---
author:
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title: Higgs Physics
---

Introduction
============

A major goal of   particle physics program at  high energy frontier, currently being pursued at  CERN Large Hadron Collider (LHC), is to unravel  nature of electroweak symmetry breaking (EWSB). While  existence of  massive electroweak gauge bosons ($W^\pm,Z$), together with  successful description of their behavior by non-abelian gauge theory, requires some form of EWSB to be present in nature, the underlying dynamics remained unknown for several decades. An appealing theoretical suggestion for such dynamics is  Higgs mechanism [@higgs-mechanism], which implies  existence of one or more Higgs bosons (depending on  specific model considered). Therefore,  search for a Higgs boson was considered a major cornerstone in  physics program of  LHC.

The spectacular discovery of a Higgs-like particle with a mass around $\MH \simeq \simMH \gev$, which has been announced by ATLAS [@ATLASdiscovery]  CMS [@CMSdiscovery], marks a milestone of an effort that has been ongoing for almost half a century and opens up a new era of particle physics. Both ATLAS  CMS reported a clear excess in  two photon channel, as well as in  $ZZ^{(*)}$ channel.  discovery was further corroborated, though not with high significance, by the $WW^{(*)}$ channel  by  final Tevatron results [@TevHiggsfinal]. Latest ATLAS/CMS results can be found in .

Many theoretical models employing  Higgs mechanism in order to account for electroweak symmetry breaking have been studied in  literature, of which the most popular ones are  Standard Model (SM) [@sm] and  Minimal Supersymmetric Standard Model (MSSM) [@mssm]. The newly discovered particle can be interpreted as  SM Higgs boson. The MSSM has a richer Higgs sector, containing two neutral $\cp$-even, one neutral $\cp$-odd  two charged Higgs bosons. The newly discovered particle can also be interpreted as  light (or the the heavy) $\cp$-even state [@Mh125]. Among alternative theoretical models beyond  SM   MSSM, the most prominent are the Two Higgs Doublet Model (THDM) [@thdm], non-minimal supersymmetric extensions of  SM (e.g. extensions of  MSSM by an extra singlet superfield [@NMSSM-etc]), little Higgs models [@lhm]   models with more than three spatial dimensions [@edm].

We will discuss  Higgs boson sector in  SM   MSSM. This includes their agreement with  recently discovered particle around $\sim \simMH \gev$    searches for  supersymmetric (SUSY) Higgs bosons at  LHC. While the LHC, after  discovery of a Higgs-like boson, will be able to measure some of its properties, a “cleaner” experimental environment, such as at  ILC, will be needed to measure all  Higgs boson characteristics [@lhcilc; @lhc2fc; @lhc2tsp].

The SM   Higgs
==============

Higgs: Why  How?
----------------

We start with looking at one of  most simple Lagrangians,  one of QED: $$\begin{aligned}
\cL_{\rm QED} &= -\ed{4} F_{\mu\nu} F^{\mu\nu} 
                 + \bar\psi (i \ga^\mu D_\mu - m) \psi~.\end{aligned}$$ Here $D_\mu$ denotes  covariant derivative $$\begin{aligned}
D_\mu &= \partial_\mu + i\,e\,A_\mu~.\end{aligned}$$ $\psi$ is  electron spinor,  $A_\mu$ is  photon vector field.  QED Lagrangian is invariant under  local $U(1)$ gauge symmetry, $$\begin{aligned}
\psi &\to e^{-i\al(x)}\psi~, \\
A_\mu &\to A_\mu + \ed{e} \partial_\mu \al(x)~.
\label{gaugeA}\end{aligned}$$ Introducing a mass term for  photon, $$\begin{aligned}
\cL_{\rm photon~mass} &= \edz m_A^2 A_\mu A^\mu~,\end{aligned}$$ however, is not gauge-invariant. Applying yields $$\begin{aligned}
\edz m_A^2 A_\mu A^\mu &\to \edz m_A^2 \KKL
A_\mu A^\mu + \frac{2}{e} A^\mu \partial_\mu \al 
+ \ed{e^2} \partial_\mu \al \, \partial^\mu \al \KKR~.\end{aligned}$$

A way out is  Higgs mechanism [@higgs-mechanism]. The simplest implementation uses one elementary complex scalar Higgs field $\Phi$ that has a vacuum expectation value $v$ (vev) that is constant in space  time. The Lagrangian of  new Higgs field reads $$\begin{aligned}
\cL_\Phi &= \cL_{\Phi, {\rm kin}} + \cL_{\Phi, {\rm pot}}\end{aligned}$$ with $$\begin{aligned}
\cL_{\Phi, {\rm kin}} &= (D_\mu \Phi)^* \, (D^\mu \Phi)~, \\
-\cL_{\Phi, {\rm pot}} &= V(\Phi) = \mu^2 |\Phi|^2 + \la |\Phi|^4~.\end{aligned}$$ Here $\la$ has to be chosen positive to have a potential bounded from below. $\mu^2$ can be either positive or negative, where we will see that $\mu^2 < 0$ yields  desired vev, as will be shown below. The complex scalar field $\Phi$ can be parametrized by two real scalar fields $\phi$ and $\eta$, $$\begin{aligned}
\Phi(x) &= \ed{\wz} \phi(x) e^{i \eta(x)}~,\end{aligned}$$ yielding $$\begin{aligned}
%V(\Phi) &\sim 
V(\phi) &= \frac{\mu^2}{2} \phi^2 + \frac{\la}{4} \phi^4~.\end{aligned}$$ Minimizing  potential one finds $$\begin{aligned}
\frac{{\rm d}V}{{\rm d}\phi}_{\big| \phi = \phi_0} &=
\mu^2 \phi_0 + \la \phi_0^3 \stackrel{!}{=} 0~.\end{aligned}$$ Only for $\mu^2 < 0$ this yields  desired non-trivial solution $$\begin{aligned}
\phi_0 &= \sqrt{\frac{-\mu^2}{\la}} \KL = \langle \phi \rangle =: v \KR~.\end{aligned}$$ The picture simplifies more by going to  “unitary gauge”, $\al(x) = -\eta(x)/v$, which yields a real-valued $\Phi$ everywhere. The kinetic term now reads $$\begin{aligned}
(D_\mu \Phi)^* \, (D^\mu \Phi) &\to 
\edz (\partial_\mu \phi)^2 + \edz e^2 q^2 \phi^2 A_\mu A^\mu~,
\label{LphiA}\end{aligned}$$ where $q$ is  charge of  Higgs field, which can now be expanded around its vev, $$\begin{aligned}
\phi(x) &= v \; + \; H(x)~.
\label{vH}\end{aligned}$$ The remaining degree of freedom, $H(x)$, is a real scalar boson, the Higgs boson.  Higgs boson mass  self-interactions are obtained by inserting into  Lagrangian (neglecting a constant term), $$\begin{aligned}
-\cL_{\rm Higgs} &= \edz \mH^2 H^2 + \frac{\kappa}{3!} H^3
                  + \frac{\xi}{4!} H^4~,\end{aligned}$$ with $$\begin{aligned}
\mH^2 = 2 \la v^2, \quad
\kappa = 3 \frac{\mH^2}{v}, \quad
\xi = 3 \frac{\mH^2}{v^2}~.\end{aligned}$$ Similarly, can be inserted in , yielding (neglecting the kinetic term for $\phi$), $$\begin{aligned}
\cL_{\rm Higgs-photon} &= \edz m_A^2 A_\mu A^\mu + e^2 q^2 v H A_\mu A^\mu
+ \edz e^2 q^2 H^2 A_\mu A^\mu\end{aligned}$$ where  second  third term describe  interaction between the photon  one or two Higgs bosons, respectively,   first term is the photon mass, $$\begin{aligned}
\mA^2 &= e^2 q^2 v^2~.
\label{mA}\end{aligned}$$ Another important feature can be observed:  coupling of  photon to the Higgs is proportional to its own mass squared.

