$N=1$ Wilsonian F-Term Nonrenormalization

By [F1], [F4], and [A2], the local Wilsonian density at every perturbative order has the same superspace and gauge-invariant decomposition into the fixed canonical matter kinetic term, an additional real D-term, a chiral F-term, and matter-independent quadratic gauge-kinetic terms. The collection of the latter terms into $P_{\mathrm{k}\mathrm{i}\mathrm{n}}$ is the algebraic separation specified in the Statement and changes no field variable or action. For a numerical constant $c$, the coefficient-extraction definition of the F bracket in [F1] gives $[c{]}_F=0$. Thus the action determines no numerical constant coefficient in $F_{\mathrm{r}\mathrm{e}\mathrm{m},\lambda }$; [A5] fixes that coefficient to its ultraviolet value. In the remaining proof apply the vertex argument to $F_{\mathrm{r}\mathrm{e}\mathrm{m}}-c_0$, which has positive dynamical-field degree. The BRST differential used to state gauge invariance is well-defined and nilpotent by [F2] and [F3]; this is a premise about the local gauge algebra, not a proof of quantum anomaly cancellation.

Work in the formal background-jet algebra of [F1] and [A3], before setting backgrounds constant. For a derivative-free chiral coefficient write $f={\sum }_{\nu \in {\mathbb{N}}^k}{a}_{\nu }{u}^{\nu }$, where each $a_{\nu }$ belongs to the remaining chiral coefficient algebra, so ${\bar{D}}_{\dot{\alpha }}{a}_{\nu }=0$. The graded chain rule gives $0={\bar{D}}_{\dot{\alpha }}f={\sum }_{j=1}^k(\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{u}_j){\bar{D}}_{\dot{\alpha }}{P}_j$. Independence of these first odd background jets forces $\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{u}_j=0$ for every $j$, coefficientwise. Indeed the coefficient of $u^{\nu -e_j}$ in this derivative is ${\nu }_j{a}_{\nu }$ whenever ${\nu }_j>0$. The coefficient algebra is over $\mathbb{C}$, hence multiplication by the nonzero integer ${\nu }_j$ is invertible, and $a_{\nu }=0$ for every nonzero multi-index $\nu$. Thus $f=a_0$ and the real D coefficients do not enter the derivative-free chiral integrand at any formal order. The same chain-rule argument with the independent first anti-chiral jets of the conjugate chiral coefficient backgrounds excludes dependence on those conjugates, so the surviving coefficients are holomorphic in the chiral coefficient backgrounds. Terms containing background superderivatives belong to the D-sector by [F1]; the matter-independent quadratic gauge chiral terms retain their topological-angle total-derivative qualification in [A1]. This argument requires neither a positive-body factorization nor an analytic extension of formal series, and makes no change of integrated field variables.

Assign the bookkeeping R charges $R({\theta }^{\alpha })=1$, $R({\bar{\theta }}^{\dot{\alpha }})=-1$, $R(\mathrm{\Phi })=R(V)=R({\tau }_a)=0$, and $R(f_r)=2-r$ to a coefficient $f_r$ multiplying a term with $r$ field-strength superfields; then $R(W_{\alpha })=1$ and the chiral density has total R charge two. To apply the topological-angle shifts, first use a real basis $R_b$ of the real-valued invariant symmetric bilinear forms on $\mathfrak{g}$. Such a basis complexifies to a basis of the space in the Statement: the real and imaginary parts of any invariant complex form restricted to $\mathfrak{g}$ are real invariant forms, because ${\mathrm{Ad}}_g$ preserves $\mathfrak{g}$. Finite-dimensional basis extraction adds no choice principle. Write $Q_a={\sum }_b{M}_{ba}{R}_b$ with an invertible complex matrix $M$, and put ${\tilde{\tau }}_b={\sum }_a{M}_{ba}{\tau }_a$; this changes only coefficient coordinates. Real shifts of ${\tilde{\tau }}_b$ change only topological-angle terms, which integrate to zero under [A1]. Consequently the change of a quadratic chiral coefficient in this real basis under such a shift is real. Its holomorphic derivative with respect to each ${\tilde{\tau }}_b$ is real-valued, hence constant: writing it as $u+iv$, the Cauchy–Riemann equations and $v=0$ make both real derivatives of $u$ vanish. Applying the same argument to every other complex scalar coupling makes these derivative coefficients constant in those couplings as well; the argument applies coefficientwise in the formal coupling expansion. A coefficient of any nonconstant nonkinetic chiral operator is unchanged by every real shift, and holomorphy therefore makes it independent of $\tilde{\mathit{\tau }}$. Transforming back gives ${\mathrm{\Delta }}_{\lambda }={\sum }_{a,b}{c}_{ab,\lambda }{\tau }_b{Q}_a(W^{\alpha },W_{\alpha })+H_{\lambda }(\mathrm{\Phi },W,f)$, where $H_{\lambda }$ is independent of every ${\tau }_b$. The uniform scaling used below commutes with this fixed linear basis change.

