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PHYSICAL REVIEW LETTERS

Natural Higgs Mass in Supersymmetry from Nondecoupling Effects Xiaochuan Lu,1,2,* Hitoshi Murayama,1,2,3,† Joshua T. Ruderman,1,2,‡ and Kohsaku Tobioka3,4,§ 1

Department of Physics, University of California, Berkeley, California 94720, USA Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan 4 Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan (Received 18 October 2013; published 14 May 2014) 2

The Higgs mass implies fine-tuning for minimal theories of weak-scale supersymmetry (SUSY). Nondecoupling effects can boost the Higgs mass when new states interact with the Higgs boson, but new sources of SUSY breaking that accompany such extensions threaten naturalness. We show that two singlets with a Dirac mass can increase the Higgs mass while maintaining naturalness in the presence of large SUSY breaking in the singlet sector. We explore the modified Higgs phenomenology of this scenario, which we call the “Dirac next-to-minimal supersymmetric standard model.” DOI: 10.1103/PhysRevLett.112.191803

PACS numbers: 14.80.Da, 12.60.Jv, 13.66.De, 13.85.Qk

Introduction.—The discovery of a new resonance at 125 GeV [1], that appears to be the long-sought Higgs boson, marks a great triumph of experimental and theoretical physics. On the other hand, the presence of this light scalar forces us to face the naturalness problem of its mass. Arguably, the best known mechanism to ease the naturalness problem is weak-scale supersymmetry (SUSY), but the lack of experimental signatures is pushing SUSY into a tight corner. In addition, the observed mass of the Higgs boson is higher than what was expected in the minimal supersymmetric standard model (MSSM), requiring finetuning of parameters at the 1% level or worse [2]. If SUSY is realized in nature, one possibility is to give up on naturalness [3]. Alternatively, theories that retain naturalness must address two problems, (I) the missing superpartners and (II) the Higgs mass. The collider limits on superpartners are highly model dependent and can be relaxed when superpartners unnecessary for naturalness are taken to be heavy [4], when less missing energy is produced due to a compressed mass spectrum [5] or due to decays to new states [6], and when R parity is violated [7]. Even if superpartners have evaded detection for one of these reasons, we must address the surprisingly heavy Higgs mass. There have been many attempts to extend the MSSM to accommodate the Higgs mass. In such extensions, new states interact with the Higgs boson, raising its mass by increasing the strength of the quartic interaction of the scalar potential. If the new states are integrated out supersymmetrically, their effects decouple and the Higgs mass is not increased. On the other hand, SUSY breaking can lead to nondecoupling effects that increase the Higgs mass. One possibility is a nondecoupling F term, as in the nextto-minimal supersymmetric standard model (NMSSM) (MSSM plus a singlet) [8,9] or λSUSY (allowing for a 0031-9007=14=112(19)=191803(5)

Landau pole) [10]. A second possibility is a nondecoupling D term that results if the Higgs boson is charged under a new gauge group [11]. In general, these extensions require new states at the few hundred GeV scale, so that the new sources of SUSY breaking do not spoil naturalness. For example, consider the NMSSM, where a singlet superfield S interacts with the MSSM Higgs bosons Hu;d through the superpotential, W ⊃ λ SHu Hd þ

M 2 S þ μ HuHd : 2

(1)

The Higgs mass is increased by m2S ; M2 þ m2S

Δm2h ¼ λ2 v2 sin2 2β

(2)

where m2S is the SUSY breaking soft mass m2S jSj2 , tan β ¼ vu =vd is the ratio of theqVEVs of the up- and down-type ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Higgs bosons, and v ¼ v2u þ v2d ¼ 174 GeV Notice that this term decouples in the supersymmetric limit M ≫ mS , which means mS should not be too small. On the other hand, mS feeds into the Higgs soft masses m2Hu;d at oneloop, requiring fine-tuning if mS ≫ mh . Therefore, with M at the weak scale, there is tension between raising the Higgs mass, which requires large mS relative to M, and naturalness, which demands small mS . In this Letter, we point out that, contrary to the above example, a lack of light scalars can help raise the Higgs mass without a cost to naturalness, if the singlet has a Dirac mass. We begin by introducing the model and discussing the Higgs mass and naturalness properties. Then, we discuss the phenomenology of the Higgs sector, which can be discovered or constrained with future collider data. We finish with our conclusions.

