Two flavour Nambu-Jona-Lasinio model: lagrangian and derivation of the mass gap equation

Question

Consulting a lot of literature about the Nambu-Jona-Lasinio model (NJL model), I have run into many different choices - even at the level of writing the Lagrangian, so I would like to clear some doubts here. Most sources (e.g. [1], eq. $(2.15)$, [4], eq. $(2.3)$)seems to agree that the Lagrangian for two flavors (up and down) is

\begin{equation} \mathcal{L}=\overline{\Psi}i\displaystyle{\not} \partial \ \Psi+G[(\overline{\Psi}\Psi)^2+(\overline{\Psi}i\gamma^5\vec{\tau}\Psi)^2]\,, \tag{L1} \label{lagrangian} \end{equation}

where $\vec{\tau}$ are the Pauli matrices acting on flavour space. Nonetheless, wikipedia writes it in a different form. Let $a$, $b$ be flavor indices (summation understood)

\begin{equation} \mathcal{L}=\overline{\Psi}i\displaystyle{\not} \partial \ \Psi+G[\overline{\Psi}^a\Psi^b\overline{\Psi}^b\Psi^a-\overline{\Psi}^a\gamma^5\Psi^b\overline{\Psi}^b\gamma^5\Psi^a]\, \tag{L2} \label{lagrangianwiki} \end{equation} Note that in \eqref{lagrangianwiki} the flavor indices are intertwined (cf. $(\overline{\Psi}\Psi)^2=\overline{\Psi}^a\Psi^a \overline{\Psi}^b\Psi^b$) and the Pauli matrix is absent, for this reason I suspect that \eqref{lagrangianwiki} was derived from \eqref{lagrangian} using the Pauli matrices completeness relation \begin{equation} \vec{\tau}_{ab}\cdot\vec{\tau}_{cd}=2\delta_{ad}\delta_{bc}-\delta_{ab}\delta_{cd}, \end{equation}

or that is it just wrong/different because I can't find it elsewhere. Nonetheless, I have seen elsewhere the chiral projection that appears on wiki, where there is only a four-fermion term and such form can be derived projecting \eqref{lagrangianwiki}. So, do you believe that \eqref{lagrangian} and \eqref{lagrangianwiki} are equivalent or not?

Moreover, when deriving the mass-gap equation, the Hartree-Fock approximation is used and I see conceptually different approaches that I can't reconcile: on the one hand [1], section C from eq. $(2.27)$ on, takes into account both the contributions of the scalar $(\overline{\Psi}\Psi)^2$ and the pseudoscalar $(\overline{\Psi}i\gamma^5\vec{\tau}\Psi)^2$ term, while [2], section 2.2 (even if the Lagrangian is a little different), only the scalar $(\overline{\Psi}\Psi)^2$ term is taken into account in the computation, since they perform mean-field linearization \begin{equation} (\overline{\Psi}\Gamma\Psi)^2\to 2\overline{\Psi}\Gamma\Psi\langle \overline{\Psi}\Gamma\Psi\rangle \end{equation} with $\Gamma\in\{1, \gamma^\mu, \gamma^5 \}$, and claim that due to Lorentz invariance and parity invariance only the scalar term survives. So, how would you reconcile such different approaches?

References

[1] S.P. Klevansky, The Nambu-Jona-Lasinio Model of quantum chromodynamics. Here.

[2] W. Weise, Hadrons in the NJL model. Here.

[3] Y. Nambu, G. Jona-Lasinio, Dynamical model of elementary particles based on analogy with superconductivity. I. Here.

[4] Y. Nambu, G. Jona-Lasinio, Dynamical model of elementary particles based on analogy with superconductivity. II. Here.

[5] Wikipedia

Answer 1

Moreover, when deriving the mass-gap equation, the Hartree-Fock approximation is used and I see conceptually different approaches that I can't reconcile: on the one hand 1, section C from eq. $(2.27)$ on, takes into account both the contributions of the scalar $(\overline{\Psi}\Psi)^2$ and the pseudoscalar $(\overline{\Psi}i\gamma^5\vec{\tau}\Psi)^2$ term, while [2], section 2.2 (even if the Lagrangian is a little different), only the scalar $(\overline{\Psi}\Psi)^2$ term is taken into account in the computation, since they perform mean-field linearization \begin{equation} (\overline{\Psi}\Gamma\Psi)^2\to 2\overline{\Psi}\Gamma\Psi\langle \overline{\Psi}\Gamma\Psi\rangle \end{equation} with $\Gamma\in\{1, \gamma^\mu, \gamma^5 \}$, and claim that due to Lorentz invariance and parity invariance only the scalar term survives. So, how would you reconcile such different approaches?

Only answering the above portion of the question.

In the general case, both the scalar term and pseudoscalar term could contribute to the gap equation, which results in a mass having both scalar and pseudoscalar parts (see more details here.) $$ m e^{\theta i\gamma_5} = m\cos\theta + m i\gamma_5 \sin\theta . $$ However, if you make a chiral/axial rotation of the fermion $$ \psi \rightarrow e^{-\frac{1}{2}\theta i\gamma_5} \psi $$ you can always make the mass a scalar $$ m\bar{\psi} e^{\theta i\gamma_5} \psi \rightarrow m\bar{\psi} \psi $$

Therefore choosing the scalar part only for gap equation does not lose any generality.

The chiral/axial rotation freedom mentioned above can indeed make a single mass a scalar, thus CP invariant. That said, this "chiral/axial rotation" has its limitation in rotating away all the CP violating (cross)-mass terms for 3 generations of fermions, leading to the weak-CP-violation in CKM matrix.

Therefore, the "parity invariance" argument about choosing scalar mass is a fake argument. It's just a conventional choice of vacuum adopted in most text books and review articles (thus this post is probably the first time you learned the shocking news that mass can be a pseudoscalar). But it's an arbitrary choice, nothing fundamental. There is nothing sacrosanct about mass being a scalar for the sake of CP invariance, since CP symmetry is violated anyway in the context of electroweak theory. It's just a matter of where to hide the CP violation, either in mass term or in the mixing angle term in CKM.

Let's take an analogy: the axial $U(1)$ symmetry enjoyed by Nambu-Jona-Lasinio Lagrangian is like a circle. The scalar part $(\overline{\Psi}\Psi)^2$ is on the x-axis, while the pseudoscalar part is on the y-axis. The gap equation breaks the axial $U(1)$ symmetry. The mass inferred by the gap equation can thus take any value on the $U(1)$ circle. Taking the real/scalar value on x-axis for the mass is an arbitrary but convenient choice. In other words, the $U(1)$ symmetry breaking vacuum on the scalar "x-axis" is not in anyway "preferred" than on the pseudoscalar "y-axis" or any point on the circle.