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Physics Letters B

www.elsevier.com/locate/physletb

Focus point gauge mediation with incomplete adjoint messengers and gauge coupling unification

Gautam Bhattacharyyaa, Tsutomu T. Yanagidab'*, Norimi Yokozakic*

a Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, India b Kavli IPMU, TODIAS, University of Tokyo, Kashiwa 277-8583, Japan c INFN, Sezione di Roma, Piazzale A. Moro 2, I-00185 Roma, Italy

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A R T I C L E I N F 0

Article history:

Received 23 June 2015

Received in revised form 19 July 2015

Accepted 20 July 2015

Available online 26 July 2015

Editor: J. Hisano

A B S T R A C T

As the mass limits on supersymmetric particles are gradually pushed to higher values due to their continuing non-observation at the CERN LHC, looking for focus point regions in the supersymmetric parameter space, which shows considerably reduced fine-tuning, is increasingly more important than ever. We explore this in the context of gauge mediated supersymmetry breaking with messengers transforming in the adjoint representation of the gauge group, namely, octet of color SU(3) and triplet of weak SU(2). A distinctive feature of this scenario is that the focus point is achieved by fixing a single combination of parameters in the messenger sector, which is invariant under the renormalization group evolution. Because of this invariance, the focus point behavior is well under control once the relevant parameters are fixed by a more fundamental theory. The observed Higgs boson mass is explained with a relatively mild fine-tuning A = 60-150. Interestingly, even in the presence of incomplete messenger multiplets of the SU(5) GUT group, the gauge couplings still unify perfectly, but at a scale which is one or two orders of magnitude above the conventional GUT scale. Because of this larger unification scale, the colored Higgs multiplets become too heavy to trigger proton decay at a rate larger than the experimentally allowed limit.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

Though still elusive, Supersymmetry (SUSY), as a class of models, continues to be the leading candidate for physics beyond the Standard Model (SM). In addition to showing the virtue of gauge coupling unification, supersymmetry provides a dynamical origin of the negative mass-square of a neutral scalar that triggers elec-troweak symmetry breaking (EWSB). As we know by now, the origin of EWSB is completely explained if the scalar top (stop) mass is around the weak scale. However, the continuing absence of SUSY signals at the CERN Large Hadron Collider (LHC) has pushed up the gluino and squark masses to larger than about 1.2-1.6 TeV [1]. Additionally, the observed Higgs boson mass around 125 GeV [2] in the SUSY framework requires large radiative corrections from the stops [3]. This in turn necessitates the average stop mass to be at least 3-5 TeV [4], which is significantly larger than the weak scale. Consequently, settling the EWSB scale at the correct value requires a large fine-tuning of the Higgs potential in general.

* Corresponding authors.

E-mail address: norimi.yokozaki@roma1.infn.it (N. Yokozaki).

Under these circumstances, the focus point SUSY [5] (see also [6] for recent discussions) deserves special attention. In this class of scenarios, one or more fixed ratios among soft SUSY breaking masses are introduced, which lessens the fine-tuning of the Higgs potential lending more credibility to the natural explanation of the EWSB scale even if the SUSY particles turn out to be very heavy.

Among focus point SUSY scenarios [7-10], the scenarios based on gauge mediation [11]1 have the advantage of suppressing the FCNC processes.2 In the context of gauge mediation, the issue of focus point has been addressed in Ref. [13], where the numbers of the weakly and strongly coupled messenger multiplets (N2 and N3, respectively) are different from each other. Thanks to sizable cancellation between soft mass parameters for particular choices of N2 and N3 during the renormalization group running, the EWSB scale is realized with milder fine-tuning. However, owing to the pres-

1 For early attempts, see also Ref. [12].

2 The focus point SUSY models based on gaugino mediation [8] and Higgs-gaugino mediation [9] also do not suffer from the SUSY FCNC problem. Moreover, the latter model can easily explain the muon g — 2 anomaly [10].

http://dx.doi.Org/10.1016/j.physletb.2015.07.052

0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.Org/licenses/by/4.0/). Funded by SCOAP3.

ence of these large number of incomplete multiplets of the grand unified theory (GUT) group, the gauge couplings do not unify.

