Department of Physics, The University of Adelaide, … · 2018-08-14 · ADP-15-4/T906 Z0mass...

ADP-15-4/T906

Z ′ mass limits and the naturalness of supersymmetry

P. Athron,1, ∗ D. Harries,2, † and A. G. Williams2, ‡

1ARC Centre of Excellence for Particle Physics at the Terascale,

School of Physics, Monash University, Melbourne VIC 3800, Australia

2ARC Centre of Excellence for Particle Physics at the Terascale,

Department of Physics, The University of Adelaide,

Adelaide, South Australia 5005, Australia

(Dated: September 28, 2018)

Abstract

The discovery of a 125 GeV Higgs boson and rising lower bounds on the masses of superpart-

ners have lead to concerns that supersymmetric models are now fine-tuned. Large stop masses,

required for a 125 GeV Higgs, feed into the electroweak symmetry breaking conditions through

renormalization group equations forcing one to fine-tune these parameters to obtain the correct

electroweak vacuum expectation value. Nonetheless, this fine-tuning depends crucially on our as-

sumptions about the supersymmetry breaking scale. At the same time, U(1) extensions provide

the most compelling solution to the µ problem, which is also a naturalness issue, and allow the

tree-level Higgs mass to be raised substantially above MZ . These very well-motivated supersym-

metric models predict a new Z ′ boson which could be discovered at the LHC, and the naturalness

of the model requires that the Z ′ boson mass should not be too far above the TeV scale. Moreover,

this fine-tuning appears at the tree level, making it less dependent on assumptions about the su-

persymmetry breaking mechanism. Here we study this fine-tuning for several U(1) supersymmetric

extensions of the Standard Model and compare it to the situation in the MSSM where the most

direct tree-level fine-tuning can be probed through chargino mass limits. We show that future LHC

Z ′ searches are extremely important for challenging the most natural scenarios in these models.

PACS numbers: 12.60.Jv, 11.30.Pb, 12.60.-i, 14.80.Bn

∗ [email protected]† [email protected]‡ [email protected]

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I. INTRODUCTION

The discovery of an approximately 125 GeV Higgs [1, 2] at the Large Hadron Collider

(LHC) has interesting implications for physics beyond the Standard Model (SM) and su-

persymmetry (SUSY). On the one hand, it provides a light Higgs boson, as expected from

supersymmetry, and can be fitted in the minimal supersymmetric standard model (MSSM).

On the other hand, the Higgs mass is slightly heavier than the constrained version of the

MSSM (cMSSM) can accommodate naturally [3, 4].

In the MSSM the Higgs mass causes a naturalness problem because at tree level it has

an upper bound of the mass of the Z boson, MZ . The dominant higher-order corrections

to the Higgs mass come from stops, and to obtain a 125 GeV Higgs they need to be rather

heavy. Heavy stops will provide a large contribution to the low-energy value of m2Hu

, the soft

breaking mass for the up-type Higgs scalar, through the evolution of the renormalization

group equations (RGEs) from the grand unification (GUT) scale to the electroweak (EW)

scale. This affects the SUSY prediction of the electroweak vacuum expectation value (VEV),

v, or MZ . This naturalness problem motivates both further examination of nonminimal

SUSY models that can raise the Higgs mass without the need for heavy stops and alternative

possibilities for how the soft breaking parameters get generated, which might set them at

lower energies, reducing the influence the stops have on m2Hu

.

In addition to that naturalness issue, often referred to as the little hierarchy problem,

the MSSM also suffers from the µ problem. This is also a naturalness problem since there

should be a natural explanation of how the µ superpotential parameter can be set to the

same scale as the soft breaking masses.

U(1) extensions of the MSSM provide a very elegant solution to this µ problem [5–12]

and also raise the Higgs mass with new F and D terms. Nonetheless, as was recently

demonstrated in the context of the exceptional supersymmetric standard model (E6SSM)

[13–15], such models can still suffer from naturalness problems with the mass of the new

Z ′ associated with the break down of the new U(1) appearing in the electroweak symmetry

breaking (EWSB) conditions at tree level [16]. Despite this the constrained version of the

E6SSM (cE6SSM) [17, 18] was still found to be significantly less tuned than the cMSSM.

Tree-level fine-tuning from the Z ′ mass was also considered previously [19].

However, this comparison of fine-tuning depends crucially upon the assumptions of these

2

gravity mediated SUSY breaking motivated constrained models and, in particular, the uni-

versality constraints being applied at the GUT scale. As mentioned above, given the findings

at the LHC, it is worth considering other possibilities, which may allow the soft masses to

be set at lower energies. As the scale at which the parameters fulfill some breaking inspired

constraints is lowered the stop masses contribute less to the fine-tuning.

At the same time in U(1) extensions, lowering the UV boundary scale for the RGE

evolution also allows even larger F -term contributions to the Higgs mass, so long as one

only requires λ, the coupling between the Singlet Higgs, S and the up- and down-type Higgs

bosons, Hu and Hd, to remain perturbative up to the UV scale and not all of the way up to

the GUT scale.

However, the tuning from the Z ′ mass limit does not disappear as the UV boundary

condition is lowered. This tuning appears in the EWSB conditions at tree level and is quite

difficult to avoid without introducing a pure gauge singlet [20].

In this paper we investigate how big this tuning is if we bring this scale all the way

down to 20 TeV, effectively minimizing the contribution from the stops. We find that the

Z ′ limit is enough to already require moderate fine-tuning in the E6SSM. We also show

this is comparable to the situation in the MSSM defined at the same scale if charginos

could be ruled out below 700 GeV. We then show how this tuning from the Z ′ mass looks

for different U(1) extensions, finding that the current severity depends upon the charges

but that Z ′ limits are important in constraining the most natural scenarios of these models.

Therefore, the Z ′ constraint is amongst the most important in terms of tuning and attacking

natural supersymmetry experimentally and the next run of the LHC will be crucial in this

respect.

Finally, we make a case study, for a few benchmarks, of the impact of raising the high-

scale boundary condition, MX , at which the SUSY breaking parameters must be fixed by

some SUSY breaking mechanism. We show that which model has less fine-tuning depends

on MX . We also see rather complicated behavior in the tuning for the E6SSM points due to

the combination of different sources of tuning.

As mentioned earlier, the-fine tuning of the cE6SSM was recently studied [16] and there

it was revealed that the associated Z ′ boson leads to a new source of fine-tuning since its

mass appears in the EWSB conditions.

However in this study we will examine this source of fine-tuning in more detail by con-

3

sidering low-energy constructions where the usual fine-tuning problem from the Higgs is

minimized. We will also consider alternate charges for the extra U(1) symmetry to relax the

focus on the E6SSM and demonstrate that this is quite a generic result.

To quantify the fine-tuning we will employ the traditional Barbieri-Giudice measure [21,

22]. This has been used extensively within the literature e.g. Refs. [16, 23–49].

A number of alternative measures have also been applied in the literature [50–65] with

varying motivations. A very different approach is to work within a Bayesian analysis. There

the concept of naturalness is automatically incorporated since in models where one must

fine-tune parameters to fit measured values of the observables, the region with high likelihood

will occupy a tiny prior volume [4, 66–70] suppressing the posterior. Indeed in the MSSM

and the next-to-MSSM (NMSSM) if one transforms GUT scale parameters to the VEVs, the

inverse of the Jacobian for this transformation looks quite like the derivatives that appear

in the traditional fine-tuning measure [66, 67, 70]. If one thinks more generally, then a

model without fine-tuning is one where the parameterization is such that all the parameters

are observables [69, 70]. This provides a quite general definition of fine-tuning as 1/|J |

where |J | is the determinant of the Jacobian for the coordinate transformation between

the parameters and the observables. Interestingly this means the tuning is the ratio of the

infinitesimal observable space volume element to the infinitesimal parameter space element

and coincides with the measure proposed in Ref. [63] when the interval of variation is taken

to zero.

While this approach has many merits here we will employ the traditional measure of

fine-tuning because it is both simple to apply and easy to compare with previous results due

to it’s widespread use. Fortunately the derivatives which appear in these tunings are also

similar to the Bayesian motivated measure so there should not be too large a discrepancy

between the two approaches.

The structure of this paper is as follows. In Sec. II we review the models we consider. In

Sec. III we specify the EWSB conditions of the models, with particular focus on how the Z ′

mass influences the prediction of MZ . Then in Sec. IV we introduce our fine-tuning measure

and our approach to evaluating it to obtain the individual sensitivities. The results are then

given in Sec. V.

4

II. U(1) EXTENSIONS AND THE E6SSM

In this paper we consider U(1) extensions of the MSSM where the gauge group at low

energies is

SU(3)C × SU(2)W × U(1)Y × U(1)′. (1)

U(1)′ is the new gauge group beyond that of the SM and MSSM. The minimal superfield

content of U(1) extensions which solve the µ problem should be ordinary left-handed quark

Qi and lepton Li (i = 1, 2, 3) superfields along with right-handed superfields uci , dci , e

ci (i =

1, 2, 3) for the up-type (s)quarks, down-type (s)quarks and charged (s)leptons respectively

and three Higgs superfields, up-type Hu, down-type Hd and a singlet under the SM gauge

group S.

Here we will refer to U(1) extensions of the MSSM, which solve the µ problem, as the

USSM [8–12]. The couplings for the U(1)′ gauge group should allow the following renormal-

izable superpotential terms required in the USSM,

WUSSM = yUij uciHu · Qj + yDij d

ciQj · Hd + yEij e

ci Lj · Hd + λSHd · Hu, (2)

with i, j ∈ {1, 2, 3}. For the SU(2) dot product we follow the convention A ·B ≡ εαβAαBβ =

A2B1 − A1B2.

The U(1)′ charges should allow for cancellations of gauge anomalies. The most elegant

way to do this is to use an extra U(1) gauge symmetry that can be obtained from the break

down of the E6 gauge symmetry which is anomaly free and have all matter fields that fill

the three generations of 27-plet representations of E6 survive down to low energies. Such

models are often referred to in the literature as E6 inspired, and we will adopt this here.

The breaking of E6 into SO(10) gives rise to E6 → SO(10) × U(1)ψ, and the subse-

quent breaking of SO(10) into SU(5) gives SO(10) → SU(5) × U(1)χ (this is reviewed in

e.g. Ref. [71]). The extra U(1) gauge symmetry at low energies should then be a linear

combination of these in the E6 inspired case,

U(1)′ = U(1)χ cos θ + U(1)ψ sin θ. (3)

In Table I the charges for several popular E6 inspired U(1) extensions are shown.

U(1) and E6 inspired extensions of the MSSM have been studied very widely in the

literature [21, 72–94] (or, for reviews, see Refs. [71, 95]). There has also been a lot of work

5

Q uc dc L ec N c S H2 H1 D D H ′ H ′√53Q

Yi

16 −2

313 −1

2 1 0 0 12 −1

2 −13

13 −1

212

2√

6Qψi 1 1 1 1 1 1 4 −2 −2 −2 −2 1 −1

2√

10Qχi −1 −1 3 3 −1 −5 0 2 −2 2 −2 3 −3

√40QNi 1 1 2 2 1 0 5 −2 −3 −2 −3 2 −2

TABLE I. The U(1)Y , U(1)ψ, U(1)χ and U(1)N charges of the chiral superfields in the E6 model.

The specific case of U(1)N , corresponding to the E6SSM, is obtained for θ = arctan√

15.

recently including investigations of the neutralino sector [96–99]; the relic density of dark

matter [100]; GUT scale family symmetries which can explain the hierarchy of masses in

the fermion sector and their associated mixings [101]; neutrino physics [102]; explanations of

the matter-antimatter asymmetry of the Universe though EW baryogenesis or leptogenesis

[93, 94, 103]; decays of the Z ′ boson [104–107]; dipole moments [108]; anomaly mediated

SUSY breaking with D-term contributions [109] and the (extended) Higgs sectors [110, 111].

Here we will focus most on the special case where the gauge symmetry is U(1)N , under

which the right-handed neutrino N c does not participate in gauge interactions. This is the

case in the E6SSM [13–15], and closely related variants [20, 112–116]. Since the right-handed

neutrino has no gauge symmetry protecting it’s mass from becoming extremely heavy such

models may explain the tiny observed masses of neutrinos via the see-saw mechanism and

the baryon asymmetry in the Universe via leptogenesis [93, 117, 118]. Recently it has also

been studied in the context of electroweak baryogenesis [119].

The gauge coupling running in the E6SSM at the two-loop level leads to unification more

precisely than in the MSSM [120] or, in slightly modified scenarios, two-step unification

can take place [112, 121]. If the exotic particles are light in these models this can open up

nonstandard decays of the SM-like Higgs boson [20, 122, 123].

The correct relic density could be obtained entirely through an almost decoupled “inert”

neutralino sector [124]. However, this is no longer phenomenologically viable due to limits

from direct detection of dark matter [125–127] and due to a significant suppression of the

decay of the lightest Higgs boson into SM states, due to a new channel into inert singlinos

opening up.

6

There are still several remaining options. One may specialize to scenarios known as the

EZSSM [115] where the inert singlinos that cause these problems are entirely decoupled and

the relic abundance is fitted with a binolike candidate with a novel mechanism involving

back-scattering into a heavier inert Higgsino. Another well motivated scenario admits two

possible dark matter candidates [116], where one will be an inert singlino and the other will

have a similar composition to MSSM neutralinos. The simplest phenomenologically viable

solution in that case is to make the singlinos extremely light hot dark matter candidates,

in which case the lightest ordinary neutralino accounts for almost all of the observed relic

abundance.

The impact of gauge kinetic mixing in the case where both of the extra U(1) symmetries

appearing from the breakdown of E6 are present at low energy was studied in Ref. [128].

The E6SSM was also included in studies looking at how first- or second-generation sfermion

masses can be used to constrain the GUT scale parameters [129] and the renormalization of

VEVs [130, 131]. The particle spectrum and collider signatures of the cE6SSM have been

studied in a series of papers, [17, 18, 106, 132]. The threshold corrections to the DR gauge

and Yukawa couplings in the E6SSM have also been calculated and their numerical impact

in the constrained version examined [133].

With three generations of matter 27-plet representations of E6 surviving to low energies,

the low-energy matter content in each generation, after integrating out the heavy right-

handed neutrinos, includes,

(Qi, uci , d

ci , Li, e

ci) + (Di, Di) + (Si) + (Hu

i ) + (Hdi ), (4)

where the Si, Hui and Hd

i have the quantum numbers of a SM singlet, and up-, down-type

Higgs fields, respectively, and the Di and Di are SU(3)C triplets that reside in the same

SU(5) multiplets as these Higgs-like states.

If one wishes to maintain gauge coupling unification this set of states should be augmented

by two extra SU(2) doublet states H ′ and H′belonging to other 27′ and 27

′multiplets that

must be incomplete at low energies.

The full superpotential for E6 inspired models coming from 27⊗ 27⊗ 27 decomposition

of the fundamental E6 representation will then be

WE6 = W0 +W1 +W2, (5)

7

where

W0 = λijkSiHdj · Hu

k + κijkSiDjDk + hNijkNci H

uj · Lk

+yUijkuciH

uj · Qk + yDijkd

ciQk · Hd

j + yEijkeci Lk · Hd

j , (6)

W1 = gQijkDiQj · Qk + gqijkˆDid

cjuck, (7)

W2 = gNijkNci Dj d

ck + gEijke

ciDju

ck + gDijkQi · Lj ˆDk. (8)

Nonetheless, while this model is very elegant so far, the superpotential of Eq. (5) contains

dangerous terms which can induce proton decay and lead to large flavor changing neutral

currents (FCNCs). There are a number of approaches to suppress these terms, involving the

use of different discrete symmetries. Here for the purposes of renormalization group running

we will simply include the following unsuppressed superpotential terms, which follows the

approach taken in work on the cE6SSM [17, 18],

W ≈ yτ L3 · Hdec3 + ybQ3 · Hdd

c3 + ytHu · Q3u

c3

+λiSHdi · Hu

i + κiSDiDi + µ′H ′ · H ′, (9)

where we denote by H3d ≡ Hd, H

3u ≡ Hu and S3 ≡ S the third-generation Higgs and

SM singlet fields that are assumed to acquire nonzero VEVs. In addition to the terms

coming from the 27⊗ 27⊗ 27 interactions given in Eq. (5), this superpotential also contains

a bilinear term µ′H ′ · H ′, arising from 27′ ⊗ 27′, which is invariant with respect to the

low-energy SM gauge group and the additional U(1)′ symmetry and also anomaly free.

This term is responsible for setting the masses of the components of the superfields H ′, H ′,

included to ensure gauge coupling unification, but it is not involved in the process of EWSB.

Consequently, the impact on the fine-tuning of the value of µ′ is much smaller than that

coming from other sectors, and so can be safely neglected in our study. In all of the scans

we present below the value of µ′ is fixed to µ′ = 5 TeV.

III. ELECTROWEAK SYMMETRY BREAKING

The Higgs scalar potential for the E6 models considered can be written as [13]

V = VF + VD + Vsoft + ∆V , (10)

where

VF = λ2|S|2(|Hd|2 + |Hu|2) + λ2|Hd ·Hu|2, (11)

8

VD =g2

8

(|Hu|2 − |Hd|2

)2+g2

2

2|H†dHu|2 +

g′212

(Q1|Hd|2 +Q2|Hu|2 +QS|S|2)2, (12)

Vsoft = m2S|S|2 +m2

Hd|Hd|2 +m2

Hu |Hu|2 +[λAλSHd ·Hu + H.c.

]. (13)

In these expressions g2, g′ =√

3/5g1, and g′1 are the SU(2), (non-GUT normalized) U(1)Y

and U(1)′ gauge couplings, respectively, and g2 = g22 + g′2. The charges Q1, Q2 and QS are

effective U(1)′ charges for Hd, Hu and S, respectively, and λ ≡ λ3. In the case of the U(1)χ

model, VF may also contain an elementary µ term, as occurs in the MSSM. The term ∆V

contains the Coleman-Weinberg contributions to the effective potential. For the purposes of

this study, we include in ∆V only the one-loop contributions from the top quark and stop

squarks,

∆V =3

32π2

[m4t1

(lnm2t1

Q2− 3

2

)+m4

t2

(lnm2t2

Q2− 3

2

)− 2m4

t

(lnm2t

Q2− 3

2

)]. (14)

Explicit expressions for the running DR top mass mt and stop masses mt1,2 are given below.