Similarly a gauge invariant Lagrangian can be defined to give mass to the chiral fermion $\psi = (\psi_L, \psi_R)^T$, $$\begin{aligned}
\cL_{\rm fermion~mass} &= y_\psi \psi_L^\dagger \, \Phi \, \psi_R + {\rm c.c.}~,\end{aligned}$$ where $y_\psi$ denotes  dimensionless Yukawa coupling. Inserting $\Phi(x) = (v + H(x))/\wz$ one finds $$\begin{aligned}
\cL_{\rm fermion~mass} &= m_\psi \psi_L^\dagger \psi_R 
                     + \frac{m_\psi}{v} H\, \psi_L^\dagger \psi_R 
                     + {\rm c.c.}~,\end{aligned}$$ with $$\begin{aligned}
m_\psi &= y_\psi \frac{v}{\wz}~.\end{aligned}$$ Again  important feature can be observed: by construction the coupling of  fermion to  Higgs boson is proportional to its own mass $m_\psi$.

The “creation” of a mass term can be viewed from a different angle (see also ). The interaction of  gauge field or  fermion field with the scalar background field, i.e.  vev, shifts  masses of these fields from zero to non-zero values. This is shown graphically in for  gauge boson (a)   fermion (b) field.

(60,50)(80,40) (0,25)(50,25)[3]{}[6]{} (55,25)(70,25) (-15,30)[$V$]{} (-15,50)[$(a)$]{}

(60,10)(40,40) (0,25)(50,25)[3]{}[6]{} (75,30)[$+$]{}

(60,10)(15,40) (0,25)(50,25)[3]{}[6]{} (25,25)(12,50)[3]{} (25,25)(38,50)[3]{} (9,53)(15,47) (9,47)(15,53) (35,53)(41,47) (35,47)(41,53) (15,80)[$v$]{} (50,80)[$v$]{} (75,30)[$+$]{}

(60,10)(-10,40) (0,25)(75,25)[3]{}[9]{} (20,25)(8,50)[3]{} (20,25)(32,50)[3]{} (55,25)(43,50)[3]{} (55,25)(67,50)[3]{} (5,53)(11,47) (5,47)(11,53) (29,53)(35,47) (29,47)(35,53) (40,53)(46,47) (40,47)(46,53) (64,53)(70,47) (64,47)(70,53) (08,80)[$v$]{} (43,80)[$v$]{} (57,80)[$v$]{} (92,80)[$v$]{} (110,30)[$+ \cdots$]{}

\

(60,80)(80,20) (0,25)(50,25) (55,25)(70,25) (-15,32)[$f$]{} (-15,50)[$(b)$]{}

(60,80)(40,20) (0,25)(50,25) (75,30)[$+$]{}

(60,80)(15,20) (0,25)(25,25) (25,25)(50,25) (25,25)(25,50)[3]{} (22,53)(28,47) (22,47)(28,53) (43,70)[$v$]{} (75,30)[$+$]{}

(60,80)(-10,20) (0,25)(25,25) (25,25)(50,25) (50,25)(75,25) (25,25)(25,50)[3]{} (50,25)(50,50)[3]{} (22,53)(28,47) (22,47)(28,53) (47,53)(53,47) (47,47)(53,53) (43,70)[$v$]{} (78,70)[$v$]{} (110,30)[$+ \cdots$]{}

\

The shift in  propagators reads (with $p$ being  external momentum and $g = e q$ in ): $$\begin{aligned}
&(a) \; &\ed{p^2} \; \to \; \ed{p^2} + 
    \sum_{k=1}^{\infty} \ed{p^2} \KKL \KL \frac{g v}{2} \KR \ed{p^2} \KKR^k
    &= \ed{p^2 - m_V^2} %\quad
    {\rm ~with~} m_V^2 = g^2 \frac{v^2}{4} ~, \\
&(b) \; &\ed{\pslash} \; \to \; \ed{\pslash} + 
    \sum_{k=1}^{\infty} \ed{\pslash} \KKL \KL \frac{y_\psi v}{2} \KR 
                                        \ed{\pslash} \KKR^k
    &= \ed{\pslash - m_\psi} %\quad
    {\rm ~with~} m_\psi = y_\psi \frac{v}{\wz} ~.\end{aligned}$$

SM Higgs Theory
---------------

We now turn to  electroweak sector of  SM, which is described by the gauge symmetry $SU(2)_L \times U(1)_Y$.  bosonic part of the Lagrangian is given by $$\begin{aligned}
\cL_{\rm bos} &= -\ed{4} B_{\mu\nu} B^{\mu\nu} 
- \ed{4} W_{\mu\nu}^a W^{\mu\nu}_a
+ |D_\mu \Phi|^2 - V(\Phi), \\
V(\Phi) &= \mu^2 |\Phi|^2 + \la |\Phi|^4~.\end{aligned}$$ $\Phi$ is a complex scalar doublet with charges $(2, 1)$ under  SM gauge groups, $$\begin{aligned}
\Phi &= \VL \phi^+ \\ \phi^0 \VR~,\end{aligned}$$ and  electric charge is given by $Q = T^3 + \edz Y$, where $T^3$  third component of  weak isospin. We furthermore have $$\begin{aligned}
D_\mu &= \partial_\mu + i g \frac{\tau^a}{2} W_{\mu\,a} 
                     + i g^\prime \frac{Y}{2} B_\mu ~, \\
B_{\mu\nu} &= \partial_\mu B_\nu - \partial_\nu B_\mu ~, \\
W_{\mu\nu}^a &= \partial_\mu W_\nu^a - \partial_\nu W_\mu^a 
               - g f^{abc} W_{\mu\,b} W_{\nu\,c}~.\end{aligned}$$ $g$  $g^\prime$ are  $SU(2)_L$  $U(1)_Y$ gauge couplings, respectively, $\tau^a$ are  Pauli matrices,  $f^{abc}$ are the $SU(2)$ structure constants.

