Supersymmetric models have traditionally been assumed to be perturbative up
to high scales due to the requirement of calculable unification. In this note I
review the recently proposed `Fat Higgs' model which relaxes the requirement of
perturbativity. In this framework, an NMSSM-like trilinear coupling becomes
strong at some intermediate scale. The NMSSM Higgses are meson composites of an
asymptotically-free gauge theory. This allows us to raise the mass of the
Higgs, thus alleviating the MSSM of its fine tuning problem. Despite the strong
coupling at an intermediate scale, the UV completion allows us to maintain
gauge coupling unification.Comment: Talk given at the 12th International Conference on Supersymmetry and
  Unification of Fundamental Interactions (SUSY 04), Tsukuba, Japan, 17-23 Jun
  2004. To appear in the proceedings. 4 page

Harnik, Roni

English

arXiv

Supersymmetric models have traditionally been assumed to be perturbative up to high scales due to the requirement of calculable unification. In this note I review the recently proposed `Fat Higgs' model which relaxes the requirement of perturbativity. In this framework, an NMSSM-like trilinear coupling becomes strong at some intermediate scale. The NMSSM Higgses are meson composites of an asymptotically-free gauge theory. This allows us to raise the mass of the Higgs, thus alleviating the MSSM of its fine tuning problem. Despite the strong coupling at an intermediate scale, the UV completion allows us to maintain gauge coupling unification