The nonminimal BRST extension is nilpotent: its square on $\bar{c},b$ is zero directly and on the original variables it is zero by [F3]. The odd Leibniz rule gives $s\mathrm{\Psi }=\int \{B(b,\mathrm{\partial }\cdot A)-\alpha B(b,b)\mathrm{/}2-B(\bar{c},{\mathrm{\partial }}^{\mu }{D}_{\mu }c)\}$. Completing the multiplier square as in [F1] gives the quadratic term $B(\mathrm{\partial }\cdot A,\mathrm{\partial }\cdot A)\mathrm{/}(2\alpha )$ and a field-independent Gaussian normalization. Hence this completion adds no interacting multiplier vertices: besides quadratic kernels it adds only $-B(\bar{c},{\mathrm{\partial }}^{\mu }[A_{\mu },c])$, whose vector end is an ordinary gauge-boson end, never a gaugino end. Uniform scaling makes each such vertex proportional to $t$ and every internal ghost or vector-multiplet propagator proportional to $t^{-1}$. The ghosts have R charge zero, so they add no R-charged coupling. By step 2.1 the additional non-FI real D coefficients and conjugate F coefficients cannot affect the derivative-free chiral sector, so graphs containing those interactions can be omitted there. A fixed bare FI insertion has only one vector-multiplet end; if attached to another vertex its internal line is a bridge, excluded by 1PI. Thus the remaining interactions are precisely the physical A, B, C types below and these explicit ghost vertices. This establishes exhaustiveness for this selected component completion without using gauge independence.

Consider a connected graph contributing to this chiral density, with $E_V$ external left-handed gaugino lines and $I_V$ internal vector lines. Uniformly scale the complete gauge-kinetic coefficient vector by a formal parameter $t$, $\mathit{\tau }\mapsto t\mathit{\tau }$, holding its coefficient ratios fixed. Each internal vector propagator then contributes $t^{-1}$. Let $A_m$ count pure-gauge vertices from the quadratic gauge-kinetic polynomial with $m\ge 3$ vector lines; let $B_{mr}$ count vertices from a remaining F-term containing $r$ field strengths and $m\ge r$ vector lines; and let $C_m$ count vertices from the fixed canonical matter kinetic term with two chiral lines and $m\ge 1$ vector lines. Let $D_{\mathrm{g}\mathrm{h}}$ count ghost vertices and $I_{\mathrm{g}\mathrm{h}}$ internal ghost propagators. There are no external ghosts in the chiral density, so $2I_{\mathrm{g}\mathrm{h}}=2D_{\mathrm{g}\mathrm{h}}$. Each $A_m$ and ghost vertex contributes one factor of $t$, and the vector-line ends satisfy $2I_V+E_V={\sum }_{m\ge 3}m{A}_m+{\sum }_r{\sum }_{m\ge r}m{B}_{mr}+{\sum }_{m\ge 1}m{C}_m+D_{\mathrm{g}\mathrm{h}}$. The R-charge condition gives ${\sum }_r{\sum }_{m\ge r}(2-r)B_{mr}=2-E_V$. Thus $N_t={\sum }_{m\ge 3}{A}_m+D_{\mathrm{g}\mathrm{h}}-I_V-I_{\mathrm{g}\mathrm{h}}={\sum }_{m\ge 3}{A}_m-I_V$; eliminating $E_V$ and $I_V$ yields $N_t=1-\frac{1}{2}[{\sum }_{m\ge 3}(m-2)A_m+{\sum }_r{\sum }_{m\ge r}(2-r+m)B_{mr}+{\sum }_{m\ge 1}m{C}_m+D_{\mathrm{g}\mathrm{h}}]$.

