Higgs mass via type II seesaw mechanism

Pavel Fileviez Perez

Particle and Astro-Particle Physics Division, Max-Planck Institute for Nuclear Physics (MPIK),Saupfercheckweg 1, 69117 Heidelberg, Germany

Sogee Spinner

International School for Advanced Studies (SISSA) and INFN, Via Bonomea 265, 34136 Trieste, Italy(Received 15 November 2012; published 11 February 2013)

We show that the simplest supersymmetric scenario where a large Higgs mass can be attained at tree

level for any ratio of the Higgs vacuum expectation values corresponds to the case where the neutrino

masses are generated through the type II seesaw mechanism. This allows a standard model-like Higgs with

mass around 125 GeV without assuming a heavy spectrum for the top squarks. We show that our results

are consistent with the bounds coming from perturbativity up to the grand unified scale and discuss gauge

coupling unification and possible signals at the Large Hadron Collider.

DOI: 10.1103/PhysRevD.87.031702 PACS numbers: 14.80.Da, 12.60.Jv, 14.60.Pq

I. INTRODUCTION

The recent discovery of a new particle [1,2] at the LargeHadron Collider (LHC) with properties similar to thestandard model (SM) Higgs is one of the most importantand exciting discoveries to hit the particle physics world inthe last several decades. Understandably, it has spurred aflurry of studies which attempt to understand the propertiesof the Higgs field. Supersymmetry provides a natural theo-retical framework to study this issue since the minimalsupersymmetric standard model (MSSM) predicts a tree-level Higgs mass no larger than the Zmass. While this fallsshort of the recently measured 125 GeV particle, loopcorrections from a heavy top squark sector can completethe difference. For a review on the predictions for theHiggs mass in the MSSM see Ref. [3].

Extending the MSSM can alleviate the tree-level Higgsmass bound. The next to minimal supersymmetric standardmodel (NMSSM) enlarges the MSSM Higgs content by anSM singlet and its coupling to the MSSM Higgs fields canincrease the tree-level mass beyondMZ. However, the newcontribution is maximized for values of the ratio of theHiggs doublet vacuum expectation values (VEVs), tan�,close to unity, while the MSSM contribution is maximizedfor tan� far from unity. For a review of the NMSSM seeRef. [4]. Furthermore, significant enhancements requirecoupling values which are not perturbative to the grandunified theory (GUT) scale. One can also extend by ahyperchargeless triplet [5,6] or a combination of the tripletand singlet [7] but in both cases the new contribution to theHiggs mass occurs for tan� close to unity and perturba-tivity is an issue, although gauge coupling unificationconsiderations can solve the latter issue [8].

The singlet and the hyperchargeless triplet are two of thethree possible fields that can introduce new quartic Higgscouplings. The third, triplets with hypercharge one, cancontribute significantly to the Higgs mass independent oftan� [5,6]. Coincidentally, these fields also facilitate the

type II seesaw mechanism for neutrino masses. In thisarticle we show that such fields allow a large tree-levelHiggs mass for any value of tan� while keeping all cou-plings perturbative to the GUT scale. We also discuss thepossibility of keeping the interesting MSSM prediction ofgauge coupling unification. The most generic signatures atthe LHC are briefly discussed.This article is organized as follows. In Sec. II we survey

the predictions for the Higgs mass in different supersym-metric models. In Sec. III we review the type II seesawmechanism for neutrino masses and the properties of theHiggs sector. In Sec. IV we discuss gauge coupling uni-fication and in Sec. V we highlight possible signals at theLHC. We summarize our main results in Sec. VI.

II. THEHIGGSMASS IN THEMSSMANDBEYOND

The Higgs sector of the MSSM is composed of twoHiggs doublets:

Hu ¼Hþ

u

H0u

!� ð1; 2; 1=2Þ;

Hd ¼H0

d

H�d

!� ð1; 2;�1=2Þ;

resulting in three neutral physical Higgs fields: h,H and A,and charged Higgs bosons H�. In the decoupling limit,M2

A � M2Z (MZ is the mass of the Z boson), the lightest

CP-even Higgs, h, is SM-like with the tree-level upperbound:

M2h � M2

Zcos22�;

requiring a large one-loop level contribution to be consis-tent with a Higgs interpretation of the new 125 GeV boson.

In the NMSSM, where a singlet superfield S allows for

the term �HSHuHd, the upper bound changes to

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M2h � M2

Zcos22�þ 1

2�2Hv

2sin22�:

This effect is relevant for tan� close to unity and for a largecontribution to the Higgs mass �H is not perturbative up tothe GUT scale. An extra contribution can be gained from

introducing a real triplet, �� ð1; 3; 0Þ, which couples as

��Hu�Hd and increases the Higgs mass upper bound to

M2h � M2

Zcos22�þ 1

2

��2H þ 1

2�2�

�v2sin22�:

Unfortunately, as in the NMSSM this effect on the Higgsmass is only relevant for small tan� and does not improveperturbativity. For the study of the Higgs sector in tripletextensions of the MSSM see Refs. [5–10]. The main goalof this work is to investigate the simplest extension of theMSSM which can generate a large Higgs mass at tree levelfor any value of tan� and to show how to keep the MSSMpredictions.

