Wess–Zumino model

The Wess–Zumino model contains chiral superfields with a polynomial superpotential. It is the simplest interacting four-dimensional supersymmetric field theory.

In a canonical Wess–Zumino model, complete the static transverse energy into $|{\varphi }^{\prime }-e^{i\alpha }\overline{W^{\prime }}{|}^2+2\mathrm{Re}(e^{-i\alpha }dW/dz)$. Choosing the phase of $\Delta W$ gives the tension bound. A solution of the displayed first-order equation attains it and preserves two of four real supercharges. For $W={\mu }^2\Phi /g-g{\Phi }^3/3$, $\varphi (z)=(\mu /g)\tanh (\mu z)$ interpolates between the two vacua and has tension $8{\mu }^3/(3g^2)$.

Localizing the rigid supersymmetry of a free Wess–Zumino model produces a derivative of the supersymmetry parameter multiplying the supercurrent. The Noether gauging procedure introduces a gravitino coupled to this current. Closure also requires a vierbein, the Einstein-Hilbert action, covariant matter kinetic terms and higher-order interactions. The leading gravitino-matter coupling contains the derivative of the scalar, not its undifferentiated value.

For canonical Kähler potential and $W=m{\Phi }^2/2+g{\Phi }^3/3$, eliminating the auxiliary field gives $V=|m\varphi +g{\varphi }^2{|}^2$. The cubic interaction Lagrangian is $-m^{\ast }g{\varphi }^{\ast }{\varphi }^2-m{g}^{\ast }\varphi ({\varphi }^{\ast }{)}^2$. Differentiating with respect to the three external fields gives the all-incoming Feynman rules $V_{\varphi \varphi {\varphi }^{\ast }}=-2i{m}^{\ast }g$ and $V_{{\varphi }^{\ast }{\varphi }^{\ast }\varphi }=-2im{g}^{\ast }$; the factor two counts identical scalar legs. For real $m,g$ and $\varphi =(A+iB)/\sqrt{2}$, the cubic Lagrangian is $-mgA(A^2+B^2)/\sqrt{2}$, giving $V_{AAA}=-3\sqrt{2}img$ and $V_{ABB}=-\sqrt{2}img$.

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.

In a canonical Wess–Zumino model with cubic superpotential $g{\Phi }^3/3$, eliminating the auxiliary field gives $V\supset |g{|}^2|\phi {|}^4$ and $\mathcal{L}\supset -g\phi \psi \psi +h.c.$. Thus the coefficient of the quartic scalar field interaction is the squared modulus of this convention for the Yukawa coupling.

In a canonical Wess–Zumino model with cubic superpotential $g{\Phi }^3/3$, auxiliary elimination gives a complex-scalar quartic interaction with two legs of each field type. With all momenta incoming and the displayed coefficient, its Feynman rule is $-i(2!2!)|g{|}^2$. For $\phi =(A+iB)/\sqrt{2}$, the same quartic scalar potential is $|g{|}^2(A^2+B^2{)}^2/4$, with real-field rules $-6i|g{|}^2$ for four identical legs and $-2i|g{|}^2$ for two of each. Factorials depend on how the coefficient is defined, not on a new physical coupling.

Supersymmetry fixes scalar and fermion couplings so that bosonic and fermionic loop contributions to scalar masses have equal quadratic ultraviolet parts with opposite signs.