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Prepared for submission to JHEP

Is Natural SUSY Natural?

Edward Hardya

aRudolf Peierls Centre for Theoretical Physics, University of Oxford,

1 Keble Road, Oxford, OX1 3NP, UK

E-mail: e.hardy12@physics.ox.ac.uk

Abstract: We study the fine tuning associated to a ‘Natural Supersymmetry’ spectrum

with stops, after RG running, significantly lighter than the first two generation sfermions

and the gluino. In particular, we emphasise that this tuning should be measured with

respect to the parameters taken to be independent at the assumed UV boundary of the

renormalisation group flow, and improve the accuracy of previous approximate expressions.

It is found that, if running begins at 1016GeV?105GeV?, decreasing the UV stop mass

tuning of the theory. In contrast, it is possible to raise the first two generation sfermion

masses out of LHC reach without introducing additional tuning. After running, regions of

parameter space favoured by naturalness and consistent with LHC bounds typically have

IR stop masses of order 1.5 TeV (0.75 TeV), and fine tuning of at least 400 (50) for high

(low) scale mediation. We also study the fine tuning of theories with Dirac gluinos. These

allow for substantial separation of the gluino and sfermion masses and, regardless of the

scale of mediation, lead to relatively low fine tuning of order 50. Hence viable models can

still favour light stops, but this requires extra structure beyond the MSSM field content.

below 0.75 (0.4) of the weak scale Majorana gluino mass does not improve the overall fine

arXiv:1306.1534v3 [hep-ph] 2 Dec 2013

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Contents

1Introduction1

2Fine Tuning to Obtain a Light Stop4

3 Electroweak Fine Tuning in Models of Natural SUSY10

4 Dirac Gauginos for Natural SUSY21

5 Summary23

A Subleading Terms from Stop Back Reaction25

1 Introduction

With the LHC giving increasingly strong limits on supersymmetric spectra with universal

sfermion masses, models of ‘natural’ supersymmetry (SUSY), where only superpartners

directly involved in the tuning of the electroweak scale are light, provide an intriguing

alternative [1, 2]. Since the parton content of the proton means many production channels

of supersymmetric particles are strongest through the first two generation sfermions, such

spectra can relax collider limits dramatically and provide hope for an electroweak sector

without significant fine tuning [3–16]. However, as was quickly realised after their initial

proposal, it is difficult to preserve a natural spectrum, which requires light stops, during

running to the weak scale [17, 18]. On one hand, the heavy first two generation sfermions

tend to drive the stops tachyonic, while on the other, a gluino above the current experi-

mental limit will tend to pull the stops to unacceptably high masses. These running effects

manifest themselves in the electroweak sector as two loop contributions to the up type

Higgs’ soft mass.

Quantifying the fine tuning of a model is a useful tool to study the viability of particular

low energy spectra [19]. This approach has been applied in a large number of studies of

supersymmetric models, for example [20–36], has been used to strongly constrain spectra

with universal sfermion masses, and has also been studied in the context of natural spectra

[37–41]. In this paper, we first derive expressions for the fine tuning required to obtain stops

significantly lighter than gluinos and the first two generation sfermions. We then extend

previous approximate results for the fine tuning of the electroweak scale introduced due to

heavy gluinos and sfermions. Applying current experimental contraints these are used to

study the extent to which fine tuning may be evaded. Our main result is that if there is a

Majorana gluino with mass > 1.5TeV there is no fine tuning benefit to decreasing the stop

masses below roughly 1 TeV if mediation is from close to the GUT scale. This is because

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such theories necessarily contain a significant amount of fine tuning in the electroweak

sector from the gluino feeding into the Higgs mass at two loops. However, while there is no

benefit to reducing the stop mass, provided the stop is not too light (? 500GeV) doing so

does not actually make the tuning of the theory worse and is not actively disfavoured. As

a result of this, we are able to put a strong lower bound on the fine tuning of theories of

natural SUSY, even though there are regions of parameter space where the LHC has not

excluded light stops.

