All Questions
Tagged with symmetry-breaking lagrangian-formalism
52
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Unitary Gauge Removing Goldstone Bosons
The Lagrangian in a spontaneously broken gauge theory at low energies looks like
$$ \frac{1}{2} m^2 ( \partial_\mu \theta - A_\mu )^2 $$
and the gauge transformations look like $\theta \rightarrow \...
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1
answer
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Goldstone theorem for classical and quantum potential
Consider a quantum theory $$\mathcal{L}(\phi^a) = \mathcal{L_{kin}}(\phi^a)-V(\phi^a),\tag{11.10}$$ depending on any type of fields $\phi^a$.
The VEV of this theory are constant fields $\phi_0^a$ such ...
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Calculation of Vertex factor from Lagrangian
I am studying spontaneous symmetry breaking of a complex scalar field $\phi(x)$ of a global $U(1)$ symmetry: $\phi(x)\to e^{i\alpha}\phi(x)$, where $\alpha$ is a real constant. I am considering the ...
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Residual Symmetry Group after Spontaneous Symmetry Breaking
I am seeking a proof of the following:
Suppose we have a theory with $n$ scalar fields $(\phi_1,...,\phi_n)$ such that the Lagrangian $L$ is invariant under the action of some group $G$.
However, $G$ ...
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1
answer
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$SU(2)$ breakdown to $U(1)$
When we break a lagrangian symmetric with $SU(2)$ with a higgs bosons being the adjoint representation, using the following v.e.v for higgs $\phi$,
$$\langle \phi \rangle = (0,0, \rho)^T.$$
Two ...
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vote
1
answer
210
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Symmetry groups breaking for the lagrangian of two complex scalar fields
Suppose we have a generic non-interacting Lagrangian of two complex scalar fields,
\begin{align}
\mathcal{L} &= (\partial^\mu \Phi^\dagger)(\partial_\mu \Phi) - \Phi^\dagger\mathbb{M}^2\Phi \tag{1}...
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1
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1
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What is the reverse operation of gauging a global symmetry?
As far as I understand, gauging a global symmetry means taking a model with a global symmetry and transforming it into a model such that the previous symmetry group is now the gauge symmetry of your ...
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1
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Do we have an analytic calculation to derive $\frac{F^2}{4}\,\text{Tr}\left\{\partial_\mu U\partial^\mu U^{\dagger}\right\}$ from the QCD Lagrangian?
I have studied the quark condensate and chiral perturbation theory. However, I am not sure where the "kinetic term" of the pion
$$\frac{F^2}{4}\operatorname{Tr}\left\{\partial_\mu U\partial^...
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2
votes
1
answer
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In QFT, are there any restrictions on spontaneous breaking $G\to H$, due to "spontaneity"?
For simplicity, let us restrict to the spontaneous breaking of global symmetries. Given any pair of groups $G\supset H$, is it always possible to find a $G$-invariant Lagrangian that gives a QFT such ...
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Decoupling in the Linear Sigma Model
In Schwartz's 'QFT and The Standard Model' the Lagrangian for the linear sigma model is given by:
$$L=\frac{1}{2}(\partial_\mu\sigma)^2+(\sqrt\frac{2m^2}{\lambda}+\frac{1}{\sqrt 2}\sigma(x))^2\frac{1}{...
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Quantization of spontaneously broken theory, which is not in the true vacuum
I wonder whether the source $J$ in QFT can make one to quantize the field when the system is in the excursion to the minimum. Precisely, I want to know that following process makes sence.
Suppose, I ...
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votes
1
answer
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How can we determine which subgroup remains unbroken after spontaneous symmetry breaking for $SU(2)\times U(1)$ symmetry?
Consider an $SU(2)$ doublet of bosons $\Phi = (\phi^+, \phi^0)^T$, where the complex scalar field $\phi^+$ destroys positively charged particles and creates negatively charged ones, and the complex ...
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1
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Peskin and Schroeder, Linear sigma model, renormalized perturbation theory
On Peskin & Schroeder's QFT pages 353-355, the book uses the Linear sigma model to illustrate the renormalization and symmetry.
We can write the Lagrangian of Linear sigma model with
$$
\begin{...
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Taylor expansion of some Lagrangian (Understanding the Blundell's Quantum field theory, Example 26.5)
I am reading the Lancaster, Blundell's Quantum field theory for the Gifted Amateur, p.243, Example 26.5 and I can't understand some sentences and I don't know how to expand some Lagrangian.
I am a ...
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Can Chiral symmetry violating term in lagrangian violate charge conversation?
The regular Lagrangian is
$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi$
If we add a chiral violating term
$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu-me^{i\theta\gamma^5})\psi$
For the ...
