Questions tagged [isospin-symmetry]

Isospin is a symmetry in nuclear physics under the interchange of neutrons and protons. The strong nuclear force conserves isospin, but the symmetry is broken by the electromagnetic force, which only operates between protons.

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Relation between isospin invariance and charge independence

I don't know if this question is better suited for this forum or math.stackexchange.com. I come from a mathematical background and I'm struggling to understand a passage in Lipkin's book "Lie groups ...
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Deuteron nucleus, what is the wavefunction (including isospin)?

What is the exact form of the wavefunction of the deuteron nucleus. Without electrons. Only neutron and proton combined to a total wavefunction. In the ket-notation I have something like this: $| \...
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Hypothetical particle disintegrates into two pions $\pi^+ \pi^-$ or $2 \pi^0$, what are the possible isospins of this particle?

I had a question that I couldn't seem to find a clear answer so there it is. Let us assume an hypothetical particle $P^0$ which decays into 2 $\pi^0$ or in a pair $\pi^+$,$\pi^-$. What are the ...
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Nuclear force behavior

I am confused about when the nuclear force is attractive and when not. Based on deuteron (the book I am following is Wong), we see that we can't have bound state with isospin T=1 (otherwise we would ...
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How to determine the isospin of a pair of nucleons?

While I was reading from the book of Brussaard on shell-model applications in nuclear spectroscopy, he mentioned that isospin of a neutron-neutron pair couple to $T=1$. On the other hand, if it was ...
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Why does total swapping symmetry arise in this linear combination of hadron wavefunctions?

When studying a proton's wavefunction in the quark model, we want a totally symmetric spin-flavour wavefunction. In most books one finds the $S^2=\frac{1}{2}, S_z = +\frac{1}{2}$ multiplets break down ...
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What's the definition of $\bar\psi$ in QCD?

This is a two part question. What is the definition of $\bar\psi$ in QCD? In QED I know that $\bar\psi=\psi^\dagger\gamma^0$, but in QCD we also have flavor and/or color space. In particular, I'm ...
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How to formulate with a tensor multiplication method $2\otimes 2\otimes 2$ in $SU(2)$ group?

In $SU(2)$ we can write $2\otimes 2=3\oplus 1$ or \begin{equation} q_iq^j=\left(q_iq^j-\frac{1}{2}\delta^i_jq^kq_k\right)+\frac{1}{2}\delta^i_jq^kq_k, \end{equation} where $q_i$ is a $SU(2)$ doublet, ...
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Expanding isospin symmetry to $SU(3)$ flavor symmetry for Baryon octet diagram

Let's consider about the Baryon octet diagram. We can identify the isospin $I_3$ symmetry by noticing the mirror symmetry with respect to the $I_3$, i.e. x-axis. However, how can we expand this into $...
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Highest and Lowest $SU(3)_F$ states

For the finite dimensional $(p,q)$-irreducible representation of $SU(3)_F$, we can label the states as $\mid T_3,Y\rangle$. Where $T_3$ is the third component of isospin and $Y$ is the hypercharge. ...
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Isovector Representation of $SU(2)$

In Electroweak Interactions by Renton (page 407) there is the calculation of matrix elements for electromagnetic interactions/weak interactions involving nuclei: A point of confusion for me is the ...
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Isospin and $\Delta$ Production

Edited to add new information I have seen the relation between cross sections postulated: \begin{align} \sigma(\nu+p\to\mu^- + \Delta^{++})=9\sigma(\nu+n\to\mu^- + \Delta^{+}) \end{align} Motivated ...
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Explaining branching ratio of $J/\!\psi \!\to \Lambda \bar\Lambda$ and $J/\!\psi \!\to \Sigma^0 \bar\Sigma^0$

I am quite puzzled by the branching ratio of $J/\!\psi \!\to \Lambda \bar\Lambda$ and $J/\!\psi \!\to \Sigma^0 \bar\Sigma^0$. The ratio is measured as ~60%, whereas I expect a ratio of $1/3$. Maybe ...
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Experimental proof for isospin conservation of strong interaction

I am having a surprisingly hard time to find direct experimental results for the (approx.) conservation of isospin in strong interactions. The canonical examples seem to be $\omega \! \to 3\pi^0$ and $...
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Pions and $SU(2)$ representation

I was reading about the pion ($\pi$) SU(2) representation and stumbled upon an expression for the isospin operator, $$ I_i=\epsilon_{ijk}\int d^3x\phi_j \dot\phi_k=-i\int d^3x \dot{\phi}^T t_i \phi ,...
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Quantum numbers of a particle system

I have a problem to understand the solution of an exercise. The statement is the following: Find the possible values of isospin, parity, charge conjugation, G-parity and totoal angular momentum J, up ...
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67 views

An invariant of the gauge group $G$ that is totally symmetric with three indices in the adjoint representation

In Ch.19 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.680 the property of a quantity $$\mathcal{A}^{abc}=\mathrm{tr}\left[t^a\{t^b,t^c\}\right]\tag{19.132}$$ ...
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What is the effect of isospin on the proton or neutron alone, i.e. not in a doublet?

