Page 1

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

Page 2

Contents

1 Introduction1

2 Fine Tuning to Obtain a Light Stop4

3 Electroweak Fine Tuning in Models of Natural SUSY10

4Dirac Gauginos for Natural SUSY21

5Summary 23

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

– 1 –

Page 3

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.

– 2 –

Page 4

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.

– 3 –

Page 5

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)

– 4 –

Page 6

Normally, the parameters˜Zifor all of the variables i are compared, and the overall fine

tuning is defined as max

{˜Zi}

through stops, gluinos and the first two generation sfermions since these are the tunings

which are relevant for considering natural spectra. In Section 3 we compute the electroweak

fine tuning with respect to the UV values of these parameters to a higher order. In partic-

ular, this is necessary because we will be interested in running from the GUT scale, and

in this case the expansion parameter isb3α3

2π2 log

In complete theories, there are other important tunings due to the µ and Bµ parameters,

and depending on the details of the electroweak sector soft terms these may be significant.

Therefore, our analysis will give a lower bound on the fine tuning.

Now we turn to the fine tuning of the gluino and first two generation masses required

to obtain a light stop at the weak scale. This is defined as

????

which receives the greatest fine tuning.

The renormalisation group equations for the stops in the presence of heavy sfermions

are well known, for example from [17, 62]. Since we are interested in the effect of the gluino

and sfermion masses and these dominate the renormalisation group equation, it is sufficient

to include only the leading effects. Later we will see the next corrections are small. The

running is given by

d

dtm2

4π

where Ciis the Casimir of the stop state (and α1is GUT normalised). We further assume

the right handed bottom sfermion and the staus remain relatively light such that they

do not have a significant effect on the running of the stops, but not so light as to be

driven tachyonic during running. Giving these states a large mass would in general make

the running faster and the fine tuning worse. This is not a large effect and does not

significantly alter any of our conclusions. We take the heavy first two generations to have a

constant mass which is a reasonable approximation if they begin fairly heavy as in natural

spectra (in our numerical analysis we include the subleading effect from their running).4

Following [17], at this level of approximation the flow can be solved exactly to give

+

πbiαi(ΛUV)

1 +bi

??

. However, we will focus on the fine tuning introduced

?

1016

103

?

∼ 0.5, which is not especially small.

Yi=

∂

?

logm2

˜t(MW)

∂ (logi)

?

????,(2.5)

where i is one of M2

3, ˜ m2

1,2or m2

˜tevaluated at the UV boundary, and˜t is the stop state

˜t= −8

?

αi(t)CiM2

i+

2

π2

??

α2

i(t)Ci

?

˜ m2

1,2,(2.6)

m2

˜t(MW) =m2

˜t(ΛUV) −

?

i

2

biCi

1

?

1 +bi

2πlog

?

ΛUV

Mi(MW)

?

αi

?2− 1

M2

− 1

i

?

i

4

1

2πlog

?

ΛUV

˜ m1,2(MW)

?

αi

Ci˜ m2

1,2,

(2.7)

4We are interested in spectra where the stops are fairly light at the UV scale and remain relatively

light during running. Hence, the overall shift in their mass during running is ? 500GeV. The first two

generation’s dominant running is the same as the stops hence these run by a similar amount, which is

negligible if they start at O (10TeV).

– 5 –

Page 7

where the gauge beta-function coefficients are defined as

contribution from the first two generation sfermion turns off at an energy scale ˜ m2

the gaugino contribution is present until the scale Mi. Equation (2.7) is written in terms

of the UV values of the gauge couplings to avoid an extra term when varying with respect

to the soft masses.

In Fig.1 we plot the weak scale lightest stop mass as a function of the weak scale

gluino and first two generation sfermion masses, after running from 1016GeV with a UV

stop mass of 200GeV. This shows that, for a given gluino mass, above a certain sfermion

mass the stops run tachyonic and there is no viable electroweak spectrum. To obtain the

light stops needed for a natural spectrum requires M3and ˜ m1,2to be such that the stop is

in the thin strip close to this boundary. The relatively small effect of the gluino increasing

the mass of the first two generation sfermions during running is visible in the lower cut off

in this plot.

Now it is straightforward to write down the fine tuning with respect to the UV gaugino

and first two generation masses. There will be two contributions to the fine tuning, one

directly from the dependence on ˜ m2

1,2, and the other from dependence inside the logarithm,

d

dt

?

1

αi

?

= −bi

2π. Note that the

1,2while

Y˜ m2

1,2=

˜ m2

m2

1,2

˜t

∂m2

∂ ˜ m2

˜t

1,2

=

˜ m2

m2

1,2

˜t

?

i

4Ci

πbiαi(ΛUV)

1

1 +bi

2πlog

?

ΛUV

˜ m1,2

?

αi

− 1

+

˜ m2

m2

1,2

˜t

?

i

Ciα2

i(ΛUV)

π2

1

1 +bi

2πlog

?

ΛUV

˜ m1,2

?

αi

2

.

(2.8)

The second term from the variation of the logarithm is typically significantly smaller than

the first and acts to reduce the fine tuning. This is expected, if the mass of the first two

generation sfermions increases then there will be slightly less running. Similarly, we find

−M2

m2

˜t

YM2

i(MW)= −M2

i

m2

˜t

2

biCi

1

?

1 +bi

2πlog

?

ΛUV

Mi(MW)

?

αi

?2− 1

1

?

i

Ci

παi

?

1 +bi

2πlog

ΛUV

Mi(MW)

?

αi

?3.

(2.9)

It is clear that the greatest fine tuning from the heavy sfermions will occur on the left

handed stop. This is because, even though the beta function coefficients b2and b3have

opposite signs, their overall contributions to (2.8) go in the same direction. For the tuning

with respect to the gauginos, we focus on the gluino, which couples equally to the left and

right handed stops, since this is clearly dominant.

Finally, there is also a fine tuning with respect to the initial stop masses. This can

be evaluated as a perturbation to the trajectory obtained already. As discussed in the

– 6 –

Page 8

Introduction, it is unclear if the left and right handed stops should be treated as indepen-

dent variables. If the masses are independent, a perturbation to the initial left handed soft

mass, ∆m2˜

d

dt

Q3

and will also feed into the right handed soft and up type Higgs mass since the renormali-

sation group includes

Q3, will satisfy

?

∆m2˜

?

⊃

2y2

16π2∆m2˜

t

Q3,(2.10)

d

dt

d

dt

?∆m2

?∆m2

˜ u3

?⊃

?⊃

4y2

16π2∆m2˜

6y2

t

16π2∆m2˜

t

Q3, (2.11)

Hu

Q3. (2.12)

Since the beta functions are linear in m2

may be obtained by integrating the full one loop renormalisation equations (assuming

MSSM field content and interactions)

?

ΛUV

?

ΛUV

˜t, the evolution of the perturbation during running

∆m2˜

Q3(MW) =1

6

5 +

?m˜

?m˜

Q3

?(3y2

?(3y2

t/4π2)?

t/4π2)?

∆m2˜

Q3(ΛUV), (2.13)

∆m2

˜ u3(MW) =1

3

−1 +

Q3

∆m2˜

Q3(ΛUV). (2.14)

Similarly, a perturbation to the right handed stop gives

∆m2

˜ u3(MW) =1

3

?

?

2 +

?m˜ u3

?m˜ u3

ΛUV

?(3y2

?(3y2

t/4π2)?

t/4π2)?

∆m2

˜ u3(ΛUV), (2.15)

∆m2˜

Q3(MW) =1

6

−1 +

ΛUV

∆m2

˜ u3(ΛUV). (2.16)

Numerically, the expressions (2.13) and (2.15) dominate. therefore, the fine tunings are

approximately

?

?

3

Ym2

˜

Q3=

m2˜

m2˜

Q3(ΛUV)

Q3(MW)

m2

˜ u3(ΛUV)

m2

˜ u3(MW)

5

6+1

?m˜ u3

6

?m˜

?(3y2

Q3

ΛUV

?(3y2

t/4π2)

t/4π2)?

,(2.17)

Ym2

˜

u3=

1

ΛUV

+2

3

?

