Vacuum stability with the effective six-pointed couplings of the Higgs bosons in the heavy supersymmetry

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1 Vacuum stability wit te effective six-pointed couplings of te Higgs bosons in te eavy supersymmetry Mikail Dubinin 1,2, and Elena Petrova 1,2, 1 Skobeltsyn Institute of Nuclear Pysics, Lomonosov Moscow State University, Moscow, Russia 2 Pysics Department, Lomonosov Moscow State University, Moscow, Russia Abstract. Consistent and penomenologically successful scenarios of te Minimal Supersymmetric Standard Model (MSSM) include te Higgs potential wic is positively defined at large fields and as two minima, one of tem is true and stable vacuum in wic we live. However, te effective terms of te MSSM potential wic appear at a loop level can spoil te vacuum stability leading to a metastable or unstable vacuum configurations. Suc a situation is analysed and te general necessary conditions of vacuum stability in te MSSM parameter space are imposed. For a majority of te MSSM "bencmark scenarios" wic are used for reconstruction of production cannels at te LHC, te effective potential is found to be always stable, except ligt stop and τ-pobic scenarios. 1 Introduction Te discovery of te Higgs boson wit mass ±0.2 GeV at te LHC [1] confirmed te fundamental concept of symmetry breaking in te scalar sector. Te properties of a new scalar, e.g. its spin, CP parity, and coupling strengts to te SM particles, are consistent witin te statistical uncertainties wit predictions of te Standard Model (SM). However, statistical uncertainties of te signal strengt and significant signal strengt errors in te production cannels are not small and sould be reduced by te future analyses of ATLAS and CMS Collaborations at te energy 13 TeV. Combined results [2] for te Higgs boson production and decay rates as well as constraints on its couplings to te SM particles leave a room for meaningful contributions of pysics beyond te SM. Precise measurements of te Higgs mass and te top quark mass are critical to determine te vacuum structure of te SM [3] wic demonstrates a metastable vacuum. In te SM te scalar sector is single-field in te unitary gauge. In te SM extensions, te situation is more complicated because te Higgs sector includes several fields. 2 Te MSSM effective Higgs potential at te one-loop In te MSSM te Higgs sector consists of five Higgs bosons. In te CP-conserving limit, tey are two CP-even neutral Higgs bosons and H, one CP-odd neutral Higgs boson A and two carged scalars dubinin@teory.sinp.msu.ru petrova@teory.sinp.msu.ru Te Autors, publised by EDP Sciences. Tis is an open access article distributed under te terms of te Creative Commons Attribution License.0 (ttp://creativecommons.org/licenses/by/.0/).

2 H ± []. As a rule, te ligtest CP-even scalar is identified wit te observable Higgs boson. Large radiative corrections at loop level ensure te -boson mass of approximately 125 GeV. Te effective Higgs potential in te Coleman-Weinberg framework [5] expanded to all orders of perturbation teory is sown scematically in figure 1 and as a form U(1 loop) = U (2) + U () + U (6) +... (1) In te literature, it is usually supposed tat all iger order potential terms of effective operators more Figure 1. Te one-loop approximation for te effective potential in te Coleman-Weinberg framework [5]. ten dimension-four in te fields are negligibly small if te following conditions are respected [6] 2 m t A t < M 2 S, 2 m tµ < M 2 S, (2) were m t =173.2 GeV is te top quark mass, A t is te trilinear Higgs t-squark interaction parameter, µ is te Higgs superfield mass parameter and M S is te squark mass scale. Tis parametrization of te MSSM soft supersymmetry breaking sector [7] is te most common in te MSSM bencmark scenarios [8]. However, te dimension-six operators may play an important role in te A t,µparameter range about/of te order of 10 1 TeV and low and moderate M S [9]. Suc a situation is rater unusual in te most of MSSM scenarios but in general parameters A t,µmay ave so large values wic respect te perturbative unitarity conditions (see e.g. [10]). Potential terms of te two-doublet scalar sector can be written as follows were U (2) = µ 2 1 (Φ 1 Φ 1) µ 2 2 (Φ 2 Φ 2) [µ 2 12 (Φ 1 Φ 2) +.c.], (3) U () = λ 1 (Φ 1 Φ 1) 2 + λ 2 (Φ 2 Φ 2) 2 + λ 3 (Φ 1 Φ 1)(Φ 2 Φ 2) + λ (Φ 1 Φ 2)(Φ 2 Φ 1) + () + [λ 5 /2(Φ 1 Φ 2)(Φ 1 Φ 2) + λ 6 (Φ 1 Φ 1)(Φ 1 Φ 2) + λ 7 (Φ 2 Φ 2)(Φ 1 Φ 2) +.c.], U (6) = κ 1 (Φ 1 Φ 1) 3 + κ 2 (Φ 2 Φ 2) 3 + κ 3 (Φ 1 Φ 1) 2 (Φ 2 Φ 2) + κ (Φ 1 Φ 1)(Φ 2 Φ 2) 2 + (5) + κ 5 (Φ 1 Φ 1)(Φ 1 Φ 2)(Φ 2 Φ 1) + κ 6 (Φ 1 Φ 2)(Φ 2 Φ 1)(Φ 2 Φ 2) + + [κ 7 (Φ 1 Φ 2) 3 + κ 8 (Φ 1 Φ 1) 2 (Φ 1 Φ 2) + κ 9 (Φ 1 Φ 1)(Φ 1 Φ 2) κ 10 (Φ 1 Φ 2) 2 (Φ 2 Φ 2) + κ 11 (Φ 1 Φ 2) 2 (Φ 2 Φ 1) + κ 12 (Φ 1 Φ 2)(Φ 2 Φ 2) κ 13 (Φ 1 Φ 1)(Φ 1 Φ 2)(Φ 2 Φ 2) +.c.], Φ i = ( ) ( ϕ + i (x) ϕ 0 i (x) = iω + i 1 2 (v i + η i + iχ i ) ), i = 1, 2 (6) Higgs doublets wit te SU(2) field states and v 1 = v cos β, v 2 = v sin β (v = 26 GeV) vacuum expectation values of tem. Te real part of µ 2 12 is fixed by zero eigenvalue of te mass matrix (wic ensures massless Goldstone boson state and defines te CP-odd scalar mass m 2 A, see details in ref. [9]) Reµ 2 12 = s β c β [m 2 A + v2 2 (2Reλ 5 + Reλ 6 cot β + Reλ 7 tan β)] + v {Reκ 9 c 3 β s β (7) + Reκ 10 c β s 3 β + 1 [Reκ 8c β + Reκ 12s β + (9Reκ 7 + Reκ 11 + Reκ 13 )s 2 β c2 β ]}, 2