Choosing $\mu^2 < 0$  minimum of  Higgs potential is found at $$\begin{aligned}
\langle \Phi \rangle &= \ed{\wz} \VL 0 \\ v \VR 
\quad {\rm with} \quad 
v:= \sqrt{\frac{-\mu^2}{\la}}~.\end{aligned}$$ $\Phi(x)$ can now be expressed through  vev,  Higgs boson and three Goldstone bosons $\phi_{1,2,3}$, $$\begin{aligned}
\Phi(x) &= \ed{\wz} \VL \phi_1(x) + i \phi_2(x) \\ 
                        v + H(x) + i \phi_3(x) \VR~.\end{aligned}$$ Diagonalizing  mass matrices of  gauge bosons, one finds that the three massless Goldstone bosons are absorbed as longitudinal components of  three massive gauge bosons, $W_\mu^\pm, Z_\mu$, while the photon $A_\mu$ remains massless, $$\begin{aligned}
W_\mu^\pm &= \ed{\wz} \KL W_\mu^1 \mp i W_\mu^2 \KR ~,\\
Z_\mu &= \cw W_\mu^3 - \sw B_\mu ~,\\
A_\mu &= \sw W_\mu^3 + \cw B_\mu ~.\end{aligned}$$ Here we have introduced  weak mixing angle $\theta_W = \arctan(g^\prime/g)$,  $\sw := \sin \theta_W$, $\cw := \cos \theta_W$.  Higgs-gauge boson interaction Lagrangian reads, $$\begin{aligned}
\cL_{\rm Higgs-gauge} &= \KKL \MW^2 W_\mu^+ W^{-\,\mu} 
                           + \edz \MZ^2 Z_\mu Z^\mu \KKR 
                      \KL 1 + \frac{H}{v} \KR^2 \non \\
&\quad - \edz \MH^2 H^2 - \frac{\kappa}{3!} H^3 - \frac{\xi}{4!} H^4~,\end{aligned}$$ with $$\begin{aligned}
\MW &= \edz g v, \quad
\MZ =  \edz \sqrt{g^2 + g^{\prime 2}} \; v, \\
(\MHSM := ) \; \MH &= \sqrt{2 \la}\; v, \quad
\kappa = 3 \frac{\MH^2}{v}, \quad
\xi = 3 \frac{\MH^2}{v^2}~.\end{aligned}$$ From  measurement of  gauge boson masses  couplings one finds $v \approx 246 \gev$. Furthermore  two massive gauge boson masses are related via $$\begin{aligned}
\frac{\MW}{\MZ} &= \frac{g}{\sqrt{g^2 + g^{\prime 2}}} \; = \; \cw~.\end{aligned}$$

We now turn to  fermion masses, where we take  top- and bottom-quark masses as a representative example.  Higgs-fermion interaction Lagrangian reads $$\begin{aligned}
\label{SMfmass}
\cL_{\rm Higgs-fermion} &= y_b Q_L^\dagger \, \Phi \, b_R \; + \;
                        y_t Q_L^\dagger \, \Phi_c \, t_R + {\rm ~h.c.}\end{aligned}$$ $Q_L = (t_L, b_L)^T$ is  left-handed $SU(2)_L$ doublet. Going to the “unitary gauge”  Higgs field can be expressed as $$\begin{aligned}
\Phi(x) &= \ed{\wz} \VL 0 \\ v + H(x) \VR~,
\label{SMPhi}\end{aligned}$$ and it is obvious that this doublet can give masses only to the bottom(-type) fermion(s). A way out is  definition of $$\begin{aligned}
\Phi_c &= i \si^2 \Phi^* \; = \; \ed{\wz} \VL v + H(x) \\ 0 \VR~,
\label{SMPhic}\end{aligned}$$ which is employed to generate  top(-type) mass(es) in . Inserting , (\[SMPhic\]) into yields $$\begin{aligned}
\cL_{\rm Higgs-fermion} &= \mb \bar b b \KL 1 + \frac{H}{v} \KR
                      + \mt \bar t t \KL 1 + \frac{H}{v} \KR\end{aligned}$$ where we have used $\bar \psi \psi = \psi_L^\dagger \psi_R + \psi_R^\dagger \psi_L$ and $\mb = y_b v/\wz$, $\mt = y_t v/\wz$.

The mass of  SM Higgs boson, $\MHSM$, is in principle a free parameter in  model. However, it is possible to derive bounds on $\MHSM$ derived from theoretical considerations [@RGEla1; @RGEla2; @RGEla3]  from experimental precision data [@lepewwg; @gfitter].

Evaluating loop diagrams as shown in  middle  right of yields  renormalization group equation (RGE) for $\la$, $$\begin{aligned}
\frac{{\rm d}\la}{{\rm d} t} &=
\frac{3}{8 \pi^2} \KKL \la^2 + \la y_t^2 - y_t^4 
     + \ed{16} \KL 2 g^4 + (g^2 + g^{\prime 2})^2 \KR \KKR~,
\label{RGEla}\end{aligned}$$ with $t = \log(Q^2/v^2)$, where $Q$ is  energy scale.

(90,80)(60,-10) (0,50)(25,25)[3]{} (0,0)(25,25)[3]{} (50,50)(25,25)[3]{} (50,0)(25,25)[3]{} (-15,65)[$H$]{} (-15,-5)[$H$]{} (70,-5)[$H$]{} (70,65)[$H$]{} (30,45)[$\la$]{}

(90,80)(10,-10) (0,50)(25,25)[3]{} (0,0)(25,25)[3]{} (75,50)(50,25)[3]{} (75,0)(50,25)[3]{} (37.5,25)(12.5,0,360)[3]{} (-15,65)[$H$]{} (-15,-5)[$H$]{} (47,57)[$H$]{} (110,-5)[$H$]{} (110,65)[$H$]{}

(50,80)(-40,2.5) (0,0)(25,25)[3]{} (0,75)(25,50)[3]{} (50,50)(75,75)[3]{} (50,25)(75,0)[3]{} (25,25)(50,25) (50,25)(50,50) (50,50)(25,50) (25,50)(25,25) (-15,100)[$H$]{} (-15,-5)[$H$]{} (53,77)[$t$]{} (110,-5)[$H$]{} (110,100)[$H$]{}

\

For large $\MH^2 \propto \la$ reduces to $$\begin{aligned}
\frac{{\rm d} \la}{{\rm d} t} &= \frac{3}{8 \pi^2} \la^2 \quad
\Rightarrow \quad \la(Q^2) = \frac{\la(v^2)}
            {1 - \frac{3 \la(v^2)}{8 \pi^2} \log \KL \frac{Q^2}{v^2} \KR}~.\end{aligned}$$ For $\frac{3 \la(v^2)}{8 \pi^2} \log \KL \frac{Q^2}{v^2} \KR = 1$ one finds that $\la$ diverges (it runs into  “Landau pole”). Requiring $\la(\La) < \infty$ yields an upper bound on $\MH^2$ depending up to which scale $\La$ the Landau pole should be avoided, $$\begin{aligned}
\la(\La) < \infty \quad \Rightarrow \quad
\MH^2 \le \frac{8 \pi^2 v^2}{3 \log \KL \frac{\La^2}{v^2} \KR}~.
\label{MHup}\end{aligned}$$