Demanding that the Higgs fields H1, H2 and the singlet S acquire real VEVs of the form

〈Hd〉 =1√2

v1

0

, 〈Hu〉 =1√2

0

v2

, 〈S〉 =s√2, (15)

at the physical minimum leads to the minimization conditions

f1 = m2Hdv1 +

λ2

2(v2

2 + s2)v1 −λAλ√

2sv2 −

g2

8(v2

2 − v21)v1 +DHdv1 +

∂∆V

∂v1

= 0, (16a)

f2 = m2Huv2 +

λ2

2(v2

1 + s2)v2 −λAλ√

2sv1 +

g2

8(v2

2 − v21)v2 +DHuv2 +

∂∆V

∂v2

= 0, (16b)

f3 = m2Ss+

λ2

2(v2

2 + v21)s− λAλ√

2v2v1 +DSs+

∂∆V

∂s= 0. (16c)

The quantities DHd , DHu and DS appearing above are U(1)′ D-term contributions that are

absent in the MSSM and NMSSM and are given by

Dφ ≡g′212

(Q1v

21 +Q2v

22 +QSs

2)Qφ. (17)

We also include these U(1)′ D-term contributions in the diagonalized stop masses,

m2t1,2

=1

2

{m2Q3

+m2u3

+g2

8

(v2

1 − v22

)+DQ +Du + 2m2

t

∓

√[m2Q3−m2

u3+

1

8(g2

2 − g21) (v2

1 − v22) +DQ −Du

]2

+ 4m2tX

2t

}, (18)

9

where m2t = y2

t v22/2, Xt = At − λsv1√

2v2, m2

Q3, m2

u3are soft breaking scalar masses and At is

a soft trilinear coupling. By definition we take mt1 to correspond to the lighter of the two

states.

As was noted in Ref. [16], the first two of the conditions in Eq. (16) may be rewritten in

the formM2

Z

2= −λ

2s2

2+m2Hd− m2

Hutan2 β

tan2 β − 1+DHd −DHu tan2 β

tan2 β − 1, (19)

sin 2β =

√2λAλs

m2Hd

+ m2Hu

+ λ2s2 +DHd +DHu

, (20)

with M2Z = g2v2/4, v2 = v2

1 + v22 and tan β = v2/v1 and where we have for convenience

absorbed the effects of the loop corrections into the soft masses,

m2Hd

= m2Hd

+1

v1

∂∆V

∂v1

,

m2Hu = m2

Hu +1

v2

∂∆V

∂v2

.

Written in the form of Eq. (19), we see the potential new source of fine-tuning alluded to

above, in the form of the third term on the right-hand side. For large values of the VEV s, the

D-term contributions can be quite a bit larger than M2Z . In particular, recent experimental

limits [134] require that the Z ′ mass be large, with for example bounds of MZ′ & 2.51 TeV

in U(1)ψ models and MZ′ & 2.62 TeV in U(1)χ models. To satisfy these limits typically

requires large values of the singlet VEV s. For example, s & 6 TeV is required in the E6SSM

with U(1)′ = U(1)N , so that |DH1|, |DH2| � M2Z for E6 models with QS 6= 0. As a result

the remaining terms on the right-hand side of Eq. (19) must be tuned to cancel this very

large contribution to MZ . Moreover, because this is a large tree-level fine-tuning, it may

negate the improvement in naturalness that is associated with having a reduced need for

heavy superpartners. In U(1) extended models for which QS 6= 0, the importance of the Z ′

mass to the fine-tuning in these models can be made even clearer by writing Eq. (19) in the

form given in Ref. [16],

c(θ, tan β)M2

Z

2= −λ

2s2

2+m2Hd− m2

Hutan2 β

tan2 β − 1+ d(θ, tan β)

M2Z′

2, (21)

where

c(θ, tan β) = 1− 4

(tan2 β − 1)

g′21g2

(Q1 −Q2 tan2 β

) (Q1 cos2 β +Q2 sin2 β

), (22)

10

d(θ, tan β) =Q1 −Q2 tan2 β

QS (tan2 β − 1). (23)

Written like so, it is evident that the fine-tuning contribution coming from the new D terms

depends both on the U(1)′ charges and the Z ′ mass, and that the tuning can be expected

to increase with MZ′ . As shall be shown below, the exact size of this tuning then depends

strongly on the choice of U(1)′ charges, via the coefficient d.

The extra U(1)′ gauge symmetry may mix with the U(1)Y gauge symmetry associated

with hypercharge through gauge kinetic mixing,

Lkinmix = −sinχ

2F YµνF

Nµν , (24)

where F Yµν and FN

µν are field strengths associated with the U(1)Y and U(1)N , respectively.

The gauge kinetic mixing can have a significant impact on the phenomenology [135–137]

and may reduce the Z ′ mass limit.

However, if the extra U(1) gauge symmetry appears from the breakdown of E6, then sinχ

should be zero at the GUT scale. Nonetheless, even if this term is zero at the outset, it will

still be radiatively generated if the trace of the U(1) charges,∑

iQYi Q′i, is nonzero. In the

cases studied here, the trace of the charges over states in the complete 27-plets vanishes,

but to be consistent with single-step gauge coupling unification, we also included H ′ and

H ′ which lead to a nonzero value for∑

iQYi Q′i. The value induced by this at the EW scale

though is rather small as can be seen1 in Fig. 3 of Ref. [135], and this was also checked with

two-loop RGEs in the E6SSM [13, 18]. For this reason and due to the huge expansion in the

number of terms in the two-loop RGEs when one allows for gauge kinetic mixing, we will

neglect this in our analysis here and throughout this paper.

In general, though, it is possible for gauge kinetic mixing to be much larger, which can

be the case if one considers an additional 5 + 5 pair of SU(5) multiplets [135] or which has

been looked at in the U(1)B−L [136, 137]. In such a case, this will impact the results in two

ways, firstly by altering the Z ′ limit from experiment and secondly by altering the charges

which appear in the EWSB condition, which can be seen from examining Eqs. (21)–(23).

1The specific incomplete multiplets we consider here correspond to the third of the four possible embeddings

referred to in Ref. [135].

11

IV. THE FINE-TUNING MEASURE

As stated above, to quantify the resulting fine-tuning we apply the traditional Barbieri-

Giudice measure [21, 22]. A specific model is characterized by a set of n model parameters

{pi} and is defined at some input scale MX . For a given parameter p in this set, one computes

an associated sensitivity,

∆p =

∣∣∣∣∂ lnM2Z

∂ ln p

∣∣∣∣ =

∣∣∣∣ pM2Z

∂M2Z

∂p

∣∣∣∣ . (25)

The coefficient ∆p measures the fractional variation in M2Z resulting from a given variation

in the parameter p. The overall fine-tuning is then taken to be ∆ = maxi{∆pi}.

The sensitivity ∆p may be calculated directly from the expression for M2Z in terms of the

pi for a particular model, which leads to a so-called master formula for calculating the fine-

tuning. A master formula for the E6SSM, obtained from the tree-level scalar potential, was

presented in Ref. [16]. In order to derive the expression presented there, the fact that s� v

was made use of to neglect certain O(v2) terms in the EWSB conditions, greatly simplifying

the final result. For the purposes of exploring a wider class of E6 inspired models, we have

derived the master formula without neglecting any O(v2) terms. The more complete tree-

level master formula is somewhat complicated. This is because, unlike in the MSSM, even at

tree level it is not possible to solve explicitly for the VEVs v1, v2 in terms of the Lagrangian

parameters. It may be written in the form

∆p = |C|−1 × |p|MZ

∣∣∣∣∑q

∆q∂q

∂p

∣∣∣∣, (26)

where the sum is over all low-energy running parameters appearing in the tree-level EWSB

conditions, i.e., q ∈ {λ,Aλ,m2Hd,m2

Hu,m2

S, g1, g2, g′1}. Expressions for the quantity C and

the ∆q appearing above are given in Appendix A. It should be noted that the effects of U(1)

mixing are neglected in deriving Eq. (26).

However, it is well known in the MSSM that radiative corrections can significantly change

(indeed, reduce) the fine-tuning [138]. It is, therefore, important when studying the fine-

tuning to include loop corrections to the effective potential in the fine-tuning measure. To

do so it is most convenient to work in terms of the EWSB conditions Eq. (16), rather than

Eq. (19). The general procedure that we use is as follows (this method has also previously

been applied in the NMSSM; see, for example, Ref. [139]). For a model in which m fields

develop real VEVs (e.g. m = 2 in the MSSM, m = 3 in the NMSSM and in the E6 models

12

considered), we require that the m minimization conditions,

f1 = f2 = · · · = fm = 0, (27)

continue to hold under an arbitrary variation in a model parameter p→ p+ δp, so that the

variations δfi satisfy

δf1 = δf2 = · · · = δfm = 0. (28)

Each fi is a function of the VEVs vj and l running parameters qk evaluated at the scale of

EWSB, fi = fi(vj, qk). Thus for each fi we find that

m∑j=1

∂fi∂vj

∂vj∂p

+l∑

k=1

∂fi∂qk

∂qk∂p

= 0. (29)

The quantities ∂fi∂vj

are the elements of the CP-even Higgs squared mass matrix M2h of the

model before rotating into the mass eigenstate basis. When evaluated for all n model

parameters, the above system of equations can be concisely expressed as

M2h

∂v1

∂p1· · · ∂v1

∂pn...

. . ....

∂vm∂p1· · · ∂vm

∂pn

= −

∂f1

∂q1· · · ∂f1

∂ql...

. . ....

∂fm∂q1· · · ∂fm

∂ql

∂q1∂p1· · · ∂q1

∂pn...

. . ....

∂ql∂p1· · · ∂ql

∂pn

. (30)

The quantities forming the first matrix on the right-hand side, along with M2h , are eas-

ily calculated by differentiating the conditions in Eq. (16) with respect to the VEVs and

the running parameters. The remaining derivatives ∂qk/∂p must be determined using the

RGEs. Once these have been obtained, it is straightforward to solve for the ∂vi/∂p. The

sensitivities ∆p are then simply linear combinations of the ∂vi/∂p and ∂qk/∂p. The effects of

radiative corrections may be easily included by including the Coleman-Weinberg potential

contributions ∆V in the EWSB conditions. Here we use the one-loop corrections given in

Eq. (14).

Evaluating the derivatives ∂qk/∂p must, in general, be done by numerically integrating

the two-loop RGEs. This is time consuming and presents an obstacle to doing large scans

of the parameter space. For studying models defined at low energies, as we do here, we can

take advantage of the fact that the running is over much smaller scales than when evolving

up to the GUT scale. This makes it possible to use approximate analytic solutions to the

RGEs that exhibit good accuracy over the range of scales considered. Given the two-loop

13

RG equation for a parameter q,

dq

dt≡ βq =

1

16π2β(1)q +

1

(16π2)2β(2)q , t ≡ ln

Q

MX

, (31)

a Taylor series expansion of the solution may be used to obtain the parameter at the scale

Q,

q(Q) = q(MX) +

∫ t

0

βq(t′) dt′ ≈ q(MX) +

t

16π2

(β(1)q +

β(2)q

16π2

)+

t2

32π2

dβ(1)q

dt+O(t2). (32)

Expanded to this order, we obtain the leading log (LL) and next-to-LL (NLL) contributions

at two-loop order. The O(t2) terms not displayed above are formally of three-loop order and

are neglected. The derivative of the one-loop β function is given by

dβ(1)q

dt=

1

16π2

∑qk

β(1)qk

∂β(1)q

∂qk, (33)

where the sum is over all running parameters appearing in β(1)q . The β functions appearing

on the right-hand side of Eqs. (32) and (33) are evaluated at the scale MX , giving a simple

analytic expression for the parameters at the scale of EWSB in terms of the model parameters

at MX . Explicit results for the relevant series expansions in the MSSM and E6 models are

presented in Appendix B.

V. RESULTS

Using the approach outlined above, we are able to scan the low-energy parameter space

of the MSSM and E6SSM and calculate the fine-tuning in each. To do so, we implemented

the above expressions for computing the fine-tuning in a modified version of the E6SSM

spectrum generator that was used in Ref. [16]. This code implemented two-loop RGEs for

all parameters except the soft scalar masses. In order to properly include the fine-tuning

impact of the SU(3) gaugino soft mass M3, we have extended the original code to make use

of the two-loop RGEs generated by SARAH [140–143] and FlexibleSUSY [144], which also

makes use of SOFTSUSY [145, 146]. The CP-even Higgs masses are calculated including

the leading one-loop effective potential contributions given in Ref. [18] and for the light

Higgs we use the leading two-loop2 contributions from Ref. [13] which are a generalization of

2Two-loop corrections calculated for a nonminimal SUSY model may now also be obtained from SARAH

[147, 148]. However, this was not available when the numerical work for this paper was carried out, and

such corrections go beyond the required precision for studying fine-tuning here.

14

the corrections in the MSSM and NMSSM calculated using effective field theory techniques

[149, 150]. To scan over the MSSM parameter space, the equivalent MSSM fine-tuning

expressions were implemented into a modified version of SOFTSUSY 3.3.10 [145]. For

consistency with the results produced in the E6 models, and for computational speed, for

our main scans only the dominant one- and two-loop corrections to the CP-even Higgs

masses were included. Finally, in all of the results below the fine-tuning was evaluated at

the scale Q = MSUSY =√mt1mt2 , where mt1,2 are the running DR stop masses evaluated at

Q = MSUSY.

As discussed in the Introduction, many recent papers interested in natural supersymmetry

have focused on light stops, with much theoretical effort to find models where it is easier

to get a 125 GeV Higgs boson and light stops simultaneously and much experimental effort

to search for light stops. This is entirely appropriate since there are many good reasons to

expect the soft masses to be set at high energies. However, that is not the only possibility

and the fine-tuning problem depends strongly on the RG evolution from the GUT scale, as

the soft Higgs masses that appear in the EWSB conditions pick up contributions from the

soft squark masses.

FIG. 1. Left panel: Scatter plot of fine-tuning in the MSSM as a function of the lightest stop mass,

mt1, for the cutoff scales (from bottom to top) MX = 20 TeV, MX = 50 TeV, MX = 100 TeV

and MX = 1016 GeV. Right panel: Scatter plot of fine-tuning in the MSSM as a function of the

lightest Higgs mass, mh1 , for the cutoff scales (from bottom to top) MX = 20 TeV, MX = 50 TeV,

MX = 100 TeV and MX = 1016 GeV.

15

To illustrate this, in the left panel of Fig. 1 we show the variation in fine-tuning for

MX = 20 TeV, 50 TeV, 100 TeV and 1016 GeV when we scan over the stop masses and

mixing, with 500 GeV ≤ mQ3 ,mu3 ≤ 10 TeV and −3810 GeV ≤ At ≤ −20 GeV. The

remaining parameters we fix such that at MSUSY they have the values µ = −97.5, B = −84.8,

M1 = 92.1, M2 = 95.9, M3 = 352, Ab = −117.9, Aτ = −7.8, mLi = 400, mei = 204,

mQ1,2 = 438, mu1,2 = 436 and mdi = 438 GeV (i = 1, 2, 3). Here we denote by M1, M2

and M3 the soft gaugino masses for U(1), SU(2) and SU(3), while Ab and Aτ are soft

trilinear couplings and the mφ are soft scalar masses for the indicated fields. The soft

bilinear coupling B is defined such that at tree level the mass of the CP-odd MSSM Higgs

boson reads m2A = 2Bµ/ sin 2β. All off-diagonal couplings and scalar masses are set to zero,

as are the first- and second-generation Yukawa couplings and soft trilinears. Although we

should stress that making this choice will lead to a spectrum which is in conflict with the

LHC limits, doing so ensures that fine-tuning due to the other parameters is small, so that

we avoid washing out the fine-tuning impact of the stops when the tuning is small3 as can

be the case when the stop masses are less than 1 TeV. Note that the Higgs mass is also

allowed to vary in this scan, as shown in the right panel of Fig. 1. This illustrates the tuning

problem which people have been worrying about since the discovery of the 125 GeV Higgs

boson as we see that raising the stop masses is also pushing up the Higgs mass, meaning

that heavier Higgs masses require more fine-tuning. However, for a low value of the UV

scale this tuning is not so severe unless the stops are very heavy, and a 125 GeV Higgs can

be obtained without much tuning in this unrealistic case where we have minimized other

sources of tuning. On the other hand, the tuning becomes more severe as we increase the

cutoff such that for MX = 1016 GeV a lightest stop mass of 1–3 TeV can result in a fine-

tuning of ≈ 100 – 1000 and the minimum tuning we find4 for a 125 GeV Higgs is ≈ 200, as

shown in Fig. 1.

Since the stop mass does not have such a large impact on the fine-tuning when the cutoff

scale is very low we can use this to see more clearly the impact of the Z ′ mass on fine-

tuning. To do so we select a fixed low cutoff of MX = 20 TeV and compare the fine tuning

between the MSSM and E6SSM for two different values of the Z ′ mass. We choose to look

at MZ′ = 2.5 TeV, which is just above the current limits, and MZ′ = 4.5 TeV, which should

3For models in which the spectrum is heavier, when the stop masses are small the fine tuning reaches a lower

bound imposed by other heavier parameters.4Note that in the calculation of the Higgs mass there is a significant theoretical error, even with leading two-

loop corrections, which should be considered when thinking about what the results imply for the minimum

fine tuning in the model consistent with the recent discovery of a 125 GeV Higgs.16

be in reach in run II at the LHC [151] and then compare the fine tuning calculated in each

case to the tuning in the MSSM. For this, we have performed a six-dimensional parameter

space scan in both the MSSM and E6SSM, varying those parameters most relevant for the

fine tuning and the Higgs mass. Therefore, the set of parameters which we vary includes µ,

B and tan β for the MSSM, and λ, Aλ and tan β, for the E6SSM, which appear at tree level

in the EWSB conditions of the models. While the RGE contribution from large stop masses

to the fine tuning is small for such a low cutoff scale, the stop contributions to the effective

potential can play a significant role in reducing the fine tuning. For this reason it is still

important to properly treat the tuning associated with stop contributions to the one-loop

effective potential, and so we also scan over the soft masses m2Q3

, m2u3

and the stop mixing

At. The relevant parameters and ranges that were scanned over are summarized in Table II.

In addition to this we also repeat each scan for three different values of M2 to allow more

variation in the chargino masses.