An important point for our study is that we assume particular renomalisation group

boundary conditions at some energy scale. The fine tuning of the theory is then measured

with respect to the parameters of the theory at this boundary, which are assumed to be

independent.1In contrast, the weak scale parameters have values which are strongly cou-

pled together by the renormalisation group equations and attempting to quantify the fine

tuning of a theory in terms of them has the capacity to miss important effects from running

(the importance of this has been emphasised in recent papers [42, 43]). Of course, choosing

the independent variables at the UV boundary automatically requires some assumptions

about the mediation of supersymmetry breaking, and in particular possible correlations

between soft terms at this scale. Additionally, we must assume there is no new physics

between the UV boundary and the weak scale that modifies the running, the possibility

that this assumption does not hold due to interactions in the SUSY breaking sector has

been studied in [44, 45]. For the majority of our study, we take the independent variables

to be the gluino mass (the other gauginos are less important for our study and we do not

need to assume a GUT structure), the stop mass and the mass of the first two generation

sfermions, which are assumed to be universal based on strong flavour constraints [46].2

Such a choice is reasonable; to obtain a natural spectrum typically requires boundary con-

ditions with heavy first two generation sfermions, an intermediate mass gluino and stops

with masses somewhat, but not too far, below the gluino. This is usually accomplished

by including several mediation mechanisms which couple to different visible sector states.

For example, the first two generation sfermions may gain their mass dominantly through

a D-term of an additional U(1) gauge group [52–57], while the gluino and stops gain their

mass either through another form of gauge mediation or gravity mediation. Hence, these

masses may be adjusted independently. Additionally, in both gravity mediation [58] and

the most general models of gauge mediation [59], the gauge fermion and sfermion masses

generated are independent.

There is an alternative scenario which is also well motivated. Suppose, the gluino and

stop masses at the UV renormalisation boundary are both generated through a single F-

term, as the result of an especially simple SUSY breaking sector and mediation mechanism.

Now, varying the gluino mass will be correlated to varying the UV stop mass, and hence

we should take the F-term to be our fundamental parameter. As we will discuss later,

this scenario actually makes the tuning of natural SUSY spectra substantially worse since

1Note however, the choice of the location of this boundary, and the set of independent parameters

there, is only physically meaningful once a complete UV theory, including all higher dimension operators,

is specified.

2Though see, for example, [47–51] for a discussion of ways in which this assumption may be relaxed.

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increasing the F-term increases the weak scale stop mass both directly though the UV

stop mass, and through the increased running from a more massive gluino. In this way

our study can be seen as providing a lower bound on the fine tuning obtained. A more

serious question is whether the left and right handed stop masses should be regarded as

one parameter, as is the case if both gain the majority of their soft mass through the same

mediation mechanism. This is expected to be the case in many models of natural SUSY,

however is not required in generic mediation. We give results for the both the case where

these are independent, and when they are not.

There is a possible proviso to our argument. It might be the case that the mediation

mechanism somehow favours UV spectra which, as the magnitude of the SUSY breaking

is varied, preserves a particular structure which minimises the running (this is the case

for focus point spectra [60, 61]). However, such a mechanism would need to couple the

stop, gluino, and first two generation sfermions in a highly non-trivial way despite their

soft masses coming from very different sources (typically R-symmetry preserving SUSY

breaking, R-symmetry breaking SUSY breaking and an additional D-term respectively),

and there seems to be no reason that SUSY breaking and mediation should know anything

at all about the MSSM renormalisation group equations. Therefore, this does not seem a

strong assumption.3

As a final caveat of our work, we have studied only the sensitivity of the electroweak

scale to the UV parameters. We make no attempt to quantify the probability, over the

theory space of SUSY breaking and mediation mechanisms, that the initial UV parameters

begin in the correct region to allow for a natural spectrum at the weak scale. Since, as

discussed, such a starting point requires multiple forms of mediation which, a priori, could

lead to a separation between the gluino and sfermion masses which is far too large to lead to

a viable natural spectrum at the weak scale. Hence, it may be thought that natural spectra

are rare over the space of models. However, there may be some hope in this direction by

linking the ratio of gluino to first two generation sfermion masses to another parameter of

approximately the correct size in the model, for example the parameter ξ2in string theory

or the ratio of fermion masses [56, 57].