For small $\MH^2 \propto \la$, on  other hand, reduces to $$\begin{aligned}
\frac{{\rm d} \la}{{\rm d} t} &= \frac{3}{8 \pi^2} 
\KKL -y_t^4 + \ed{16} \KL 2 g^4 + (g^2 + g^{\prime 2})^2 \KR \KKR \\
\Rightarrow \quad \la(Q^2) &= \la(v^2) \frac{3}{8 \pi^2}
\KKL -y_t^4 + \ed{16} \KL 2 g^4 + (g^2 + g^{\prime 2})^2 \KR \KKR
\log\KL\frac{Q^2}{v^2}\KR~.\end{aligned}$$ Demanding $V(v) < V(0)$, corresponding to $\la(\La) > 0$ one finds a lower bound on $\MH^2$ depending on $\La$, $$\begin{aligned}
\la(\La) > 0 \quad \Rightarrow \quad
\MH^2 \; > \; \frac{v^2}{4 \pi^2}
\KKL  -y_t^4 + \ed{16} \KL 2 g^4 + (g^2 + g^{\prime 2})^2 \KR \KKR
\log\KL\frac{\La^2}{v^2}\KR~.
\label{MHlow}\end{aligned}$$

The combination of  upper bound in   lower bound in on $\MH$ is shown in . Requiring  validity of  SM up to  GUT scale yields a limit on the SM Higgs boson mass of $130 \gev \lsim \MHSM \lsim 180 \gev$.

![image](higgs_sm_bounds1){height="6cm"} ![image](higgs_sm_bounds2){height="5cm"}

Predictions for a SM Higgs-boson at  LHC {#sec:SMHiggs}
----------------------------------------

In order to efficiently search for  SM Higgs boson at  LHC precise predictions for  production cross sections   decay branching ratios are necessary. To provide most up-to-date predictions in 2010 the “LHC Higgs Cross Section Working Group” [@lhchxswg] was founded. Two of  main results are shown in , see for an extensive list of references.  left plot shows  SM theory predictions for  main production cross sections, where  colored bands indicate  theoretical uncertainties. The right plot shows  branching ratios (BRs), again with  colored band indicating  theory uncertainty (see for more details). Results of this type are constantly updated  refined by  Working Group.

![image](Higgs_XS_8TeV_LM200){width=".45\textwidth"}   ![image](Higgs_BR_LM){width=".45\textwidth" height=".45\textwidth"}

Discovery of an SM Higgs-like particle at  LHC {#sec:SMHiggsLHC}
----------------------------------------------

On 4th of July 2012 both ATLAS [@ATLASdiscovery] and CMS [@CMSdiscovery] announced  discovery of a new boson with a mass of $\sim 125.5 \gev$. This discovery marks a milestone of an effort that has been ongoing for almost half a century and opens up a new era of particle physics. In one can see  $p_0$ values of  search for the SM Higgs boson (with all search channels combined) as presented by ATLAS (left)  CMS (right) in July 2012.  $p_0$ value gives the probability that  experimental results observed can be caused by background only, i.e. in this case assuming  absense of a Higgs boson at each given mass. While  $p_0$ values are close to $\sim 0.5$ for nearly all hypothetical Higgs boson masses (as would be expected for the absense of a Higgs boson), both experiments show a very low $p_0$ value of $p_0 \sim 10^{-6}$ around $\MH \sim 125.5 \gev$. This corresponds to a discovery of a new particle at  $5\,\si$ level by each experiment individually.

Another step in  analysis is a comparison of  measurement of production cross sectinos times branching ratios with  respective SM prediction. Two examples, using LHC data of about $5\,\ifb$ at $7 \tev$  about $20\,\ifb$ at $8 \tev$ are shown in . Here ATLAS [@ATLAS_5p20ifb] (left) and CMS [@CMS-Higgs-WWW] (right) compare their experimental results with  SM prediction in various channels. It can be seen that all channels are, within  theoretical  experimental uncertainties, in agreement with the SM.

![image](CombinedResults-ATLAS-040712){width=".48\textwidth" height=".45\textwidth"}   ![image](Combined-CMS-040712){width=".45\textwidth"}

![image](ATLAS_5p20ifb_allchannels){width=".45\textwidth" height=".45\textwidth"}   ![image](CMS_5p20ifb_allchannels){width=".45\textwidth"}

In this discussion it must be kept in mind that a measurement of the total width  thus of individual couplings is not possible at the LHC (see, e.g.,  references therein). In  SM, for a fixed value of $\MH$, all Higgs couplings to other (SM) particles are specifed. Consequently, it is in general not possible to perform a fit to experimenal data within  SM, where  Higgs couplings are treated as free parameters. Therefore, in order to test  compatibility of  predictions for the SM Higgs boson with  (2012) experimental data,  LHC Higgs Cross Section Working Group proposed several benchmark scenarios for “coupling scale factors” [@HiggsRecommendation; @YR3] (see for a recent review on Higgs coupling extractions). Effectively,   predicted SM Higgs cross sections  partial decay widths are dressed with scale factors $\kappa_i$ (and $\kappa_i=1$ corresponds to  SM). Several assumptions are made for this $\kappa$-framework: there is only one state at $\simMH \gev$ responsible for  signal,  coupling structure is  same as for  SM Higgs (i.e. it is a $\cp$-even scalar),   zero width approximation is assumed to be valid, allowing for a clear separation  simple handling of production and decay of  Higgs particle. The most relevant coupling strength modifiers are $\kappa_t$, $\kappa_b$, $\kappa_\tau$, $\kappa_W$, $\kappa_Z$, $\kappa_\gamma$, $\kappa_g$, …[^1]

One limitation at  LHC (but not at  ILC) is  fact that the total width cannot be determined experimentally without additional theory assumptions. In  absence of a total width measurement only ratios of $\kappa$’s can be determined from experimental data. An assumption often made is $\kappa_{W,Z} \le 1$ [@Duhrssen:2004cv]. A recent analysis from CMS using  Higgs decays to $ZZ$ far off-shell yielded an upper limit on  total width about four times larger than the SM width [@CMS:2014ala]. However, here  assumption of the equality of on-shell  off-shell couplings of  Higgs boson plays a crucial role. It was pointed out that this equality is violated in particular in  presense of new physics in  Higgs sector [@Englert:2014aca].

In  left plot of we compare  results estimated for  HL-LHC (with $3 \iab$  an assumed improvement of 50% in  theoretical uncertainties) with  various stages of  ILC under  theory assumption $\kappa_{W,Z} \le 1$ [@HiggsCouplings]. This most general fit includes $\kappa_{W,Z}$ for  gauge bosons, $\kappa_{u,d,l}$ for up-type quarks, down-type quarks  charged leptons, respectively, as well as $\kappa_\gamma$  $\kappa_g$ for the loop-induced couplings of  Higgs to photons  gluons. Also the (possibly invisible) branching ratio of  Higgs boson to new physics ($\br(H \to {\rm NP})$) is included. One can observe that  HL-LHC and  ILC250 yield comparable results. However, going to higher ILC energies, yields substantially higher precisions in  fit for the coupling scale factors. In  final stage of  ILC (ILC1000 LumiUp), precisions at  per-mille level in $\kappa_{W,Z}$ are possible.  $1-2\%$ range is reached for all other $\kappa$’s. The branching ratio to new physics can be restricted to  per-mille level.

