if 3 0, the massive scalar excitation disappears, and there are again terms in the propagator which behave like p 2 . However, even if we avoid the special cases 0 and 3 0, and if we use the propagator derived from (5.1.7), we still do not obtain a clean renormalization of the Green's functions. We now turn to the implications of gauge invariance.Before we write down the BRS transformations for gravity, let us first establish the commutation relation for gravitational gauge transformations, which reveals the group structure of the theory. Take the gauge transformation (5.1.4) of h , generated by and perform a second gauge transformation, generated by , on the h fields appearing there. Then antisymmetrize in and .The result is where the repeated indices denote both summation over the discrete values of the indices and integration over the spacetime arguments of the functions or operators indexed. The BRS transformations for gravity appropriate for the gauge-fixing term (5.1.6) are [13] where is an infinitesimal anticommuting constant parameter.The importance of these transformations resides in the quantities which they leave invariant. Note that BRS C C 0 5. 1. 10 and BRS D C 0.

1. 11
As a result of Eq. (5.1.11), the only part of the ghost action which varies under the BRS transformations is the antighost C . Accordingly, the transformation (2.2.9c) has been chosen to make the variation of the ghost action just cancel the variation of the gauge-fixing term. Therefore, the entire effective action is BRS invariant: Equations (5.1.9), (5.1.10), and (5.1.12) now enable us to write the Slavnov identities in an economical way. In order to carry out the renormalization program, we will need to have Slavnov identities for the proper vertices.

Slavnov identities for Green's functions
First consider the Slavnov identities for Green's functions.

1. 13
Anticommuting sources have been included for the ghost and antighost fields, and the effective action has been enlarged by the inclusion of BRS invariant couplings of the ghosts and gravitons to some external fields K (anticommuting) and L (commuting),

1. 21
Another identity which we shall need is the ghost equation of motion. To derive this equation, we shift the antighost integration variable C to C C , again with no resulting change in the value of the generating functional: We define now the generating functional of connected Green's functions as the logarithm of the functional (5.1.13), W T , , , K , L i ln Z T , , , K , L .

1. 23
and make use of the couplings to the external fields K and L to rewrite (5. We have chosen to denote the expectation values of the fields by the same symbols which were used for the fields in the effective action (5.1.14). The Since the external fields K and L do not participate in the Legendre transformation (5.1.26), for them we have the relations Finally, the Slavnov identity for the generating functional of proper vertices is obtained by transcribing (5.1.24) using the relations (5.1.26), (5.1.28), and (5.1.29) We also have the ghost equation of motion, The Slavnov identity (5.1.33) is quadratic in the functional . This nonlinearity is reflected in the fact that the renormalization of the effective action generally also involves the renormalization of the BRS transformations which must leave the effective action invariant.
The canonical approach uses the Slavnov identity for the generating functional of proper vertices to derive a linear equation for the divergent parts of the proper vertices. This equation is then solved to display the structure of the divergences. From this structure, it can be seen how to renormalize the effective action so that it remains invariant under a renormalized set of BRS transformations [13].
Suppose that we have successfully renormalized the reduced effective action up to n 1 loop order; that is, suppose we have constructed a quantum extension of which satisfies Eqs. (5.1.17) and (5.1.18) exactly, and which leads to finite proper vertices when calculated up to order n 1. We will denote this renormalized quantity by n 1 . In general, it contains terms of many different orders in the loop expansion, including orders greater than n 1. The n 1 loop part of the reduced generating functional of proper vertices will be denoted by n 1 .
When we proceed to calculate n , we find that it contains divergences. Some of these come from n-loop Feynman integrals. Since all the subintegrals of an n-loop Feynman integral contain less than w loops, they are finite by assumption. Therefore, the divergences which arise from w-Ioop Feynman integrals come only from the overall divergences of the integrals, so the corresponding parts of n are local in structure. In the dimensional regularization procedure, these divergences are of order 1 d 4 1 , where d is the dimensionality of spacetime in the Feynman integrals.
There may also be divergent parts of n which do not arise from loop integrals, and which contain higher-order poles in the regulating parameter . Such divergences comes from n-loop order parts of n 1 which are necessary to ensure that (5.1.17) is satisfied. Consequently, they too have a local structure. We may separate the divergent and finite parts of n :

1. 37
Since each term on the right-hand side of (5.1.37) remains finite as 0, while each term on the left-hand side contains a factor with at least a simple pole in e, each side of the equation must vanish separately. Remembering the Eq.