.(2.18)

The behaviour of these expressions is interesting. If there is a small separation between

the mediation scale and the weak scale then

?m˜

Q3

ΛUV

?(3y2

t/4π2)

∼ 1 and the fine tuning

Ym2

˜

Q3∼

m2

˜

Q3(ΛUV)

m2

˜

Q3(MW)as is the leading order expectation. However if there is a large separation

between these scales then running proceeds for sufficiently long that the stop back reaction

from a perturbation suppresses the initial perturbation, reducing the fine tuning. For a

mediation scale of 1016GeV,

?m˜

Q3

ΛUV

?(3y2

t/4π2)

∼ 0.1, (2.19)

– 7 –

Page 9

so this can be a significant effect in the models we are interested in. Not surprisingly the

tuning of the left handed stop is greater since it is less strongly damped by the renormali-

sation.

In the case where these two stop masses are linked, the renormalisation group equations

for the perturbation are modified since the left handed stop perturbation feeds into the

right handed stop perturbation and vice versa. These are easily integrated to obtain

??m˜ t3

?

ΛUV

∆m2˜

Q3(MW) =1

3ΛUV

?m˜ t3

?(3y2

?(3y2

t/4π2)

+ 2

?

∆m2

˜ t3(ΛUV),(2.20)

∆m2

˜ u3(MW) =1

3

2

t/4π2)

+ 1

?

∆m2

˜ t3(ΛUV). (2.21)

Therefore,

Ym2

˜ t3=

m2

m2˜

˜ t3(ΛUV)

Q3(MW)

??m˜ t3

ΛUV

?(3y2

t/4π2)

+ 1

?

. (2.22)

As before, for ΛUV not too large, the damping is not significant and these expressions

reduce to the leading order expectation Ym2

˜ t3∼

m2

m2

˜ t3(ΛUV)

Q3(MW). However if ΛUV is close to the

˜

GUT scale the difference can be significant.

It is useful to gain some physical insight by finding approximate expressions in various

limits. For ΛUV = 1016GeV, (2.8) and (2.9) reduce to

Y˜ m2

1,2? 0.03˜ m2

1,2

m2

˜t

(2.23)

YM2

3? 0.74M2

3(MW)

m2

˜t

, (2.24)

where the sfermion and stop masses are evaluated at the UV boundary. For a low scale

model with ΛUV = 106GeV we obtain,

Y˜ m2

1,2? 0.0079˜ m2

1,2

m2

˜t

(2.25)

YM2

3? 0.36M2

3(MW)

m2

˜t

. (2.26)

Of course, the fine tuning is significantly smaller in the low scale case. In the high

scale case the stop will tend to be pulled up to within

while in the low scale case the stop will be pulled to ∼ 0.6 of the gluino mass. The first

two generation sfermions have a smaller effect, typically decreasing the stop masses by

an amount given by ∼ 0.2 and ∼ 0.08 of their mass in the high and low scale mediation

respectively. Also, the subleading correction from the variation of the logarithm is more

important in models of low scale mediation, which matches intuition. These are found to

agree within ∼ 20% with the variations evaluated numerically using the code SOFTSUSY

[66]. The next correction term is due to the back reaction from the contribution to m˜t

√0.74 ∼ 0.9 of the gluino mass,

– 8 –

Page 10

Stop

Mass /

GeV

Figure 1. The stop mass obtained at the weak scale as a function of the weak scale gluino and

first two generation sfermion masses, after running from the GUT scale at 1016GeV assuming an

initial mass of 200GeV. The lower cutoff is due to the gluino increasing the the first two generation

sfermions masses during running, while the upper cutoff is due to the stop running tachyonic above

this line.

to its own renormalisation group equation. In Appendix A, we compute this effect and

include it in our numerical work.

In order to gauge the severity of these fine tunings, recall the expression for the tuning

of the electroweak scale (we take cos(2β) = 1 which gives a minimum value for the tuning),

˜Z˜ Q3∼˜Z˜ u3∼ 9.1 × 10−6m2

˜ Q3/GeV2log

?ΛUV

m˜t

?

. (2.27)

A minimal SUGRA spectrum with sfermions and gluinos at 2500GeV would have a Z˜ Q3∼

350 for ΛUV = 106GeV and Z˜ Q3∼ 1500 for ΛUV = 1016GeV. In contrast, a reasonable

natural spectra has m˜t= 200GeV, ˜ m1,2∼ 104GeV and M3∼ 2500GeV at the weak scale.

If ΛUV = 1016GeV, we obtain

Y˜ m2

1,2∼ 80,YM2

3∼ 115,Ym2

˜ t3∼ 15,Ym2

˜

Q3∼ 20.(2.28)

– 9 –

Page 11

500100015002000 2500 3000

0

2000

4000

6000

8000

10000

12000

14000

Gluino Mass ê GeV

Generation 1 & 2 Sfermion Masses ê GeV

500 10001500200025003000

0

2000

4000

6000

8000

10000

12000

14000

Gluino Mass ê GeV

Generation 1 & 2 Sfermion Masses ê GeV

Fine

Tuning

20

60

100

140

Figure 2. The fine tuning required to obtain a stop mass of 200GeV at the weak scale, with Left:

A UV boundary of 1016GeV and Right: A UV boundary of 106GeV, as a function of the weak

scale gluino mass and the UV value of the first two generation sfermion masses.

For a low scale model, with ΛUV = 106GeV

Y˜ m2

1,2∼ 20,YM2

3∼ 50,Ym2

˜ t3∼ 20,Ym2

˜

Q3∼ 25.

?

(2.29)

We define the overall fine tuning Y to be given by Y = max

Fig.2 we show the fine tuning required to obtain a stop of mass 200GeV at the weak scale

in the plane M3(MW), ˜ m1,2evaluated at the weak scale for high and low scale mediation,

assuming the two stop masses are not independent in the UV. These examples demonstrate

our first result, it is possible to obtain a fairly light stop in the presence of a gluino mass

> 2TeV and first two generation sfermions with mass > 5TeV with a tuning of order

5 ÷ 100 depending on the scale of mediation. By itself this is not a large tuning compared

to that found in the electroweak sector of typical MSSM models or extensions, hence a

light stop should not be regarded as a particularly tuned scenario in itself.

{YM2

3,Y˜ m2

1,2,Ym2

t3}

?

. In

3Electroweak Fine Tuning in Models of Natural SUSY

We now turn to the question of whether a natural SUSY scenario, compatible with current

limits, can lead to an electroweak sector with low fine tuning. For simplicity, we assume

the MSSM parameters and interactions and that tanβ is fairly large, in which case the

electroweak scale is given by

M2

Z= −2

?

m2

Hu+ |µ|2?

+ O

?

1

tan2β

?

.(3.1)

In particular, we want to know the variation in the electroweak VEV as the UV parameters

of the theory are varied in a natural SUSY theory.

– 10 –

Page 12

Consider the dependence on the first two generation sfermion masses. If we take just

the 2-loop expression for the beta function of the up type Higgs mass, arising from SU(2)

and U(1) gauge interactions,

dm2

dt

Hu

⊃

2

π2

??

α2

i(t)Ci(Hu)

?

˜ m2

1,2,(3.2)

we would obtain a contribution containing a single logarithm, giving a fairly small (but not

completely negligible) tuning if ˜ m1,2is of order a few TeV.5However, we know that the

electroweak VEV has a strong dependence on the stop mass, which itself has a significant

dependence on ˜ m1,2, hence is clearly associated to a tuning. The appropriate way to

measure this is through a total derivative (where Z˜ m2

respect to the UV value of ˜ m2

1,2)

d?logM2

d

1,2is defined as the fine tuning with

Z˜ m2

1,2=

Z

?

?

log ˜ m2

1,2

?,(3.3)

which includes the effect of the sfermions feeding into the stops which then feed into the

up type Higgs. This gives a three loop, logarithm squared contribution which can be

significant, especially if the mediation scale is high. The shift in the stop mass as a result

of a change in sfermion mass depends on the energy scale. Therefore it is necessary to

integrate over all energy scales to obtain the weak scale fluctuation in the Higgs mass,

?tM3

?tM3

∆m2

Hu(MW)|M3=

tΛ

∆βm2

Hu(t)|M3dt,

∂βm2

∂m2

˜t(t)

(3.4)

=

tΛ

Hu(t)

∆m2

˜t(t) +

∂βm2

∂ ˜ m2

Hu(t)

1,2(t)∆˜ m2

1,2(t),(3.5)

where the first term is the contribution through the stop, and the second is the direct

two loop contribution. The fine tuning is then given by (using the approximate relation

between the weak scale up type Higgs mass and the Z mass (2.3))

d?logM2

dlog ˜ m2

1,2

∂ log˜ m2

1,2

Z

?