III. TYPE II SEESAWAND THE HIGGS MASS

In order to implement the type II seesaw mechanism[11] for neutrino masses in the MSSM, the Higgs contentmust be extended with two SUð2Þ triplets:

�1 ¼1ffiffi2

p ��1 �0

1

���1 � 1ffiffi

2p ��

1

0@

1A� ð1; 3;�1Þ; (1)

�2 ¼1ffiffi2

p �þ2 �þþ

2

�02 � 1ffiffi

2p �þ

2

0@

1A� ð1; 3; 1Þ: (2)

In this case the relevant superpotential reads as

W II ¼ yuQHuuc þ ydQHdd

c þ yeLHdec ��HuHd

þ��Tr�1�2 þ �1Hu�1Hu þ �2Hd�2Hd

þ f�L�2L; (3)

where the last term generates the neutrino mass matrix:

M� ¼ f�v2ffiffiffi2

p ; (4)

where v2 is the VEVof �02. Notice that if v2 � 1 GeV, f�

must be small, about 10�9. The �1 and �2 terms in thesuperpotential allow for quartic Higgs couplings, whichmodify the upper bound on the lightest CP-even scalartree-level mass to

M2h � cos22�M2

Z þ 2sin4��21v

2 þ 2cos4��22v

2: (5)

The new contributions can be sizable for any value of tan�,but interestingly the �1 and MSSM contributions bothincrease with tan� allowing for constructive interference.This is in contrast to the scenarios reviewed in the previoussection where the additional quartic term (proportional tosin22�) has the inverse tan� dependence of the MSSM

contribution. The corresponding soft supersymmetrybreaking Lagrangian is

�LSoft ¼ m2HujHuj2 þm2

HdjHdj2 þm2

�1j�1j2 þm2

�2j�2j2

þ ð�bHuHd þ b�Tr�1�2 þ a1Hu�1Hu

þ a2Hd�2Hd þ afL�2Lþ H:c:Þ þ � � � ; (6)

where the remaining MSSM soft terms have been left out.The scalar potential coming from the D term is given by

VD ¼ 1

8g21ðjHuj2 � jHdj2 þ 2Tr�y

2�2 � 2Tr�y1�1Þ2

þ 1

8g22

X3a¼1

ðHyu�aHu þHy

d�aHd þ 2Tr�y1�a�1

þ 2Tr�y2�a�2Þ2: (7)

The VEVs of the fields are defined to be

hH0ui ¼ vuffiffiffi

2p ; hH0

di ¼vdffiffiffi2

p ;

h�01i ¼

v1ffiffiffi2

p ; h�02i ¼

v2ffiffiffi2

p :

The VEVs of the triplets break custodial symmetry and aretherefore constrained by the � parameter to be less thanabout 2–3 GeV [10]. Therefore, working to zeroth order inv1 and v2 the following minimization conditions can bederived:

0 ¼ m2Hd

þ�2 þ 1

2c2�M

2Z þ t�bþ c2��

22v

2; (8)

0 ¼ m2Hu

þ�2 � 1

2c2�M

2Z þ

1

t�bþ s2��

21v

2; (9)

0 ¼ m2�1

þ�2� þ c2�M

2Z þ 2s2��

21v

2 þ v2

v1

b�

þ v2ffiffiffi2

pv1

ðc2��2�� � s2��1�� a1s2�Þ; (10)

0 ¼ m2�2

þ�2� � c2�M

2Z þ 2c2��

22v

2 þ v1

v2

b�

þ v2ffiffiffi2

pv2

ða2c2� þ s2��2�� s2��1��Þ: (11)

Defining v1 ¼ v� cos� and v2 ¼ v� sin�we can solve forv� and one finds

v� ¼ v2ffiffiffi2

pcos�M2

1

ð�c2��2�� þ s2��1�� a1s2�Þ; (12)

where

M21 ¼ m2

�1þ�2

� þ c2�M2Z þ 2s2��

21v

2 þ tan�b�: (13)

Notice that if M1 is around the TeV scale and the trilinearterm a1 is small, as in gauge mediation, one can get

