# Why coupling constants with negative mass dimensions lead to non-renormalizable theories?

can somebody explain or point to the relating mathematics showing Why coupling constants with negative mass dimensions lead to non-renormalizable theories?

• – Qmechanic May 20 at 12:58

For an amputated one-particle irreducible (1PI) Feynman diagram the superficial degree of divergence $$D$$ is equal to$$^1$$ \begin{align} D&~=~\sum_f [\widetilde{G}_{0f}]I_f + Ld +\sum_i V_i d_i \cr &~=~\sum_f(2[\phi_f]-d) I_f + \left(\sum_f I_f -(\sum_i V_i -1)\right)d +\sum_i V_i d_i \cr &~=~d+2\sum_f[\phi_f] I_f - \sum_i(d-d_i) V_i \cr &~=~d+\sum_f[\phi_f] \left(\sum_i V_i n_{if}-E_f\right) - \sum_i(d-d_i) V_i \cr &~=~d - \sum_i \left(d - d_i - \sum_f [\phi_f] n_{if}\right) V_i - \sum_f [\phi_f] E_f\cr &~=~d - \sum_i [\lambda_i] V_i - \sum_f [\phi_f] E_f,\tag{*} \end{align} where

• $$d$$ is the number of spacetime dimensions;

• $$[\cdot]$$ denotes the mass dimension in units where $$\hbar=1=c$$;

• $$L$$ is the number of independent loops;

• $$I_f$$ is the number of internal lines with a free propagator $$\widetilde{G}_{0f}$$ in the Fourier momentum space of a field $$\phi_f$$ of type $$f$$;

• $$V_i$$ is the number of vertices of $$i$$'th interaction type with coupling constant $$\lambda_i$$, $$d_i$$ number of spacetime derivatives, and $$n_{if}$$ legs of type $$f$$;

• $$E_f$$ is the number of amputated external lines with a field $$\phi_f$$ of type $$f$$.

Let us now return to OP's question. If an interaction vertex, say of type $$i_0$$, has $$[\lambda_{i_0}]<0$$, then eq. (*) indicates that we can build infinitely many superficially divergent Feynman diagrams with $$D\geq 0$$ by using more and more vertices of type $$i_0$$. This render the theory non-renormalizable in the old Dyson sense.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; eq. (12.1.8).

2. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; eqs. (10.11) + (10.13).

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$$^1$$ It is implicitly assumed that the coefficients in front of the kinetic terms in the action are dimensionless. The quantity $$[\phi_f]$$ is non-negative for $$d\geq 2$$.