The gauge coupling unification may be achieved non-trivially in a framework where the messenger particles of gauge mediation transform in the adjoint representation of the GUT group. First, in Ref. [14], it was shown that the presence of adjoint matter multiplets with mass around 1013-1014 GeV can lead to gauge coupling unification around the string scale [15], which is one or two orders of magnitude above the conventional GUT scale, even if the adjoint matters do not form complete GUT multiplets. Subsequently, it was noticed that these adjoint multiplets can be employed as messenger superfields [16] for gauge mediation that would generate soft SUSY breaking masses. Such adjoint gauge mediation scenarios naturally lead to mass splitting among colored and uncolored particles right at the messenger scale [16,17].

We note at this point that in the context of SU(5) GUT, working with adjoint messengers, namely, SU(3) octet and SU(2) triplet, which are incomplete multiplets of SU(5), has a certain advantage over using messengers of complete multiplets, e.g. SU(3) triplet and SU(2) doublet. In the latter case, the requirement of precise gauge coupling unification demands that some colored Higgs multiplets weigh around 1015-1016 GeV [18], which would prompt unacceptably large proton decay. On the other hand, gauge coupling unification with adjoint messengers would necessitate the colored Higgs multiplets to hover around 1017-1018 GeV due to the larger unification scale, which is consistent with proton lifetime [19,20].

In this Letter, we exhibit how the fine-tuning of the EWSB can be reduced by utilizing the mass splitting of the adjoint representation messengers, more specifically, between the SU(3) octet and SU(2) triplet messengers. The focus point behavior is controlled by fixing one single combination of the superpotential parameters. Remarkably, this combination is invariant under the renormaliza-tion group evolution, i.e. it is stable against radiative corrections. Thus the focus point behavior in adjoint messenger gauge mediation model is more robust (assuming that the value of this combination is fixed by some more fundamental physics) than other SUSY breaking scenarios in the general class of minimal supersym-metric standard model (MSSM). In the latter scenarios, to reach the focus point region, various relations among soft SUSY breaking and/or preserving (like ¡) parameters need to be assumed which are neither invariant under renormalization group evolution nor independent of the SUSY breaking scale. This lends a substantial credibility to the attainment of focus point in adjoint messenger gauge mediation models.

2. Adjoint messenger gauge mediation (AMGMSB)

In the present scenario SUSY breaking is accomplished by gauge mediation with messengers transforming in the adjoint representation of the gauge group [16,17,21]. These messengers transform as (8,1) and (1, 3) under SU(3)c x SU(2)l gauge group, and may have originated from the non-Goldstone modes of the 24 dimensional Higgs multiplet in the SU(5) GUT gauge group. The resultant soft masses of weakly and strongly interacting supersymmetric particles, which are significantly different from those in minimal GMSB, allow for a significant reduction of fine-tuning [21]. The superpotential in the messenger sector is:

W mess = (M8 + A8 Z ) Tr(s8 ) + (M3 + A3 Z ) Tr^2),

where Z is a spurion field whose F-term vacuum expectation value (VEV) FZ breaks supersymmetry, whose effects are transmitted to the observable sector via messenger loops.

Even though the messenger multiplets in our model are incomplete SU(5) multiplets, the gauge coupling unification is still

M3 = 5 • 1013GeV M8/M3 = 1/9

103 10s 107 1 09 1 011 1013 1 015 1 017 renormalization scale (GeV)

Fig. 1. Unification of the three gauge couplings at the two-loop level with SU(3)c octet and SU(2)i triplet messengers with their masses around 1013 GeV. Here, as (MZ) = 0.1185 and mSUSY = 3 TeV.

achieved for M3 ~ M8 ~ 10-10 GeV at a scale somewhat higher than the conventional MGut — 1016 GeV [14], being around Mstr ^ 5 • 1017 GeV, which we call the string scale [15]. It is expected that at this scale the gauge and gravitational couplings are unified.

