---
address: |
    Institute of Field Physics, Department of Physics and Astronomy,\
    University of North Carolina, Chapel Hill, NC 27599-3255, USA\
    E-mail: ng@physics.unc.edu
author:
- 'Y. JACK NG'
title: MAGNETIC CATALYSIS OF CHIRAL SYMMETRY BREAKING AND THE PAULI PROBLEM
---

=cmr8

1.5pt

\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}

Let me begin with a joke which some of you may have heard before. One of Wolfgang Pauli’s life-long dreams was to understand why the fine structure constant in electrodynamics is 1/137 (in the infrared regime). Pauli was also known to be a difficult person, very hard to please. As the joke goes, the first thing Pauli asked God after his death was to explain why $\alpha$ = 1/137. As God went on with His explanation, Pauli grew more and more dissatisfied. After five minutes, Pauli was seen storming out of Heaven’s Gates mumbling, “Ridiculous!”

Like Pauli, I also would like to understand why $\alpha$ = 1/137. To dignify this problem, I will call it the Pauli problem. It is possible that chiral symmetry breaking by an external field in QED may provide some insight on this old problem by giving a critical value of $\alpha$ close to 1/137.[@JBW] Admittedly, nothing close to that magic value has arisen in the results we have obtained so far [@Lee], but our analysis is not yet complete.

My interest in chiral symmetry breaking by an external field dates back a dozen years ago, to the time when a multiple correlated and narrow-peak structures in electron and positron spectra[@GSI] was observed in heavy-ion experiments at GSI. Kikuchi and I [@newphase] interpreted the $e^+e^-$ peaks as decay products of a new type of positronium, which is formed in a new QED phase induced by the electromagnetic fields of the colliding heavy ions. The theoretical underpinning of this scenario was provided by earlier works [@Mir] which indicated that QED might have a non-perturbative strong-coupling phase, characterized by spontaneous chiral symmetry breaking, in addition to the familar weak-coupling phase. The negative results in recent heavy-ion collision experiments at Argonne [@Argonne] have rendered our interpretation moot. Nevertheless, the problem of chiral symmetry breaking by an external field is still interesting as it may shed light on the Pauli problem, and as it provides an example of vacuum engineering by manipulating external fields to alter the symmetry properties of the vacuum. But more concretely, our study provides a new non-perturbative phenomenon in (3+1)-dimensional quantum field theories and a new method to study it.

First, what kind of external fields can induce chiral symmetry breaking in gauge theories?[@Ng] Lessons gained from studying the Nambu-Jona-Lasinio model [@KL] lead us to believe that uniform magnetic fields are prime candidates. To put our problem in as general a setting as possible, we want an approach that treat both the coupling and the external field non-perturbatively. The former criterion is met by using the Schwinger-Dyson equations (or equivalently, the Nambu-Bethe-Salpeter equations [@GMS]); the latter condition is satisfied by applying the strong-field techniques introduced by Schwinger and others.