Using ILC data  theory assumption $\kappa_{W,Z} \le 1$ can be dropped, since  “$Z$-recoil method” (see and references therein) allows for a model independent determination of  $HZZ$ coupling. The corresponding results are shown in  right plot of , where  HL-LHC results are combined with  various stages of  ILC.  results from  HL-LHC alone continue to very large values of  $\kappa$’s, since  fit cannot be done without theory assumptions. Including  ILC measurements (where  first line corresponds to the inclusion of [*only*]{}  $\si_{ZH}^{\rm total}$ measurement at the ILC) yields a converging fit. In  final ILC stage $\kappa_{W,Z}$ are determined to better than one per-cent, whereas  other coupling scale factors are obtained in  $1-2\%$ range.  branching ratio to new physics is restricted to be smaller than one per-cent.

![image](HLLHC_vs_ILC_KVle1){width="45.00000%"} ![image](HLLHC_ILC_mod_indep_withLHConly){width="45.00000%"}

The Higgs in Supersymmetry
==========================

Why SUSY?
---------

Theories based on Supersymmetry (SUSY) [@mssm] are widely considered as  theoretically most appealing extension of  SM. They are consistent with  approximate unification of  gauge coupling constants at  GUT scale and provide a way to cancel  quadratic divergences in  Higgs sector hence stabilizing  huge hierarchy between  GUT   electroweak (EW) scale. Furthermore, in SUSY theories  breaking of  electroweak symmetry is naturally induced at  EW scale,   lightest supersymmetric particle can be neutral, weakly interacting and absolutely stable, providing therefore a natural solution for  dark matter problem.

The Minimal Supersymmetric Standard Model (MSSM) constitutes, hence its name,  minimal supersymmetric extension of the SM.  number of SUSY generators is $N=1$,  smallest possible value. In order to keep anomaly cancellation, contrary to  SM a second Higgs doublet is needed [@glawei]. All SM multiplets, including  two Higgs doublets, are extended to supersymmetric multiplets, resulting in scalar partners for quarks and leptons (“squarks”  “sleptons”)  fermionic partners for the SM gauge boson   Higgs bosons (“gauginos”, “higgsinos” and “gluinos”). So far,  direct search for SUSY particles has not been successful. One can only set lower bounds of to on their masses [@SUSYMoriond14; @SUSYMoriond14-QCD].

The MSSM Higgs sector
---------------------

An excellent review on this subject is given in .

### The Higgs boson sector at tree-level {#sec:Higgstree}

Contrary to  SM, in  MSSM two Higgs doublets are required. The Higgs potential [@hhg] $$\begin{aligned}
V &= m_{1}^2 |\cHe|^2 + m_{2}^2 |\cHz|^2 
      - m_{12}^2 (\epsilon_{ab} \cHe^a\cHz^b + \mbox{h.c.})  \non \\
  &  + \frac{1}{8}(g^2+g^{\prime 2}) \left[ |\cHe|^2 - |\cHz|^2 \right]^2
        + \frac{1}{2} g^2|\cHe^{\dag} \cHz|^2~,
\label{higgspot}\end{aligned}$$ contains $m_1, m_2, m_{12}$ as soft SUSY breaking parameters; $g, g'$ are as before  $SU(2)$  $U(1)$ gauge couplings,   $\epsilon_{12} = -1$.

The doublet fields $\cHe$  $\cHz$ are decomposed in  following way: $$\begin{aligned}
\cHe &= \VL \cHe^0 \\[0.5ex] \cHe^- \VR \; = \; \VL v_1 
        + \frac{1}{\sqrt2}(\phi_1^0 - i\chi_1^0) \\[0.5ex] -\phi_1^- \VR~,  
        \non \\
\cHz &= \VL \cHz^+ \\[0.5ex] \cHz^0 \VR \; = \; \VL \phi_2^+ \\[0.5ex] 
        v_2 + \frac{1}{\sqrt2}(\phi_2^0 + i\chi_2^0) \VR~.
\label{higgsfeldunrot}\end{aligned}$$ $\cHe$ gives mass to  down-type fermions, while $\cHz$ gives masses to  up-type fermions. The potential (\[higgspot\]) can be described with  help of two independent parameters (besides $g$  $g'$): $\Tb = v_2/v_1$  $M_A^2 = -m_{12}^2(\Tb+\CTb)$, where $M_A$ is  mass of  $\cp$-odd Higgs boson $A$.

Which values can be expected for $\tb$? One natural choice would be $\tb \approx 1$, i.e. both vevs are about  same. On  other hand, one can argue that $v_2$ is responsible for  top quark mass, while $v_1$ gives rise to  bottom quark mass. Assuming that their mass differences comes largely from  vevs, while their Yukawa couplings could be about  same.  natural value for $\tb$ would then be $\tb \approx \mt/\mb$. Consequently, one can expect $$\begin{aligned}
\label{tbrange}
1 \lsim \tb \lsim 50~.\end{aligned}$$

The diagonalization of  bilinear part of  Higgs potential, i.e. of  Higgs mass matrices, is performed via  orthogonal transformations $$\begin{aligned}
\label{hHdiag}
\VL H^0 \\[0.5ex] h^0 \VR &= \ML \Ca & \Sa \\[0.5ex] -\Sa & \Ca \MR 
\VL \phi_1^0 \\[0.5ex] \phi_2^0~, \VR  \\
\label{AGdiag}
\VL G^0 \\[0.5ex] A^0 \VR &= \ML \Cb & \Sbe \\[0.5ex] -\Sbe & \Cb \MR 
\VL \chi_1^0 \\[0.5ex] \chi_2^0 \VR~,  \\
\label{Hpmdiag}
\VL G^{\pm} \\[0.5ex] H^{\pm} \VR &= \ML \Cb & \Sbe \\[0.5ex] -\Sbe & 
\Cb \MR \VL \phi_1^{\pm} \\[0.5ex] \phi_2^{\pm} \VR~.\end{aligned}$$ The mixing angle $\al$ is determined through $$\begin{aligned}
\al = {\rm arctan}\KKL 
  \frac{-(\MA^2 + \MZ^2) \Sbe \Cb}
       {\MZ^2 \CQb + \MA^2 \SQb - m^2_{h,{\rm tree}}} \KKR~, ~~
 -\frac{\pi}{2} < \al < 0
\label{alphaborn}\end{aligned}$$ with $m_{h, {\rm tree}}$ defined below in .\
One gets  following Higgs spectrum: $$\begin{aligned}
\mbox{2 neutral bosons},\, {\cal CP} = +1 &: h, H \non \\
\mbox{1 neutral boson},\, {\cal CP} = -1  &: A \non \\
\mbox{2 charged bosons}                   &: H^+, H^- \non \\
\mbox{3 unphysical Goldstone bosons}      &: G, G^+, G^- .\end{aligned}$$