In this case, we now consider realistic scenarios, where the parameters that are not

scanned over are set to values which keep the associated states comfortably above their

experimental limits. So in both the MSSM and E6SSM, all other soft scalar masses are set

to 5 TeV. We require a valid spectrum with no tachyonic states to exclude points which

would have an unrealistic minimum, for example due to the appearance of charge or color

breaking (CCB) minima. We work in the third family approximation, taking the first- and

second-generation Yukawa couplings to be zero, and we also assume that their associated

soft trilinears vanish. Similarly, we take Ab = Aτ = 0 GeV. The U(1) gaugino soft mass M1

was fixed to M1 = 300 GeV, and we fix M3 = 2000 GeV. Additionally, in the E6SSM the

U(1)N gaugino soft mass M ′1 is held fixed with M ′

1 = M1 = 300 GeV, and µ′ = 5 TeV.

In Fig. 2, results from the scan are plotted showing the tuning for each case against the

lightest Higgs mass. As expected, the dependence on the Higgs mass is now quite weak,

while the minimum tuning in the model for the E6SSM is increased by the mass of the Z ′

boson. So in the case of a very low cutoff the tuning required to get a 125 GeV Higgs is

not so large. However, the tuning from the Z ′ mass appears already at tree level and is,

therefore, not suppressed when the cutoff scale is low. In our scan we find that, for the

points satisfying the current limit on the mass of the Z ′ boson and having an approximately

125 GeV Higgs, the minimum fine-tuning that can be achieved is ∆min ≈ 121. If run II of

the LHC further pushes up the limit on the Z ′ mass to be above 4.5 TeV then the fine-tuning

17

MSSM E6SSM

2 ≤ tanβ ≤ 50 2 ≤ tanβ ≤ 50

−1 TeV ≤ µ ≤ 1 TeV −3 ≤ λ ≤ 3

−1 TeV ≤ B ≤ 1 TeV −10 TeV ≤ Aλ ≤ 10 TeV

200 GeV ≤ mQ3 ≤ 2000 GeV 200 GeV ≤ mQ3 ≤ 2000 GeV

200 GeV ≤ mu3 ≤ 2000 GeV 200 GeV ≤ mu3 ≤ 2000 GeV

−10 TeV ≤ At ≤ 10 TeV −10 TeV ≤ At ≤ 10 TeV

M2 = 100, 1050, 2000 GeV M2 = 100, 1050, 2000 GeV

TABLE II. The parameters scanned over and the ranges of values used in the MSSM and the

E6SSM models.

in the model will be greater than at least ∆min ≈ 394 for a Higgs mass between 124.5 and

125.5 GeV.

This demonstrates two important points about these U(1) extensions–first, that limits

on the Z ′ mass play an incredibly important role in constraining natural scenarios in such

models and, second, that the tuning from the Z ′ limits in these models depends less on

assumptions about SUSY breaking than the tuning required by the 125 GeV Higgs mea-

surement which concerns people in the MSSM.

There is another limit which plays a similar role. Chargino limits directly constrain the

µ parameter (or effective µ parameter in these U(1) extensions). The LEP bound [152] on

chargino masses, excluding mχ±1. 104 GeV, implies that |µ| should only be greater than

∼ 100 GeV, which is not substantially larger than MZ . Consequently the bound from LEP is

not high enough to have an impact on the fine-tuning obtained in the models and parameter

space regions that we have studied, as we have checked explicitly. Significantly larger lower

bounds on the µ parameter, and therefore on the fine-tuning, may arise from chargino limits

coming from LHC searches. However, the chargino limits from the LHC depend on whether

there are light sleptons or sneutrinos and the mass difference between the lightest chargino

and lightest neutralino. Current limits placed by CMS and ATLAS extend up to mχ±1≈

700–740 GeV if there are light sleptons [153, 154] with much weaker bounds if there are no

18

FIG. 2. Scatter plot of fine-tuning vs lightest Higgs mass for the MSSM (light blue, bottom band),

E6SSM with MZ′ = 2.5 TeV (dark blue, middle band) and E6SSM with MZ′ = 4.5 TeV (dark

yellow, top band). Note that there are points for which the fine-tuning in the MSSM and E6SSM

with MZ′ = 2.5 TeV is larger than is visible on this plot and those below; however, these points

are obscured by the overlaid data for the E6SSM with MZ′ = 4.5 TeV, and it is the lower bound

on the achievable tuning that is of interest here.

light sleptons or sneutrinos.5

Nonetheless, for the MSSM the impact of potential chargino mass limits is shown in

Fig. 3. There we see that if the full parameter space with mχ±1< 700 GeV was excluded, the

impact would be to make the tuning in the MSSM with a 20 TeV cutoff similar to that of the

E6SSM with the same cutoff and a Z ′ mass just larger than current limits. In the E6SSM,

while raising the chargino limit can have the same impact in principle, due to current limits

on the Z ′ mass already imposing a significant degree of tuning, chargino masses do not make

much of a noticeable change.

5Useful summary plots of these limit may be found on the public pages of ATLAS, https:

//atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/SUSY/ATLAS_SUSY_EWSummary/

ATLAS_SUSY_EWSummary.png and CMS http://cms.web.cern.ch/sites/cms.web.cern.ch/files/

styles/large/public/field/image/Image_03_exclusion_Combined.png?itok=8FMBpu_1.

19

https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/SUSY/ATLAS_SUSY_EWSummary/ATLAS_SUSY_EWSummary.png



http://cms.web.cern.ch/sites/cms.web.cern.ch/files/styles/large/public/field/image/Image_03_exclusion_Combined.png?itok=8FMBpu_1

http://cms.web.cern.ch/sites/cms.web.cern.ch/files/styles/large/public/field/image/Image_03_exclusion_Combined.png?itok=8FMBpu_1

FIG. 3. Scatter plot of fine-tuning vs lightest Higgs mass in the MSSM with 200 GeV ≤ mχ±1≤ 400

GeV shown in light blue (bottom band), 500 GeV ≤ mχ±1≤ 600 GeV in dark blue (middle band),

and 700 GeV ≤ mχ±1≤ 800 GeV in dark yellow (top band).

The exact level of tuning from the Z ′ depends on the charges of the extra U(1) gauge

symmetry it is associated with. In Fig. 4 we look at the fine-tuning for other U(1) extensions

for the same Z ′ masses as we did for the E6SSM. To simplify the analysis we fix tan β = 10,

but scan over the remaining parameters as in Table II and fix the rest to the same values

we did in the scan carried out for Fig. 2. In order to more clearly identify the lower bound

on the obtainable tuning in each model, the parameter values for points in these main grid

scans with a low fine-tuning were then used as the starting points for smaller scans about

those values. In these smaller scans the parameters were more finely varied to populate the

low fine-tuning regions.

As can be seen in Fig. 4, the severity of the tunings varies quite a bit. This is because

the charges appear as coefficients in front of the Z ′ mass in the EWSB condition. These

charges change the value of the coefficient d in Eq. (21). The values of the coefficient d in

each model, for tan β = 10, is {−0.01, 0.40, 0.50, 0.81} for {U(1)I , U(1)N , U(1)ψ, U(1)η} and

20

FIG. 4. Top left panel: Scatter plot of the fine-tuning vs lightest Higgs mass in the U(1)I model.

Top right panel: Scatter plot of the fine-tuning vs lightest Higgs mass in the U(1)ψ model. Bottom

panel: Scatter plot of the fine-tuning vs lightest Higgs mass in the U(1)η model. In each plot points

with MZ′ = 2.5 TeV are shown in dark blue (bottom band), and points with MZ′ = 4.5 TeV are

shown in dark yellow (top band).

this determines which of the models are most tuned.

Interestingly, the coefficient d is very small (and negative) in the case of the U(1)I . This

allows a dramatic reduction in the fine-tuning from the U(1)I symmetry. This is a result

of the Hu charge associated with U(1)I vanishing, which means that the D terms to the

lightest Higgs which is predominantly Hu at large tan β are suppressed, making it difficult

to raise the Higgs mass in the same way as happens in the other models and explaining why

heavier Higgs values in this model can’t be obtained. Therefore, the fine-tuning behaviour

in this model is closer to that of the MSSM, and in this case raising the Z ′ mass limit to 4.5

21

TeV will have little impact on naturalness. From naively estimating the tuning, using the d

coefficient one can estimate that Z ′ limits need to be around 15 TeV before they will raise

the tuning in this model.

Finally we want to emphasize that while in Fig. 2 the E6SSM looks more fine-tuned than

the MSSM this depends on the high scale boundary, MX , where the parameters are assumed

to be set by some SUSY breaking mechanism. Indeed in Ref. [16] a constrained version of

the E6SSM, with the high scale boundary at the GUT scale, is considered and there the

cE6SSM was found to be less tuned than the cMSSM. Since a 125 GeV Higgs can be achieved

in the E6SSM with lighter stops, then if the cutoff is large, the larger stop masses of the

MSSM can make that model more fine-tuned due to large RGE effects.

To further illustrate this point, we looked at how the tuning varies with MX for low

tuning benchmarks in the MSSM and E6SSM. These benchmarks are defined in Table III

and the results are shown in Fig. 5. Since the behavior is quite complicated we now discuss

these in detail as it provides some insight into the many differences in the tuning between

the two models.

In the top panel one can see that the MSSM BM1 tuning (dotted curve) steadily climbs

as the cutoff scale is increased, as one would expect when the tuning originates from large

soft masses entering from the RGEs. The panel on the middle left confirms this, showing

that the largest tuning contributions come from ∆At and ∆m2Hu

with the former being the

larger sensitivity until MX ≈ 108 GeV at which point ∆m2Hu

takes over, leading to the small

kink in overall tuning which can be seen in the dotted curve in the top panel. In this case

we have chosen a point with large mixing, which is known to reduce the MSSM tuning. We

found this does not eliminate the tuning as there is still a strong sensitivity to At, but we

did find that large mixing lead to less fine-tuning overall for the points we examined.

Comparing the MSSM tunings to the E6SSM tunings one can see that which point is

more fine-tuned depends on the scale at which the parameters are defined. This illustrates

that any statement about which model is more tuned depends on the high scale boundary,

MX .

For E6SSM BM1 the fine-tuning is shown by the solid curve in the top panel of Fig. 5

and the individual sensitivities are given in the middle right panel. The tuning actually

reduces initially as the cutoff is is increased from 20 TeV. This occurs because the largest

sensitivity is initially ∆λ (shown in solid light blue in the middle right panel). This contains

22

FIG. 5. Top panel: Scatter plot of the fine-tuning as a function of the cutoff scale MX for the four

benchmark points given in Table III. Middle left panel: Individual sensitivities for MSSM BM1

plotted against the high scale MX which give the overall tuning shown by the dotted line in the

top panel. Middle right panel: Individual sensitivities for E6SSM BM1 plotted against the high

scale MX which give the overall tuning shown by the solid line in the top panel. Bottom left panel:

Individual sensitivities for MSSM BM2 plotted against the high scale MX which give the overall

tuning shown by the dash-dotted line in the top panel. Bottom right panel: Individual sensitivities

for E6SSM BM2 plotted against the high scale MX which give the overall tuning shown by the

dashed line in the top panel.

23

MSSM BM1 MSSM BM2 E6SSM BM1 E6SSM BM2

tanβ(MZ) 10 10 10 10

s(MSUSY) [GeV] · · · · · · 6700 6700

κ1,2,3(MSUSY) · · · · · · 0.6 0.52

λ1,2(MSUSY) · · · · · · 0.2 0.13

µeff(MSUSY) [GeV] 689.7 1013.5 1093.3 1313.0

Beff(MSUSY) [GeV] 345.7 1032.5 3792.7 817.8

Aτ (MSUSY) [GeV] 0 −5057.9 0 −88.5

Ab(MSUSY) [GeV] 0 −5707.2 0 −1720.7

At(MSUSY) [GeV] −3335.7 −2734.8 −1100 −1103.2

m2L1,2

(MSUSY) [GeV2] 2.5× 107 6.35× 106 2.5× 107 4.94× 106

m2L3

(MSUSY) [GeV2] 2.5× 107 6.22× 106 2.5× 107 4.90× 106

m2e1,2

(MSUSY) [GeV2] 2.5× 107 6.27× 106 2.5× 107 5.21× 106

m2e3

(MSUSY) [GeV2] 2.5× 107 6.03× 106 2.5× 107 5.11× 106

m2Q1,2

(MSUSY) [GeV2] 2.5× 107 7.37× 106 2.5× 107 5.76× 106

m2Q3

(MSUSY) [GeV2] 4.45× 106 3.97× 106 4.50× 105 3.61× 106

m2u1,2

(MSUSY) [GeV2] 2.5× 107 7.30× 106 2.5× 107 5.54× 106

m2u3

(MSUSY) [GeV2] 4.0× 106 6.60× 105 5.86× 105 2.04× 106

m2d1,2

(MSUSY) [GeV2] 2.5× 107 7.30× 106 2.5× 107 5.88× 106

m2d3

(MSUSY) [GeV2] 2.5× 107 7.03× 106 2.5× 107 5.78× 106

m2Hd

(MSUSY) [GeV2] 1.82× 106 8.96× 106 4.06× 107 1.04× 107

m2Hu

(MSUSY) [GeV2] −3.60× 105 −9.35× 105 5.0× 105 −2.66× 105

m2S(MSUSY) [GeV2] · · · · · · −3.10× 106 −3.17× 106

M1(MSUSY) [GeV] 300 260.8 300 173.4

M2(MSUSY) [GeV] 2000 479.2 1050 281.4

M3(MSUSY) [GeV] 2000 1312.3 2000 1200

M ′1(MSUSY) [GeV] · · · · · · 300 175.2

MZ′ [GeV] · · · · · · 2473.2 2512.7

mh1[GeV] 124.3 124.4 125.0 126.2

mt1 [GeV] 1942.1 861.6 993.8 1665.0

mt2 [GeV] 2220.1 2023.9 1174.8 2094.4

mg [GeV] 2259.8 1472.9 2290.0 1407.4

∆(MX = 20 TeV) 157.3 242.8 165.3 402.1

∆(MX = 1016 GeV) 1089.0 949.0 1722.3 546.7

TABLE III. Parameters for the MSSM and E6SSM benchmark points. In the E6SSM, we define

µeff ≡ λs/√

2 and Beff = Aλ. The soft masses m2Hd

, m2Hu

and m2S are those that satisfy the

EWSB conditions including one-loop Coleman-Weinberg corrections involving the top and stops.

For E6SSM BM1 (BM2) we also set µ′ = 5000.0 (897.9) GeV, Bµ′ = 5000.0 (−4.21× 105) GeV2,

Aκ1,2,3 = 0 (−1389.2) GeV, Aλ1,2 = 0 (−52.9) GeV, m2D1,2,3

= 2.5 × 107 (4.81 × 106) GeV2,

m2D1,2,3

= 2.5 × 107 (4.90 × 106) GeV2, m2Hd

1,2= 2.5 × 107 (4.46 × 106) GeV2, m2

Hu1,2

= 2.5 × 107

(4.81× 106) GeV2, m2S1,2

= 2.5× 107 (5.28× 106) GeV2, m2H′ = 2.5× 107 (4.94× 106) GeV2 and

m2H′

= 2.5× 107 (4.87× 106) GeV2.24

some terms proportional to M ′ 2Z , which provide the dominant contribution to this sensitivity

at very low MX . However, as MX is increased contributions from the soft masses become

more important and these actually start to cancel the large contribution to ∆λ coming from

MZ′ until ∆λ passes through zero. At the same time though these large soft masses also

cause other sensitivities to grow, in particular ∆M3 . The fine-tuning rises with MX once

MX & 105 − 106 GeV, but remains lower than that of the other points, until MX ≈ 108

GeV. Eventually the ∆M3 sensitivity leads to this point being the most fine-tuned of the

four shown in Fig. 5.

Although the gluino mass and M3(MSUSY ) have similar values to those in the MSSM

BM1 point, in the E6SSM M3(MX) is larger due to the altered RGE running from exotic

matter6. This is why this E6SSM BM1 has a larger tuning at larger values of MX , coming

from ∆M3 .

Interestingly other sensitivities are suppressed by this effect since at the same time larger

M3 at higher scales reduces the soft squark masses at MX . Therefore, the stop mass contri-

butions are ameliorated, compared to the MSSM, both by allowing lighter stops at MSUSY

and by the modified RGE running. Nonetheless the stops still do lead to ∆m2Hu

increasing

with the cutoff through the usual mechanism7.

By contrast the tuning for E6SSM BM2 is very different, as is shown by the dashed line in

the top panel, with the individual sensitivities given in the bottom right panel. This point

was chosen as it had a much lighter gluino mass that is just above the experimental limit of

1.4 TeV [155]. At 20 TeV this benchmark is not amongst the lowest tuned points, since at

that scale the tree-level tuning from MZ′ dominates. However, the reduction in M3 means

that ∆M3 is substantially lower and only becomes the dominant tuning at a much larger

scale of MX & 1012 − 1013 GeV, giving a tuning at 1016 GeV of ≈ 546, which is far below

that of the other three benchmark points.

In addition to this, the soft parameters in E6SSM BM2 follow a pattern similar to that

found in the constrained model. With the exception of the parameters m2Q3

, m2u3

, m2Hd

,

m2Hu

, and M3, the values of which are given in Table III, the soft masses at the SUSY scale

correspond to the values that result in the cE6SSM with m0 = 2.2 TeV, M1/2 = 1003 GeV,

6This altered RGE running is a result of the exotic matter introduced to keep the extra U(1) anomaly free.7Wherein m2

Hu(MSUSY ) receives a positive contribution to it’s mass from m2

Hu(MX) and a negative con-

tribution from m2Q3

(MX) and m2u3

(MX), allowing heavy stop masses to cause fine-tuning. In this case

m2Hu

(MSUSY ) is held fixed so as the cutoff increases the values of these soft masses at MX will be larger

and there will be a bigger cancellation between them, increasing the sensitivity of MZ to both m2Hu

and the

soft scalar masses for the stops.25

A0 = 500 GeV, κ1,2,3(MX) = 0.1923, λ(MX) = 0.2646 and λ1,2(MX) = 0.1. This leads to a

significant reduction in the contributions to the RG running of m2Hu

and m2Q3

coming from

terms of the form g21Σ1 and, to a lesser extent, g′21 Σ′1. Here we define for the E6SSM (see

also Eqs. (B8-B9) for general U(1) inspired models)

Σ1 =3∑i=1

(m2Qi− 2m2

ui+m2

di+m2

ei−m2

Li+m2

Hui−m2

Hdi

+m2Di−m2

Di

)−m2

H′ +m2H′,

Σ′1 =3∑i=1

(6m2

Qi+ 3m2

ui+ 6m2

di+m2

ei+ 4m2

Li− 4m2

Hui− 6m2

Hdi

+ 5m2Si− 9m2

Di− 6m2

Di

)+ 4m2

H′ − 4m2H′.