?? =

∂

??

?tm1,2

tΛ

∂?d

dt

?log?M2

Z

???

∂m2

˜t(t)

m2

˜t(t) +∂?d

dt

?log?M2

Z

???

∂ ˜ m2

1,2(t)

˜ m2

1,2(t)dt

(3.6)

=2˜ m2

1,2

M2

Z

∂

∂

?

˜ m2

1,2

?

?tm1,2

tΛ

∂?d

dtm2

∂m2

Hu

?

˜t(t)

m2

˜t(t) +∂?d

dtm2

∂ ˜ m2

1,2(t)

Hu

?

˜ m2

1,2(t)dt. (3.7)

Using the expressions (2.4), (2.7) and (3.2), and including a factor to two to account for

5The other two loop contributions are all proportional to the yukawas of the first two generation fermions

squared, and are completely negligible.

– 11 –

Page 13

the fact that the coupling occurs through both the left and right handed stops, we obtain

?tm12

1 +biαi

Z˜ m2

1,2=

˜ m2

M2

1,2

Z

∂

∂

?

˜ m2

1,2

?

tΛ

3m2

t

4π2v2cos(2β)

?

i

8Ci

πbiαi

?

1

1 +biαi

2π(tΛ− t)− 1

?

˜ m2

1,2(3.8)

+

2

π2

?

?

α2

iCi(Hu)

2πlog

?

ΛUV

˜ m1,2

?

˜ m2

?

1,2dt(3.9)

=

˜ m2

M2

1,2

Z

∂

∂

?

˜ m2

1,2

?

?

i

?

A8Ci

πbiαi

log

?

Λ

˜ m1,2

?

−

2π

αibilog1 +biαi

2π

log

?

Λ

˜ m1,2

???

(3.10)

+4Ci

πbiαi

1

1 +bi

2πlog

?

ΛUV

˜ m1,2(MW)

?

αi

− 1

˜ m2

1,2,(3.11)

where A =

the fine tuning. The largest is from the direct variation of the initial sfermion masses,

while the second contribution comes from varying the end point of the logarithm, and is

somewhat smaller.

Intuitively, it is clear what is occurring in the first term. There are two fine tunings

occurring at different levels in the theory, the electroweak VEV is tuned by the mass of

the stop, which is itself tuned by the first two generations. The first term in (3.11) is

effectively the result of multiplying these together, and weighing them by a factor less than

1 to take into account that the gluino only generates a change in the stop mass after some

running has occurred. Numerical evaluation shows that the second term (the two loop

direct contribution) typically gives a shift in the mass squared of ∼ 10 ÷ 50% of the first

term, and acts in the opposite direction reducing the overall fine tuning. This is because

the direct contribution decreases the Higgs mass squared, while the indirect contribution

decreases the stop mass squared resulting in a less negative Higgs mass squared.

For the natural spectra we are interested in, the shift in the Higgs mass directly from

the gluino is completely negligible compared to the logarithm squared contribution that

occurs through the stop mass, hence we focus on the later.6This gives

=M2

M2

Z

3

1 +b3α3

2πlog

3m2

t

4π2v2cos(2β). As in the previous section, each term gives two contributions to

ZM2

3=M2

3

M2

Z

A

∂

∂?M2

∂

∂?M2

3

?

?4

?tM3

tΛ

4

b3C3

1

?

1 +b3α3

2π(tΛ− t)

?

Λ

M3

?2− 1

M2

3dt(3.12)

3

A

b3C3

b3α3

2πlog2?

Λ

M3

?

?M2

3.(3.13)

Next, we turn to the tuning with respect to the initial stop mass. Since the renormalisation

group equation governing the behaviour of a perturbation at the UV boundary of the stop

6The direct gluino contribution is two loop but only enhanced by a single logarithm compared to the

two loop, two log enhanced contribution we study.

– 12 –

Page 14

500 1000150020002500

0

100

200

300

400

Mass ê GeV

Fine Tuning

ué3 & Q

té3

é3

M3

mé12

5001000 150020002500

0

200

400

600

800

1000

1200

1400

Mass ê GeV

Fine Tuning

ué3 & Q

té3

é3

M3

mé12

Figure 3. The fine tuning in the electroweak sector as a function of the soft parameters, for low

scale mediation with ΛUV = 106GeV (top) and high scale mediation ΛUV = 1016GeV (bottom).

The plots are a function of the weak scale gluino mass since its running is fairly independent of the

other parameters in the theory. The other masses are the values at the mediation scale, which may

run to smaller or larger values when evolved to the weak scale.

mass may be solved exactly (at one loop order), as in (2.14), we can evaluate the shift in

the low energy Higgs soft mass directly. This leads to

??m˜

m2˜

Q3

M2

Z

∂m2˜

Q3

∆m2

Hu(MZ) =1

2

Q3

ΛUV

?(3y2

t/4π2)

− 1

?

∆m2˜

Q3(ΛUV) (3.14)

Zm2

˜

Q3=

∂

??

??m˜

Q3

ΛUV

?(3y2

t/4π2)

− 1

?

∆m2˜

Q3(ΛUV), (3.15)

– 13 –

Page 15

for the left handed stop. The expression for the right handed stop is given by

??m˜ u3

Zm2

˜

u3=

m2

M2

˜ u3

Z

∂

∂

?

m2

˜ u3

?

ΛUV

?(3y2

t/4π2)

− 1

?

∆m2

˜ u3(ΛUV).(3.16)

Alternatively, if we regard the UV masses of the left and right handed stops as one variable

a similar computation easily gives

??m˜ t3

Zm2

˜ t3= 2m2

˜ t3

M2

Z

∂

∂

?

m2

˜ t3

?

ΛUV

?(3y2

t/4π2)

− 1

?

∆m2

˜ t3(ΛUV). (3.17)

If we assume the Higgs sector of the MSSM, solving the same set of renormalisation group

equations, gives the tuning from a variation in the initial soft mass m2

??mHu

Huof

Zm2

Hu=m2

Hu

M2

Z

∂

∂?m2

Hu

?

ΛUV

?(3y2

t/4π2)

+ 1

?

∆m2

Hu(ΛUV).(3.18)

Assuming tanβ is moderately sized, it is straightforward to check that the tuning with

respect to m2

complicated, for example in the NMSSM, the exact expression here will be modified however

it is still expected to still take the form Zm2

ΛUV ∼ mHuso there is very little running.

Finally, we turn to the µ and Bµ parameters. These do not feed strongly into other

soft masses, and the tuning with respect to them is given by

Hdis negligible compared to that from m2

Hu. If the Higgs sector is more

Hu? 2m2

Hu

M2

Z, with the equality satisfied if

Zµ2 = 2µ2(ΛUV)

M2

Z

∂µ2(MZ)

∂µ2(ΛUV),

∂Bµ(MZ)

∂Bµ(ΛUV),

(3.19)

ZBµ= 2Bµ(ΛUV)

M2

Z

(3.20)

where the dependence of MZ on Bµ arises from the terms which are higher order in

1

tanβ. Assuming an MSSM Higgs sector and solving the renormalisation group equations

numerically, we find that for µ = 400GeV and Bµ = 200GeV at the weak scale

Zµ2 ∼ 40,

ZBµ∼ 10,

(3.21)

(3.22)

(3.23)

for both high and low scale mediation. Since these values of µ and Bµ are allowed by

collider constraints, and it will turn out that the tunings are less than those from the

stops, gluinos, and sfermions, the tunings from these parameters may be neglected from

this point onwards. Once these parameters are fixed, the Higgs soft mass in the IR, and

therefore after renormalisation flow at the UV boundary, is also fixed by (3.1).7

7An alternative but equivalent approach would be to fix the UV boundary stop soft masses at a relatively

small value in which case µ and Bµ would be determined by the same relation.

