Naturalness made easy: two-loop naturalness bounds on minimal SM extensions
Received: August
Published for SISSA by Springer
to a Barbieri-Giudice-like 0 1 2 3
Open Access 0 1 2 3
c The Authors. 0 1 2 3
0 Melbourne , 3010 , Australia
1 School of Physics, University of Melbourne
2 ARC Centre of Excellence for Particle Physics at the Terascale
3 eld theory of the Standard
The main result of this paper is a collection of conservative naturalness bounds on minimal extensions of the Standard Model by (vector-like) fermionic or scalar gauge multiplets. Within, we advocate for an intuitive and physical concept of naturalness built upon the renormalisation group equations. In the e ective Model plus a gauge multiplet with mass M , the low scale Higgs mass parameter is a calculable function of MS input parameters de ned at some high scale h > M . If the Higgs mass is very sensitive to these input parameters, then this signi es a naturalness problem. To sensibly capture the sensitivity, it is shown how a sensitivity measure can be rigorously derived as a Bayesian model comparison, which reduces in a relevant limit ne-tuning measure. This measure is fully generalisable to any perturbative EFT. The interesting results of our two-loop renormalisation group study are as follows: for h =
Beyond Standard Model; E ective eld theories; Renormalization Group
-
Naturalness
made easy: two-loop naturalness bounds
minimal SM
extensions
Pl we nd \10%
ne-tuning" bounds on the masses of various gauge
weaker than for scalars; these bounds remain
nite in the limit
M +, weakening
Contents
1 Introduction 2 Physical Naturalness 2.1
Sensitivity measure
Fermion-like case
Scalar-like case
Naturalness bounds
Vector-like fermion
Complex scalar
Comment on the Planck-weak hierarchy
A Sensitivity measure as a Bayesian model comparison
Introduction
logical shortcomings.
Many of these can be addressed with minimal extensions of the
(88 GeV)2, appearing in the SM potential 2HyH + (HyH)2, is
naturalness problem.
paper: a parameter in a quantum
eld theory is \natural" if its measured value at low scale
a measure which quanti es sensitivity of 2(mZ ) to the high scale physics.
a physical Higgs naturalness problem?
Vector-like fermionic and scalar GMs of various
low scale to a high scale input parameter of the model.
We show that this sensitivity
on the masses of the various GMs.
and could in principle be applied to any model.
are discussed in section 5, and we conclude in section 6.
Physical Naturalness
cussion, let us appeal to an illuminating toy model.
Toy model
a renormalisation scale R > M takes the form
C2( R) might be comprised of SM and/or beyond-SM couplings. The RGEs allow
a naturalness problem.
M ) C2M 2 log
the observed low scale
under-shooting.
is the step function. It is now easy to see when a naturalness problem arises. If
either of C2M 2 or CT M 2 is
2(mZ ), then the input parameter
( h) must be
small change in 2
( h) ruins this cancellation, and thus the Higgs mass is unnatural, i.e.
( R) RG trajectory, whereby only a very particular input 2
( h) will lead to
2(mZ ); a small change in this value leads to signi cant over- or
cancellation between the unmeasurable bare mass and the cuto
regulator contribution
bounding the nite loop corrections to
2. Second, since the parameters which appear in
Sensitivity measure
description; more details can be found in appendix A.
perfectly measured set of m
evidence B for a model M is a function of the unconstrained input parameters I
0 =
(Im+1; : : : ; In):
(M; I0) ' t
where J is the m
I1 = log 2
can then be written as a function of the unconstrained parameters,
(M; I0) =
B(M0; I0)
B(M; I0)
input parameter at low scale. In our context, a large value of
essentially tells us that,
given a at prior density in log 2
( h), the observed value
2(mZ ) is unlikely [speci cally
with respect to a
at probability density in log 2(mZ )], i.e.
2(mZ ) is sensitive to the
2(mZ ), are approximately insensitive to the unconstrained inputs, B(M0; I0)
becomes independent of I0 and eq. (2.4) reduces to
is exact. This is clearly reminiscent of the Barbieri-Giudice
ne-tuning measure. A value
M) on the Je reys scale [40], or a 10%
ne-tuning from the Barbieri-Giudice perspective.
the renormalised mass parameter M k( h)
in the model. Our SM+GM models M are de ned by MS inputs at the high scale
We minimise over all unconstrained parameters apart from M k( h) to obtain a
sensitivity measure which depends only on M and
(M; h) =
unconstrained inputs in the vicinity of the minimum.
C2M 2; CT M 2
Fermion-like case
constrained by experiment so that Oi =
log 2(mZ ); log C1; log C2 and Ij =
(M; h) =
This is just a Barbieri-Giudice-like
ne-tuning measure comparing percentage changes
C1 log( h=mZ )
1 and taking
h > M , we see that
( h) is made up of two pieces:
@ log M k
= 1 +
trajectory (with slope C2M 2), the CT M 2 piece is due to the
nite threshold correction,
if C2M 2 or CT M 2 is
2(mZ ), as expected. This even holds in the limit where the high
scale approaches the heavy p (...truncated)