Every coefficient inside the square brackets is a positive integer. A graph contributing to the $\mathit{\tau }$-independent part $H_{\lambda }$ must have $N_t=0$, so the bracket equals two. The bracket-two possibilities containing ghosts are $D_{\mathrm{g}\mathrm{h}}=2$ alone, or $D_{\mathrm{g}\mathrm{h}}=1$ with one $A_3$ or one $C_1$. In the first case the R-charge equation requires $E_V=2$, but both vector ends are gauge-boson ends of ghost vertices and cannot be external left-handed gauginos. In either mixed case the single ghost vertex contracts its own ghost ends. Its unique vector end must connect to the other vertex because it cannot be an external gaugino; that connecting line is a bridge, violating 1PI. Thus none contributes to this chiral coefficient, leaving $D_{\mathrm{g}\mathrm{h}}=0$. The remaining positive-integer solutions are two cubic or one quartic pure-gauge vertices, one $B_{mr}$ vertex with $m=r$, two $C_1$ vertices, one $C_2$ vertex, or the mixed case with one cubic gauge vertex and one $C_1$ vertex. For the $B_{mr}$ case, [step 5.1]'s R-charge equation gives $E_V=r=m$; the vector-end equation then gives $I_V=0$. It is an original single tree insertion in $F_{\mathrm{r}\mathrm{e}\mathrm{m}}$. In the $A_3=2$ pure-gauge case the R-charge equation gives $E_V=2$ and the vector-end equation is $2I_V+2=6$, so $I_V=2$ and $L=2-2+1=1$. In the $A_4=1$ case it is $2I_V+2=4$, so $I_V=1$ and $L=1-1+1=1$. Both are one-particle-irreducible one-loop corrections to the matter-independent quadratic gauge-kinetic polynomial. For either matter-only case, the R-charge equation gives $E_V=2$ and the vector-end equation gives $2I_V+2=2$, so $I_V=0$. Each $C_m$ vertex contains one chiral and one antichiral matter line. A contribution to a holomorphic F-term cannot have an external antichiral line; with no other vertex type present, the two $C_1$ vertices therefore contract their two antichiral ends with their two chiral ends, giving $I_{\mathrm{\Phi }}=2$, while the one $C_2$ vertex contracts its own chiral/antichiral pair, giving $I_{\mathrm{\Phi }}=1$. Thus in each case the total internal-line count equals the vertex count and $L=1$; no matter line is external, so these graphs correct only the matter-independent quadratic gauge-kinetic polynomial, including any allowed Abelian mixing term. In the mixed case the end equation is $2I_V+2=3+1=4$, so $I_V=1$; the single $C_1$ vertex contracts its chiral/antichiral pair, giving $I_{\mathrm{\Phi }}=1$. Hence $I=2$, $N_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}=2$, and $L=1$. Its sole vector line is a bridge separating the gauge vertex from the matter vertex, so it is one-particle-reducible and excluded by the vertex convention in [F1]. Therefore every non-tree contribution to $H_{\lambda }$ is a one-loop shift of the matter-independent quadratic gauge-kinetic coefficient vector; no nonkinetic term in $H_{\lambda }$ receives a radiative correction. A contribution linear in $\mathit{\tau }$ would require $N_t=1$, hence a zero bracket and no interaction vertices; it is absent. This proves equality for every positive-field-degree coefficient of $F_{\mathrm{r}\mathrm{e}\mathrm{m},\lambda }$ and $F_{\mathrm{r}\mathrm{e}\mathrm{m}}$; their constant coefficients agree by [A5]. Thus $F_{\mathrm{r}\mathrm{e}\mathrm{m},\lambda }=F_{\mathrm{r}\mathrm{e}\mathrm{m}}$ in the specified representatives, and there are no higher-loop gauge-kinetic corrections.