– 14 –

Page 16

Having given expressions for the individual parameters at the renormalisation bound-

ary scale, the overall fine tuning is taken to be ∆ = max({Zi}). Initially, we focus on the

fine tuning introduced by the gluino mass, stop mass, and sfermion masses which are fairly

independent of the exact details of the Higgs sector. In contrast, the fine tuning from the

Higgs soft mass m2

Huis dependent on both the µ/Bµ parameters, and whether the theory

is the MSSM, the NMSSM, or some other extension (which is required in order to obtain

the correct physical Higgs mass in some regions of parameter space). As a result, we study

the fine tuning from m2

Huin a typical MSSM Higgs sector seperately at the end of this

Section. There it is seen that the tuning introduced is typically slightly smaller, but of the

same order of magnitude, as that due to the other parameters. In addition, the regions

with the lowest stop, gluino and sfermion fine tuning coincide with the regions where the

Higgs has the lowest fine tuning. Therefore, the conclusions we draw about the overall

tuning of the theory in the discussion that follows are valid, despite the omission of this

important parameter.

Returning to the stop, gluino and sfermion soft masses, expanding the fine tuning

expressions (3.11), (3.13) in the parameterb3α3

2πlog

dependence recovers the expressions used in previous papers such as [38].8However, since

α3 is fairly large over all energy scales, and we are potentially interested in high scale

models which can have large logarithms, we retain the full dependence in our numerical

studies. In Fig.3 we plot the fine tuning obtained as a result of the UV soft parameters

for low and high scale mediation. We include both the cases where the stop masses are

independent in the UV and when they are not. Not surprisingly, when they are both set

by one parameter the fine tuning is rather worse since both feed into the up type Higgs

mass simultaneously.

The physics of these expressions is clear, for a given UV stop mass a larger gluino

or sfermion mass is never actively favoured since they lead to greater fine tuning of the

electroweak scale though their effect on the running of the stop.9However, provided ZM2

and Z˜ m2

worse (at least with the measure of fine tuning adopted here). Hence, collider bounds can

be somewhat alleviated without introducing fine tuning in the style of natural SUSY. It

is interesting to ask what is the ratio of m2

tuning. In particular, suppose we fix the gluino mass to be 2TeV at the weak scale, we

wish to know the maximum UV masses the stop and first two generation sfermions may

have before they dominate the fine tuning. In Fig.4 we plot the UV masses of the stops

and first two generation sfermions for this scenario, both for the case of the left and right

handed stops being independent, and when they are not. Hence, if the gluino is at 2TeV,

there is no fine tuning benefit to having UV stop masses below 1 ÷ 1.5TeV for GUT scale

mediation, and 0.5 ÷ 1TeV for very low scale mediation. From the bottom panel of Fig.4

it is clear that a gluino of this mass forces the tuning of the electroweak scale to be at

?

ΛUV

MW

?

and retaining only the leading

3

1,2remain smaller than Zm2

˜ t, increasing them does not actually make the fine tuning

˜t, M2

3, and ˜ m2

1,2which saturates a given fine

8Note this leads to a factor 2 difference in some expressions since we have included the finite energy

range required for the gluino to shift the stop mass.

9It will be seen later that for a given weak scale stop mass this does not necessarily hold as the UV stop

mass is then a function of the gluino and sfermion masses.

– 15 –

Page 17

68 10 121416

0

100

200

300

400

Log10HLUVê GeVL

Fine Tuning

6810 121416

800.

1000.

1200.

1400.

1600.

1800.

2000.

12000.

14000.

16000.

18000.

20000.

22000.

Log10HLUVê GeVL

UV Stop Mass ê GeV

UV 1st & 2nd Gen. Sfermion Masses ê GeV

Figure 4. Top: The UV stop (red) and sfermion masses (blue) that lead to the same fine tuning

of the electroweak scale as a gluino with weak scale mass of 2TeV as a function of the mediation

scale. We show both the case where the left and right handed stops are independent parameters

(solid lines) and when they are fixed equal (dashed). Lowering the stop or sfermion masses below

these masses does not improve the fine tuning of the theory, and hence this graph limits the extent

to which a natural spectrum can be obtained. Bottom: The fine tuning corresponding to a 2TeV

gluino as a function of mediation scale. By construction, this is the same as the fine tuning generated

by stops at the masses in the top panel. If fine tuning better than 1% is imposed then the mediation

scale is limited to ΛUV < 107GeV.

least ∼ 400 if running begins at the GUT scale. In contrast, we see it is easily possible to

separate the first two generation sfermions significantly from the gluino and stops without

increasing the fine tuning of the theory, which is clearly beneficial for collider limits.

Of course, the relevant quantities for collider physics are the weak scale masses, and

– 16 –

Page 18

40006000

Sfermion Mass ê GeV

80001000012000

0

500

1000

1500

2000

Stop Mass ê GeV

500

550

600

650

500010000

Sfermion Mass ê GeV

1500020000

0

200

400

600

800

1000

1200

1400

Stop Mass ê GeV

75

100

125

150

175

Fine

Tuning

Fine

Tuning

Figure 5. The electroweak fine tuning as a function of the weak scale sfermion and stop masses

(assuming the two stops are not independent) with a weak scale gluino mass of 2TeV for Left:

ΛUV = 106GeV, and Right: ΛUV = 1016GeV. The regions below the dashed black line have a

tachyonic stop mass at the UV boundary. Since sfermion masses > 3TeV are not constrained by

collider limits, it is clear that for low scale mediation there is no improvement in fine tuning through

decreasing the stops below ∼ 1.4TeV. For high scale mediation, especially if we demand the stop is

not tachyonic at the UV boundary, the majority of the region with the lowest fine tuning actually

has a fairly heavy weak scale stop ∼ 1.5TeV.

the running of the stops depends on the masses of the sfermions and the gluinos. As a

result, the regions in Fig.4 which minimise fine tuning with a given gluino mass can be

somewhat shifted. Therefore we plot the electroweak fine tuning as a function of the weak

scale stop mass and first two generation sfermion masses with the weak scale gluino mass

fixed at 2TeV, for low and high scale mediation, in Fig.5.10The conversion is carried out

by numerically solving the renormalisation group equations between their UV boundary

and the weak scale. It is assumed the two stops are not independent, however this does

not qualitatively affect our conclusions. In these plots, due to the fixed gluino mass, the

smallest possible electroweak fine tuning is ∼ 60 and ∼ 450 for low and high scale mediation

respectively in agreement with Fig.4. Hence, the large areas of parameter space with the

lowest fine tuning in the centre of both plots have their fine tuning dominated by the gluino.

It is clear that for low scale mediation, there is no particular preference for the weak

scale stop mass to be much lighter than ? 1.5TeV. For high scale mediation, the largest

region of parameter space with low fine tuning actually has relatively large stop masses,

∼ 1.5TeV. In this case, heavy stop masses are even further favoured if we demand the

stop is non-tachyonic at the boundary. This is a reasonable restriction since such boundary

conditions can lead to deep colour breaking vacua in the early universe.11As the sfermions

10The weak scale masses here are actually ¯

correction to convert to the physical stop mass, but this is a small correction.

11Although the existence of colour charge breaking vacua is not necessarily problematic if the color

preserving vacua is metastable on timescales longer than the age of the universe [67–69], but there is still

the problem of why our universe settled in the metastable vacuum during its early evolution.

MS masses and not pole masses. There is an additional finite

– 17 –

Page 19

10001500200025003000

0

500

1000

1500

2000

2500

Gluino Mass ê GeV

Stop Mass ê GeV

100

200

300

400

500

10001500200025003000

0

500

1000

1500

2000

2500

Gluino Mass ê GeV

Stop Mass ê GeV

250

500

750

1000

1250

Fine

Tuning

Fine

Tuning

Figure 6. The electroweak fine tuning as a function of the weak scale gluino and stop masses

(assuming the two stops are not independent) with the sfermion mass fixed at 6 TeV for the cases

Left: ΛUV = 1016GeV and Right: ΛUV = 106GeV. The regions below the dashed black line have

a tachyonic stop mass at the UV boundary.

tend to increase the stop mass during running up in energy, the maximum weak scale stop

mass that results in a tachyonic stop in the UV is decreased as the first two generation

sfermions are made heavier.

In Fig.5 contours of constant UV stop mass are approximately circle arcs concentric

with the tachyonic contour. Starting from the line of tachyonic UV stops and moving

outwards, the fine tuning starts to increase as new fine tuning contours are reached. This

is the transition from the region dominated by the gluino tuning, to a region where the

UV stop mass is the largest fine tuning. From Fig.4, this occurs when the UV stop mass

is roughly ∼ 700GeV and ∼ 1000GeV for low and high scale mediation respectively.

