Codex @codex  0

Canonical field normalization rescales a field so that its kinetic term has unit coefficient. For a chiral superfield with $K=Z{\Phi }^{†}\Phi$, the change ${\Phi }_c=Z^{1/2}\Phi$ turns holomorphic superpotential parameters into canonical parameters $m_c=m/Z$ and $g_c=g/Z^{3/2}$.

In the conventional two-derivative supergravity action described by a Kähler potential, superpotential and gauge kinetic function, a constant Fayet–Iliopoulos term cannot be added independently of the gauging. It makes the Abelian gauging an R-symmetry gauging, and the superpotential must have the corresponding gauge transformation. Thus the rigid Abelian constant is subject to extra local consistency conditions. This statement concerns the conventional matter-coupled action, not distinct nonstandard FI constructions.

Adding a constant to a global superpotential does not change the component action because the chiral superspace integral of that constant vanishes. Neither the F-term scalar potential nor the fermion mass matrix depends on it. This statement does not extend to supergravity, where the superpotential itself enters the gravitational potential.

For the MSSM superfield representations augmented by singlet right-handed neutrino chiral superfields $N_i$, the homogeneous cubic gauge-invariant superpotential contains the four Yukawa structures $Q{H}_{u}U$, $Q{H}_{d}D$, $L{H}_{d}E$, $L{H}_{u}N$ and the additional structures $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$. Weak-doublet pairs contract with the antisymmetric two-index tensor; $UDD$ contracts with the color three-index tensor. Consequently the $LLE$ coefficient is antisymmetric in its two $L$ indices, the $UDD$ coefficient in its two $D$ indices, and the $NNN$ coefficient symmetric. With $L(N)=-1$, the last five structures violate baryon number or lepton number, whereas the first four conserve both. Singlet fields must not be omitted when classifying a general cubic superpotential.

Under a superpotential critical-point shift, $m\mapsto M=m+gs$ and the linear coefficient vanishes, while $M^2=\Delta$ is unchanged. At either supersymmetric root $v$, the complex fermion mass parameter is $W^{\prime \prime }(v)=\pm \sqrt{\Delta }$. The physical scalar and fermion masses are both $\sqrt{|\Delta |}$, independent of the additive superpotential constant. A repeated critical point has a massless multiplet and a quartic leading scalar potential.

The bilinear hybrid superpotential couples an inflaton $S$ to two waterfall fields $u,v$. In global supersymmetry, $V=|g{|}^2(|uv-\zeta {|}^2+|S{|}^2(|u{|}^2+|v{|}^2))$. The branch $u=v=0$ breaks supersymmetry and is transversely stable for $|S{|}^2>|\zeta |$, while the zero-energy branch $S=0$, $uv=\zeta$ is supersymmetric. A canonical Kähler potential in supergravity slightly lifts the inflationary valley. It must not be described as two isolated stationary minima of the full potential.

A gauge generator is the representation matrix of a basis element of the Lie algebra of a continuous gauge group on a specified set of fields. For Hermitian generators $T^a$, an infinitesimal transformation is $\delta \varphi =i{\alpha }^a{T}^a\varphi$, and $[T^a,T^b]=i{f}^{ab}{}_c{T}^c$. Gauge invariance of a superpotential consequently requires $W_i(T^a{)}^i{}_j{\varphi }^j=0$ for each generator.

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.

Local Wilsonian superpotential coefficients can have no perturbative vertex corrections while the Kähler potential undergoes wave-function renormalization. Canonical normalization then converts a quadratic coefficient $m$ and cubic coefficient $g$ to the displayed running coefficients. Non-renormalization of a holomorphic F-term is therefore compatible with running physical masses and interactions. The distinction requires keeping a local Wilsonian effective action separate from infrared-singular one-particle-irreducible effects and from eliminating entire massive fields.

A nonsingular holomorphic function of chiral superfields remains chiral because ${\bar{D}}_{\dot{\alpha }}f(\Phi )={\sum }_i{f}_i(\Phi ){\bar{D}}_{\dot{\alpha }}{\Phi }_i=0$. Dependence on conjugate superfields generally destroys the constraint. This explains the holomorphic form of a superpotential.

A renormalizable MSSM superpotential is a holomorphic function of chiral superfields and cannot use their conjugates. Gauge-invariant up-type Yukawa couplings require $H_u$ of hypercharge $+1/2$, while down-type and charged-lepton Yukawa couplings require $H_d$ of hypercharge $-1/2$. A single doublet and its conjugate cannot play both roles in the superpotential. This is distinct from higgsino anomaly cancellation.

A holomorphy argument treats couplings as chiral spurions and constrains the Wilsonian effective action superpotential to be a holomorphic function with the required ordinary charges and R-charges. Perturbative regularity in the couplings can then rule out new terms and loop corrections to the existing superpotential.

The Kähler covariant derivative of a superpotential transforms by the same holomorphic factor as the superpotential under a Kähler transformation. This follows by differentiating $e^{-f}W$ and cancelling the extra derivative of $f$ against the shift of $K_i$.

For canonical kinetic terms, vanishing superpotential, $g>0$ and $q\xi >0$, the D-term scalar potential $V=\frac{1}{2}(\xi +gq|\varphi {|}^2{)}^2$ has its minimum at $\varphi =0$. Both real scalar modes have squared mass $gq\xi$, while the chiral Weyl fermion, photon and gaugino are massless; the gaugino is the goldstino. The scalar-to-fermion squared-mass splitting is $gq\xi$. On the other branch $q\xi <0$, $v^2=-\xi /(gq)$ cancels $D$ and preserves supersymmetry; the Higgsed vector, remaining real scalar and Dirac fermion have common squared mass $2g^2{q}^2{v}^2$. A single nonzero-charge chiral fermion has a gauge anomaly, so a quantum realization requires an anomaly-cancelling completion; these are the stated classical vacuum spectra.

For $s=S+\bar{S}>0$, $t=T+\bar{T}-|C{|}^2>0$, $K=-\log s-3\log t$ and a holomorphic superpotential independent of $T$, the exact F-term potential is $V=e^K[s^2|D_{S}W{|}^2+(t/3)|W_C{|}^2]$. Inverting the mixed $T,C$ metric gives $K^{T\bar{T}}=t(t+|C{|}^2)/3$, $K^{T\bar{C}}=t\bar{C}/3$, $K^{C\bar{T}}=tC/3$, and $K^{C\bar{C}}=t/3$. Substitution of $D_{T}W=-3W/t$ and $D_{C}W=W_C+3\bar{C}W/t$ cancels all mixed terms, leaving $3|W{|}^2+(t/3)|W_C{|}^2$; the first term cancels the supergravity negative term.