Ghost number and power counting
Structure of the effective action (5.1.14) shows that we may define the following conserved quantity, called ghost number [13]: we require of the functional X that N ghost X 1.

1. 47
In order to complete analysis of the structure of div n , we must supplement the symmetry equations (5.1.42), (5.1.43), and (5.1.47) with the constraints on the divergences which arise from power counting. Accordingly, we introduce the following notations: n E number of graviton vertices with two derivatives, n G number of antighost-graviton-ghost vertices, n K number of K-graviton-ghost vertices, n L number of L-ghost-ghost vertices, I G number of internal-ghost propagators, E C number of external ghosts, E C number of external antighosts. Since graviton propagators behave like p 4 , and ghost propagators like p 2 , we are led by standard power counting to the degree of divergence of an arbitrary diagram, The last term in (5.1.48) arises because each external antighost line carries with it a factor of external momentum. We can make use of the topological relation to write the degree of divergence as Together with conservation of ghost number,Eq. (5.1.50) enables us to catalog three different types of divergent structures involving ghosts. These are illustrated in Fig.5

1. 52
The breakup between the gauge-invariant divergences S and the rest of (5.1.52) is determined only up to a term of the form [13] which can be generated by adding to P a term proportional to h g g . The profusion of divergences allowed by (5.1.52) appears to make the task of renormalizing the effective action rather complicated. Although the many divergent structures do pose a considerable nuisance for practical calculations, the situation is still reminiscent in principle of the renormalization of Yang-Mills theories. There, the non-gauge-invariant divergences may be eliminated by a number of field renormalizations. We shall find the same to be true here, but because the gravitational field h carries no weight in the power counting, there is nothing to prevent the field renormalizations from being nonlinear, or from mixing the gravitational and ghost fields. The corresponding renormalizations procedure considered in [13].
Remark 5.1.2.We assume now that: (i) The local Poincaré group of momentum space is deformed at some fundamental high-energy cutoff [9], [10]. (ii) The canonical quadratic invariant p 2 ab p a p b collapses at high-energy cutoff and being replaced by the non-quadratic invariant:

1. 58
Corresponding Green's functions are then given by a generating functional

1. 59
Remark 5.1.4.(I)The renormalizable models which we have considered in this section many years regarded only as constructs for a study of the ultraviolet problem of quantum gravity. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Minkowski space-time. In canonical case they do have only some promise as phenomenological models.
(II) However, for their unphysical behavior may be restricted to arbitrarily large energy scales mentioned above by an appropriate limitation on the renormalized masses m 2 and m 0 . Actually, it is only the massive spin-two excitations of the field which give the trouble with unitarity and thus require a very large mass. The limit on the mass m 0 is determined only by the observational constraints on the static field.

5.2.Cleaner methods
The renormalization procedure described in the last section is sufficiently complicated to make practical calculations unappealing. We now turn to other choices of the gauge-fixing term which greatly simplify matters by eliminating the need for the field and transformation renormalizations.
A. Unweighted gauge condition Explicit calculations of samples of the nongauge-invariant divergences allowed by (6.17) reveal that they depend upon the gauge-fixing parameter which was introduced into the effective action by the weighting functional (3.6). This suggests that if we take the limit 0, all the field and transformation renormalizations may disappear. This limit as 0 returns us to the unweighted gauge condition with the same Feynman rules as those obtained using the simple Gaussian representation (3.4) of the gauge-fixing -function.The graviton propagator in the limit 0 maybe calculated as suggested above, setting 0 in the propagator calculated for finite (cf. sec.6.4), or by substituting the gauge condition (8.1) into the linearized classical field equations and then inverting. The resulting propagator is constructed entirely from projectors which are transverse in all their indices:

2. 2
The definitions of the projectors P 2 and P 0 s are given in sec.6.4.The antighost-graviton-ghost interaction is The first two terms in this expression contain the gauge condition (5.2.1), and consequently do not connect to the graviton propagator (5.2.2). Similarly, integration by parts in the remaining term can be used to move the derivatives onto the ghost field C . When these derivatives fall on h they form the gauge condition (5.2.1) again,so we have effectively The symbol is used to indicate that terms containing h hpo or 2 h have been dropped, since they do not connect to the graviton propagator.
The power-counting rule given in Sec. 5.1 must be modified as a results of (5.2.4). In one-particle-irreducible (1PI) diagrams, there is a separate vertex V ChC for each external ghost and antighost line. Consequently, each of these lines carries with it two factors of external momentum. The resulting degree of divergence of an arbitrary 1PI diagram is This result would hold even if we had not chosen (2.2) as our definition of the gravitational field variable. However, the simple relation (5.2.4) is dependent upon that choice, which accords with the harmonic gauge condition (5.2.1). Otherwise there would be a complicated cancellation between vertices.
From the power-counting rule (5.2.5), we see that each of the three types of diagrams shown in Fig. 5 Together with (6.2.6a), this implies that div n 0 is gauge invariant. All the divergences may therefore be eliminated by renormalizations of the parameters , and in I sym and by the addition of a cosmological counterterm. The field variables and the BRS transformations do not need to be renormalized.The contrast between the complicated renormalization procedure which one must use when the quantum theory is defined with the gauge -fixing term (3.7) and the much simpler procedure for the unweighted gauge condition is reminiscent of the situation in the axial gauge in Yang-Mills theory. There, the ghosts decouple entirely from the Yang-Mills fields if one uses the unweighted axial gauge condition. However, if one smears the axial gauge with a weighting functional, the resulting propagator does connect to the ghosts, and then there arise non-gauge-invariant divergences. These Yang-Mills divergences are similar to those we would have obtained in the gravitational theory had we kept the two-derivative gauge-fixing term derived from (3.5). In both cases, the part of the propagator which depends upon the gauge-fixing parameter has a bad asymptotic behavior for large momenta, leading to nongauge-invariant divergences of progressively higher order as the calculation proceeds in perturbation theory.
Taking the limit 0 is necessary for the axial gauge quantization of Yang-Mills theory to avoid these artifactual divergences. However, this limit is less useful in other gauges: Although one obtains an improvement in the power counting just as we have found for gravitation, the improvement is not sufficient to eliminate all the nongauge-invariant divergences, and one must still renormalize the Yang-Mills gauge transformation. Thus, although taking the limit 0 is perfectly acceptable in Yang-Mills theory, it is generally of no particular advantage, and has not been much used in the literature.

B. Third-derivative gauge
Since we are dealing with theories in which the classical field equations involve fourth derivatives, the Cauchy data which must be initially specified to determine the classical evolution of the field include the values of the field and up to its third derivatives on some spacelike hyper surface. Accordingly, we should also be prepared to use gauge conditions which involve up to third derivatives. A gauge condition of this type which has the same structure as the harmonic gauge condition (5.2.1) is If we weight the gauge condition (6.2.8) with the Gaussian functional (3.5), we get the gauge-fixing term Another way to arrive at (5.2.9) is to start from the usual harmonic gauge condition (5.2.1) and to weight it with the functional When we obtain (5.2.9) this second way, it is clear that the ghost action which we must use is exactly the same that we had before in the generating functional (3,2). This also follows from the first method of arriving at (5.2.9), because we may always redefine the antighost field: 2 C C . The gauge-fixing term (5.2.9) requires us to change the BRS transformation of the antighost field C . The new transformation is The Slavnov identities for the generating functionals of Green's functions and of proper vertices must be changed too, but the identity for the reduced generating functional of proper vertices, 6 6 1 2 remains the same as (5.20). Consequently, the renormalization equation is the same as (6.3).The Feynman rules which we obtain using (5.2.9) differ from those obtained using (3.7) only in the replacement of the factors of 2 k 4 in the graviton propagator by 4 k 6 . This change brings about a reduction in the degree of divergence of those parts of diagrams which depend on the parameter . The degree of divergence is reduced by 2 for each factor of , so that once again all three types of diagram involving ghosts shown in Fig. 6.1.2 are convergent. The renormalization equation then implies that all the divergences in 6 div n are gauge invariant. 5.3.Coupling to fields of standard matter and to fields of physical ghost matter. Now that we know how to carry out the renormalization procedure for a purely gravitational model, it is straightforward to include coupling to other renormalizable fields. As an example, we discuss a massive scalar field in interaction with the gravitational field alone, adding to the action (5.1.1) the additional term The BRS transformations must now include a transport term for the scalar field, This transformation is nilpotent: In order to write the Slavnov identities, we make use of (5.3.3) by adding a term coupling the scalar and ghost fields to a new anticommuting external field B x : In the generating functional of Green's functions, the scalar field is coupled to a source J x ; the Legendre transformation then trades this dependence on J x for a dependence on х W/ J x in the generating functional of proper vertices. The Slavnov identity for the reduced generating functional of proper vertices reads As before, this identity leads to the renormalization equation for div n .Power counting, using the unweighted gauge condition, gives the degree of divergence of an arbitrary 1PI diagram, where n B is the number of B-scalar-ghost vertices and E S is the number of external scalar lines. The external scalar lines are counted twice in (5.3.6) because of the linkage of scalar fields and derivatives in the interaction between scalars and gravitons (the mass term is super-renormalizable and is not included in the power counting). This linkage is similar to the linkage of ghosts and derivatives which we have already encountered.The power-counting rule (5.3.6), together with the conservation of ghost number, shows that all 1PI diagrams with external ghost lines are convergent, so that