On the far right side of the right panel, there is also a region where the sfermion

controls the fine tuning, indicated by the vertical contours. Additionally, in the top right

of both plots of Fig.5, there is a region where, if the sfermion mass was fixed, decreasing

the stop mass would reduce the theory’s fine tuning. However, since the LHC does not

strongly constrain the first two generation sfermion masses in the ranges we are considering

(in contrast to the gluino), we are not forced into this region of parameter space. Hence,

the region with intermediate scale sfermion masses and relatively heavy stops, which has

the lowest fine tuning possible for a fixed gluino mass, is favoured.

We also show the electroweak fine tuning as a function of the weak scale stop mass and

gluino mass, with the first two generation sfermions fixed, in Fig.6. In these plots there is

a lower bound on the fine tuning as a result of the fixed sfermion mass, which is reached in

the dark blue region in the left plot and the white region of the right plot. Considering a

gluino mass of ∼ 2TeV with high scale mediation, the contours are vertical for small stop

masses. This is yet another sign that the gluino is dominating the fine tuning in this region,

and the weak scale stop mass is unimportant provided it is ? 1.5TeV. In the tachyonic

regions it can be seen that increasing the weak scale stop mass can actually improve the fine

– 18 –

Page 20

500010000

Sfermion Mass ê GeV

1500020000

0

200

400

600

800

1000

1200

1400

Stop Mass ê GeV

75

100

125

150

175

200

4000 6000

Sfermion Mass ê GeV

??Z2

80001000012000

0

500

1000

1500

2000

Stop Mass ê GeV

500

600

700

800

900

1000

Fine

Tuning

Fine

Tuning

Figure 7. The electroweak fine tuning, using the measure δ =

scale sfermion and stop masses (assuming the two stops are not independent) with a weak scale

gluino mass of 2TeV for Left: ΛUV = 106GeV, and Right: ΛUV = 1016GeV. The regions below

the dashed black line have a tachyonic stop mass at the UV boundary. As a result of the tuning

introduced by the sfermions using this measure, lighter sfermions which correspond to heavier weak

scale stops are favoured.

i, as a function of the weak

tuning. This occurs since increasing the stop mass leads to a less tachyonic UV boundary

????

of both plots in Fig.5.

For gluino masses of ? 1.5TeV, the 6 TeV sfermions are far below their critical mass

where significant tuning is introduced (for example by examining Fig.4). As discussed, this

is the most plausible scenario for natural spectra given LHC bounds, and the tuning is

dominated by the stops or gluino. In these regions the upward pull of the gluino during

running leads to fairly large stop masses, especially if we require the stop is not tachyonic

at the UV boundary.

stop mass. As a result the ratio

m2

m2

˜ t(ΛUV)

˜ t(MW)

????is smaller, and the electroweak fine tuning is

decreased if this is dominated by the stop mass. This can also be seen in the bottom left

If instead an alternative definition of fine tuning, δ =

results are obtained. In this case, increasing the gluino or first two generation soft masses

always increases the fine tuning, however until the critical points obtained above are reached

this increase only makes the fine tuning worse by a modest amount. Above the critical

soft masses, the fine tuning increases quickly as the soft masses are increased. In Fig.7

we plot the tuning as a function of the weak scale stop and first two generation sfermion

masses, with the weak scale gluino fixed at 2TeV. The fine tuning pressure from the

sfermion masses actually results in the regions with the smallest fine tuning having large

stop masses.

Additionally, we briefly consider an alternative scenario discussed in the Introduction,

where both the gluino and stop masses depend on the same F-term in the theory. In this

case, F2is the fundamental parameter we should measure fine tuning with respect to. It

??

iZ2

i, is used, broadly similar

– 19 –

Page 21

40006000

Sfermion Mass ê GeV

80001000012000

0

500

1000

1500

2000

Stop Mass ê GeV

50

100

150

200

250

300

40006000

Sfermion Mass ê GeV

80001000012000

0

500

1000

1500

2000

Stop Mass ê GeV

500

1000

1500

2000

2500

3000

Fine

Tuning

Fine

Tuning

Figure 8. The electroweak fine tuning from the initial up type Higgs mass, as a function of the

weak scale sfermion and stop masses (assuming the two stops are not independent) with a weak

scale gluino mass of 2TeV for Left: ΛUV = 106GeV, and Right: ΛUV = 1016GeV. The regions

below the dashed black line have a tachyonic Higgs mass at the UV boundary.

is easy to see this scenario will be more fine tuned for a given gluino mass than our main

case. Parametrically, the gluino mass is given by M3∼

m2

˜t∼

UV

F2generates a 1% increase in both the gluino and stop mass squared. As a result the fine

tuning is worse than if the gluino and stop were independent variables.

Finally, we return to the issue of the tuning as a result of the soft mass m2

µ = 400GeV, in Fig.8 we plot this fine tuning as a function of the weak scale stop and

sfermion masses, with the weak scale gluino fixed at 2TeV in exact analogy to Fig.5. The

fine tuning is calculated by numerically running the IR soft Higgs mass which gives the

correct electroweak scale, to the UV boundary and evaluating (3.18).12Clearly, the tuning

from the Higgs soft mass is not especially small. This is to be expected since the Higgs soft

mass, of course, appears at tree level in the electroweak VEV. However, in the regions of

lowest fine tuning, the tuning from the Higgs mass is typically slightly smaller than that

from the other parameters. The plot also shows that in the regions of lowest fine tuning the

UV Higgs mass is not far near zero, and there are large parts of parameter space with small

fine tuning where the Higgs soft mass squared (at the UV boundary of the renormalisation

group flow) is positive. For low scale mediation, the part of parameter space with tuning

less than 50 has??m2

low fine tuning actually coincide quite closely with the regions where the other parameters

have low fine tuning. Therefore, our previous estimates of the fine tuning and favoured

regions can be valid even when the details of a Higgs sector are included.

F2

M2

UV

and the stop mass also by

F2

M2

where MUV is a typical mass at the mediation scale. Hence, a 1% increase in

Hu. Taking

Hu

??? (500GeV)2at the UV boundary of the RG flow, and for high scale

mediation the region with tuning less than 500 has??m2

Hu

??? (1000GeV)2. The regions with

12We assume vanishing A-terms although these may be important for generating the correct physical

Higgs mass in some theories, and if large modify the running slightly.

– 20 –

Page 22

4Dirac Gauginos for Natural SUSY

In this section we consider an interesting extension of the MSSM, Dirac gluinos. As is well

known these provide an effective way of shielding the stop from gaining large corrections

compared to the usual Majorana case. More precisely, in such theories there is an N = 2

supersymmetry in the gauge sector which means there are no infinite log enhanced correc-

tions to the stop mass from the gluino. The only term is a finite piece generated below

the scale where the heaviest part of the effective N = 2 multiplet is integrated out, which

is typically the sgluon (the new scalar octet partner of the gluon), and above the mass of

the gluino. As noted by many authors, Dirac gluinos provide a compelling mechanism for

maintaining a supersymmetric spectrum without significant fine tuning [70–76]. Hence, it

is interesting to quantify the fine tuning obtained in such models.

We focus on a simple model, following [70, 75]. There is an additional U(1) gauge

group which obtains a D-term expectation value, and has field strength W?. This couples

to the visible sector N = 2 gauge multiplet, which can be written in N = 1 notation as a

vector multiplet with field strength W, and a chiral multiplet A in the adjoint of the gauge

group, only through a term

?

It can be shown that this operator also induces a mass for the real component of the sgluon,

˜ m2

is no direct coupling between the SUSY breaking sector and the sfermions. Instead these

are generated only by radiative corrections from the gauge sector as discussed in detail in

[70]. The induced stop soft mass is given by

d2θ

√2W?

MUV

α

Wα

jAj.(4.1)

i, of size ˜ m3= 2M3, where Miis the Dirac gaugino mass.13In this minimal model there

∆m2

˜t=

?

i

Ciαi

π

M2

ilog

?˜ m2

i

M2

i

?

,(4.2)

=C3α3M2

3

π

log(4),(4.3)

where we have included only the dominant gluino contribution. The up type Higgs receives

a contribution to its mass from the stop which is only present in the running between the

stop soft mass and the scale at which this mass is generated. Since the stop mass is

generated only in the small energy range between the sgluon and gluino masses, it is a

reasonable approximation to assume it is tuned on instantaneously at the gluino mass.14

Then the mass shift in the up type Higgs is given by

∆?δm2

Hu

?= −3λ2

t

8π2m2

˜tlog

?

