Questions tagged [self-energy]
Quantum-mechanical object that captures some of the characteristics of particles associated to interactions with themselves and with other particles, such as the change in its mass as a function of its energy. Particularly useful in quantum field theory, either for relativistic particle physics or condensed matter physics. Although it also makes sense as a classical concept, it mostly has fallen into disuse nowadays.
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Wick rotation from Minkowski Dirac theory to Euclidean Dirac theory: $\gamma^{0} = -i\gamma^{4}$
I am reading Path Integrals and Quantum Anomalies by Kazuo Fujikawa and Hiroshi Suzuki. In chapter 4.2 they calculate the self-energy of photon for QED and say that the actual calculation is performed ...
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48 views
BCS and Feynman Diagrams
I am currently studying the strong coupling theory (by Eliashberg) of superconductivity, which is derived by BCS by considering self-energy diagrams and the following interaction potential
$$ V(q, i\...
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This is a question from renormalized perturbation theory that appeared in my exam. I have very little idea about renormalized perturbation theory [closed]
Derive the one-loop renormalized $ψ$ selfenergy to O($h^{2})$ in the theory $L_{int} = −h|ψ|^{2}φ$. For what
kinematical region does this self energy become complex? Do you have any physical ...
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Dimensional analysis and more: Order of divergences in the $\phi^4$ theory
On page 324 of his book on qft, Peskin analyzes the superficial degree of divergence of the 1-loop self-energy correction of the $\phi^4$ scalar theory. Realizing that $D=2$, he concludes that the ...
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1answer
70 views
Gauge invariance of loop Diagrams
Say we have a gauge-fixed QED Lagrangian: $$\mathcal{L} = - \frac{1}{4}{F}_{\mu\nu}F^{\mu\nu}+ \frac{1}{2a}\left(\partial_\mu A^\mu\right)^2+\bar\psi_1(i\gamma^\mu D_\mu - m_1)\psi_1.$$
My question ...
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Classical point-charge self-energy in terms of momentum cut-off of QED
When computing the one-loop correction $\Sigma_2$ of the electron self-energy, Peskin & Schroeder on page 218 use Pauli-Villars regularization on a certain divergent momentum integral by making ...
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1answer
81 views
What are the sources of infinities in (unrenormalized) quantum field theory calculations?
The PSE questions and answers about this question I've found don't answer it to my satisfaction, so I am asking my own version, with the principal options I am aware of listed. Although one ...
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1answer
51 views
From 1PI function to cross section
The full propagator in QFT is the inverse of this two-point 1PI function defined as $$
\tilde{\Gamma}^{(2)}(p)=p^{2}-m^{2}-\Sigma(p)
$$
where $\Sigma(p)$ is the self-energy. I am have calculated the ...
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38 views
QCD quark self-energy, is the propagator momentum in the right direction?
For the quark self-energy diagram the amplitude is giving by:
The fermion propagator is given by $\frac{i}{\not p + l}$ in the lecture. It is not supposed to be $\frac{i}{\not p - l}$? It's seems ...
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1answer
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Does a vanishing self-energy imply a Gaussian fixed point in a $\phi^4$ theory below the upper critical dimension?
Assuming a QFT description of a second-order phase transition. From the free theory, one obtains some critical exponents and one performs an $\epsilon$-expansion below the upper critical dimension. ...
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How can the connected Greens 2pt function be summed as a geometric series of the self-energy if the self-energy contains divergent terms?
In Ryder's book on QFT page 341 we can see
$$\begin{align}
D_{\mu\nu}'=D_{\mu\nu}-D_{\mu\alpha}\big(k^\alpha k^\beta-g^{\alpha\beta}k^2\big)\Pi(k^2)D_{\beta\nu}
\end{align}$$
and hence putting $D_{\mu\...
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1answer
74 views
How to derive $\imath q^\mu\mathcal{M}_ \mu(k;q;p)=0$?
\begin{equation}
\imath q^\mu\mathcal{M}_ \mu(k;q;p)=-\imath\tilde{e}\mathcal{M}_0(p;k-q)+\imath\tilde{e}\mathcal{M}_0(p+q;k)
\end{equation}
This is exactly the Ward-Takahashi identity for two ...
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2answers
66 views
Dimensional regularization of Electron self-energy from Ryder's book
I am Studying Electron self-energy using Ryder's textbook, In page 334 we can see
Defining $k'=k-pz$ and avoiding the term linear in $k'$(because it integrates to zero) gives
\begin{equation}
\Sigma(...
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Why we insert self-energy subgraphs in external lines to cancel infrared divergences completely (like in QED)?
Here we need to keep terms originating from the same line.
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Is it correct this formula for electrostatic energy?
In CGS: I know that:
$$
U_E = \frac12\sum_{i=1}^Nq_i\sum_{j\neq i}\frac{q_j}{|\mathbf{r}_i-\mathbf{r}_j|}.
$$
In the continuous. Is it correct this conclussion?
$$
U_E=\frac12\int \rho(\mathbf{r})\...