3. 8
These divergences may be eliminated by renormalizations of the appropriate coefficients in I sym and I , and by the addition of a cosmological counterterm. It should be noted that the absense of a term like R 2 g in (6.3.8) is due to the linkage of scalars and derivatives. If this linkage were broken by the inclusion in (6.3.1) of a scalar self-interaction 4 g , then it would be nec-essary to include as well the nonminimal gravitational-scalar interaction.
The scalar field example shows that once renormalizability has been established for a purely gravitational model, the inclusion of couplings to other renormalizable fields poses no further problems (except possibly the necessity for a nonminimal gravitational-scalar interaction). In particular, the Faddeev-Popov ghost machinery remains unrenormalized just as it did in the purely gravitational case.The allowed divergences may be summarized by assigning a power-counting weight to each field, and then requiring that divergent structures be gauge invariant and of power-counting weight four or less. It is necessary to take into account any linkages of fields and derivatives in the interactions by augmenting the weight of a field by the number of derivatives linked to it. The weight of the gravitational field is zero, and before linkages with derivatives are taken into account, the weights of other fields are simply given by their canonical dimensions.

5.4.The graviton propagator.
The inversion of the gravitational kinetic matrix which is necessary to calculate the graviton propagator involves a substantial amount of Lorentz algebra on symmetric rank-two tensors. To organize the calculation, it is convenient to use a set of orthogonal projectors in momentum space. We choose a set of projectors which emphasises transversality,16 since this is important in Sec. 5.3.These projectors are constructed using the transverse and longitudinal projectors for vector quantities, The four projectors for symmetric rank-two tensors then reads a P 2 1 2

4. 2
For a massive tensor field in the rest frame, the projectors (5.2.2.a)-(5.2.2.d) select out the spin-two, spin-one, and two spin-zero parts of the field.However, the projectors (5.2.2) do not span the operator space of the gravitational field equations. In order to have a complete basis, we must also include the two spin-zero transfer operators, The orthogonality relations of the projectors ( where i and j run from 0 to 2, and a and b take on the values w and s.In order to calculate the graviton propagator, we must first write out the part of the effective action (5.1.14) which is purely quadratic in the gravitational field h . Going over to momentum space and using ( To determine the propagator (5.4.6) completely, we must specify how the k 0 integration contour is to skirt the poles in calculating Feynman integrals. We do this in the customary way by including ie terms in the denominators of the individual poles, which must first be obtained by separating (5.4.6) into partial fractions. Ignoring for the moment the terms proportional to , we find

4. 7
Normally, one requires that quantum states have positive-definite norm and energy. Such states give rise to poles in the propagator with positive residues. Since both the massless pole and the pole at i .

4. 9
On the other hand, the negative residue of the massive spin-two pole at k 2 1 faces us with a choice between two unfortunate alternatives: to give up either the positive definiteness of the norm or of the energy of the corresponding quantum states.
Both choices give the required negative residue, but they differ in the way the pole must be shifted.If the massive spin-two states are taken to have negative norm, the situation is analogous to a Pauli-Villars regularized theory. We recall that in the usual derivation of the propagator, one starts from 0|T h x h x |0 , transforms to momentum space, and sums over a complete set of momentum eigenstates inserted between the two field operators. The only difference in the present case is that the negative-norm states must be accompanied by a vector space metric factor of 1 in the sum over states. This gives rise to a negative residue for the massive spin-two pole, but does not affect the location of the pole, whose denominator is consequently given by k 2 1 i .

4. 10
As the Pauli-Villars analogy leads us to expect, the choice (5.4.10), together with (5.4.8) and (5.4.9), gives a high-energy behavior of the total propagator which is like k 4 . To see this, one may, for example, perform a Wick rotation into Euclidean space and then drop the ге terms. This is allowedbecause (5.4.10), (5.4.8), and (5.4.9) all shift the poles in the same way. If the massive spin-two states are taken to have negative energy, the pole in the propagator acquires a negative residue for a different reason.In this case, there are no vector space metric factors in the sum over states, but the expansion of the field operators into creation and annihilation operators involves normalization factors 2|k 0 | 1/2 2k 0 1/2 . These contribute an overall minus sign to the massive spin-two part of the propagator. In addition, the sign of the energy flow for a given time ordering is opposite to that for a positive-energy field, so the denominator of the pole is now given by k 2 1 i .