In the unconstrained case, this contribution acts to drive up the values of m2Q3

and m2Hu

,

and thus the associated tuning sensitivities, at the cutoff scale MX . In the case of E6SSM

BM2, on the other hand, the reduced splitting between the soft masses leads to a much

smaller contribution from these terms. Together with the reduction in M3 described above,

this allows to maintain the observed low fine-tuning at very large values of MX .

MSSM benchmark BM2 (dash-dotted in top panel, individual sensitivities in bottom left

panel) is designed to be similar to E6SSM BM2, for a reasonable comparison. However, from

the individual sensitivities one can see that the behavior is quite similar to MSSM BM1,

though in this case ∆m2Hu

becomes the largest tuning at a higher MX and does not reach

such large values, since more of the tuning is from the mixing in this case.

VI. CONCLUSIONS

Prior to stringent experimental constraints on the mass of the lightest Higgs boson and

squarks in supersymmetric models, a simple picture of a natural SUSY model emerged

from theoretical reasoning, with soft masses set to similar values at the GUT scale through

local gravity interactions with the hidden sector. Through the use of renormalization group

running, one can then see that at the EW scale the stops enter the EWSB condition for

MZ ; therefore, it was expected that these masses should not be much bigger than 100 GeV.

However, to disturb this elegant picture, first LEP placed constraints on the Higgs mass,

requiring it to be above 114.4 GeV [156, 157], which already introduced significant tuning

for constrained models since heavy stops are required to raise the lightest Higgs mass above

26

its tree-level bound. Then, recently, this problem got much worse since the LHC measured

the Higgs mass to be around 125 GeV.

U(1) extensions motivated by the µ problem, E6 GUT theories and the connection to

string theory contain both F - and D- term contributions to the light Higgs mass which

can raise the tree-level mass, evading the need for large radiative corrections to increase it.

However, such models come with their own fine-tuning problem, where the Z ′ mass appears

in the EWSB condition for MZ at tree level. While in a previous study of the constrained

E6SSM it was found that the tuning is less severe than the MSSM, it was still significant.

In light of such difficulties it is worth considering whether the simple picture which

emerged is wrong in some way and if there are other possibilities that allow naturalness. Or

to phrase this in a more challenging manner are there ways to constrain the naturalness of

these models that do not rely upon assumptions about how SUSY is broken?

We have investigated this question here in the context of the MSSM and U(1) extensions.

Since the RG evolution links the soft masses together and causes these problems from stop

and gluino masses the most conservative approach to placing naturalness limits is to choose

a low cutoff. We find that in the MSSM the most direct way to constrain naturalness in the

model without making assumptions about the SUSY breaking scale is through limits on the

chargino masses. Current LHC limits on charginos are not model independent and thereby

leave many gaps where one can have light charginos.

In contrast we find that in U(1) extensions of the MSSM there is an additional way to

constrain the naturalness of the models, which is through the Z ′ mass limit. We find when

we impose a low cutoff of 20 TeV for setting the soft masses, the lowest tuning in the E6SSM

compatible with a Z ′ mass of 2.5 TeV was ∆ ≈ 121, while if the LHC run II can place a limit

of 4.5 TeV on M ′Z then the tuning would be approximately 394. By comparison the current

situation in the MSSM only requires a tuning of around 38. This should be interpreted as

saying that in the most conservative limits one can place on naturalness in these models,

the tuning in the E6SSM is worse. However, if there are no charginos below 700 GeV then

the situation in the two models would be the same.

This should also be contrasted with what happens as we raise the high scale boundary,

MX . We showed that for our benchmark points, which one is more tuned depends very

strongly on MX . The E6SSM tuning is sufficiently complicated by the interplay of these

different sources of tension in the EWSB conditions that a small reduction in fine-tuning

27

can even occur for a moderate increase in MX . However, as MX increases towards the scale

where the gauge couplings unify, the familiar tunings do dominate, though with tunings from

the gluino mass appearing to be more significant relative to those from soft scalar masses.

We also looked at the tuning in different U(1) extensions for fixed tan β = 10. We found

that in every case except for the U(1)I the fine-tuning was much worse for the larger Z ′ mass,

further emphasizing the importance of this in U(1) extensions. The U(1)I model showed

the least tuning due to the vanishing charge of the Hu state. This model is quite interesting

in the sense that it provides a solution to the µ problem while avoiding the large tuning

(with current limits) from the Z ′ mass. However, one should remember we are looking at

conservative limits on naturalness here and there is no solution to the usual tuning coming

from the large stops needed to get a 125 GeV Higgs in this model, which will be a problem

as the UV scale is raised.

ACKNOWLEDGMENTS

We would like to thank R. Nevzorov for many useful discussions and comments and

F. Staub for very helpful responses regarding the gauge kinetic mixing contributions in

SARAH. DH would also like to thank A. Ismail and T. Rizzo for their helpful comments

regarding fine-tuning in the MSSM. PA would like to thank J.H. Park, A. Voigt and

D. Stockinger for useful discussion regarding the use of FlexibleSUSY in this project. This

work was supported by the University of Adelaide and by the ARC Centre of Excellence for

Particle Physics at the Terascale.

Appendix A: FINE-TUNING MASTER FORMULA

To write down the tree-level master formula, it is convenient to define the quantities

zi = εijk∂fj∂s

∂fk∂ tan β

(A1)

with f1, f2, f3 as given in Eq. (16). The relevant partial derivatives are

∂f1

∂ tan β= −2MZ

gcos2 β

{λAλs√

2cos β + sin β

[m2Hd

+s2

2

(λ2 + g′21 Q1QS

)+M2

Z

(5

2− 4λ2

g2− 4g′21

g2Q1Q2 +

6g′21g2

Q21

)]+ 3M2

Z sin3 β

[2λ2

g2− 1

28

+2g′21g2

(Q1Q2 −Q2

1

)]},

∂f1

∂s=

2MZ

g

[s(λ2 + g′21 Q1QS

)cos β − λAλ√

2sin β

],

∂f2

∂ tan β=

2MZ

gcos2 β

{λAλs√

2sin β + cos β

[m2Hu +

s2

2

(λ2 + g′21 Q2QS

)+M2

Z

(5

2− 4λ2

g2− 4g′21

g2Q1Q2 +

6g′21g2

Q22

)]+ 3M2

Z cos3 β

[2λ2

g2− 1

+2g′21g2

(Q1Q2 −Q2

2

)]},

∂f2

∂s=

2MZ

g

[s(λ2 + g′21 Q2QS

)sin β − λAλ√

2cos β

],

∂f3

∂ tan β=

2M2Z

g2cos2 β

[g′21 QSs (Q2 −Q1) sin 2β −

√2λAλ cos 2β

],

∂f3

∂s= m2

S +2λ2M2

Z

g2+g′212QS

[4M2

Z

g2

(Q1 cos2 β +Q2 sin2 β

)+ 3QSs

2

].

For a running parameter q appearing in the tree-level EWSB conditions, the corresponding

contribution to the individual sensitivity can then be written

∆q = z1∂f1

∂q+ z2

∂f2

∂q+ z3

∂f3

∂q. (A2)

It is straightforward to compute the appropriate derivatives directly from the EWSB con-

ditions, Eq. (16). Similarly, the quantity C appearing in Eq. (26) is given by

C =1

2

(z1∂f1

∂MZ

+ z2∂f2

∂MZ

+ z3∂f3

∂MZ

), (A3)

with

∂f1

∂MZ

=2

gcos β

(m2Hd

+λ2s2

2+g′212Q1QSs

2 +6g′21g2

Q21M

2Z

)−√

2λAλs

gsin β

+3M2

Z

gcos β cos 2β +

6

g3M2

Z sin β sin 2β[λ2 + g′21

(Q1Q2 −Q2

1

)],

∂f2

∂MZ

=2

gsin β

(m2Hu +

λ2s2

2+g′212Q2QSs

2 +6g′21g2

Q22M

2Z

)−√

2λAλs

gcos β

−3M2Z

gsin β cos 2β +

6

g3M2

Z cos β sin 2β[λ2 + g′21

(Q1Q2 − Q2

2)],

∂f3

∂MZ

=4MZ

g2

[λ2s− λAλ√

2sin 2β + g′21 QSs

(Q1 cos2 β +Q2 sin2 β

)].

29

Appendix B: RGE CONTRIBUTIONS

Provided that one does not run over too large a range of scales, the solutions to the RG

equations for a model can be reasonably well approximated by a Taylor series, Eq. (32). For

a parameter p, this reads

q(Q) = q(MX) +t

16π2

(β(1)q +

β(2)q

16π2

)+

t2

(16π2)2b(2)q (MX),

where we have for convenience defined

b(2)q (MX) =

1

2!

∑qk

β(1)qk

∂βq∂qk

∣∣∣∣∣MX

.

We have constructed the necessary series solutions in both the MSSM and the U(1) extended

models. Due to the smallness of the first- and second-generation Yukawa couplings, we

neglect them in our calculations. The corresponding soft SUSY breaking trilinears are

likewise omitted. Additionally, all soft mass matrices are assumed diagonal, and the gaugino

masses are taken to be real.

In the MSSM, the relevant parameters for the fine-tuning calculation are µ, B, m2Hu

, m2Hd

at tree level. For the renormalization group running of the relevant parameters SOFTSUSY

uses the one- and two-loop RGEs from [158, 159]. The corresponding O(t2) contributions

are

b(2)µ =

µ

2

[45y4

t + 45y4b + 9y4

τ + 30y2t y

2b + 6y2

t y2τ + 18y2

by2τ − 32g2

3(y2t + y2

b )

− 12g22(3y2

t + 3y2b + y2

τ )−4

5g2

1(11y2t + 8y2

b + 6y2τ ) + 3g4

2 −189

25g4

1 +18

5g2

1g22

], (B1a)

b(2)B = 72y4

tAt + 72y4bAb + 16y4

τAτ + 12y2t y

2b (At + Ab) + 12y2

τy2b (Ab + Aτ )

− 32g23y

2t (At −M3)− 32g2

3y2b (Ab −M3)

− 18g22y

2t (At −M2)− 18g2

2y2b (Ab −M2)− 6g2

2y2τ (Aτ −M2)

− 26

5g2

1y2t (At −M1)− 14

5g2

1y2b (Ab −M1)− 18

5g2

1y2τ (Aτ −M1)

+ 12g42M2 +

396

25g4

1M1, (B1b)

b(2)

m2Hd

= 72y4b

(m2Hd

+m2Q3

+m2d3

+ 2A2b

)+ 6y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3+ (At + Ab)

2)

30

+ 12y2τy

2b

(2m2

Hd+m2

Q3+m2

d3+m2

L3+m2

e3+ (Aτ + Ab)

2)

+ 16y4τ

(m2Hd

+m2L3

+m2e3

+ 2A2τ

)− 32g2

3y2b

(m2Hd

+m2Q3

+m2d3

+ A2b − 2M3Ab + 2M2

3

)− 18g2

2y2b

(m2Hd

+m2Q3

+m2d3

+ A2b − 2M2Ab + 2M2

2

)− 6g2

2y2τ

(m2Hd

+m2L3

+m2e3

+ A2τ − 2M2Aτ + 2M2

2

)− 14

5g2

1y2b

(m2Hd

+m2Q3

+m2d3

+ A2b − 2M1Ab + 2M2

1

)− 18

5g2

1y2τ

(m2Hd

+m2L3

+m2e3

+ A2τ − 2M1Aτ + 2M2

1

)− 18g4

2M22 −

198

25g4

1

(S + 3M2

1

), (B1c)

b(2)

m2Hu

= 72y4t

(m2Hu +m2

Q3+m2

u3+ 2A2

t

)+ 6y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3+ (At + Ab)

2)

− 32g23y

2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2AtM3 + 2M23

)− 18g2

2y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2AtM2 + 2M22

)− 26

5g2

1y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2AtM1 + 2M21

)+

198

25g4

1

(S − 3M2

1

)− 18g4

2M22 . (B1d)

In these expressions the quantity S is defined by

S = m2Hu −m

2Hd

+3∑i=1

(m2Qi−m2

Li− 2m2

ui+m2

di+m2

ei

). (B2)

If, in addition, the one-loop contributions to the effective potential from top and stop loops

are included, it is also necessary to construct the expansions for m2Q3

, m2u3

and At. The

coefficients read

b(2)

m2Q3

= 24y4t

(m2Hu +m2

Q3+m2

u3+ 2A2

t

)+ 24y4

b

(m2Hd

+m2Q3

+m2d3

+ 2A2b

)+ 4y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3+ (At + Ab)

2)

+ 2y2by

2τ

(2m2

Hd+m2

Q3+m2

L3+m2

d3+m2

e3+ (Ab + Aτ )

2)

− 32

3g2

3y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2M3At + 2M23

)− 32

3g2

3y2b

(m2Hd

+m2Q3

+m2d3

+ A2b − 2M3Ab + 2M2

3

)− 6g2

2y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2M2At + 2M22

)31

− 6g22y

2b

(m2Hd

+m2Q3

+m2d3

+ A2b − 2M2Ab + 2M2

2

)− 26

15g2

1y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2M1At + 2M21

)− 14

15g2

1y2b

(m2Hd

+m2Q3

+m2d3

+ A2b − 2M1Ab + 2M2

1

)+ 96g4

3M23 − 18g4

2M22 +

66

25g4

1(S −M21 ), (B3a)

b(2)

m2u3

= 48y4t

(m2Hu +m2

Q3+m2

u3+ 2A2

t

)+ 4y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3+ (At + Ab)

2)

− 64

3g2

3y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2M3At + 2M23

)− 12g2

2y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2M2At + 2M22

)− 52

15g2

1y2t

(m2Hu +m2

Q3+m2

u3+ A2

t − 2M1At + 2M21

)+ 96g4

3M23 −

264

25g4

1

(S + 4M2

1

), (B3b)

b(2)At

= 144y4tAt + 24y4

bAb + 14y2t y

2b (At + Ab) + 2y2

by2τ (Ab + Aτ )

− 64g23y

2t (At −M3)− 36g2

2y2t (At −M2)− 52

5g2

1y2t (At −M1)

− 32

3g2

3y2b (Ab −M3)− 6g2

2y2b (Ab −M2)− 14

15g2

1y2b (Ab −M1)

− 64g43M3 + 12g4

2M2 +572

25g4

1M1 (B3c)

We can similarly obtain the two-loop β functions and coefficients b(2)p for a general set of

U(1)′ charges. Two-loop RGEs for the gauge and Yukawa couplings, gaugino masses and soft

trilinears, along with the one-loop RGEs for the soft scalar masses, were originally obtained

in Ref. [18] for the particular case of the E6SSM. FlexibleSUSY uses full one- and two-loop

RGEs from SARAH, which for the models considered here are based on Ref. [159] and the

recent extension8 in Ref. [160] to include models with multiple U(1) gauge groups, in the

most general case where the trace of the matrix formed from the charges QYi of the U(1)Y

gauge symmetry and Qi of the extra U(1)′ symmetry does not vanish, i.e.∑

iQYi Qi 6= 0.

When this trace is nonzero it will also induce gauge kinetic mixing to be generated during

RGE evolution and this is the case in the models we consider here.9 However, when these

models are evolved down from the GUT scale, the radiatively generated gauge kinetic mixing

8In the version of SARAH which we used the extra terms from this extension were included for all terms

except the trilinear and bilinear soft masses. We have been in contact with the SARAH author about this

and understand they will be included in future versions.9In E6 inspired models with only complete 27-plet matter multiplets this trace would vanish. However since

we assume some incomplete multiplets so that our models are consistent with gauge coupling unification

this trace doesn’t vanish. 32

gives an off-diagonal gauge coupling, g11, of just ≈ 0.02 at the electroweak scale [13] and so

it does not play a large role. At the same time if gauge kinetic mixing is included the RGE

expressions become very large and unmanageable, so we neglected the gauge kinetic mixing

by setting the SARAH flag NoU1Mixing to true.

At tree level in the EWSB conditions the parameters that must be considered are λ, Aλ,

m2Hu

, m2Hd

, m2S and g1, g2 and g′1. Neglecting kinetic mixing, the two-loop β functions for

the relevant gauge couplings read

β(1)g1

=48

5g3

1, (B4a)

β(2)g1

=2

25g3

1

(30g′21 ΠY

Q + 117g21 + 135g2

2 + 300g23 − 10Σκ − 15Σλ − 65y2

t − 35y2b − 45y2

τ

),

(B4b)

β(1)g2

= 4g32, (B4c)

β(2)g2

=2

5g3

2

(5g′21 ΠL

Q + 9g21 + 115g2

2 + 60g23 − 5Σλ − 15y2

t − 15y2b − 5y2

τ

), (B4d)

β(1)

g′1= g′31 ΣQ, (B4e)

β(2)

g′1=

2

5g′31(−15Σκ

(Q2D +Q2

D +Q2S

)− 30y2

b

(Q2d +Q2

1 +Q2Q

)+ 120g2

3ΠCQ

− 10y2τ

(Q2e +Q2

L +Q21

)+ 15g2

2ΠLQ + 10g′21 ΠQ + 6g2

1ΠYQ

− 10Σλ

(Q2S +Q2

1 +Q22

)− 30y2

t

(Q2u +Q2

2 +Q2Q

)). (B4f)

In order to keep these expressions compact, we have used the notation

ΣQ =∑i

Q2i =

321

40cos2 θ +

217

24sin2 θ +

27

8√

15sin 2θ

to denote the trace over the U(1)′ charges, along with10

ΣYQ =

∑i

√5

3QYi Qi = − 3√

10cos θ − 1√

6sin θ,

ΠQ =∑i

Q4i

=2049

1600cos4 θ +

483

80√

15cos3 θ sin θ +

681

160cos2 θ sin2 θ +

9

16√

15cos θ sin3 θ +

1297

576sin4 θ,

10The first of these is the trace which is assumed to vanish in Ref. [159]. Although we use the NoU1Mixing

flag to neglect gauge kinetic mixing, SARAH does this by removing the RGE for the off diagonal gauge

couplings and effectively setting them to zero at all scales by removing all terms involving them from the

RGEs. Therefore, some terms with this trace remain and the RGEs shown here do not reduce to those

which one would obtain from Ref. [159] or Ref. [18] unless ΣYQ = 0. Note, however, these contributions do

not appear in the corresponding trilinears due to the version of SARAH used.