At tree level  mass matrix of  neutral $\cp$-even Higgs bosons is given in  $\Pe$-$\Pz$-basis in terms of $\MZ$, $\MA$,  $\Tb$ by $$\begin{aligned}
M_{\rm Higgs}^{2, {\rm tree}} &= \ML \mpe^2 & \mpez^2 \\ 
                           \mpez^2 & \mpz^2 \MR 
= \ML \MA^2 \SQb + \MZ^2 \CQb & -(\MA^2 + \MZ^2) \Sbe \Cb \\
    -(\MA^2 + \MZ^2) \Sbe \Cb & \MA^2 \CQb + \MZ^2 \SQb \MR,
\label{higgsmassmatrixtree}\end{aligned}$$ which by diagonalization according to yields the tree-level Higgs boson masses $$\begin{aligned}
M_{\rm Higgs}^{2, {\rm tree}} 
   \stackrel{\al}{\longrightarrow}
   \ML m_{H,{\rm tree}}^2 & 0 \\ 0 &  m_{h,{\rm tree}}^2 \MR\end{aligned}$$ with $$\begin{aligned}
m_{H,h, {\rm tree}}^2 &= 
\edz \KKL \MA^2 + \MZ^2
         \pm \sqrt{(\MA^2 + \MZ^2)^2 - 4 \MZ^2 \MA^2 \CQZb} \KKR ~.
\label{mhtree}\end{aligned}$$ From this formula  famous tree-level bound $$\begin{aligned}
m_{h, {\rm tree}} \le \mbox{min}\{\MA, \MZ\} \cdot |\CZb| \le \MZ\end{aligned}$$ can be obtained. The charged Higgs boson mass is given by $$\begin{aligned}
\label{rMSSM:mHp}
\mHp^2 = \MA^2 + \MW^2~.\end{aligned}$$ The masses of  gauge bosons are given in analogy to  SM: $$\begin{aligned}
M_W^2 = \frac{1}{2} g^2 (v_1^2+v_2^2) ;\qquad
M_Z^2 = \frac{1}{2}(g^2+g^{\prime 2})(v_1^2+v_2^2) ;\qquad M_\gamma=0.\end{aligned}$$

The couplings of  Higgs bosons are modified from  corresponding SM couplings already at  tree-level. Some examples are $$\begin{aligned}
g_{hVV} &= \sin(\be - \al) \; g_{HVV}^{\rm SM}, \quad V = W^{\pm}, Z~, \\
g_{HVV} &= \cos(\be - \al) \; g_{HVV}^{\rm SM} ~,\\
g_{h b\bar b}, g_{h \tau^+\tau^-} &= - \frac{\sin\al}{\cos\be} \; 
                         g_{H b\bar b, H \tau^+\tau^-}^{\rm SM} ~, \\
g_{h t\bar t} &= \frac{\cos\al}{\sin\be} \; g_{H t\bar t}^{\rm SM} ~, \\
g_{A b\bar b}, g_{A \tau^+\tau^-} &= \ga_5\tb \; 
             g_{H b\bar b, H \tau^+\tau^-}^{\rm SM}~.\end{aligned}$$ The following can be observed:  couplings of  $\cp$-even Higgs boson to SM gauge bosons is always suppressed with respect to  SM coupling. However, if $g_{hVV}^2$ is close to zero, $g_{HVV}^2$ is close to $(g_{HVV}^{\rm SM})^2$  vice versa, i.e. it is not possible to decouple $\cp$-even Higgs bosons from  SM gauge bosons. The coupling of  $h$ to down-type fermions can be suppressed [*or enhanced*]{} with respect to  SM value, depending on  size of $\Sa/\Cb$. Especially for not too large values of $\MA$  large $\tb$ one finds $|\Sa/\Cb| \gg 1$, leading to a strong enhancement of this coupling.  same holds, in principle, for  coupling of  $h$ to up-type fermions. However, for large parts of  MSSM parameter space the additional factor is found to be $|\Ca/\Sbe| < 1$. For  $\cp$-odd Higgs boson an additional factor $\tb$ is found. According to this can lead to a strongly enhanced coupling of the $A$ boson to bottom quarks or $\tau$ leptons, resulting in new search strategies at  LHC for  $\cp$-odd Higgs boson, see below.

For $\MA \gsim 150 \gev$  “decoupling limit” is reached. The couplings of  light Higgs boson become SM-like, i.e.  additional factors approach 1.  couplings of  heavy neutral Higgs bosons become similar, $g_{Axx} \approx g_{Hxx}$,   masses of  heavy neutral  charged Higgs bosons fulfill $\MA \approx \MH \approx \MHp$. As a consequence, search strategies for  $A$ boson can also be applied to  $H$ boson,  both are hard to disentangle at hadron colliders (see also below).

### The scalar quark sector {#sec:squark}

Since  most relevant squarks for  MSSM Higgs boson sector are the $\Stop$ and $\Sbot$ particles, here we explicitly list their mass matrices in  basis of  gauge eigenstates $\StopL, \StopR$  $\SbotL, \SbotR$: $$\begin{aligned}
\label{stopmassmatrix}
{\cal M}^2_{\Stop} &=
  \ML \MstL^2 + \mt^2 + \CZb (\edz - \frac{2}{3} \sw^2) \MZ^2 &
      \mt \Xt \\
      \mt \Xt &
      \MstR^2 + \mt^2 + \frac{2}{3} \CZb \sw^2 \MZ^2 
  \MR, \\
& \non \\
\label{sbotmassmatrix}
{\cal M}^2_{\Sbot} &=
  \ML \MsbL^2 + \mb^2 + \CZb (-\edz + \frac{1}{3} \sw^2) \MZ^2 &
      \mb \Xb \\
      \mb \Xb &
      \MsbR^2 + \mb^2 - \frac{1}{3} \CZb \sw^2 \MZ^2 
  \MR.\end{aligned}$$ $\MstL$, $\MstR$, $\MsbL$  $\MsbR$ are  (diagonal) soft SUSY-breaking parameters. We furthermore have $$\begin{aligned}
\mt \Xt = \mt (\At - \mu \CTb) , \quad
\mb\, \Xb = \mb\, (\Ab - \mu \Tb) .
\label{eq:Xtb}\end{aligned}$$ The soft SUSY-breaking parameters $\At$  $\Ab$ denote  trilinear Higgs–stop  Higgs–sbottom coupling, and $\mu$ is  Higgs mixing parameter. $SU(2)$ gauge invariance requires  relation $$\begin{aligned}
\MstL = \MsbL .\end{aligned}$$ Diagonalizing ${\cal M}^2_{\Stop}$  ${\cal M}^2_{\Sbot}$ with the mixing angles $\tst$  $\tsb$, respectively, yields  physical $\Stop$ and $\Sbot$ masses: $\mste$, $\mstz$, $\msbe$  $\msbz$.