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© 2014 American Physical Society

The model.—To illustrate this possibility, we consider a modification of Eq. (1) where S receives a Dirac mass with ¯ another singlet S, W ¼ λSH u Hd þ MSS¯ þ μHu Hd : Uð1ÞPQ Uð1ÞS¯

(3)

Hu

Hd

S

S¯

μ

M

1 0

1 0

−2 0

−2 1

−2 0

4 −1

We call this model the Dirac NMSSM. The absence of various dangerous operators (such as large tadpoles for the singlets) follows from a Uð1ÞPQ × Uð1ÞS¯ Peccei-Quinnlike symmetry, here, Uð1ÞS¯ has the effect of differentiating ¯ u Hd. Because μ and S and S¯ and forbidding the operator SH M explicitly break the Uð1ÞPQ × Uð1ÞS¯ symmetry, we regard them to be spurions, originating from chiral superfields (“flavons” [12]) so that holomorphy is used to avoid certain unwanted terms (“SUSY zeros” [13]). By classifying all possible operators induced by these spurions, we see that a tadpole for S¯ is suppressed adequately, ¯ W ⊃ cS¯ μM S;

(4)

where cS¯ is a Oð1Þ coefficient. Other terms involving only singlets are forbidden by the symmetries or suppressed by the cutoff. The following soft supersymmetry breaking terms are allowed by the symmetries, ΔV soft ¼

m2Hu jHu j2

þ

m2Hd jHd j2

þ

m2S jSj2

þ

¯2 m2S¯ jSj

þ λAλ SHu Hd þ MBS SS¯ þ μBHu Hd þ c:c: þ tS¯ S¯ þ tS S þ c:c:

(5)

The last tadpole arises from a nonholomorphic term μ† S. The small hierarchy, which we consider later, between the soft masses of S¯ and others can be naturally obtained in gauge mediation models if S¯ couples to the messengers in the superpotential. It is easy to write down a model where m2S¯ is positive at the one-loop level, while the soft masses for squarks and sleptons arise at the two-loop level via gauge mediation [14]. The tadpole for S¯ is generated at one-loop and, hence, tS¯ ≃ μMmS¯ =4π, while soft masses and the tadpole for S are generated at higher order. We checked these singlet tadpoles do not introduce extra tuning even with large M and mS¯ . We would like to understand whether the new quartic term, jλHu Hd j2 , can naturally raise the Higgs mass. Integrating out S and S¯ we find the following potential for the doublet-like Higgs bosons, V eff

2 ¼ jλHu Hd j 1 −

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M2 λ2 − M 2 þ m2S¯ M 2 þ m2S

× jAλ H u H d þ μ ðjH u j2 þ jHd j2 Þj2 ;

where we keep the leading ðM2 þ m2S;S¯ Þ−1 terms and neglect the tadpole terms for simplicity. The additional Higgs quartic term does not decouple when m2S¯ is large, as in the NMSSM at large m2S . The SM-like Higgs mass becomes, m2S¯ 2 2 2 2 2 mh ¼ mh;MSSM ðm~t Þ þ λ v sin 2β M 2 þ m2S¯ −

λ2 v 2 jAλ sin 2β − 2μ j2 ; M þ m2S

(7)

2

in the limit where the VEVs and mass eigenstates are aligned. The Higgs sector is natural when there are no large radiative corrections to m2Hu;d . The renormalization group (RG) of the up-type Higgs contains the terms, μ

d 2 1 mHu ¼ 2 ð3y2t ½m2Q~ þ m2~t þ λ2 m2S Þ þ … R 3 dμ 8π

(8)

While heavy top squarks or m2S lead to fine-tuning, we find that m2S¯ does not appear. In fact, the RGs for m2Hu;d are independent of m2S¯ to all orders in mass-independent schemes, because S¯ couples to the MSSM+S sector only through the dimensionful coupling M. There is logarithmic sensitivity to m2S¯ from the one-loop finite threshold correction, δm2H ≡ δm2Hu;d ¼