4. 11
The difference between the poles given by (5.4.10) and (5.4.11) is a term proportional to k 2 1 . While the choice of (5.4.10) leads to the desired behavior, this additional term effectively spoils the high-energy behavior of (5.4.11). Thus, our power-counting requirements lead us to adopt an indefinite-metric state vector space, following the analogy to Pauli-Villars regularization.The pure k 4 terms in (5.4.6), proportional to , may be handled by confluence, replacing them by 1 k 2 i 1 k 2 i 1 , and then letting 0 at the end of the calculation.

6.1.Fractional Integration in negative dimensions.
Let H D be a Hausdorff measure [33][34] and X n , D n is measurable set. Let s x be a function s : X such that is symmetric with respect to some centre x 0 X, i.e. s x constant for all x satisfying d x, x 0 r for arbitrary values of r. Then the integral in respect to Hausdorff measure over n-dimensional metric space X is then given by [33]: The integral in RHS of the Eq.(3.1.1) is known in the theory of the Weyl fractional calculus where, the Weyl fractional integral W D f x , is given by 6. 1. 2 Remark 6.1.1. In order to extend the Weyl fractional integral (6.1.1) in negative dimensions we apply the Colombeau generalized functions [21][22][23][24][25] and Colombeau generalized numbers [23].Recall that Colombeau algebras G of the Colombeau generalized functions defined as follows [21][22].
Let be an open subset of n . Throughout this paper, for elements of the space C 0,1 of sequences of smooth functions indexed by 0, 1 we shall use the canonical notation u so u C , 0, 1 . Notice that G is a differential algebra.Equivalence classes of sequences u will be denoted by cl u . is a differential algebra containing D as a linear subspace and C as subalgebra.  The bare photon propagator is renormalised by the formal summation of vacuum polarisation diagrams, whose lowest-order contribution given by Eq.(6.2.36).Again, q can be expanded around the mass shell q 2 0, yielding q P q 2 q 2 .

2. 50
The full photon propagator can be written as q q q 2 g q 2 , with q 2 1 P q 2 q 2 1

2. 51
The term in brackets contributes to the renormalisation of the bare charge e Bare , which relates to the renormalised charge e by All other contributions to the renormalisation of the electric charge cancel, as can be explicitly seen by a summation of the lowest-order radiative corrections to the charge 1 B L P e Bare .

2. 56
As can be shown from (6.2.38) and (6.2.41), L equals B and only the Z 3, , factor remains for the renormalisation of the electric charge.
Let us consider corrections to the magnetic moment due to vertex corrections as (6.2.40).In particular, the term proportional to q remains finite for Hausdorff dimensions smaller than six. It gives rise to low-order contributions to the anomalous magnetic moment as well as the l 0 splitting of energy levels in atoms (Lamb shift).Utilising the expansion of the gamma function into a polynomial 1 z i 0 c i z i with coefficients c 0 1, c n 1 n 1 1 Here, e is the form factor of the electromagnetic current proportional to q . Presently the difference between experimental and theoretical values of e suggests where x 1 x 2 and k 1 k 2 , is lower negative dimensional with D _ k 2 and apperer negative dimensional with D _ k 1 . Remark 6.3.5.In this subsection we define generalized Hausdorff-Colombeau measure.In subsection VI.4 we will prove that negative dimensions of fractals equipped with the Hausdorff-Colombeau measure in natural way.
Let be an open subset of n , let X be metric space X n and let F be a set F E i i of subsets E i of X. Let E, x, x be a function : F . Let C F be a set of the all functions E, x, x such that E, x, x C whenever E F. Throughout this paper, for elements of the space C F 0,1 of sequences of smooth functions indexed by 0,

3. 17
Notice that G F is a differential algebra.Equivalence classes of sequences E, x, x will be denoted by cl or simply .  The Colombeau-Lebesgue-Stieltjes integral over continuous functions f : X can be evaluated similarly as in subsection III.3,(but using the limit in sense of Colombeau generalized functions) of infinitesimal covering diameter when E i i is a disjoined covering and x i E i :