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Physics Letters B 773 (2017) 534–543

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Global aspects of the renormalization group and the hierarchy problem

Andrei T. Patrascu

University College London, Department of Physics and Astronomy, London, WC1E 6BT, UK

a r t i c l e i n f o a b s t r a c t

Article history:

Received

6 August 2017

Received

in revised form 28 August 2017

Accepted

5 September 2017

Available

online 7 September 2017

Editor: N.

Lambert

The discovery of the Higgs boson by the ATLAS and CMS collaborations allowed us to precisely determine

its mass being 125.09 ± 0.24 GeV. This value is intriguing as it lies at the frontier between the regions

of stability and meta-stability of the standard model vacuum. It is known that the hierarchy problem

can be interpreted in terms of the near criticality between the two phases. The coeﬃcient of the Higgs

bilinear in the scalar potential, m

, is pushed by quantum corrections away from zero, towards the

extremes of the interval [−M

, M

] where M

is the Planck mass. In this article, I show that demanding

topological invariance for the renormalisation group allows us to extend the beta functions such that the

particular value of the Higgs mass parameter observed in our universe regains naturalness. In holographic

terms, invariance to changes of topology in the bulk is dual to a natural large hierarchy in the boundary

quantum ﬁeld theory. The demand of invariance to topology changes in the bulk appears to be strongly

tied to the invariance of string theory to T-duality in the presence of H-ﬂuxes.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The discovery of the Higgs boson and the measurement of its

mass as lying at the very edge between EW stability and metasta-

bility

regions, not accompanied by the discovery of supersym-

metric

particles, already casts shadows of doubt upon a potential

supersymmetric solution to the hierarchy problem. This state of

affairs asks for the search of new principles of nature that may

allow us to understand the origins of the standard model param-

eters

without having to rely on symmetries and without having

to give up calculability as is the case with multiverse interpreta-

tions.

As recent experimental results suggest, the standard model

vacuum appears to be metastable. Understanding the parameters

of the standard model that lead to such a conclusion would there-

fore

reveal information about the future fate of our universe. In

this article, I show that naturalness can be restored if one imposes

invariance of the physical theory with respect to topology changes

associated to the renormalisation group. Let BG be the classifying

space of the (in this case renormalisation) group G constructed as

the quotient EG/G, where EG is a contractible space on which G

acts

freely.

E-mail address: andrei.patrascu.11@ucl.ac.uk.

By means of the isomorphism in cohomology

(BG, Z) = H

(G, Z) (1)

it is clear that the topological properties (as described by cohomol-

ogy)

of the renormalisation group G, are equivalent to the topolog-

ical

properties of the space on which G acts. According to the du-

ality

between renormalisation group equations of the low energy

theory and the evolution equations along the new directions in a

higher dimensional effective theory (e.g. 5-d Supergravity) [1–3]

the

cohomology of the renormalisation group is to be associated

to the cohomology of the subspace induced by the evolution equa-

tions

along the additional directions in a higher energy effective

theory. Therefore, imposing invariance with respect to the change

in the renormalisation group topology is equivalent to imposing

invariance with respect to topology changes of the high energy ef-

fective

theory. This prescription is the low energy “remnant” of the

string theoretical T-duality.

Therefore,

understanding how the principle of topological in-

variance,

observed initially in string theory, affects low energy ef-

fective

ﬁeld theories like the standard model or 5-d supergravity

could herald new ways of connecting low energy physics to the

physics determined by extended fundamental objects expected to

arise at or near the Planck scale. While the observation that the

string theoretical T-duality may be related to the hierarchy prob-

lem

is not new [1], the fact that the topological invariance implied

by T-duality is the key to restoring naturalness, is.

http://dx.doi.org/10.1016/j.physletb.2017.09.010

0370-2693/

SCOAP

A.T. Patrascu / Physics Letters B 773 (2017) 534–543 535

The beta functions (generators of the renormalisation group)

allow us to probe high energies starting from our low energy ef-

fective

ﬁeld theory. However, at some point they must enter the

regime where string geometry effects become relevant. In order to

connect lower energy scales to string geometry effects involving

topology changing dualities, such beta functions must not discern

topologies connected by means of dualities. This requirement alters

the topological structure of the renormalisation group such that it

compensates for the topology changes that may occur at the string

level. The construction of topology invariant theories with a focus

on group quantisation was performed in [4]. The methods intro-

duced

there are particularly relevant because they deal with group

topology.