So I start with a proton $p$. I extend my "physical" space by means of the internal degree of freedom of isospin, so that I know write $p$ in a higher dimensional space: $$ p = \left( \begin{array}{...
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Why is neutron slightly heavier than the proton? [duplicate]

With latest knowledge of QCD, is there any explanation for why the neutron is slightly heavier than the proton? Can it be boiled down to a simple formula?
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When is $Q= B/2 + I_3$ true in particle physics?

I have seen stated that for unflavoured quarks, the relation between baryon number, electric charge and third component of isospin can be written as $Q= B/2 + I_3$. However, I have also seen this ...
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496 views

Do electrons have third component of isospin?

From what I've read, it seems like there are two types of isospin: the strong and the weak. I guess this means that leptons have no strong isospin, but they do have weak isospin. Am I right? I am a ...
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Why does the pion live in a representation of isospin SU(2) and is the mediator of the strong force generated by color SU(3)?

Why does the pion live in a representation of isospin $\rm SU(2)$ and is the mediator of the strong force generated by color $\rm SU(3)$? I somehow find strange that this is the case. Given that $\rm ...
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A question about isospin of pion and kaon

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.611 when discussing QCD renormalization of the weak interaction, two operators ${\cal O}^{1/2}$ and ${\...
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Clebsch-Gordan coefficient for 1x0

I'm trying to work out the combination of $|1\ 0 \rangle|0\ 0 \rangle$ (in this case they represent isospin, $|I\ I_3 \rangle$) using Clebsch-Gordan coefficients, but the table for $j_1\times j_2=1\...
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Why does the upper component of a $SU(2)$ doublet has $T^3=1/2$ and lower component $T^3=-1/2$ and not the opposite?

For a $SU(2)$ doublet, why does the upper component have $T^3=1/2$ and lower component $T^3=-1/2$? I know that this can be answered in the Standard Model by using $Q=T^3+Y/2$. But that is because we ...
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How to determine isospin $T$ (not just $T_z$) of a nuclear ground state

I’m trying to work out the total isospin and the $z$-component of the isospin for specific elements like $^{20}O$ and $^{20}F$ in the ground state. I’ve worked out the $z$-component, as I concluded ...
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156 views

What is meant by “the symmetry of an interaction”?

My understanding of a symmetry is this: apply an operation (e. g. parity inversion) to a system. If it behaves the same afterwards, it is symmetric under that operation. Now, quite often I see ...
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Two-nucleon coupling: symmetrization postulate

I have some trouble understanding isospin symmetry. If we consider a system of a proton and a neutron as two different iso-states of a nucleon and we wish to write the wave function, do we have to ...
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What are the other components of the weak isospin?

Sorry for a lazy question, I was just looking this up on the internet for a basic grasp of Standard Model. However, I couldn’t find anything on Wikipedia or just somehow managed to miss it on my ...
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Kanamori hamiltonian, rotational invariance and isospin

The Kanamori hamiltonian, if the coefficients satisfy a certain relationship, can be seen to be rotationally invariant. Its symmetry is $U(1)_C\times SU(2)_S\times SO(3)_O$ (I add the subscripts $C,S,...
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Determining conservation of isospin

I am trying to understand a worked example. The problem is to determine if all conservation laws are conserved in $$ \pi^-+n \rightarrow K^++\Lambda^0 $$ The answer says that isospin is conserved, ...
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What is the symmetry of the pion triplet ($\pi^{-}, \pi^{0}, \pi^{+}$)?

Under the entry "Isospin" in Wikipedia, it is written: The pions are assigned to the triplet (the spin-1, $\mathbf{3}$, or adjoint representation) of $SU(2)$ Why is the symmetry not $SU(3)$ since ...
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What is the symmetry of the doublet ($u$, $d$)?

The up quark $u$ and down quark $d$ form a doublet ($u$, $d$). I know that they have a SU(2) symmetry. However, they also have a U(1) symmetry because they are regarded as two states of the same type ...
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Why do we regard ($e^{-}, \nu_{e}$) as a doublet?