M2

m2

3

˜t

?

,(4.4)

13As discussed in [70], there actually exists another, independent, supersoft term coupling W?and A

which gives a mass to the sgluon. For simplicity we assume this operator is absent from the theory.

14This assumption leads to an error in the size of the logarithm in (4.2) of ∼

factor of

2is due to the finite energy range taken for the stop mass to be generated from the gluino mass.

Since the typical value of the logarithm is log

m˜ t

are working.

1

2log2 ∼ 0.3, where the

1

?

M3

?

∼ 2.5 this is negligable at the accuracy to which we

– 21 –

Page 23

which is clearly very suppressed relative to the MSSM case. Since the sgluon is heavier than

the gluino, the energy range where the gluino mass feeds into the stop mass is separated

from that in which the stop mass feeds into the Higgs mass, hence there is no need to

carry out an integration over energies as we were required to do previously. The overall

dependence of the Higgs mass on the gluino mass is then given by

?

=3λ2

8π2

π

m2

Hu|gluino=−3λ2

t

8π2m2

˜tlog

M2

m2

3

˜t

?

(4.5)

t

C3α3M2

3

log(4)log

?C3α3

π

log(4)

?

.(4.6)

Hence, the fine tuning is

˜ZM2

3=M2

3

m2

Z

3λ2

2π2

t

C3α3

π

log(4)log

?C3α3

π

log(4)

?

,(4.7)

? 0.0282M2

3

m2

Z

.(4.8)

In this expression the two stop masses are not treated as independent variables since they

are both generated through the gaugino masses, and cannot be adjusted independently.

Therefore we regard the weak scale gluino mass as the only independent variable (or equiv-

alently the stop mass). While, as previously discussed, using the weak scale value is an

approximation, it is sufficient since there is very little running in such a theory. Further,

since the running all occurs over a very small range of energies there is no need to account

for the running of gauge couplings. Importantly, these expressions are independent of the

mediation scale. This is due to the N = 2 structure cutting off the running at a lower

scale, and this is what allows for the improvement in fine tuning.

The indirect fine tuning of the Higgs by the gluino through the stop mass, which was

found to be the dominant contribution in the Majorana case, still appears as a logarithm

squared, however now goes as

?M3

which of course is much suppressed. In effect, the scale where a full N = 2 spectrum appears

is acting as a UV boundary. This is a desirable alternative to a conventional model with a

very low cutoff since it is still compatible with a string theory completion [77], and avoids

problematic higher dimension operators from a SUSY breaking and mediation sector which

is not far separated in energy scale from the weak scale. Dirac models may also appear

naturally out of models with spontaneous supersymmetry breaking [78].

Since we are dealing with logarithms of O(1), such terms now no longer necessarily

dominate over other finite, non-log enhanced corrections. To obtain an accurate measure

of fine tuning these should be included. In particular, these are the reason that it is

not possible to make the fine tuning arbitrarily small for heavy superpartners by taking

˜ m3= M3and M3= m˜t. The threshold corrections from the gluino can be calculated from

∼ log

m˜t

?

log

?˜ mi

Mi

?

,(4.9)

– 22 –

Page 24

from [79]. These make an O(1) difference to the stop masses generated. While it would

be interesting to evaluate the full threshold corrections including those from the sgluon, in

this section we consider only the log enhanced pieces as an approximation, leaving a full

calculation to future work.

As the logarithms are now small, it is worthwhile to check the one loop contribution

of the electroweakinos to the Higgs mass does not dominate the fine tuning. These give a

contribution to the Higgs mass

δm2

Hu= δm2

Hd=α2(M2)C2M2

2

π

log

?˜ m2

2

M2

2

?

, (4.10)

which leads to a tuning of the electroweak VEV of approximately

∆M2=M2

2

m2

Z

2α2(M2)C2

π

?

log

?˜ m2

2

M2

2

?

− 1

?

,(4.11)

=M2

m2

= 0.0062M2

2

Z

2α2(M2)C2

π

(log(4) − 1), (4.12)

2

m2

Z

.(4.13)

Since the wino is typically significantly less massive than the gluino, this is only a small

contribution to the fine tuning.

In Fig.9 we plot the fine tuning as a function of the gluino mass and also the stop

mass which is fixed by the gluino mass (solid lines). However, as discussed, the accuracy

of this is compromised by the small logarithms involved. The logarithm involved in the

gluino generating the stop masses, log

M2

3

the neglect of the finite pieces. As a result, Fig.9 should be regarded as giving a rough ap-

proximation for the fine tuning as a function of the gluino mass. In contrast, the logarithm

?

˜ m2

3

?

∼ 1.3, is certainly not large enough to justify

involving the Higgs masses generated from the stops is given by log

?

M2

m2

3

˜ t

?

∼ 5. Since this

is somewhat larger, the fine tuning as a function of stop mass in this figure is expected to

be more accurate.

By comparison with the expressions found in the previous section, we find the fine

tuning as a function of stop mass is comparable to an MSSM model with a very low

cutoff of ΛUV = 105GeV (with both stops masses fixed by one parameter). However,

as the cutoff ΛUV is raised, Dirac gauginos quickly lead to a benefit in reducing the fine

tuning. Hence, for string models, a Dirac gluino provides a very strong option to retain as

natural a spectrum as possible, as well as being well motivated theoretically. Of course,

a disadvantage of such models is that the N = 2 scalar partners spoil traditional SUSY

unification unless other new states are also present, requiring more model building.

5Summary

We have carried out a careful study of the fine tuning in theories of natural supersymmetry,

in particular concentrating on the tuning due to the UV masses of the gluino, first two gen-

eration sfermions, and the stops. In doing so, we have improved previous approximations

– 23 –

Page 25

02000400060008000

0

50

100

150

200

Mass ê GeV

Fine Tuning

Gluino

Stop

Figure 9. The electroweak fine tuning of the minimal Dirac model as a function of gluino and stop

masses (solid lines). Note that, in this model, the stop mass is a function of the gluino mass, hence

these are not independent variables. For comparison we also plot the fine tuning for the MSSM,

obtained in Section 3, for the cases ΛUV = 105GeV, dashed lines, and for ΛUV = 106GeV, dotted

lines. It is seen that while the Dirac model gives comparable fine tuning to a very low scale MSSM

model, it quickly leads to an improvement in fine tuning as the UV boundary is increased.

which can introduce an O(1) correction to the results obtained. From these expressions,

we have obtained limits on the extent to which it is beneficial to raise the gluino mass

above the stop masses, and a lower bound on the fine tuning of theories for a given weak

scale gluino mass.

For models with high scale mediation, if there is a Majorana gluino mass of 2TeV the

fine tuning is at least ? 400, and only constrains the UV stop mass to be ? 1.5TeV. After

running to the weak scale, the stop mass can be up to 2TeV without affected the fine

tuning, and in fact the largest regions of parameter space with the lowest fine tuning have

fairly heavy IR stop masses of ∼ 1.5TeV. Models with low scale mediation and a 2TeV

Majorana gluino have a fine tuning of at least ? 50, and the UV stop mass is constrained

to be ? 500GeV. After running, the regions with the lowest fine tuning have IR stop

masses up to 1400GeV. In both high and low scale mediation models, the masses of the

first two generation sfermions may be made very large, far out of reach of the LHC, without

introducing additional fine tuning to the theory.

Finally, we have discussed an attractive alternative to the MSSM, Dirac gluinos. These

allow for spectra with moderate fine and significant separation of the gluino and stops,

comparable to that in theories with low scale mediation and a Majorana gluino, even if

the scale of mediation is high. These are therefore a very attractive proposition for string-

motivated models.

– 24 –

Page 26

Acknowledgements

We are grateful to Asimina Arvanitaki, Masha Baryakhtar , Savas Dimopoulos, Saso Groz-

danov, Ulrich Haisch, Xinlu Huang, Ken Van Tilburg, and James Unwin for useful discus-

sions, and especially John March-Russell for discussions and comments on the manuscript.

We are also grateful to the JHEP referee for a number of very useful comments, including

on the distinction between an assumed UV boundary and a genuine UV cut-off and the

importance of the direct two loop contribution from the sfermions to the Higgs.