### Higher-order corrections to Higgs boson masses

A review about this subject can be found in . In  Feynman diagrammatic (FD) approach  higher-order corrected $\cp$-even Higgs boson masses in  MSSM are derived by finding the poles of  $(h,H)$-propagator matrix.  inverse of this matrix is given by $$\left(\Delta_{\rm Higgs}\right)^{-1}
= - i \ML p^2 -  \mHtree^2 + \hSi_{HH}(p^2) &  \hSi_{hH}(p^2) \\
     \hSi_{hH}(p^2) & p^2 -  \mhtree^2 + \hSi_{hh}(p^2) \MR~.
\label{higgsmassmatrixnondiag}$$ Determining  poles of  matrix $\Delta_{\rm Higgs}$ in is equivalent to solving the equation $$\left[p^2 - \mhtree^2 + \hSi_{hh}(p^2) \right]
\left[p^2 - \mHtree^2 + \hSi_{HH}(p^2) \right] -
\left[\hSi_{hH}(p^2)\right]^2 = 0\,.
\label{eq:proppole}$$ The very leading one-loop correction to $\Mh^2$ is given by $$\begin{aligned}
\De\Mh^2 &\sim \GF \mt^4 \log\KL\frac{\mste \mstz}{\mt^2}\KR~,
\label{DeltaMhmt4}\end{aligned}$$ where $\GF$ denotes  Fermi constant. shows two important aspects: First,  leading loop corrections go with $\mt^4$, which is a “very large number”. Consequently,  loop corrections can strongly affect $\Mh$  push  mass beyond  reach of LEP [@LEPHiggsSM; @LEPHiggsMSSM]  into  mass regime of  newly discovered boson at $\sim 125.5 \gev$. Second,  scalar fermion masses (in this case  scalar top masses) appear in  log entering  loop corrections (acting as a “cut-off” where  new physics enter). In this way  light Higgs boson mass depends on all other sectors via loop corrections. This dependence is particularly pronounced for  scalar top sector due to  large mass of  top quark  can be used to constrain  masses  mixings in  scalar top sector [@Mh125], see below.

The status of  available results for  self-energy contributions to can be summarized as follows. The complete one-loop result within  MSSM is known [@ERZ; @mhiggsf1lA; @mhiggsf1lB; @mhiggsf1lC]. The by far dominant one-loop contribution is   term due to top  stop loops ($\alt \equiv h_t^2 / (4 \pi)$, $h_t$ being the top-quark Yukawa coupling).  computation of  two-loop corrections has meanwhile reached a stage where all  presumably dominant contributions are available [@mhiggsletter; @mhiggslong; @mhiggsFD2; @bse; @mhiggsEP0; @mhiggsEP1; @mhiggsEP1b; @mhiggsEP2; @mhiggsEP3; @mhiggsEP4; @mhiggsEP4b; @mhiggsRG1; @mhiggsRG1a]. In particular,   contributions to  self-energies – evaluated in the FD as well as in  effective potential (EP) method – as well as  , ,   contributions – evaluated in  EP approach – are known for vanishing external momenta. An evaluation of  momentum dependence at  two-loop level in a pure  calculation was presented in . A (nearly) full two-loop EP calculation, including even  leading three-loop corrections, has also been published [@mhiggsEP5].  calculation presented in Ref. [@mhiggsEP5] is not publicly available as a computer code for Higgs-mass calculations. Subsequently, another leading three-loop calculation of , depending on  various SUSY mass hierarchies, has been performed [@mhiggsFD3l], resulting in  code [H3m]{} (which adds  three-loop corrections to  [FeynHiggs]{} result). Most recently, a combination of  full one-loop result, supplemented with leading  subleading two-loop corrections evaluated in the Feynman-diagrammatic/effective potential method  a resummation of the leading  subleading logarithmic corrections from  scalar-top sector has been published [@Mh-logresum] in  latest version of the code  [@feynhiggs; @mhiggslong; @mhiggsAEC; @mhcMSSMlong; @Mh-logresum] (also including  leading $p^2$ dependend two-loop corrections [@mhiggs2Lp2]). While previous to this combination   remaining theoretical uncertainty on  lightest $\cp$-even Higgs boson mass had been estimated to be about $3 \gev$ [@mhiggsAEC; @PomssmRep],  combined result was roughly estimated to yield an uncertainty of about $2 \gev$ [@Mh-logresum; @ehowp]; however, more detailed analyses will be necessary to yield a more solid result. Taking  available loop corrections into account,  upper limit of $\Mh$ is shifted to [@mhiggsAEC], $$\begin{aligned}
\label{Mh135}
\Mh \le 135 \gev~\end{aligned}$$ (as obtained with  code [FeynHiggs]{} [@feynhiggs; @mhiggslong; @mhiggsAEC; @mhcMSSMlong; @Mh-logresum]). This limit takes into account  experimental uncertainty for  top quark mass as well as  intrinsic uncertainties from unknown higher-order corrections. Consequently, a Higgs boson with a mass of $\sim 125.5 \gev$ can naturally be explained by  MSSM.

The charged Higgs boson mass is obtained by solving  equation $$\begin{aligned}
\label{rMSSM:mHpHO}
p^2 - \mHp^2 - \ser{H^-H^+}(p^2) = 0~.\end{aligned}$$ For  charged Higgs boson self-energy  full one-loop corrections are known [@chargedmhiggs; @markusPhD] as well as leading two-loop corrections at  [@chargedmhiggs2L].

MSSM Higgs bosons at  LHC {#sec:MSSMHiggsLHC}
-------------------------

The “decoupling limit” has been discussed above for  tree-level couplings  masses of  MSSM Higgs bosons. This limit also persists taking into account radiative corrections.  corresponding Higgs boson masses are shown in for $\tb = 5$ in   benchmark scenario [@benchmark2; @benchmark4] obtained with [FeynHiggs]{}. For $\MA \gsim 180 \gev$  lightest Higgs boson mass approaches its upper limit (depending on  SUSY parameters),   heavy Higgs boson masses are nearly degenerate. Furthermore, also  light Higgs boson couplings including loop corrections approach their SM-value for. Consequently, for $\MA \gsim 180
\gev$ an SM-like Higgs boson (below $\sim 135 \gev$) can naturally be explained by  MSSM. On  other hand, deviations from a SM-like behavior can be described in  MSSM by deviating from  full decoupling limit.

![image](decoupling){width=".99\textwidth"}

$\phantom{0}$

An example for the various productions cross sections at  LHC is shown in (for $\sqrt{s} = 14 \tev$). For low masses the light Higgs cross sections are visible,  for $\MH \gsim 130 \gev$ the heavy $\cp$-even Higgs cross section is displayed, while  cross sections for  $\cp$-odd $A$ boson are given for  whole mass range. As discussed above  $g_{Abb}$ coupling is enhanced by $\tb$ with respect to  corresponding SM value. Consequently,  $b\bar b A$ cross section is  largest or second largest cross section for all $\MA$, despite   relatively small value of $\tb = 5$. For larger $\tb$, see , this cross section can become even more dominant. Furthermore,  coupling of the heavy $\cp$-even Higgs boson becomes very similar to  one of the $A$ boson,   two production cross sections, $b \bar b A$   $b \bar b H$ are indistinguishable in  plot for $\MA > 200 \gev$.