For illustration of gauge unification, we display the one-loop beta-functions of the gauge couplings. The gauge couplings at Mstr are given by

_i -1 b1 Mstr

a- (Mstr) = a- (msusv) - — ln-,

1 1 2n msusv

-v™ ^ -v ^ b2 + 2, Mstr a- (Mstr) = a2 (msusv)--r-ln

_1 _1 b3 + 3 Mstr a3 (Mstr) = a3 (msusv)--;-ln

+ f in-

msusv 2n msusv

-+ — ln-,

msusv 2n msusv

where bj = (33/5,1, -3) is the coefficient of one-loop beta-function for the gauge coupling gj, and mSUSY is the typical mass scale of strongly interacting SuSY particles, defined here more specifically as the stop mass scale mSUSY = (mQ3mj3)1/2. It turns out that a-2 3 — (57, 31, 13) at mSUSY = 3 TeV.

From Eq. (2), we can write (following the discussion in Ref. [22])

1 1 1 6

(5a, - 3a- - 2a- )(msusv) = — ln 123 n

i Mmess \ / Mstr Y / \msusv/

(a-1 - 3a2 1 + 2a3 1)(msusv):

= -A/ + 3 ln M.

5n y msusv/ n M 8

Using the above it is straightforward to obtain M2trMmess ~ MGUT, where Mmess = (M3M8)1/2. Requiring Mstr < 1018 GeV, it follows that Mmess > 1012 GeV. From the second equation of Eq. (3), we see that the larger Mstr, or equivalently smaller Mmess, requires a larger ratio of M3/M8 for the gauge coupling unification. For instance, for Mstr = 1017 (1018) GeV, one requires M3/M8 ~ 7(18) at the one-loop level. We, however, employ two-loop renormaliza-tion group equations (RGE) for the running of the gauge couplings, which is displayed in Fig. 1.

It is appropriate at this stage to highlight an advantage of using adjoint messengers for gauge mediation in GUT framework, more specifically, with SU(5) as the GUT group. The high scale spectra invariably contain colored Higgs multiplets, namely, Hc and Hc, which belong to 5h(= (Hu, Hc)) and 5h(= (Hd, Hc)) of SU(5), where Hu (Hd) denotes the up-type (down-type) weak doublet Higgs multiplets. The mass of the colored Higgs multiplet MHc is

predicted to be around the unification scale.3 Adjoint messenger gauge mediation has the distinct advantage of pushing the unification scale beyond the conventional GUT scale to Mstr (~ MHc ) = 1017-1018 GeV, which can easily accommodate the experimental constraints from the proton lifetime [20]. This is because the proton decay rate (p ^ K +v) is suppressed by 1/MjjC [19].

On the contrary, if the messengers are complete multiplets of SU(5), the unification scale is MGUT ~ 1016 GeV (the conventional scale), and then MHc ~ MGUT. Moreover, the precise gauge coupling unification requires MHc to be 1015-1016 GeV. This necessitates inclusion of threshold corrections to the gauge couplings, namely, —(1/5n) ln (MGUT/MHc) to a—1 and —(1/2n) x ln (Mgut/Mh^ to a—1. Then the proton decay rate would overshoot the experimental limit for (sub-)TeV scale SUSY [18].

With these messenger multiplets, the gaugino masses from the messenger loops at the scale Mmess are

MB -0' mww ~ T&(2A3^ Mg-tIi(3a)

where A3 = X3 <FZ) /M3 and A8 = X8 <FZ) /M8, provided that X3 <Z) and X8 <Z) are much smaller than M3 and M8, respectively. The sfermion masses at Mmess are given by

mQ —

(16n 2)2

22 mD= mû~

3 g34 (3A8) + 4 s4( 2 A3)

2 2 2 mL = mHu = mHd

(16n2)2 3 ^^ 2 3

(16n2)2-4 g4(2A2)'

mi - 0.