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v� � 1 GeV, thereby agreeing with the � parameter con-straints. It is important to mention that in the limit when �1,�2, a1, a2 ! 0 B� L is an exact global symmetry.Therefore, one can say that the VEVs of the seesaw tripletsare protected by the global B� L symmetry. Notice that inthis model the minimization conditions are more involvedand on top of the usual fine-tuning present in the MSSMone has a fine-tuning in the new minimization conditions.Figure 1 shows the values of the Higgs mass at tree levelversus �1 and a scan over the remaining parameters asfollows:

tan�� ð2–45Þ; j�2j � ð�1;1Þ;j�j; j��j � ð100;500Þ GeV; jv1j; jv2j � ð0:5;1Þ GeV;

jbj; jb�j � ð502;10002Þ GeV2; ja1j; ja2j � ð�1;1Þ TeV:Included is the upper bound on the Higgs mass [Eq. (5)]with �2 ¼ 0 and tan� ¼ 10 (in orange). The area outsidethe vertical red dashed lines is not perturbative up to theGUT scale under the assumptions discussed in the nextsection. Given the values from the scan, one can useEqs. (10) and (11) to calculate the soft triplet masses.They are in the following ranges: m�1

�ð273;8900ÞGeVand m�2

� ð272; 4630Þ GeV.Figure 1 demonstrates that it is possible to achieve a

large Higgs mass at tree level even when �1 is small. Inprinciple, one can even saturate the value of the 125 GeVHiggs mass at tree level. Here we have assumed that allother Higgs fields are heavier than 300 GeV, the

decoupling limit where the lightest Higgs is SM-like.The contribution from �2 is proportional to cos4� anddoes not play a major role. Focusing on the two mostrelevant parameters, we show in Fig. 2 a simple tree-levelMh isoplot in the �1 � tan� plane, where the dashed redlines once more delineate the nonperturbative region. Onecan appreciate that consistency with a 125 GeV Higgsmass is possible at tree level with all coupling perturbativeto the GUT scale.Now, let us discuss the effect of loop corrections to

understand how light the top squarks can be when thetree-level mass is large. Curves of constant Higgs mass atthe two-loop level with zero left-right mixing in the topsquark sector are displayed in Fig. 3 in the plane of right-handed versus left-handed top squark soft masses. Curvesfor �1 ¼ 0:16, 0.18, 0.20 are solid red, dashed green anddotted blue, respectively. Loop calculations were per-formed using FEYNHIGGS [12] and we use 1.5 TeV forthe gluino mass. As one can appreciate from Fig. 3 thetop squarks can be very light even when the left-rightmixing is zero. In summary, the type II seesaw mechanismprovides the simplest scenario which easily accommodatesa 125 GeV Higgs mass for generic values of tan� and isperturbative up to the GUT scale.

IV. UNIFICATION AND PERTURBATIVITY

It is well known that the triplet seesaw fields alonedestroy the gauge coupling unification present in theMSSM and in order to keep this interesting prediction of

Mh upper bound:

2 0, tan 10

Mh 110 GeV

Mh 125 GeV

Nonperturbative:

0.6 0.4 0.2 0.0 0.2 0.4 0.650

100

150

200

1

Mh

GeV

FIG. 1 (color online). The tree-level Higgs mass versus �1

with parameters scanned as described in the text. The orangecurve represents the upper bound on the tree-level mass, Eq. (5),with �2 ¼ 0 and tan� ¼ 10. To aid the eye, horizontal lines at110 and 125 GeV are drawn in green and blue, respectively. Thearea outside the vertical red dashed lines becomes nonperturba-tive before the GUT scale under the assumptions discussed in thenext section.

125115105

95

125115

105

95

Nonperturbative:

5 10 15 20

0.4

0.2

0.0

0.2

0.4

tan

1

FIG. 2 (color online). Curves of constant tree-level Mh in the�1 � t� plane. The area outside the vertical red dashed lines

becomes nonperturbative before the GUT scale under the as-sumptions discussed in the next section.

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the MSSM (assuming the desert) one must add extra fieldsat the TeV scale. In our case one could add fields to

complete a 15H and 15H of SUð5Þ. However, this possi-bility is not very appealing since the gauge coupling willnot be perturbative up to the GUT scale. A second possi-bility was proposed in Ref. [13]:

D c�ð�3;1;1=3Þ; �Dc�ð3;1;�1=3Þ; ��ð8;1;0Þ;

(14)

where these fields are called ‘‘magic’’ fields since they donot complete an SUð5Þ representation with the seesawtriplets but still allow for gauge coupling unification.

