Properties of a proposed background independent exact renormalization group Vlad-Mihai Mandric and Tim R. Morris

2025-05-02 0 0 611.85KB 52 页 10玖币

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Properties of a proposed background

independent exact renormalization group

Vlad-Mihai Mandric and Tim R. Morris

STAG Research Centre & Department of Physics and Astronomy,

University of Southampton, Highﬁeld, Southampton, SO17 1BJ, U.K.

V.M.Mandric@soton.ac.uk, T.R.Morris@soton.ac.uk

Abstract

We explore the properties of a recently proposed background independent exact renormaliza-

tion group approach to gauge theories and gravity. In the process we also develop the machinery

needed to study it rigorously. The proposal comes with some advantages. It preserves gauge

invariance manifestly, avoids introducing unphysical ﬁelds, such as ghosts and Pauli-Villars

ﬁelds, and does not require gauge-ﬁxing. However, we show that in the simple case of SU(N)

Yang-Mills it does not completely regularise the longitudinal part of vertex functions already

at one loop, invalidating certain methods for extracting universal components. Moreover we

demonstrate a kind of no-go theorem: within the proposed structure, whatever choice is made

for covariantisation and cutoﬀ proﬁles, the two-point vertex ﬂow equation at one loop cannot

be both transverse, as required by gauge invariance, and fully regularised.

arXiv:2210.00492v3 [hep-th] 22 Mar 2023

Contents

1 Introduction 2

2 A proposal for a background independent exact RG 3

2.1 Notation and basic ingredients for regularisation . . . . . . . . . . . . . . . . . . . . 3

2.2 Themainidea ....................................... 8

2.3 Regularisationstructure .................................. 9

2.4 Flowequation........................................ 11

3 Regularisation failure: a sketch 13

4 Perturbative expansion 14

4.1 Loopexpansion....................................... 15

4.2 Vertexexpansion...................................... 17

4.2.1 Action vertex properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.2 Kernel vertex properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.3 Two-point action vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.4 Zero-point kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.5 Higher-pointvertices................................ 22

4.2.6 Large momentum behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Regularisation of higher point classical vertices 26

6 One loop beta function 28

6.1 Classicalﬂowequations .................................. 29

6.2 Traceterms......................................... 31

6.3 Two-point vertex at zeroth order in momentum . . . . . . . . . . . . . . . . . . . . . 33

6.4 Two-point vertex at second order in momentum . . . . . . . . . . . . . . . . . . . . . 35

7 Summary and Conclusions 41

A Interleave identities for compound kernels 42

B Why preregularisation is necessary 46

1 Introduction

Understanding the Wilsonian renormalization group (RG) structure of quantum gravity is surely

of importance, see e.g. [1–10], and central to this is the rˆole of diﬀeomorphism invariance. One

approach is to try to generalise the exact RG [11,12] to gravity, in such a way that it is manifestly

diﬀeomorphism invariant. On the one hand this would allow computations to be done whilst keeping

exact diﬀeomorphism invariance at every stage, i.e. without gauge ﬁxing, and on the other hand,

these computations would be background independent, that is performed without ﬁrst choosing the

space-time manifold and background metric.

At the classical level this can be done [13]. However in order to compute quantum corrections,

extra ultraviolet regularisation has to be incorporated into the exact RG so that the integration is

properly cut oﬀ in some diﬀeomorphism invariant way at the eﬀective cutoﬀ scale Λ. For the simpler

case of gauge invariance in SU(N) Yang-Mills theory this problem was solved by incorporating

gauge invariant PV (Pauli-Villars) ﬁelds arising from a spontaneously broken SU(N|N) gauge

theory [14–36]. It is not clear whether one can generalise such a scheme to gravity however, in

particular it is not clear what should play the rˆole of SU(N|N), although a kind of supergravity

has been suggested in [37].

Recently a diﬀerent approach to the problem of regularisation has been pursued [38]. Explicit

PV ﬁelds are avoided and instead replaced by functional determinants which, if constructed as

squares of simpler determinants, can be shown to work in the framework of standard perturbation

theory [39–41]. Furthermore a geometric approach is followed where the determinants can be

regarded as deﬁning a regularised volume element on the orbit space of the gauge theory [41]. The

ﬂow equation is then formulated in a way that is manifestly invariant under ﬁeld redeﬁnitions, and

applies equally well to both gravity and gauge theory [38].

Although this new proposal has elegant features, it is not immediately evident that the reg-

ularisation is successfully implemented in the ﬂow equation once we delve into the details of the

relevant Feynman diagrams, ﬁrst at one loop and then also at higher loops. In this paper we put the

proposal to the test by constructing explicit expressions for the relevant vertices and then carefully

analyse their UV (ultraviolet) behaviour, as a function of loop momentum, in the form that they

appear in quantum corrections. For this purpose it is suﬃcient to focus on Yang-Mills since if the

regularisation fails for Yang-Mills it most certainly fails for quantum gravity (given the latter’s

poor UV behaviour). Working with Yang-Mills also means we can take over methods used for these

investigations in the earlier successful construction [14–36]. As we will see, the proposal of ref. [38]

unfortunately fails to fully regularise already at one loop, but in a rather subtle way, which in

particular invalidates powerful techniques previously used to extract universal information [23,50].