One can see that the bino and right-handed sleptons are mass-less, since there is no messenger field charged under the U(1)y gauge group. In order to give masses to the right-handed sleptons, we consider the minimal Kahler for the MSSM matter multiplets and the spurion Z. Then the MSSM matter fields receive a common mass m0 from the supergravity scalar potential, which is equal to the gravitino mass m3/2 = <FZ) /(*/3MP).

In this setup, the gluino mass at the soft SUSY breaking mass scale (~ TeV) is

-> a3 (msusY) . , M g (msusY) =-73-(3A8)

~ 4.0 TeV •

¿8 0.001

/ m3/2 \ V500 GeV/

1013 GeV

The bino can get a mass from the gauge kinetic function:

L d2o( 1 — ^W a Wa 1 + h.c. (7)

4 g\J V MP) a

k {Fz> /-MB (Mstr ) = "MT = ^3km3/2.

Alternatively, one can consider the sequestered form of the Kahler potential, which ensures the absence of FCNC. In this case, the right-handed slepton masses are generated by the bino loop, which is nothing but the gaugino mediation mechanism. Another

3 The contributions of these colored Higgs states to gauge coupling evolution can-

not be ignored if MHc is smaller than Mstr. To account for their contributions,

one must add -(1/5n) ln (Mstr/MHc) to a-1 and -(1/2n) ln (Mstr/MHc) to a-1

in Eq. (2).

option is to introduce a pair of 5 and 5 messengers to generate the bino and right-handed slepton masses, which would also contribute to other masses. In this Letter, for simplicity, we work with only SU(3) octet and SU(2) triplet adjoint messengers and stick to the case of the minimal Kahler potential, as mentioned above.

3. Focus point in the AMGMSB

We consider the fine-tuning of the EWSB scale with soft masses generated from these adjoint messengers. The EWSB conditions are given by

g2 + g2 2 r 2

—^4—2 v = J

(m2Hu + îk A ) tan2 P

tan2 p - 1

m2 +-L dAV mHd + 2Vd

(4) tan2 p + 1 r

tan p LBj

tan2 P - 1 J msusY 1 dAV 2Vu dVu

1 dAV 2Vd dVd

where AV denotes a one-loop correction to the Higgs potential, and m2Hu and m2Hi are the soft masses for the up-type and down-type Higgs, respectively. The Higgsino mass parameter is denoted by j, and B x is a soft susY breaking parameter of the Higgs bilinear term. The above equations tell us that the EWsB scale is determined dominantly by ¡j? and [mH + 1/(2vU)(dAV/dvu)] for large tanp (= (H°)/ (H°)). "

Now we consider renormalization group running of m2H from the high scale to weak scale. To understand the behavior of this running intuitively, we first demonstrate it using approximate analytic solutions of RGEs. The Higgs soft masses receive negative contributions from stop and gluino loops and positive contributions from the wino loop. The dominant negative contributions induced by the top-Yukawa coupling yt are4

(mHu ( Qr))neg -

Q 3 (M mess ) + mU 3 (M mess

- kgMg (Mmess),

where Qr is the renormalization scale taken to be the stop mass scale and

k = exp

_0 v '

kg = dt 0

y2(t/)g32 (Mmess) ^ 2n 4

, with t = ln(Qr/Mmess),

' 1 - mt'/2 '

_(1 - V3t')2_

y2(t')g43 (Mmess) t

6n6 (1 - q3t')2 '

4 The difference of the coefficients in front of mQ and mU between Eq. (11) and Eq. (15) arises from U(1)y contributions:

-, dmH 3 -, (16n 2)-H- 3 - g2 [Tr(mQ - 2m? ) + ...].

However, the above U(1)y contributions are eventually canceled out in the most of the gauge mediation models when their effects on each individual soft masses are summed up.

r 3 2.7 -

r3 -2.5 -

120 140 160 180 200 220 240 260 280 300

As (TeV)

-2.4 -

-2.3 -

160 180 200 220 240 260 280 300 320 340

As (TeV)

Fig. 2. Contours of A (solid) and mh/GeV (dashed). In the gray region, the EWSB does not occur. The messenger scale is taken as Mmess = 10 GeV. Here, tanp = 15, mt(pole) = 173.34 GeV and as(MZ) = 0.1185.