33

ΠYQ =

∑i

(√5

3QYi

)2

Q2i =

59

40cos2 θ +

31

24sin2 θ +

3

8√

15sin 2θ,

ΠLQ = 3Q2

1 + 3Q22 +Q2

H′ +Q2H′

+ 3Q2L + 9Q2

Q =39

20cos2 θ +

19

12sin2 θ +

3

4√

15sin 2θ,

ΠCQ = Q2

d +Q2D +Q2

D+ 2Q2

Q +Q2u =

1

2.

Note that in these expressions the U(1)Y and U(1)′ charges are assumed to be GUT-

normalized. The expressions in terms of the E6 mixing angle θ follow from the charge

assignments given in Table I and hold provided that U(1) mixing is neglected. Similarly, we

write

Σλ = λ21 + λ2

2 + λ23, Σκ = κ2

1 + κ22 + κ2

3,

Πλ = λ41 + λ4

2 + λ43, Πκ = κ4

1 + κ42 + κ4

3.

The corresponding O(t2) coefficients for the gauge couplings are simply

b(2)g1

=3456

25g5

1, (B5a)

b(2)g2

= 24g52, (B5b)

b(2)

g′1=

3

2g′51 Σ2

Q. (B5c)

The one- and two-loop contributions to the β function for λ and the O(t2) coefficient in the

series expansion are

β(1)λ = λ

[2λ2 + 2Σλ + 3Σκ + 3y2

t + 3y2b + y2

τ − 3g22 −

3

5g2

1 − 2(Q2

1 +Q22 +Q2

S

)g′21

], (B6a)

β(2)λ = λ

{− 2λ2

(λ2 + 2Σλ + 3Σκ

)− 4Πλ − 6Πκ − 3λ2

(3y2

t + 3y2b + y2

τ

)− 3

(3y4

t + 3y4b + 2y2

t y2b + y4

τ

)+ 6g2

2Σλ +2

5g2

1

(2y2

t − y2b + 3y2

τ + 2Σκ + 3Σλ

)+g′21

[4Q2

Sλ2 − 6

(Q2

2 −Q2Q −Q2

u

)y2t − 6

(Q2

1 −Q2Q −Q2

d

)y2b

−2(Q2

1 −Q2L −Q2

e

)y2τ − 6

(Q2S −Q2

D −Q2D

)Σκ − 4

(Q2S −Q2

1 −Q22

)Σλ

]+ 16g2

3

(y2t + y2

b + Σκ

)+

33

2g4

2 +297

50g4

1 + 2g′41[2Q4

1 + 2Q42 + 2Q4

S +(Q2

1 +Q22 +Q2

S

)ΣQ

]+

9

5g2

1g22 + 6g′21 g

22

(Q2

1 +Q22

)+

6

5g2

1g′21

[Q2

1 +Q22 + (Q2 −Q1) ΣY

Q

]}, (B6b)

b(2)λ = λ

{2λ2

(3λ2 + 4Σλ + 6Σκ

)+ 4Πλ + 6Πκ + 6

(Σλ +

3

2Σκ

)2

+ 7λ2(3y2

t + 3y2b + y2

τ

)34

+(3y2

t + 3y2b + y2

τ

)(2Σλ + 3Σκ) + 3

(15

2y4t +

15

2y4b +

3

2y4τ + 5y2

by2t + 3y2

by2τ + y2

t y2τ

)− 1

5g2

1

(12λ2 + 16y2

b + 22y2t + 12y2

τ + 12Σλ + 13Σκ

)− 3g2

2

(4λ2 + 6y2

t + 6y2b + 2y2

τ + 4Σλ + 3Σκ

)− 16g2

3

(y2t + y2

b + Σκ

)− 2g′21

[4(Q2

1 +Q22 +Q2

S

) (λ2 + Σλ

)+ 3Σκ

(Q2

1 +Q22 +Q2

S +Q2D +Q2

D

)+ 3y2

b

(2Q2

1 +Q22 +Q2

d +Q2Q +Q2

S

)+ 3y2

t

(Q2

1 + 2Q22 +Q2

Q +Q2S +Q2

u

)+y2

τ

(2Q2

1 +Q22 +Q2

e +Q2L +Q2

S

)]− 15

2g4

2 −279

50g4

1

+ 2g′41(Q2

1 +Q22 +Q2

S

) (Q2

1 +Q22 +Q2

S − ΣQ

)+

9

5g2

1g22 + 6g′21 g

22

(Q2

1 +Q22 +Q2

S

)+

6

5g2

1g′21

(Q2

1 +Q22 +Q2

S

)}. (B6c)

It is sufficient for our purposes to consider the trilinear coupling aλ ≡ λAλ, rather than Aλ,

for which the relevant expressions read

β(1)aλ

= aλ

[2λ2 + 2Σλ + 3Σκ + 3y2

t + 3y2b + y2

τ − 3g22 −

3

5g2

1 − 2(Q2

1 +Q22 +Q2

S

)g′21

]+ λ

[4λaλ + 4Σaλ + 6Σaκ + 6ytat + 6ybab + 2yτaτ + 6g2

2M2 +6

5g2

1M1

+ 4g′21 M′1

(Q2

1 +Q22 +Q2

S

) ], (B7a)

β(2)aλ

= aλ

{− 2λ2

(λ2 + 2Σλ + 3Σκ

)− 4Πλ − 6Πκ − 3λ2

(3y2

t + 3y2b + y2

τ

)− 3

(3y4

t + 3y4b + 2y2

t y2b + y4

τ

)+ 6g2

2Σλ +2

5g2

1

(2y2

t − y2b + 3y2

τ + 2Σκ + 3Σλ

)+g′21

[4Q2

Sλ2 − 6

(Q2

2 −Q2Q −Q2

u

)y2t − 6

(Q2

1 −Q2Q −Q2

d

)y2b

−2(Q2

1 −Q2L −Q2

e

)y2τ − 6

(Q2S −Q2

D −Q2D

)Σκ − 4

(Q2S −Q2

1 −Q22

)Σλ

]+ 16g2

3

(y2t + y2

b + Σκ

)+

33

2g4

2 +297

50g4

1 + 2g′41[2Q4

1 + 2Q42 + 2Q4

S +(Q2

1 +Q22 +Q2

S

)ΣQ

]+

9

5g2

1g22 + 6g′21 g

22

(Q2

1 +Q22

)+

6

5g2

1g′21

(Q2

1 +Q22

)}+ λ

{− 4λaλ

(λ2 + 2Σλ + 3Σκ

)− 4λ2 (λaλ + 2Σaλ + 3Σaκ)− 16Πaλ − 24Πaκ − 6λaλ

(3y2

t + 3y2b + y2

τ

)− 6λ2 (3ytat + 3ybab + yτaτ )− 12

[3y3

t at + 3y3bab + ytyb (ytab + ybat) + y3

τaτ]

+ 32g23

[ytat + ybab + Σaκ −

(y2t + y2

b + Σκ

)M3

]+ 12g2

2 (Σaλ − ΣλM2)

+2

5g2

1

[4ytat − 2ybab + 6yτaτ + 4Σaκ + 6Σaλ − 2

(2y2

t − y2b + 3y2

τ + 2Σκ + 3Σλ

)M1

]+ 4g′21

[3yt(Q2Q −Q2

2 +Q2u

)(at − ytM ′

1) + 3yb(Q2Q −Q2

1 +Q2d

)(ab − ybM ′

1)

35

+ yτ(Q2L −Q2

1 +Q2e

)(aτ − yτM ′

1) + 2Q2Sλ (aλ − λM ′

1)

+ 2(Q2

1 +Q22 −Q2

S

)(Σaλ − ΣλM

′1) + 3

(Q2D +Q2

D−Q2

S

)(Σaκ − ΣκM

′1)]

− 66g42M2 −

594

25g4

1M1 − 8g′41 M′1

[2(Q4

1 +Q42 +Q4

S

)+(Q2

1 +Q22 +Q2

S

)ΣQ

]− 18

5g2

2g21 (M2 +M1)− 12g2

2g′21

(Q2

1 +Q22

)(M2 +M ′

1)

− 12

5g2

1g′21

(Q2

1 +Q22

)(M1 +M ′

1)

}, (B7b)

b(2)aλ

= λ

[4λaλ + 4Σaλ + 6Σaκ + 6ytat + 6ybab + 2yτaτ + 6g2

2M2 +6

5g2

1M1

+ 4g′21 M′1

(Q2

1 +Q22 +Q2

S

) ]×[2λ2 + 2Σλ + 3Σκ + 3y2

t + 3y2b + y2

τ − 3g22

− 3

5g2

1 − 2(Q2

1 +Q22 +Q2

S

)g′21

]+ aλ

{2λ2

(3λ2 + 4Σλ + 6Σκ

)+ 4Πλ + 6Πκ

+ 6

(Σλ +

3

2Σκ

)2

+ 7λ2(3y2

t + 3y2b + y2

τ

)+(3y2

t + 3y2b + y2

τ

)(2Σλ + 3Σκ)

+ 3

(15

2y4t +

15

2y4b +

3

2y4τ + 5y2

by2t + 3y2

by2τ + y2

t y2τ

)− 1

5g2

1

(12λ2 + 16y2

b + 22y2t + 12y2

τ + 12Σλ + 13Σκ

)− 3g2

2

(4λ2 + 6y2

t + 6y2b + 2y2

τ + 4Σλ + 3Σκ

)− 16g2

3

(y2t + y2

b + Σκ

)− 2g′21

[4(Q2

1 +Q22 +Q2

S

) (λ2 + Σλ

)+ 3Σκ

(Q2

1 +Q22 +Q2

S +Q2D +Q2

D

)+ 3y2

b

(2Q2

1 +Q22 +Q2

d +Q2Q +Q2

S

)+ 3y2

t

(Q2

1 + 2Q22 +Q2

Q +Q2S +Q2

u

)+y2

τ

(2Q2

1 +Q22 +Q2

e +Q2L +Q2

S

)]− 15

2g4

2 −279

50g4

1

+ 2g′41(Q2

1 +Q22 +Q2

S

) (Q2

1 +Q22 +Q2

S − ΣQ

)+

9

5g2

1g22 + 6g′21 g

22

(Q2

1 +Q22 +Q2

S

)+

6

5g2

1g′21

(Q2

1 +Q22 +Q2

S

)}+ λ

[16λ3aλ + 16Πaλ + 24Πaκ + 8λ

3∑i=1

λi (λiaλ + aλiλ)

+ 12λ3∑i=1

κi (κiaλ + aκiλ) + 83∑i=1

3∑j=1

λiλj(aλiλj + λiaλj

)+ 18

3∑i=1

3∑j=1

κiκj(κiaκj + aκiκj

)+ 24

3∑i=1

3∑j=1

λiκj(λiaκj + aλiκj

)+ 72y3

t at + 72y3bab

+ 16y3τaτ + 12ytyb (atyb + ytab) + 12ybyτ (abyτ + ybaτ ) + 30λyt (aλyt + λat)

+ 30λyb (aλyb + λab) + 10λyτ (aλyτ + λaτ )− 32g23yt (at − ytM3)− 32g2

3yb (ab − ybM3)

− 32g23 (Σaκ −M3Σκ)− 12g2

2λ (aλ − λM2)− 18g22yt (at − ytM2)− 18g2

2yb (ab − ybM2)

36

− 6g22yτ (aτ − yτM2)− 12g2

2 (Σaλ −M2Σλ)−12

5g2

1λ (aλ − λM1)− 26

5g2

1yt (at − ytM1)

− 14

5g2

1yb (ab − ybM1)− 18

5g2

1yτ (aτ − yτM1)− 12

5g2

1 (Σaλ −M1Σλ)−8

5g2

1 (Σaκ −M1Σκ)

− 8g′21 λ(Q2

1 +Q22 +Q2

S

)(aλ − λM ′

1)− 12g′21 yt(Q2

2 +Q2Q +Q2

u

)(at − ytM ′

1)

− 12g′21 yb(Q2

1 +Q2Q +Q2

d

)(ab − ybM ′

1)− 4g′21 yτ(Q2

1 +Q2L +Q2

e

)(aτ − yτM ′

1)

− 8g′21(Q2

1 +Q22 +Q2

S

)(Σaλ −M ′

1Σλ)− 12g′21(Q2S +Q2

D +Q2D

)(Σaκ −M ′

1Σκ)

+ 48g42M2 +

576

25g4

1M1 + 8g′41 M′1ΣQ

(Q2

1 +Q22 +Q2

S

) ], (B7c)

where

Σaλ = λ1aλ1 + λ2aλ2 + λ3aλ3 , Σaκ = κ1aκ1 + κ2aκ2 + κ3aκ3 ,

Πaλ = λ31aλ1 + λ3

2aλ2 + λ33aλ3 , Πaκ = κ3

1aκ1 + κ32aκ2 + κ3

3aκ3 ,

Note that aλi ≡ λiAλi , aκi ≡ κiAκi , at ≡ ytAt, ab ≡ ybAb and aτ ≡ yτAτ . Defining

Σ1 =3∑i=1

(m2Qi− 2m2

ui+m2

di+m2

ei−m2

Li+m2

Hui−m2

Hdi

+m2Di−m2

Di

)−m2

H′ +m2H′,

(B8)

Σ′1 =3∑i=1

(6QQm

2Qi

+ 3Qum2ui

+ 3Qdm2di

+Qem2ei

+ 2QLm2Li

+ 2Q2m2Hui

+ 2Q1m2Hdi

+QSm2Si

+ 3QDm2Di

+ 3QDm2Di

)+ 2QH′m

2H′ + 2QH′m

2H′, (B9)

the one- and two-loop β functions and the O(t2) coefficients for m2Hd

are

β(1)

m2Hd

= 2λ2(m2Hd

+m2Hu +m2

S

)+ 2a2

λ + 6y2b

(m2Hd

+m2Q3

+m2d3

)+ 6a2

b

+ 2y2τ

(m2Hd

+m2L3

+m2e3

)+ 2a2

τ − 6g22M

22 −

6

5g2

1M21 − 8Q2

1g′21 M

′21

− 3

5g2

1Σ1 + 2Q1g′21 Σ′1, (B10a)

β(2)

m2Hd

= −36y4b

(m2Hd

+m2Q3

+m2d3

)− 12y4

τ

(m2Hd

+m2L3

+m2e3

)− 72y2

ba2b − 24y2

τa2τ

− 6y2t y

2b

(m2Hd

+m2Hu + 2m2

Q3+m2

u3+m2

d3

)− 6 (ytab + ybat)

2

− 4λ4(m2Hd

+m2Hu +m2

S

)− 8λ2a2

λ − λ2

3∑i=1

[4λ2

i

(m2Hd

+m2Hu + 2m2

S +m2Hdi

+m2Hui

)+ 6κ2

i

(m2Hd

+m2Hu + 2m2

S +m2Di

+m2Di

)]−

3∑i=1

[4 (λiaλ + λaλi)

2

37

+6 (κiaλ + λaκi)2]− 6λ2y2

t

(m2Hd

+ 2m2Hu +m2

S +m2Q3

+m2u3

)− 6 (λat + ytaλ)

2

+ 32g23y

2b

(m2Hd

+m2Q3

+m2d3

+ 2M23

)+ 32g2

3

(a2b − 2ybabM3

)+

6

5g2

1y2t

(3m2

Hu +m2Q3− 4m2

u3

)− 2

5g2

1y2b

(11m2

Hd−m2

Q3− 4m2

d3+ 4M2

1

)− 4

5g2

1

(a2b − 2ybabM1

)+

6

5g2

1y2τ

(m2Hd

+m2L3

+ 4m2e3

+ 4M21

)+

12

5g2

1

(a2τ − 2yτaτM1

)+

6

5g2

1

3∑i=1

[λ2i

(m2Hdi−m2

Hui

)+ κ2

i

(m2Di−m2

Di

)]+ 12g′21 y

2b

(Q2Q +Q2

d −Q21

)×(m2Hd

+m2Q3

+m2d3

+ 2M ′21

)+ 12g′21

(Q2Q +Q2

d −Q21

) (a2b − 2ybabM

′1

)− 24Q1g

′21 y

2b

(Q1m

2Hd

+QQm2Q3

+Qdm2d3

)+ 4g′21 y

2τ

(Q2L +Q2

e −Q21

)×(m2Hd

+m2L3

+m2e3

+ 2M ′21

)+ 4g′21

(Q2L +Q2

e −Q21

) (a2τ − 2yτaτM

′1

)− 8Q1g

′21 y

2τ

(Q1m

2Hd

+QLm2L3

+Qem2e3

)− 24Q1g

′21 y

2t

(Q2m

2Hu +QQm

2Q3

+Qum2u3

)+ 4g′21 λ

2(Q2

2 +Q2S −Q2

1

) (m2Hd

+m2Hu +m2

S + 2M ′21

)+ 4g′21

(Q2

2 +Q2S −Q2

1

)×(a2λ − 2λaλM

′1

)− 4Q1g

′21

3∑i=1

[2λ2

i

(Q1m

2Hdi

+Q2m2Hui

+QSm2S

)+3κ2

i

(m2S +m2

Di+m2

Di

)]− 16

5g2

3g21

3∑i=1

(m2Qi− 2m2

ui+m2

di+m2

Di−m2

Di

)+ 32Q1g

23g′21

3∑i=1

(2QQm

2Qi

+Qum2ui

+Qdm2di

+QDm2Di

+QDm2Di

)+ 3g4

2

[29M2

2 +m2H′ +m2

H′+

3∑i=1

(3m2

Qi+m2

Li+m2

Hdi

+m2Hui

)]

+9

5g2

2g21

[2(M2

1 +M1M2 +M22

)+m2

H′ −m2H′−

3∑i=1

(m2Qi−m2

Li+m2

Hui−m2

Hdi

)]