The paper is organised as follows. In section 2we review the proposal, and in the process,

set out our notation and conventions. In sec. 3we sketch why the regularisation can fail. In

section 4we build the machinery needed to rigorously test the structure of the renormalization

scheme at the perturbative level. In sec. 5we conﬁrm that the higher point classical vertices

incorporate the assumed regularisation. Then in sec. 6we apply the techniques extensively to an

analysis of the simplest one-loop correction namely that for the eﬀective action two-point vertex.

We show that the regularisation is suﬃcient for the momentum independent part, giving a vanishing

result as it should by gauge invariance, only if the Λ-derivative of quadratically divergent constant

part is discarded. (Recall that Λ is the eﬀective cutoﬀ scale.) The part that is second order in

momentum ought to give the one-loop beta function, if properly regulated. However we show that

the result cannot both be completely regularised and transverse. It can be taken to be transverse

only if the Λ-derivative of a linearly divergent part is discarded. Since this holds for all choices of

covariantisation and cutoﬀ proﬁles, it points to some inherent limitations in the structure of the

proposed ﬂow equation. In sec. 7we summarise and draw our conclusions.

2 A proposal for a background independent exact RG

2.1 Notation and basic ingredients for regularisation

The ﬂow equation proposed in ref. [38] incorporates a geometric approach to the quantisation of

gauge theories [42,43] and as such it is useful to use DeWitt notation. On the other hand, as we will

see, a detailed understanding of the UV properties can only be reached by working with explicit

expressions for the vertices. In the case of Yang-Mills theories, the ﬂow equations then take their

simplest form if we regard the gauge ﬁelds as valued in the Lie algebra, i.e. contracted into the

generators [14,15]. We will therefore work with both notations as appropriate.

We work in Euclidean signature and, therefore, we will keep all gauge group indices as su-

perscripts and Lorentz indices as subscripts for convenience. For position and momentum space

integrals we introduce the shorthand

ZdD

x=Zx

,ZdD

(2π)D=Zp

,(2.1)

respectively, where Dis the number of dimensions we are working in. We will frequently switch

between position and momentum space representations throughout the paper and this will help us

keep everything clean and concise. For similar reasons we will adopt the following convention:

δ(p)≡(2π)Dδ(D)(p),(2.2)

where δ(D)(p) is the standard D-dimensional Dirac delta function. Our convention for Fourier

transforms is then

φ(p) = Zx

φ(x)e−ipx .(2.3)

We will use the DeWitt compact notation and its explicit representation interchangeably, so for

example

φaJa≡X

aZx

µ(x)Ja

µ(x).(2.4)

Last, but not least, we will adopt the following shorthands for momenta and scalar functions of

momenta, respectively:

Λ≡˜p , (2.5)

Kp2

Λ2≡Kp,(2.6)

where Λ is the eﬀective cutoﬀ energy scale.

We work with the gauge group SU(N). We use DeWitt Latin indices from the start of the

alphabet to label gauge ﬁelds, thus φa, but as already mentioned, when we need more explicit

expressions it will be convenient to regard the gauge ﬁelds Aµ(x) as contracted into the generators:

φa≡Aµ(x) = Aa

µ(x)Ta.(2.7)

With appropriate deﬁnitions for the vertex functions in either language, the expressions will of

course be equal, and the two representations are thus equivalent. The generators Taare taken

to be hermitian, in the fundamental representation, and orthonormalized as tr(TaTb) = 1

2δab. A

second set of DeWitt Greek indices from the start of the alphabet is used to label gauge parameters;

the map from the two languages is thus:

α≡ω(x) = ωa(x)Ta.(2.8)

If we adopt the geometric approach of [42,43] to the quantisation of gauge theories, we regard

the ﬁelds φaas coordinates on an inﬁnite dimensional ‘manifold’ Φ, the space of all possible ﬁeld

conﬁgurations. This has the structure of a ﬁbre bundle, and the ﬁbres are the gauge orbits G.

However, all the physics happens on the quotient space Φ/G, where each point belongs to a unique

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PropertiesofaproposedbackgroundindependentexactrenormalizationgroupVlad-MihaiMandricandTimR.MorrisSTAGResearchCentre&DepartmentofPhysicsandAstronomy,UniversityofSouthampton,Higheld,Southampton,SO171BJ,U.K.V.M.Mandric@soton.ac.uk,T.R.Morris@soton.ac.ukAbstractWeexplorethepropertiesofarecentlypropose...

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Properties of a proposed background independent exact renormalization group Vlad-Mihai Mandric and Tim R. Morris

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