ASubleading Terms from Stop Back Reaction

In this Appendix we calculate the back-reaction from the stop, when it is perturbed by

a change in the gluino or sfermion masses. While this is a small effect, we include it in

our numerical simulations. It occurs due to a term ∼ ytm2˜

equation for the stops which tends to suppress any change in the stop mass.

First, the effect on the gaugino fine tuning. If, at a scale t, a gaugino has led to a

change in the left handed stop mass of ∆m2˜

Q3in the renormalisation group

Q3, this will feed back into the running as

d

dt

?

∆m2˜

Q3

?

=

2y2

16π2∆m2˜

t

Q3(t). (A.1)

This can be integrated by using the expression for ∆m2˜

Q3(t) in (2.7)

∆

?

∆m2˜

Q3

?

=

2y2

16π2

t

2Ci

bi

αi(Λ)log2?

1 +biαi(Λ)

2π

?t

tΛ

?M2

i(t) − M2

i(Λ)?dt

?M2

(A.2)

=Ciy2

8π2

t

Λ

m˜

?

?

?

Q3

?

Λ

)

log

m˜

Q3

i.(A.3)

The leading expression for ∆

A similar procedure gives the back reaction from the first two generation sfermions as

?

m˜

Q3

?

∆m2

˜ u3

?

= 2 × ∆

?

∆m2˜

Q3

.

∆

?

∆m2˜

Q3

?

?

=

?

i

y2

π3biCiαi

t

log

?

Λ

?

−

2π

biαilog1 +biαi

2π

log

?

Λ

m˜

Q3

???

˜ m2

1,2,

(A.4)

and again ∆

˜ m2

1,2, their contribution to the fine tuning with respect to these variables is straightforward.

∆m2

˜ u3

?

= 2×∆

?

∆m2˜

Q3

?

. Since these corrections depend linearly on M2

iand

References

[1] S. Dimopoulos and G. Giudice, Naturalness constraints in supersymmetric theories with

nonuniversal soft terms, Phys.Lett. B357 (1995) 573–578, [hep-ph/9507282].

[2] A. G. Cohen, D. Kaplan, and A. Nelson, The More minimal supersymmetric standard model,

Phys.Lett. B388 (1996) 588–598, [hep-ph/9607394].

– 25 –

Page 27

[3] G. D. Kribs, A. Martin, and A. Menon, Natural Supersymmetry and Implications for Higgs

physics, arXiv:1305.1313.

[4] K. Krizka, A. Kumar, and D. E. Morrissey, Very Light Scalar Top Quarks at the LHC,

arXiv:1212.4856.

[5] R. Auzzi, A. Giveon, S. B. Gudnason, and T. Shacham, A Light Stop with Flavor in Natural

SUSY, JHEP 1301 (2013) 169, [arXiv:1208.6263].

[6] J. R. Espinosa, C. Grojean, V. Sanz, and M. Trott, NSUSY fits, JHEP 1212 (2012) 077,

[arXiv:1207.7355].

[7] Z. Han, A. Katz, D. Krohn, and M. Reece, (Light) Stop Signs, JHEP 1208 (2012) 083,

[arXiv:1205.5808].

[8] H. M. Lee, V. Sanz, and M. Trott, Hitting sbottom in natural SUSY, JHEP 1205 (2012) 139,

[arXiv:1204.0802].

[9] Y. Bai, H.-C. Cheng, J. Gallicchio, and J. Gu, Stop the Top Background of the Stop Search,

JHEP 1207 (2012) 110, [arXiv:1203.4813].

[10] B. Allanach and B. Gripaios, Hide and Seek With Natural Supersymmetry at the LHC, JHEP

1205 (2012) 062, [arXiv:1202.6616].

[11] G. Larsen, Y. Nomura, and H. L. Roberts, Supersymmetry with Light Stops, JHEP 1206

(2012) 032, [arXiv:1202.6339].

[12] X.-J. Bi, Q.-S. Yan, and P.-F. Yin, Probing Light Stop Pairs at the LHC, Phys.Rev. D85

(2012) 035005, [arXiv:1111.2250].

[13] C. Brust, A. Katz, S. Lawrence, and R. Sundrum, SUSY, the Third Generation and the

LHC, JHEP 1203 (2012) 103, [arXiv:1110.6670].

[14] E. Arganda, J. L. Diaz-Cruz, and A. Szynkman, Decays of H0/A0in supersymmetric

scenarios with heavy sfermions, Eur.Phys.J. C73 (2013) 2384, [arXiv:1211.0163].

[15] E. Arganda, J. L. Diaz-Cruz, and A. Szynkman, Slim SUSY, Phys.Lett. B722 (2013) 100,

[arXiv:1301.0708].

[16] J. Cao, C. Han, L. Wu, J. M. Yang, and Y. Zhang, Probing Natural SUSY from Stop Pair

Production at the LHC, JHEP 1211 (2012) 039, [arXiv:1206.3865].

[17] N. Arkani-Hamed and H. Murayama, Can the supersymmetric flavor problem decouple?,

Phys.Rev. D56 (1997) 6733–6737, [hep-ph/9703259].

[18] J. Hisano, K. Kurosawa, and Y. Nomura, Natural effective supersymmetry, Nucl.Phys. B584

(2000) 3–45, [hep-ph/0002286].

[19] R. Barbieri and G. Giudice, Upper bounds on supersymmetric particle masses, Nuclear

Physics B 306 (1988), no. 1 63 – 76.

[20] R. Kitano and Y. Nomura, Supersymmetry, naturalness, and signatures at the LHC,

Phys.Rev. D73 (2006) 095004, [hep-ph/0602096].

[21] R. Kitano and Y. Nomura, A Solution to the supersymmetric fine-tuning problem within the

MSSM, Phys.Lett. B631 (2005) 58–67, [hep-ph/0509039].

[22] S. Antusch, L. Calibbi, V. Maurer, M. Monaco, and M. Spinrath, Naturalness of the

Non-Universal MSSM in the Light of the Recent Higgs Results, JHEP 01 (2013) 187,

[arXiv:1207.7236].

– 26 –

Page 28

[23] J. L. Feng, Naturalness and the Status of Supersymmetry, arXiv:1302.6587.

[24] A. Arvanitaki, N. Craig, S. Dimopoulos, and G. Villadoro, Mini-Split, JHEP 1302 (2013)

126, [arXiv:1210.0555].

[25] A. Strumia, Naturalness of supersymmetric models, hep-ph/9904247.

[26] G. L. Kane and S. King, Naturalness implications of LEP results, Phys.Lett. B451 (1999)

113–122, [hep-ph/9810374].

[27] Z. Kang, J. Li, and T. Li, On Naturalness of the MSSM and NMSSM, JHEP 1211 (2012)

024, [arXiv:1201.5305].

[28] S. Cassel, D. Ghilencea, and G. Ross, Testing SUSY at the LHC: Electroweak and Dark

matter fine tuning at two-loop order, Nucl.Phys. B835 (2010) 110–134, [arXiv:1001.3884].

[29] S. Cassel, D. Ghilencea, and G. Ross, Testing SUSY, Phys.Lett. B687 (2010) 214–218,

[arXiv:0911.1134].

[30] D. Ghilencea, A new approach to Naturalness in SUSY models, arXiv:1304.1193.

[31] P. Athron, M. Binjonaid, and S. F. King, Fine Tuning in the Constrained Exceptional

Supersymmetric Standard Model, arXiv:1302.5291.

[32] M. Cabrera, J. Casas, and R. Ruiz de Austri, Bayesian approach and Naturalness in MSSM

analyses for the LHC, JHEP 0903 (2009) 075, [arXiv:0812.0536].

[33] J. Casas, J. Espinosa, and I. Hidalgo, The MSSM fine tuning problem: A Way out, JHEP

0401 (2004) 008, [hep-ph/0310137].

[34] B. de Carlos and J. Casas, One loop analysis of the electroweak breaking in supersymmetric

models and the fine tuning problem, Phys.Lett. B309 (1993) 320–328, [hep-ph/9303291].

[35] M. Badziak, E. Dudas, M. Olechowski, and S. Pokorski, Inverted Sfermion Mass Hierarchy

and the Higgs Boson Mass in the MSSM, JHEP 1207 (2012) 155, [arXiv:1205.1675].

