Supersymmetry breaking

Supersymmetry breaking occurs when the action or vacuum is not invariant under the supercharges. Explicit breaking destroys the exact superalgebra, while spontaneous breaking preserves the action but gives a vacuum not annihilated by every supercharge and produces a Goldstino.

Dynamical supersymmetry breaking occurs when quantum dynamics produces a vacuum that does not preserve supersymmetry. An asymptotically free hidden sector can generate the displayed exponentially small scale through dimensional transmutation. This supplies a possible origin of a hierarchy, but strong coupling by itself does not prove breaking: the vacuum equations and dynamics must exclude a supersymmetric vacuum.

// Target: supersymmetry.bigb

With one canonical ungauged chiral superfield and $\kappa \ne 0$, the auxiliary expectation is nonzero at every scalar value. The potential is flat and positive, so supersymmetry is broken without an isolated minimum. The fermion is a massless Goldstino and the scalar is massless at tree level. Equality of these zero masses does not imply an unbroken supersymmetric vacuum.

Explicit breaking changes the action so that the former supercharges are not conserved symmetries of the full Hamiltonian. Spontaneous breaking preserves the action and algebra but has a vacuum that some supercharge does not annihilate. Energy positivity in global supersymmetry then implies positive vacuum energy density, and a Goldstino accompanies the broken symmetry. That exact-algebra energy argument does not apply to an arbitrarily explicitly broken Hamiltonian. The physical particle spectrum in a broken vacuum need not assemble into the finite equal-mass supermultiplets of an invariant vacuum.

A hidden sector contains fields whose auxiliary fields acquire expectation values that break supersymmetry. Loop effects or suppressed interactions communicate the breaking to visible fields as effective soft supersymmetry breaking. This can avoid the canonical visible-sector tree-level supertrace mass sum rule.

A goldstino is the massless spin-one-half mode associated with spontaneously broken global supersymmetry. For F-term breaking, the fermion in a multiplet with nonzero auxiliary field transforms by an inhomogeneous term, $\delta G\supset \sqrt{2}ϵF$. In supergravity, the corresponding mode supplies the longitudinal degrees of freedom of a massive gravitino.

For canonical global four-dimensional N=1 supersymmetry, define $F^i=\overline{W_i}$, $H_{ij}=W_{ij}$ and $G_{ia}=g^a{\bar{\varphi }}_j(T^a{)}^j{}_i$ at a stationary vacuum. Gauge invariance of the superpotential gives $G^{T}F=0$, while stationarity of $V={\sum }_i|W_i{|}^2+\frac{1}{2}{\sum }_a(D^a{)}^2$ gives $HF-GD=0$. Therefore the canonically normalized fermion mass matrix, after a harmless gaugino phase choice, obeysIf supersymmetry is broken, this vector is nonzero and specifies a massless Goldstino. The matrix with off-diagonal blocks $-G$ instead acts on $(F,D)$; it is related to the canonical matrix by a change of normalization, not by equality of canonical entries. Additional zero modes can coexist with the Goldstino.

Stationarity of a canonical global F-term scalar potential gives ${\partial }_{i}V={\sum }_j{W}_{ij}\overline{W_j}=0$. If some $F_j=-\overline{W_j}$ is nonzero, this vector is in the kernel of the chiral-superfield fermion mass matrix. It supplies the massless goldstino direction. If every $F_j$ vanishes, this argument gives no broken-symmetry mode.

Soft supersymmetry breaking adds operators of positive mass dimension that break supersymmetry without reintroducing new ultraviolet quadratic divergences.

Soft supersymmetry breaking removes quadratic ultraviolet sensitivity but can leave scalar-mass corrections proportional to the soft squared masses and logarithms of scale ratios. The high-momentum difference of a paired scalar and fermion propagator falls as their squared-mass splitting divided by momentum to the fourth power. Heavy soft partners can therefore still create a tuning problem through logarithmic and finite thresholds even though the original quadratic hierarchy problem has been controlled.