The superpotential relevant for the new fields is given by

W C ¼ 1

2��Tr�

2 þ 1

3Tr�3 þ Dc� �D

c þ �dc� �Dc

þ yDQHdDc þ�DD

c �Dc þ�bb

c �Dc: (15)

It is important to mention that in this model we haveassumed generic mass terms for all fields and the conser-vation of R parity. However, it is also possible to introducea singlet and a Z3 symmetry (a la NMSSM) in which allsupersymmetry mass terms are generated after symmetrybreaking (this was done without the colored fields inRef. [10]). Therefore, we could replace the mass termsby a new coupling and the singlet field as

� ! �HS; �� ! ��S; �D ! �DS;

thereby solving the � problem. The renormalization groupequations were used to derive the red dashed lines inFigs. 1 and 2, which delineate the nonperturbative regime.While the �1 beta function receives a large self-contribution the increased size of the gauge couplingshelps to keep it perturbative. Beyond �1 � 0:5, the systembecomes nonperturbative due to this large contribution.The unification of the gauge couplings assuming that allnew particles are at the TeV scale is shown in Fig. 4 as well

1 0.161 0.181 0.20

125

128

122

125

128

122

600 800 1000 1200 1400

600

800

1000

1200

1400

MQ3

GeV

Mt R

GeV

FIG. 3 (color online). Values of constant Mh at the two-looplevel in the M~tR �M ~Q3

plane for zero left-right top squark

mixing and for �1 ¼ 0:16, 0.18, 0.20 in red solid, green dashedand blue dotted curves, respectively.

11

21

31

2 4 6 8 10 12 14 160

10

20

30

40

50

60

Log10 1 GeV

1

tan 10

yt

2

1

4 6 8 10 12 14 160.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Log10 1 GeV

y

FIG. 4 (color online). Unification of the gauge couplings in thecomplete model with the colored octet and the new vectorlikepair (upper) and the running of the Yukawa couplings relevant tothe Higgs mass (lower). Yukawa couplings not shown are 0.1 atthe TeV scale except for f�, which is insignificantly small due toneutrino masses.

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as an example of the running of the Yukawa couplingsrelevant for the Higgs mass discussion. For the latter plot,Yukawa couplings not shown are assumed to be 0.1 at theTeV scale except for f�, which is insignificantly small dueto neutrino masses. As one can see, all couplings areperturbative.

V. PHENOMENOLOGICAL ASPECTS

Here we discuss some of the relevant phenomenologicalaspects of the new physical Higgses and matter fields.

(i) Doubly charged Higgses.—Because of the relativelylarge VEVs of the triplets (about one GeV) and smallvalue of the triplet-lepton coupling (f�), the decay���i ! W�W� dominates over ���

i ! e�j e�kassuming that the channel ���

i ! W�H�i is kine-

matically suppressed. It is therefore not possible tosearch for lepton number violation in this context.The decays of the doubly charged Higgses into W’shave been investigated in detail in Ref. [14]. Themost interesting signals in this case correspond to thepair production of the doubly charged Higgs fields:

pp ! H��H�� ! 4jþ e�i e�j þ EmissT : (16)

Such states can be identified at the LHC with highluminosity.The decays of the doubly charged Higgsinos into alepton and slepton are also suppressed for the samereason mentioned above for the doubly chargedHiggs into two leptons. Then, the main decays areinto a neutralino and doubly charged Higgs or thedecay into a W and a chargino. These channels arequite sensitive to the supersymmetric spectrum butare unique for these type of models.

(ii) Colored octets.—The field �� ð8; 1; 0Þ can decayinto two heavy quarksDc and �Dc (a heavy quark �Dc

and a down-type quark) through the Yukawa cou-pling (�) or into two gluons at the one-loop level

through the cubic term Tr�3. Therefore, pair pro-duction of these fields through QCD interactionsyields final states with four jets, where the invariantmass of two jets corresponds to the mass of thecolored octet. The octetino has the same quantumnumbers as the gluino and it can decay into acolored octet and a gluino, giving rise to signalswith multileptons and missing energy if we assumeR-parity conservation.

(iii) Heavy quarks.—The new quarks Dc and �Dc arevectorlike and could modify the Higgs decays.These can mix with the bottom quarks and suppressthe Higgs branching ratio into two bottom quarks.As any colored field, they can be produced withlarge cross sections and decay into a Higgs and aSM quark through the Yukawa coupling, or into agauge boson and a quark.

VI. SUMMARY

In this article we have shown that the fields necessary forthe type II seesaw mechanism for neutrino masses can alsoraise the SM-like Higgs mass up to 125 GeV even at treelevel. This can be accomplished independently of the valueof tan� and all couplings remain perturbative up to theGUT scale. While the fields that complete GUT represen-tations with the triplets would lead to Landau poles for thegauge couplings, preservation of gauge coupling unifica-tion can be accomplished through so-called magic fields: avectorlike pair of down-type quarks and a color octet.Some of the collider signals of these fields were brieflydiscussed and a more detailed analysis will be published inthe near future.

ACKNOWLEDGMENTS

P. F. P. thanks M. Lindner and Mark B. Wise for com-ments and discussions. S. S. would also like to thankA. Romanino for a discussion.

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