3 were s β = sin β, c β = cos β and so on. Tree level boundary conditions for Higgs self-couplings at te scale M S are [11] 1,2 (M S ) = g2 1 + g2 2, λ tree 3 (M S ) = g2 2 g2 1, λ tree (M S ) = g2 2 2, (8) λ tree 5,6,7 (M S ) = 0, κ1,...,13 tree (M S ) = 0, λ tree were g 1 and g 2 are te standard gauge couplings, wile at te loop-level and below M S scale tese couplings acquire radiative corrections λ 1,2 (M) = λ tree 1,2 (M S ) λ 1,2 (M)/2, λ 3,...,7 (M) = λ tree 3,...,7 (M S ) λ 3,...,7 (M), κ 1,...13 (M) = κ 1,...13 (M). Radiative corrections to parameters λ i (i=1,...7) in te effective field teory framework ave been analysed in ref. [6, 7, 12, 13]. Radiative corrections to te parameters κ i (i =1,...13) in te approximation of degenerate squark masses ave been obtained in ref. [9]. An example of te one-loop renormalization group improved (RG-improved) tresold correction for λ 1 and te tresold correction for κ 1 can be written as λtr 1 2 = + κ tr 1 = + 3 [ A b 2 32π 2 b 2 A b 2 µ MS 2 6MS 2 t + 2 6MS b l + g2 2 + g2 1 ( 2 MS 2 t µ 2 2 b A b 2 ) ] (9) 1 ( 11g 768π g 2 36 (g2 1 + g2 2 ) ) 2 b l, 6 b 2 3 A b 2 + A b A b 6 g g2 2 32MS 2π2 MS 2 MS 10MS 6 b 3 3 A b 2 + A b (10) 128MS 2π2 MS 2 2MS 2 ( ) b 5 3 g 1 + 2g2 1 g g 2 1 A b 2 6 µ 6 t 512M 2 S π2 + (g g2 2 ) µ t 2 t 256M 6 S π2 2M 2 S (17g 1 6g2 1 g g 2 ) µ M S π M 8 S π2 g M 2 S π2 (g 1 g 2 ), g 2 m t 2mW sin β and b = g 2 m b 2mW cos β ( M 2 were l ln S ), σ is te renormalization scale, σ 2 t = are Yukawa couplings. One can notice inspecting tese explicit forms tat radiative corrections κ tr begin to play an important role if te following conditions are true µ m t cot β M 2 S, µ m b tan β M 2 S, A t m t M 2 S, A b m b M 2 S, (11) µa t m 2 t cot β M S, µa b m 2 b tan β M S. In tis case, te corrections coming from te dimension-six operators of te one-loop potential can be appreciable even if te inequalities (2) are respected. 3 Vacuum stability wit respect to κ 1,...13 terms Te local minimum v =26 GeV of te Higgs potential exists if te following conditions are respected U U = 0, = 0, = U U ϕ 0 1 ϕ 0 i =v i ϕ 0 i =v i ϕ0 1 ϕ 0 1 ϕ0 2 > 0, U ϕ > 0. (12) 0 1 ϕ0 1 ϕ 0 ϕ 0 i =v i i =v i ϕ 0 1 ϕ 0 2 U ϕ 0 2 ϕ0 1 U ϕ 0 2 ϕ0 2 3