![image](mhmax_tb05_LHC.cl.eps){width=".85\textwidth" height="8cm"}

More precise results in  most important channels, $gg \to \phi$  $b \bar b \to \phi$ ($\phi = h, H, A$) have been obtained by  LHC Higgs Cross Section Working Group [@lhchxswg], see also  references therein. Most recently a new code, [SusHi]{} [@sushi] for  $gg \to \phi$ (and $bb\phi$) production mode(s) including  full MSSM one-loop contributions as well as higher-order SM  MSSM corrections has been presented, see for more details.

Of particular interest is  “LHC wedge” region, i.e.the region in which only  light $\cp$-even MSSM Higgs boson, but none of  heavy MSSM Higgs bosons can be detected at  LHC. It appears for $\MA \gsim 200 \gev$ at intermediate $\tb$  widens to larger $\tb$ values for larger $\MA$. Consequently, in  “LHC wedge” only a SM-like light Higgs boson can be discovered at  LHC,  part of the LHC wedge (depending on  explicit choice of SUSY parameters) can be in agreement with $\Mh \sim 125.5 \gev$. This region, bounded from above by the 95% CL exclusion contours for  heavy neutral MSSM Higgs bosons can be seen in  [@CMS-HAtautau].

![image](CMS_5p20ifb_MSSM_Phitautau){width=".99\textwidth"}

$\phantom{0}$

Agreement of  MSSM Higgs sector with a Higgs at [$\sim 125.5 \gev$]{} {#sec:125MSSM}
---------------------------------------------------------------------

Many investigations have been performed analyzing  agreement of the MSSM with a Higgs boson at $\sim 125.5 \gev$. In a first step only the mass information can be used to test  model, while in a second step also  rate information of  various Higgs search channels can be taken into account. Here we briefly discuss  results in two of the new benchmark scenarios [@benchmark4], devised for  search for heavy MSSM Higgs bosons. In  left plot of the $\mhmax$ scenario is shown.  red area is excluded by LHC searches for the heavy MSSM Higgs bosons,  blue area is excluded by LEP Higgs searches,   light shaded red area is excluded by LHC searches for a SM-like Higgs boson.  bounds have been obtained with [HiggsBounds]{} [@higgsbounds] (where an extensive list of original references can be found).  green area yields $\Mh = 125 \pm 3 \gev$, i.e.  region allowed by  experimental data, taking into account  theoretical uncertainty in  $\Mh$ calculation as discussed above. Since   scenario maximizes  light $\cp$-even Higgs boson mass it is possible to extract lower (one parameter) limits on $\MA$  $\tb$ from  edges of  green band. By choosing  parameters entering via radiative corrections such that those corrections yield a maximum upward shift to $\Mh$,  lower bounds on $\MA$  $\tb$ that can be obtained are general in  sense that they (approximately) hold for [*any*]{} values of  other parameters. To address  (small) residual $\msusy (:= \MstL = \MstR = \MsbR)$ dependence of  lower bounds on $\MA$  $\tb$, limits have been extracted for  three different values $\msusy=\{0.5, 1, 2\}\tev$, see  [@Mh125]. For comparison also the previous limits derived from  LEP Higgs searches [@LEPHiggsMSSM] are shown, i.e.  before  incorporation of  Higgs discovery at  LHC. The bounds on $\MA$ translate directly into lower limits on $\MHp$, which are also given in  table. More recent experimental Higgs exclusion bounds shift these limits to even higher values, see  left plot in . Consequently,  experimental result of $\Mh \sim \simMH \pm 3 \gev$ requires $\MHp \gsim \mt$ with important consequences for  charged Higgs boson phenomenology.

In  right plot of we show  $\mh^{\rm mod+}$ scenario that differs from   scenario in  choice of $\Xt$. While in   scenario $\Xt/\msusy = +2$ had been chosen to maximize $\Mh$, in  $\mh^{\rm mod+}$ scenario $\Xt/\msusy = +1.5$ is used to yield a “good” $\Mh$ value over  nearly  entire $\MA$-$\tb$ plane, which is visible as  extended green region.

![image](benchmark_402-hb400-det){width=".48\textwidth"} ![image](benchmark_414-hb400){width=".48\textwidth"}

  ---------------- ------- ------------- -------------- ------- ------------- --------------
                                                                              
   $\msusy$ (GeV)   $\tb$   $\MA$ (GeV)   $\MHp$ (GeV)   $\tb$   $\MA$ (GeV)   $\MHp$ (GeV)
        500         $2.7$      $95$          $123$       $4.5$      $140$         $161$
        1000        $2.2$      $95$          $123$       $3.2$      $133$         $155$
        2000        $2.0$      $95$          $123$       $2.9$      $130$         $152$
  ---------------- ------- ------------- -------------- ------- ------------- --------------

![image](h_Xt_mstop1){width=".48\textwidth"} ![image](h_mstop1_mstop2){width=".48\textwidth"}

It is also possible to investigate what can be inferred from   assumed Higgs signal about  higher-order corrections in  Higgs sector. In a scan over seven relevant MSSM parameters has been performed: $\MA$, $\tb$, $\mu$, $\msusy$, $M_{\tilde l_3}$ (the soft SUSY-breaking parameter for  third generation of scalar leptons), $A_f$ (a “universal” trilinear coupling),  $M_2$ (the soft SUSY-breaking parameter for gauginos).  measurement of  Higgs boson mass as well as (the then current) results for Higgs boson production  decay rates were taken into account. In we show  results in the $\Xt/\msusy$-$\mste$ plane (left, with $\msusy \equiv M_{\tilde q_3}$) and in  $\mste$-$\mstz$ plane (right). Blue (gray) points were accepted (rejected) by [HiggsBounds]{}. Red (yellow) points have $\De\chi^2 \le 2.30 (5.99)$, thus they constitute  “favored” part of  parameter space. One can see that values of $\Xt/\msusy \approx +2$ are preferred.  light scalar top mass can be as low as $\sim 200 \gev$, while  heavier scalar top mass starts at $\sim 650 \gev$. A clear correlation between  two masses can be observed (where  scan stopped around $\msusy \sim 1.5 \tev$). While no absolute value for any of  stop masses can be obtained, very light masses are in agreement with $\Mh \sim \simMH$, with interesting prospects for  LHC   ILC.

### Acknowledgments {#acknowledgments .unnumbered}

I thank  organizers for creating a stimulating environment  for their hospitality, in particular when  hotel bar ran out of beer.

[99.]{}

[^1]: We do not discuss here  triple Higgs coupling, see for a recent review.