A sparticle is the supersymmetric partner of a Standard Model particle. It differs in spin by one half and has the opposite R-parity.

In the MSSM, electroweak symmetry breaking mixes the neutral bino, wino and two neutral higgsinos into four neutral Majorana spinor mass eigenstates, the neutralinos. A neutralino can be the lightest supersymmetric particle; with conserved R-parity it then provides a stable possible dark matter candidate.

The lightest supersymmetric particle is the least massive sparticle in a given spectrum. Exact R-parity makes it stable: a lighter final state cannot contain another odd particle, while a final state of ordinary particles is even. A neutral weakly interacting lightest supersymmetric particle is a possible dark matter constituent, but its cosmological abundance must be computed separately.

A slepton is a scalar sparticle partner of a lepton. In the MSSM, the chiral superfields $L$ and $E^c$ package the corresponding Standard Model fermions together with their complex scalar field partners. Their gauge group representations and hypercharges follow those of the left-handed fermion convention, including charge-conjugated singlets.

A higgsino is the Weyl spinor partner of a Higgs scalar field in a chiral superfield. After a supersymmetric Higgs mechanism, its combination along a broken gauge direction mixes with the gaugino in the resulting massive N=1 vector multiplet.

A gaugino is the spin-one-half superpartner of a gauge boson and transforms in the adjoint representation of its gauge group. Gaugino loops modify gauge-coupling beta functions above the supersymmetry-breaking threshold.

The bino is the spin-one-half gaugino of the hypercharge $U(1{)}_Y$ gauge group. Its adjoint representation is the singlet $(1,1,0)$, so it has no hypercharge of its own. After electroweak symmetry breaking, it can mix with the neutral wino and higgsinos into neutralinos.

A wino is a spin-one-half gaugino of the weak $SU(2{)}_L$ gauge group. Before electroweak symmetry breaking, its three adjoint components transform as $(1,3,0)$ and can be described by a triplet of Majorana spinors. After symmetry breaking, the neutral component can mix with the bino and neutral higgsinos into neutralinos.

The gluino is the spin-one-half sparticle paired with a gluon. In the MSSM, the eight adjoint color components form a Majorana spinor in $(8,1,0)$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Its gauge representation is real, so the opposite-helicity antiparticle states are already included in the adjoint multiplet.

A squark is the spin-zero supersymmetric partner of a quark. It carries the same colour, weak-isospin and hypercharge quantum numbers as the corresponding quark chirality.

The top squark, or stop, is the scalar superpartner of the top quark. Its mass controls the leading residual top-sector correction to the Higgs mass after soft supersymmetry breaking.

An O'Raifeartaigh model is a theory of chiral superfields whose F-term equations cannot all vanish. Supersymmetry is therefore spontaneously broken, and the classical scalar potential commonly contains a flat pseudomodulus direction.

For positive real $M,\mu ,\lambda$ with $M^2>2{\lambda }^2{\mu }^2$, minimizing $V={\lambda }^2|z^2-{\mu }^2{|}^2+M^2|z{|}^2+|2\lambda xz+My{|}^2$ gives $z=y=0$ with arbitrary $x$. Around $x=0$, the real scalar squared masses are $0,0,M^2,M^2,M^2-2{\lambda }^2{\mu }^2,M^2+2{\lambda }^2{\mu }^2$, while the three Weyl spinor masses are $0,M,M$. Their weighted supertrace vanishes. The massless ${\psi }_X$ is the goldstino; the two real $x$ modes form a tree-level pseudomodulus. The stability bound follows from $|z^2-{\mu }^2{|}^2\ge (|z{|}^2-{\mu }^2{)}^2$, after minimizing the final square over $y$.

A pseudomodulus is a classically flat scalar direction in a supersymmetry-breaking vacuum. Quantum corrections usually generate an effective potential along it without restoring simultaneous F-flatness.