M 2 þ m2S¯ ðλMÞ2 log ; ð4πÞ2 M2

(9)

which still allows for very heavy m2S¯ without fine-tuning. One may wonder if there are dangerous finite threshold ¯ In corrections to m2Hu at higher order, after integrating out S. fact, there is no quadratic sensitivity to m2S¯ to all orders. This follows because any dependence on m2S¯ must be proportional to jMj2 [since S¯ becomes free when M → 0 and by conservation of Uð1ÞS¯ ], but jMj2 m2S¯ has too high mass dimension. The mass dimension cannot be reduced from other mass parameters appearing in the denominator because threshold corrections are always analytic functions of IR mass parameters [15]. It may seem contradictory that naturalness is maintained in the limit of very heavy mS¯ , since removing the S¯ scalar from the spectrum constitutes a hard breaking of SUSY. The reason is that the effective theory, with the S¯ fermion but no scalar present at low energies is actually equivalent to a theory with only softly broken supersymmetry, where the MSSM is augmented by the Kähler operators, Keff ¼ S¯ † S¯ − θ2 θ¯ 2 ðMDα SDα S¯ þ c:c: þ M 2 jS þ cS¯ μj2 Þ; (10)

(6)

and where the scalar and F term of S¯ are reintroduced at low energy, but completely decoupled from the other states.

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FIG. 1 (color online). The tuning Δ, defined in Eq. (11), for the Dirac NMSSM is shown on the left as a function of M and mS¯ . For comparison, the tuning of the NMSSM is shown on the right as a function of M and mS . The red region has high fine-tuning, Δ > 100, and the purple region requires m~t > 2 TeV, signaling severe fine-tuning ≳Oð103 Þ.

We call this mechanism semisoft supersymmetry breaking. It is crucial that S¯ couples to the other fields only through dimensionful couplings. Note that Dirac gauginos are a different example of where adding new fields can lead to improved naturalness properties [16]. The most natural region of parameter space has mS and M at the hundreds of GeV scale, to avoid large corrections to mHu , and large mS¯ ≳ 10 TeV, to maximize the second term of Eq. (7). The tree-level contribution to the Higgs mass can be large enough such that m~t takes a natural value at the hundreds of GeV scale. We have performed a quantitative study of the finetuning in the Dirac NMSSM, shown to the left of Fig. 1 as a function of (M, mS¯ ). We computed the radiative corrections from the top sector to the Higgs mass at RG-improved leading-log order, analogous to [17]. We have confirmed that our results match the FEYNHIGGS software [18], for the MSSM, within Δmh ≃ 1 GeV in the parameter regime of interest. We fix At ¼ 0 for simplicity, and other parameters are fixed according to the table, shown below. Here, we adopt a parameter μeff ≡ μ þ λhSi for convenience. We have chosen λ to saturate the upper limit such that it does not reach a Landau pole below the unification scale [19]. For each value of (M, mS¯ ), the top squark soft masses, m~t ¼ m~tR ¼ mQ~ 3 , are chosen to maintain the lightest scalar mass at 125 GeV. The degree of fine-tuning is estimated by dm2Hu dm2Hd 2 2 2 2 Δ ¼ 2 max mHu ; mHd ; L; L; δmH ; beff ; d ln μ d ln μ mh (11) ¯ where beff ¼ μB þ λðAλ hSi þ MhSiÞ and we take L ≡ logðΛ=m~t Þ ¼ 6, corresponding to low-scale SUSY breaking. We assume that contributions through gauge couplings to the RGs for m2Hu;d are subdominant. Benchmark Parameters λ ¼ 0.74 beff ¼ ð190 GeVÞ2 M ¼ 1 TeV