While the main focus of our work is on conventional Majorana gauginos, an interesting

alternative is to introduce additional fields that allow the generation of Dirac gaugino mass

term. We study the electroweak fine tuning in a simple example of such a model, and find

that, independent of the mediation scale, it is comparable to a MSSM theory with very low

cutoff. Hence, this is a good option for reducing fine tuning in models where the mediation

scale is required to be high, for example if attempting to build a string-motivated UV

completion.

As is well known, there is also a tension between light stops and the observed Higgs

mass of ∼ 125GeV. At tree level in the MSSM, the Higgs mass is bounded by the mass of

the Z boson, and radiative corrections from fairly heavy stops are required to raise its mass

to the observed value (see, for example, [62]). For the purposes of this work, we assume

3In contrast focus point scenarios typically only involve one, simple, form of mediation to all MSSM

fields, hence can occur as a result of single numerical coincidence in the structure of the mediators which

seems far less artificial than would be required for a natural SUSY spectrum.

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this can be evaded through an NMSSM like model, in which an additional singlet is present

giving an extra tree level contribution to the Higgs mass. The extra field content of such

a model does not alter the leading dependence of the Higgs mass on the gluino, stops, and

sfermions during running so will not affect our fine tuning results, and is independently

motivated for its ability to solve the µ problem [63]. The extra field content will somewhat

change the fine tuning with respect to the soft Higgs mass, which we calculate within the

MSSM, however the parametric form will be unchanged, and ultimately we will find this

is not typically the dominant tuning. Even in the NMSSM very light stops are potentially

problematic, since in this case, the coupling, λ, of the singlet, S, to the Higgs through

the term λSHuHdmust be large at the weak scale [64]. Typically such values, lead to λ

running to a strong coupling regime before 1016GeV, although this does not necessarily

ruin the successful prediction of gauge unification [65]. In contrast, we will find that the

most natural regions of parameter space not yet excluded by LHC limits may have relatively

heavy stop masses, which allow λ to be small or the Higgs mass to be generated directly

in the MSSM without additional structure.

Turning to the structure of this paper, in Section 2 we discus the fine tuning of the

UV parameters required to obtain a light stop after running. Section 3 contains the main

results on the tuning of the electroweak VEV in natural scenarios, while Section 4 contains

our discussion of Dirac gauginos.

2 Fine Tuning to Obtain a Light Stop

We begin by briefly reviewing the fine tuning of the electroweak scale introduced by stops,

as was defined in the early phenomenological studies of supersymmetry [19]. The fine

tuning due to the weak scale values of the stops is given by

????

where, for future convenience, the tilde denotes that this is a fine tuning with respect to

the theory’s weak scale parameters. We will generally use the convention that soft terms

without their scale specified are evaluated at the UV boundary of the renormalisation flow

of the theory, ΛUV, which is typically the scale at which SUSY breaking is mediated.

It is straightforward to estimate˜Z˜t. Stops give a contribution to the up type Higgs

through running, which is given at leading log level by

˜Z˜t=

∂?logM2

logm2

Z

?

∂

?

˜t(MW)

?

????=

m2

M2

˜t

Z

∂M2

˜t(MW),

Z

∂m2

(2.1)

δm2

Hu(MW) =−3y2

t

8π2

?m2

u3(MW) + m2

Q3(MW) + A2

t(MW)?log

?ΛUV

m˜t

?

. (2.2)

Additionally, to a good approximation,

˜Zm˜

Q3=

????

2δm2

M2

Hu

Z

????. (2.3)

Hence the fine tuning parameter at this order is

˜Zm˜

Q3=

3

4π2cos(2β)

m2

v2M2

t

Z

log

?ΛUV

m˜t

?

m2˜

Q3(MW). (2.4)

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