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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 ${\cal O}^{3/2}$ were introduced. These two operators carry isospin $1/2$ and $3/2$ respectively. Subsequently, the textbook reads

This operator (${\cal O}^{1/2}$) can mediate decays of the $K$ meson that change the isospin by $1/2$ unit, such as $K^0\to\pi^+\pi^-$, but not processes that change the isospin by $3/2$, such as $K^+\to\pi^+\pi^0$.

I understand that the three pions form a triplet of isospin, while the kaons form two doublets of isospin. For instance, $K^0$ is composed of $d\bar s$ quarks with $I=1/2, I_3=-1/2$, $K^+$ is composed of $u\bar s$ with $I=1/2, I_3=1/2$, $\pi^+$ is composed of $u\bar d$ with $I=1, I_3=1$ etc. So it seems for $\pi^+\pi^0$ the total $I_3=1$, whey the isospin is changed by $3/2$?

I might have mistaken some basic concepts in a terrible way. Many thanks for the help in advance!

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(After this answer was posted, a duplicate was noted. This answer is equivalent to the answers by Paganini and Franklin in that earlier post, but with more detail.)

As stated in the question:

  • $K^+$ has $(I,I_3) = (1/2,1/2)$

  • $\pi^+$ has $(I,I_3)=(1,1)$

  • $\pi^0$ has $(I,I_3)=(1,0)$

Therefore, $\pi^+\pi^0$ must have either $I=0$, $1$, or $2$.

The case $I=0$ is not consistent with $I_3=1+0=1$.

To rule out $I=1$, consider the ways to get $I_3=1$ by combining $I_3=1$ ($\pi^+$) with $I_3=0$ ($\pi^0$). The possibilities with definite values of $I$ are:

  • Case 1: $1\otimes 0 - 0\otimes 1$, which has $I=1$.

  • Case 2: $1\otimes 0 + 0\otimes 1$, which has $I=2$,

Both cases have $I_3=1$. The $I$ values can be inferred from the fact that applying the $I_3$-raising operator gives zero ($1\otimes 1 - 1\otimes 1$) in case 1 and gives a non-zero result ($1\otimes 1$) in case 2. This shows that if the final two-pion state had $I=1$, then it would need to be antisymmetric in the isospin labels. But pions are bosons, and the total angular momentum is zero (because the initial kaon has spin zero). So, since the pions are distinguished from each other only by their isospin, the state must be symmetric with respect to the isospin labels. This rules out the case $I=1$. (See appendix 1 for more detail.)

The only remaining possibility is that $\pi^+\pi^0$ has $I=2$. Therefore, the process $K^+\rightarrow \pi^+\pi^0$ would have $\Delta I = 3/2$, as the textbook claimed — assuming the textbook was referring to the change in total isospin $I$, not the component $I_3$.


Appendix 1

Here is a more detailed explanation of why the two-pion state must be symmetric with respect to the isospin labels. (This was used above to rule out $I=1$.) For simplicity, work in a non-relativistic model in which $a(x)$ and $b(x)$ are the annihilation operators for $\pi^+$ and $\pi^0$, respectively, at a spatial point $x$, and $a^*(x)$ and $b^*(x)$ are the corresponding creation operators. Consider the two states $$ |\pm\rangle = \int d^3x\,d^3y\ f(x,y)\big(a^*(x)b^*(y)\pm b^*(x)a^*(y)\big) |0\rangle $$ where $|0\rangle$ is the vacuum state. The signs $+$ and $-$ correspond to $I=2$ and $I=1$, respectively, as explained before. The goal here is to show that the antisymmetric case ($-$) cannot have zero angular momentum.

Pions are bosons, so the operators $a$ and $b$ commute with each other. Use this to get $$ |\pm\rangle = \int d^3x\,d^3y\ \big(f(x,y)\pm f(y,x)\big)a^*(x)b^*(y) |0\rangle. $$ Working in the center-of-mass frame, the total momentum must be zero, so the wavefunction $f(x,y)$ can only depend on the difference $x-y$. Then, to have zero angular momentum about the origin, the wavefunction must be invariant under rotations about the origin, so it can only depend on $|x-y|$. This implies $f(x,y)-f(y,x)=0$, because $|x-y|=|y-x|$. This implies that $|-\rangle$ must be the zero vector, which does not represent a physical state. Therefore, as claimed, the angular momentum cannot be zero in the antisymmetric case.


Appendix 2

Everything in this answer assumes that the "kaon" in question is the charged kaon of lowest mass, usually denoted $K^+$. According to the Particle Data Group (http://pdg.lbl.gov/2018/tables/contents_tables_mesons.html), the $K^+$ has total angular momentum zero. More-massive and shorter-lived particles/resonances with the same valence quark content ($u\bar s$) can also be produced, with non-zero total angular momentum. Whether the non-zero total angular momentum is due to orbital or spin angular momentum (or both) is an interesting quesiton that I won't try to answer here, but see https://arxiv.org/abs/0803.2775 for a related question applied to the proton.

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    $\begingroup$ I also looked at the other answer physics.stackexchange.com/q/166331, but I still do not understand why I=1 is ruled out? $\endgroup$ – gamebm Nov 20 '18 at 17:42
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    $\begingroup$ @gamebm That's an excellent question -- I completely forgot to address that in my answer. I edited the answer to address the $I=1$ case. It's the same as the reasoning that Paganini and Jerrold Franklin used in their answers to the earlier question, but for whatever it's worth, I spelled it out again here in my own words. $\endgroup$ – Chiral Anomaly Nov 21 '18 at 0:05
  • $\begingroup$ Really appreciated for the detailed updated answer! it seems much clearer to me (comparing to the other question). However, I still have the following (silly) doubts. (1) is it possible the initial kaon is in l=1 orbital state, so that the total angular momentum is 1 and wave function is anti-symmetric? $\endgroup$ – gamebm Nov 22 '18 at 0:35
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    $\begingroup$ (2) more importantly, so the final two-pion state is considered as the wave function of identical particles? In other words, the two pions, namely, $\pi^+$ and $\pi^-$ should be viewed as two identical particles with different isospin? If not, why should we even bother with the symmetry of the total wave function? As we should only worry about the conserved quantities like total angular momentum, but why the total wave function turning from a symmetric one into an anti-symmetric one matters? Is it related to parity conservation? I am still quite confused at this point. :( $\endgroup$ – gamebm Nov 22 '18 at 0:43
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    $\begingroup$ @gamebm I added a couple of appendices to the main answer related to both (1) and (2). Appendix 2 provides a couple of links related to (1) but doesn't really answer it. Appendix 1 tries to answer (2), which as you said is the more important issue. Hope this helps! I deleted one of my earlier comments because it's covered in the main post now. $\endgroup$ – Chiral Anomaly Nov 22 '18 at 16:30

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