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    LBNL- 56564      The Supersymmetric Fat Higgs  RONI HARNIK  Physics Department, UC Berkeley, Berkeley, CA 94720, USA Theoretical Physics Group, LBNL, UC Berkeley, Berkeley, CA 94720, USA                        This work was supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.  DISCLAIMER  This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California. arXiv:hep-ph/0410366v1  27 Oct 2004The Supersymmetric Fat HiggsRONI HARNIKPhysics Department, UC Berkeley, Berkeley, CA 94720, USATheoretical Physics Group, LBNL, UC Berkeley, Berkeley, CA 94720, USAE-mail: roni@socrates.berkeley.eduABSTRACTSupersymmetric models have traditionally been assumed to be perturbative up to highscales due to the requirement of calculable unification. In this note I review the recentlyproposed ‘Fat Higgs’ model which relaxes the requirement of perturbativity. In thisframework, an NMSSM-like trilinear coupling becomes strong at some intermediatescale. The NMSSM Higgses are meson composites of an asymptotically-free gaugetheory. This allows us to raise the mass of the Higgs, thus alleviating the MSSM ofits fine tuning problem. Despite the strong coupling at an intermediate scale, the UVcompletion allows us to maintain gauge coupling unification.1. Motivation - The Little Hierarchy Problem(s)The post-LEP era has forced models of electroweak symmetry breaking (EWSB) to confrontdata. For example, in models with a low cutoff one is expected to write all of the operatorsallowed by symmetries, suppressed by the appropriate power of the cutoff, Λ, in the spirit ofEFT. Assuming these operators have coefficients of O(1), one can place a lower bound on thecutoff from EW measurements Λ ∼> 10TeV [1]. However, since it is natural to tie the EW to thedynamical Λ, one then needs to explain the little hierarchy between them. This is the ‘LittleHierarchy Problem’ or ‘LEP Paradox’.In supersymmetric models, where physics is weakly coupled at a TeV, one naively does notrun into this problem because there is no low cutoff at which we must introduce all of theoperators allowed by symmetry. However, a little hierarchy problem emerges once the MSSMis required to fulfill the LEP bound for the mass of the Higgs, mh > 115 GeV. In the MSSMthe Higgs’ quartic coupling is determined by SUSY and is given by the D-term potential. Thisplaces an upper-bound of mZ on the tree level Higgs mass. One can raise this value by radiativecorrections from top-stop loops, but this requires the stop to be quite heavy, mt˜ > 500 GeV.However, top-stop loops are also responsible for EWSB in the MSSM, thus the stop mass isinvolved in setting the EW scale. The need to address the little hierarchy between mt˜ and theEW scale is the ‘Supersymmetric Little Hierarchy problem’ a. Within the MSSM it may onlybe solved by fine tuning other parameters in the Higgs potential.The tree level bound on the Higgs mass may be raised by going beyond the MSSM. In theNMSSM an additional singlet is added along with a superpotentialW = λNHuHd −k3N3, (1)which gives an additional quartic coupling to the Higgs, λ2|HuHd|2. This contributes to raisethe Higgs mass to m2h ∼ λ2v2 +O(m2Z) which can easily be above 115 GeV.aFor a more detailed discussion see e.g. [2].However, we cannot raise the mass arbitrarily by increasing λ. The reason is that the effectiveλ(µ) grows in the UV and eventually hits a Landau pole at a scale Λ. Above this scale we losecontrol of the theory and even the relevant degrees of freedom are unknown. Encountering thisscale below the GUT scale would be a disaster for unification. Demanding that the Landaupole be above MGUT has yielded an upper-bound on λ(v) and thus on the Higgs mass. In theNMSSM this gives mh < 150 GeV.However, in principle, nothing can prevent us from giving up unification and bringing Λdown to increase the mass of the Higgs. This model is now is an EFT below a cutoff whichwe imagine as some dynamical scale of compositeness. As we have seen, this model solvesthe supersymmetric version of the little hierarchy problem. Since we are now dealing with alow-cutoff model we may wonder if the “regular” little hierarchy problem was re-introduced i.e.explaining the little hierarchy between the cutoff and the EW scale. However this problem istrivially solved since the little hierarchy is stabilized by supersymmetry.It is interesting to draw an analogy between the models of Little Higgs [3] and the stronglycoupled version of the NMSSM peresented above. Both models are EFTs with a low cutoffsuplemented by additional symmetries that protect the Higgs mass. In the former the symmetryis a global symmetry that is collectively broken and in the later it is supersymmetry. Themain advantage of the supersymmetric model is that due to exact results in strongly coupledsupersymmetric gauge theories, it is much easier to UV complete.2. UV Completion – The Fat HiggsIn [2] we UV completed a cousin theory of the NMSSM given by the superpotentialW = λN(HuHd − v20). (2)This superpotential is similar to the NMSSM in that it raises the tree level Higgs mass above theLEP bound for a sufficiently large λ(v) but then requires a UV completion above the Landaupole of λ.In this brief note, I will only present the essential components of our model and advertisesome of its features. A more complete model and its analysis is presented in [2]. The UV dy-namics of our model consists of a new strongly coupled gauge group, SU(2)H . The we introducesix doublets, T 1...6, under SU(2)H which corresponds to Nf = 3. The SU(2)H is asymptoticallyfree and thus becomes strong at some scale, ΛH . The T ’s are also carry electroweak charge. Thecharge assignments under SU(2)H × SU(2)L × U(1)Y is(T 1, T 2) ≡ T = (2,2, 0), (T 3, T 4) = (2,1,±12), (T 5, T 6) = (2,1, 0). (3)We also write a tree-level superpotentialW = −mT 5T 6 +Wdecouple (4)where the meaning of Wdecouple will become clear presently. Below the scale ΛH this theoryconfines and low energy degrees