To my understanding, particles in a multiplet have similar properties (similar masses and so on) and slight or symmetric differences (in mass, electric charge, spin, and so on). They can be regarded ...
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What are the $\pi^1$, $\pi^2$ and $\pi^3$?

I found this site to give a nice explanation of isospin. Apparently, the charged pions can be described by a superposition of $\pi^1$ and $\pi^2$. Let us now consider the $\pi$'s in the ...
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What is the difference between $\Lambda$ and $\Sigma$ baryons? [duplicate]

The Lambda (uds) - 1115 MeV The Sigma (uds) - 1192 MeV Other than the 77 MeV, why are these 2 different particles? Is it the spin of 1 quark? If so, is it possible to determine which one? If not ...
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$\tau \rightarrow \rho \nu $ relation with $ \rho \rightarrow e^+ e^-$

I'm reading "Okun, Leptons and quarks.", specifically subsection 13.3 "Semihadronic decays. General remarks." Okun says "The amplitude of the $\tau \rightarrow \rho \nu $ decay is related directly to ...
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Can the reaction ${\rm n}+{\rm p} \rightarrow {\rm d} + \pi^0$ give a deuteron in excited state?

I'm trying to find the isospin quantum numbers of the deuteron on the right hand side of this equation. I know that $|\rm p\rangle = |{1\over2},{1\over2}\rangle$, $|\rm n\rangle = |{1\over2},-{1\over2}...
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Are both Gell-Mann Nishijima formulas true for the same reason?

The Gell-Mann Nishijima formula states that $$Q = I_3 + \frac{Y}{2}$$ where $Q$ is the electric charge, $I_3$ is the third component of isospin, and $Y$ is the hypercharge. This was an empirical fact ...
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What is the isospin of the photon?

As already mentioned in this Phys.SE post, the PDG booklet says that the isospin of a photon is $0,1$. Is it referring to the strong or to the weak isospin of the photon? What would be the 3rd ...
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Calculating matrix elements for quarkonium decay amplitudes

In this paper (Appendix A) several decay amplitudes are calculated. How does this work? Note that $$ |n \bar n \rangle = \frac{1}{\sqrt{2}} (| u \bar u \rangle + |d \bar d\rangle ) \\ \eta = \cos \...
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Up and down quark content of neutral pion

If isospin were a perfect symmetry, then the neutral pion would have equal $u\bar u$ and $d\bar d$ content, but since up quarks are slightly lighter than down quarks, the neutral pion, being the ...
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Does the parity operator commute with the isospin T?

I don't know if the parity operator commute with the isospin operator T for any representation, i.e., [P,T_i]=0 ? i=1,2,3.
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What is the isospin of the neutral pion?

The neutral pion, π0, should have an isospin of 0. To be certain, I am referring to the third component of isospin, $I_3$. The reason for this is very simple: I3 is an additive quantum number. The ...
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Why can a coupled wavefunction of 2 nucleons seemingly arbitrarily give two different expressions?

When looking at the Isospin representation of a proton-neutron system (with the notation $|I,I_3\rangle$), you can go from an uncoupled to a coupled representation like this: $$\textstyle|\frac{1}{2},...
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Calculation of Feynman invariant amplitude with internal global symmetry indices: trace over\and isospin

This is a complete rewriting of the older post, making more clear the problem. The issue here is to compute the $$|M|^2 =4a^2(\delta_{ad}\delta_{bc} - \frac{1}{2} \delta_{ab}\delta_{cd}) ^2 (u_{1b}^{...
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The $SU(2)$ isospin for quark v.s. anti-quark: Fix $\bar{d}=-|1/2,1/2>$?

The isospin for two quarks $u,d$, it is chosen that $$ u=|1/2,1/2>, d=|1/2,-1/2>, $$ in fundamental Rep of $SU(2)$. the anti-quarks have anti-fundamental Rep of $SU(2)$ [thus same as the ...
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Do Halzen and Martin use p and n to represent complex numbers?

In Halzen and Martin's Quarks and Leptons, on page 42, the $SU(2)$ isospin transformation represented by $e^{i\boldsymbol{\theta}\cdot\boldsymbol{\tau}/2}$ is said to act on the column represented by $...
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How nuclear force is scalar in isospin space?

In Introductory Nuclear Physics by Samuel Shaw Ming Wong (Pg. 75), it's given that nuclear force is two body in nature and scalar in isospin space. What is meant by scalar in isospin space?
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Why is the strong interaction blind to the third component of Isospin?

Neutron and the proton have different values of $I_3$, the third component of the Isospin. Yet the strong interaction cannot distinguish between them. Why is this so? I can't understand this. The ...