TheMinimal Supersymmetric extension of the Standard Model (MSSM) is obtained by elevating
theSM matter and gauge fields to super-fields and extending the Higgs mechanism. As hinted at in the previous section, the gives a pair of super-partner bosons corresponding to each of the SM
quarks and leptons (squarksandsleptons). The gauge vector fields are similarly extended to include fermiongauginos inheriting similar gauge interactions as theSMbosons. The Higgs SU(2) doublet of theSMmust however be extended to a Higgs sector, including both “up-type” and “down-type” complex scalar pairs. This is because the na¨ıve addition to theMSSMLagrangian involving a single Higgs superfield and its complex conjugate does not respect supersymmetry, so that separate fields are required to give masses to the up- and down-type particles. Table3.1summarizes the resulting super-field content of theMSSM.
The Lagrangian of theMSSMconsists of a Higgs super-potential added to the chiral and vector field kinematic terms. The super-potential consists of Yukawa couplings to the quarks and leptons as well as the “mu term” responsible for electroweak symmetry breaking
W= ¯uyuQHu−dy¯dQHd−ey¯ eLHd+µHuHd, (3.1)
where super-fields XS are written X and all suppressed indices are summed over. Here Yukawa
couplings are represented as matrices in generation-space so that information on particle masses as well as mixing matrices (i.e. CKM) are encoded. In addition to providing particle masses, these terms are responsible for mediating the t¯th interaction, stop decays via the t˜t˜h vertex, and stop
3. Supersymmetry 20
self-interactions. Each process is governed by the universal top Yukawa coupling, a fact critical to the cancellations providing theSUSYsolution of the hierarchy problem.
Each Higgs doublet separately attains a vacuum expectation value, denoted vu and vd, which
are usually re-expressed as the ratio of the two values tanβ =vu/vd constrained by the measured
value ofv2
SM=vu2+v2d. Similar to theSMcase, the Goldstone degrees of freedom give rise to gauge
boson masses with those remaining leading to physical Higgs bosons. However, the two complex SU(2) doublets correspond to eight degrees of freedom, leading to five new massive scalar bosons. These are two charged HiggsH±, two scalarsh0andH0 (withh0the lighter of the two, most often assumed to be theSMHiggs boson), and a pseudoscalarA0. The relation between the parameters of the Lagrangian and the physical masses may be found in Ref [49].
The remaining interactions added to the theory by enforcing SUSY are each proportional to the SM gauge couplings. For example, gluinos may be directly produced via the ggg˜˜g coupling arising from kineticQCDterms. Color factors associated with such vertices lead to largepp→˜g˜g∗1
cross sections, making these sparticles early targets of new, record-energy particle colliders. Kinetic terms for the chiral quark multiplets lead to t˜t˜g vertices (and similar) allowing gluinos to decay. The structure of the interactions of the electroweak bosons and their superpartners is similar, with
Table 3.1: The super-field content of the MSSM. All but generational indices are suppressed.
Type Super-field Particle content Sparticle content
Chiral
QS i
LH up-, down-type quarks LH up-, down-type squarks (e.g. tL, bL) (e.g. ˜tL,˜bL)
uS i
RH up-type quarks RH up-type squarks
(e.g. tR) (e.g. ˜tR)
dS i
RH down-type quarks RH down-type squarks
(e.g. bR) (e.g. ˜bR)
LS i
LH charged and neutral leptons LH charged and neutral sleptons (e.g. τL, νLτ) (e.g. ˜τL,ν˜Lτ)
eSi
RH charged leptons RH charged sleptons
(e.g. τR) (e.g. ˜τR)
HS u
Charged, neutral up-type Higgs Charged, neutral up-type Higgsinos (h0
u, h+u) (˜h0u,˜h+u)
HS d
Charged, neutral down-type Higgs Charged, neutral down-type Higgsinos (h0 d, h−d) (˜h0d,˜h−d) Vector gS Gluonsg Gluinos ˜g WS W bosonsW0,W± Winos ˜W0, ˜W± BS B boson B0 Bino ˜B0
1In this documentSUSYanti-particles will be denoted ˜s∗ in place of the usual ˜sconvention to achieve more
the complete set obtained from theSMgauge vertices, exchanging pairs of SMparticles with their supersymmetric partners.