tan β ¼ 2 Aλ ¼ 0 mS¯ ¼ 10 TeV

μeff ¼ 150 GeV Bs ¼ 100 GeV mS ¼ 800 GeV

For comparison, the right of Fig. 1 shows the tuning in the NMSSM, which corresponds to the Dirac NMSSM replacing S¯ → S [which removes the Uð1ÞS¯ symmetry]. The superpotential of the NMSSM corresponds to Eq. (1), plus the tadpole cS μMS. We treat mS as a free parameter instead of mS¯ and use the same fine-tuning measure of Eq. (11), except the threshold correction δm2H is absent and beff ¼ μB þ λhSiðAλ þ MÞ. We see that the least-tuned region of the Dirac NMSSM corresponds to M ∼ 2 TeV and mS¯ ≳ 10 TeV, where the tree-level correction to the Higgs mass is maximized. The fine-tuning is dominated by δm2H in the large M region, and by the contribution of m~t to mHu in the rest of the plane. On the other hand, the NMSSM becomes highly tuned when mS is large (since it radiatively corrects mHu;d ), and then mS ≲ 1 TeV is favored. Note that the region of low tuning in the NMSSM extends to the supersymmetric limit, mS → 0. In this region the Higgs mass is increased by a new contribution to the quartic coupling proportional to λ2 ðMμ sin 2β − μ2 Þ=M2 (see Ref. [9] for more details). Higgs phenomenology.—We now discuss the experimental signatures of the Dirac NMSSM. The phenomenology of the NMSSM is well studied [2,20]. The natural region of the Dirac NMSSM differs from the NMSSM in that the singlet states are too heavy to be produced at the LHC. The low-energy Higgs phenomenology is that of a two Higgs doublet model, and we focus here on the nature of the SM-like Higgs h and the heavier doubletlike Higgs H [21]. The properties of the two doublets differ from the MSSM due to the presence of the nondecoupling quartic coupling jλH u Hd j2 , which raises the Higgs mass by the semisoft SUSY breaking, described above. We consider the potential with radiative corrections from the top squark sector and we find that the couplings of the SM-like Higgs boson to leptons and down-type quarks are lowered, while couplings to the up-type quarks are slightly increased compared to those in the SM, which results in deviations to the cross sections and decay patterns shown to the left of Fig. 2. These effects decouple in the limit mH ≫ mh , which corresponds to large beff . We also show, to the right of Fig. 2, the decay branching ratios of H.

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, BR BRSM

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gg

0.9 0.8 200

300

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C C

3

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mH GeV

FIG. 2 (color online). The branching ratios and production cross sections of the SM-like Higgs boson are shown, normalized to the SM values [22,23], on the left as a function of the heavy doubletlike Higgs mass, mH . On the right, we show several branching ratios of the heavy doubletlike Higgs boson as a function of its mass. Note that the location of the chargino and neutralino thresholds depend on the -ino spectrum. Here we take heavy gauginos and μeff ¼ 150 GeV.

Because of the nondecoupling term, di-Higgs decay, H → 2h, becomes the dominant decay once its threshold is opened, mH ≳ 250 GeV. There are now two relevant constraints on the Higgs sector of the Dirac NMSSM. The first comes from measurements of the couplings of the SM-like Higgs from the ATLAS [24,25] and CMS [26,27] Collaborations. The second comes from direct searches for the heavier state decaying to dibosons, H → ZZ; WW [25,27]. The former excludes mH ≲ 220 GeV at 95%, while the latter extends this limit to mH ∼ 260 GeV by the CMS Collaboration Future Reach from Coupling Measurements

5

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7TeV

20fb

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LHC, 300fb 1 at 14TeV LHC, 3000fb 1 at 14TeV ILC250 LHC ILC500 LHC ILC1000 up

4l

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H ZZ

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Higgs Couplings CMS ATLAS

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5fb

3 2 400

600

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FIG. 3 (color online). The shaded regions show current constraints on our model by the SM-like Higgs coupling measurements and direct searches of H. The curves show the expected Δχ 2 from combined measurements pﬃﬃﬃ of the Higgs-like couplings at the high-luminosity LHC at s ¼ 14 TeV [28] and pﬃﬃﬃ the ILC at s ¼ 250, 500, and 1000 GeV.The optimistic [29,30] (conservative [31]) ILC reach curves are solid (dashed) and neglect (include) theoretical uncertainties in the Higgs branching ratios. The ILC analyses include the expected LHC measurements. For comparison, we show the present limits and also the expected limit of the current ATLAS measurements (solid, black).