+ 12Q1g22g′21

[2Q1

(M ′2

1 +M ′1M2 +M2

2

)+QH′m

2H′ +QH′m

2H′

+3∑i=1

(3QQm

2Qi

+QLm2Li

+Q1m2Hdi

+Q2m2Hui

)]+

1

25g4

1

[891M2

1 + 18m2H′

+ 23∑i=1

(m2di

+ 5m2Di

+m2Di− 9m2

ei+ 9m2

Hdi

+ 9m2Li

+m2Qi

+ 28m2ui

)]− 4

5g2

1g′21

[6Q1

(3Qd + 3QD − 3QD + 3Qe − 4Q1 + 3Q2 +QH′ −QH′ − 3QL

+ 3QQ − 6Qu

) (M2

1 +M1M′1 +M ′2

1

)+ 3Q2

H′m2H′− 3Q2

H′m2H′

38

+ 33∑i=1

(Q2dm

2di

+Q2Dm2Di−Q2

Dm2Di

+Q2em

2ei−Q2

1m2Hdi

+Q22m

2Hui−Q2

Lm2Li

+Q2Qm

2Qi− 2Q2

um2ui

)+ 3Q1QH′m

2H′− 9Q1QH′m

2H′ +Q1

3∑i=1

(4Qdm

2di

+ 4QDm2Di

− 8QDm2Di− 9Q1m

2Hdi

+ 3Q2m2Hui− 9QLm

2Li

+ 5QQm2Qi− 20Qum

2ui

)]+ 8Q1g

′41

[3Q1M

′21

(9Q2

d + 9Q2D

+ 9Q2D + 3Q2

e + 8Q21 + 6Q2

2 + 2Q2H′

+ 2Q2H′ + 6Q2

L

+ 18Q2Q + 3Q2

S + 9Q2u

)+ 2Q3

H′m2H′

+ 2Q3H′m

2H′ +

3∑i=1

(3Q3

dm2di

+ 3Q3Dm2Di

+ 3Q3Dm

2Di

+Q3em

2ei

+ 2Q31m

2Hdi

+ 2Q32m

2Hui

+ 2Q3Lm

2Li

+ 6Q3Qm

2Qi

+Q3Sm

2Si

+ 3Q3um

2ui

)+ 2Q1Q

2H′m2H′

+ 2Q1Q2H′m

2H′ +Q1

3∑i=1

(3Q2

dm2di

+ 3Q2Dm2Di

+ 3Q2Dm

2Di

+Q2em

2ei

+ 2Q21m

2Hdi

+ 2Q22m

2Hui

+ 2Q2Lm

2Li

+ 6Q2Qm

2Qi

+Q2Sm

2Si

+ 3Q2um

2ui

)], (B10b)

b(2)

m2Hd

= 72y4b

(m2Hd

+m2Q3

+m2d3

)+ 144y2

ba2b + 16y4

τ

(m2Hd

+m2L3

+m2e3

)+ 32y2

τa2τ

+ 8λ4(m2Hd

+m2Hu +m2

S

)+ 16λ2a2

λ + 6y2t y

2b

(m2Hd

+m2Hu + 2m2

Q3+m2

u3+m2

d3

)+ 6 (ytab + ybat)

2 + 12y2by

2τ

(2m2

Hd+m2

Q3+m2

d3+m2

L3+m2

e3

)+ 12 (ybaτ + yτab)

2

+ 6λ2y2t

(m2Hd

+ 2m2Hu +m2

S +m2Q3

+m2u3

)+ 6 (λat + ytaλ)

2

+ 12λ2y2b

(2m2

Hd+m2

Hu +m2S +m2

Q3+m2

d3

)+ 12 (λab + ybaλ)

2

+ 4λ2y2τ

(2m2

Hd+m2

Hu +m2S +m2

L3+m2

e3

)+ 4 (λaτ + yτaλ)

2

+ 43∑i=1

[λ2λ2

i

(m2Hdi

+m2Hui

+m2Hd

+m2Hu + 2m2

S

)+ (λaλi + λiaλ)

2]

+ 63∑i=1

[λ2κ2

i

(m2Hd

+m2Hu + 2m2

S +m2Di

+m2Di

)+ (λaκi + κiaλ)

2]

− 32g23y

2b

(m2Hd

+m2Q3

+m2d3

+ 2M23

)− 32g2

3

(a2b − 2ybabM3

)− 18g2

2y2b

(m2Hd

+m2Q3

+m2d3

+ 2M22

)− 18g2

2

(a2b − 2ybabM2

)− 6g2

2y2τ

(m2Hd

+m2L3

+m2e3

+ 2M22

)− 6g2

2

(a2τ − 2yτaτM2

)− 6g2

2λ2(m2Hd

+m2Hu +m2

S + 2M22

)− 6g2

2

(a2λ − 2λaλM2

)− 14

5g2

1y2b

(m2Hd

+m2Q3

+m2d3

+ 2M21

)− 14

5g2

1

(a2b − 2ybabM1

)− 18

5g2

1y2τ

(m2Hd

+m2L3

+m2e3

+ 2M21

)− 18

5g2

1

(a2τ − 2yτaτM1

)39

− 6

5g2

1λ2(m2Hd

+m2Hu +m2

S + 2M21

)− 6

5g2

1

(a2λ − 2λaλM1

)− 12g′21 y

2b

(Q2

1 +Q2Q +Q2

d

) (m2Hd

+m2Q3

+m2d3

+ 2M ′21

)− 12g′21

(Q2

1 +Q2Q +Q2

d

)×(a2b − 2ybabM

′1

)+ 6g′21 y

2b (Q1 +QQ +Qd)

(2Q1m

2Hd

+ 2Q1m2Q3

+ 2Q1m2d3

+ Σ′1)

+ 12Q1g′21 a

2b (Q1 +QQ +Qd)− 4g′21 y

2τ

(Q2

1 +Q2L +Q2

e

) (m2Hd

+m2L3

+m2e3

+ 2M ′21

)− 4g′21

(Q2

1 +Q2L +Q2

e

) (a2τ − 2yτaτM

′1

)+ 2g′21 y

2τ (Q1 +QL +Qe)

×(2Q1m

2Hd

+ 2Q1m2L3

+ 2Q1m2e3

+ Σ′1)

+ 4Q1g′21 a

2τ (Q1 +QL +Qe)

+ 12Q1g′21 y

2t (Q2 +QQ +Qu)

(m2Hu +m2

Q3+m2

u3

)+ 12Q1g

′21 a

2t (Q2 +QQ +Qu)

− 4g′21 λ2(Q2

1 +Q22 +Q2

S

) (m2Hd

+m2Hu +m2

S + 2M ′21

)− 4g′21

(Q2

1 +Q22 +Q2

S

)×(a2λ − 2λaλM

′1

)+ 2g′21 λ

2 (Q1 +Q2 +QS) Σ′1 + 4Q1g′21 (Q1 +Q2 +QS)

×3∑i=1

[λ2i

(m2Hdi

+m2Hui

+m2S

)+ a2

λi

]+ 6Q1g

′21 (QS +QD +QD)

×3∑i=1

[κ2i

(m2S +m2

Di+m2

Di

)+ a2

κi

]− 96Q1g

23g′21 M

23 (2QQ +Qu +Qd +QD +QD)

− 72g42M

22 − 12Q1g

22g′21 M

22 (9QQ + 3QL + 3Q1 + 3Q2 +QH′ +QH′)

− 288

25g4

1

(Σ1 + 3M2

1

)− 3

5g2

1g′21

[4Q1M

21

(2Qd + 2QD + 2QD + 6Qe + 3Q1 + 3Q2

+QH′ +QH′ + 3QL +QQ + 8Qu

)− 4M ′2

1

(3Q2

d + 3Q2D− 3Q2

D + 3Q2e − 3Q2

1 + 3Q22

+Q2H′−Q2

H′ − 3Q2L + 3Q2

Q − 6Q2u

)+ (Σ′1 − 2Q1Σ1) ΣY

Q

]− 4Q1g

′41

[2M ′2

1

(9Q3

d

+ 9Q3D

+ 9Q3D + 3Q3

e + 6Q31 + 6Q3

2 + 2Q3H′

+ 2Q3H′ + 6Q3

L + 18Q3Q + 3Q3

S + 9Q3u

)+(6Q1M

′21 − Σ′1

)ΣQ

]. (B10c)

Similarly, those for m2Hu

read

β(1)

m2Hu

= 2λ2(m2Hd

+m2Hu +m2

S

)+ 2a2

λ + 6y2t

(m2Hu +m2

Q3+m2

u3

)+ 6a2

t

− 6g22M

22 −

6

5g2

1M21 − 8Q2

2g′21 M

′21 +

3

5g2

1Σ1 + 2Q2g′21 Σ′1, (B11a)

β(2)

m2Hu

= −36y4t

(m2Hu +m2

Q3+m2

u3

)− 6y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3

)− 72y2

t a2t − 6 (ytab + ybat)

2 − 4λ4(m2Hd

+m2Hu +m2

S

)− 8λ2a2

λ

− λ2

3∑i=1

[4λ2

i

(m2Hd

+m2Hu + 2m2

S +m2Hdi

+m2Hui

)+ 6κ2

i

(m2Hd

+m2Hu + 2m2

S +m2Di

+m2Di

)]−

3∑i=1

[4 (λiaλ + λaλi)

2

40

+ 6 (κiaλ + λaκi)2 ]− 6λ2y2

b

(2m2

Hd+m2

Hu +m2S +m2

Q3+m2

d3

)− 2λ2y2

τ

(2m2

Hd+m2

Hu +m2S +m2

L3+m2

e3

)− 6 (λab + ybaλ)

2 − 2 (λaτ + yτaλ)2

+ 32g23y

2t

(m2Hu +m2

Q3+m2

u3+ 2M2

3

)+ 32g2

3

(a2t − 2ytatM3

)+

2

5g2

1y2t

(−5m2

Hu +m2Q3

+ 16m2u3

+ 8M21

)+

8

5g2

1

(a2t − 2ytatM1

)+

6

5g2

1y2b

(3m2

Hd−m2

Q3− 2m2

d3

)+

6

5g2

1y2τ

(m2Hd

+m2L3− 2m2

e3

)+

6

5g2

1

3∑i=1

[λ2i

(m2Hdi−m2

Hui

)+ κ2

i

(m2Di−m2

Di

)]+ 12g′21 y

2t

(Q2Q +Q2

u −Q22

) (m2Hu +m2

Q3+m2

u3+ 2M ′2

1

)+ 12g′21

(Q2Q +Q2

u −Q22

) (a2t − 2ytatM

′1

)− 24Q2g

′21 y

2t

(Q2m

2Hu +QQm

2Q3

+Qum2u3

)− 24Q2g

′21 y

2b

(Q1m

2Hd

+QQm2Q3

+Qdm2d3

)− 8Q2g

′21 y

2τ

(Q1m

2Hd

+QLm2L3

+Qem2e3

)+ 4g′21 λ

2(Q2

1 −Q22 +Q2

S

) (m2Hd

+m2Hu +m2

S + 2M ′21

)+ 4g′21

(Q2

1 −Q22 +Q2

S

) (a2λ − 2λaλM

′1

)− 4Q2g

′21

3∑i=1

[2λ2

i

(Q1m

2Hdi

+Q2m2Hui

+QSm2S

)+ 3κ2

i

(QSm

2S +QDm

2Di

+QDm2Di

)]+

16

5g2

3g21

3∑i=1

(m2Qi− 2m2

ui+m2

di+m2

Di−m2

Di

)+ 32Q2g

23g′21

3∑i=1

(2QQm

2Qi

+Qum2ui

+Qdm2di

+QDm2Di

+QDm2Di

)+ 3g4

2

[29M2

2 +m2H′ +m2

H′+

3∑i=1

(3m2

Qi+m2

Li+m2

Hdi

+m2Hui

)]

+9

5g2

2g21

[2(M2

1 +M1M2 +M22

)+m2

H′−m2

H′ +3∑i=1

(m2Qi−m2

Li+m2

Hui−m2

Hdi

)]

+ 12Q2g22g′21

[2Q2

(M ′2

1 +M ′1M2 +M2

2

)+QH′m

2H′ +QH′m

2H′

+3∑i=1

(3QQm

2Qi

+QLm2Li

+Q1m2Hdi

+Q2m2Hui

)]+

1

25g4

1

[891M2

1 + 18m2H′

+3∑i=1

(10m2

di+ 2m2

Di+ 10m2

Di+ 54m2

ei+ 18m2

Hui

+ 4m2Qi− 8m2

ui

)]+

4

5g2

1g′21

[6Q2

(3Qd +QD − 3QD + 3Qe − 3Q1 + 4Q2 +QH′ −QH′ − 3QL

+ 3QQ − 6Qu

) (M2

1 +M1M′1 +M ′2

1

)+ 3Q2

H′m2H′− 3Q2

H′m2H′

41

+ 33∑i=1

(Q2dm

2di

+Q2Dm2Di−Q2

Dm2Di

+Q2em

2ei−Q2

1m2Hdi

+Q22m

2Hui−Q2

Lm2Li

+Q2Qm

2Qi− 2Q2

um2ui

)+ 9Q2QH′m

2H′− 3Q2QH′m

2H′ +Q2

3∑i=1

(8Qdm

2di

+ 8QDm2Di

−4QDm2Di

+ 12Qem2ei− 3Q1m

2Hdi

+ 9Q2m2Hui− 3QLm

2Li

+ 7QQm2Qi− 4Qum

2ui

)]+ 8Q2g

′41

[3Q2M

′21

(9Q2

d + 9Q2D

+ 9Q2D + 3Q2

e + 6Q21 + 8Q2

2 + 2Q2H′

+ 2Q2H′ + 6Q2

L

+ 18Q2Q + 3Q2

S + 9Q2u

)+ 2Q3

H′m2H′

+ 2Q3H′m

2H′ +

3∑i=1

(3Q3

dm2di

+ 3Q3Dm2Di

+ 3Q3Dm

2Di

+Q3em

2ei

+ 2Q31m

2Hdi

+ 2Q32m

2Hui

+ 2Q3Lm

2Li

+ 6Q3Qm

2Qi

+Q3Sm

2Si

+ 3Q3um

2ui

)+ 2Q2Q

2H′m2H′

+ 2Q2Q2H′m

2H′ +Q2

3∑i=1

(3Q2

dm2di

+ 3Q2Dm2Di

+ 3Q2Dm

2Di

+Q2em

2ei

+ 2Q21m

2Hdi

+ 2Q22m

2Hui

+ 2Q2Lm

2Li

+ 6Q2Qm

2Qi

+Q2Sm

2Si

+ 3Q2um

2ui

)], (B11b)

b(2)

m2Hu

= 72y4t

(m2Hu +m2

Q3+m2

u3

)+ 144y2

t a2t + 8λ4

(m2Hd

+m2Hu +m2

S

)+ 16λ2a2

λ

+ 6y2t y

2b

(m2Hd

+m2Hu + 2m2

Q3+m2

u3+m2

d3

)+ 6 (ytab + ybat)

2

+ 12λ2y2t

(m2Hd

+ 2m2Hu +m2

S +m2Q3

+m2u3

)+ 12 (λat + ytaλ)

2

+ 6λ2y2b

(2m2

Hd+m2

Hu +m2S +m2

Q3+m2

d3

)+ 6 (λab + ybaλ)

2

+ 2λ2y2τ

(2m2

Hd+m2

Hu +m2S +m2

L3+m2

e3

)+ 2 (λaτ + yτaλ)

2

+ 43∑i=1

[λ2λ2

i

(m2Hdi

+m2Hui

+m2Hd

+m2Hu + 2m2

S

)+ (λaλi + λiaλ)

2]

+ 63∑i=1

[λ2κ2

i

(m2Hd

+m2Hu + 2m2

S +m2Di

+m2Di

)+ (λaκi + κiaλ)

2]

− 32g23y

2t

(m2Hu +m2

Q3+m2

u3+ 2M2

3

)− 32g2

3

(a2t − 2ytatM3

)− 18g2

2y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

2

)− 18g2

2

(a2t − 2ytatM2

)− 6g2

2λ2(m2Hd

+m2Hu +m2

S + 2M22

)− 6g2

2

(a2λ − 2λaλM2

)− 26

5g2

1y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

1

)− 26

5g2

1

(a2t − 2ytatM1

)− 6

5g2

1λ2(m2Hd

+m2Hu +m2

S + 2M21

)− 6

5g2

1

(a2λ − 2λaλM1

)− 12g′21 y

2t

(Q2

2 +Q2Q +Q2

u

) (m2Hu +m2

Q3+m2

u3+ 2M ′2

1

)− 12g′21

(Q2

2 +Q2Q +Q2

u

)×(a2t − 2ytatM

′1

)+ 6g′21 y

2t (Q2 +QQ +Qu)

(2Q2m

2Hu + 2Q2m

2Q3

+ 2Q2m2u3

+ Σ′1)

42

+ 12Q2g′21 a

2t (Q2 +QQ +Qu) + 12Q2g

′21 y

2b (Q1 +QQ +Qd)

(m2Hd

+m2Q3

+m2d3

)+ 12Q2g

′21 a

2b (Q1 +QQ +Qd) + 4Q2g

′21 y

2τ (Q1 +QL +Qe)

(m2Hd

+m2L3

+m2e3

)+ 4Q2g

′21 a

2τ (Q1 +QL +Qe)− 4g′21 λ

2(Q2

1 +Q22 +Q2

S

) (m2Hd

+m2Hu +m2

S + 2M ′21

)− 4g′21

(Q2

1 +Q22 +Q2

S

) (a2λ − 2λaλM

′1

)+ 2g′21 λ

2 (Q1 +Q2 +QS) Σ′1

+ 4Q2g′21 (Q1 +Q2 +QS)

3∑i=1

[λ2i

(m2Hdi

+m2Hui

+m2S

)+ a2

λi

]+ 6Q2g

′21 (QS +QD +QD)

3∑i=1

[κ2i

(m2S +m2

Di+m2

Di

)+ a2

κi

]− 96Q2g

23g′21 M

23 (2QQ +Qu +Qd +QD +QD)− 72g4

2M22

− 12Q2g22g′21 M

22 (9QQ + 3QL + 3Q1 + 3Q2 +QH′ +QH′) +

288

25g4

1

(Σ1 − 3M2

1

)− 3

5g2

1g′21

[4Q2M

21

(2Qd + 2QD + 2QD + 6Qe + 3Q1 + 3Q2 +QH′ +QH′ + 3QL

+QQ + 8Qu

)+ 4M ′2

1

(3Q2

d + 3Q2D− 3Q2

D + 3Q2e − 3Q2

1 + 3Q22 +Q2

H′−Q2

H′

− 3Q2L + 3Q2

Q − 6Q2u

)− (2Q2Σ1 + Σ′1) ΣY

Q

]− 4Q2g

′41

[2M ′2

1

(9Q3

d + 9Q3D

+ 9Q3D

+ 3Q3e + 6Q3

1 + 6Q32 + 2Q3

H′+ 2Q3

H′ + 6Q3L + 18Q3

Q + 3Q3S + 9Q3

u

)+(6Q2M

′21 − Σ′1

)ΣQ

], (B11c)

while those for m2S are

β(1)

m2S

=3∑i=1

[4λ2

i

(m2Hdi

+m2Hui

+m2S

)+ 4a2

λi+ 6κ2

i

(m2S +m2

Di+m2

Di

)+ 6a2

κi

]− 8Q2

Sg′21 M

′21 + 2QSg

′21 Σ′1, (B12a)