Let us start with the motion of a massless fermion of charge $e$ in an external electromagnetic field. It is described by the Green’s function that satisfies the modified Dirac equation proposed by Schwinger: $$\gamma \cdot \Pi(x) G_A(x,y) + \int d^4x' M(x,x') G_A(x',y) = 
\delta^{(4)}(x-y),
\label{Greeneq}$$ where $\Pi_\mu(x) = - i \partial_\mu - e A_\mu(x)$, and $M(x,x')$ is the mass operator $M$ in the coordinate representation. For a constant magnetic field of strength $H$, we may take $A_2 = Hx_1$ to be the only nonzero component of $A_\mu$. We adopt the method due to Ritus[@Ritus], which is based on the use of the eigenfunctions of the mass operator and the diagonalization of the latter. As shown by Ritus, $M$ is diagonal in the representation of the eigenfunctions $E_p(x)$ of the operator $(\gamma \cdot \Pi)^2$: $$- (\gamma \cdot \Pi)^2  E_p(x) = p^2 E_p(x).
\label{eigeneq}$$ The advantage of using this representation is obvious: $M$ can now be put in terms of its eigenvalues, so the problems arising from its dependence on the operator $\Pi$ can be avoided. In the chiral representation in which $\sigma_3$ and $\gamma_5$ are diagonal with eigenvalues $\sigma = \pm 1$ and $\chi = \pm 1$, respectively, the eigenfunctions $E_{p\sigma\chi}(x)$ take the form $$E_{p\sigma\chi}(x) = N {\rm e}^{i (p_0x^0 + p_2x^2 + p_3x^3)} D_n(\rho) 
\omega_{\sigma\chi} \equiv \tilde{E}_{p\sigma\chi} \omega_{\sigma\chi},
\label{eigenfcn}$$ where $D_n(\rho)$ are the parabolic cylinder functions with indices $$n = n(k,\sigma) \equiv k + \frac{e H \sigma}{2 |e H|} - \frac{1}{2},
~~~~k = 0, 1, 2, ...,
\label{index}$$ and argument $\rho = \sqrt{2 |e H|} (x_1 - \frac{p_2}{e H})$. Note that $n = 0,~1,~2,~...~$. The normalization factor is $N = (4 \pi |eH|)^{1/4}/\sqrt{n!}$; $p$ stands for the set $(p_0, p_2, p_3, k)$; and $\omega_{\sigma\chi}$ are the bispinors of $\sigma_3$ and $\gamma_5$.

Following Ritus, we form the orthonormal and complete eigenfunction-matrices $E_p = {\rm diag}(\tilde{E}_{p11},~
\tilde{E}_{p-11},~\tilde{E}_{p1-1},~\tilde{E}_{p-1-1})$. They satisfy $$\gamma \cdot \Pi~E_p(x) = E_p(x)~\gamma \cdot \bar{p}$$ and $$M(x,x') E_p(x') = E_p(x) \delta^{(4)}(x-x') {\Sigma}_A(\bar{p}),
\label{masseigeneq}$$ where ${\Sigma}_A(\bar{p})$ represents the eigenvalues of the mass operator, and $\bar{p}_0 = p_0,~\bar{p}_1 = 0,~\bar{p}_2 
= - {\rm sgn}(eH) \sqrt{2|eH|k},~\bar{p}_3 = p_3$. These properties of the $E_p(x)$ allow us to express the Green’s function in the $E_p$-representation as $(\bar{E}_p \equiv \gamma^0 E_p^\dagger \gamma^0)$ $$G_A(x,y) = \Sigma \!\!\!\!\!\! \int \frac{d^4p}{(2 \pi)^4} E_p(x) \frac{1}
{\gamma \cdot \bar{p} + {\Sigma}_A(\bar{p})} \bar{E}_p(y),
~~~\Sigma \!\!\!\!\!\! \int d^4p \equiv \sum_{k} \int dp_0 dp_2 dp_3.
\label{Greenfcn}$$

We work in the ladder quenched approximation. In terms of the notations: $\bar{p"}_{\!\!\!\!_0} = p_0 - q_0$, $\bar{p"}_{\!\!\!\!_1} = 0$, $\bar{p"}_{\!\!\!\!_2} = -~{\rm sgn}(eH) \sqrt{2|eH|k"}$, $\bar{p"}_{\!\!\!\!_3} = p_3 - q_3$, the Schwinger-Dyson equation takes the form $$\Sigma_A(\bar{p}) \simeq \frac{i e^2}{(2 \pi)^3} |eH| \int dq_0 dq_3 
\int_0^\infty dr^2 {\rm e}^{- r^2} \frac{-2}{q^2} \frac
{\Sigma_A(\bar{p"})}{\bar{p"}^2 + \Sigma_A(\bar{p"})}
\label{fermass}$$ where $q^2 = - q_0^2 + q_3^2 + 2 |eH| r^2$ and $\bar{p"}^2 = 
- (p_0 - q_0)^2 + (p_3 - q_3)^2 + 2 |eH| k"$, and where summation over $k'' = k, ~k \pm 1$ is understood. We have assumed that, due to the factor e$^{- r^2}$ in the integrand, contributions from large values of $r$ are suppressed.