V3 = -3

g3 (Mmess) 8n2 '

In addition to the above negative contributions, there are positive contributions arising from the wino loop and tree-level Higgs soft mass:

2 k +1 2 2 (m2Hu (Qr))pos — —— m2Hu (M mess) + kWMw (Mmess), (13)

where we show the only dominant contributions and

3g22(Mmess)

kw =--8nm-t

1 - V2t/2

(1 - mt)2

9y2 (t')g2 (Mmess)

1 - n2t'/2

(1 - mt')2

, with

g2 (Mmess) 8n2 '

The sizes of the coefficients are k ~ 0'4, kg ~ 0J and k

Therefore in the case of the minimal GMSB with 5 and 5 messengers, the negative contributions substantially dominate over the positive contributions, leading to only a small cancellation. One thus needs larger Mw (Mmess) and/or mHu (Mmess) to obtain a sizable cancellation leading to small m2H at the soft mass scale. We will see below how it is achieved in our scenario.

Now we evaluate the value of m2H at mSUSY more precisely by numerically solving two-loop RGEs [23]. By taking Mmess = 1013 GeV, tan p = 15, mt(pole) = 173'34 GeV and as (mZ ) = 0' 1185, we obtain

m2Hu (3 TeV) = 0.704mHu + 0.019m2Hd

- 0.336mQ - 0.167m? - 0.056m|

+ 0.055mf - 0.054m?

+ 0.011M2 + 0.192M2~ - 0.727M2

- 0.003MBMW - 0.062MWMg

- 0.010MBMg, (15)

where soft SUSY breaking mass parameters in the right-hand side of Eq. (15) are defined at Mmess. By using Eqs. (4) and (5), we obtain (r3 = A3/A8)

mH (3 TeV) — [0'165- 0'035r3 - 1'222]Mg '

Note that m2H (3 TeV) nearly vanishes for r3 — 2'8, -2'6, i.e. we reach a focus point region. Here, we have neglected the contribution from the universal scalar mass. In fact, this contribution is rather small as m2H (3 TeV) 3 0'164m^. Here we make a crucial observation that the ratio r3 — X3M8/(X8M3) is RGE invariant:

X(3,8)(t) = X(3,8)(t0) exp

dt'(yz + 2 ye (3,8))

M(3,8) (t) = M(3,8)(t0) exp

f dt'(2yS(3,8 t0

where yi is the anomalous dimension of the field i. It immediately follows that

ks(t)M8(t) _ ks(t0)M8(t0) X8(t)Ms(t) = X8(t0)Ms(t0).

Once the ratio r3 is fixed by some fundamental physics, it is stable against radiative corrections, i.e. invariant under RGE running. This unique property lends significant reliability and robustness to our scenario over other competitive focus point SUSY models.

Now, we estimate the fine-tuning of the EWSB scale using the following measure [24]:

A = max{| Aa |},

d ln V d ln V

d ln v d ln v d ln v

d ln | Fz r d ln d ln Bd' d ln Mt'd ln m0

V=Vobs

where vobs ~ 174.1 GeV and B0 is the scalar potential B-term at the messenger scale, which may, for example, be generated by the Giudice-Masiero mechanism [25] or from a constant term in the superpotential.

In Fig. 2, we show the contours of A and mh. The Higgs boson mass is calculated using FeynHiggs 2.10.3 [26], and A is evaluated utilizing SOFTSUSY 3.6.1 [27]. To avoid the tachyonic stau, we take the universal scalar masses at Mmess as5

m0 (Mmess) =

180 TeV

500 GeV'

Also, the bino mass is regarded as an input parameter at Mmess, and taken as Mb (Mmess) = 250 GeV. The sign of the ^-parameter is taken to be positive. In the gray region with large |r3|, the

