I have heard (numerous times) that topologically nontrivial pion field configurations carry baryon number. It's clear the the field configurations can carry a topological quantum number (the winding number or homotopy class). However, what I do not understand is how this can be identified with the baryon number. Is it a consequence of some conservation law that applies to a linear combination of the quark number and winding number?

  • $\begingroup$ @Buzz- Not sure about pions. But the baryon (and lepton) number current is anomalous in Standard model and baryon number violation is related to the winding number of Yang-Mills instantons. The anomaly term proportional to the winding number. $\endgroup$
    – SRS
    Dec 22, 2016 at 17:46
  • $\begingroup$ @SRS Yes, that I understand. But I've heard something similar about pion field configurations carrying baryon number, and it's not clear to me how this can occur for Goldsone boson fields. $\endgroup$
    – Buzz
    Dec 22, 2016 at 17:47
  • $\begingroup$ Pions knot up into Skyrmions. Brown & Zahed 1986 Review. Witten's original paper on Skyrmion fermi statistics should have answered your question fully. $\endgroup$ Dec 22, 2016 at 23:23

1 Answer 1


First, the pseudo-scalar mesons themself are excitations above the vacuum having zero baryon number. The topological configurations typically are extrema of the classical action (or the energy, which typically is related); we choose them as the ground state of the theory, and then study perturbations (i.e., excitations) around it. Therefore it is incorrect to say about "topological excitations of the pion field": the pions are quantum excitations around the vacuum, and of course not topological; they are just coordinates of elements of the coset space $SU_{L}(2)\times SU_{L}(2)/SU_{V}(2)$, where $G \simeq SU_{L}(2)\times SU_{R}(2)$ is the approximate global symmetry of the $(u,d)$ quarks sector, and $SU_{V}(2)$ is the symmetry to which $G$ is spontaneously broken down.

The skyrmions as the classical vacuum of the chiral theory

As I said, the choice of the vacuum is typically dictated by the classical theory. Let's treat the chiral kinetic theory with non-minimal 4-derivative term proposed by Skyrme: $$ S = \int d^{4}x \left(\frac{f_{\pi}^{2}}{16}\text{tr}\big[L_{\mu}L^{\mu}\big]+\frac{\epsilon^{2}}{4}\text{tr}\big[ [L_{\mu}, L_{\nu}]^{2}\big] \right), \quad L_{\mu} = U^{\dagger}(x)\partial_{\mu}U(x) $$ The static skyrmion solution $U_{\text{sc}}(x)$ existing in $R^{3}$ minimizes the expression for the energy functional: $$ E \geqslant 12\sqrt{2}f_{\pi}\epsilon n, $$ where $n$ is winding number characterizing the given homotopic class of the homotopic group $\pi_{3}(SU(n))=Z$. It can be given by the integral over group manifold on the sphere $S^{3}$: explicitly, $$ \tag 1 n = \frac{i}{24\pi^{2}}\int \limits_{R^{3}}d^{3}\mathbf r\epsilon^{ijk}\text{tr}\big[ L_{i}L_{j}L_{k}\big] $$ Once the skyrmion solution is found, we can set it as the vacuum of our theory, and then to expand the theory around the vacuum. In our theory, these excitations are called pions. We do that by using the anzats $$ U(x) = \tilde{U}(x)U_{\text{skyrmion}}(x)\tilde{U}^{-1}(x), \quad \tilde{U}(x) = \text{exp}\big[ i\frac{t_{a}\pi_{a}}{f_{\pi}}\big] $$

Relation between the winding number and the baryon number

What You want to know is how $(1)$ is related to the baryon number. This requires us to know the underlying theory of quarks $q$. The classical (conserved) expression for the quark baryon number current is associated with the global symmetry $U_{B}(1)$: $$ J^{\mu}_{B} = \sum_{i}\bar{q}_{i}\gamma^{\mu}Bq_{i}, \quad q_{i} = \{u,d,s\}, $$ where $B = \frac{1}{N_{c}}\text{diag}(1,1,1)$ is the $U_{B}(1)$ global transformations generator, and $N_{c} = 3$ is the number of quark colors.

In presence of external gauge fields $A_{\mu}^{a}$ (in the Standard Model these fields are $SU_{L}(2)$) it acquires a non-local contribution (i.e., the contribution which can't be written as the derivative in coordinate space) which generates the chiral anomaly: $$ \partial_{\mu}\langle J^{\mu}_{B}(x)\rangle_{A_{\mu}} = \frac{N_{c}}{16\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\text{tr}\big[ F_{\mu\nu}F_{\alpha\beta}\big] $$ It turns out that it is possible to find the construction in terms of fields $U$ which reproduces the anomalous piece of the baryon current (it was realized by Witten in his article about the Wess-Zumino term); it was found by gauging the Wess-Zumino term with respect to $U_{B}(1)$ group. It has the form $$ \langle J^{\mu}_{\text{B}}(x)\rangle_{\text{anomalous}} = \frac{i}{24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\text{tr}\big[ L_{\nu}L_{\alpha}L_{\beta}\big] $$ The baryon charge which is carried by this piece of current is $$ \langle Q_{B} \rangle = \frac{i}{24\pi^{2}}\int d^{3}\mathbf{r} \epsilon^{ikl}\text{tr}\big[L_{i}L_{j}L_{k}\big], $$ which coincides with the skyrmion topological number $(1)$. The only thing which is remained for showing that the skyrmion can play the role of nucleons is to show is that the lowest energy skyrmion (i.e., the unit winding number skyrmion) has spin one half. This was done by Witten in another paper.

See also the perfect white paper on the Skyrme model given by Cosmas Zachos in the comment section.


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