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Photon mass Infrared divergence regulization in the one-loop electron self-energy in QED
So basically I'm trying to calculate the one-loop mass and field strength counterterms from the electron's self-energy in QED using Pauli-Villars regularization (i.e. some heavy particle of mass $\...
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1answer
60 views
Self-energy of a $D$-dimensional statistical mechanical model
We consider a $D$ dimensional statistical mechanical model whose partition function is given by
$$\begin{aligned}
\mathcal{Z} &=\int \mathcal{D} \phi(x) \exp \left(-\mathcal{S}_{\phi}\right) \\
\...
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Tadpoles and photon self-energy in QED
Tadpoles diagrams are zero in QED for many reasons, one of them is the Lorentz invariance of the vacuum expectation value of the current.
Without considering the external photon line (as Peskin &...
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Scalar self-energy in $\mathcal{N}=4$ SYM in position space
In this paper (BMN Correlators and Operator Mixing in $\mathcal{N}=4$ Super Yang-Mills Theory, 2002), the authors give the following expression for the self-energy of the scalar propagator in $\...
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In electron transport calculations, should lead self-energies always be diagonal in a mode energy eigenbasis?
In electron transport calculations for semiconductor devices, the non-equilibrium Green's function method is often used. The Green's function takes the form
$$G = \left[EI-H-\Sigma_L-\Sigma_R\right]^{-...
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1answer
81 views
One-loop diagram self-energy
I need to find the diagram contributing to the self energy of $\varphi$. Say I had an interaction term in the Lagrangian 2 $\varphi$ real scalar fields, and one $\psi$ real scalar field, $\varphi^2\...
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1answer
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Iterative Greens function calculation
I have a Hamiltonian which has an interactive and non-interactive parts.
$H = H_0 + H_I$
$H_I$ comes from the non-local electron-electron interaction and must be calculated self-consistently.
I ...
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Why is the self-energy for quarks in $d=2$ Large $N$ QCD only order $g^2$?
In an interesting article by 't Hooft , he is able to find the exact quark propagator, in the large $N$ limit of QCD. He finds that the full 1PI self-energy is given by:
$$\Gamma(p)=-\frac{g^2}{2\pi} \...
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54 views
Is the issue of self force on point charges solved by QED?
I know that classically self force is not a very tractable problem for point particles, although some attempts have been made in various ways over the years.
I also know that even in QED ...
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99 views
Using second order perturbation to calculate electron self-energy caused by electron-phonon interaction
Assume the electron-phonon interaction is given by
$$
H_{ep} = \sum_{k q} D(q) c_{k+q}^\dagger c_k (a_q + a_{-q}^\dagger).
$$
In the theory of superconductivity, the electron-electron effective ...
1
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1answer
111 views
Why do the Euler-Mascheroni constant $\gamma$ and $\ln 4\pi$ not show up in observables (renormalisation of electric charge)?
The one-loop contribution of the vacuum polarisation of the photon after using dimreg is given by
$$\Pi_2^{\mu\nu}= e^2 J(q) \left(\eta^{\mu\nu} - \frac{q^\mu q^\nu}{q^2}\right),$$
with the metric ...
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Feynman self-energy diagrams
In the Feynman picture, I don't understand how virtual photons in the self-energy diagram for a rest-frame electron can have energies that exceed $2m_e$. Aren't negative energy states of the electron ...
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Why isn't a quark-antiquark loop included in the photon self-energy corrections?
In QED, the Lagrangian has a term $\bar{\psi}A^\mu\psi$, which gives a correction to the photon propagator, where the loop is made of a pair electron-positron, with the 1st order diagram:
In the ...
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votes
1answer
232 views
First-order Contribution to the Self-energy Operator
In Altand and Simons' book 'Condensed Matter Field Theory,' on page 225 they claim that the first-order contribution to the self-energy (effective mass) operator reads
$$\big[\Sigma_p^{(1)}\big]^{ab} =...
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Self-energy series expression in terms of unperturbed Green function for exited states
I would like to understand how to arrive at the series in equation (36) in this paper https://arxiv.org/abs/cond-mat/0506438, specifically
$$\Sigma(E) = V+VG'_0(E)V+VG'_0(E)VG'_0(E)V$$
where $G'_0(E)$ ...
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2answers
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Symmetry factor of gluon self-energy
In Peskin & Schroeder, p.523, they give the diagram contributing to the gluon self-energy that arises from the 3-gluon vertex, and they claim that the $1/2$ factor is a symmetry factor:
How can ...
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1answer
161 views
Self-energy in two scalar Yukawa interaction
Considering the Lagrangian of two scalar fields in $d=4$:
$$\mathcal{L}=\frac{1}{2}(\partial\phi)^2-\frac{1}{2}m^2\phi^2+\frac{1}{2}(\partial\chi)^2-\frac{1}{2}M^2\chi^2-g\phi^2\chi$$
What would be ...