search for H → ZZ (except for a small gap near mH ≈ 235 GeV). We also estimate the future reach to probe mH with future Higgs coupling measurements at the LHC and ILC [28–31], as shown in Fig. 3. The increased sensitivity at the ILC is dominated by the improved measurements projected for the bb¯ and τþ τ− couplings [29,30]. Discussion.—The LHC has discovered a new particle, consistent with the Higgs boson, with a mass near 125 GeV.Weak-scale SUSY must be reevaluated in light of this discovery. Naturalness demands new dynamics beyond the minimal theory, such as a nondecoupling F term, but this implies new sources of SUSY breaking that themselves threaten naturalness. In this Letter, we have identified a new model where the Higgs couples to a singlet field with a Dirac mass. The nondecoupling F term is naturally realized through semisoft SUSY breaking, because large mS¯ helps raise the Higgs mass but does not threaten naturalness. The first collider signatures of the Dirac NMSSM are expected to be those of the MSSM fields, with the singlet sector naturally heavier than 1 TeV. The key feature of semisoft SUSY breaking in the Dirac NMSSM is that S¯ couples to the MSSM only through the dimensionful Dirac mass, M. We note that interactions between S¯ and other new states are not constrained by naturalness, even if these states experience SUSY breaking. Therefore, the Dirac NMSSM represents a new type of portal, whereby our sector can interact with new sectors, with large SUSY breaking, without spoiling naturalness in our sector. We thank Lawrence Hall, Matthew McCullough, Satyanarayan Mukhopadhyay, Tilman Plehn, Filippo Sala, Satoshi Shirai, and Neal Weiner for helpful discussions. We especially thank Yasunori Nomura for discussions and for pointing out that the Dirac mass can be thought of as a new type of portal. The work of H. M. was supported in part by the U.S. DOE under Contract No. DEAC03-76SF00098, by the NSF under Grant No. PHY-1002399, by the JSPS

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PHYSICAL REVIEW LETTERS

Grant (C) No. 23540289, by the FIRST program Subaru Measurements of Images and Redshifts (SuMIRe), CSTP, and by WPI, MEXT, Japan. J. T. R. is supported by a fellowship from the Miller Institute for Basic Research in Science. The work of K. T. is supported in part by the Grantin-Aid for JSPS Fellows.

*

[email protected] [email protected]; [email protected] ‡ [email protected] § [email protected] [1] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716, 1 (2012); S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 716, 30 (2012). [2] L. J. Hall, D. Pinner, and J. T. Ruderman, J. High Energy Phys. 04 (2012) 131. [3] J. D. Wells, arXiv:hep-ph/0306127; N. Arkani-Hamed and S. Dimopoulos, J. High Energy Phys. 06 (2005) 073. [4] S. Dimopoulos and G. F. Giudice, Phys. Lett. B 357, 573 (1995); A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Phys. Lett. B 388, 588 (1996); M. Papucci, J. T. Ruderman, and A. Weiler, J. High Energy Phys. 09 (2012) 035; C. Brust, A. Katz, S. Lawrence, and R. Sundrum, J. High Energy Phys. 03 (2012) 103. [5] T. J. LeCompte and S. P. Martin, Phys. Rev. D 85, 035023 (2012); H. Murayama, Y. Nomura, S. Shirai, and K. Tobioka, Phys. Rev. D 86, 115014 (2012). [6] J. Fan, M. Reece, and J. T. Ruderman, J. High Energy Phys. 11 (2011) 012; 07 (2012) 196; M. BaryakhtarN. Craig, and K. Van Tilburg, J. High Energy Phys. 07 (2012) 164. [7] For a review, R. Barbier et al., Phys. Rep. 420, 1 (2005); For recent studies, see, for example, C. Csaki, Y. Grossman, and B. Heidenreich, Phys. Rev. D 85, 095009 (2012); P. W. Graham, D. E. Kaplan, S. Rajendran, and P. Saraswat, J. High Energy Phys. 07 (2012) 149; J. T. Ruderman, T. R. Slatyer, and N. Weiner, J. High Energy Phys. 09 (2013) 094. [8] J. R. Espinosa and M. Quiros, Phys. Lett. B 279, 92 (1992); U. Ellwanger, C. Hugonie, and A. M. Teixeira, Phys. Rep. 496, 1 (2010). [9] Y. Nomura, D. Poland, and B. Tweedie, Phys. Lett. B 633, 573 (2006); A. Delgado, C. Kolda, J. P. Olson, and A. de la Puente, Phys. Rev. Lett. 105, 091802 (2010); G. G. Ross and K. Schmidt-Hoberg, Nucl. Phys. B862, 710 (2012). [10] R. Harnik, G. D. Kribs, D. T. Larson, and H. Murayama, Phys. Rev. D 70, 015002 (2004); R. Barbieri, L. J. Hall, Y. Nomura, and V. S. Rychkov, Phys. Rev. D 75, 035007 (2007); M. Dine, N. Seiberg, and S. Thomas, Phys. Rev. D 76, 095004 (2007). [11] P. Batra, A. Delgado, D. E. Kaplan, and T. M. P. Tait, J. High Energy Phys. 02 (2004) 043; A. Maloney, A. Pierce, and J. G. Wacker, J. High Energy Phys. 06 (2006) 034; C. Cheung and H. L. Roberts, J. High Energy Phys. 12 (2013) 018. [12] N. Arkani-Hamed, C. D. Carone, L. J. Hall, and H. Murayama, Phys. Rev. D 54, 7032 (1996). †

