# How to find the gamma function for a fermion vacuum energy calculation?

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• The black vegetable
In summary, the conversation discusses the method of calculating the vacuum energy of a fermion using the Lagrangian and expanding it around the classical field. The next step is to find the gamma function, but the formula for it does not match the equation. The conversation also mentions the method used in Peskin and Schroder P374, where they use a scalable field Lagrangian and get the Klein Gordon operator instead of the Dirac operator. The conversation ends with the question of how to find the gamma function for this scenario.
The black vegetable
TL;DR Summary
I am trying to calculate one loop contribution to the vacuum energy from a fermion.
Following the method by Peskin and Shroesder 11.4 Trying to calculate the vacuum energy of a fermion. If my method is correct so far the next step is to find gamma function , the formula I have for gamma fuctions doesn't match this equation. Can anyone help with the next step?
Starting with the Lagrangian $$L=i \bar{\Psi} \partial / \Psi-m_{e} \bar{\Psi} \Psi-\lambda \Psi \bar{\Psi} \phi$$Expanding about the classical field

$$\Psi_{c l}+\zeta \quad \bar{\Psi}=\bar{\Psi}_{c l}+\bar{\zeta} \quad \phi \rightarrow \phi_{c l}+\rho$$

The only terms quadratic with with ##\zeta\bar{\zeta}##

$$\bar{\zeta}i \gamma^{\mu} \partial_{\mu} \zeta-m_{e} \bar{\zeta} \zeta-\lambda \bar{\zeta} \zeta\left(\phi_{c l}+\rho\right)$$When comparing this to the formula for the effective action this coincides with

$$\left[-\frac{\delta^{2} L_{1}}{\delta\bar{\Psi}(x) \delta\Psi(y)}\right]=i \gamma^{\mu} \partial_{\mu}-m_{e}-\lambda\left(\phi_{c l}+\rho\right)=i \gamma^{\mu} \partial_{\mu}-M_{e}$$

In Peskin and Schroder P374 they are doing this with a scalable field Lagrangian, where they get the Klein Gordon operator instead of the dirac operator. If I follow the method the next stage is to find the Gamma function for

$$\operatorname{Tr} \log \left(\gamma^{\mu} \partial_{\mu}+m\right)=\sum_{p} \log \left(\gamma^{\mu} p_{\mu}+m\right)$$

Where after a wicks rotation they get something similar to this but for a scaler field.

$$=V T \int \frac{d^{4} p}{(2 \pi)^{4}} \log \left(\gamma^{\mu} p_{\mu}+m\right)=V T \frac{\partial}{\partial a} \int \frac{d^{4} p}{(2 \pi)^{4}} \frac{1}{\left(\gamma^{\mu} p_{\mu}+m\right)^{a}}|_{a=0}$$

How do I find the gamma function for this, it doesn't fit my equation?

To answer my own question, I think it turns out to be the same as the scaler field loop but with the opposite sign ,Klein Gordon equation is just Dirac Equation squared, So just replace it with the square root Klein Gordon equation, then because it's log bring the exponent (1/2) in front and proceed as you would with Dim reg scaler field.

## 1. What is a 1 loop Fermion Vacuum energy?

The 1 loop Fermion Vacuum energy is a concept in quantum field theory that describes the energy of the vacuum state in a system of fermions. It takes into account the effects of virtual particles, which constantly pop in and out of existence in the vacuum, and their contribution to the overall energy of the system.

## 2. How is the 1 loop Fermion Vacuum energy calculated?

The 1 loop Fermion Vacuum energy is calculated using perturbation theory, which involves breaking down the energy into smaller, more manageable calculations. These calculations take into account the interactions between the fermions and the virtual particles in the vacuum state.

## 3. What is the significance of the 1 loop Fermion Vacuum energy in physics?

The 1 loop Fermion Vacuum energy is significant because it helps us understand the behavior of quantum systems, particularly in relation to the vacuum state. It also has implications for the cosmological constant and the overall energy density of the universe.

## 4. How does the 1 loop Fermion Vacuum energy differ from other types of vacuum energy?

The 1 loop Fermion Vacuum energy differs from other types of vacuum energy, such as the Casimir effect or the vacuum energy predicted by the Standard Model, because it takes into account the effects of virtual particles and their interactions with fermions. This makes it a more accurate and comprehensive calculation of the energy of the vacuum state.

## 5. What are the potential applications of the 1 loop Fermion Vacuum energy?

The 1 loop Fermion Vacuum energy has potential applications in quantum field theory, cosmology, and particle physics. It can help us better understand the behavior of quantum systems and the properties of the universe, and may also have implications for the development of new technologies based on quantum principles.

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