4 (a) (b) (c) Figure 2. Te allowed regions defined by Eq. (12) for m =125 GeV and (a) tan β=1, M S =1.5 TeV, (b) tan β=10, M S =1.5 TeV and (c) tan β=10, M S =5 TeV. Yellow and green areas correspond to an allowed parameter range for te Higgs potential U = U (2) + U () + U (6) ; yellow area corresponds to an allowed parameter range for te pottential U (2) + U () in te one-loop expansion of te effective Higgs potential. For example if κ i =0, ten = [ 2Reµ v2 (3λ 6 c 2 β + 3λ 7c 2 β + λ 35s 2β )] 2 (13) [v 2 ( λ 7 + 3λ 6 cot 2 β + λ 1 cot 3 β) + 2Reµ 2 12 csc2 β] [v 2 ( λ 2 3λ 7 cot β + λ 6 cot 3 β) 2Reµ 2 12 cot β csc2 β]s β tan β>0, were Reµ 2 12 is defined by eq. (7). Allowed domains on te plane (µ, A t) for fixed values of M S and tan β are sown in figure 2. Tey are reconstructed for te case wen te ligtest CP-even scalar mass m =125 GeV and masses of oter Higgs bosons are positively defined. Several examples of te mass spectrum of scalars can be found in [9]. Te combination of green and yellow regions includes an allowed A t and µ range if te Higgs potential terms U (2) + U () are extended by U (6) operators, eq. (5), and te yellow area represents te allowed region for U (2) + U () terms only. Visible deviations of te contours for te dimension-four potential are observed if te values of A t respect te conditions of eq. (11). Configuration for U (2) + U () + U (6) in te case of "good" (A t, µ) parameters for te effective potential decomposition is sown in figure 3 (a). Te potential as two minima and is positively defined at a large ϕ 0 i. So, in tis case, te vacuum in wic we live is true and stable. But if A t and µ are cosen from te yellow area of figure 2 (a), two minima degenerate to a gully wic forms a saddle configuration in te origin, see figure 3 (b). Te conditions defined by eq. (12) are not respected in tis case and corrections coming from dimension-six operators spoil te vacuum stability. In te case U ij =0, te gully forms a flat direction wen te stationary point at te origin is degenerate and nonisolated [1]. For te LHC analyses te MSSM multidimensional parameter space is reduced to specific bencmark parameter sets wic demonstrate different collider penomenology. Six bencmark sets wic are specified in [8] are usually denominated as m max, m mod+, m mod, ligt stop, ligt stau and τ-pobic scenarios. In tese bencmark scenarios M S < 1.5 TeV, µ<m S (except τ-pobic scenario) and A t = X t + µ/ tan β, were X t O(1)M S, so large values of A t were te difference between various radiative corrections is appreciable can be acieved only for extremely small tan β region (see eq. 11), were one can expect an evidence of additional contributions. Te minimum of te potential

5 (a) (b) Figure 3. Higgs potential forms, see (1), for M S =1.5 TeV, tan β=1 and (a) A t = µ=1.5 TeV, (b) A t =7.5 TeV, µ=2 TeV, see also figure 2 (a). Te plane U =0 is sown in green colour. disappears only at low tan β for ligt stop and τ-pobic scenarios wile oter demonstrate remarkable stability wit respect to radiative corrections. Summary Te problem of te SM vacuum stability looks less alarming in te MSSM extension. Te best enquiry would be if te beyond SM parameters can be cosen in suc a way tat te vacuum would be always stable. In te case of two-doublet Higgs sector, te potential as two identical minima, one of tem is a vacuum in wic we live. Te conditions defined by eq. (12) guarantee te stable vacuum, but modification of Higgs potential demonstrates tat some parameter sets respecting te minimization in te dimension-four version are not acceptable for te dimension-six extension. Instead of te stable minimum wit vev s one can ave a saddle configuration at te origin. New terms of dimensionsix in te fields are significant if eq. (11) are respected. In te majority of te MSSM bencmark scenarios te local minimum is stable wit respect to dimension-six terms, but for scenarios ligt stop and τ-pobic te low tan β region is excluded. Te autors are grateful to I. Volobuev for useful remarks. Tis work was supported by Grant No. NS References [1] G. Aad et al. (ATLAS Collaboration), Pys. Lett. B 716, 1(2012); S. Catrcyan et al. (CMS Collaboration), Pys. Lett. B 716, 30(2012) [2] Aad G. et al. (ATLAS and CMS Collaborations), JHEP 1608, 05 (2016); Aad G. et al. (ATLAS and CMS Collaborations), Pys. Rev. Lett. 11, (2015) [3] F. Bezrukov, M. Kalmykov, B. Kniel, and M. Saposnikov, JHEP 1210, 10 (2012); J. Elias- Miro et al, Pys. Lett. B 709, 222 (2012); J. Casas, J. Espinosa, and M. Quiros, Pys. Lett. B 382, 37 (1996); A. Linde, Pys. Lett. B 92, 209 (1980); H. D. Politzer and S. Wolfram, Pys. Lett. B 82, 22 (1979); ibid, B (1979); N. V. Krasnikov, Yad. Fiz., 28, 59 (1978); L. Maiani, G. Parisi, and R. Petronzio, Nucl. Pys. B 136, 115 (1978) 5

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