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[13] M. Leurer, Y. Nir, and N. Seiberg, Nucl. Phys. B420, 468 (1994). [14] For example, G. R. Dvali, G. F. Giudice, and A. Pomarol, Nucl. Phys. B478, 31 (1996). [15] H. Georgi, Annu. Rev. Nucl. Part. Sci. 43, 209 (1993). [16] P. J. Fox, A. E. Nelson, and N. Weiner, J. High Energy Phys. 08 (2002) 035. [17] H. E. Haber and R. Hempfling, Phys. Rev. D 48, 4280 (1993); M. S. Carena, J. R. Espinosa, M. Quiros, and C. E. M. Wagner, Phys. Lett. B 355, 209 (1995); H. E. Haber, R. Hempfling, and A. H. Hoang, Z. Phys. C 75, 539 (1997); M. S. Carena, J. R. Ellis, A. Pilaftsis, and C. E. M. Wagner, Nucl. Phys. B586, 92 (2000). [18] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein, J. High Energy Phys. 02 (2007) 047. [19] R. Barbieri, L. J. Hall, A. Y. Papaioannou, D. Pappadopulo, and V. S. Rychkov, J. High Energy Phys. 03 (2008) 005. [20] For recent studies, see, for example, U. Ellwanger, J. High Energy Phys. 03 (2012) 044; K. Agashe, Y. Cui, and R. Franceschini, J. High Energy Phys. 02 (2013) 031; R. T. D’Agnolo, E. Kuflik, and M. Zanetti, J. High Energy Phys. 03 (2013) 043; T. Gherghetta, B. von Harling, A. D. Medina, and M. A. Schmidt, J. High Energy Phys. 02 (2013) 032; C. Cheung, S. D. McDermott, and K. M. Zurek, J. High Energy Phys. 04 (2013) 074; R. Barbieri, D. Buttazzo, K. Kannike, F. Sala, and A. Tesi, Phys. Rev. D 87, 115018 (2013). [21] For recent studies, see, for example, N. Craig, J. A. Evans, R. Gray, C. Kilic, M. Park, S. Somalwar, and S. Thomas, J. High Energy Phys. 02 (2013) 033; N. Craig, J. Galloway, and S. Thomas, arXiv:1305.2424. [22] S. Dittmaier et al. (LHC Higgs Cross Section Working Group Collaboration), Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables (CERN, Geneva, 2011). [23] S. Dittmaier et al. (LHC Higgs Cross Section Working Group Collaboration), Handbook of LHC Higgs Cross Sections: 2. Differential Distributions (CERN, Geneva, 2012). [24] ATLAS Collaboration, Reports No. ATLAS-CONF-2013034; No. ATLAS-CONF-2013-012; No. ATLAS-CONF2013-030; No. ATLAS-CONF-2012-160. [25] ATLAS Collaboration, Reports No. ATLAS-CONF-2013013; No. ATLAS-CONF-2013-067. [26] CMS Collaboration, Reports No. CMS-PAS-HIG-13-001; No. CMS-PAS-HIG-13-004; No. CMS-PAS-HIG-12-044. [27] CMS Collaboration, Reports No. CMS-PAS-HIG-13-002; No. CMS-PAS-HIG-13-003. [28] ATLAS Collaboration, Report No. ATL-PHYS-PUB-2012004. [29] M. E. Peskin, arXiv:1207.2516. [30] http://www.snowmass2013.org/tiki‑index.php?page=The +Higgs+Boson HiggsSnowmassReport_Sep3.pdf. [31] M. Klute, R. Lafaye, T. Plehn, M. Rauch, and D. Zerwas, Europhys. Lett. 101, 51001 (2013).

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