# From vertex function to anomalous dimension

• In a $d$ dimensional space-time, how does one argue that the mass dimension of the $n-$point vertex function is $D = d + n(1-\frac{d}{2})$?

• Why is the following equality assumed or does one prove that a function $f$ will exist such that, $\Gamma^n(tp,m,g,\mu) = f(t)\Gamma^n(p,m(t),g(t),\mu)$?

..in writing the above I guess one is assuming that all the inflowing momentum are equal to $p$ but if that is so then what is the meaning of again specifying the "renormalization scale" of $\mu$?..Does this somehow help fix the values of the functions, $m(t)$ and $g(t)$ at some value of $t$?..If yes, how?..

If the above equality is assumed as a property of the vertex function then it naturally follows that it satisfies the differential equation,

$[-t\frac{\partial}{\partial t} + m[\gamma_m(g)-1]\frac{\partial}{\partial m} + \beta(g)\frac{\partial}{\partial g} + [D-n\gamma(g)]]\Gamma^n(tp,m,g,\mu)=0$

where one defines,

$\beta(g) = t\frac{\partial g(t)}{\partial t}$

$m[\gamma_m(g)-1] = t\frac{\partial m(t)}{\partial t}$

$D-n\gamma(g) = \frac{t}{f}\frac{\partial f}{\partial t}$

and the last equality integrates to give, $f(t) = t^De^{-n\int_1^t \frac{\gamma(g(t))}{t}dt}$ and $-n\int_1^t \frac{\gamma(g(t))}{t}dt$ is defined as the "anomalous dimension".

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