β(2)

m2S

=3∑i=1

[− 16λ4

i

(m2Hui

+m2Hdi

+m2S

)− 24κ4

i

(m2Di

+m2Di

+m2S

)− 32λ2

i a2λi− 48κ2

i a2κi

]− 12λ2y2

t

(2m2

Hu +m2Hd

+m2S +m2

Q3+m2

u3

)− 12λ2y2

b

(m2Hu + 2m2

Hd+m2

S +m2Q3

+m2d3

)− 4λ2y2

τ

(m2Hu + 2m2

Hd+m2

S +m2L3

+m2e3

)− 12 (λat + ytaλ)

2 − 12 (λab + ybaλ)2

− 4 (λaτ + yτaλ)2 + 32g2

3

3∑i=1

[κ2i

(m2S +m2

Di+m2

Di+ 2M2

3

)+ a2

κi− 2κiaκiM3

]+ 12g2

2

3∑i=1

[λ2i

(m2Hdi

+m2Hui

+m2S + 2M2

2

)+ a2

λi− 2λiaλiM2

]

43

+4

5g2

1

3∑i=1

[3λ2

i

(m2Hdi

+m2Hui

+m2S + 2M2

1

)+ 2κ2

i

(m2S +m2

Di+m2

Di+ 2M2

1

)+ 3

(a2λi− 2λiaλiM1

)+ 2

(a2κi− 2κiaκiM1

) ]+ 4g′21

3∑i=1

[2λ2

i

(Q2

1 +Q22 −Q2

S

) (m2Hdi

+m2Hui

+m2S + 2M ′2

1

)− 2QSλ

2i

(Q1m

2Hdi

+Q2m2Hui

+QSm2S

)+ 3κ2

i

(Q2D +Q2

D−Q2

S

)×(m2S +m2

Di+m2

Di+ 2M ′2

1

)− 3QSκ

2i

(QSm

2S +QDm

2Di

+QDm2Di

)+ 2

(a2λi− 2λiaλiM

′1

) (Q2

1 +Q22 −Q2

S

)+ 3

(a2κi− 2κiaκiM

′1

) (Q2D +Q2

D−Q2

S

) ]− 24QSg

′21 y

2t

(Q2m

2Hu +QQm

2Q3

+Qum2u3

)− 24QSg

′21 y

2b

(Q1m

2Hd

+QQm2Q3

+Qdm2d3

)− 8QSg

′21 y

2τ

(Q1m

2Hd

+QLm2L3

+Qem2e3

)+ 32QSg

23g′21

3∑i=1

(2QQm

2Qi

+Qum2ui

+Qdm2di

+QDm2Di

+QDm2Di

)+ 12QSg

22g′21

[QH′m

2H′

+QH′m2H′

+3∑i=1

(3QQm

2Qi

+QLm2Li

+Q1m2Hdi

+Q2m2Hui

)]+

4

5QSg

21g′21

[3QH′m

2H′

+ 3QH′m2H′

+3∑i=1

(2Qdm

2di

+ 2QDm2Di

+ 2QDm2Di

+ 6Qem2ei

+ 3Q1m2Hdi

+ 3Q2m2Hui

+ 3QLm2Li

+QQm2Qi

+ 8Qum2ui

)]+ 8QSg

′41

[3QSM

′21

(9Q2

d + 9Q2D

+ 9Q2D + 3Q2

e + 6Q21 + 6Q2

2

+ 2Q2H′

+ 2Q2H′ + 6Q2

L + 18Q2Q + 5Q2

S + 9Q2u

)+ 2Q3

H′m2H′

+ 2Q3H′m

2H′

+3∑i=1

(3Q3

dm2di

+ 3Q3Dm2Di

+ 3Q3Dm

2Di

+Q3em

2ei

+ 2Q31m

2Hdi

+ 2Q32m

2Hui

+ 2Q3Lm

2Li

+ 6Q3Qm

2Qi

+Q3Sm

2Si

+ 3Q3um

2ui

)+ 2QSQ

2H′m2H′

+ 2QSQ2H′m

2H′

+QS

3∑i=1

(3Q2

dm2di

+ 3Q2Dm2Di

+ 3Q2Dm

2Di

+Q2em

2ei

+ 2Q21m

2Hdi

+ 2Q22m

2Hui

+ 2Q2Lm

2Li

+ 6Q2Qm

2Qi

+Q2Sm

2Si

+ 3Q2um

2ui

)], (B12b)

b(2)

m2S

= 83∑i=1

[2λ4

i

(m2Hdi

+m2Hui

+m2S

)+ 4λ2

i a2λi

+ 3κ4i

(m2S +m2

Di+m2

Di

)+ 6κ2

i a2κi

]+ 8

3∑i=1

3∑j=1

[λ2iλ

2j

(m2Hdi

+m2Hui

+m2Hdj

+m2Huj

+ 2m2S

)+(λiaλj + λjaλi

)2]

44

+ 243∑i=1

3∑j=1

[λ2iκ

2j

(m2Hdi

+m2Hui

+ 2m2S +m2

Di+m2

Di

)+(λiaκj + κjaλi

)2]

+ 183∑i=1

3∑j=1

[κ2iκ

2j

(2m2

S +m2Di

+m2Di

+m2Dj

+m2Dj

)+(κiaκj + κjaκi

)2]

+ 12λ2y2t

(2m2

Hu +m2Hd

+m2S +m2

Q3+m2

u3

)+ 12λ2y2

b

(m2Hu + 2m2

Hd+m2

S

+m2Q3

+m2d3

)+ 4λ2y2

τ

(2m2

Hd+m2

Hu +m2S +m2

L3+m2

e3

)+ 12 (λat + ytaλ)

2

+ 12 (λab + ybaλ)2 + 4 (λaτ + yτaλ)

2 − 32g23

3∑i=1

[κ2i

(m2S +m2

Di+m2

Di+ 2M2

3

)+a2

κi− 2κiaκiM3

]− 12g2

2

3∑i=1

[λ2i

(m2Hdi

+m2Hui

+m2S + 2M2

2

)+ a2

λi− 2λiaλiM2

]− 4

5g2

1

3∑i=1

[3λ2

i

(m2Hdi

+m2Hui

+m2S + 2M2

1

)+ 3a2

λi− 6λiaλiM1

+ 2κ2i

(m2S +m2

Di+m2

Di+ 2M2

1

)+ 2a2

κi− 4κiaκiM1

]+ 2g′21

3∑i=1

[− 4λ2

i

(Q2

1 +Q22 +Q2

S

) (m2Hdi

+m2Hui

+m2S + 2M ′2

1

)+ 2λ2

i (Q1 +Q2 +QS)(QSm

2Hdi

+QSm2Hui

+QSm2S + Σ′1

)− 4

(Q2

1 +Q22 +Q2

S

)×(a2λi− 2λiaλiM

′1

)+ 2QSa

2λi

(Q1 +Q2 +QS)− 6κ2i

(Q2S +Q2

D +Q2D

)×(m2S +m2

Di+m2

Di+ 2M ′2

1

)+ 3κ2

i (QS +QD +QD)(QSm

2S +QSm

2Di

+QSm2Di

+ Σ′1

)− 6

(Q2S +Q2

D +Q2D

) (a2κi− 2κiaκiM

′1

)+ 3QSa

2κi

(QS +QD +QD)]

+ 12QSg′21 y

2t (Q2 +QQ +Qu)

(m2Hu +m2

Q3+m2

u3

)+ 12QSg

′21 a

2t (Q2 +QQ +Qu)

+ 12QSg′21 y

2b (Q1 +QQ +Qd)

(m2Hd

+m2Q3

+m2d3

)+ 12QSg

′21 a

2b (Q1 +QQ +Qd)

+ 4QSg′21 y

2τ (Q1 +QL +Qe)

(m2Hd

+m2L3

+m2e3

)+ 4QSg

′21 a

2τ (Q1 +QL +Qe)

− 96QSg23g′21 M

23 (2QQ +Qu +Qd +QD +QD)− 12QSg

22g′21 M

22

(9QQ + 3QL + 3Q1

+ 3Q2 +QH′ +QH′)− 6

5QSg

21g′21

[2M2

1

(2Qd + 2QD + 2QD + 6Qe + 3Q1 + 3Q2 +QH′

+QH′ + 3QL +QQ + 8Qu

)− Σ1ΣY

Q

]− 4QSg

′41

[2M ′2

1

(9Q3

d + 9Q3D

+ 9Q3D + 3Q3

e + 6Q31

+ 6Q32 + 2Q3

H′+ 2Q3

H′ + 6Q3L + 18Q3

Q + 3Q3S + 9Q3

u

)+(6QSM

′21 − Σ′1

)ΣQ

]. (B12c)

If the one-loop contributions to the effective potential from top and stop loops are also

included, it is necessary to consider the expansions for yt, at, m2Q3

and m2u3

. The required

45

expressions for yt read

β(1)yt = yt

[λ2 + 6y2

t + y2b −

16

3g2

3 − 3g22 −

13

15g2

1 − 2g′21(Q2

2 +Q2Q +Q2

u

)], (B13a)

β(2)yt = yt

{− 22y4

t − 5y4b − 5y2

t y2b − y2

by2τ − λ2

(λ2 + 3y2

t + 4y2b + y2

τ + 2Σλ + 3Σκ

)+ 2g′21

[λ2(Q2

1 −Q22 +Q2

S

)+ 2y2

t

(2Q2

Q +Q2u

)+ y2

b

(Q2

1 −Q2Q +Q2

d

)]+ 16g2

3y2t + 6g2

2y2t + g2

1

(6

5y2t +

2

5y2b

)+

128

9g4

3 +33

2g4

2 +3913

450g4

1

+ 2g′41[2(Q4

2 +Q4Q +Q4

u

)+(Q2

2 +Q2Q +Q2

u

)ΣQ

]+ 8g2

3g22 +

136

45g2

3g21

+32

3g2

3g′21

(Q2Q +Q2

u

)+ g2

2g21 + 6g2

2g′21

(Q2

2 +Q2Q

)+

2

5g2

1g′21

[3Q2

2 +1

3Q2Q +

16

3Q2u + (3Q2 +QQ − 4Qu) ΣY

Q

]}, (B13b)

b(2)yt = yt

{54y4

t +13

2y4b + 13y2

t y2b + y2

by2τ + λ2

(5

2λ2 + 15y2

t + 5y2b + y2

τ + 2Σλ + 3Σκ

)− 16

3g2

3

(λ2 + 2y2

b + 12y2t

)− 6g2

2

(λ2 + y2

b + 6y2t

)− g2

1

(22

15λ2 +

4

3y2b +

52

5y2t

)− 2g′21

[λ2(Q2

1 + 2Q22 +Q2

S +Q2Q +Q2

u

)+ y2

b

(Q2

1 +Q22 + 2Q2

Q +Q2u +Q2

d

)+12y2

t

(Q2

2 +Q2Q +Q2

u

)]+

128

9g4

3 −15

2g4

2 −143

18g4

1

+ 2g′41(Q2

2 +Q2Q +Q2

u

) (Q2

2 +Q2Q +Q2

u − ΣQ

)+ 16g2

3g22 +

208

45g2

3g21

+32

3g2

3g′21

(Q2

2 +Q2Q +Q2

u

)+

13

5g2

2g21 + 6g2

2g′21

(Q2

2 +Q2Q +Q2

u

)+

26

15g2

1g′21

(Q2

2 +Q2Q +Q2

u

)}, (B13c)

and those for at read

β(1)at = at

[λ2 + 6y2

t + y2b −

16

3g2

3 − 3g22 −

13

15g2

1 − 2g′21(Q2

2 +Q2Q +Q2

u

)]+ yt

[2λaλ + 12ytat + 2ybab +

32

3g2

3M3 + 6g22M2 +

26

15g2

1M1

+ 4g′21 M′1

(Q2

2 +Q2Q +Q2

u

) ], (B14a)

β(2)at = at

{− 22y4

t − 5y4b − 5y2

t y2b − y2

by2τ − λ2

(λ2 + 3y2

t + 4y2b + y2

τ + 2Σλ + 3Σκ

)+ 2g′21

[λ2(Q2

1 −Q22 +Q2

S

)+ 2y2

t

(2Q2

Q +Q2u

)+ y2

b

(Q2

1 −Q2Q +Q2

d

)]+ 16g2

3y2t + 6g2

2y2t + g2

1

(6

5y2t +

2

5y2b

)+

128

9g4

3 +33

2g4

2 +3913

450g4

1

46

+ 2g′41[2(Q4

2 +Q4Q +Q4

u

)+(Q2

2 +Q2Q +Q2

u

)ΣQ

]+ 8g2

3g22 +

136

45g2

3g21

+32

3g2

3g′21

(Q2Q +Q2

u

)+ g2

2g21 + 6g2

2g′21

(Q2

2 +Q2Q

)+

2

5g2

1g′21

[3Q2

2 +1

3Q2Q +

16

3Q2u

]}+ yt

{− 88y3

t at − 20y3bab − 10ytyb (ybat + ytab)

− 2ybyτ (ybaτ + yτab)− 2λaλ(2λ2 + 3y2

t + 4y2b + y2

τ + 2Σλ + 3Σκ

)− 2λ2 (3ytat + 4ybab + yτaτ + 2Σaλ + 3Σaκ) + 32g2

3yt (at − ytM3)

+ 12g22yt (at − ytM2) +

2

5g2

1

[6ytat + 2ybab −

(6y2

t + 2y2b

)M1

]+ 4g′21

[λ(Q2

1 −Q22 +Q2

S

)(aλ − λM ′

1) + 2yt(2Q2

Q +Q2u

)(at − ytM ′

1)

+ yb(Q2

1 −Q2Q +Q2

d

)(ab − ybM ′

1)]− 512

9g4

3M3 − 66g42M2 −

7826

225g4

1M1

− 8g′41 M′1

[2(Q4

2 +Q4Q +Q4

u

)+(Q2

2 +Q2Q +Q2

u

)ΣQ

]− 16g2

3g22 (M3 +M2)

− 272

45g2

3g21 (M3 +M1)− 64

3g2

3g′21

(Q2Q +Q2

u

)(M3 +M ′

1)− 2g22g

21 (M2 +M1)

− 12g22g′21

(Q2

2 +Q2Q

)(M2 +M ′

1)− 4

15g2

1g′21

(9Q2

2 +Q2Q + 16Q2

u

)(M1 +M ′

1)

}, (B14b)

b(2)at = yt

[2λaλ + 12ytat + 2ybab +

32

3g2

3M3 + 6g22M2 +

26

15g2

1M1

+ 4g′21 M′1

(Q2

2 +Q2Q +Q2

u

) ]×[λ2 + 6y2

t + y2b −

16

3g2

3 − 3g22 −

13

15g2

1

− 2g′21(Q2

2 +Q2Q +Q2

u

) ]+ at

{54y4

t +13

2y4b + 13y2

t y2b + y2

by2τ

+ λ2

(5

2λ2 + 15y2

t + 5y2b + y2

τ + 2Σλ + 3Σκ

)− 16

3g2

3

(λ2 + 2y2

b + 12y2t

)− 6g2

2

(λ2 + y2

b + 6y2t

)− g2

1

(22

15λ2 +

4

3y2b +

52

5y2t

)− 2g′21

[λ2(Q2

1 + 2Q22 +Q2

S +Q2Q +Q2

u

)+ y2

b

(Q2

1 +Q22 + 2Q2

Q +Q2u +Q2

d

)+12y2

t

(Q2

2 +Q2Q +Q2

u

)]+

128

9g4

3 −15

2g4

2 −143

18g4

1

+ 2g′41(Q2

2 +Q2Q +Q2

u

) (Q2

2 +Q2Q +Q2

u − ΣQ

)+ 16g2

3g22 +

208

45g2

3g21

+32

3g2

3g′21

(Q2

2 +Q2Q +Q2

u

)+

13

5g2

2g21 + 6g2

2g′21

(Q2

2 +Q2Q +Q2

u

)+

26

15g2

1g′21

(Q2

2 +Q2Q +Q2

u

)}+ yt

[144y3

t at + 24y3bab + 14ytyb (ytab + atyb)

+ 2ybyτ (ybaτ + abyτ ) + 18λyt (λat + aλyt) + 8λyb (λab + aλyb) + 2λyτ (λaτ + aλyτ )

47

+ 8λ3aλ + 4λ3∑i=1

λi (λaλi + aλλi) + 6λ3∑i=1

κi (λaκi + aλκi)− 64g23yt (at − ytM3)

− 32

3g2

3yb (ab − ybM3)− 36g22yt (at − ytM2)− 6g2

2yb (ab − ybM2)− 6g22λ (aλ − λM2)

− 52

5g2

1yt (at − ytM1)− 14

15g2

1yb (ab − ybM1)− 18

15g2

1λ (aλ − λM1)

− 24g′21 yt(Q2

2 +Q2Q +Q2

u

)(at − ytM ′

1)− 4g′21 yb(Q2

1 +Q2Q +Q2

d

)(ab − ybM ′

1)

− 4g′21 λ(Q2

1 +Q22 +Q2

S

)(aλ − λM ′

1) + 48g42M2 +

832

25g4

1M1

+ 8g′41 M′1ΣQ

(Q2

2 +Q2Q +Q2

u

) ]. (B14c)

The one- and two-loop β functions and the resulting O(t2) coefficient for m2Q3

are

β(1)

m2Q3

= 2y2t

(m2Hu +m2

Q3+m2

u3

)+ 2a2

t + 2y2b

(m2Hd

+m2Q3

+m2d3

)+ 2a2

b

− 32

3g2

3M23 − 6g2

2M22 −

2

15g2

1M21 − 8Q2

Qg′21 M

′21 +

1

5g2

1Σ1 + 2QQg′21 Σ′1, (B15a)