For an unbroken U(1) factor, the prepotential shifts by $V_1\mapsto V_1-{\mathrm{\Lambda }}_1-{\bar{\mathrm{\Lambda }}}_1$. A constant coefficient $\xi$ multiplying $\int d^{4}x[V_1{]}_D$ is gauge invariant because the Berezin integral of the chiral and anti-chiral shifts vanishes under the boundary condition in [F1]. For a prospective field-independent correction with real coefficient superfield $S$, the physical variation contains $-[P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(S){\mathrm{\Lambda }}_1{]}_F$ and its Hermitian conjugate, where $P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=-\frac{1}{2}{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{2}}}^L{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{1}}}^L{\mathrm{\mid }}_{\bar{\theta }=0}=+\frac{1}{4}{\bar{D}}^{\dot{\alpha }}{\bar{D}}_{\dot{\alpha }}$ at fixed chiral coordinates is the normalized D-to-F projector in [F1]. Since compactly supported chiral ${\mathrm{\Lambda }}_1$ is arbitrary, its chiral pairing and the conjugate pairing vanish for all tests only if ${\bar{D}}^{2}S=0=D^{2}S$. Write the derivative-free real coefficient as a formal function of the real coefficient spurions $P_j$ and the chiral/anti-chiral coefficient spurions $f_r,{\bar{f}}_r,{\tau }_a,{\bar{\tau }}_a$. In ${\bar{D}}^{2}S$, the coefficients of the independent second anti-chiral jets ${\bar{D}}^2{P}_j$, ${\bar{D}}^2{\bar{f}}_r$, and ${\bar{D}}^2{\bar{\tau }}_a$ are the corresponding formal partial derivatives; their independence forces $S$ to be independent of all $P_j,{\bar{f}}_r,{\bar{\tau }}_a$. Reality then makes it independent of $f_r,{\tau }_a$ as well, so $S$ is constant. A separate field-dependent coupling $[L{V}_1{]}_D$ with $L$ as in [A4] remains distinct D-term data, not the field-independent FI coefficient. Therefore a correction to the constant FI coefficient is independent of the coupling spurions in [A3]. A bare FI insertion, held constant in [A3], is one-valent. If attached to another vertex its unique internal line is a bridge, so it cannot enter an interacting one-particle-irreducible correction. All remaining mass and interaction coefficients are variable spurions; a correction independent of them is computed by setting them to zero. Besides the canonical matter vertices, retain the pure gauge-kinetic vertices and scale all gauge-kinetic coefficients by $t$ as in step 5.1. For a connected coupling-independent correction with one external $V_1$, $N_t={\sum }_m{A}_m-I_V=0$ and $2I_V+1={\sum }_{m\ge 3}m{A}_m+{\sum }_{m\ge 1}m{C}_m+D_{\mathrm{g}\mathrm{h}}$, hence the full condition is $\sum (m-2)A_m+\sum m{C}_m+D_{\mathrm{g}\mathrm{h}}=1$. Besides one $A_3$ or one $C_1$, this allows one ghost vertex. Its external central Abelian gauge-boson leg vanishes since $[Z,c]=0$ for a central generator $Z$; it has no auxiliary leg either, so cannot correct the FI coefficient. The remaining possibilities are one $A_3$ or one $C_1$ vertex. The $A_3$ case cannot have an external Abelian prepotential: its curvature is linear, and invariance of each quadratic form gives $Q(Z,[X,Y])=Q([Z,X],Y)=0$ for a central Abelian generator $Z$. The $C_1$ case has $I_V=0$ and its matter ends contract, giving precisely the one-loop tadpole. By the common-kernel hypothesis in [A4], its integral is the same for each chiral component, with the representation entering only through its charge. Writing its common value as $C_{\lambda }$, the correction is $\delta {\xi }^{(1)}=C_{\lambda }{\sum }_i{q}_i$. This is a one-loop graph, so the correction vanishes when the charge sum is zero and there are no higher-loop FI graphs.

The argument constrains the remaining holomorphic integrand, matter-independent quadratic gauge-kinetic coefficients, and FI tadpole only; it places no equality requirement on the arbitrary real function $G_{\lambda }$ in [F1] or on the real Kähler data in [F5]. Since the high-mode split retains momenta below $\lambda$ and [A1] excludes nonperturbative sectors, no conclusion follows for an arbitrary one-particle-irreducible functional, instanton corrections, coefficients after field rescaling, or a continuum limit. These are exactly the qualifications in the statement.