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Let $g(E)$ denote the total density of states, including both spin directions. The field lowers one spin energy by ${\mu }_{B}B$ and raises the other by the same amount. To first order in $B$, each spin population changes by the one-spin density $g(E_F)/2$ times the displacement of its Fermi edge. The population imbalance is thereforeMultiplying by the Bohr magneton carried by each excess aligned electron givesThus, with this total-density convention,The zero-temperature magnetic susceptibility isThis weak positive response is Pauli paramagnetism. If susceptibility per unit volume is required, both $M$ and $g$ are divided by $V$.

The hierarchy problem is visible in the one-loop top-quark contribution to the Higgs boson mass parameter,In a supersymmetric theory the corresponding stop loop has the opposite quadratic term. If supersymmetry is softly broken, the remaining correction is roughlyThe two relevant one-loop Feynman diagrams are a Higgs two-point function with a closed top-quark loop and the analogous stop loop. Requiring the residual correction not to exceed the electroweak scale by many orders of magnitude gives the naturalness expectation $m_{\tilde{t}}$, and broadly other sparticle masses, at a few TeV or below. This is an order-of-magnitude argument rather than a mass bound.

R-parity iswhere $B$ is baryon number, $L$ is lepton number and $s$ is spin angular momentum. Standard Model particles have $R_p=+1$, while their superpartners have $R_p=-1$. The renormalizable R-parity-violating superpotential of the Minimal supersymmetric Standard Model iswith ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$.

Combining ${\lambda }_{i1k}^{\prime }{L}_i{Q}_1{D}_k^c$ with ${\lambda }_{11k}^{\prime \prime }{U}_1^c{D}_1^c{D}_k^c$ produces proton decay through exchange of a virtual right-handed down-type squark. The tree diagram has $u$ and $d$ entering the baryon-number-violating vertex, an internal ${\tilde{d}}_{Rk}$ line, and the lepton-number-violating vertex emitting ${\bar{\nu }}_i$ and a light quark. Hadronization givesThe invisible particle is an antineutrino of any family $i=1,2,3$, and electric-charge conservation requires a positively charged pion. The antisymmetry of ${\lambda }_{11k}^{\prime \prime }$ requires $k=2$ or $3$.

Integrating out a squark of mass $m_{\tilde{q}}$ gives a dimension-six interaction with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/m_{\tilde{q}}^2$. Ignoring order-one hadronic and phase-space factors,For $m_p\sim 1\mathrm{GeV}$ and $m_{\tilde{q}}\sim 10^4\mathrm{GeV}$, the factor before the couplings is about $10^{-18}$--$10^{-17}\mathrm{GeV}$. A lifetime above $10^{34}$ years is about $10^{65}{\mathrm{GeV}}^{-1}$, so ${\Gamma }_p\lesssim 10^{-65}\mathrm{GeV}$ andat order-of-magnitude accuracy. Hadronic matrix elements and flavour choices alter the numerical coefficient, not the conclusion that simultaneous baryon- and lepton-number violation is extraordinarily constrained.

The matter-parity factor $(-1{)}^{3(B-L)}$ is odd for every quark or lepton superfield and even for either Higgs superfield. Each term in $W_{\mathrm{RPV}}$ contains an odd number of matter superfields, so R-parity forbids all four types of term. Standard MSSM Yukawa interactions contain two matter superfields and remain allowed.

Finally, the requested four-point interaction contains a squark, an antisquark, a $W$ boson and a gluon. It comes from the squark gauge-covariant kinetic termThe cross term isbecause the colour and weak generators commute. With all fields incoming, its Feynman rule iswith the conventional $1/\sqrt{2}$ conversion when $W^I$ is replaced by $W^{\pm }$.

A meson with zero spin angular momentum and negative parity. The pion and kaon are familiar examples. Between two pseudoscalar mesons, parity conservation in the strong interaction forces the axial current matrix element to vanish, while the vector current has pseudoscalar-to-pseudoscalar form factors. The weak decay as a whole need not conserve parity.

Write $U=A{U}_0{A}^{-1}$, $A\in SU(2)$, and $A^{-1}\dot{A}=i\mathbf{\omega }\cdot \mathbf{\sigma }/2$. The rotational kinetic energy is $\Lambda |\mathbf{\omega }{|}^2/2$. Since $A$ and $-A$ give the same classical field, the physical orientation space is $SO(3)$, covered by $SU(2)$. For fermionic unit baryons the Finkelstein-Rubinstein constraints require $\Psi (-A)=-\Psi (A)$. The SU(2) representations then give half-integer $j$, with spin angular momentum and isospin both of magnitude $j$. The rigid-rotor energies are $M+j(j+1){\hslash }^2/(2\Lambda )$; the lowest multiplets model the nucleon and Delta baryon. Large rotational energies can excite deformation and radiation, beyond the rigid approximation.

The components of spin angular momentum satisfy $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$. In units $\hslash =1$, the spin ladder operators obey $[S^{+},S^{-}]=2S^z$ and $[S^z,S^{\pm }]=\pm S^{\pm }$.

In units $\hslash =1$, spin one-half is the two-dimensional irreducible spin angular momentum representation with components $S_i={\sigma }_i/2$. The Pauli matrices give eigenvalues $\pm 1/2$ along every unit axis. The normalized $x$-axis states are $(|{\uparrow }_z⟩\pm |{\downarrow }_z⟩)/\sqrt{2}$.

A spin sum adds a squared scattering amplitude over the unobserved spins of external particles. For an on-shell Dirac spinor, the completeness relations ${\sum }_s{u}_s(p){\bar{u}}_s(p)=/p+m$ and ${\sum }_s{v}_s(p){\bar{v}}_s(p)=/p-m$ turn the sum into a gamma-matrix trace.