- 925
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Gaussian Propagator and Symmetry Breaking
Regarding the propagator $\mathcal{G}(k,i\omega,r)$ of a Euclidean scalar real Gaussian quantum field theory
$$\mathcal{Z_0}=\int\mathcal{D}[\phi]e^{-\mathcal{S}[\phi]}$$
$$\mathcal{S[\phi]}=\int d{\...
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2
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149
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Does the number of broken generators in SSB depend on the choice of VEV?
I take the Lagrangian,
$$\mathcal{L}=\frac{1}{2}\partial_\mu \phi^T\,\partial^\mu\phi\,-\, \frac{1}{2}\mu^2\phi^T\phi-\frac{\lambda}{4}(\phi^T\phi)^2~,$$
where $\phi=(\phi_1,\,\phi_2,\,\phi_3)$ (real ...
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Lagrangian density: What assumption are we making and how could it change?
I'm having a hard time picturing the physical system under consideration in QFT. Scattering and propagators are understandable since we a priori assume we have certain particles given by the ...
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votes
2
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393
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How can one physically describe imaginary mass in the Higgs mechanism?
I studied the Higgs mechanism a couple of times now and one question that always comes to my mind is the imaginary part of the mass in the Higgs potential.
The Higgs potential can be written as $$V = -...
- 437
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2
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212
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Spontaneous Chiral Symmetry Breaking in Schwartz
I am reading M. Schwartz's book on QFT, equation (28.24)/(28.22). They say that a set scalar fields will transform as, where $g_L$ belongs to $SU(2)_{L}$ and $g_R$ belongs to $SU(2)_R$:
$$\Sigma\...
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How does the Higgs mechanism generate mass for the $W$ and $Z$ gauge bosons?
I came across this discussion point about how the Higgs mechanism generates mass for the $W$ and $Z$ gauge bosons (see attached problem below). Regarding the Higgs field factor $$\Phi^2 = \frac{1}{2}(...
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Is electroweak symmetry breaking equivalent to the non-zero Higgs VEV?
In the Wikipedia article on the electroweak interaction, they show the Lagrangian before and after symmetry breaking, and to my untrained eye they look totally different, in part because the 4 gauge ...
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2
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442
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Spontaneous symmetry breaking with 1 massive scalar but three unbroken generators?
Consider some theory of four real scalar fields, $\phi_1$, $\phi_2$, $\phi_3$ and $\phi_4$, that is invariant under a global $$SO(4)\cong SU(2)_L\times SU(2)_R$$ symmetry. We can rewrite the real ...
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Spontaneous symmetry breaking of $SU(2)$ in real and complex scalar fields
Here is a toy problem whose aim is to help me understand breaking global symmetries into subgroups and under which occasions that is possible. My question is: does a field being real or complex affect ...
- 2,282
2
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1
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254
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VEV implying symmetry breaking, but unable to pick out specific subgroup?
Let's say we have a scalar theory with an $O(N)$ symmetry, for which the scalar fields $\phi_{nm}$ transform as a rank $2$ tensor. I can write down an action which spontaneously breaks the symmetry
$$...
- 3,514
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What does a broken symmetry mean for the Lagrangian?
I am a little confused about symmetry breaking - in particular, what I see to be two different interpretations of it.
First, what I have seen taken to be the definition of a broken symmetry - we start ...
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161
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Simplest potential to demonstrate Symmetry Breaking of $\rm SU(4)\times SU(2)_L\times SU(2)_R$ into $SU(3) \times U(1)_Q$?
In a 1974 Pati and Salam published the paper "Lepton Number as the Fourth Color", which suggested the gauge group $\rm SU(4)\times SU(2)_L\times SU(2)_R$ could be the fundamental symmetry ...
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181
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Chiral symmetry in massless QCD
The QCD Lagrangian for two flavors is:
$-\frac{1}{4} G\tilde{G}+i\bar{u}\displaystyle{\not} D u+i\bar{d} \displaystyle{\not} D d-m_u\bar{u} u-m_d\bar{d}d$
or alternaively
$-\frac{1}{4} G\tilde{G}+i\...
user255856
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Two-field Symmetry Breaking unitary gauge
Let's consider the following theory:
$$L= -\frac{1}{4}F_{\mu \nu}F^{\mu\nu} +{1\over 2} |D_\mu \Phi|^2 +{1\over 2}|D_\mu \chi|^2 + \lambda_1\bigl(|\Phi|^2-\frac{v_1^2}{2}\bigr) +\lambda_2\bigl(|\chi|^...
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What's the difference and connection between symmetry breaking and anomaly?
I'm just wondering what's the difference between symmetry breaking and anomaly.
From my understanding, symmetry breaking means: there is a symmetry in the action, but in the ground state of the ...
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