5 Strictly speaking, the universal scalar masses should be taken at Mstr. However, it makes only a small difference.

Table 1

Sample mass spectra. We take Mmess =

1013 GeV.

tan ß

Ml (Mmess) mo(Mmess)

mgluino msquark mstop

mh (m ßl )

(mßr )

180 TeV 2.8 15

250 GeV 450 GeV

123.1 GeV

538 GeV

3.6 TeV 3.4-4.5 TeV 2.2, 4.1 TeV 3.1 TeV 473 GeV 221 GeV 128 GeV 550 GeV 2.6 TeV

tan ß

M1(Mmess) m0(Mmess)

|A | /<

mgluino msquark mstop

miL (mßL)

m~es (m ßR)

280 TeV

250 GeV 700 GeV

125.1 GeV

156 156

850 GeV

5.4 TeV 5.1-6.7 TeV 3.4, 6.2 TeV

4.5 TeV 727 GeV 399 GeV 124 GeV 870 GeV 3.8 TeV

tan ß

M1(Mmess) m0(Mmess)

|A | /'

mgluino msquark mstop

mh (m ßL) m~es (m ßs) mü

230 TeV

250 GeV 600 GeV

123.0 GeV

652 GeV

4.5 TeV 4.2-5.5 TeV 3.1, 5.1 TeV

3.6 TeV 618 GeV 394 GeV 131 GeV 670 GeV 3.1 TeV

mx 0 X1

mx0 X1

mx0 X1

mx ± X1

mx ± x2

Fig. 3. Contours of A (solid) and mh/GeV (dashed) in minimal GMSB (i.e. with 5 and 5 messengers of SU(5) GUT). We set tanp = 25 and N5 = 3. Other parameters are same as in Fig. 2.

EWSB does not occur. One can see the observed Higgs boson mass is explained with A = 60-150 for r3 ~ 2.8 (Fig. 2, left panel). When r3 is negative (Fig. 2, right-panel), the required fine-tuning to reach the correct Higgs boson mass is slightly larger than the positive r3 case. These results can be compared to the minimal GMSB case (with only 5 and 5 messengers), shown in Fig. 3. Demanding mh > 123 GeV, the required A is around 750-1500 for Mmess > 109 GeV. For this plot we have taken the number of 5 and 5 pairs to be N5 = 3, though the required A does not significantly depend on this choice. Comparing Fig. 3 with Fig. 2 it is clear that our adjoint messenger model in the focus point region for r3 ~ 2.8 is significantly less tuned than minimal GMSB.

Finally we show some sample spectrum in Table 1. One can see that the stau can be light as 200-400 GeV, which may be testable at the LHC depending on the bino and Higgsino masses. Admittedly, the bino-like lightest neutralino may give rise to too large relic density causing over-closure of the universe. This can be avoided by tuning on a tiny amount of R-parity violation. In this case, axion could become a potential dark matter candidate.

4. Conclusions

We have considered a gauge mediated SUSY breaking scenario with messengers transforming in the adjoint representation of the gauge group as color octet and weak triplet. We have shown that focus point exists in this framework. The fine-tuning of the

EWSB scale is considerably reduced in the focus point region: A = 60-150, while explaining the observed Higgs boson mass around 125 GeV. In fact, the fine-tuning is considerably reduced in our scenario compared to that in minimal gauge mediation. Two distinctive features attribute a substantial credibility to our scenario: (i) a single combination of messenger sector parameters, which is RGE invariant, controls the focus point. This means that the focus point behavior is stable once a more fundamental theory fixes that combination; (ii) the special feature of color octet and weak triplet adjoint messengers triggering late gauge unification renders consistency of the scenario with colored Higgs mediated proton decay constraints.

Acknowledgements

We thank Hajime Fukuda and Hitoshi Murayama for useful discussions. G.B. thanks Kavli IPMU for hospitality when the work was done. This work is supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, No. 26104009, Grant-in-Aid No. 26287039 from the Japan Society for the Promotion of Science (JSPS), and the World Premier International Research Center Initiative (WPI), MEXT, Japan (T.T.Y.). The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 279972 "NPFlavour" (N.Y.).

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