One-loop renormalization of $\mathcal{N}=\frac{1}{2}$ supersymmetric gauge theory with a superpotential

We construct a superpotential for the general $\mathcal{N}=\frac{1}{2}$ supersymmetric gauge theory coupled to chiral matter in the fundamental and adjoint representations, and investigate the one-loop renormalizability of the theories.

Figure 1

Diagrams with one gaugino, one scalar and one chiral fermion line; the dot represents the position of a $C$.

Figure 2

Diagrams with one gaugino, one scalar, one chiral fermion, and one gauge line; the dot represents the position of a $C$.

Figure 3

Diagrams with one gauge, one scalar and one auxiliary line; the dot represents the position of a $C$.

Figure 4

Diagrams with two gauge, one scalar and one auxiliary line; the dot represents the position of a $C$.

Figure 5

Diagrams with two gaugino, one scalar and one auxiliary line; the dot represents the position of a $C$ or a $|C{|}^2$.

Figure 6

Diagrams with two scalar and two chiral fermion lines; the dot represents the position of a $C$.

Figure 7

Diagrams with two scalar, one gauge line; a dot denotes a $C$, a cross a mass and a crossed circle a vertex with both a mass and a $C$.

Figure 8

Diagrams with two scalar, two gauge lines; a dot denotes a $C$, a cross a mass and a crossed circle a vertex with both a mass and a $C$.

Figure 9

Diagrams with two scalar, two gaugino lines; a dot denotes a $C$, a cross a mass and a crossed circle a vertex with both a mass and a $C$.

Figure 10

Diagrams with three scalar lines; a dot represents the position of a $C$, a cross a Yukawa vertex and a crossed circle a Yukawa vertex with a $C$.

Figure 11

Diagrams with three scalar lines and one gauge line; a crossed circle represents a Yukawa vertex with a $C$. Figures 11(e-v) are not depicted explicitly. Figures 11(e-o) are obtained from Figs. 7a, 7b, 7c, 7d, 7e, 7f, 7g, 7h, 7i, 7j, 7k by adding an external scalar $\overline{\varphi }$ line at the position of the cross. Figures 11(p-v) are obtained from Figs. 7l, 7m, 7n, 7o, 7p, 7q, 7r by adding an external scalar $\overline{\varphi }$ line at the position of the crossed circle.

Figure 12

Diagrams with two chiral fermion lines and one auxiliary line; the dot represents the position of a $C$ and the cross a Yukawa vertex.

Figure 13

Diagrams with one gauge line and one auxiliary line; a cross represents a mass insertion or Yukawa vertex and a crossed circle a vertex with both a mass and a $C$.

Figure 14

A diagram with one chiral fermion line and one gaugino line; a cross represents a mass insertion or Yukawa vertex.

Figure 15

Diagrams with two chiral fermion lines and one scalar; a dot represents the position of a $C$, a cross represents a mass insertion or Yukawa vertex, a crossed circle a vertex with both a mass and a $C$, and a box a mass-Yukawa vertex.

Figure 16

Diagrams with two chiral fermion lines and two scalars; a dot represents the position of a $C$, a cross a Yukawa vertex and a crossed circle a Yukawa vertex with a $C$.

Figure 17

A diagram with one scalar and one gauge line; a crossed circle represents a vertex with both a mass and a $C$, and a box a mass-Yukawa vertex.