β(2)

m2Q3

= −20y4t

(m2Hu +m2

Q3+m2

u3

)− 20y4

b

(m2Hd

+m2Q3

+m2d3

)− 2y2

τy2b

(2m2

Hd+m2

Q3+m2

L3+m2

d3+m2

e3

)− 40y2

t a2t − 40y2

ba2b

− 2 (ybaτ + yτab)2 − 2λ2y2

t

(2m2

Hu +m2Hd

+m2S +m2

Q3+m2

u3

)− 2λ2y2

b

(m2Hu + 2m2

Hd+m2

S +m2Q3

+m2d3


2 − 2 (λab + ybaλ)2

+2

5g2

1y2t

(m2Hu + 3m2

Q3+ 8m2

u3+ 8M2

1

)+

8

5g2

1

(a2t − 2ytatM1

)+

2

5g2

1y2b

(5m2

Hd+m2

Q3+ 4M2

1

)+

4

5g2

1

(a2b − 2ybabM1

)+

2

5g2

1y2τ

(m2Hd

+m2L3− 2m2

e3

)+

2

5g2

1

3∑i=1

[κ2i

(m2Di−m2

Di

)+ λ2

i

(m2Hdi−m2

Hui

)]+ 4g′21 y

2t

(Q2

2 −Q2Q +Q2

u

) (m2Hu +m2

Q3+m2

u3+ 2M ′2

1

)+ 4g′21

(Q2

2 −Q2Q +Q2

u

) (a2t − 2ytatM

′1

)− 24QQg

′21 y

2t

(Q2m

2Hu +QQm

2Q3

+Qum2u3

)+ 4g′21 y

2b

(Q2

1 −Q2Q +Q2

d

) (m2Hd

+m2Q3

+m2d3

+ 2M ′21

)+ 4g′21

(Q2

1 −Q2Q +Q2

d

) (a2b − 2ybabM

′1

)− 24QQg

′21 y

2b

(Q1m

2Hd

+QQm2Q3

+Qdm2d3

)− 8QQg

′21 y

2τ

(Q1m

2Hd

+QLm2L3

+Qem2e3

)− 4QQg

′21

3∑i=1

[3κ2

i

(QSm

2S +QDm

2Di

+QDm2Di

)+ 2λ2

i

(Q1m

2Hdi

+Q2m2Hui

+QSm2S

)]+

16

3g4

3

[10M2

3 +3∑i=1

(2m2

Qi+m2

ui+m2

di+m2

Di+m2

Di

)]+ 32g2

3g22

(M2

2 +M2M3 +M23

)48

+16

45g2

3g21

[2(M2

1 +M1M3 +M23

)+ 3

3∑i=1

(m2Qi

+m2di− 2m2

ui−m2

Di+m2

Di

)]

+32

3QQg

23g′21

[4QQ

(M ′2

1 +M ′1M3 +M2

3

)+ 3

3∑i=1

(2QQm

2Qi

+Qum2ui

+Qdm2di

+QDm2Di

+QDm2Di

)]+ 3g4

2

[29M2

2 +m2H′

+m2H′ +

3∑i=1

(3m2

Qi+m2

Li+m2

Hdi

+m2Hui

)]

+1

5g2

2g21

[2(M2

1 +M1M2 +M22

)+ 3m2

H′− 3m2

H′ + 33∑i=1

(m2Qi−m2

Li+m2

Hui−m2

Hdi

)]

+ 12QQg22g′21

[2QQ

(M ′2

1 +M ′1M2 +M2

2

)+QH′m

2H′

+QH′m2H′

+3∑i=1

(3QQm

2Qi

+QLm2Li

+Q1m2Hdi

+Q2m2Hui

)]+

1

75g4

1

[289M2

1 + 12m2H′− 6m2

H′

+3∑i=1

(6m2

di− 2m2

Di+ 6m2

Di+ 42m2

ei− 6m2

Hdi

+ 12m2Hui− 6m2

Li+ 2m2

Qi− 24m2

ui

)]+

4

15g2

1g′21

[2QQ

(9Qd + 9QD − 9QD + 9Qe − 9Q1 + 9Q2 + 3QH′ − 3QH′ − 9QL

+ 10QQ − 18Qu

) (M2

1 +M1M′1 +M ′2

1

)+ 3Q2

H′m2H′− 3Q2

H′m2H′

+ 33∑i=1

(Q2dm

2di

+Q2Dm2Di−Q2

Dm2Di

+Q2em

2ei−Q2

1m2Hdi

+Q22m

2Hui

−Q2Lm

2Li−Q2

Qm2Qi− 2Q2

um2ui

)+ 15QQQH′m

2H′

+ 3QQQH′m2H′

+QQ

3∑i=1

(12Qdm

2di

+ 12QDm2Di

+ 24Qem2ei

+ 3Q1m2Hdi

+ 15Q2m2Hui

+ 3QLm2Li

+ 15QQm2Qi

+ 12m2ui

)]+ 8QQg

′41

[3QQM

′21

(9Q2

d + 9Q2D

+ 9Q2D + 3Q2

e + 6Q21

+ 6Q22 + 2Q2

H′+ 2Q2

H′ + 6Q2L + 20Q2

Q + 3Q2S + 9Q2

u

)+ 2Q3

H′m2H′

+ 2Q3H′m

2H′

+3∑i=1

(3Q3

dm2di

+ 3Q3Dm2Di

+ 3Q3Dm

2Di

+Q3em

2ei

+ 2Q31m

2Hdi

+ 2Q32m

2Hui

+ 2Q3Lm

2Li

+ 6Q3Qm

2Qi

+Q3Sm

2Si

+ 3Q3um

2ui

)+ 2QQQ

2H′m2H′

+ 2QQQ2H′m

2H′

+QQ

3∑i=1

(3Q2

dm2di

+ 3Q2Dm2Di

+ 3Q2Dm

2Di

+Q2em

2ei

+ 2Q21m

2Hdi

+ 2Q22m

2Hui

+ 2Q2Lm

2Li

+ 6Q2Qm

2Qi

+Q2Sm

2Si

+ 3Q2um

2ui

)], (B15b)

b(2)

m2Q3

= 24y4t

(mH2

u+m2

Q3+m2

u3

)+ 24y4

b

(m2Hd

+m2Q3

+m2d3

)+ 48y2

t a2t + 48y2

ba2b

49

+ 4y2t y

2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3

)+ 4 (ytab + ybat)

2

+ 2y2by

2τ

(2m2

Hd+m2

Q3+m2

d3+m2

L3+m2

e3

)+ 2 (yτab + ybaτ )

2

+ 2λ2y2t

(m2Hd

+ 2m2Hu +m2

S +m2Q3

+m2u3

)+ 2 (λat + ytaλ)

2

+ 2λ2y2b

(2m2

Hd+m2

Hu +m2S +m2

Q3+m2

d3

)+ 2 (λab + ybaλ)

2

− 32

3g2

3y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

3

)− 32

3g2

3

(a2t − 2ytatM3

)− 32

3g2

3y2b

(m2Hd

+m2Q3

+m2d3

+ 2M23

)− 32

3g2

3

(a2b − 2ybabM3

)− 6g2

2y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

2

)− 6g2

2

(a2t − 2ytatM2

)− 6g2

2y2b

(m2Hd

+m2Q3

+m2d3

+ 2M22

)− 6g2

2

(a2b − 2ybabM2

)− 26

15g2

1y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

1

)− 26

15g2

1

(a2t − 2ytatM1

)− 14

15g2

1y2b

(m2Hd

+m2Q3

+m2d3

+ 2M21

)− 14

15g2

1

(a2b − 2ybabM1

)− 4g′21 y

2t

(Q2

2 +Q2Q +Q2

u

) (m2Hu +m2

Q3+m2

u3+ 2M ′2

1

)− 4g′21

(Q2

2 +Q2Q +Q2

u

)×(a2t − 2ytatM

′1

)+ 2g′21 y

2t (Q2 +QQ +Qu)

(6QQm

2Hu + 6QQm

2Q3

+ 6QQm2u3

+ Σ′1)

+ 12QQg′21 a

2t (Q2 +QQ +Qu)− 4g′21 y

2b

(Q2

1 +Q2Q +Q2

d

) (m2Hd

+m2Q3

+m2d3

+ 2M ′21

)− 4g′21

(Q2

1 +Q2Q +Q2

d

) (a2b − 2ybabM

′1

)+ 2g′21 y

2b (Q1 +QQ +Qd)

×(6QQm

2Hd

+ 6QQm2Q3

+ 6QQm2d3

+ Σ′1)

+ 12QQg′21 a

2b (Q1 +QQ +Qd)

+ 4QQg′21 y

2τ (Q1 +QL +Qe)

(m2Hd

+m2L3

+m2e3

)+ 4QQg

′21 a

2τ (Q1 +QL +Qe)

+ 2QQg′21

3∑i=1

[2λ2

i (Q1 +Q2 +QS)(m2Hdi

+m2Hui

+m2S

)+ 2a2

λi(Q1 +Q2 +QS)

+ 3κ2i (QS +QD +QD)

(m2S +m2

Di+m2

Di

)+ 3a2

κi(QS +QD +QD)

]− 96QQg

23g′21 M

23 (2QQ +Qu +Qd +QD +QD)− 72g4

2M22

− 12QQg22g′21 M

22 (9QQ + 3QL + 3Q1 + 3Q2 +QH′ +QH′) +

96

25g4

1

(Σ1 −M2

1

)− 1

5g2

1g′21

[12QQM

21

(2Qd + 2QD + 2QD + 6Qe + 3Q1 + 3Q2 +QH′ +QH′ + 3QL

+QQ + 8Qu

)+ 4M ′2

1

(3Q2

d + 3Q2D− 3Q2

D + 3Q2e − 3Q2

1 + 3Q22 +Q2

H′−Q2

H′ − 3Q2L

+ 3Q2Q − 6Q2

u

)− (6QQΣ1 + Σ′1) ΣY

Q

]− 4QQg

′41

[2M ′2

1

(9Q3

d + 9Q3D

+ 9Q3D + 3Q3

e + 6Q31

+ 6Q32 + 2Q3

H′+ 2Q3

H′ + 6Q3L + 18Q3

Q + 3Q3S + 9Q3

u

)+(6QQM

′21 − Σ′1

)ΣQ

]. (B15c)

50

Finally, the relevant expressions for the soft mass m2u3

read

β(1)

m2u3

= 4y2t

(m2Hu +m2

Q3+m2

u3

)+ 4a2

t −32

3g2

3M23 −

32

15g2

1M21 − 8Q2

ug′21 M

′21

− 4

5g2

1Σ1 + 2Qug′21 Σ′1, (B16a)

β(2)

m2u3

= −32y4t

(m2Hu +m2

Q3+m2

u3

)− 4y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3

)− 64y2

t a2t − 4 (ytab + ybat)

2 − 4λ2y2t

(2m2

Hu +m2Hd

+m2S +m2

Q3+m2

u3


2 + 12g22y

2t

(m2Hu +m2

Q3+m2

u3+ 2M2

2

)+ 12g2

2

(a2t − 2ytatM2

)− 4

5g2

1y2t

(−5m2

Hu −m2Q3

+ 9m2u3

+ 2M21

)− 4

5g2

1

(a2t − 2ytatM1

)− 8

5g2

1y2b

(3m2

Hd−m2

Q3− 2m2

d3

)− 8

5g2

1y2τ

(m2Hd

+m2L3− 2m2

e3

)− 8

5g2

1

3∑i=1

[κ2i

(m2Di−m2

Di

)+ λ2

i

(m2Hdi−m2

Hui

)]+ 8g′21 y

2t

(Q2

2 +Q2Q −Q2

u

) (m2Hu +m2

Q3+m2

u3+ 2M ′2

1

)+ 8g′21

(Q2

2 +Q2Q −Q2

u

) (a2t − 2ytatM

′1

)− 24Qug

′21 y

2t

(Q2m

2Hu +QQm

2Q3

+Qum2u3

)− 24Qug

′21 y

2b

(Q1m

2Hd

+QQm2Q3

+Qdm2d3

)− 8Qug

′21 y

2τ

(Q1m

2Hd

+QLm2L3

+Qem2e3

)− 12Qug

′21

3∑i=1

κ2i

(QSm

2S +QDm

2Di

+QDm2Di

)− 8Qug

′21

3∑i=1

λ2i

(Q1m

2Hd

+Q2m2Hu +QSm

2S

)+

16

3g4

3

[10M2

3 +3∑i=1

(m2di

+m2Di

+m2Di

+ 2m2Qi

+m2ui

)]

+64

45g2

3g21

[8(M2

1 +M1M3 +M23

)− 3

3∑i=1

(m2di−m2

Di+m2

Di+m2

Qi− 2m2

ui

)]

+8

3Qug

23g′21

[16Qu

(M ′2

1 +M ′1M3 +M2

3

)+ 12

3∑i=1

(Qdm

2di

+QDm2Di

+QDm2Di

+ 2QQm2Qi

+Qum2ui

)]

+12

5g2

2g21

[m2H′ −m2

H′+

3∑i=1

(m2Hdi−m2

Hui

+m2Li−m2

Qi

)]

+ 12Qug22g′21

[QH′m

2H′ +QH′m

2H′

+3∑i=1

(Q1m

2Hdi

+Q2m2Hui

+QLm2Li

+ 3QQm2Qi

)]

+4

75g4

1

[1261M2

1 + 21m2H′ + 3m2

H′

51

+3∑i=1

(4m2

di+ 12m2

Di+ 4m2

Di− 12m2

ei+ 21m2

Hdi

+ 3m2Hui

+ 21m2Li

+ 3m2Qi

+ 64m2ui

)]+

4

15g2

1g′21

[− 8Qu

(9Qd + 9QD − 9QD + 9Qe − 9Q1 + 9Q2 + 3QH′ − 3QH′ − 9QL

+ 9QQ − 22Qu

) (M2

1 +M1M′1 +M ′2

1

)+ 12Q2

H′m2H′ − 12Q2

H′m2H′

+ 123∑i=1

(−Q2

dm2di−Q2

Dm2Di

+Q2Dm

2Di−Q2

em2ei

+Q21m

2Hdi−Q2

2m2Hui

+Q2Lm

2Li

−Q2Qm

2Qi

+Q2um

2ui

)− 15QH′Qum

2H′

+ 33QH′Qum2H′ +Qu

3∑i=1

(− 18Qdm

2di

− 18QDm2Di

+ 30QDm2Di− 6Qem

2ei

+ 33Q1m2Hdi− 15Q2m

2Hui

+ 33QLm2Li

− 21QQm2Qi

+ 84Qum2ui

)]+ 8Qug

′41

[3QuM

′21

(9Q2

d + 9Q2D

+ 9Q2D + 3Q2

e + 6Q21

+ 6Q22 + 2Q2

H′+ 2Q2

H′ + 6Q2L + 18Q2

Q + 3Q2S + 11Q2

u

)+ 2Q3

H′m2H′ + 2Q3

H′m2H′

+3∑i=1

(3Q3

dm2di

+ 3Q3Dm2Di

+ 3Q3Dm

2Di

+Q3em

2ei

+ 2Q31m

2Hdi

+ 2Q32m

2Hui

+ 2Q3Lm

2Li

+ 6Q3Qm

2Qi

+Q3Sm

2Si

+ 3Q3um

2ui

)+ 2QuQ

2H′m

2H′ + 2QuQ

2H′m2H′

+Qu

3∑i=1

(3Q2

dm2di

+ 3Q2Dm2Di

+ 3Q2Dm

2Di

+Q2em

2ei

+ 2Q21m

2Hdi

+ 2Q22m

2Hui

+ 2Q2Lm

2Li

+ 6Q2Qm

2Qi

+Q2Sm

2Si

+ 3Q2um

2ui

)], (B16b)

b(2)

m2u3

= 48y4t

(m2Hu +m2

Q3+m2

u3

)+ 4y2

t y2b

(m2Hu +m2

Hd+ 2m2

Q3+m2

u3+m2

d3

)+ 96y2

t a2t

+ 4 (ytab + ybat)2 + 4λ2y2

t

(2m2

Hu +m2Hd

+m2S +m2

Q3+m2

u3

)+ 4 (λat + ytaλ)

2

− 64

3g2

3y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

3

)− 64

3g2

3

(a2t − 2ytatM3

)− 12g2

2y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

2

)− 12g2

2

(a2t − 2ytatM2

)− 52

15g2

1y2t

(m2Hu +m2

Q3+m2

u3+ 2M2

1

)− 52

15g2

1

(a2t − 2ytatM1

)− 8g′21 y

2t

(Q2

2 +Q2Q +Q2

u

) (m2Hu +m2

Q3+m2

u3+ 2M ′2

1

)− 8g′21

(Q2

2 +Q2Q +Q2

u

) (a2t − 2ytatM

′1

)+ 4g′21 y

2t (Q2 +QQ +Qu)

(3Qum

2Hu + 3Qum

2Q3

+ 3Qum2u3

+ Σ′1)

+ 12Qug′21 a

2t (Q2 +QQ +Qu) + 12Qug

′21 y

2b (Q1 +QQ +Qd)

(m2Hd

+m2Q3

+m2d3

)+ 12Qug

′21 a

2b (Q1 +QQ +Qd)

52

+ 4Qug′21 y

2τ (Q1 +QL +Qe)

(m2Hd

+m2L3

+m2e3

)+ 4Qug

′21 a

2τ (Q1 +QL +Qe)

+ 2Qug′21

3∑i=1

[3κ2

i (QS +QD +QD)(m2S +m2

Di+m2

Di

)+ 3a2

κi(QS +QD +QD)

+ 2λ2i (Q1 +Q2 +QS)

(m2Hdi

+m2Hui

+m2S

)+ 2a2

λi(Q1 +Q2 +QS)

]− 96Qug

23g′21 M

23 (2QQ +Qu +Qd +QD +QD)

− 12Qug22g′21 M

22 (9QQ + 3QL + 3Q1 + 3Q2 +QH′ +QH′)−

384

25g4

1

(Σ1 + 4M2

1

)+

2

5g2

1g′21

[8M ′2

1 (3Q2d + 3Q2

D− 3Q2

D + 3Q2e − 3Q2

1 + 3Q22 +Q2

H′−Q2

H′ − 3Q2L

+ 3Q2Q − 6Q2

u) + 6QuM21 (2Qd + 2QD + 2QD + 6Qe + 3Q1 + 3Q2 +QH′ +QH′

+ 3QL +QQ + 8Qu) + (3QuΣ1 − 2Σ′1) ΣYQ

]− 4Qug

′41

[2M ′2

1 (9Q3d + 9Q3

D + 9Q3D

+ 3Q3e + 6Q3

1 + 6Q32 + 2Q3

H′+ 2Q3

H′ + 6Q3L + 18Q3

Q + 3Q3S + 9Q3

u)

+(6QuM

′21 − Σ′1

)ΣQ

]. (B16c)

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