The

creation of explicitly T-duality invariant string theories re-

solved

the questions regarding the high energy domain. Represen-

tative

articles are [27], [33]. Transferring such invariance to the

renormalisation group is the focus of this work. Due to the dual-

ity

between RG ﬂux and extra dimensional equations of motion,

the goal of this article is to explore the ﬁeld theoretical, bound-

ary

effects of the invariance of the theory to topology changes in

the bulk. Of course, after the methodology is established, new in-

sights

upon the cosmological constant, the vacuum stability, or the

nature of dark matter may be obtained. I will only discuss here

applications to the hierarchy problem, leaving the other subjects

to a future, cosmology oriented article. Of course the standard

model hierarchy problem and the cosmological constant problem

are most likely related.

quantum ﬁeld theory, energy scales are regarded as linearly

ordered, from the lowest to the highest. Moving from high en-

ergy

scales to low energy scales leads to decoupling of the heavy

states and to effective theories. The domain where such quantum

ﬁeld theories are valid is also where the renormalisation group is

useful, allowing us to move between different energy scales. As

we move towards higher energies, in supergravity, we may iden-

tify

renormalisation group ﬂows with equations of motion on the

additional directions. In string theory however, distinguishing the

scales is not perfectly deﬁned. T-duality itself must arise by means

of a “conspiracy” between physics at all scales simultaneously [5].

T-duality in string theory, together with Buscher’s rules for the tar-

get

space [6,7], is to be associated with the low energy demand for

invariance of physics under a change of topology of the renormali-

sation

group. The plan of this article is as follows. First, I introduce

the renormalisation group from an algebraic perspective, connect-

ing

the form of the generators (beta functions) with the global

topological structure of the group. Next, I impose the invariance

of physical results with respect to group topology changes by suit-

ably

altering the group laws. In order to take this new principle

into account a modiﬁcation of the beta function will be required.

Finally, I will show on a simple quantum ﬁeld theoretical model

how these modiﬁcations explain the naturalness without the need

of additional symmetries or particles. Of course, the presence of

new symmetries at higher scales is not excluded by these results.

Supersymmetry may still be a symmetry of nature at higher en-

ergies

but as experiments suggest, it may not be able to explain

naturalness on its own.

2. Bulk space and the renormalisation group

Given holography and, particularly, the AdS/CFT duality, quan-

tum

gravity can be written in terms of a non-gravitational quan-

tum

ﬁeld theory on a boundary, allowing the application of

Wilson’s renormalisation group. In this context, T-duality is seen

as a duality in the bulk space involving string theory. Invariance

with respect to T-duality has been made explicit at this level, for

example by using a type of space that looks locally like a Rieman-

nian

manifold but which is glued together from these local patches

not just by diffeomorphisms but also by T-duality transformations

along some torus ﬁbres (T-folds). Such a T-fold would be a target

space for a string sigma-model that is only locally a Riemannian

manifold but globally would look like a more general geometry.

T-folds have a description in terms of spaces that locally are ﬁbre

products of a torus ﬁbre bundle with its T-dual [34]. Among the

ﬁrst to create an explicitly T-duality invariant formulation for the

string world sheet action was Tseytlin [27]. Here, I show that by

holographic duality, a demand for topological invariance resulting

from T-duality on the bulk, results in the restoration of naturalness

in the hierarchy of the boundary quantum ﬁeld theory.

The

gauge/gravity duality can be interpreted, according to [20]

a geometrisation of the quantum ﬁeld theoretical renormalisa-

tion

group. While the connection between a higher dimensional

equation of motion and the lower dimensional renormalisation

group ﬂow was known already in [1], it was [20] who reformulated

this connection in modern, holographic terms. The exact renormal-

isation

group equations can be expressed as Hamilton equations

for the radial evolution in a model space–time with one higher di-

mension.

The

jet-bundle structure corresponding to the quantum ﬁeld

theory is employed by [20] in order to show how the geometry

corresponding to the RG equations emerges. These equations are

being interpreted as higher spin equations of motion. In this article

I emphasise mostly the topological aspects of the RG and how du-

alities

connecting topologically distinct string geometries manifest

themselves at lower energies as invariances to topological changes

of the renormalisation group. To understand such a topological ap-

proach

to the bulk-space structure, it is important to understand

how the generators of the renormalisation group, the beta func-

tions,

emerge as geometrical objects. The scale transformations in

the ﬁeld theory correspond to movement in the additional ra-

dial

dimension [20] and the renormalisation group trajectories are

associated to speciﬁc geometries. If the RG ﬂow begins or ends

near a ﬁxed point, the corresponding geometries are anti-de Sitter

and connections between gauge/gravity dualities and RG ﬂows are

known [21].

The

picture that emerges is that we should formulate the holo-

graphic

duals of exact RG equations in terms of connections and

sections of certain bundles over a model space–time. Indeed, once

such a formulation has been constructed the next step will be

to establish the topological invariance at low energies that results

from the string theoretical T-duality. Following ref. [20] in the ﬁrst

step of the Wilsonian RG process, we integrate out a shell of fast

modes and analyse the effect on the sources. Lowering M → λM,

where λ = 1 −  is interpreted as an integration of the fast modes,

we obtain the variation of the sources



B =−M

δU =−M

(2)

We demand that for the overall partition function

= Z

−1



[

dφdφ

∗

]{(M

+ e

) − Z

−1

+iS

}=0(3)

where the action has been regulated and split into S = S

+ S

with S

containing the B(x, y) sources. Following the calculations

of [20] we obtain



B =−M

B = B



U =−M

U =−i NTr(

(4)

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