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Hirerarchy Problem and Higgs mass Correction in One Loop

Physics Asked by Mohammad Khan on August 17, 2021

I want to calculate Higgs mass correction in SM to show that:
$$
delta m_{H} = frac{Lambda^{2}}{32 pi^{2}} [6 lambda + frac{1}{4}(9g^{2}+ 3 g^{‘ 2}) – y_{t}^{2}]
$$

but as far as I know From Field Theory ,first we need to make Higgs Lagrangian and after renormalize bare quantities in it we can found the Higgs mass correction.so I did From electroweak SSB ,and I got :
$$
ell_{H}=frac{1}{2} partial_{mu}H_{0}(x) partial^{mu}H_{0}(x)+ frac{g^{2}_{0}}{4} W_{mu 0}^{+} W^{ – mu}_{0} H_{0}(x)H_{0}(x) + frac{g_{0} ^{2} nu_{0}}{2} W_{mu 0 }^{+} W_{0}^{ – mu} H_{0}(x) + frac{1}{8} (g_{0}^{2} + g_{0}^{‘ 2})Z_{mu 0}Z_{0}^{mu} H_{0}(x)H_{0}(x) + frac{nu_{0}}{4}(g_{0}^{2} + g_{0}^{‘ 2})Z_{0 mu}Z^{mu }_{0} H_{0}(x) – frac{1}{2}m_{H 0}^{2} H^{2}_{0}(x) – frac{lambda_{0}}{4}H_{0}^{4}(x) – lambda_{0}nu_{0} H_{0}^{3}(x) – m_{f_{i}0}bar{f}_{0i}f_{0i} – frac{G_{f H 0}}{sqrt{2}} bar{f}_{0i}H_{0}(x)f_{0i} + frac{1}{2}M_{W0}^{2} W_{mu0}^{+} W_{0}^{ – mu}+frac{1}{2} M_{Z0}^{2} Z_{0mu} Z^{mu}_{0}
$$

where "f",denote Fermion Fields such as electrons or quarks $dots$ , and all quantities are "bare".but I have many doubts about choosing proper counter terms for achieve final result.as far as I know Higgs self interaction will give us this correction So I choose :
$$
delta m_{H} = m_{0H}^{2} Z_{2}- m_{H}^{2} qquad H_{0}(x) = sqrt{Z_{2}} H_{R}(x)
$$

And I certain about this ,because make Higgs mass correction appear in it’s self interaction.But other quantities are very and I can not be certain about any things.Can anyone help me or tell me where I can see this calculation?

He has done me a great favor

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