of freedom are described by the anti-symmetric meson matrixMij = TiTj . Let us relabel part of the meson matrix as followsN =M56,(H+uH0u)=(M13M23),(H0dH−d)=(M14M24), (5)0.01 0.1 1 10 100 100024681012ΛHHλ αΗcomposites asymptoticfreedom4pi∼4picouplingm     ( Λ )1/2v0~EWSB: mµ/ΛFigure 1: The renormalization of the couplings in the model presented. When 4piv0 ≪ ΛH the mesons condenseat weak coupling and the theory is calculable.noting that the mesons labeled as Hu and Hd indeed transform as the Higgs fields of the MSSM,and N is a singlet, as needed in Eq. (2). The superpotential Wdecouple in Eq. (4) involvesadditional new fields that marry all of the other meson fields in Mij . This can be done naturallyby enforcing an additional Z3 symmetry [2]. The theory below ΛH is thus an effective theory ofHu, Hd and N alone.This theory generates a dynamical superpotentialWdyn =PfMΛ3⊃1Λ3NHuHd (6)Once Hu,d and N are canonically normalized to dimension one fields this will produce therenormalizable trilinear term in Eq. (2). The mass for the third flavor of Eq. (4) becomes alinear term for the singlet N . Putting these two contributions together gives a superpotential ofthe form of Eq. (2). This superpotential breaks EW symmetry even in the SUSY limit, whichreleases the stop from its role in EWSB in the MSSM. The scale v0 at which EWSB occurs inthe SUSY limit may be estimated in NDA as v20 =mΛH(4pi)2 .The dynamics in this model are heuristically summarized in figure 1. At high energy thetheory is a weekly couple gauge theory with the T ’s as the relevant degrees of freedom. Aswe run down the theory becomes strong at ΛH at which point we substitute the degrees offreedom to the meson fields and the coupling λ for the gauge coupling. Below ΛH the coupling λrenormalizes quickly to weak coupling. If the scale m is somewhat below ΛH , EWSB will occurat weak coupling. Electroweak observables in this model, such as the oblique parameters S andT or the Higgs spectrum, are thus calculable.In [2] the Higgs spectrum was calculated for a range of SUSY breaking parameters. Thelightest Higgs was indeed found to be heavier than conventional MSSM values, easily reaching350 GeV or higher. A distinct signal for the Higgs spectrum in our models is that the pseudo-scalar Higgs is always heavier than the charged one whereas in the MSSM the converse is alwaytrue. The S and T parameters were also calculated and were found to be within the 1σ allowedregion for a wide range of parameters.EWSB may be communicated to the matter sector by a scalar version of an ETC sector. Weadd heavy fundamental Higgses, ϕu,d and ϕ¯u,d, that couple both to the T ’s and to the MSSMWf = Mf (ϕuϕ¯u + ϕ¯dϕd) + ϕ¯d(TT4) + ϕ¯u(TT3)+hijuQiujϕu + hijd Qidjϕd + hije Liejϕd. (7)At the scale Mf , the heavy Higgses can be integrated out, leaving a direct coupling betweenthe MSSM and the right combination of T ’s that becomes the composite Higgs once the theoryconfines. The effective Yukawa to the fat Higgses ishu,d4piΛHMf. If Mf ∼ ΛH this may present aproblem in generating a large top mass.In order to avoid fine tuning the SUSY breaking scale in this theory must be of order λv0which is set by supersymmetric scales. This problem is reminiscent of the µ problem of theMSSM. A more complete model was introduced in [2] in order to relate the SUSY breakingscale to m, ΛH and also Mf . This was done by adding a fourth flavor of T ’s which results innear conformal behavior. The new walking dynamics is also beneficial to enhancing the effectiveMSSM Yukawa couplings by anomalous dimensions as in walking technicolor. However, in thiscase the anomalous dimensions are calculable due to their relation to anomaly free R-charges.Finally, lets revisit the possibility of unification in this model. Below the scale ΛH thematter content of our model is simply that of the NMSSM. The running of the 3-2-1 gaugecouplings follow their regular trajectories. At ΛH we must match the low energy theory to theUV completion. Due to holomorphy, the matching equation between the low and high energytheories is set by bare parameters [4]. This leads us to believe that the threshold correctionsat ΛH are small. Above ΛH the T ’s must substitute the composite Higgses in the running.It is amusing that the T ’s contribute to the one-loop beta functions of the MSSM exactly astwo Higgs doublets do, which leaves us on the MSSM trajectory. However, we have also addedthe fundamental Higgses ϕu,d, ϕ¯u,d, as well as new fields that take part in the Wdecouple, all ofwhich contribute as three more pairs of Higgs doublets. This deviation from the MSSM runningmay be corrected. For example, adding three pairs of heavy triplets with Y = ±1/3, completesthe “SU(5) multiplets” and brings us back on track. Though this remedy may seem somewhatcontrivedb, we note that the possibility of even talking about unification above a compositenessscale may be viewed as significant progress.3. AcknowledgementsI would like to thank my collaborators- Graham Kribs, Daniel Larson and Hitoshi Murayama. Manythanks to Zacharia Chacko who suggested I go to SUSY and for pointing out how many 10 minuteintervals there were in February. This work was supported in part by the DOE under contracts DE-FG02-90ER40542 and DE-AC03-76SF00098 and in part by NSF grant PHY-0098840.4. References[1] R. Barbieri and A. Strumia, arXiv:hep-ph/0007265.[2] R. Harnik, G. D. Kribs, D. T. Larson and H. Murayama, Phys. Rev. D 70, 015002 (2004)[arXiv:hep-ph/0311349].[3] N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Lett. B 513, 232 (2001)[arXiv:hep-ph/0105239] and references thereof.[4] N. Arkani-Hamed and H. Murayama, Phys. Rev. D 57, 6638 (1998) [arXiv:hep-th/9705189].[5] S. Chang, C. Kilic and R. Mahbubani, arXiv:hep-ph/0405267.bFor a possibly more attractive solution with a slimmer Higgs see [5].

The Supersymmetric Fat Higgs

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The Supersymmetric Fat Higgs

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