[36] G. G. Ross and K. Schmidt-Hoberg, The Fine-Tuning of the Generalised NMSSM,

Nucl.Phys. B862 (2012) 710–719, [arXiv:1108.1284].

[37] H. Baer, V. Barger, P. Huang, A. Mustafayev, and X. Tata, Radiative natural SUSY with a

125 GeV Higgs boson, Phys.Rev.Lett. 109 (2012) 161802, [arXiv:1207.3343].

[38] M. Papucci, J. T. Ruderman, and A. Weiler, Natural SUSY Endures, JHEP 1209 (2012)

035, [arXiv:1110.6926].

[39] K. Agashe and M. Graesser, Supersymmetry breaking and the supersymmetric flavor problem:

An Analysis of decoupling the first two generation scalars, Phys.Rev. D59 (1999) 015007,

[hep-ph/9801446].

[40] C. Wymant, Optimising Stop Naturalness, Phys.Rev. D86 (2012) 115023,

[arXiv:1208.1737].

[41] D. Ghilencea and G. Ross, The fine-tuning cost of the likelihood in SUSY models, Nucl.Phys.

B868 (2013) 65–74, [arXiv:1208.0837].

[42] H. Baer, V. Barger, P. Huang, D. Mickelson, A. Mustafayev, et. al., Post-LHC7 fine-tuning

in the mSUGRA/CMSSM model with a 125 GeV Higgs boson, arXiv:1210.3019.

[43] H. Baer, V. Barger, and M. Padeffke-Kirkland, Electroweak versus high scale finetuning in

the 19-parameter SUGRA model, arXiv:1304.6732.

– 27 –

Page 29

[44] M. Dine, P. Fox, E. Gorbatov, Y. Shadmi, Y. Shirman, et. al., Visible effects of the hidden

sector, Phys.Rev. D70 (2004) 045023, [hep-ph/0405159].

[45] A. G. Cohen, T. S. Roy, and M. Schmaltz, Hidden sector renormalization of MSSM scalar

masses, JHEP 0702 (2007) 027, [hep-ph/0612100].

[46] R. Contino and I. Scimemi, The Supersymmetric flavor problem for heavy first two generation

scalars at next-to-leading order, Eur.Phys.J. C10 (1999) 347–356, [hep-ph/9809437].

[47] G. D. Kribs, E. Poppitz, and N. Weiner, Flavor in supersymmetry with an extended

R-symmetry, Phys.Rev. D78 (2008) 055010, [arXiv:0712.2039].

[48] M. Abdullah, I. Galon, Y. Shadmi, and Y. Shirman, Flavored Gauge Mediation, A Heavy

Higgs, and Supersymmetric Alignment, JHEP 1306 (2013) 057, [arXiv:1209.4904].

[49] M. J. Perez, P. Ramond, and J. Zhang, Mixing supersymmetry and family symmetry

breakings, Phys.Rev. D87 (2013), no. 3 035021, [arXiv:1209.6071].

[50] R. Mahbubani, M. Papucci, G. Perez, J. T. Ruderman, and A. Weiler, Light non-degenerate

squarks at the LHC, arXiv:1212.3328.

[51] I. Galon, G. Perez, and Y. Shadmi, Non-Degenerate Squarks from Flavored Gauge Mediation,

arXiv:1306.6631.

[52] G. Dvali and A. Pomarol, Anomalous U(1) as a mediator of supersymmetry breaking,

Phys.Rev.Lett. 77 (1996) 3728–3731, [hep-ph/9607383].

[53] A. E. Nelson and D. Wright, Horizontal, anomalous U(1) symmetry for the more minimal

supersymmetric standard model, Phys.Rev. D56 (1997) 1598–1604, [hep-ph/9702359].

[54] D. E. Kaplan, F. Lepeintre, A. Masiero, A. E. Nelson, and A. Riotto, Fermion masses and

gauge mediated supersymmetry breaking from a single U(1), Phys.Rev. D60 (1999) 055003,

[hep-ph/9806430].

[55] D. E. Kaplan and G. D. Kribs, Phenomenology of flavor mediated supersymmetry breaking,

Phys.Rev. D61 (2000) 075011, [hep-ph/9906341].

[56] N. Craig, M. McCullough, and J. Thaler, Flavor Mediation Delivers Natural SUSY, JHEP

1206 (2012) 046, [arXiv:1203.1622].

[57] E. Hardy and J. March-Russell, Retrofitted Natural Supersymmetry from a U(1),

arXiv:1302.5423.

[58] A. Brignole, L. E. Ibanez, and C. Munoz, Soft supersymmetry breaking terms from

supergravity and superstring models, hep-ph/9707209.

[59] P. Meade, N. Seiberg, and D. Shih, General Gauge Mediation, Prog.Theor.Phys.Suppl. 177

(2009) 143–158, [arXiv:0801.3278].

[60] J. L. Feng, K. T. Matchev, and T. Moroi, Focus points and naturalness in supersymmetry,

Phys.Rev. D61 (2000) 075005, [hep-ph/9909334].

[61] D. Horton and G. Ross, Naturalness and Focus Points with Non-Universal Gaugino Masses,

Nucl.Phys. B830 (2010) 221–247, [arXiv:0908.0857].

[62] S. P. Martin, A Supersymmetry primer, hep-ph/9709356.

[63] U. Ellwanger, C. Hugonie, and A. M. Teixeira, The Next-to-Minimal Supersymmetric

Standard Model, Phys.Rept. 496 (2010) 1–77, [arXiv:0910.1785].

– 28 –

Page 30

[64] R. Barbieri, L. J. Hall, Y. Nomura, and V. S. Rychkov, Supersymmetry without a Light Higgs

Boson, Phys.Rev. D75 (2007) 035007, [hep-ph/0607332].

[65] E. Hardy, J. March-Russell, and J. Unwin, Precision Unification in lambda SUSY with a 125

GeV Higgs, JHEP 1210 (2012) 072, [arXiv:1207.1435].

[66] B. Allanach, SOFTSUSY: a program for calculating supersymmetric spectra,

Comput.Phys.Commun. 143 (2002) 305–331, [hep-ph/0104145].

[67] A. Riotto and E. Roulet, Vacuum decay along supersymmetric flat directions, Phys.Lett.

B377 (1996) 60–66, [hep-ph/9512401].

[68] A. Riotto, E. Roulet, and I. Vilja, Preheating and vacuum metastability in supersymmetry,

Phys.Lett. B390 (1997) 73–79, [hep-ph/9607403].

[69] A. Kusenko, P. Langacker, and G. Segre, Phase transitions and vacuum tunneling into

charge and color breaking minima in the MSSM, Phys.Rev. D54 (1996) 5824–5834,

[hep-ph/9602414].

[70] P. J. Fox, A. E. Nelson, and N. Weiner, Dirac gaugino masses and supersoft supersymmetry

breaking, JHEP 0208 (2002) 035, [hep-ph/0206096].

[71] T. Kikuchi, A Solution to the little hierarchy problem in a partly N=2 extension of the

MSSM, arXiv:0812.2569.

[72] L. M. Carpenter, Dirac Gauginos, Negative Supertraces and Gauge Mediation, JHEP 1209

(2012) 102, [arXiv:1007.0017].

[73] R. Davies, J. March-Russell, and M. McCullough, A Supersymmetric One Higgs Doublet

Model, JHEP 1104 (2011) 108, [arXiv:1103.1647].

[74] K. Benakli, Dirac Gauginos: A User Manual, Fortsch.Phys. 59 (2011) 1079–1082,

[arXiv:1106.1649].

[75] G. D. Kribs and A. Martin, Supersoft Supersymmetry is Super-Safe, Phys.Rev. D85 (2012)

115014, [arXiv:1203.4821].

[76] K. Benakli, M. D. Goodsell, and F. Staub, Dirac Gauginos and the 125 GeV Higgs,

arXiv:1211.0552.

[77] R. Davies, Dirac gauginos and unification in F-theory, JHEP 1210 (2012) 010,

[arXiv:1205.1942].

[78] S. Abel and M. Goodsell, Easy Dirac Gauginos, JHEP 1106 (2011) 064, [arXiv:1102.0014].

[79] D. M. Pierce, J. A. Bagger, K. T. Matchev, and R.-j. Zhang, Precision corrections in the

minimal supersymmetric standard model, Nucl.Phys. B491 (1997) 3–67, [hep-ph/9606211].

– 29 –