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1answer
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Computation of the self-energy term of the exact propagator for $\varphi^3$ theory in Srednicki
In M. Srednicki "Quantum field theory", Section 14 -Loop corrections to the propagator-, the exact propagator $\mathbf {\tilde \Delta} (k^2)$ is stated as
$$\frac{1}{i} \mathbf {\tilde \Delta} (k^2)...
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1answer
144 views
How to know how the self-energy changes the mass?
Suppose we have a Green's function of the typical form
\begin{equation}
G(k)=\frac{1}{k^2-m^2-\Sigma(k)}
\end{equation}
where $\Sigma(k)$ is the self energy of that particle. How exactly can we ...
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1answer
372 views
How to obtain the quasiparticle equation from Dyson equation?
The problem is formulated as follows:
Dyson equation for zero temperature Green's function:
\begin{equation}
\left[
i\dfrac{\partial}{\partial t_1} - h(\vec{r}_1)
\right]
G(1,2)-\int d3 \Sigma(1,3)G(...
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votes
1answer
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Self-energy of conducting shell
While calculating self energy of a conducting shell we integrate $Vdq$,where $V=\frac{q}{4π\epsilon r}$, but when the $dq$ charge is bought close to the shell it changes the charge distribution on the ...
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votes
1answer
327 views
Combinatorics geometric series two-point function
In this answer Proof of geometric series two-point function it is said:
Now what about the coefficients in front of each Feynman diagram? Due to the combinatorics/factorization involved it ...
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votes
1answer
103 views
Are problems with self-energy of point charge in classical electrodynamics solved by field quantization?
Classical electrodynamics gives strange results when considering a moving charge in its self generated field (Abraham-Lorentz equation).
Some 50 years ago there were many efforts and publications ...
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1answer
253 views
2PI-effective action and functional derivatives
I'm trying to work out the 2PI-effective action for complex scalar fields.
Introducing a multi field index $(a,b,c...)$ the complex conjugation and all other degrees of freedoms are suppressed, and ...
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104 views
Self-energy that does not obey sum rule
Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that
1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
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1answer
74 views
Problem with converting Integral to Gamma functions (from HQET heavy quark self-energy diagram)
In the calculation of HQET radiative correction, I came across the Equation:
$$\int_0^{\infty}d\lambda ~ \lambda^{-\epsilon}(\lambda+\omega)^{-\epsilon} = \frac{1}{2\sqrt{\pi}}\Gamma(\epsilon-\frac{1}{...
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90 views
Geometric series of two point function and self energy
This question is related to this question Proof of geometric series two-point function.
Suppose we have a graph $A$ with a symmetry factor $s_1$. According to Srednicki (chapter 9, eq. (9.13)) for a ...
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votes
1answer
320 views
Proof of geometric series two-point function
In deriving the expression for the exact propagator
$$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$
for $\phi^4$ theory all books that i know use the following argument:
$$G_c^{(2)}(x_1,x_2)=G_0^{(2)}...
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1answer
193 views
Proof of 1-particle irreducible (1PI) diagrams
If we split the effective action into
$$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]$$
we can show that the full propagator is given by
$$G= i[iG − Σ]^{-1}$$
With
$$Σ=-Γ_{ΦΦ}^{int} [Φ]$$
Here $Γ_{...
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176 views
Question about Gluon self-energy diagram/symmetry factor
I am a qft beginner and have a question:
In Peskin & Schroeder chapter 16.5 there is given an expression for a gluon self energy diagram:
According to the feynman rules we get
$$\frac{1}{2} \int ...
1
vote
1answer
811 views
Diagrams involved in 1-loop electron self-energy in QED
I'm following the derivation of electron self-energy at 1-loop in QED on Peskin-Schroeder, page 216. To second order in the coupling the considered diagram (7.15) is
The 2-point correlator at second ...
7
votes
1answer
421 views
Where are the poles of the one-particle Green's function located in the complex plane?
This post is a followup question to: How to get an imaginary self energy?
In the cited post, the two following representations for the one-particle Green's function are shown:
$$G(k,\omega) = \...
2
votes
0answers
103 views
Infrared cutoff in the Kramers-Kronig relation for the marginal Fermi liquid
I am going through Andre-Marie Tremblay's derivation of the real part of the self energy in his lecture notes on the many-body problem. On page
254, if we take the imaginary $\Sigma''(k,\,\omega)\sim \...
5
votes
1answer
195 views
Asymptotic relation of Green's function for diverging self energy
I am considering the derivation on pages 64 to 66 of Zagoskin's Quantum Theory of Many-Body Systems. They consider a Green's function in the Lehmann representation:
$$ G(p,\,\omega)=(2\pi)^3 \sum_s \...
2
votes
1answer
63 views
Integrals involving Bose-functions (Computational)
In short, I'm looking for some advice/literature how to deal numerically with Bose function.
My physical problem is to calculate a coupled set of Self-energies, thermal loop integrals, self ...
