- #1

Ramtin123

- 22

- 0

Using the Feynman rules, we can write the amplitude as follows:

$$ \Gamma_f \sim - tr \int \frac{i}{\displaystyle{\not}\ell -m_f}

\frac{i^2}{(\displaystyle{\not}\ell+ \displaystyle{\not}k -m_f)^2}

\frac{d^4 \ell}{(2 \pi)^4} $$

where ##k## and ##\ell## are external Higgs and loop momenta, and ##m_f## being the fermion mass. I have ignored the coefficients coming from vertex factors.

After evaluating the loop integral functions using dimensional regularisation, I ended up with the following expression:

$$ \Gamma_f \sim \frac{m_f}{4 \pi^2}

\left( 3\Delta -6\ln \frac{m_f}{\mu} +4

-6 \sqrt{1-4x} \ \ln \frac{1 +\sqrt{1-4x}}{2\sqrt{x}} \right)$$

The loop integral is clearly divergent due to ##\Delta \equiv \ln \pi - \gamma_E + 2/\epsilon## term, when ##\epsilon = 4 -d \to 0##.

Here, ##x \equiv (m_f/k)^2##, ##d## is dimension, ##\mu## regularisation scale, and ##\gamma_E## being Euler-Mascheroni constant.

How should I introduce the counter-terms to make the expression for the amplitude finite?