Let us make a Wick rotation to Euclidean space: $p_0 \rightarrow i p_4$, $q_0 \rightarrow i q_4$, and consider the case with $p = 0$, i.e., $p_0 = p_3 
= k = 0$. We assume that the dominant contributions to the integral in Eq.(\[fermass\]) come from the infrared region of small $q_3$ and $q_4$, and that the $k'' = 0$ term dominates over the $k'' = 1$ term (this assumption has been checked to be self-consistent). Then, it is reasonable to replace $\Sigma_A(\bar{p"})$ in the integrand by $\Sigma_A(0) = m \times 
{\bf 1}$. The SD equation yields the nonzero dynamical mass as $$m \simeq a~\sqrt{|eH|}~{\rm e}^{- b\sqrt{\frac{\pi}{\alpha}}},
\label{result}$$ where $a$ and $b$ are constants of order 1.

Eq.(\[result\]) clearly demonstrates the nonperturbative nature of the result. It also shows that our approximations break down when $\alpha > O(1)$. Our earlier assumption that effectively $r \ll 1$ is translated to the physical assumption that $m/\sqrt{|eH|}  \ll 1$, which requires that $\alpha \ll O(1)$; in other words, the dynamical chiral symmetry breaking solution we have found applies to the weak-coupling regime of QED! We have checked that indeed the infrared region of $q_3$ and $q_4$ gives the dominant contributions to the integrals.

To establish that the above solution to the SD equation for the fermion self-energy does indeed correspond to a dynamical chiral symmetry-breaking solution, it is necessary to demonstrate the existence of the corresponding Nambu-Goldstone boson. One way to establish this is by studying the Nambu-Bethe-Salpeter equation of the bound-state NG boson, as was done by Gusynin [*et al.*]{} [@GMS], who found a solution consistent with our Eq.(\[result\]). As a consistency check on our approach, we [@Lee] have recovered the same result from the NBS equation, using the $E_p$-representation of the fermion propagator. The chiral condensate, the order parameter for dynamical symmetry breaking, can be easily computed, based on our formalism. We obtain ($\psi$ is the fermion field) $$\langle \bar{\psi} \psi \rangle
~\simeq~
-~\frac{|eH|}{2 \pi^2} ~m ~\ln\left(\frac{|eH|}{m^2}\right).
\label{psibpsi}$$

Thermal effects and the effects of a chemical potential can be readily incorporated into our study of chiral symmetry breaking in an external magnetic field. [@Lee] One finds that chiral symmetry is restored above a critical temperature which is of the order of the dynamical fermion mass (given by Eq.(\[result\])), and the corresponding phase transition is of second order. In contrast, the chiral symmetry restoration above a critical chemical potential (also of the order of the dynamical fermion mass) is a first order phase transition. The chiral condensates for both cases have the same form as given above in Eq.(\[psibpsi\]).

To summarize, we have shown that chiral symmetry is dynamically broken in quenched, ladder QED when an external magnetic field is present. So, the external magnetic field acts as a catalysis for chiral symmetry breaking. Furthermore, in the lowest Landau level approximation, this chiral symmetry breaking is generated in the infrared region where the QED gauge coupling is weak. As pointed out by Gusynin [*et al.*]{} [@GMS], the dynamics of LLL is (4 minus 2)-dimensional \[see the second propagator factor with $k'' = 0$ in Eq.(\[fermass\])\]. But this effective dimensional reduction is only for charged states. Propagators for neutral states (like the photon and NG boson) are still 4-dimensional \[see the first propagator factor in Eq.(\[fermass\])\]. Thus the Mermin-Wagner-Coleman theorem (which stipulates that there can be no spontaneous breakdown of continuous symmetries in dimensions less than three) is successfully evaded by our chiral symmetry breaking solution. In fact, it is the interplay between the 2-dimensional dynamics of the charged particles (in the LLL approxiamtion) and the 4-dimensional dynamics of the neutral particles that is partly responsible for some of the characteristics of our chiral symmetry-breaking solution. For the magnetic catalysis to take place, the Landau energy $(\sqrt{|eH|})$ must be much greater than the fermion mass. If the fermion mass is taken to be zero, the critical magnetic field is zero. Thus chiral symmetry is broken at an arbitrarily weak magnetic field. In the LLL approximation, the fermion pairing dynamics responsible for chiral condensates is (1 + 1)-dimensional, and is hence strong in the infrared region. Thus chiral symmetry breaking takes place even at the weakest attractive interactions between the fermions, yielding a zero critical gauge coupling.