However, asSUSYis apparently not perfectly realized in nature, the completely supersymmetric Lagrangian must be augmented by SUSY-breaking terms. In order to not re-introduce similar problems to the ones that we have previously advertisedSUSYas solving, only a subset of terms are considered, which break SUSYsoftly (do not introduce new quadratic divergences). The possible soft breaking terms include gaugino and scalar mass terms as well as (dimensionful) scalar trilinear self-couplings. Such soft-breaking parameters cannot be too close to the electroweak scale in order to evade experimental limits from a wide array of sparticle searches. However, pushing the the soft masses to too high a scale would produce a new hierarchy problem (the addition of soft stop mass produces a correctionδm2
h∼yt2m2˜t), motivating soft masses not far beyond the TeV-scale.
With the introduction of soft masses, the spectra of physical sparticle masses becomes more complicated, with the dependence on various parameters of the theory encoded in the mass matrices. In the case of top squarks, the is given by [50]:
M˜t= m 2 ˜ tL+m 2 t+ 12− 2 3sinθ 2 W m2 Zcos 2β mt(At−µ/tanβ) mt(At−µ/tanβ) m2˜tR+m 2 t+23sinθ 2 Wm2Zcos 2β (3.2)
in the basis of chiral states. Here m˜tL and m˜tR are the soft masses and At is the stop trilinear
coupling. Similar relations hold for the remaining fermions and can be simplified by neglecting first- and second-generation Yukawa couplings.
The case of neutralinos and charginos (oftenelectroweakinos) is similarly complex, with mixing among each of the Bino, Wino, and Higgsino states. The neutralino mass matrix is
Mχ0 =
M1 0 −cosβsinθWmZ sinβsinθWmZ
0 M2 cosβcosθWmZ −sinβcosθWmZ
−cosβsinθWmZ cosβcosθWmZ 0 −µ
sinβsinθWmZ −sinβcosθWmZ −µ 0
(3.3)
and the chargino matrix is more easily diagonalized to give eigenvalues [49]
m2χ± 1,2= 1 2 |M2|2+|µ|2+ 2m2W ∓ q (|M2|2+|µ|2+ 2m2W)2−4|µM2−m2W sin 2β|2 . (3.4) In the decoupling limit, the neutralino mass matrixMχ0 describes four mass eigenstates specified by three mass parameters. The Bino-like state and Wino-like states are controlled byM1 and M2, respectively, while two near-degenerate Higgsino-like states have masses near µ. In the case of a
3. Supersymmetry 22
HiggsinoLSPwheremZ µM1, M2, the mass of these states may be approximated as mH˜0
1,2=|µ|+ m2
Z(sgn(µ)∓sin 2β)(µ±M1cos2θW ±M2sin2θW)
2(µ±M1)(µ±M2) . (3.5)
This leads to the interesting prediction of two new neutral states with ∆M ≈m 2 Wtan2θW(±|µ|sin 2β+M1) (M2 1 − |µ|2) , (3.6)
along with an intermediate charged state (with mass given by Equation3.4) [51].
While theMSSMprovides a rich framework to understand the signatures ofSUSYthat one might expect to observe near the electroweak scale, in practice experimental results are not often inter- preted using this model. One the one hand, theMSSM Lagrangian contains many free parameters that have already been tightly constrained by low-energy physics. For example, off-diagonal mass terms relating the first two generations could lead to appreciableµ→eγ rates in the lepton sector andK0−K¯0 mixing in the quark sector. Given these bounds and the relative lack of sensitivity to the remaining phase space at high-energy colliders, it is reasonable to make simplifying assumptions when interpreting LHC results. On the other hand there is motivation to go beyond the MSSM, adding new Lagrangian terms that may result from explicit manifestations ofSUSY-breaking, ex- plain the neutrino mass hierarchy, and solve further theoretical problems. Both approaches will be explored in the following sections.