As for applications, chiral symmetry breaking by an external magnetic field may be relevant to condensed matter systems (e.g., in quantum Hall effect and in superconductivity [@KM]). It may also find applications in astrophysics (e.g., in the cooling process of neutron stars). It has been suggested in the literature [@GMS] that the chiral symmetry-breaking solution may play a role in the electroweak phase transition during the early evolution of the Universe since a huge magnetic field (with the Landau energy of the order of electroweak breaking scale) was purportedly generated by the phase transition. Unfortunately, the electroweak phase transition took place at a temperature also of the order of the electroweak scale, which is much higher than the critical temperature for a magnetic field of such a magnitude. Therefore, it is very unlikely that such a magnetic field could change the character of the electroweak phase transition in any way. But, if (for reasons not yet understood) magnetic fields of such extraordinarily large magnitudes as to overwhelm the thermal effects were generated in the early epoch, then the chiral symmetry-breaking solution reported here could be very relevant to the early evolution of the Universe.

There remain several interesting questions which we intend to investigate. For instance, how do other background field configurations affect chiral symmetry breaking? (As an example, it is expected that an electric field would tend to break up the condensate and destabilize the vacuum, thus inhibiting chiral symmetry breaking [@DW]). Are there strong-coupling solutions of chiral symmetry breaking in an external magnetic field? If so, are the weak-coupling solutions we have found and the new strong-coupling solutions related by some kind of duality? How does one extend the chiral symmetry-breaking results to inhomogenous configurations? Can one understand the chiral symmetry breaking by a simple (perhaps topological) argument? [@Sem]

But, perhaps the most urgent task is to have a better understanding of this phenomenon of chiral symmetry breaking by a magnetic field. After all, if dynamical symmetry breaking occurs when particles interact strongly as in QCD, then how can [*weak*]{} electromagnetic interactions in an arbitrarily [*weak*]{} magnetic field lead to chiral symmetry breaking? Above, we have given a partial answer to this question by arguing that the dimensional reduction in the LLL approximation effectively enhances such weak interactions in the infrared region. But can the LLL approximation be the whole story?

For a better perspective, it is useful to recall that in strong-coupling QED [@Mir], where chiral symmetry is broken, the anomalous dimension of the fermion mass operator is one, thus four-fermion interactions are relevant operators. In fact, it is the dynamical running of the four-fermion couplings that leads to a finite dynamical fermion mass, and hence an infrared-meaningful theory in the continuum limit. Now, returning to the phenomenon of magnetic catalysis, one is naturally led to ask: what are the effects of four-fermion interactions in the chiral symmetry breaking triggered by an external magnetic field? A recent work by Hong [@Hong] suggests that quantum effects of fermions in higher Landau levels induce marginal four-fermion couplings (along with other interactions). In turn, the four-fermion interactions give rise to chiral symmetry breaking when the external magnetic field is super strong. Hong’s analysis, though incomplete, is intriguing. It hints at a non-zero critical magnetic field in the magnetic catalysis of chiral symmetry breaking. The next question to ask is this: Will a complete analysis of this phenomenon yield a non-zero critical gauge coupling as well as a non-zero critical magnetic field? Indications from the analysis of strong-coupling QED are encouraging: in strong-coupling QED, supplemented by the appropriate four-fermion interactions, both the critical gauge coupling and the critical four-fermion coupling are non-zero. This brings us back to the question posed at the beginning of this talk: Does chiral symmetry breaking by an external field shed light on the Pauli problem by giving a critical value of $\alpha$ close to 1/137?

Acknowledgements {#acknowledgements .unnumbered}
================

This talk is based on works done in collaboration with C.N. Leung, D.-S. Lee, and A. Ackley, supported in part by the U.S. Department of Energy under Grant No. DE-FG05-85ER-40219 Task A.

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==========

[99]{}

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